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About the book:
This second volume of strategic classroom activities is designed to develop, through a hands-on approach, a basic mathematical understanding and appreciation of fractals. The concepts presented on fractals include self-similarity, the chaos game, and complexity as it relates to fractal dimension. These strategic activities have been developed from a sound instructional base, stressing the connections to the contemporary curriculum as recommended in the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards for School Mathematics. Where appropriate, the activities take advantage of the technological power of the graphics calculator. These activities make excellent extensions to many of the topics that are already taught in the current curriculum. Together, they can be used as a complete unit or as the beginning for a semester course on fractals. The slide package that accompanies this first volume includes some of the highest quality fractal images available anywhere. Students and educators alike will find these materials most stimulating.
Softcover, ISBN 0387975543 Publisher: Springer, 1993 Used - Good, Usually ships in 1-2 business days, Good clean copy with no missing pages might be an ex library copy; may contain some notes and or highlighting
Softcover, ISBN 0387975543 Publisher: Springer387975543 Publisher: Springer75543 Publisher: Springer, 1993 Used - Very Good, Usually ships in 1-2 business days, All pages like new. Paperback cover has been folded over. JH All proceeds benefit local libraries.
Softcover, ISBN 0387975543 Publisher: Springer75543 Publisher: Springer-Verlag New York Inc., |
Bensenville Algebra 1Prealgebra focuses primarily on arithmetic. The most fundamental skills are reading and writing of whole numbers. From there, basic arithmetic operations of addition, subtraction, multiplication and division are defined for the whole numbers |
Mathematics 2 (H) 40 hours (Mandatory) - Topics include quadratic theory and the remainder theorem, basic integration, use of the addition formulae in trigonometry and the equation of a circle and tangency.
Mathematics 3 (H) 40 hours - Topics include vectors in three dimensions and the scalar product, further differentiation and integration, properties of exponential and logarithmic functions and further trigonometric relationships |
Algebra for College Students
9780136129080
ISBN:
0136129080
Edition: 3 Pub Date: 2007 Publisher: Prentice Hall
Summary: "Algebra for College Students "is typically used in a very comprehensive 1-semester Intermediate Algebra course serving as a gateway course to other college-level mathematics courses. The goal of the Intermediate Algebra course is to provide students with the mathematical skills that are prerequisites for courses such as College Algebra, Elementary Statistics, Liberal-Arts Math and Mathematics for Teachers. This Alge...bra for College students text may also be used in a 1-semester, lower-level College Algebra course as a prerequisite to Precalculus.
Angel, Allen R. is the author of Algebra for College Students, published 2007 under ISBN 9780136129080 and 0136129080. Four hundred eighty five Algebra for College Students textbooks are available for sale on ValoreBooks.com, one hundred sixty five used from the cheapest price of $33.94, or buy new starting at $188.00 |
chieve the best possible standard with this essential companion to the bestselling book of traditional practice and guidance. This supporting book contains all the answers to the exercises in the bestselling First Aid in Mathematics. This series provides all the help and support for learning and practising Mathematics, with comprehensive coverage of core mathematical topics in clear and accessible language. First Aid in Mathematics: - Develops a strong basis of understanding with core topics covered in clear and accessible language - Improves student's ability to work through problems with plenty of practice exercises and revision tests - Reflects its international readership with terms and information that are appropriate for students worldwide |
Casio's latest and most advanced scientific calculator features new Natural Textbook Display and improved math functionality. The FX 115ES Plus
is designed to be the perfect choice for high school and college students learning general math, trigonometry, statistics, algebra I |
introducing logic and by emphasizing the structure and nature of the arguments used, this book helps readers transition from computationally oriented mathematics to abstract mathematics with its emphasis on proofs. Uses clear expositions and examples, helpful practice problems, numerous drawings, and selected hints/answers. Offers a new boxed review of key terms after each section. Rewrites many exercises. Features more than 250 true/false questions. Includes more than 100 practice problems. Provides exceptionally high-quality drawings to illustrate key ideas. Provides numerous examples and more than 1,000 exercises. A thorough reference for readers who need to increase or brush up on their advanced mathematics skills.Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
19 reviews
60 of 60 people found the following review helpful
Definitely a good first textSept. 4 2002
By
Charlie Johnson
- Published on Amazon.com
Format: Hardcover52 of 53 people found the following review helpful
This book was surprisingly goodJuly 2 2002
By
Zachary Turner
- Published on Amazon.com
Format: Hardcover
I didn't think this book was going to be very good, but the author has "proved" me wrong ;-) This book starts out so basic that in my class (which was the first analysis course in our math department) we actually skipped the first 1/3 or so of the book. The first 9 or 10 sections consist of stuff like basic set theory, logic, definition of a function, etc. I would think that even the most elementary Analysis books would completely leave this out and expect that the reader is already familiar with this. So if you need it, this book will be a good resource for you. Then the book goes into a very nice introduction to topology. Basic concepts like open/closed sets, accumulation points, compact sets, etc. Topology can be a little intimidating simply because it's _so_ abstract, but this book makes the basic concepts very easy to understand, and prepares one for a more advanced course in topology. Alot of (good) Elementary Analysis books leave topology out, but I'm glad this book contained it. It is a very interesting subject. All the material in the book is explained probably about as easily as the concepts CAN be explained. If you still have trouble with it, you might consider a different major. Not to say that this book transforms a very difficult subject into a pathetically easy piece of cake because that's impossible, but the material is presented probably as easily as it can be in order to maintain precision and detail (which is the whole point of Analysis). The book is definitely not running short in the examples or end-of-section problems department, so that is another plus. The problems at the end of each section range in difficulty from problems that almost exactly match an example worked in detail in the section, to fairly challenging problems. With enough time though the average student could probably do every problem at the end of every section. I'd recommend this book for self study as well as a supplement to any introductory analysis course. If you have already have exposure to rigorous proof of calculus theorems, then this book will probably be too basic for you. The reason this book got 4 stars instead of 5 is because of its utterly ridiculous price. Just as good is Elementary Analysis: The Theory of Calculus, ISBN: 038790459X, except that it doesn't include the section on Topology ...
24 of 24 people found the following review helpful
Acceptable but could have been better.April 20 2008
By
Gregory E. Hersh
- Published on Amazon.com
Format: Hardcover
Verified Purchase
This is fairly basic introduction to Principles of Analysis, on intermediate undergrad level, strictly in R^1. The only other similar book I'm familiar is Kirkwood. The books of Rudin, Apostol, etc present the subject on much higher level.
My original intention was to take a course with Rudin, but after I've realized I had a hard time digesting his style, I've decided to take more elementary course. I knew the course would be using Lay, so I got this textbook and tried to learn it on my own, but wasn't sure how I was doing and ended up taking the course (still with Lay) anyway. So I'm quite familiar with this textbook. The only topics we didn't cover is "series" and "sequences and series of functions".
Now overall I would say it's a mixed bag. First, the good things. The first few introductory sections on sets and proof techniques are excellent, highly recommended, that's how I learned how to prove. I found exercises very useful.
Now things I don't like. First, lots of typos. I think I had 4th edition, and still I've managed to find over 20 misprints, incorrect references, etc, etc, all were reported directly to author. Second, and that's probably more important, in several instances the proofs are too convoluted and not self-motivating. To be more specific, the proof of Heine-Borell theorem is less than adequate. It is correct, but that's the kind of proof you read and then entirely forget how it went. I remember on the first reading I didn't feel comfortable with this proof at all. When I discussed this book with professor I was going to take that course with, he (surprisingle) agreed with me and told me he would present a different proof (and he did, much better one). Another example: proof that the modified Dirichlet function is Riemann-integrable. The proof can be substantially simplified. In fact, I've managed to simplify it. Finally, the same professor told me Lay's presentation of Riemann integrals had some holes in them, so he used Kirkwood instead. In fact he told me he was making choice between Kirkwood and lay (but ended up choosing Lay because he didn't like Kirkwood's book layout. Kind of funny reason, I think.)
In any case, I think Kirkwood is a bit better for self-study. Unfortunately it doesn't have intro to proofs, logic and sets. Ideally you should have both books, if you plan for self-study.
(note: I did took the Principles of analysis, after I've finished that one with Lay, and did quite well.)
8 of 9 people found the following review helpful
Great Book for Intro to AnalysisMarch 13 2008
By
Charles Saunders
- Published on Amazon.com
Format: Hardcover
This is a very good book for someone to look at before going into an analysis class with Rudin. If you have never done proofs or seen metric spaces or uniform continuity, etc., this is a nice, but brief, intro. This book will NOT teach you analysis - you have to use Rudin for that. But it is great for acquainting/preparing you for Rudin.
6 of 7 people found the following review helpful
Real Anal.July 21 2011
By
:)
- Published on Amazon.com
Format: Hardcover
Verified Purchase
I love this book. It so easy to understand and will be a helpful guide with my encounters with Anal. this coming semester. |
Blogs
What's happening in the world of private tutoring?
Calculus BlogsKeith K.
All too often, calculus textbooks misrepresent the proof this formula:
\frac{d}{dx} e^{x}= e^{x}
The texts by Finney, Demana, et al. usually introduce, without explanation or proof, the limit below:
\lim_{h \rightarrow 0} \frac{ (e^{h}-1) }{h}=1
The problem with this approach is that it deprives the student of key concepts regarding the exponential function, ex . The student often thinks of e as the number that is 2.71828...... because Precalculus and Calculus teachers define it as
such. However, that definition is not the logical definition, but a mere incidental byproduct. The logical definition of e is the exponential base in the function whose tangent line has a slope of 1 at x=0 in the function f(x)=ex . The calculation
of the numerical value of this base to be 2.71828... is a later development that results from the definition. It is not the logical definition.
Khan Academy sidesteps this...
read more
Ellen S. if you're chugging through a problem and suddenly find yourself confronted with
x = √(-9)
It seems like to finish this problem we'll need to take the square root of a negative number – but we can't, so what do we do? Drop the sign and hope nobody notices? Mark it as 'undefined' like dividing by zero? Give up? Cry?
Well, actually, we don't have to do any of that, because we've got an imaginary friend to help us.
Meet i.
i is a mathematical constant, whose sole definition is that i2 = -1. Or, in other words,
i = √(-1). i is an imaginary number...
read more
Keith M.
When it comes to using a legitimate online resource to help with tutoring mathematics, or answering mathematical questions I use Wolfram.com.
This website is very diverse and allows the user to input any mathematical equation, formula etc.
With subject areas of mathematics, such as calculus, Wolfram.com has proved to be extremely beneficial, especially when working with difficult integrals and derivatives.
With the Pro version of this website, which is well worth its value, you will be provided step-by-step instructions on how to solve the particular problem that you have inputted.
Check out this website and explore the countless benefits it has to offer.
KeithEmily B.
I am a University of Utah mathematics major and I love the word FREE. (cheap is good too)
I don't have a lot of money so any Free resources to help me study are worth it to me. Since I know a lot about mathematics that is what I will be posting here.
The key to Mathematics is Learning, Practicing, Learning, Practicing, and sometimes it goes in the opposite order: Practicing, Learning, Practicing, Learning. But either way a good resource to me has a bit of both: they teach you how and why you do something
and they make you do it as well. A really good resource will teach you how and why, make you try it, and then will show you why you got it wrong and what you should have done, and then make you do more problems of the same type. So then, without further ado,
here are the resources:
Paul's online notes (type it in google it will be one of the first to pop up)
his notes are free, come with worked out...
read more
Martin R.
I've started brushing up on Calculus. I studied Calculus in high school and took two semesters in college, bu that was forty years ago. It's really interesting how persponal memories pack themselves in along with Diffrential Equations and Integrals.
Skyler H.
Hello everyone,
One of my Calculus students had an interesting Related Rates problem that I had to go home and think about for a while in order to figure out. The problem was set up as such:
A 25 inch piece of rope needs to be cut into 2 pieces to form a square and a circle. How should the rope be cut so that the combined surface area of the circle and square is as small as possible?
Here's what we'll need to do:
1. We will have to form equations that relate the length of the perimeter and circumference to the combined surface area.
2. We will then differentiate to create an equation with the derivative of the surface area with respect to lengths of rope.
3. Wherever this derivative equals 0 there will be a maxima or minima, and so we will set the derivative = to 0 and determine which critical points are minima...
read more
Jeffrey J.
You can find some really good resources for math test prep in the used bookstores in a college town. Some examples that I like are: (1) Humongous Book of ______________ Problems (fill in the blank with your math topic); (2) the REA Problem Solvers series;
and (3) the Schaum's Outlines. If you don't live near a college town it might be worth a Saturday trip just to buy books. Alternately, all of these are available (used) through the Amazon Marketplace sellers at really low prices.
You should preview each title of these book series that you might be considering to be sure you like the authors style. Each one is different. You may like one series' treatment of Pre-Calc but prefer a different series for Calculus.
So how do you use these books ?
They are an alternate resource for explanations of basic concepts and problem solving techniques. You should use them as 'hint mills' and sources of problems to...
read more
Jeffrey J.
You'd think that, "If I'm paying for tutoring, he should be answering MY questions. Not the other way around."
While I can sympathize with the general sentiment, I'd say,"you're way off base there!"
I think that the tutor/teacher/coach should never ask the student directly,"Do you understand __________ ?" Not knowing the subject matter, how would the student know/evaluate/determine if they understood or not ? Generally they can't, that's why the need
a tutor. Rather than ask about specific content, directly, I ask questions to determine if the student understands the material and how the pieces fit together. Sometimes that's five or six questions.
Here's my general GAME PLAN: Find out where they are. Tell them, show them, then see what they heard and saw.
When your tutor's asking you questions, he/she is probably working the same kind of plan. You can help them help you by always providing the syllabus...
read more?
It was like a door opened and light flooded in. I knew how to do it! I wrote my steps down, checked the answer in the back of the book, and there it was. My answer matched! It was one of those moments when your confidence soars. It seems silly now
that I got so excited about solving that one problem, but I consider that moment a defining moment when I knew I could be good at math.
The rest of the year was still challenging, but I felt like I knew how to get better at solving math problems: do as many problems as I could from the...
read more
Margery W.
I have many students tell me that they are afraid to ask a stupid question in class. I tell them that there are NO stupid questions, only stupid mistakes because you didn't ask the question! Too many smart students in Calculus think that by asking a
question they will appear weak. What most students don't know is that probably there are many other students with the same question in their head that they are afraid to ask! The look of relief on other students' faces when some else asks a question is amazing.
Another thing smart students have a problem with is writing down each step. Newsflash: by the time you get to Calculus you can no longer do the problems in your head. Calculus problems generally are difficult because it is not just a matter of memorizing
a formula and applying it. In Calculus you are expected to extrapolate the knowledge you have learned to problems you have never seen before. This is...
read moreJustisse M.
Naturally, Anything new can be challenging. For example, Calculus, now my favorite math topic, once was something somewhat confusing. How did I master Calculus? By asking people around me to explain it. The trick, at least for me, isn't how you explain
it, it's how you define it. When someone finally stated "the change in y with respect to x" I finally understood. It was an immediate understanding of all concepts of calculus.
So what my secret? It's learning everyone elses secret until I find one to make mine!
Jenna G.
I've heard this sentiment over and over--sometimes from students, and sometimes, I'll admit, in my own head.
Last night, I was working on my own math homework, and there was one problem I just couldn't get my head around. I read the book, looked back at my class notes, and even sat down with a tutor for a while, and still, when I tried a new problem of the same type
on my own, it just didn't work!
"Maybe I'm not as good at math as I thought," I told myself. "Am I REALLY smart enough for bioengineering?"
It was hard, but I told myself "YES!" And I kept working. I laid the assigned problems aside and started doing other problems of the same type from the book. I checked my work every time. Each problem took at least ten minutes to solve, and the first three
were ALL wrong! I kept going. I got one right, and it made sense! I did another, and it was half right, but there was still a problem. I did another, and it was right...
read more
Raymond V.
Developing a grounded understanding of numbers, and number operations provides the firmest foundation for learning math. Touching, seeing, and manipulating physical objects are perhaps the surest way to accomplish that in the beginning.
Developing the practice of drawing pictures to reflect an arithmetic or story problem is the next step and soon becomes a central tool for thinking through a math problem whether represented in math and science, or encountered in life.
Finally, talking about, through, and around math, arithmetic, problems, and solutions is equally important to proficiency in math and any other area of education, socialization, and life.
It is important to recognize the preferred learning style of each student in order to achieve the best opportunity to that student's learning and performance. Yet, excellent teaching includes multiple approaches and learning styles on the way to each student's
full facility, proficiency, and confidence. This necessaryJohn D.
As summer approaches this Friday, thoughts turn to fun in the sun, relaxing on the beach, swimming in the surf and the sand between your toes, strolls on the boardwalk, and just enjoying time with friends and family. This summer may be a little different
at my beloved Jersey Shore, where many towns and beachside communities have been impacted by the fury of Super Storm Sandy. Much rehabilitation and reconstruction has been done to both homes and hearts, but make no mistake that much work still needs to be
initiated and completed.
As a tutor, I have been much more aware of the importance of keeping the academic skills of students sharp during the summer months. Studies have indicated that students actually regress academically during the months, with a loss of knowledge previously
acquired. I have a ten year old daughter just completing fourth grade, and we always spend time in the summer on Math skills and drills, reading comprehension, and writing. From a business point...
read more
Becky W.
Before I go run a marathon, play with my family at the pool, ride a roller coaster, head to the beach, or eat some serious amounts of ice cream, I will look back on the successful school year I have had.
I got to tutor over 15 students in Middle School, High School, SAT, and College Math...and even Chemistry! I watched GPAs rise for everybody-some were happy just to pass that College Math course to graduate, others enjoyed their hard earned As that were
brought up from the D level. I must say, that things did get crazy with exams at the end of the year, but it all worked out amidst our busy-ness.
My tutoring schedule is light for the summer, and I am hoping nobody waits until December exams to contact me! I want things to be done the right way...after all of the swimming, adventure, and ice cream, of course! :)
-Becky |
College Algebra Essentials - 14 edition
Summary: When Julie Miller began writing her successful developmental math series, one of her primary goals was to bridge the gap between preparatory courses and college algebra. For thousands of students, the Miller/O'Neill/Hyde (or M/O/H) series has provided a solid foundation in developmental mathematics. With the Miller College Algebra series, Julie has carried forward her clear, concise writing style; highly effective pedagogical features; and complete author-created technological packag...show moree to students in this course area. The main objectives of the college algebra series are three-fold: - Provide students with a clear and logical presentation of the basic concepts that will prepare them for continued study in mathematics. - Help students develop logical thinking and problem-solving skills that will benefit them in all aspects of life. - Motivate students by demonstrating the significance of mathematics in their lives through69.99 +$3.99 s/h
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College Trigonometry
This course provides an integrated technological approach to trigonometric applications used in problem solving. Emphasis is placed on applications involving trigonometric ratios, right triangles, oblique triangles, trigonometric functions, graphing, vectors, and complex numbers. Upon completion, students should be able to apply the above principles of trigonometry to problem solving and communication. This course has been approved to satisfy the Comprehensive Articulation Agreement general education core requirement in natural sciences/mathematics for the Associate in Arts Degree |
Algebra 1
9780078651137
ISBN:
0078651131
Pub Date: 2005 Publisher: Glencoe/McGraw-Hill School Pub Co
Summary: A flexible program with the solid content students need "Glencoe Algebra 1" strengthens student understanding and provides the tools students need to succeed--from the first day your students begin to learn the vocabulary of algebra until the day they take final exams and standardized tests.
Holliday, Berchie is the author of Algebra 1, published 2005 under ISBN 9780078651137 and 0078651131. Five hundred fif...ty five Algebra 1 textbooks are available for sale on ValoreBooks.com, five hundred fifty one used from the cheapest price of $5.15, or buy new starting at $45.00.[read more [more2005. 1" tear at the bottom of the spine, otherwise a good copy with clean pages. School markings inside the front. Book only-does not include additional resources. Booksaver [more]
2005. 1" tear at the bottom of the spine, otherwise a good copy with clean pages. School markings inside the front |
The key is knowing the facts in pre-algebra, algebra, geometry, and basic trigonometry, and doing some critical thinking, to use all the information that you are given, even when the information is in the answer choices. Elementary math is focused on introducing students to the basic operations ... |
Mastering Essential Math Skills This exercise book is an excellent resource to practice and review math skills youŽll need to establish a strong foundation and smooth transition into Algebra and other higher math courses. Workbooks are available for 4th – 5th grade and middle school / high school. |
This book provides a concrete and readable text for the traditional course in elementary differential equations that science, engineering, and mathematics students take following calculus. This is a strongly algebraic-oriented text with some computer enhancements for numerical methods.
Matters of definition, classification, and logical structure deserve (and receive here) careful attention for the first time in the mathematical experience of many of the students. While it is neither feasible nor desirable to include proofs of the fundamental existence and uniqueness theorems along the way in an elementary course, students need to see precise and clear-cut statements of these theorems and understand their role in the subject. Appropriate existence and uniqueness proofs in the Appendix are included and referenced where appropriate in the main body of the |
gebra for College StudentsHardcover– Jan 12 2011
The Dugopolski series in developmental mathematics has helped thousands of students succeed in their developmental math courses.Algebra for College Students, 6e is part of the latest offerings in the successful Dugopolski series in mathematics. In his books, students and faculty will find short, precise explanations of terms and concepts written in clear, understandable language that is mathematically accurate. Dugopolski also includes a double cross-referencing system between the examples and exercise sets, so no matter where the students start, they will see the connection between the two. Finally, the author finds it important to not only provide quality but also a wide variety and quantity of exercises and applications. |
books.google.com - This... to Solve Word Problems in Algebra
How to Solve Word Problems in Algebra: A Solved Problem Approach
This and college students take algebra annually, and almost all are stumped by word problems-- Features Solution methods that are easy to learn and remember, plus a self-test-- Stands alone or can be used with any standard Algebra I or II textbook |
Logarithmic Functions - Problem 2 of 3 This video is a continuation and reviews the properties of logarithms. It also provides an example that demonstrates how to graph logarithmic functions. (4:43) Author(s): No creator set
Properties of Logarithms This video explains two important properties of logarithms and the methods used to get to and solve for variables that are in exponents or inside logarithms. (5:09) Author(s): No creator set
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Properties of Logarithms - Problem 1 of 3 This video is a continuation and reviews the properties of logarithms. It also provides examples that illustrate how the properties of logarithms are used in various calculations. (5:45) Author(s): No creator set
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No related items provided in this feedNatural Logarithm This video explains how the natural log is the logarithm to the base of the number "e" and is the inverse function of an exponential function. Natural logarithms are special types of logarithms and are used in solving time and growth problems. (6:05) Author(s): No creator set
Natural Logarithm - Problem 2 of 3 This video is a continuation and describes some properties of logarithms as they apply to the natural log. It also provides examples of how to apply these properties. (2:10) Author(s): No creator set
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Natural Logarithm - Problem 3 of 3 This video is a continuation and describes the family of functions. It demonstrates how any exponential function can be described with base "e" and provides examples. (4:08) Author(s): No creator set
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Exponential Growth This video explains how exponential growth is a type of exponential function where instead of having a variable in the base of the function, it is in the exponent. It works through an example problem that demonstrates exponential growth. (4:27) Author(s): No creator set
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Exponential Growth - Problem 1 of 3 This video is a continuation and demonstrates how to use the exponential growth formula to solve an example problem. (4:31) |
As I'm sure every math student has experienced, I'm often infuriated by the inaccessibility of advanced math. I know there are a lot of people on YouTube and other places like Khan Academy or iTunes U who make brilliant video lectures, and I enjoy them a lot.
I've thought for a while about making my own series of video lectures, and I would probably do them on Quantum Logic (the C*-algebra approach), but I thought I would ask you - do you think people would use them?
Any suggestions at all would be welcome.
EDIT: Thank you for the overwhelming feedback. I will make the first lesson as soon as possible, which might be upwards of a week or two depending on the resources I can find. Please feel free to leave suggestions as to the format of these lectures if you have any thoughts on the matter.
Yes, it is always the best to go ahead and do this: for yourself, for the people of the future, and for the people who might just say "wow thank goodness for that guy who helped make my life that much better" plus, do it for me.
What is Quantum Logic. Never mind this, I felt like I got a decent beginners grasp of Quantum Logic by reading the Wikipedia article. Still, I am sure others would appreciate the first video to be about what Quantum Logic is and it's applications.
The only prerequisite is Linear Algebra - a knowledge of vectors and linear operators is needed, although I would hope to review what is needed. Any experience in abstract algebra would be helpful as well, but not necessary.
This would not help me much now because I am probably not at the point where I could fully understand appreciate it, but it would be great to come back to it. That is the great thing about lectures online because even 20 years down the line they can still be accessed in some way (hopefully).
I may stick with the writing tablet approach used in Khan Academy simply because of time issues. Chalkboard talks are terrific, but I'm not yet convinced they're the best for the video medium. That being said, I'm interested in hearing why people might prefer other options!
Here's why I prefer chalkboard over Khan's style. First, I can actually see you talk, gesture, etc., and this makes it easier to pay attention to your voice. Second, I can see the whole chalk board at once. It's nice to glance back to an earlier part of the board mid-lecture to gain my bearings. Third, I guess it's more aesthetically pleasing than just an electronic display.
I love quantum and I love logic, so quantum logic would be one of the best things ever.
Seriously, if you really have time, resources and knowledge to do this, please, please, pretty please with sugar on top, do it!
And if I might ask: HD, or at least good quality 480p, good sound (this one is the hardest one), do not write to much (if it's faster and easier to say something, don't bother with writing it down), but on the other hand, write everything needed so that a guy whose maths education stopped on linear & abstract algebra, statistics, multivariate calculus and computability theory, with only basic understanding of QM and quantum computing will understand it.
If you manage to do it, kudos, karma, and every other internet equivalent of those for you.
I am working on it. The first step is to re-outline my current notes on the topic, and then get to work on the video lessons themselves. My first Wacom tablet is on its way, and once it arrives I'll be ready to start producing videos.
I'm curious because I'm not clear on what you mean specifically by "quantum" logic. Quantum is like the religion of physics. Every time I come across it, I'm like, "what?? I don't know if I believe that..."
Here's the short version. In the classical world, whether we're rolling dice or talking about the weather, we have ways of making "logical" conclusions which follow "common sense." In quantum physics, many phenomena appear to defy common sense, which is the very root of why quantum theory feels weird or mystical. This all traces back to the fact that common sense is not enough to describe the real world.
This is where quantum logic steps in. We get a new way of describing events and their probabilities so that the mystery of quantum physics is a bit more tangible.
Gotcha. I wasn't sure if you meant quantum logic as far as the logic behind something like quantum computing or "Cooper pairs" or if you were referring to the idea of logic backing up the notions behind quantum phenomena. |
Used Condition. Multiple Copies Available for immediate shipment. Cover may have minor imperfections. Binding tight Contents clean and intact. May contain minimal ...highlighting or writing.Read moreShow Less
Florence, Kentucky, U.S.A. 1998 Hardcover Very Good Very goodHelping students grasp the why of algebra through patient explanations, Goodman and Hirsch gradually build students' confidence without sacrificing rigor. To help students move beyond the "how" of algebra (computational proficiency) to the "why" (conceptual understanding), the authors introduce topics at an elementary level and return to them at increasing levels of complexity. Their gradual introduction of concepts, rules, and definitions through a wealth of illustrative examples—both numerical and algebraic—helps students compare and contrast related ideas and understand the sometimes subtle distinctions among a wide |
If for any reason you are not happy to continue with your course, just return it within 7 days for a full refund.
Course Overview
FAST TRACK
A Level Mathematics (Full AS + A2) Course
A distance learning course is the ideal way to gain an A Level in Mathematics. Whether you're looking to go on to further education, improve your job prospects or expand your knowledge, distance learning A Level Mathematics is a flexible and convenient course, which allows you to comprehensively prepare for the A Level Mathematics exam through home study. What's more, because the distance learning A Level Maths course is a fully comprehensive course, no prior knowledge is required.
Our Fast Track A Level courses are 1 year courses with which you will receive priority tutor support throughout the duration. Upon completion of the course you will receive a certificate awarded by a national awarding body such as Edexcel, AQA and OCR. We also offer UCAS points with these courses which will vary depending on the grade you receive.
By the end of your course, you will be able to work with complex functions and understand how to calculate forces in mechanics. You will cover topics such as geometry, calculus, trigonometry and algebra and gain the ability to manipulate figures, and use abstract reasoning and logic in problem solving.
There are many modules that make up A-Level Mathematics. The main modules are as follows:
Core 1 Core 2 Mechanics 1 Core 3 Core 4 Mechanics 2
The first three modules constitute AS Level and the later three modules are A2 Level. Each module is presented separately. The content of the modules have references to Edexcel Modular Mathematics textbooks, which will need to be purchased in conjunction with the course. The course builds up understanding of key mathematical concepts gradually and sequentially.
Topics such as differentiation and integration are covered in all the Core modules and form part of the essential mathematical toolkit needed for higher study.
Mathematics has a distinct vocabulary, notation and terminology all of its own – for many mathematics is a language. You are expected not only to recognise and understand notation and vocabulary in examination papers, but use them correctly in your answers. If you have problems remembering these it is a good idea to write them on index cards or to develop a file of key words as you go along in order to help you learn them.
HOME STUDY SUPPORT
___________________________________________________________
You will be provided with comprehensive online materials designed to provide you with everything required to complete your course of study. You will have your own personal tutor helping you with your course work and with any questions you may have. Plus you can contact a Student Advisors by email or phone for all the practical advice you may need – so we really are with you 100%.
Your tutor will give you various resource sites to visit and there is extra material in the student support portal. In addition there are several supporting texts books for this course but again this depends on which approach you are following and which book best suits your individual needs
Further information can be obtained through Edexcel. The final assessment criteria and structure is the same regardless of which approach you choose.
For each module there are two or three formal tutor-marked assessments which, when completed, should be sent to your tutor, via e-mail or conventional post. All of the information within the units is written in line with the requirements of the Edexcel examination board specification and the demands of the examination.
All units are broken down into clear subject topics, and students should spend the amount of time studying for each topic as advised by their course tutor. Each topic module has been written for ease of understanding and topic coverage may be of different length and difficulty depending on the level of detail and information required. You will need to study the modules in the sequence provided.
The AS and A2 modules link together and assessment is all done by examination as there is no coursework option.
Students are required to arrange and pay for examinations, students should note exam dates for the syllabus via the exam board website. Details of examination dates and examination enrolment deadlines are included in the 'Student Welcome Pack'.
We recommended you spend approximately 300 hours of your time studying for the Mathematics Course. The pace of study is completely up to you. To give an example, if you dedicate 6 hours a week to the course it would take you a year to complete but if you could spare 12 hours a week you could complete it in six months.
Get started today, and you can earn your A Level in Mathematics in as little as six months, or take up to a year, if you prefer – the choice is yours.
The enrolment fee for the Fast Track A' Level Mathematics (Full AS + A2) home study course is £456.25, though for a limited time we are offering you the opportunity to pay only £365 which is a 20% discount if you enrol online and pay in full.
You can also opt for our Easy Payment Plan and enrol online today by paying a deposit of £91.25 and then 4 equal payments of £91.25 per month. The first instalment is paid about a month after you receive your course.
Private Practice Accountant
As an accountant working in private practice, it would be your job to provide accountancy services for fee-paying clients. You would handle clients' tax and accounts, carry out audits, and advise clients how to improve their business finances. Your clients could range from small businesses that do not employ their own in-house management accountant, to large companies or wealthy individuals. Your main tasks could include: preparing financial statements, business plans and budget reports, producing annual and monthly accounts, carrying out independent audits of company accounts, managing clients' spending, costs, credit, payroll and investments, filing tax returns and providing tax advice, forecasting future profits and financial performance, helping businesses that may be in financial difficulties, dealing with insolvencies, advising companies how to improve their financial systems and profitability and finding and preventing fraud (known as forensic accounting). You might deal with all of these tasks if you worked freelance or for a small accountancy practice, whilst in a larger practice you might specialise in one area such as tax or insolvency.
WHAT'S INCLUDED
___________________________________________________________
Your course fee covers the full syllabus that you will need to successfully complete the Fast Track A' Level Mathematics (Full AS + A2) home study course with 12 months full tutor support.
1. Online study materials to enable the student to successfully complete the A' Level.
2. A full range of student services, including:
Tutor marked assessments (TMAs)
Access to the online student portal (Student Chat, Forums and Online Support Resources)
In the student 'On Campus' you are also able to take part in the student chat room and forums as part of our online student community. After enrolling online you will receive your username and password to access the On Campus area within 5 working days. You will also receive your personal student number via email.
Students are required to arrange and pay for their examinations and manage the course work element if the subject requires this. Students must check the relevant examination board website for further information and final examination sitting dates for the specification.
From time to time we may enrol our students with our partner sites; this is dependent on the number of students enrolling on a particular course and course material availability. If this happens, nothing changes for you other than the name of the college administering your course. We will continue to be your point of contact; you will get the exact same course you have enrolled on with the same high level of quality content and support.
The course can be enrolled upon by students Internationally. There are no deadlines for enrolments.
WHAT TO DO NOW?
___________________________________________________________
Step One: It's simple - all you need to do now is choose whether you want to pay in instalments or in full and then click on the relevant enrolment button at the top of this page.
Step Two: Once you have selected your payment option to enrol you will be redirected (this can take a few seconds) to our PayPal payment page (you do not need a PayPal account) for you to select your payment method and complete your enrolment.
Step Three: Within 7 days (normally 48 hours) your enrolment papers and course materials will be with you. A tutor and a dedicated support advisor will also be allocated to you.
That's it..... Thanks and wishing you the best of luck with your studies.
___________________________________________________________
live chat
Still got a questions? Talk to a specialist A' Level advisor...
Course Brochure
Download our course brochure for all the information you may need about this course and it's awarding body. You can enrol online, or by post using the enclosed form.
At eDistance Learning we fully understand the need to support our students beyond their course completion to help them achieve their desired goals. To help you to succeed in your chosen career, we will be pleased to provide the following services on successful completion of your course. |
Mathematical Excursions - Text Only - 04 edition
Summary: A new text for the liberal arts math course by a seasoned author team, Mathematical Excursions, is uniquely designed to help students see math at work in the contemporary world. Using the proven Aufmann Interactive Method, students learn to master problem-solving in meaningful contexts. In addition, multi-part Excursion exercises emphasize collaborative learning. The text's extensive topical coverage offers instructors flexibility in designing a course that meets the...show moreir students' needs and curriculum requirements. ...show less Lockwood, Joanne S.Section 4.1 Early Numeration Systems Excursion: A Rosetta Tablet for the Traditional Chinese Numeration System Section 4.2 Place-Value Systems Excursion: Subtraction Via the Nines Complement and the End Around Carry Section 4.3 Different Base Systems Excursion: Information Retrieval Via a Binary Search Section 4.4 Arithmetic in Different Bases Excursion: Subtraction in Base Two Via the Ones' Complement and the End Around Carry Section 4.5 Prime Numbers and Selected Topics from Number Theory Excursion: The Distribution of the Primes Section 4.6 Additional Topics from Number Theory Excursion: A Sum of the Divisors Formula0395727790 |
In this calculus worksheet, students use integration to solve word problems they differentiate between integration and anti derivatives, and between definite and indefinite integrals. There are 3 questions with an answer key.
Eleventh and twelfth graders find the volume of figures using cross sections. They use their Ti-Nspire to find the volume of a solid formed by cross sections of a function. Pupils find the integrals of the shapes using cross sections.
Twelfth graders explore an application of the definite integrals. In this calculus lesson, 12th graders graph three functions on the same domain and each goes through the same three points. Students use the symbolic capacity of their calculator and calculus to find the shortest of each of the paths through these points. Students explore the piece-wise linear function which also goes through the three points.
Twelfth graders investigate an application of definite integrals. In this Calculus instructional activity, 12th graders use the symbolic capacity of the TI-89 calculator and the concept of the definite integral to explore the area bounded by a function and the x-axis.
Watch this video with your junior engineers or your higher-level mathematicians. In it, Drew builds half of an arch bridge with building blocks. There happens to be a magnetic board behind the blocks, on which he affixes a magnet for each block. When the blocks are removed, what appears to be a graph of an integral function, a natural logarithm, is left behind. Really revealing!
In this college level calculus worksheet students apply various integrations techniques and use integration to determine the area bounded by curves. The three page worksheet contains eight problems Answers are not provided.
Students explore the concept of finding the area between two curves. In this finding the area between two curves lesson, students model the logos of McDonalds, Nike, and Motorola on grid paper. Students find functions to represent the logos. Students find the area between two curves of each logo by taking the integrals of the functions.
Students calculate area under a curve using Riemann Sums. In this calculus lesson plan, students investigate the integral through estimation and calculation. They compare their approximate answer to their true answer.
For this college level calculus worksheet, students evaluate the given integrals and compute the given limits or show that the limit does not exist. The three page worksheet contains ten problems. Answers are not provided. |
This lesson from Illuminations looks at exponential decay. The example of how kidneys filter blood is used. The material asks students to determine the amount of a drug that remains in the body over a period of time....
This lesson from illuminations helps to illustrate quadratic equations. Students will determine the maximum value of a quadratic equation and compare different equations. The lesson also asks students to move between...
This lesson uses the example of successive discounts at a retail store to demonstrate numeric, algebraic, and graphical representations of compositions of mathematical functions. Students will get the opportunity to...
This algebra lesson from Illuminations helps students learning to graph real world data. Students will collect, graph and analyze data, and choose an appropriate mathematical model for a particular situation. They will...
This algebra lesson from Illuminations has students collect data for rolling objects of differing sizes in order to further understand periodic phenomena. They will then create two sinusoidal graphs of the data.... |
More About
This Textbook
Overview
Based on recent research on the adolescent brain, Active Algebra presents a living, working example of how teachers can use active learning techniques to make linear relationships more meaningful for students. In addition to the 10 reproducible, sequenced lessons, this award-winning resource includes seven chapters of guidance in teaching algebra. It offers comprehensive coverage of active learning strategies, mental math, student presentations, graphing calculators, classroom management and discipline, authentic classroom experiences, and brain research and mathematics. It also provides connections to NCTM's Principles and Standards for School Mathematics |
MiLearn High School Course Complements
Curriculum-based high school level math resources in an interactive, "concept, example, application" web design. Courses contain over 500 web pages of step-by step examples and solutions; free demos are available on the site. MiLearn also provides consulting
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MindPlay
The three-part series Fraction-Oids helps children understand, in a visual way, what happens when a denominator gets smaller or larger; discover for themselves what a common denominator is; understand why division of fractions makes numbers bigger; and
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mrs. tinashe blanchet - Tinashe Blanchet
Courses, collaborations, and other content by Blanchet, who teaches high school math in Marrero, Louisiana. Freely-accessible Moodle classes have included International Baccalaureate (IB) Math Studies and Advanced Math/Pre-Calculus; professional development
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MyiMaths - MyiMaths Ltd
A teaching resource comprised of hundreds of interactive lessons, each with dynamically generated activities, games, tools, and questions. See samples and reviews before purchasing a subscription.
Oliver Knill
Research, teaching, and media by this preceptor in Harvard's math department. Knill began in dynamical systems, tackling first ergodic and spectral theoretical questions, then probability theory and elementary number theory. This led to his "passion for
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OnlineCollegePrep.com - Robert Hackworth
Distance learning courses in reading, writing, math, and study skills. A free online arithmetic course is available for adults needing help to prepare for pre-college and college-level math instruction. A real instructor instructs and leads the course.
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Penn State University Mathematics Department
Course home pages and instructional material; seminars, colloquia, and conferences; and subject area pages for Penn State research centers, with preprints and links to other resources on the Web: Algebra and Number Theory; Dynamical Systems; Mathematical
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Philippe Flajolet
Philippe Flajolet researches the analysis of algorithms, analytic combinatorics, computer algebra, asymptotic analysis, special functions, random structures, and natural languages. An extensive collection of his articles, and the first chapters of
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Philosophy of Science - David Banach
A syllabus and collected resources for a course in the philosophy of science, including excerpts from various books. Topics include ancient Greek science and mathematics: the Golden Section, Pythagoras, infinity and continuity, and Plato and Aristotle,
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The PostCALC Project - Duke University
The PostCALC Project is a branch of the Connected Curriculum Library that presents interactive, mathematically-based modules designed for high school students who have finished a year-long course in calculus. These modules, each appropriate for several
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Probability Tutorials - Noel Vaillant
An online course on measure theory, lebesgue integration and probability, with tutorials (in PDF format) designed as a set of simple exercises, leading gradually to the establishment of deeper results. Proved theorems, as well as clear definitions, are
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Project NExT: New Experiences in Teaching - MAA, Exxon
Project NExT (New Experiences in Teaching) is a program for new or recent Ph.D.s in the mathematical sciences who are interested in improving the teaching and learning of undergraduate mathematics. See, in particular, Classroom Materials: Writing Intensive
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Project VISIT
An online professional development collaboratory for high school teachers of science, technology, and social science, supported by the National Science Foundation. VISIT seeks teacher leaders to participate in leading this program during February - May,
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An Introduction to Hyperbolic 3-Manifolds
0521839742
9780521839747
Outer Circles:We live in a three-dimensional space; what sort of space is it? Can we build it from simple geometric objects? The answers to such questions have been found in the last 30 years, and Outer Circles describes the basic mathematics needed for those answers as well as making clear the grand design of the subject of hyperbolic manifolds as a whole. The purpose of Outer Circles is to provide an account of the contemporary theory, accessible to those with minimal formal background in topology, hyperbolic geometry, and complex analysis. The text explains what is needed, and provides the expertise to use the primary tools to arrive at a thorough understanding of the big picture. This picture is further filled out by numerous exercises and expositions at the ends of the chapters and is complemented by a profusion of high quality illustrations. There is an extensive bibliography for further study.
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Rent Outer Circles 1st edition today, or search our site for A. textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Cambridge University Press. |
study of sequences, although seen as an incipient numerical progression, is the foundation of mathematical analysis. This...
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The study of sequences, although seen as an incipient numerical progression, is the foundation of mathematical analysis. This concept in turn is based on set theory and Dedekind cuts, and is studied by mathematical analysis"iLumina is a digital library of sharable undergraduate teaching materials for chemistry, biology, physics, mathematics, and...
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"iLumina is a digital library of sharable undergraduate teaching materials for chemistry, biology, physics, mathematics, and computer science. It is designed to quickly and accurately connect users with the educational resources they need. These resources range in type from highly granular objects such as individual images and video clips to entire courses. Resources in iLumina are cataloged in the MARC and NSDL metadata formats, which capture both technical and education-specific information about each resource. iLumina contains thousands of educational resources and several virtual collections. Please feel free to contribute your own resources to iLumina by following the contribute link in the header."
Complete Classroom InterActivity for exploring the fact: If the median to the longest side of a triangle and the two halves...
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Complete Classroom InterActivity for exploring the fact: If the median to the longest side of a triangle and the two halves of this side are equal, then the triangle is a right-triangle.Includes 1 page worksheet for student; 1 page answer sheet for educator.Estimated class time: 45 minutesCan be connected to Thale's Theorem on circles, diameters and right-triangles.Can be done online or downloaded for offline use. |
Union Square, NJ Prealgebra teach you to prepare animated slides for your presentations in physical sciences courses. In some curricula, pre-calculus would be more informatively labeled "Algebra 3." It is a last-minute maintenance check-up to ensure that students can perform algebraic operations involvi |
Official Course Information
Mathematics Courses:
Catalog Description:
Focuses on foundational concepts of number theory, arithmetic, and algebra that underlie the elementary school mathematics curriculum. Emphasizes problem solving and the historical development of numeration systems and other mathematical concepts. Prerequisite: MAT092 with a grade of C or better, or equivalent by placement.
Catalog Description:
Covers fundamental concepts of probability, statistics, measurement and geometry and the historical development of Mathematics. Develops understanding of mathematical concepts that form the foundation for topics included in the elementary mathematics curriculum and further develops creative problem-solving and mathematical communication skills. Prerequisite: MAT116.
Catalog Description:
Topics include absolute value equations and inequalities; solving systems of linear equations using determinants; review of rational exponents, radical expressions and complex numbers; quadratic equations and inequalities; solving equations reducible to quadratic form; quadratic, exponential and logarithmic functions and applications; graphs of functions; algebra of functions; inverse functions; conic sections; nonlinear systems of equations and inequalities; right triangle trigonometry; trigonometric functions and the unit circle. Not open to students with credit in MAT 140 or higher. Prerequisite: MAT 102 with a grade of 'C' or higher or by placement.
Catalog Description:
Covers the principal concepts of differential and integral calculus as they relate to business applications. Students will study functions (including exponential and logarithmic), limits, differentiation, and integration. Emphasis will be placed upon the use of calculus in solving problems from the fields of business & economics. Prerequisite: MAT 136 or MAT 140 with a grade of 'C' or higher or by placement or instructor permission.
Catalog Description:
Studies polynomial, rational, exponential, logarithmic, and trigonometric functions and their inverses.
Four class hours. Not open for credit to students who have credit in MAT 141 or higher.
Prerequisite: MAT 136 (with a minimum grade of C) or equivalent by placement.
Catalog Description:
Studies functions of a single variable with regard to limits, continuity, differentiation, anti-differentiation, and applications of these topics. Concludes with a study of the definite integral and the fundamental theorem of calculus.
Four class hours. Prerequisite: MAT 140 (with a minimum grade of C) or equivalent by placement.
Catalog Description:
Covers infinite series, curves in parametric and polar forms, vectors, partial differentiation, and multiple integrals. Applications of these topics focus on analysis of functions and surfaces in 3 dimensional space. Fall only. Prerequisite: MAT 142 with a grade of 'C' or higher or by placement. May be taken concurrently with MAT 142 with instructor permission.
Catalog Description:
Covers solutions of first order differential equations, Euler's Method, linear differential equations with constant coefficients, non-linear equations, LaPlace transforms, numerical solutions, and series solutions. Applications focus on modeling with first- and second-order linear equations. CAS Software (such as Derive or Maple) will be used for solving a variety of application problems. Spring only. Prerequisite: MAT 142 with a grade of 'C' or higher or by placement. |
Mathematics Problems And Solutions
Tools for Solving Mathematics problems. Tools for Solving Mathematics problems. Mathematics Tools is a tools that help people in solving Mathematical problems such as: Solving quadratic equation and cubic equationSolving System of equations (2 or 3 unknowns)Working in the Base Number...
Magic Problems Creator for Mathematics has a powerful Wizard to help you Create your own custom problems collections. Magic Problems Creator for Mathematics has a powerful Wizard to help you Create your own custom problems collections.ALGEBRA: - Matrices, determinants - Matricial equations - Study of system of equations- Easy and fast to use.- User's manual in...
Visual Mathematics is a highly interactive visualization software (containing -at least- 67 modules) addressed to High school, College and University students. This is a very powerful tool that helps to learn and solve problems by the hundreds in a very short time. Included areas: Arithmetic, Algebra, Geometry, Trigonometry, Analytic Geometry and miscellaneous.Visual Mathematics, a member of the Virtual Dynamics Mathematics Virtual Laboratory,...
Numerical Mathematics is the branch of mathematics that develops, analyzes, and applies methods to compute with finite-precision numbers. Numerical Mathematics is the branch of mathematics that develops, analyzes, and applies methods to compute with finite-precision numbers. Numerical mathematics is a vast field whose importance cannot be over-emphasized. The solution of real-life...
Interpolation and Regression are fundamental and important calculations in mathematics. Interpolation and Regression are fundamental and important calculations in mathematics. Mr. Newton and Mr. Gauss were engaged in-depth with numerical solutions for these problems. Today, there are improved algorithms, that can solve such tasks....
An game that teaches mathematics whilst being fun and rewarding. The interactive content of the computer game teaches the student to really see how mathematic skills can help to solve daily problems and how it helps to play along in the game. The game mix the classic platform genre with interactive problems and practice sessions. The software contains simulations of a calculator, sketch pad, graphs and adds the ability to construct objects.A simple mathematics program for students. A simple mathematics program for students. FMFS stands for FMFS may free students, it is a small collection of programs related to mathematics. It consists of a 2D graph plotter, a 3D graph plotter, a evaluator for basic mathematics expressions at...
Implementation done in Mathematics for a quadratic sieve. Implementation done in Mathematics for a quadratic sieve. A very basic Quadratic Sieve implementation done for a BSc in Mathematics, using MPIR. Experienced programmers may groan when seeing the code - however it should give a simple enough...
Approximates tabulated functions in 1 to 4 independent variables by finite power and/or trigonometric series. Approximates tabulated functions in 1 to 4 independent variables by finite power and/or trigonometric series. The program fits the function exactly at the grid points. |
Precalculus - With 2 CDS - 4th edition
Summary: Bob Blitzer's background in mathematics and behavioral sciences, along with his commitment to teaching, inspired him to develop a precalculus series that gets students engaged and keeps them engaged. Presenting the full scope of mathematics is just the first step. Blitzer draws students in with applications that use math to solve real-life problems. is most energized by teaching mathematics and has taught a variety of mathematics courses at Miami-Dade College for nearly 30 years. He has received numerous teaching awards, including Innovator of the Year from the League for Innovations in the Community College, and was among the first group of recipients at Miami-Dade College for an endowed chair based on excellence in the classroom. Bob has written Intermediate Algebra for College Students, Introductory Algebra for College Students, Essentials of Intermediate Algebra for College Students, Introductory and Intermediate Algebra for College Students, Essentials of Introductory and Intermediate Algebra for College Students, Algebra for College Students, Thinking Mathematically, College Algebra, Algebra and Trigonometry, and Precalculus, all published by Pearson Prentice Hall.
Absolutely BRAND NEW ORIGINAL US HARDCOVER STUDENT 4th Edition / Mint condition / Never been read / ISBN-10: 0321559843. Shipped out in one business day with free tracking.169.95 +$3.99 s/h
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100% BRAND NEW ORIGINAL US HARDCOVER STUDENT 4th Edition / Mint condition / Never been read / ISBN-10: 9780321559845. Shipped out in one business day with free tracking.
$175.00 +$3.99 s/h
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bluehouse acton, MA
Brand new.
$20903215598 |
Calculus Demystified - 03 edition
Summary: Here's an innovative shortcut to gaining a more intuitive understanding of both differential and integral calculus. In Calculus Demystified an experienced teacher and author of more than 30 books puts all the math background you need inside and uses practical examples, real data, and a totally different approach to mastering calculus. With Calculus Demystified you ease into the subject one simple step at a time -- at your own speed. A user-friendly, accessible style ...show moreincorporating frequent reviews, assessments, and the actual application of ideas helps you to understand and retain all the important concepts33.32 +$3.99 s/h
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Books Revisited Chatham, NJ
As new.
$42.64 |
More About
This Textbook
Overview
Calculus teachers recognize Calculus as the leading resource among the "reform" projects that employ the rule of four and streamline the curriculum in order to deepen conceptual understanding. The fifth edition uses all strands of the "Rule of Four" - graphical, numeric, symbolic/algebraic, and verbal/applied presentations - to make concepts easier to understand. The book focuses on exploring fundamental ideas rather than comprehensive coverage of multiple similar cases that are not fundamentally unique. Readers will also gain access to WileyPLUS, an online tool that allows for extensive drills and practice. Calculus teachers will build on their understanding in the field and discover new ways to present concepts to their students.
Editorial Reviews
Booknews
Focusing on a limited number of topics in more depth than is perhaps customary in a calculus text, this volume emphasizes the meaning, in practical, graphical, and numerical terms, of the symbols used. Chapters cover functions, the derivative and the definite integral, short-cuts to differentiation, using the derivative, constructive antiderivatives, integration, using the definite integral, approximations and series, differential equations, functions of several variables, vectors, differentiating functions of many variables, optimization, integrating functions of many variables, parameterized curves, vector fields, line integrals, flux integrals, and calculus of vector fields. Includes short answers to odd-numbered problems at the back of the book |
Algebra 2, Student Edition
From the first day your students begin to learn the vocabulary of algebra until the day they take final exams and standardized tests, these programs ...Show synopsisFrom the first day your students begin to learn the vocabulary of algebra until the day they take final exams and standardized tests, these programs strengthen student understanding and provide the tools students need to succeed 2A 3E.
Description:Fair. 0028251369 MULTIPLE COPIES AVAILABLE-COVER IS TATTERED...Fair. 0028251369 |
reference source for advanced high school and college math students. Also useful to professionals who use math on the job. Approximately 700 math terms are defined. Includes illustrative diagrams. |
Study & Master Mathematics Grade 9 is an exciting new course that covers and integrates all the Learning Outcomes for Mathematics stated in the National Curriculum Statement. The material is presented in a user-friendly way that will not only boost learners' confidence, but will also show them how to enjoy Mathematics. |
slight shelf wear. NO observed tears, highlighting or writing"[Kline] is unfalteringly clear in explaining mathematical ideas; he is learned but not pedantic; he has historical discernment, a sympathetic social outlook and a nice sense of fun and irony.... The beauty and fascination and rare excellence of mathematics emerge from his story. It is an exciting, provocative book."--Scientific American
"Still the best textbook for the history and philosophy of mathematics for undergraduate liberal arts students. Especially good for the age of the Scientific Revulution."--Janet A. Fitzgerald, Molloy College, NY
About the Author
Morris Kline is Professor Emeritus at the Courant Institute of Mathematical Sciences, New York University.
Most Helpful Customer Reviews
This book is geared to the general reader who has a cursory knowledge of mathematics. The chapters are organized around physical phenomena and the math behind their explanation. The result is a charming and VERY useful book. I have the 1970 edition which is quite worn from frequent use. The chapter titled, Differential Equations - The Heart Of Analysis, is exceptionally beautiful and pertinent. Reading this book is akin to a treasure hunt. There is page after page of mathematical discovery. Reading the chapter on Motion Of Projectiles made me terribly angry at the banal way in which this topic is handled in high school texts. Things such as quadratic equations and the law of gravitation are explained very well. I sincerely believe that this book should be a required text for High School math students. Highly recommended. The Dover edition is very affordable so even if it means foregoing a meal, do it. Buy this book! Well worth your time.
Kline's book is a surprisingly accessible history of math, equations and all. It's a perfect balance for those who know their history and want to know more about the math behind scientific ideas, and those who know their math and want to delve into the practical applications of mathematical ideas. I loved the examination of how Eratosthenes calculated the circumference of the earth pretty accurately for a guy without so much as a telescope. I had terrific "aha" experiences reading the book, and the students I tutor in math are getting a bunch of interesting background stories with their algebra lessons. Galileo's and Newton's calculations are a great way for the theoretically inclined to get their feet wet in physics. For all of us mathematicians who actually thought math developed in a vacuum, this book has excellent examples of how inextricably linked math, science and history are.
I still have a hard time deciding if this is a book about math or physics, but actually, it's about both. Kline follows the history of physics beginning with the Greeks as a way to describe the evolution of mathematics up to the 20th century, and he does it brilliantly.
Kline is a master teacher, and his enthusiasm for his subject is evident. This is probably the clearest writing on mathematics (and physics) that I have ever seen. His chapters on the differential and integral calculus make their basic principles understandable to anyone with a high-school mathematics education -- not an easy task.
He shows how again and again through history, problems in physics led to mathematical discoveries which not only allowed the physical problems to be solved, but also advanced mathematics itself. And yet nowhere in the book is anything beyond simple algebra and geometry required to understand what he's saying. A remarkable achievement.
I agree with the above review and would simply like to add my own thoughts. The book illustrates the fascinating way in which mathematics, society, religion, politics and of course physics have affected each other (it goes both ways!) through out the ages. Furthermore, the author nicely illustrates the processes by which people think and how those processes have also changed through the ages (i.e., The Age of Reason versus The Renisance). This book left me with real insights as to the nature and limitations of the current state of mathematics and physics. Things are not as they seem, my friend! Lastly, the author displays an appreciation for the humor and irony of the history which makes this book hard to put down at times. I never thought a math/history book could be a "page turner"... Read it.
In most mathematics classes, students are presented with a completed edifice, and given a floor plan to help them navigate the halls. While this approach works for many people, others need a little more basic information. In this book, Morris Kline builds the building, starting with the mud and straw of the bricks. "Mathematics in Western Culture" shows that the history of mathematics is one of hundreds of years of people sitting in the sand, drawing shapes and lines, scratching their heads, and trying to figure things out. This is not necessarily Dr. Kline's intention for the book, but this is certainly one of the many messages to be derived from it. A fascinating, exciting book which makes mathematics more understandable and accessible.
This book is excellent! Have you ever wondered, where did math come from? What caused/ how has math to developed? If so this book will hit the nail on the head for you. Dr. Kline is fabulous in this book, he explains things very clearly and gives the reader an overview of some of the more practical uses of math. After reading this book you will look at the world with a much better understanding of how math is used in the real world.
Kline also explains why math is so abstract (think of the way American schools teach math). Along with this he explains why math is so precise (due to it's being limited to using inductive reasoning only).
In fact, this book is a humanities course mixed in with the practical usage of mathematics, which all add up to a brilliant text. But don't be mislead, the book is not absent of the actual equations to help you understand some of the math. It's just simplified so as to be short of a textbook on how to do mathematics.
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books.google.com - In... Mathematics
In us partly defined problems needing only moderately accurate solutions. This engaging book is an antidote to the rigor mortis brought on by too much mathematical rigor, teaching us how to guess answers without needing a proof or an exact calculation.In Street-Fighting Mathematics, Sanjoy Mahajan builds, sharpens, and demonstrates tools for educated guessing and down-and-dirty, opportunistic problem solving across diverse fields of knowledge--from mathematics to management. Mahajan describes six tools: dimensional analysis, easy cases, lumping, picture proofs, successive approximation, and reasoning by analogy. Illustrating each tool with numerous examples, he carefully separates the tool--the general principle--from the particular application so that the reader can most easily grasp the tool itself to use on problems of particular interest. Street-Fighting Mathematics grew out of a short course taught by the author at MIT for students ranging from first-year undergraduates to graduate students ready for careers in physics, mathematics, management, electrical engineering, computer science, and biology. They benefited from an approach that avoided rigor and taught them how to use mathematics to solve real problems.Street-Fighting Mathematics will appear in print and online under a Creative Commons Noncommercial Share Alike license.
User ratings
A fantastic book which only requires some mathematical wherewithal. Covers various methods for estimation and informal proofs. I really liked the intuitive way he describes the topics, which range ...Read full review
Not to brag, but I was a math whiz in high school and college, which is why I'm a fan of pop-math books that frame real world issues through the lens of numbers. Given the title of this book (and its ...Read full review |
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Number theory, an ongoing rich area of mathematical exploration, is noted for its theoretical depth, with connections and applications to other fields from representation theory, to physics, cryptography, and more. While the forefront of number theory is replete with sophisticated and famous open problems, at its foundation are basic, elementary ideas that can stimulate and challenge beginning students. This lively introductory text focuses on a problem-solving approach to the subject.Key features of Number Theory: Structures, Examples, and Problems:* A rigorous exposition starts with the natural numbers and the basics.* Important concepts are presented with an example, which may also emphasize an application. The exposition moves systematically and intuitively to uncover deeper properties.* Topics include divisibility, unique factorization, modular arithmetic and the Chinese Remainder Theorem, Diophantine equations, quadratic residues, binomial coefficients, Fermat and Mersenne primes and other special numbers, and special sequences. Sections on mathematical induction and the pigeonhole principle, as well as a discussion of other number systems are covered.* Unique exercises reinforce and motivate the reader, with selected solutions to some of the problems.* Glossary, bibliography, and comprehensive index round out the text.Written by distinguished research mathematicians and renowned teachers, this text is a clear, accessible introduction to the subject and a source of fascinating problems and puzzles, from advanced high school students to undergraduates, their instructors, and general readers at all levels5
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With a focus on the needs of educators and students, Making Sense of Data presents the steps and issues that need to be considered in order to successfully complete a data analysis or data mining project. This Second Edition focuses…
An essential library of basic commands you can copy and paste into R The powerful and open-source statistical programming language R is rapidly growing in popularity, but it requires that you type in commands at the keyboard rather than use…
A visual way to easily access the strategies and tactics in Book Yourself Solid Learning new concepts is easier when you can see the solution. Book Yourself Solid Illustrated, a remarkable, one-of-a-kind work of art, transforms the Book Yourself Solid system…
The Bittinger Worktext Series changed the face of developmental education with the introduction of objective-based worktexts that presented math one concept at a time. This approach allowed readers to understand the rationale behind each concept before practicing the associated skills…
This book is Part I of the fourth edition of Robert Sedgewick and Kevin Wayne's Algorithms , the leading textbook on algorithms today, widely used in colleges and universities worldwide. Part I contains Chapters 1 through 3 of the book…. |
Jun 27, 2008
The National Institute of Standards and Technology (NIST) has released a five-chapter preview of the much-anticipated online Digital Library of Mathematical Functions (DLMF). In development for over a decade, the DLMF is designed to be a modern successor to the 1964 "Handbook of Mathematical Functions," a reference work that is the most widely distributed NIST publication (with over a million copies in print) and one of the most cited works in the mathematical literature (still receiving over 1,600 yearly citations in the research literature). The preview of the new DLMF is a fully functional beta-level release of five of the 36 chapters.
The DLMF is designed to be the definitive reference work on the special functions of applied mathematics. Special functions are "special" because they occur very frequently in mathematical modeling of physical phenomena, from atomic physics to optics and water waves. These functions have also found applications in many other areas; for example, cryptography and signal analysis. The DLMF provides basic information needed to use these functions in practice, such as their precise definitions, alternate ways to represent them mathematically, illustrations of how the functions behave with extreme values and relationships between functions.
The DLMF provides various visual aids to provide qualitative information on the behavior of mathematical functions, including interactive Web-based tools for rotating and zooming in on three-dimensional representations. These 3-D visualizations can be explored with free browsers and plugins designed to work in virtual reality markup language (VRML). Mouse over any mathematical function, and the DLMF provides a description of what it is; click on it, and the DLMF goes to an entire page on the function. The DLMF adheres to a high standard for handbooks by providing references to or hints for the proofs of all mathematical statements. It also provides advice on methods for computing mathematical functions, as well as pointers to available software.
The complete DLMF, with 31 additional chapters providing information on mathematical functions from Airy to Zeta, is expected to be released in early 2009. With over 9,000 equations and more than 500 figures, it will have about twice the amount of technical material of the 1964 Handbook. An approximately 1,000-page print edition that covers all of the mathematical information available online also will be published. The DLMF, which is being compiled and extensively edited at NIST, received initial seed money from the National Science Foundation and resulted from contributions of more than 50 subject-area experts worldwide. The NIST editors for the DLMF are Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert and Charles W. Clark.
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A classic online mathematical reference offered by the National Institute of Standards and Technology (NIST) now features a better way for users to view its most complicated illustrations—three-dimensional ...
The National Institute of Standards and Technology has released the Digital Library of Mathematical Functions (DLMF) and its printed companion, the NIST Handbook of Mathematical Functions, the much-anticipated successors |
Mathematics All Around, Fourth Edition, is the textbook for today's liberal arts mathematics students. Tom Pirnot presents math in a way that is accessible, interesting, and relevant. Like having a teacher on call, its clear, conversational writing style is enjoyable to read and focuses on helping students understand the math, not just get the correct answers on the test. Useful features throughout the book enable students to become comfortable with thinking about numbers and interpreting the numerical world around them. Problem Solving: Strategies and Principles; Set Theory: Using Mathematics to Classify Objects; Logic: The Study of What's True or False or Somewhere in Between; Graph Theory (Networks): The Mathematics of Relationships; Numeration Systems: Does It Matter How We Name Numbers?; Number Theory and the Real Number System: Understanding the Numbers All Around Us; Algebraic Models: How Do We Approximate Reality?; Modeling with Systems of Linear Equations and Inequalities: What's the Best Way to Do It?; Consumer Mathematics: The Mathematics of Everyday Life; Geometry: Ancient and Modern Mathematics Embrace; Apportionment: How Do We Measure Fairness?; Voting: Using Mathematics to Make Choices; Counting: Just How Many Are There?; Probability: What Are the Chances?; Descriptive Statistics: What a Data Set Tells Us For all readers interested in mathematics. |
Prealgebra for Two-Year Colleges/To the instructor
This is a prealgebra text for adult students at a two-year college. This text is meant to facilitate teaching via the Socratic method. The book also endeavors to connect topics through themes that are useful to adults, as opposed to covering fractions, decimals, and percents in three separate chapters, as if they were unconnected concepts.
This is a prealgebra text for adult students at a two-year college. Most of these students have passed a prealgebra class (or higher) in the past, but do not remember enough of it to accurately solve linear equations in one variable that contain fractions and decimals, without the aid of a calculator.
These students remember many mathematical procedures and rules, but they do not remember them correctly, or they do not know when to use which procedure. For example, these students remember that "two negatives make a positive," so they will tell you that (-3) + (-5) = +8. Also, they will tell you that
,
because they are misapplying the rule
.
When asked to draw a circle and shade in two fifths, these students are likely to draw one of the pictures below.
This text is meant to facilitate teaching via the Socratic method. Our premise is that use of the Socratic method will help students understand the reasons behind the procedures and rules. Students will then be able to figure out which rule to use, or (if they have forgotten the rule altogether) they will be able to reason their way to an answer.
In order to facilitate the Socratic method, this text is in the style of a workbook. There is a teacher's edition, with both halves of the Socratic dialogue (suggestions for teacher's half are in red italics). The student edition has white space for the student's work in place of the italic font. |
Find a GoldenEven though these are considered to be "pre-calculus" topics, they are important in their own right. They are often encountered in business, medicine, engineering - even in the legal arena! So even if you aren't planning to go on to calculus, being aware of them can help you in your future |
Algebra 1/ 2 : Incremental Development - 3rd edition
Summary: Algebra 1/2 is made up of five instructional components: Introduction of the New Increment, Examples with Complete Solutions, Practice of the Increment, Daily Problem Set, and Cumulative Tests. Algebra 1/2 covers all topics normally taught in pre-algebra, as well as additional topics from geometry and discrete mathematics. It is recommended for seventh-graders who plan to take first-year algebra inthe eighth grade, or for eighth-graders who plan to take first-year al...show moregebra in the ninth grade. Algebra 1/2 represents the culmination of the study of pre-algebra mathematics.
This is a worn, used book that definitely shows wear. Considered a reading copy only!
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Basic concepts and properties of algebra are introduced early to prepare for equation solving. Abundant exercises graded by difficulty level address a wide range of abilities. Concepts and skills are introduced algebraically, graphically, numerically, and verbally-often in the same lesson to help your child make the connections and to address diverse learning styles. Focused on developing algebra concepts and skills
Book Description:PRENTICE HALL, 2008. Hardcover. Book Condition: New. MULT. COPIES! This is the NJ edition! Same as the national except for the first few pages with state specific material in the beginning of the book. The national edition begins thereafter - all page#'s match up. Brand new, never used. We ship daily!. Bookseller Inventory # AMAN0005763 |
This 10 page document has three beginner word problems per page for your child/students to practice. It covers skills like addition, subtraction, multiplication, division, time and money among others. Wether it is to prepare for a state test, or just fun practice, these word problems will work out great. Great for homeschoolers!
Challenging Genetics: Study Guide is a short course that covers all chapters of genetics. It is the companion book for Challenging Genetics: Test Bank, which can be used by university, college and high school students who want to learn the concepts of genetics.
Module XI of the 1040 Exam Prep e-Books Series summarizes the several topics included in the RTRP competency examination outline related to ethics and tax preparers practice obligations before the Internal Revenue Service. The main rules set in the sub-parts A, B, and C are examined.This module also includes the basic concepts of the alternative minimum tax.
Preparing for any competency examination require of a holistic plan. Mastering the study materials is not enough. A high percentage of exam takers fail for lack of preparation for the emotional challenge that an exam represents. A successful exam experience requires a balanced effort to obtain the right mix of knowledge, confidence, and stamina. This e-book provides strategies to help exam takersModule X of the 1040 Exam Prep e-Books Series covers the principal aspects of the income tax regulation for self-employed taxpayers. Most items of income and expenses of a small business, declared in Schedule C, are examined in this module.
Module VIII of the 1040 Exam Prep e-Books Series covers the rules for the treatment of capital gains and losses for individual taxpayers. Individuals and small business income tax return are the focus of the Registered Tax Return Preparer competency examination.
Module VII of the 1040 Exam Prep e-Books Series analyses the tax credits for individual taxpayers. Tax credits are one of the components of the Form 1040 formula used to determine the tax due or the refund for a tax period. There are several types of credits that can directly reduce the tax liability. The tax liability minus tax credits equals payable taxes, or tax refund
Module VI of the 1040 Exam Prep e-Books Series summarizes what tax preparers need to know about standard and itemized deductions for the RTRP exam. Most taxpayers have the choice to use the method of deduction that gives them the lowest tax liability by choosing between standard and itemized deduction.
Module V of the 1040 Exam Prep e-Books Series focuses on the analysis of deductions. The Form 1040 formula, works with two main groups of deductions: "For AGI" or "above the line" and "From AGI" or "below the line".Both types of deductions are examined in this module.
Module IV of the 1040 Exam Prep e-Books Series focuses on the analysis of items of income that Congress has specifically authorized the taxpayers to exclude form Gross Income. Non-taxable incomes identification and analysis is covered in this module.
Module III of the 1040 Exam Prep e-Books Series presents a description and analysis of the most common elements taxpayers have to report as income and their corresponding tax treatment. As a general rule, all types of income are taxable unless they are specifically exempted by law.
Module II of the 1040 Exam Prep e-Books Series covers general considerations to take into account in filing tax returns such as taxpayers filing status and filling requirements. Questions such as which forms should be used in each case, when, where, and how should the return be filed, what records should taxpayers keep and how long, how to amend a return and much more are answered in this module |
Monday, November 4, 2013
Saturday, November 2, 2013
Reading: if you have not done it already, read p. 88, including trying example #5.
Exercises: pp. 94-96
#51, 53, 55-60 - see p. 88 & Example 5 for help.
#61-64 - see Example 3, but write out your answer Example 1 or Exercises 9 & 10. Use {}.
#65-66
#67-70 - hint: each of the functions is used only once
#77-78, 81 - You will have to...
Tuesday, October 29, 2013
Friday, October 25, 2013
READ pp. 83-87 (through Example 3)
- Reading a math book means read the text, but do it in small chunks - read a couple sentences or a paragraph, which may include explanations and definitions. Make sure you understand it reasonably well. If you do not, reread it until you do before moving on.
-When you come to an example, the text has... |
More About
This Book
Overview
This workbook's fifth edition has been updated to reflect questions and question types appearing on the most recent tests.
Hundreds of math questions in both multiple-choice and grid-in formats with worked out solutions
Math strategies to help test-takers approach and correctly answer question types that might be unfamiliar to them
All questions answered and explained
Here is intensive preparation for the SAT's all-important Math section, and a valuable learning tool for college-bound students who need extra help in math and feel the need to raise their math scores |
More About
This Textbook
Overview
Self-contained and suitable for undergraduate students, this text offers a working knowledge of calculus and statistics. It assumes only a familiarity with basic analytic geometry, presenting a coordinated study that develops the interrelationships between calculus, probability, and statistics.
Starting with the basic concepts of function and probability, the text addresses some specific probabilities and proceeds to surveys of random variables and graphs, the derivative, applications of the derivative, sequences and series, and integration. Additional topics include the integral and continuous variates, some basic discrete distributions, as well as other important distributions, hypothesis testing, functions of several variables, and regression and correlation. The text concludes with an appendix, answers to selected exercises, a general index, and an index of symbols |
Elementary Linear Algebra
9780132296540
ISBN:
0132296543
Edition: 9 Pub Date: 2007 Publisher: Prentice Hall
Summary: This text presents the basic ideas of linear algebra in a manner that offers students a fine balance between abstraction/theory and computational skills. The emphasis is on not just teaching how to read a proof but also on how to write a proof.
Kolman, Bernard is the author of Elementary Linear Algebra, published 2007 under ISBN 9780132296540 and 0132296543. Four hundred seventeen Elementary Linear Algebra ...textbooks are available for sale on ValoreBooks.com, one hundred thirty six used from the cheapest price of $79.80, or buy new starting at $149 free 14 day return policy. Contact Customer Service for questions.[less] |
9780321654274
ISBN:
0321654277
Edition: 3 Pub Date: 2010 Publisher: Addison Wesley
Summary: Beckmann, Sybilla is the author of Mathematics for Elementary Teachers with Activity Manual (3rd Edition), published 2010 under ISBN 9780321654274 and 0321654277. Four hundred fifty Mathematics for Elementary Teachers with Activity Manual (3rd Edition) textbooks are available for sale on ValoreBooks.com, two hundred forty two used from the cheapest price of $4.46, or buy new starting at $28 |
This algebra lesson from Illuminations helps students learning to graph real world data. Students will collect, graph and analyze data, and choose an appropriate mathematical model for a particular situation. They will
This math lesson from Illuminations helps students improve their skills in identifying equivalent trigonometric expressions. Students will be able to practice using trigonometric functions and equivalent expressions.... |
Algebra 1
Algebra 1 is a comprehensive course in the elements of algebra. This course stresses an understanding not only of basic algebraic principles and techniques, but also of how to model and solve real-world problems. Students will study the writing, graphing, and solving of linear equations and inequalities, both individually and in systems; the basics of quadratic, polynomial, and exponential functions and their graphs; and the fundamentals of probability and data analysis.
This course provides students with a variety of study materials to complement the textbook. The course guidebook offers regular checkpoints for students to test their skills; recommended problems for further practice; and over five hundred additional worked-out solutions. The course website offers animations, games, and other activities so students can further explore course concepts.
Students must have access to a graphing calculator for this subject. Popular among high school and college students are graphing calculators manufactured by Texas Instruments, such as the TI-84+ Silver edition and the TI-nSpire CX as well as the older TI-84+ or TI-83+ in the event you wish to purchase used. The American School is not affiliated with nor endorses these products. The information is provided here as a courtesy to our students and is not to be construed as an endorsement.
Among the activities students do in completing this subject are applying properties of rational numbers and units to solve problems; creating equations that describe numbers or relationships; representing and solving equations and inequalities graphically; describing and correcting errors; performing arithmetic operations on polynomials; interpreting key features of functions and their graphs; constructing and applying mathematical and graphic models to solve problems; and using the rules of probability to compute theoretical probabilities of events.
There are 13 examinations in Algebra 1.
Larson et al., LarsonAlgebra 1, Holt McDougal.
What others are saying:
Everything in Planning Your Career was awesome. I'm glad I chose the American School to make up my credit. — Steve, Michigan |
Intermediate Algebra
9780495108405
ISBN:
0495108405
Edition: 8 Pub Date: 2007 Publisher: Thomson Learning
Summary: Algebra is accessible and engaging with this popular text from Charles "Pat" McKeague! INTERMEDIATE ALGEBRA is infused with McKeague's passion for teaching mathematics. With years of classroom experience, he knows how to write in a way that you will understand and appreciate. McKeague's attention to detail and exceptionally clear writing style help you to move through each new concept with ease. Real-world applicatio...ns in every chapter of this user-friendly book highlight the relevance of what you are learning. And studying is easier than ever with the book's multimedia learning resources, including ThomsonNOW for INTERMEDIATE ALGEBRA, a personalized online learning companion.
McKeague, Charles P. is the author of Intermediate Algebra, published 2007 under ISBN 9780495108405 and 0495108405. Three hundred fifty nine Intermediate Algebra textbooks are available for sale on ValoreBooks.com, one hundred nineteen used from the cheapest price of $2.33, or buy new starting at $45 |
This very accessible guide offers a thorough introduction to the basics of differential equations and linear algebra. Expertly integrating the two topics, it explains concepts clearly and logically -without sacrificing level or rigor - and supports material with a vast array of problems of varying levels for readers to choose from. Promotes in-depth understanding (vs. rote memorization) - enabling readers to fully comprehend abstract concepts and finish with a solid and working knowledge of linear mathematics. Offers one of the most lucid and clearly written narratives on the subject |
25327","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":5.52,"ASIN":"1572225351","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":5.3,"ASIN":"1572229098","isPreorder":0}],"shippingId":"1572225327::h2uYuSkfO7rZhSCMWH%2F7fyrdhvyayZA4oDxwXuNvUufIp9SSsBcX03gmmeVI2y%2Fwrq2EXWTTPrxmkYE7GfXBA%2BJoMUB6bt72BjsRB4p2o6s%3D,1572225351::3BSQMI67UHulRGkBlEa2xHjQd9hGd44nMVdAgcqCuAP4Z4QUFzA8ZDFBN83MEqXASku0IJY8Sp%2BRLnMpwJuy3dgVh%2BDfnnzQnQQYT8dNUlo%3D,1572229098::k09ehPuWRnlertqoKarka1oLnNDUyXNvBRQ2ABN1RDQMb9r8X3kYXMjK3NWyd8QaqPGtCIIVjfA4ku3WalYvNi0cb1rAc4XP7xCNMpwqcWtCJ2cGcOnAT real help for kids taking Geometry - if they use it. The format is VERY dense, and it takes some study itself to gain facility in using it well. I do find that one really need both part 1 and part 2 of this guide if s/he wants complete coverage of a high school geometry course, so plan on investing in both. Also, the organization is somewhat odd or random it seems between part 1 and 2. These are the only reasons I did not give this study aid a 5 star rating, even though it clearly does say "part 1" -- organizationally, this is one of the weaker Quickstudy Guides. You'll need both parts even for the start of Geometry.
Once you figure out where information is on the PACKED pages of the guide, it really is helpful, so I would definitely recommend it. It is ESPECIALLY good if you have a textbook like the one used at my son's school, where all of the key concepts are in bold lettering, but the key words are left out with big question marks in blanks for the kids to work out/learn in class through "inductive reasoning" and/or lecture. It might be the dumbest textbook idea yet, since the information just is NOT THERE to be read by kids who miss a day of school, or if a teacher skips lessons, etc. This guide is GREAT to fill in the book full of question marks!
So, buy Part 1 AND Part 2, and you will have a complete picture of Geometry and a truly GOOD study aid.
This is a keeper for Geometry when you are solving problems and do not have a tablet or laptop at hand to search for math formulas. I gave it a four because it is kind of freshman high school basic level. It is not college level.
Best purchased I ever made. They say you get what you paid for, but in this case I got a whole lot for a little price! Great reference sheet for the kids while doing homework. Easy to take along. It's laminated and very durable. I have since bought several for different subjects. It's hard to remember all the formulas it and it helps during review time for exams |
Murphy, TX AlgebraReal world applications are presented within the course content and a function's approach is emphasized. Prealgebra instruction includes a review of the basics of mathematics and a thorough introduction to integers, basic equations and word problems. This course is designed to develop the skills and understanding to perform the fundamental operations on whole numbers, fractions and decimals. |
Course 2 Applications and Concepts
Setting the standard in middle school mathematics "Mathematics: Applications and Concepts" is a three-course middle school series intended to bridge ...Show synopsisSetting the standard in middle school mathematics "Mathematics: Applications and Concepts" is a three-course middle school series intended to bridge the gap from elementary mathematics to Algebra 1. The program is designed to motivate your students, enable them to see the usefulness of mathematics in the world around them, enhance their fluency in the language of mathematics, and prepare them for success in algebra and geometry 0078652634 Well used book with very heavy cover wear and...Fair. 0078652634Used book / Good condition / May have minimal highlighting,...Used book / Good condition / May have minimal highlighting, underlining or writing / Cover may have scuffed or bent corners / may be missing dust jacket / pages may have creases. Multiple copies available! |
Linear Algebra: An Introduction
In this appealing and well-written text, the author gives readers a substructure for a firm understanding of the abstract concepts of linear algebra ...Show synopsisIn this appealing and well-written text, the author gives readers a substructure for a firm understanding of the abstract concepts of linear algebra and its applications. Covers matrices, vector spaces, linear transformations, as well as applications to Jordan canonical forms, differential equations, and Markov chains. Includes exercises with answers and hints20887843 Brand New International edition. 100% Same...New. 0120887843 Brand New International edition. 100% Same Content as US Edition. ISBN and Cover might be different in some cases20887843 USED BOOK in good condition| No supplements|...Good. 01208878 |
Math for Health Care Professionals
9781401858032
ISBN:
1401858031
Pub Date: 2004 Publisher: Thomson Learning
Summary: Math for Health Care Professionals is a comprehensive, foundational resource that is equally effective in the classroom or for self-study. It assumes no prior knowledge of mathematics or health care but merges the two topics into the capstone of a complete learning package, including a student workbook. While the fundamentals of mathematics are a foundation to this book, their application to health care is emphasized.... Drug dosages, intake and output, weights and measures, temperatures, IV drip rates, and conversions are a focus, and illustrations of syringes, prescriptions, medication labels, IV bags, and I and O charts allow the reader to practice real-life health care skills requiring mathematics.
Kennamer, Michael is the author of Math for Health Care Professionals, published 2004 under ISBN 9781401858032 and 1401858031. Five hundred eighty one Math for Health Care Professionals textbooks are available for sale on ValoreBooks.com, one hundred eighty three used from the cheapest price of $5.45, or buy new starting at $2301858031 new book, never used, Has slight shelf wear due to storage. Has school name and number in book. All books directly from Avitar Books, we never use a 3rd party. Wil [more]
1401858031 |
The course provides an introduction to the mathematical analysis and linear algebra. The course starts with the real numbers and the related one-variable real functions by studying limits, and continuity.
Highlights of Calculus is a series of short videos that introduces the basics of calculus, how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject.
Calculus can be thought of as the mathematics of CHANGE. Because everything in the world is changing, calculus helps us track those changes. Calculus is among the most important and useful developments of human thought.
Differential equations are, in addition to a topic of study in mathematics, the main language in which the laws and phenomena of science are expressed. In basic terms, a differential equation is an expression that describes how a system changes from one moment of time to another, or from one point in space to another.
The main purpose of this course is to bridge the gap between introductory mathematics courses in algebra, linear algebra, and calculus on one hand and advanced courses like mathematical analysis and abstract algebra, on the other hand, which typically require students to provide proofs of propositions and theorems.
Multivariable Calculus is an expansion of Single-Variable Calculus in that it extends single variable calculus to higher dimensions. You may find that these courses share many of the same basic concepts, and that Multivariable Calculus will simply extend your knowledge of functions to functions of several variables. |
Discipline summary
Mathematics is the study of the structure, order and relationships of and between numbers, quantities, forms and space. Considered to be the underlying language of science, it seeks to establish truth through abstraction and formal logic. Pure mathematics, or speculative mathematics, solves abstract problems as an end in itself. Applied mathematics solves practical problems in science, engineering and business by studying and applying mathematical principles and sometimes results in the formulation of new disciplines, such as statistics or game theory.
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Communicating in the Language of Mathematics
Information Age Education (IAE) is an Oregon non-profit corporation created by David Moursund in July, 2007. It works to improve the informal and formal education of people of all ages throughout the world. A number of people have contributed their time and expertise in developing the materials that are made available free in the various IAE publications. Click here to learn how you can help develop new IAE materials.
Mathematics consists of content and know-how. What is know-how in mathematics? The ability to solve problems. (George Polya, How to Solve It, a New Aspect of Mathematics.)
Extra Short Summary
Communication in math involves making use of the processes of reading, writing, speaking, listening, and thinking as one communicates with one's self, other people, computers, books, and other aids to the storage, retrieval, and use of the collected mathematical knowledge of the world. Current precollege math education systems have substantial room for improvement in helping students learn to communicate effectively in the "language" of mathematics.
Short Summary
Math is a huge discipline with great breadth and great depth. The discipline existed before the development of written languages. As you know, reading and writing—including the development of math notation and math vocabulary—have contributed immensely to the discipline of math.
Many people consider math to be a language. It is not a general purpose language, such as English or Spanish. Rather, it is a discipline-specific language. Each discipline has developed specialized vocabulary and its own ways of communication that are specific to the discipline. Consider, for example, music notation and a person learning to sight read music. Perhaps you have seen and heard examples of the language of basketball or the language of football. On televised broadcasts of games one sometimes gets to see diagrams of plays and hear the language used by the coach to communicate plays and give directions to players.
A summary of arguments supporting the idea that math is a language is available in Logan (2000) and a more recent discussion is available on the Web. Certainly, math is an area in which one can learn oral and written communication and can learn to think using the vocabulary, symbols, and ideas of the language of math.
Now, add to this concept the word communication in the field called Information and Communication Technology (ICT). Note that in the United States this field is more often called Information Technology (IT), while the rest of the world tends to use the term ICT. The communication aspects of ICT open up a whole new dimension in every academic discipline.
Logan (2000) argues that the Internet is a language and that computer programming languages (collectively) are language. Clearly the whole field of Computer and Information Science and its applications in ICT strongly overlap with the discipline of math. Indeed, in a number of colleges, Computer and Information Science is organized within the Department of Mathematics. Thus, it is appropriate to consider the language aspects of Computer Science and ICT as we consider communication in math.
Here are some of the more important ideas in this document:
Although one can spend a lifetime studying math and still learn only a modest part of the discipline, young children can gain a useful level of math knowledge and skill via "oral tradition" even before they begin to learn to read and write. Oral and tangible, visual communication in and about math is an important part of the discipline.
Reading and writing are a major aid to accumulating information and sharing it with people alive today and those of the future. This has proven to be especially important in math, because the results of successful math research in the past are still valid today.
Reading and writing (including drawing pictures and diagrams) are powerful aids to one's brain as it attempts to solve challenging math problems. Reading and writing also help to overcome the limitations of one's short-term memory.
The language of mathematics is designed to facilitate very precise communication. This precise communication is helpful in examining one's own work on a problem, drawing upon the previous work of others, and in collaborating with others in attempts to solve challenging problems.
Our growing understanding of brain science is contributing significantly to our understanding of how one communicates with one's self in gaining increased expertise in solving challenging problems and accomplishing challenging tasks in math (and in other disciplines).
Information and Communication Technology (ICT) has brought new dimensions to communication, and some of these are especially important in math. Printed books and other "hard copy" storage are static storage media. They store information, but they do not process information. ICT has both storage and processing capabilities. It allows the storage and retrieval of information in an interactive medium that has some machine intelligence (artificial intelligence). Even an inexpensive handheld, solar-battery 6-function calculator illustrates this basic idea. There is a big difference between retrieving a book that explains how to solve certain types of equations and making use of a computer program that can solve all of these types of equations.
We all understand the idea of a native language speaker of a natural language. Students learning an additional language will often progress better when taught by a native language speaker who can fluently listen, read, talk, write, and think in the language, and who is skilled in teaching the language. We prefer that this teacher be fluent in a "standard" version of the language and not have a local accent and vocabulary that would give pause to many native speakers of the same language. The same idea holds in math education. The math educational system in the United States is significantly hampered because so many of the people teaching math do not have the math knowledge, skills, and math pedagogic knowledge—and, most important, level of fluency—that would classify them as being math education native language speakers.
The last bulleted item points to a major weakness in our current math educational system. A great many of our children are being taught math by teachers who are ill prepared to deal with the complexities of being a successful teacher in this discipline. This topic is addressed in more detail throughout the remainder of this document.
Reading & Writing Across the Curriculum
This document is intended mainly for teachers of math teachers and for preservice and inservice teachers of math. Oral and written communication are recognized as being an important part of the core of a modern education. All preservice teachers learn about the need for students to learn to read and write—both in general, and within each discipline they study. Reading across the curriculum is a common theme. Sometimes this expression is taken to mean both reading and writing, but often writing does not receive as much emphasis as reading.
Many students find it is quite difficult to reach or exceed contemporary standards in reading and writing in a natural language such as English. Even for those who graduate from high school and go on to higher education, reading and writing can still be major challenges. Many students find that they have to undertake remedial work in this area when they enter college, trade school, or an occupation.
"Contemporary standards" are usually set so that a significant percentage of students do not meet the standards. Failure to meet standards is often then considered to be the fault of the teacher not teaching well enough or the student being lazy.
I am particularly interested in students learning to read and write in the discipline of mathematics. For example, I am interested in whether typical students learn to read math well enough so they can use this skill to learn math by reading their textbooks and other books. I am interested in whether students learn to write in the discipline of math well enough so they can communicate mathematically (in writing) with themselves, their teachers, and others. An immediate complication is that many people learn more easily through discussion, demonstration, or guided practice than by reading.
As I have explored this topic, I have done some comparing and contrasting with reading and writing in other areas, such as music and chess. I have also thought about how Information and Communication Technology (ICT) is affecting or should be affecting reading and writing across the disciplines. ICT allows experiences akin to personal discussion, demonstration, and guided practice.
While this document focuses on communication in math, it draws on ideas from other curriculum disciplines and lays groundwork for other people to explore the topic of communication in other disciplines. It also explores current and potential impacts of ICT on communication in math and in other disciplines.
Writing Math to Learn Math
The general idea of writing to learn cuts across the curriculum. There is a substantial amount of research and practitioner knowledge about having students write math in order to learn math.
A 5/19/08 search of "writing to learn" returned abut 90,000 hits. A search on writing to learn math returned about 20,000 hits. Here are some examples of available articles.
The paper is based on a survey study of 104 teacher education programs spread across the United States. The focus is on all preservice teachers learning more about the process of writing across the curriculum. Here is a paragraph quoted from the paper:
The fact that only four respondents require all preservice candidates to take a separate course in process writing indicates that faculty of many colleges of education do not see the value in a course that focuses on writing and writing-to-learn strategies. That another four require such a course only for English teachers points to a belief that a separate course cannot be justified for teachers across the curriculum. Some faculties may still be uninformed about how writing is an integral tool for assisting students to comprehend more deeply and clearly what they are studying; they may be unaware of the research that underscores the value of incorporating writing-to-learn strategies in every discipline. [Bold added for emphasis.]
El-Rahman, Madiha (n.d.) The Effects of Writing-to-Learn Strategy on the Mathematics Achievement of Preparatory Stage Pupils in Egypt. Retrieved 5/19/08: Here are a couple of examples of activities quoted from the paper:
Pair Share This is a very simple activity to use when the teacher senses that the student does not understand the lesson. He stops and asks them to explain what is giving them trouble. After the students "Free - write" for a couple of minutes, they share their writing with their classmates. This can help to remove their confusion (Burchfield and others, 1993).
Journal Writing This is a diary-like series of writing assignments. Each assignment is short and written in prose rather than in the traditional mathematical style. The students can write in their journals: daily goals, rational for learning any concepts, and the strategies used to solve problems (Bagley, 1992: 660). It can give both the teachers and the students great insight into a student's progress (Potter, 1996: 184).
This research study looked at the teaching of high school math in three different schools over a period of several years. The researchers were particularly interested in the effects of not grouping students by math ability or previous math performance. Thus, in one of the high schools, all first year high school students began with first year high school algebra and then all proceeded to geometry. Based on the data from the three high schools that were studied, the argument is that this is a good thing to do.
Here is an interesting tidbit quoted from the Boaler and Staple study:
The correlation between students' scores on the language arts and mathematics sections of the AYP tests, across the whole state of California, was a staggering 0.932 for 2004. This data point provides strong indication that the mathematics tests were testing language as much as mathematics. This argument could not be made in reverse as the language tests do not contain mathematics. [Bold added for emphasis.]
This very high correlation seems to be supportive of Keith Devlin's assertion (in his book, The Math Gene) that anybody who has the brain power to become literate in a natural language can learn math.
New research shows a lack of language skills can hamstring a student's ability to understand the most fundamental concepts in mathematics. A series of studies led by Susan Goldin-Meadow, a psychology professor at the University of Chicago, found that profoundly deaf adults in Nicaragua who had not learned a formal sign language could not accurately describe or understand numbers greater than three. While hearing adults and those who used formal sign language easily counted and distinguished groups of objects, those who used only self-created "homesigning" gestures could not consistently extend the proper number of fingers to count more than three objects at a time, nor could they match the number of objects in one set to those in another set.
Quoting from the Susan Goldin-Meadow article:
Does learning language change the way we think about number? The exact quantities to which words like "seven," "eight," and "nine" refer seem so basic it is hard to imagine that we might need the word "seven" to have the concept seven. But evidence from groups who have not been exposed to conven- tional numerical systems suggests that language, particularly the numeral list in a count routine, may be importantly involved in the ability to represent the exact cardinal values of large sets.
The Mundurukú (1) and Pirahã (2) are Amazonian people in rural Brazil whose languages do not contain words for exact numbers larger than five (the Mundurukú) or any exact number words at all (the Pirahã).* Adults in these cultures have not been reported to invent ways to communicate about the large numbers for which they do not have words. In addition, these groups do not display a robust ability to match sets exactly with respect to number, except when a one-to-one correspondence strategy is readily available (e.g., pairing each object in one visible row with a corresponding object in a second visible row) (3). The absence of a linguistic model for representing exact number (in this case, a count list) could explain the difficulties Pirahã and Mundurukú adults have representing large exact numbers (2, 4–6). However, their difficulties could just as easily be explained by the absence of culturally supported contexts in which exact number must be encoded (7, 8). To disentangle these possibilities, we investigated the numerical abilities of individuals who lack a linguistic model for number but who live in the numerate culture of Nicaragua: "homesigners."
Specific Problem
The specific problem situation being addressed is that U.S. precollege math education is not as good as most American taxpayers would like it to be. It definitely is not as successful as math education in a number of other countries. Possible reasons are many, and many people are dedicated to improving the American system.
Considerable literature addresses the problems of math education and how to improve math education. This IAE-pedia contains some of these documents.
Relevant substantial research has been done in many disciplines such as math, music, chess, etc. K. Anders Ericsson is a world leader in this research field. The link is to a short article on expert's long-term working memory that summarizes some of the key ideas in teaching and learning. All teachers (and, indeed, all students) can benefit by having some knowledge of this field.
The document you are currently reading focuses specifically on communication in math. This includes looking at some related aspects of ICT, brain science (chunking and expertise especially), and empowering students and their teachers.
This document is written specifically for use in preservice and inservice math education courses and workshops. The focus is on the idea that the discipline of math includes a language that we call the language of mathematics. We want students to learn to read, write, speak, listen, and think creatively in the language of mathematics. In essence, the goal is for students to become mathematicians at the level of the math they have studied. We want them to learn to use effectively their math content knowledge and skill to solve challenging problems and accomplish challenging tasks that are amenable to effective use of the math they have studied—not to mention their learning the math skills they will need to navigate through all the financial and life-decision hazards and opportunities they will face.
The goal of this document is to encourage and support discussion and deep thought followed by constructive action. A good use of this document in a preservice or inservice teacher education course would be to have students read it in advance of a class meeting and form their personal opinions on some of the ideas. During class, students would then share their insights and ideas in small group and whole class discussion. A follow-up activity might be having students continue the discussion in an online environment, write about this article in their math journals, do research on one specific idea in the article that came up during the in-class discussions, or develop some instructional materials that could be used to help implement their ideas.
To support the intended use, from time to time this document contains a question suitable for personal reflection or for discussion in a workshop or class. Here is an example:
For reflection and discussion: Drawing upon your knowledge of yourself and other people you know, analyze your levels of expertise in the areas of reading, writing, speaking, listening, thinking, and problem solving in mathematics. You might find it helpful to use the terms fluency and/or expertise in doing this analysis. Identify your relative strengths and weaknesses. Think about how our math education system contributed to your relative strengths weaknesses. What does your analysis suggest in terms of possible ways to improve our math education system?
Be tolerant of your first draft; you will gain many ideas when you and your colleagues share your first attempts.
Introduction to Reading and Writing
The development of reading and writing about 5,200 years ago was a major milestone in human history. From then on, there has been an accelerating pace of change in societies of the world brought on by the accumulation and sharing of data, information, knowledge, and wisdom. Major accelerating inventions in this process are the printing press, electronic communication (telegraph etc.), the computer, and the Internet and Web. The totality of this accumulation is huge and currently is perhaps doubling every five to ten years. The Web, all by itself, is a virtual library with many times the content of the largest physical libraries on our planet. It is continuing a rapid pace of growth. More content is added each day than a typical person can read in a lifetime.
The development of general-purpose written languages brought with it a start in the development of a written language for the discipline of mathematics. Over thousands of years, the discipline of mathematics and its language have grown and matured. Math educators support the idea of students learning to read, write, speak, listen, and think creatively in the language of mathematics. All of these aspects of communication contribute to representing and solving math problems.
Developing fluency (read, write, speak, listen, understand, and think) in the language of mathematics certainly has some similarities to doing the same thing in a natural language such as English. However, there are considerable differences, and some are discussed in this article.
An example of a similarity is provided by the challenge a high school student or adult faces trying to learn a foreign language. It is not too hard to memorize a large number of words and phrases. However, many second language learners of high school age and older find it is hard to learn to think and gain verbal fluency in a foreign language. Research strongly supports the value of starting to learn a second language at a much younger age and being taught by native language speakers who are skilled teachers.
For reflection and discussion: You know that children are exposed to math as they gain oral communication skills well before they start school. Our school system starts formal math instruction at the earliest grade levels. Thus, we cannot attribute our lack of success in math education to not starting early enough! Why do you suppose that this large amount of math instruction over many years produces such poor results for many students?
For reflection and discussion: There are many disciplines that school students are capable of learning. What is there about mathematics that justifies the many years of math coursework required of all students? Think of possible arguments for having less required coursework in math, thereby freeing up time for more coursework in other areas that might be of more specific interest and importance to some students. In thinking about and discussing this topic, try to give examples of recent times when it has been beneficial to you to draw upon your knowledge of high school algebra and geometry to deal with problems outside of a school setting.
Learning to Read and Write in a Natural Language
The undamaged human brain is genetically "wired" for learning oral and visual communication. Children learn to understand, talk, and think in spoken language supplemented by gestures long before they reach kindergarten. They learn whatever language or languages commonly used in their environment. Thus, children growing up in a bilingual or trilingual home and community environment will become orally bilingual or trilingual.
When children start on the process of learning to read and write, they already have a substantial level of oral fluency. Young children are displaying a high level of creativity and intelligence as they communicate orally. Learning to read and write draws heavily on the ability to create meaningful utterances and understand spoken language.
Young students also have a substantial and growing knowledge of the world. This often provides help in discerning the meaning of a sequence of words or a sentence. Students have considerable ability to extract meaning from context and from pictures. Pictures in story books help students in extracting meaning from the written presentation.
As a child moves through the first few grades of elementary school, the child continues to gain verbal fluency. A combination of informal learning outside of school and the formal schooling adds thousands of words per year to the child's oral fluency repertoire. This steadily growing oral fluency provides a growing foundation for building fluency in reading and writing.
Being around those whose oral and written language fluency is quite a bit greater than the child's substantially aids the process. Think of this as role modeling. The child can observe and hear oral communication being routinely used. As a child attempts to imitate and participate in this oral communication; immediate feedback is provided by proficient speakers of the language.
In many homes, young children are read to frequently. Research strongly supports that this and other adult role modeling in reading and writing makes a major contribution to children's future linguistic development.
Even with a strong supportive background, most students take many years of instruction and practice to develop a level of reading and writing expertise that meets contemporary standards. Thus, most colleges and universities require entering freshman to take a year sequence in writing. In the U.S., this course is often called English Composition.
The idea of "contemporary standards" is important. magazines and newspapers are written at or below 10th grade reading level. A great many adults who graduated from high school have considerable difficulty reading above this level. (This document's Flesch-Kincaid readability is approximately 10th grade level. The Gunning Fog index suggests high school completion is needed for reading the document.)
Similar observations have been made about average adult performance in other areas. For example, many adults who graduated from high school function in math at about the 6th to 7th grade level.
Such observations point to the major difference between standards that governments and others want to set, and what is readily achievable by our current educational system. The educational leaders in each academic discipline have created "standards" that they feel students should achieve. While each discipline's standards may appear to be reasonable or desirable when viewed individually, the collected set of standards far exceed what an ordinary student can achieve, possibly in part because the experts may overestimate the needs of the general public. For example, consider the terms a capella (music), undecidability (math), and zugzwang (chess)—all of which have 'real-life' implications.
Moreover, while students demonstrate they have achieved a standard by passing a particular test, the reality is that forgetting occurs (in many cases, quite rapidly) so that even in a test-based standards system, relatively few people continue to meet the standards as they become adults.
For reflection and discussion: Why do you think it is so hard to learn to be a good writer, when it is relatively easy to learn to talk in a manner that meets contemporary standards? (Hmm. Does an average high school graduate meet the oral fluency standards that our schools would like to set?) Next, think about the same question for learning math.
For reflection and discussion:We know that a person's knowledge and skills in an area degradates over time if the knowledge and skills are not being used. A different way of saying this is that students forget much of what they (supposedly) learn in school. Think about some personal examples. In what ways does our educational system acknowledge that people forget, and attempt to accommodate the forgetting?
Some Brain Theory: Seven Plus or Minus Two
Written and oral language are aids to thinking. Thinking is sometimes described as "talking silently to oneself." Such thinking allows a person to contemplate various actions and possible outcomes of the actions—without actually carrying out the actions. You probably know some people who "think out loud." In addition, having s research subject talk aloud while solving a problem is a useful research technique to gain insight into a person's thinking process during problem solving.
This section provides some general information about the human brain and some roles of language in thinking and problem solving.
Humans have three types of memory:
Sensory memory stores data from one's senses, and for only a short time. For example, visual sensory memory stores an image for less than a second, and auditory sensory memory stores aural information for less than four seconds.
Working memory (short-term memory) can store and actively process a small number of chunks. It retains these chunks for less than 20 seconds.
Long-term memory has a large capacity and stores information for a long period of time. Over time, information stored in long-term memory tends to become more and more difficult to remember—that is, to retrieve—if it is not used very often. However, traces of these stored memories continue to exist, and they can be an aid as one relearns what was learned in the past.
When you work to solve a problem, you bring information and ideas about the problem into your working memory. You consciously manipulate this information and ideas. Research on working memory indicates that for most people the size of this memory is about 7 ± 2 chunks (Miller, 1956).
This means, for example, that a typical person can read or hear a seven-digit telephone number and remember it long enough to key it into a telephone keypad. The word chunk is very important. For example, the sequence of four digits 1 4 9 2 can be thought of as four distinct chunks. However, it can also be thought of as one chunk—the year when Columbus discovered America. It can also be represented as two chunks-14 and 92. The point is that appropriate chunking of ideas and information is a powerful aid to overcoming limitations of short-term memory.
The names of the number words in Chinese are, on average, shorter than the corresponding names of the number words in English. In terms of digit recall, 7 English digits are about the same length as 9 Chinese digits. Native language speakers of Chinese have a greater short term memory digit span than native language speakers of English. The referenced article is about the research work of Stanislas Dehaene.
Your brain is very good at learning meaningful chunks of information. Think about some of your personal chunks such as constructivism, multiplication, democracy, complex numbers, transfer of learning, and Mozart. Undoubtedly these chunks have different meanings for me than for you. Moreover, our chunks are of different size. Research indicates that experts in a discipline have more chunks and much larger chunks (in their discipline) than do novices.
As a personal example, my chunk "multiplication" covers multiplication of positive and negative integers, fractions, decimal fractions, irrational numbers, complex numbers, functions (such as trigonometric and polynomial), matrices, and so on. My breadth and depth of meaning and understanding were developed through years of undergraduate and graduate work in mathematics. Others might connect "multiplication" with the pressure of having to learn the multiplication table before a test or recall "go forth and multiply—a paraphrase of various biblical phrases—and related jokes.
Here is another example. You "know" what the number line is. When you think about the number line, your mind probably conjures up some sort of picture, perhaps a line with equally spaced marks on it, and the marks labeled with digits such as … -4, -3, -2, -1, 0, 1, 2, 3, 4, …. You can think of this as a chunk. My number line mental chunk is not the same as yours. Through years of studying and using math, my mental math number line chunk has grown to include rational numbers and irrational numbers. It has grown to include irrational numbers that are called transcendental numbers. Moreover, my number line chunk is closely tied in with chunks about numbers in different bases, different sizes of infinity, some results from the area of math called number theory, complex numbers, and other components of math. You probably modified your conception of "number line" as soon as you were reminded of these other numbers; I remind myself of these numbers as I'm visualizing a number line.
It is useful to think of a chunk as a label or representation (perhaps a word, phrase, visual image, sound, smell, taste, or touch sensation) and a collection of pointers. A chunk has four important characteristics:
It can be used by short-term memory in a conscious, thinking, problem-solving process.
It can be used to retrieve more detailed information from long-term memory.
It serves as an anchor for constructing new knowledge and skills. (It lies at the root of constructivism learning theory.)
It is a key to higher levels of expertise in a discipline. High-level experts in a discipline have a large repertoire of chunks in that discipline. They think and solve problems making use of these chunks. Furthermore, as proficiency in a disciple increases, chunks become bigger. Such a larger chunk can include both information about a problem situation and possible actions to take in attempting to solve the problem.
In terms of communication, chunks and chunking are a critical aspect of how one's working memory communicates with one's long-term memory. A chunk may have a name. As indicated above, the name "multiplication" allows my short-term memory to access (retrieve) a large "multiplication" chunk in my long-term memory. For another example, the name of one of your friends allows your short-term memory to access a chunk of information about your friend.
However, seeing your friend in person, seeing a picture of your friend, of smelling a particular smell can also trigger this information retrieval. No words are used in this retrieval process. This indicates we all have and use an extensive non-verbal language. And, of course, you are familiar with the language of gestures. This language can be very extensive. Think, for example, of American Sign Language.
There has been substantial research on roles of building and using chunks in gaining a high level of expertise in a discipline. One idea that has emerged is that, in some sense, high-level experts are able to use such chunks as a kind of an extension of their working memory. That is, within the area of high-level expertise, these experts are able to function as if they had a working memory that is much larger than "normal." For instance, driving a route new to you in a city requires concentration. As you become familiar with the route, you are able to mentally construct the route so that you automatically allow for speeds, lane changes, possible trouble, etc., and you are able to devote much of your consciousness to other matters.
In brief summary, creating, storing, and using chunks of information are essential to building a high level of expertise in an area. Such chunking ties in with oral, written, and nonverbal communication and thinking, and it is applicable in every academic discipline. Expertise in a discipline is dependent on having an extensive repertoire of large chunks specific to that discipline.
However, there are many chunks that have interdisciplinary use and value. Suppose, for example, that a person develops a high level of expertise in understanding and making use of careful, logical, rigorous arguments in a discipline such as math. Many of the chunks involved in this type of problem solving in math carry over to other disciplines such as the sciences and law.
For reflection and discussion: Think about some discipline in which you have a reasonably high level of expertise. Identify some chunks in your brain that you use in this discipline. You know that many other people who have lower expertise in this discipline lack entirely or have much less robust chunks. Also, think about whether you make use of this chunk is other disciplines.
Learning Mathematics
The healthy human brain is genetically wired for learning some math and math-related knowledge and skills. Howard Gardner has identified logical/mathematical and spatial as two types of intelligences. Spatial intelligence can be quite important in attempting to solve some types of math problems. Learn more about Howard Gardner at
Very young infants have a little number sense, such as being able to distinguish between two of an object and three of an object. Recent research suggests that perhaps this is an innate ability of infants to sense that something is wrong when they are expecting to see two objects and are presented with one or three of the objects. Stanislas Dehaene, who was mentioned earlier in this article, is a world leader in this type of math-related brain research.
A research experiment might involve showing an infant two objects set on a small stage. A screen comes down in front of the stage and then goes up. There seems to be an innate expectation that the number of objects will not have changed. Researchers can time the increased eye time fixation of a viewer when a change occurs.
Toddlers who can crawl readily learn to orient themselves in their spatial environment, finding their way around different parts of a house. Such spatial skills are essential to a hunter-gatherer life style in which people had to forage for food and then find their way back to their clan.
Now, think about a child learning words for numbers. As an example, I have a young grandson who is quite bright. At an early age he could say in order the words one, two, three, … up to about sixteen. However, his understanding of these words was quite limited. At the time, he had some working understanding of one and two, and perhaps three.
There is a large difference between being able to say words and having an understanding of what they mean. This, of course, is true for both math words and non-math words. The above example suggests that quantity is a relatively abstract idea that is a challenge to learners.
Attributes such as color, size, shape, numerosity (number, quantity) and so on are all learning challenges. Numerosity and other math-related words and concepts have the added challenge in that our contemporary standards tend to expect a high level of perfection.
By the time an average child enters the first grade, the child has developed a reasonable level of skill in using the number counting words to be able to determine and say the number of objects in a small set. The child can do simple additions, such as 2 + 5 through a process of counting. Quite a few children have learned counting on either through their own discovery or through being explicitly taught by the time they begin the first grade.
This is a major step in learning math and allows relatively young children to do math at a higher level than people growing up in a hunter-gatherer society whose natural language is mathematically quite limited.
Stanislas Dehaene has shown that the ability to estimate amounts—an innate 'number sense' that human beings have in common with various other species—forms the basis for our mathematical (abstract reasoning) and arithmetic (calculation) abilities. The latter ability does, however, require a well-developed system of symbols—a language system. Evidence for this duality has been found not only in scientific experiments but also in anthropological research. One example is the language of the Amazonian Mundurukú tribe, which has words for numbers only up to five. The Mundurukú are not able to perform precise calculations with larger numbers, but they can approximate and compare larger amounts.
Thus, the average child starting school has a beginning level of understanding of the number line. However, the number line is a quite complex math concept. We expect students to learn about fractions and decimal fractions. We expect students to learn about both positive and negative numbers. We expect students to learn to perform arithmetic on the various types of numbers on the number line. If we go back 4,000 years, only the most learned mathematicians of their time could effectively handle the range of math we are expecting grade school students to learn.
For reflection and discussion: One difference between natural language and the language of mathematics is the degree of precision required in communication. In many situations, small errors in the use of natural language do not destroy the overall correctness or effectiveness of a communication. Explore this idea and its math education implications.
Oral Tradition
This section is a work in progress.
To a very large extent, math is taught using methods that might be described as "oral tradition." Students learn to recognize some math symbols and math words. However, many do not learn to read math at a level that allows them to learn math by reading math. It is only when students reach the more advanced high school math courses that there is a significant emphasis on learning the math by reading the math book and other resource materials. Thus, a great many students graduate from high school with a very limited ability to learn math by reading a math book or other math resource materials.
Math education makes extensive use of "word" or "story" problems. Here, a problem that can perhaps be represented and solved mathematically is presented in a natural language statement that may contain few or perhaps no math symbols and vocabulary. A student must meet the challenge of understanding the problem, translating it into math notation and vocabulary, solving the math problem, and translating the results back into the context of the original natural language statement of the word problem.
This is an important component of math education, since many of the problems that can be addressed using math are not explicitly stated in math notation. Indeed, we now have very powerful Computer Algebra Systems that can solve a very wide range of problems that are stated in math notation. Thus, a major challenge in math education is to prepare students to deal with the tasks of determining when math might be useful in solving a problem and in representing such a problem using math notation.
Comment by Jen Jensen 4/28/09
The following is quoted from an email message sent to the National Council of Supervisors of Mathematics on 4/28/09:
One of the most difficult components to today's problem-based texts such as CMP or Core Plus is the reading component. I have run numerous collaborative coaching cycles with math teachers over the last two years with the goal of understanding how to implement effective reading strategies in a math classroom. These usually involve reading teachers and their expertise is always beneficial. I also look back to one of the original goals of public education and see that our charge is to create an educated citizenry that can participate intelligently in the democratic process.
This being said, I believe that English and Social Studies teachers are critically important in the education of our children and should be paid at the same level of math and science teachers. To me the issue is not the subject being taught but the quality of the teacher doing the teaching. Highly qualified does not mean quality. We need a process for raising the quality of our teaching force-such as the coaching model, and then we need to remove those teachers who refuse to participate and improve their practice.
Ken Jensen
Instructional Math Coach
Aurora Public Schools
Native Natural Language Speakers and Native Math Language Speakers
When children grow up in a bilingual or trilingual natural language environment, they grow up bilingual or trilingual. This idea is often incorporated into schools. Some students get to attend a bilingual elementary school in which the content areas are taught in the student's second or third language. It is highly desirable that the teachers teaching the content areas be native language speakers of the language(s).
We all understand the idea of a native language speaker of a "standard version" of a natural language. We expect the native language speaker to think in the language, know the culture of the people who speak the language, and have a native accent. Moreover, we prefer that this person not have a strongly regional accent and vocabulary. We want learners to be learning a relatively standard version of the language and with a relatively standard accent.
Now, take this idea and carry it over into math education. What might we mean by a "native math language speaker" of mathematics?
First, consider the following quote from George Polya, a world-class math educator and math researcher. In a talk to elementary school teachers, Polya said:
To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems. For the higher aims about which I am now talking are some general tactics of problems—to have the right attitude for problems and to be able to attack all kinds of problems, not only very simple problems, which can be solved with the skills of the primary school, but more complicated problems of engineering, physics and so on, which will be further developed in the high school. But the foundations should be started in the primary school. And so I think an essential point in the primary school is to introduce the children to the tactics of problem solving. Not to solve this or that kind of problem, not to make just long divisions or some such thing, but to develop a general attitude for the solution of problems. (Polya, 1969)
In summary, the term native math language speaker means someone who has a high level of fluency in reading, writing, speaking, listening, thinking, and creative problem solving in the discipline of mathematics. A native math language speaker knows the culture of mathematicians. In short, a native math language speaker is a mathematician.
In the remainder of this article, I will use the term mathematician interchangeable with a native math language speaker.
However, think about an average 6-year-old, an average 12-year-old, and an average 18-year-old native speaker of a language such as "standard" English. The 6-year-old has substantial oral communication and thinking capability in the language, but has a relatively limited vocabulary and is just getting started in learning reading and writing. The 12-year-old has a much larger vocabulary, is still better at oral communication, and has made significant progress in reading and writing. The 18-year-old is still more skilled at reading, writing, speaking, listening, and thinking in his or her native language.
When applied to math, observation leaves us with the situation in which a person might be a high school graduate but mathematically function like an average 12-year-old. In some cases we have 6-year-olds who mathematically function like average 12-year-olds.
Here is one more piece of the math education puzzle. A person who is certified as a teacher has attended 16 or more years of schooling starting at the first grade or earlier. This person knows a tremendous amount about general pedagogy (how to teach), only a small fraction of which was learned in teacher education courses. This person also knows a tremendous amount about how to be a math teacher, and yet may have taken only one or two courses in math pedagogy. Learning on the job is a very important part of becoming a good teacher.
The typical elementary school teacher is a "math teacher" as well as being a teacher of other disciplines. How can we tell if this person is a qualified "math educator?"
Let's take the specific case of an average elementary school teacher who is responsible for teaching math as well as a number of other subjects. We can talk about the number of years of coursework (precollege and college) experience the teacher has had being explicitly taught in math content, being explicitly taught in math pedagogy, being exposed to general pedagogy, and being exposed to math pedagogy.
This is a work in progress, and the question just asked is a difficult one to address. We might be able to quantify the situation with a statement such as, "This teacher functions at the level of an 8th grade mathematician, an average 10th grader in math pedagogy, and an average graduate of an elementary teachers' college program in general pedagogy."
The levels of the three different measures of qualification will change as the teacher learns on the job through classroom experience and staff development. We know that, on average, teachers gain considerably in their overall levels of effectiveness during their first half-dozen years on the job.
I leave this topic for now, still not having given a specific definition of what we mean by an appropriately qualified (well-qualified) math educator for some specific category of students. For example, a person who is a well-qualified math educator for learning disabled students might not be as well qualified in working with talented and gifted students, and vice versa. I have suggested (recommended) that this person needs to be a "native language speaker" in math content, math pedagogy, and pedagogy. But I have not specified the grade levels or age levels or other measures we want to use in each area. I have not specified what might be meant by "standard math," or "standard math education."
Substantial research supports the contention that a major weakness in our precollege math educational system is the relative weakness in the math pedagogy and math problem-solving capabilities of many teachers of math.
A Personal Story
Being a native math language speaker means that one can "do" mathematics. It means having a high level of expertise in solving math problems, recognizing problem situations in which math is apt to be useful, having quite a bit of math knowledge and skills, being able to use one's math knowledge and skills, and being comfortable in the culture of mathematicians. It takes many years of time and a considerable amount of effort to become a native math language speaker (that is, to be come a mathematician).
Here is a personal example. I grew up in a home where both my mother and father taught math in a university. Each was a mathematician. Thus, merely by growing up in this environment, I was given a large boost in moving toward becoming a native math language speaker. However, much more was required.
I did well in math in elementary and secondary school. I then went on to college and majored in math. By the time I finished a bachelor's degree in math, I had a good start on being a mathematician. Four years of graduate work, resulting in a doctorate in math, certainly qualified me to be considered a native math language speaker. I was fluent, with a high level of expertise, in reading, writing, speaking, listening, creatively thinking, and problem solving in math. My four years of graduate work essentially constituted a math immersion program, with all of the teaching being done by highly-qualified mathematicians with a high level of fluency and competence in math.
However, I was not a mathematics educator. I had very little teaching experience during my graduate work, and I had no specific instruction on how to teach math. I had some math teachers who were more effective than others. Indeed, some of them satisfy my definition of math educator. Others were clearly much more engrossed in their research. Some were both excellent math educators and excellent math researchers.
Nor was I a math historian. Sure, I had learned a little math history as I studied math. However, my level of expertise in this component of the discipline of math was minimal.
One Major Math Education Problem
We have now come to one crux of a difficult education problem. Most children do not grow up in a home environment of native math language speakers (that is, mathematicians). Moreover, most students do not have their elementary school math taught by native math language speakers. Indeed, even in their middle school and high school math courses, many students are not being taught by people who would be considered to be native math language speakers. It tends to be the students who take the more advanced math courses who are taught by mathematicians.
One way to attack this problem is by departmentalizing the teaching of math at the grade school level, and requiring that teachers of math at every grade level have at least a bachelor's degree in mathematics. Some of the countries that do well in international math education comparisons do take such an approach.
Another approach is to have math teachers at all grade levels take rigorous and demanding math education workshops and summer courses, year after year after year. Some teachers do this, and indeed develop into native math language speakers.
Note that the goal is not to make such teachers into research mathematicians. Instead, it is to make them into "expert level" math education mathematicians who specialize in teaching math to specific groups of students.
A third approach would be to place much greater emphasis in the math curriculum on students learning to read, write, speak, listen, and creatively think and solve challenging problems in math. Such ideas are often emphasized in both preservice and inservice education for math teachers.
There is a strong parallel between good math communication and the teaching of reading and writing in a natural language such as English. In teaching writing in a natural language, there is a great need for the teacher to read what a student is writing and provide feedback on the content. How well is the intended "message" being conveyed by what the student has written? Of course, teachers also provide feedback on spelling and grammar. But feedback on the content is essential. So it is with math. But, especially at the elementary school level, the math feedback is usually focused only on the correctness of an answer.
The following reference presents research on the mathematics that a third grade teacher needs to know.
Compare and Contrast with Music Education
I find it helpful, when thinking about educational strategies in one discipline, to compare and contrast that approach with strategies in other disciplines. I often have selected music and chess as disciplines to compare and contrast with math. This section looks at the language of music and the next section looks at the language of chess.
Music is one of the nine multiple intelligences identified by Howard Gardner. See While people vary in their musical "IQ," most people can gain a relatively high level of expertise in music if they have appropriate opportunities and interest. Most people gain a useful and enjoyable level of music expertise just through their informal exposures to music and opportunities to sing. A "professional" high level of expertise takes many thousands of hours of good instruction and practice.
This same situation exists for the multiple intelligence that Howard Gardner labels as logical/mathematical. Both music and math are, at some level, built into our genes. But it takes a great deal of study and practice to develop this innate capacity to meet contemporary standards of expertise.
Introduction
Making music includes singing, chanting, humming, whistling, rhythmic clapping, use of drums and other musical instruments, creating music, composing new music, and so on. Music existed long before the development of reading and writing. Music was part of the environment that children grew up in tens of thousands of years ago. Musical knowledge and skills passed from generation to generation via children growing up in a musical environment of role models and being expected to participate.
Oral tradition and making and using musical instruments served the discipline of music well for many thousands of years, up until quite recent times. Even after reading and writing were invented, very few people learned to read and write. Moreover, even for people who did learn to read and write, a decent system of musical notation for the reading and writing of music had not yet been invented. The invention of a good written language for music took many thousands of years. Quoting from the Wikipedia:
Scholar and music theorist Isidore of Seville, writing in the early 7th century, famously remarked that it was impossible to notate music. By the middle of the 9th century, however, a form of notation began to develop in monasteries in Europe for Gregorian chant, using symbols known as neumes; the earliest surviving musical notation of this type is in the Musica disciplina of Aurelian of Réôme, from about 850. There are scattered survivals from the Iberian peninsula [sic] before this time of a type of notation known as Visigothic neumes, but its few surviving fragments have not yet been deciphered.
The ancestry of modern symbolic music notation originated in the Roman Catholic Church, as monks developed methods to put plainchant (sacred songs) to paper. The earliest of these ancestral systems, dating from the 8th century, did not originally utilize a staff, and used neum (or neuma or pneuma), a system of dots and strokes that were placed above the text. Although capable of expressing considerable musical complexity, the dots and strokes could not exactly express pitch or time. This system served mainly as a reminder to one who already knew the tune, rather than a means by which one who had never heard the tune could sing it exactly at sight.
We are all used to the term "musician" to describe a person who has a high level of knowledge and skill in some parts of the discipline of making music. Music is such a large discipline that no one can gain a high level of expertise over the entire discipline. A world class opera singer need not know how to play a violin, compose music, or direct an orchestra. However, world class singers, violinists, composers, and conductors are all immersed in the culture of music and each can communicate effectively in the language of music.
Many musicians have a high level of expertise as music teachers. Indeed, perhaps because music instruction tends to be performed by good music educators, many musicians learn a great deal about how to teach music as they learn music. In any case, many musicians learn to teach music and come to depend on this knowledge and skill as a source of income.
Our educational system accepts the idea that a person teaching music should be a native language speaker of music (that is, a musician) and a music educator. In telling contrast, American schools have a large percentage of people teaching math who are ill-prepared in the content and the teaching of math.
Most children grow up in a home environment is which there is singing and other music. They may learn jingles and other music from radio, television, or Internet ads, from computerized games, from music storage and playback devices, and so on. If parents and other caregivers have a reasonably high level of music interest and fluency, their children will learn a lot of music by being immersed in such an environment.
However, learning to play a musical instrument, learning to read music, learning to compose music, learning to sing well individually and in a group, and so on, all require many years of informal and formal instruction plus lots of practice.
Three important characteristics of informal and formal music education are:
The music learner is frequently given the opportunity to observe (listen to) the performance of others who have a relatively high level of expertise. Math has this characteristic for very few children—or adults.
Music is a human endeavor that is often performed in a group setting. This setting promotes and facilitates communication and sharing. This sharing—making music together—is an important aspect of music. There are large intrinsic and extrinsic rewards in such musical sharing. Again, this is seldom true for math.
Music provides a learning environment in which the learner can readily tell that he or she is gaining in expertise and can demonstrate to others this increasing expertise. Math exhibits some of this characteristic, but not nearly as strongly as music.
Each of these characteristics of music education offers us some insights into math education.
The first item suggests that we can improve math education by improving the general math environment. For example, we currently teach math during a specific math period in school, and math may receive very little attention during the rest of the day. A child's math-related environment at home may be quite limited. Our educational system has accepted the idea of reading and writing across (throughout) the curriculum. How about making this into "the three R's across the curriculum"?
The second item points to a major challenge in math education. While math, like music, is a human endeavor, it mainly lacks the group learning and performance aspects of music. The intrinsic and extrinsic reward structure in math is quite different from that in music.
The third item suggests we can improve math education by creating better feedback mechanisms and by helping students gain expertise in sensing and assessing their own progress. Research in math eduction supports these ideas.
For reflection and discussion. Think about the three ideas listed above in terms of your personal math education experiences. What does your personal analysis suggest to you in terms of things you can do to improve the math education experiences of students?
Doing and Consuming Music and Math
From an educational point of view, it seems important to distinguish between those who learn to "do" music (perform at some level) and those who merely consume (listen to) music. Both categories of people existed before the development of electronic technology. We can educate for each of these endeavors, and most people have some level of interest in and expertise in each of them. Participation in the second category has been greatly influenced by the development of technology for the broadcasting, recording, and playback of sound, and new developments in electronic music composition and performance.
We all "do" math. When my stomach rumbles and I glance at my watch, I do a mental calculation of how long it has been since my last meal and how long before my next meal. When I fill my car's gas tank, I estimate the cost and the miles per gallon since the last fill up. In a store, I pay some money and get some change, then I check the correctness of the change.
We all consume math, but not in the same way that we consume music. For example, I use my cell phone to make a call. The cell phone system makes quite sophisticated use of computers and a variety of other equipment. People designing the equipment and the overall phone system made extensive use of math. So, when I use my cell phone, I am making extensive use of math. However, that does not give me the same sense of feeling contentment that I get by "making use of music" when I am listening to a live or high-quality recording of an expert musical performance.
ICT and Music
The overall discipline of music has been strongly impacted by Information and Communication Technology (ICT). The history of this impact certainly dates back to the development of the telephone (so music could be transmitted over a wire) and recording devices (so music could be stored, edited, and widely distributed).
We now have relatively inexpensive electronic hardware and software for creating, editing, storing, playback, sharing, and performing music. With such technology, a grade school student can learn to compose and edit music, and can use a computer as a performance instrument. This is an important idea. When grade school students compose music and use a computer in the editing and playback process,they can hear what they produce, improve it by editing, and share their music with others. This is in marked contrast to what typically happens in math instruction.
Computer technology allows people to build personal libraries of music. Computer technology can "notate" music—that is produce a written musical score from a live or recorded music performance. Artificial intelligence music generation systems have generated music "in the style of" various world class composers, music deemed comparable in quality to human-created compositions. These electronic musical instruments and a wide range of storage and editing tools have greatly changed the music recording and performance industries.
Relating back to math education, one of the key ideas given above is that ICT has brought us a number of new electronic musical instruments to "do" music. In math, ICT has brought us 6-function, scientific, graphing, and equation-solving calculators, and computers with vastly greater powers. These new math tools are instruments that can be used to "do" math. Thus, math education is faced by the problem of determining the extent to which it wants to facilitate students learning to use these electronic tools (instruments) and pay less or no attention to other more traditional tools. We can ask, "What is so special about paper and pencil computational algorithms?"
Here is an idea that might interest you. Solving a complex math problem or producing a proof in math is somewhat akin to composing in music. A musical composer does not have to have a high level of expertise in playing each of the instruments needed to perform a composition. That is, composing and performing are two different skills. Of course, the composer needs to know the capabilities and limitations of the various musical instruments and the human limitations in playing the instruments.
In math, for example, a person working to understand and solve a math problem can imagine having a three-dimensional picture of a particular three-dimensional geometric figure, and being able to readily view this picture from different directions. We have long had computer graphics software that can produce such three-dimensional representations.
Somewhat similarly, a person working to solve a math problem may decide that it would be helpful to fit a mathematical function (perhaps a quadratic, cubic, or higher order polynomial function) to some data, and then find the places where this function crosses the x axis. Computer programs have long existed that readily accomplish such tasks.
To continue this example, the math problem solver may decide that it would be helpful to perform various statistical computations on some data. Computer programs that can carry out such computations incorporate a huge amount of collected knowledge from a great many mathematical statisticians.
Perhaps you see a pattern emerging here. Think of a person attempting to solve a complex math problem as a composer, developing a set of instructions that can be carried out to solve the problem. The math problem-solving composer can draw upon the performance capabilities of calculators and computers. The math problem-solving composer does not need to have a high level of expertise in areas in which calculators and computers can readily produce very high levels of performance.
Here is another math education idea coming from music. One of the reasons math education is part of the core curriculum is that math is a powerful aid to solving problems in many different disciplines. What computers and other aspects of ICT have done is make it possible to automate finding the solution to many of these problems. Consider an analogy between having an automated tool that solves a particular problem, and having a music storage and playback device. A music consumer gets to listen to and have the benefits of recorded music. A person using a computerized tool that incorporates many mathematical functions gets the advantage of a mathematical performance.
In summary, ICT has strongly affected doing and consuming music. It has equally strong potentials in math. However, our math educational system has made only modest progress in realizing (making use of) these potentials.
Compare and Contrast with Chess
This section examines the game and language of chess. Extensive research on good chess players has given us good insight into the role of chunking (see above) in learning to be a good chess player. There is considerable similarity between this chunking used in playing chess (solving the problem of making a good move) and chunking used in solving problems in math.
Chess (just like checkers)is played on an 8-square by 8-square board. The game has a long history and is played by millions of people. At the higher levels of play there are masters grandmasters, international grandmasters, and world champions. At lower levels, grade school children can learn the game and there are tournaments for players at all levels. Many websites explain the rules and some of the details of the game.
Chess is a game of skill. This is in marked contrast to card games such as bridge and poker, where the "luck of the draw" makes a big difference in the short run. In a chess game, two players compete against each other. The game ends in a win for one of the players, or a draw. There is some advantage to being the first player to make a move, but in a chess match between two players, they take turns in going first. The "luck" that chess players talk about is when their opponent overlooks a possible good move or fails to see that a move s/he is about to make is a bad move.
Chess has an oral and written language. However, chess is not a significant part of the everyday life of most people. Nor is chess one of the nine areas of multiple intelligence identified by Howard Gardner. See Children who grow up in a "chess-playing" family may learn the game when they are quite young and gain from the collected oral history of chess that is part of the family conversation.
Chess has been extensively studied to help understand problem solving and how humans can improve their problem-solving skills. It has also been extensively studied by people working in the field of artificial intelligence. How does one go about developing a computer program that is good at playing chess? What can we learn about human intelligence and the education of children through the study of how humans and computers learn and become better at playing chess?
You don't have to be a skilled chess player to understand the chess section in this Communication in the Language of Mathematics document. Here is enough background to get you going.
Chess Notation
The columns (files) of the 8 x 8 board are lettered a, b, … h, and the rows (ranks) are numbered 1, 2, … 8. In chess, the person playing the White pieces always moves first. The lettering and numbering notation used to identify the spaces on the board is convenient and natural from the point of view of the person playing the White pieces.
The names of the pieces (in English) are abbreviated as follow: K=King, Q=Queen, R=Rook, B=Bishop, N=Knight, and P=Pawn. This board coordinate system and the piece name abbreviations make it quite easy to record all of the moves in a game.
For example, here are the first few moves of a game. White always moves first, and White's moves are in the left column. The sequence of moves given below indicates that White's Bishop captures Black's Knight on White's fourth move.
Pe2 to e4 — Pe7 to e5
Ng1 to f3 — Nb8 to c6
Bf1 to b5 — Pa7 to a6
Bb5 x Nc6 —
This notation can be tightened up considerably. Here is a tighter notation that conveys the same information. The notation assumes that the reader knows the legal moves. Thus, the first move of Pe4 means that White's pawn that is at e2 is moved to the e4 location. It is the only pawn that can legally move to that location at this point in the game.
Pe4 Pe5
Nf3 Nc6
Bb5 Pa6
BxN
Notice how easy it is to make an exact record of a chess game and to learn to read such a record. Contrast this with musical notation and learning to read music, or the notations used in math. From a notational point of view, music and math are far more complex than chess. It takes only s few minutes to learn the written language used to store a record of a game of chess.
Communication is More than Just Notation
This section discusses some of the learning that takes place as one develops into a good chess player. If you have not played chess, or not played it much, the ideas will probably pass you by. But, for each idea, think about whether it might be applicable in learning to being better at math problem solving. Keep asking yourself, "What 'big ideas' in math problem solving do I know that I use regularly and that I can help my students learn to use?"
Here is a piece of information useful for understanding the example in the following paragraph. In chess, a Knight's move must be either two horizontal and one vertical square or one horizontal and two vertical squares. This allows a Knight to attack various pieces that cannot, in turn, be attacking back.
Like any well-developed discipline, chess has an extensive vocabulary. Also, as in any discipline-specific vocabulary, many chess terms are adapted from natural language vocabularies. For example, you might think of a fork as an eating utensil. Of course, you have heard of a tuning fork used in music. You have heard of a fork in a road. In the diagram, White has just moved the Knight to d7, actually forming a triple fork. This particular fork of King, Queen, and Rook is also known in chess as a "family fork."
You might be able to guess meanings of terms such as open file and Queenside. Other terms such as check, gambit, castle, and fianchetto (Italian for "on the flank") are more challenging.
This special vocabulary and notation are important for communicating about and thinking about chess. However, there is more to such endeavors than just vocabulary and notation. Your brain stores images that represent emotions, sounds, smells, pictures, and so on. Your brain draws upon these mental images as it works to solve problems and accomplish tasks. Good chess players have stored many thousands of chess patterns (chess chunks) in their brains. For them, a short look at a chess game in progress provides information needed to retrieve mental chunks of information related to possible future outcomes of the game in progress.
In this aspect of communication with one's self, there are clear similarities among chess, math, and music. In each discipline, one learns chunks, stores them, and learns to think in terms of these chunks. Some of the chunks have names, while others are mental patterns that one accesses through other means such as mental pictures, sounds, "gut-level feelings," and so on.
Learning a discipline-specific chunk and how to make effective use of it is a step toward increased expertise in a discipline. However, accumulating a large number of chunks in and by itself does not make one into a high-level expert in a discipline. It is learning to "see," "sense," "hear," "feel," "recognize," etc., the relationships among chunks, and to make use of appropriate combinations of these chunks, that is key to having a high level of expertise in a discipline.
The discipline-specificity issue is worth repeating. Substantial research supports the need for discipline-specific knowledge and skill (discipline-specific chunks) in order to have a high level of expertise in that discipline. That is why, for most people, it takes so many years of effort in order to become a high-level expert in a discipline. But it takes more than just rote memorization of chunks to achieve this high level of expertise.
Chess Strategy
There are many chess websites available. On the Web one can read about chess, see the rankings of the best players, follow tournaments in progress, play against human opponents, play against a computer, or try your hand at solving challenging chess problems. These websites can be used just for fun and can also be used to gain increased expertise as a chess player.
Here is a personal story. I learned to play chess when I was relatively young. That is, I learned the legal moves and to play well enough so that it was fun to play with other kids my age.
A number of years later I became interested enough in the game to read a couple of chess books. One was a "how to" book that explained some of the strategies that good players find useful. Another was a chess history book, looking at some of the great players and games from the past. I found both types of book enjoyable. The "how to" book substantially increased my level of playing ability.
This is an "aside." Notice that I read these chess books for fun and to learn to be a better chess player. At about the same time I read some books such as The World of Mathematics that had little to do with the math I was being taught in school. Our educational system places a lot of emphasis on students learning to read well enough so they can learn by reading. In math education, however, we do not take much advantage of a student's steadily improving expertise in reading. Think back over your own math education. Did you ever read a math book for fun or to further your math knowledge and skills beyond what was being taught in school?
Here is a simple example. In chess, one of the key ideas is to maintain the mobility of your pieces. That is, to keep as many move options available for your pieces as possible. Another strategy is to gain control of the center of the board. Among other things, control of the center tends to increase your own level of mobility and decrease your opponent's level of mobility.
Suppose that you had studied a book discussing these two strategies, and that it contained some examples of how to make use of the strategies. You then play a game against an opponent who (up until now) was your equal, but who had not received formal instruction (from a teacher, book, or opponent) on these two strategies. The chances are quite good that you will now be the superior player. This little bit of formal instruction gives you a large advantage over an "unschooled" opponent.
This is an important idea. It might well be that, as you play chess, you will discover some of these strategies for yourself. However, there are many strategies that are useful at various points in a typical game. Many have been discovered and carefully analyzed by world class chess players. The accumulated knowledge in this area is far more than one person could discover (unaided by the previous work of others) in a lifetime.
Now, consider my opponent who is consistently losing to me because I have been making use of these two strategic concepts. My opponent may carefully analyze these (losing) games and eventually discover the concepts of mobility and center control. Alternatively, I might mention the two ideas and illustrate them in a game that we have recently completed. In both cases, I am assuming that we have written down the moves from the games, so that we have a written record that allows us to analyze games we have played in the past.
Chess is both a fun game for children and a discipline of fierce, ego-involved competition. With few exceptions, it takes ten thousand or more hours of study and practice to become an international grandmaster, assuming inherent talent. Much of this time and effort is spent studying games that have been played by exceptionally good players in the past, and games one has played in the past.
These insights into learning chess strategies provide some useful insights into learning math. There are many different strategies for attempting to solve math problems. Many of these are designed to aid in communicating with one's self, such as by drawing a diagram, making a table, creating a mental model or image, and so on.
Determining a Chess Player's Strength
Chess is a competitive game. If two players of approximately equal chess-playing strength play against each other a number of times, they will each win about half of the games. If one player is much stronger than the other, this player will win almost all the time.
Over the years, the discipline of chess has developed a relatively accurate means for determining a player's strength. The method is somewhat like that used in rating teams in competitive sports. Careful records are kept of how well players do against each other in different tournaments. Even if two players have never played against each other, they will have played against players whose strength or rank has been determined through tournament play.
As in competitive sports, there are chess tournaments pitting the top players in the world against each other, and there are world championship matches in which two players compete against each other, with the winner designated as the world champion.
While there are competitions in both math and music, there is essentially nothing like the level of competition one finds in chess. There are many world class mathematicians, and there are many world class musicians. However, there is no world champion mathematician determined by head-to-head competition.
Computers have brought an additional approach to chess rankings. One can compare (and rank) humans in how well they do playing against various computer chess programs. People throughout the world can compare themselves in terms of how well they do against a particular chess program, set at a particular level of difficulty.
Without a competition and ranking system, math students have no easy way to compare their own math strengths against each other. Let me share a personal example. For students in college, there is a national math competition called the Putnam competition. Throughout the United States, on one specific day, entrants spend the day working on 12 problems. This is done on their own campuses, and the test is carefully proctored.
I was the best math undergraduate at my university. I knew this because of having taken course with the other top students—in some sense, competing with these students in math classes and the tests given in the classes. I thought of myself as being quite good at math.
I competed in the Putnum contest in both my junior and senior years. In both years, a quarter to a third of all entrants scored better than me! Objectively, I was quite good. Only the better students would enter the competition, and I was better than about seven of ten other such students. Even so, my ego's feeling of "being quite good" suffered significantly.
One of the goals in the No Child Left Behind Act is to move toward a ranking system in math education that can be used to measure the relative strengths of schools. The people supporting this type of "competition" believe that it will help to improve the precollege math educational system in this country.
It is not at all obvious that making math into a competitive "sport" will lead to improved math learning and performance for students as a whole. Indeed, it might well do just the opposite. Those who are not highly talented and highly motivated in math (as well as those who are not basically competitive in what they do) may well choose not to compete. One might well see widespread implementation of the sentiment: "I'm not very good at math. Why should I compete, when I will always come out in the bottom half?"
Contrast this with a person learning that the knowledge and skills they are gaining in math empower them to do various things they need or want to do. Through study and practice, they get better at doing those things. This suggests that math education can become more successful through helping students, individually as necessary, to grasp the personal advantages (empowerment) they accrue through their math studies.
For reflection and discussion: Suppose we had a computer program that could "play the game" of math at different difficulty levels. A student studying math could play against this game to determine his or her current math ranking. Here, we are assuming that this "game of math" is good enough to be used throughout the world to determine a student's math level of learning, understanding, and overall "math strength." How do you think this would affect math education? To help your thinking on this question, you might want to read David Moursund's short article Chesslandia: A Parable.
Rote Memory
It may feel to you that a discussion of rote memory is a far reach from a discussion of communication in math and math education. Here is the way I see it. Much of what a person does when attempting to solve problems and accomplish tasks in any discipline involves communicating with her or him self. One consciously communicates with data, information, knowledge, and wisdom stored in one's brain. One carries on a mental conversation. Indeed, you probably know people who verbalize—talk out loud to themselves—during this thinking.
One can memorize with little or no understanding of what is being memorized. One way to think about this is in a stimulus/response setting. A person's brain (or, some other animal's brain) is trained to respond in a specified manner to a specified stimulus. The stimulus elicits the response, and the responder does not need to have an "understanding" of the meaning of what is stored in the brain and produces the response.
Of course, we can also have stimulus/response learning in which the response has meaning to the learner. You may be able to respond quite rapidly to the stimulus 8 x 7 =, and produce a response of 56. Upon further reflection, you realize that you have done a "multiplication fact" problem-solving task, producing an answer of 56.You may realize that this answer is a little bigger than 50 and a lot less than 100. You may realize that a score of 56 on a hundred point test may not lead correspond to a good grade on the test.
Rote memory, with or without understanding, can be used in the storage and retrieval of part of the collected knowledge with a particular area. This can be quite helpful in solving some of the frequently occurring problems within that area.
Here is an illustration from the game of chess. A chess game begins with White and Black each having 16 playing pieces. It is possible to carefully analyze the board situations that result after all possible sequences of one move by each player, two moves by each player, and so on. Of course, the number of possible sequences grows exponentially, and soon becomes so huge that no person (indeed, even all the past and present chess players in the world) can analyze all of them.
However, what has been done is that many of the interesting and potentially good opening sequences of moves have been carefully analyzed by high level chess experts. A huge amount of this collected chess knowledge is available in books and in other media. Any person who has learned to read chess notation can access this collected chess knowledge.
A chess player gains a considerable advantage by studying these well-analyzed sequences of opening moves and by memorizing a large number of them. Rote memory of the results of work done by others is a good substitute for "reinventing the wheel." In a game between reasonably highly ranked chess players, the first half dozen or more moves by each player tend to be made quite quickly, using rote memory.
After that, the thinking and chess problem-solving begins. Each player soon encounters a position (a chess problem) that he or she has never encountered before. However, even here having a large repertoire of memorized chunks is very important. In essence, such chunks correspond to parts of a game position. The good chess player recognizes parts of the problem as being similar to or even exactly the same as parts of board positions that he or she has carefully analyzed in the past.
Math education can be approached via rote memory. We can have a student memorize facts, definitions, and algorithms. Rote memory is useful in solving frequently occurring problems. Moreover, math problem solving makes use of chunks much in the same way as chess playing does.
It turns out, however, that math is played on a much larger playing board (many more playing pieces) than is chess. In dealing with the math people encounter in their everyday lives, they quickly move beyond the point where rote memory suffices. In novel problems, problem solvers quickly move beyond being able to succeed from rote memory and enter the mode of attempting to make effective use of chunks of information stored in their long-term memories. A large repertoire of such chunks and lots of experience in drawing on such chunks is essential in dealing with challenging math problems.
Artificial Intelligence and Chess Playing
Consider the following type of competition. Well before the competition begins, competitors are given a copy of a very comprehensive dictionary. During the contest, competitors are given a definition from the dictionary. Their task is to say and spell the word that has been defined.
By dint of considerable study and practice, a person can get very good at this rote memory game. But a computer can become letter perfect in a very short period of time. Computers are very good at rote memory, and computers can store the spelling and pronunciation of all of the words in a dictionary.
When researchers in artificial intelligence went looking for a game to study in which (at the time) humans were much better than computers, many decided on chess. One can become better at chess by rote memory, but chess is far more than a rote memory game.
In chess, rote memory is quite helpful at the beginning of a game. It can also be quite helpful near the end of a game in which each player has lost a number of pieces. In that situation, there are relatively few pieces left on the board. Many such end games have been carefully analyzed by chess experts and computers. The results are available in books and databases.
It is in the mid game—after use of the memorized openings and before use of memorized end games—that intelligence is needed. How does a human chess player analyze possible moves in order to select the one that improves their situation and/or damages the opponents' position? One way to gain insight into this is through working with skilled chess players. Get them to "think out loud" as they analyze chess problems. Of course, there are also many books full of the written analyses of games played by good chess players.
This interaction with expert problem solvers has been used in many different disciplines. It has led to the development of expert systems (computer programs that are good at solving challenging problems) in many different disciplines.
This is part of the challenge for educators in the information age. Thousands of researchers are working on developing computer programs that make use of computer capabilities (machine intelligence, artificial intelligence) to solve or help solve problems in various disciplines. Sometimes the artificial intelligence methods parallel human intelligence methods. Often they don't. Rather, they make use of methods that take advantage of the large memory and great speed of computer systems.
The first chess-playing computer programs were very weak compared to humans. However, over the years, computers got much more capable, and chess-playing programs got much better. By 1997, an IBM computer named Deep Blue beat Garry Kasparov, the world's human chess champion! A dedicated chess machine called Hydra was programmed to take advantage of a combination of rote memory and a type of artificial intelligence relevant to chess playing—and became as good at the world's best human chess players.
The success of chess-playing computers did not lead to the game of chess gradually going away. Nowadays there are chess matches that pit computer against computer. There are chess matches that pit human plus computer against human plus computer. Many chess players practice their skills against computer programs. With all of this, chess remains a game that many people learn to play and enjoy playing against human and computer opponents.
Artificial intelligence has been used in other game-playing computer programs. IBM undertook the challenge of developing a computer program that could play the popular TV game show Jeopardy. In this game, opponents are given an answer to a quite specific question, and compete to see who can first correctly state the question. In February of 2011, an IBM computer system named Watson defeated two human champions in this game. See
Artificial Intelligence and Solving Math Problems
Artificial intelligence, rote memory, and steadily increasing computer memory size and speed have been applied in mathematics. For many years, there have been high quality Algebra Systems. Such computer programs can solve a wide range of math problems.
You know that an inexpensive 6-function calculator can add, subtract, multiply, divide, and compute square roots. That is, it can "do" some of the things we are teaching grade school students to do through rote memory and through use of memorized algorithms. A modern Computer Algebra System (CAS) has this same level of capability up through calculus and linear algebra. That is, in every part of the math curriculum where rote memory and use of memorized algorithms is useful, artificially intelligent CAS systems can do (typically, faster and more accurately) what we are teaching students to do by hand.
The math education community needs to think carefully about the steadily growing "intelligence" of computer systems. If a computer system can solve a certain category of math problems, what "by hand and by brain" knowledge and skills in this specific area do we want students to acquire?
Our math educational system has been struggling with this situation for years. To the extent that math resembles a competitive game, computers are far better than humans at many aspects of it. It seems evident that this math education quandary will continue to exist for the foreseeable future.
For reflection and discussion. In chess, the development of computer programs that can outplay even the best of human players has not resulted in the demise of the game. Chess players enjoy the head-to-head competition with each other and the social aspects of being part of the chess community. Within in certain areas, computers are far better at math than humans. How is this affecting what we are doing in math education? Are there aspects of math that are very large numbers of people want to learn because they are fun—personally and socially rewarding—independently of whether computers can do them better than humans? For example, to what extent is Sudoku a math game?
For reflection and discussion. Suppose that our education system decided that all math education above the 8th grade was elective. Any course requiring a higher level of math knowledge and skill could clearly specify the higher level of math prerequisite that was required. However, students could well graduate from high school and college without taking math courses above the 8th grade level—or, by only taking such higher level math courses when they had a clear personal need to do so. What are your thoughts on how this would affect our overall educational system?
Final Remarks
For me, the ideas that I have discovered and explored while writing this article are quite thought-provoking. While my initial focus was on communication and math education, many of the ideas apply to learning in every academic discipline. For example, brain research on chunking is applicable in any discipline, including disciplines as diverse as carpentry, dancing, and Texas hold 'em. Skill in creating and using personal chunks is an essential component of self-talking and planning in solving challenging problems.
Information and Communication Technology (ICT) brings a new dimension to communication. One way to think about this is that a computer is an artificially intelligent machine that one can communicate with and make use of in solving problems, accomplishing tasks, and learning. Thus, within each discipline in our educational system, educators are now faced by the challenge of helping their students communicate effectively with computers and other artificially intelligent aids to solving problems and accomplishing tasks.
This challenge is especially large in math and in other disciplines where computers are especially useful (powerful, capable) in solving or helping to solve problems and accomplish tasks. That is, many chunks include procedures that one can learn to carry out "by hand" but that computers can carry out faster and (often) more accurately. For each of these, a student could have an option of learning all details of the chunk, or of learning that such a chunk exists and that a computer can accurately and rapidly carry out the details of the procedure(s) associated with the chunk.
For reflection and discussion: What are your thoughts on education, including having a student learn about (including how to retrieve and how to make use of) a number of computerized chunks?
In my opinion, our math educational system spends far too much time helping students to learn (memorize, often with little understanding) to do things that computers can do faster and more accurately. This uses up so much of the math education time, that relatively little time is spent on understanding, creative thinking, problem posing, and other activities in which human intelligence far exceeds computer intelligence.
Some of this memory work is important. Often a person is called upon to make real time decisions (quick decisions) based on using math knowledge and skills. As computerized processing and information retrieval systems get better, and as more computer intelligence (including math-related computer intelligence) is built into machines, we will need to continually reexamine those aspects of math that need to be stored in one's head.
For reflection and discussion: Think back over this article. Identify one or two ideas that you found particularly interesting and that you tend to agree with. Find one or two that you fond uninteresting and/or that you strongly disagree with. Do a compare and contrast, working to increase your insight into communication aspects of improving math education in our information age world.
With the release of every new international mathematics assessment, concern about U.S. students' mathematics achievement has grown. Each mediocre showing by American students makes it plain that the teaching and learning of mathematics needs improvement. Thus, the country, once more, has begun to turn its worried attention to mathematics education. Unfortunately, past reform movements have consisted more of effort than effect. We are not likely to succeed this time, either, without accounting for the disappointing outcomes of past efforts and examining the factors that contribute to success in other countries. Consider what research and experience consistently reveal: Although the typical methods of improving U.S. instructional quality have been to develop curriculum, and—especially in the last decade—to articulate standards for what students should learn, little improvement is possible without direct attention to the practice of teaching. Strong standards and quality curriculum are important. But no curriculum teaches itself, and standards do not operate independently of professionals' use of them. To implement standards and curriculum effectively, school systems depend upon the work of skilled teachers who understand the subject matter. How well teachers know mathematics is central to their capacity to use instructional materials wisely, to assess students' progress, and to make sound judgments about presentation, emphasis, and sequencing.
That the quality of mathematics teaching depends on teachers' knowledge of the content should not be a surprise. Equally unsurprising is that many U.S. teachers lack sound mathematical understanding and skill. This is to be expected because most teachers—like most other adults in this country—are graduates of the very system that we seek to improve. Their own opportunities to learn mathematics have been uneven, and often inadequate, just like those of their non-teaching peers. Studies over the past 15 years consistently reveal that the mathematical knowledge of many teachers is dismayingly thin.1 Invisible in this research, however, is the fact that the mathematical knowledge of most adult Americans is as weak, and often weaker. We are simply failing to reach reasonable standards of mathematical proficiency with most of our students, and those students become the next generation of adults, some of them teachers. This is a big problem, and a challenge to our desire to improve.
Test Scores Were Not Significantly Higher in Classrooms Using Selected Reading and Mathematics Software Products. Test scores in treatment classrooms that were randomly assigned to use products did not differ from test scores in control classrooms by statistically significant margins.
Effects Were Correlated With Some Classroom and School Characteristics. For reading products, effects on overall test scores were correlated with the student-teacher ratio in first grade classrooms and with the amount of time that products were used in fourth grade classrooms. For math products, effects were uncorrelated with classroom and school characteristics.
The Mathematics Course : Surprising as it may sound the learning of mathematics is not just about learning to 'get the right answer'. It is also (amongst other things) about being able to think mathematically and read mathematically, and then being able to show how you develop your ability, reading and thinking. Consequently, as part of the coursework you will need not only to be able to do the mathematics set but also be able to describe exactly the process by which you went about doing such mathematics.
Hence, throughout the course you will need to demonstrate your developing mathematical thinking, technical and reading ability by:
solving specific mathematical problems
adopting the approach of reading mathematics. This will be done by interpreting technical text, mathematical expressions, solutions to mathematical problems, diagrams, etc...
studying and learning how you go about working on, solving and hence, learning mathematics
Specifically, the course aims to help you develop the following abilities :
the ability to solve appropriate mathematical problems
the ability to construct appropriate mathematical proofs
the ability to read mathematically by interpreting/describing mathematical text, expressions, solutions and/or proofs as appropriate, and demonstrate this through written and/or oral work
think mathematically by identifying mathematical patterns and use these to extend given mathematics
the ability to critically analyze and discuss issues in mathematics, as well as your learning of mathematics
the ability to work individually and in groups on the topic of mathematics
Mazur assigns reading and expects his students to email him questions about what they do not understand.
Mazur assigns reading and gives an online quiz to see what they do not understand.
Mazur make sues of "clickers" (hand held student response units) in class to get feedback from students.
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MA: … Despite the fact that it is possible to accomplish all of these objectives and many more with a single question, do you think that it is useful to have a specific primary objective when designing and planning the delivery of a question?
EM: Oh, yes ! I often actually use students' questions. I actually use this now with a teaching technique called "Just-in-Time Teaching" …. Basically, the students read before class and then they tell me in an e-mail what they find difficult or confusing. I use that to prepare my lecture. In other words rather than lecture on what I find difficult, I will take some of their confusion and bounce it straight back at them.
Miller, G.A. (1956). The magical number seven, plus or minus two: Some limits on our capacity for processing information. Originally published in The Psychological Review, 1956, vol. 63, pp. 81-97. Retrieved 2/29/08:
Because people forget much of what they learn, students could benefit from learning strategies that provide long-lasting knowledge. Yet surprisingly little is known about how long-term retention is most efficiently achieved. Here we examine how retention is affected by two variables: the duration of a study session and the temporal distribution of study time across multiple sessions. Our results suggest that a single session devoted to the study of some material should continue long enough to ensure that mastery is achieved but that immediate further study of the same material is an inefficient use of time. Our data also show that the benefit of distributing a fixed amount of study time across two study sessions – the spacing effect – depends jointly on the interval between study sessions and the interval between study and test. We discuss the practical implications of both findings, especially in regard to mathematics learning. [Bold added for emphasis.]
Last month's column described how mirror neurons provide us with a mental template of the active motor neurons of someone who is speaking. The person's comments create an analogous template of the content of the speaker's thoughts. So if a person says cat, it activates the mirror neurons our brain uses to say cat, but also the neurons that process our memories and images of cat as a concept.
…
When I begin to write an article, I have a general sense but no set outline of what I hope to write. I explore the concept on my keyboard, and the article gradually begins to emerge. As in conversation, the focus may shift from the original idea. At one point, though, everything becomes clearer, and then considerable rewriting sharpens the text. This often also occurs in a conversation or meeting, when a consensus suddenly occurs, and the issue is then quickly resolved.
What's odd is that when things are most confusing, I'll often suddenly wake up from sleep with the mental clarity that had eluded me while writing during the day. I have no explanation for this, except that my thoughts about current tasks seem to continue at a subconscious level, whether awake or asleep. We've all experienced this when we can't recall a familiar name. We go on with other thoughts, and then hours later the name suddenly pops up in our mind.
This suggests that while thought and language are perhaps two sides of a single coin, thought can occur without language—and alas, a lot of language occurs without thought.
In UMERC, several members are active in research that focuses on aspects of language and communication. This page describes the research agenda of this group.…
One goal of mathematics education is for students to develop the ability to communicate using the forms of language and forms of representations that are typical for mathematics. Aspects of language are thus important parts of mathematics. However, sometimes there is a wish to separate language from mathematics, e.g. when constructing mathematical test items and wanting them not to measure reading ability but mathematical ability. Therefore, there is a need to investigate more thoroughly properties and functions of language in relation to mathematical content and to knowledge and learning of mathematics.
This article looks at possible futures of the Web. It focuses specifically on increasing linguistic "intelligence" of the Web. Web 3.0 will have a much better ability to "read" the content of websites, extract meaning, and link this meaning to that stored in other Websites. |
MATH Course
Mathematics for System Safety Analysis
Course Description:
This course is focused on the mathematics used in system safety analysis. The purpose of this course is to provide the trainees with a working understanding of the mathematical theories underlying system safety analysis. From this course, the trainees will be able to properly interpret the results of a system safety analysis and use it in their intended applications. The course will begin with the fundamentals of probability theory and will cover the uses of that theory for solving various system safety problems. Statistical methods will also be covered in relations to establishing equipment failure frequencies. System safety examples will be used throughout the course. Each student should bring a calculator with statistical functions.
Objectives: To provide a level of understanding of the mathematical concepts used in conducting system safety analyses.
Who Should Attend: Individuals who intend to take the system safety course or would like to enhance their understanding of the fundamental mathematical theories used in system theory.
Course Outline
Probability Theory
Permutations and Combinations
Bernoullie Process and Binomial and Multinomial Distributions
Normal Distribution
Poisson Process and Distribution
Boolean Algebra
Statistics and Failure Data Analysis
Uncertainty Analysis Using Bayesian Method
Prerequisite: Attendees should have an engineering or hard science background. |
Galois Theory, Third Edition (Chapman Hall/CRC Mathematics Series) for an Amazon Gift Card of up to £3.86, which you can then spend on millions of items across the site. Trade-in values may vary (terms apply). Learn more
Book Description"This edition preserves and even extends one of the most popular features of the original edition: the historical introduction and the story of the fatal duel of Evariste Galois. These historical notes should be of interest to students as well as mathematicians in general. [A]fter more than 30 years, Ian Stewart's Galois Theory remains a valuable textbook for algebra undergraduate students." - Zentralblatt MATH, 1049 "The penultimate chapter is about algebraically closed fields and the last chapter, on transcendental numbers, contains 'what-every-mathematician-should-see-at-least-once', the proof of transcendence of pi. The book is designed for the second and third year undergraduate courses. I will certainly use it." - EMS Newsletter
First Sentence
In the first part of this book, Chapters 1 to 15, we present a (fairly) modern version of Galois's ideas in the same setting that he used, namely, the complex numbers. Read the first page
I have to admit, I struggled through my Maths Degree, but Galois Theory was something I developed a bit of a fondness for, probably because of this book! This beautiful area of mathematics is presented in a clear and helpful manner, and unusually for a textbook, explains some of the definitions and theorems that you may have missed earlier in your course. Textbook writers tend to forget that refreshers are sometimes needed on elementary points - not so here. Some interesting details about Galois himself are included, and it's funny how this seems to aid understanding of the subject. It also gives useful examples and applications of what you have learnt, and throughout the text is a clear sense of where you are heading. If you're stuck at Uni and baffled with Galois Theory, this book will be indispensible.
This book introduces the subject in a very approachable manner. The book assumes that the basics of group theory are already understood by the reader and the first few chapters gently explains the field theory required before the Galois theory starts. Some chapters are very theoretic (and therefore dull) but this is circumvented by a number of very good examples to motivate the theory. Also, later chapters give applications, such as the solubility of equations and the fundamental theorem of algebra, in which you can see the theory in action and also how beautiful the mathematics is. I would recommend this book to anyone studying abstract algebra as it is an excellent introduction to not only Galois theory but field theory as well.
This is a good book but it could have been much better. There are quite a few mistakes but most are not that confusing. The main problem is that a lot of the explanations are not as detailed as I would have liked (Maybe I'm just thick). This book is best for those who don't know much abstract algebra since most of the book deals with polynomials over the complex numbers rather than a general ring. This makes some of the proofs much messier than necessary. I recommend getting the Second Edition instead since it starts straight away with general rings and also has a few solutions in the back unlike the third edition.
You might have thought that by the third edition most of the mistakes would have been corrected, but every chapter seems to contain a few. The prose is annoying and contains a lot of waffle which is confusing. Also the book is cheaply bound and badly laid out. There are many good galois theory textbooks to be read so don't spend £40 on this. |
realistic practice questions and exercises to prepare you for the Math portion of the GRE.Kaplan's Math Workbook for the GRE, 9th Edition, comprehensively addresses the math section of the GRE Revised General Test. This workbook is a highly effective way to prepare for the math section of the GRE Revised General Test.Kaplan's Math Workbook for the GRE, 9th Edition includes:• 6 full-length Quantitative Reasoning practice sets • Diagnostic tool for even more targeted Quantitative practice • Review of crucial math skills and concepts (including arithmetic, algebra, data interpretation, geometry, and probability) • Key strategies for all Quantitative Reasoning question types on the revised GREKaplan is dedicated to helping our students score higher. We guarantee that students will raise their scores–or get their money back. |
Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more |
0521798MP Interact Book S1 (SMP Interact Key Stage 3)
SMP Interact is a new generation mathematics course for key stages 3 and 4. It aims to meet teachers' professional needs and provide an enriched learning experience for pupils. This is one of three books for pupils at the end of year 7/beginning of Year 8. There is an associated Teacher's Guide, Practice Book and Resource sheets. The S series leads to the level 4-6 SAT |
This project uses PHP Code to programing mathematics and theoritical physics it begins with finding the roots of nonlinear equation and Lagrange interpolation we will develop it step by step you can also download it from
My personal computing suite provides some programs I developed as a hobbyist and currently include a lambda calculus interpreter and a calculator which solves arithmetic expressions; both are written in C++ Function Wizard is designed to provide a lightweight, fast, and easy to use application that will integrate, differentiate, calculate maxima and minima, etc. basic polynomials, as well as assist in other tedious polynomial tasks.
GraphIte is a mathematical graphing program for Windows platforms built in C++ on the MFC libraries by Mr Soo Reams and Mr Stephen Bennett. Many of the features are not understood by either of the aforementioned, or Tim Cook. |
A Guide to MATLAB: For Beginners and Experienced Users
This is a short, focused introduction to MATLAB, a comprehensive software system for mathematical and technical computing. It contains concise explanations of essential MATLAB commands, as well easily understood instructions for using MATLAB's programming features, graphical capabilities, simulation models, and rich desktop interface. Written for MATLAB 7 it can also be used with earlier (and later) versions of MATLAB.This book teaches how to graph functions, solve equations, manipulate images, and much more. It contains explicit instructions for using MATLAB's companion software, Simulink, which allows graphical models to be built for dynamical systems. MATLAB's new "publish" feature is discussed, which allows mathematical computations to be combined with text and graphics, to produce polished, integrated, interactive documents.
Customer Reviews:
can be used for most versions of Matlab
By W Boudville - January 27, 2007
[A review of the 2nd edition.]
The 2nd edition includes descriptions of Matlab 7. But in fact, much of the text seems unchanged from the 1st edition. Because while Matlab continues to add refinements, it is by now very stable in its core functionality. Thus, at least for beginners, the book is useful, regardless of which precise version of Matlab is on your machine.
Many of the examples are indeed quite easy. The underlying maths knowledge is at the undergraduate level. Say for 1st or 2nd year undergrad maths courses. One example uses Lagrange's equation in classical mechanics. That seems to be about as advanced as matters get.
The graphical capability of Matlab given by the book is equivalent to what Mathematica provides. Though perhaps the latter has somewhat spiffier graphics, the graphical functionality is about the same.
Unorganized and unintuitive
By jontell3 - February 3, 2013
The book is ordered in a way that makes intuitive learning very difficult. The built in functions are breezed over offering little explanation as to there actual mechanisms. Overall, gives little insight into MATLAB.
Good small textbook
By M. Ercolino "thenakedeye" - November 21, 2010
This is more of a textbook than a guide. It has good examples and a little code, but I was hoping for more code examples and found it only a good companion to my normal textbook,
The development and business communities recently have experienced a tremendous resurgence of interest in promoting value chains development as a way to add value, lower transaction costs, diversify ... |
More About
This Textbook
Overview
This book provides a progressive math background for individuals entering or currently involved in professional health care. Carefully developed examples are followed by problems directly related to a range of health care services. With both educational expectations and vocabulary for each unit, this book uses examples, problems and tests to evaluate student's understanding of section objectives. The use of a calculator is presented to both the student and instructor as an optional means to approach problems. Supplements Instructor's Manual 0-8273-6689-2 7 3/4" x 9 1/4", 288 pages, 1 color, paperbound |
Math SL is the median level math in IB Program. It is a cross
between Pre-Calculus andCalculus AB. It is a two year course divided into Math SL I and Math
SL II. At the end of the two year course there is a final exam. The exam plus your portfolios combine to give you
a final score.
Math SL I - This course is designed to meet
the syllabus requirements for the IB Math Methods SL 1 or as pre-calculus for those students not intending to take the IB
exam. It is the beginning of a demanding two year study of advanced mathematical topics including functions, exponents and
logarithms, trigonometry, vectors, sequences and series, limits, probability and statistics. IB portfolio projects are a required
part of the course. Graphing calculators are required for the course with the TI-83, 83+ strongly recommended. Students taking
this course should have above average math skills.
Math SL II - This course is designed to complete
the syllabus requirements for the IB Standard Level mathematics. Techniques and applications of differential and integral
calculus will be studied and a review of topics covered in pre-calculus will be included. IB portfolio projects are a required
part of the course. Graphing calculators are required with the TI-83,83+ strongly recommended. This is a demanding course
and should be taken by students who have an interest in mathematics and a strong background in mathematical techniques. |
About AppShopper
MathCast: College Algebra
iOS Universal
Limited Introductory Price!
These video tutorials are presented by Roderick V. James PhD. EE, Adjunct Professor at Houston Community College. Dr. James has been involved with education for the past ten years. His past teaching experience at the Keller Graduate School of Management of DeVry University included the duties of Curriculum Manager for Project Management.
Currently, Dr. James teaches a range of math courses including Fundamentals of Mathematics I & II, College Algebra, Trigonometry, Pre-Calculus and Calculus and Finite Mathematics. "These video tutorial courses provide a way for students to compliment their classroom experience." said Dr. James. He continues, "The videos also provide the busy professional with a quick reference to a particular topic." Dr, James also stresses that this video is not a substitute for attending class, but a compliment to the classroom experience. As in all aspects of life "Practice Makes Perfect".
In math Practice means working problems! During this comprehensive 8 hour and 23 minute tutorial, College Algebra students will be presented with all the knowledge, tools and techniques necessary to make an "A" in the course. |
Incorporating new problems and examples, the second edition of Linear Systems features MATLAB material in each chapter and at the back of the book. It gives clear descriptions of linear systems and uses mathematics not only to prove axiomatic theory, but also to enhance physical and intuitive understanding.
Most Helpful Customer Reviews
I am a third year physics student. while taking a course on signal analysis that was very electronics orientated, i found most signals books unhelpful in the way they present their material. this book is clear and the examples explain key concepts perfectly. i particularly like the coverage of convolution. once again, great book. very helpful.
This introductory volume is very well-written and well-explained and covers the field of signals and linear systems comprehensively. This book offers rigorous but clear explanations, vivid examples in a plausible and easy to understand manner. This book, Linear Systems and Signals, provides a comprehensive treatment of the subject and encourages students to discover information and principles on their own. Author uses mathematics to enhance physical and intuitive understanding, instead of merely employing it to prove axiomatic theory. The book is conveniently organized into five parts that allow flexibility in teaching discrete-time and continuous-time systems. This is an excellent book for undergraduates.
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
38 reviews
33 of 34 people found the following review helpful
One of the finest books I've seen11 Mar 2004
By
"idreamofjeani"
- Published on Amazon.com
Format: Hardcover
The author is really solid. He writes his text carefully. You won't find a higher quality, more coherent book on the subject when it comes to systems. The best part is the fist half, the first 6 chapters on system analysis. The second half, signal analysis, is also quite good, but a bit harder to understand and follow than the first half. It is still far better than just about every other book out there... except for Peter Kraniauskas' book, "Transforms in Signals and Systems," which is the best choice for understanding signal analysis. I highly recommend getting both Lathi's book and Peter's book if you can afford it. If you can only afford one book, you are a first time student in systems and signals, or you are interested mainly in circuits, filters, transfer functions, frequency and phase responses and so forth, go with Lathi's book. If you can only afford one book, and you are interested in DSP with Fourier analysis, and discrete Fourier analysis, as well as spectrum analysis in general, go with Peter Kraniauskas' book. In Summary: If you are in a class called "Systems and Circuits" or something like that, get Lathi's book. If you are in a class called "Discrete Time Signals and Systems" or "Frequency Analysis", go with Peter's book.
13 of 14 people found the following review helpful
The Best College Text book for Linear Systems and Signals10 Oct 2003
By
A Customer
- Published on Amazon.com
Format: Hardcover
In 1996, I bought this book for a class on Linear Systems. After reading the first three chapters of this book, I quickly became very impressed with Author's Style. Lathi's book is well writen and it is very easy to understand. When I took my communication class, I decided to buy Lathi's Modern Digital Communication book even though we were assigned a different class book. Lathi's books are fun to read and learn. I use this book every day as a Reference for my Digital Signal Processing projects.
11 of 12 people found the following review helpful
This is truly an excellent book.26 Nov 1999
By
Sadhak
- Published on Amazon.com
Format: Hardcover
"This is truly an excellent book...Lathi's writing is of high caliber...I believe the strongest point of the book is prof Lathi's ability to describe very difficult material in a very clear and simple way."
19 of 24 people found the following review helpful
An excellent textbook14 July 2004
By
James Fett
- Published on Amazon.com
Format: Hardcover
Lathi's book is an excellent, easy-to-read and easy-to-understand text on systems and signals. He spends a lot of time explaining in-depth examples, doing so without assuming the student has insight he or she probably does not-a common problem in other textbooks. Lathi goes out of his way to explain every nuance, greatly helping the learning processs Unlike most other textbooks, this one is easy to read, without a glut of technical jargon, or pages and pages of mathematical formulae without text explaining where the author is heading. Lathi also includes a very comprehensive background section, polishing and reviewing the mathematical techniques necessary for solving the problems. It is handy to have this in one book, rather than require trigonometry, matrix algebra, and differential equations books at the standby for review. One complaint is that this book does not have solutions to some of the homework problems, helping with additional study. This isn't quite the problem it is in other books since the examples are very thorough and insightful. Simply put, not only is this probably the best textbook on systems and signals, it is one of the best textbooks I have ever used.
8 of 9 people found the following review helpful
This book stands head and shoulders above the rest.26 Nov 1999
By
Sadhak
- Published on Amazon.com
Format: Hardcover
"This book stands head and shoulders above the rest of the books in the field... A pedagogically sound approach... I like the author's style of writing. All the chapters are uniformly well written and reflect the author's experience in teaching the subject to students with varying degrees of interest and abilities. He recognizes the common source of student difficulty and addresses each one." |
AAT Chapter 1 Project Options
Your project should add up to 100 points and is due on Tuesday, October 26 in
class.
If there are any parts of the exam in which you did not do well, you should attempt
to make your project cover that aspect of the chapter. If you can give me definitive
evidence that you now understand how to do a part of the chapter that you did not
understand when you took the test, I will raise your exam grade. Choose from the
following options wisely:
100 points:
a.) Write an essay, rap, poem, or letter that interconnects at least 25 of the terms
in the vocabulary list on page 58 of your textbook. This should not be a
simple list of terms and their definitions, but a cohesive draft that shows how
the concepts relate and build upon one another. You will be graded on your
mathematical knowledge rather than technical writing skills, but examples of
your knowledge and definitions should be weaved in carefully and in your
own words.
b.) Complete the chapter study guide and write 2-3 sentences to sum up the
"point" of each section.
c.) Write questions for next year's AAT jeopardy!—you should have a jeopardy
round, a double jeopardy round, and a final jeopardy question. See me for the
template.
50 points:
a.) Come up with an experiment, gather at least 10 points of data, and write an
algebraic model to describe what happens in real life. Be sure to explain
whether your data is best described by a linear equation or not. Use formulas
if necessary and create a graph of the data.
b.) Explain each step of the order of operations and demonstrate what happens
in problems when the order of operations is not followed. Then, create
poster of phrases that mean "add" "subtract" "multiply" "divide" and "group"
to help your classmates better decrypt the ACT.
c.) Take a practice ACT exam (get it from me) and highlight the questions that
cover topics from algebra 1 and chapter 1 of our book. Then create a worked-
out solution set that explains why each incorrect answer doesn't work and
why the correct answer is the best choice. Write it in student-friendly
language, not in "math-geek speak," and turn it in with the completed test.
Use the resources in the library to help you.
d.) Create a 20-minute "mini lesson" and 5-10 practice problems for your peers
to complete on one of the sections in unit 1. You may choose to create a
powerpoint or write out a script of how it should be explained and come up
with a list of questions that your peers may have and answers to those
questions. These questions may be shared with the next class for their
quarter review.
25 points:
a.) Look at question 76 on page 55. Get the height of 10 people in inches and use
the conversion factor (1in=2.54cm) to find their height in centimeters. Then
set up an absolute value inequality to describe how long their femur is likely
to be. Solve to find how long each person's femur is.
b.) Look at question 50 on page 46. Explain how the inequality changes when
the entrance fee and food costs rise and fall. Write an inequality for each of
the following situations: entrance fees of $20 and $30, food costs of $10 and
$20. Graph the inequalities.
c.) Set up the problem and write an explanation of your problem-solving model
for question 22 on page 38. Your explanation should be approximately 1
paragraph in length and should use appropriate mathematical terminology.
d.) Rewrite the common formulas found on page 28 to solve for alternate
variables. Then write one paragraph explaining why this skill is necessary.
e.) Use the formula given to explain and solve problems 43 and 44 on page 23.
Write a paragraph to explain your answers. (1 paragraph can cover both
problems)
f.) Create a chart to model the population of Hawaii from 1980 until 2010. Then
check this with current population data. Is the model still accurate? Why or
why not?
g.) Complete problems 64 or 65 on page 9. Write a paragraph explaining how
you have solved your problems. (1 paragraph can solve both problems |
Elementary Linear Algebra
9780132296540
ISBN:
0132296543
Edition: 9 Pub Date: 2007 Publisher: Prentice Hall
Summary: This text presents the basic ideas of linear algebra in a manner that offers students a fine balance between abstraction/theory and computational skills. The emphasis is on not just teaching how to read a proof but also on how to write a proof.
Kolman, Bernard is the author of Elementary Linear Algebra, published 2007 under ISBN 9780132296540 and 0132296543. Five hundred eighty Elementary Linear Algebra tex...tbooks are available for sale on ValoreBooks.com, one hundred forty seven used from the cheapest price of $81.90, or buy new starting at $149.59.[read more [more |
books.google.com - "From modern-day challenges such as balancing a checkbook, following the stock market, buying a home, and figuring out credit card finance charges to appreciating historical developments by Pythagoras, Archimedes, Newton, and other mathematicians, this engaging resource addresses more than 1,000 questions... Handy Math Answer Book |
9780201704: Graphs and Models
Appropriate for a one-term course in intermediate algebra, this text is intended for those students who have completed a first course in algebra. By requiring the use of the graphing calculator, this text joins the leading trend in the developmental math market. The authors' state their view on their use of technology as, "We use the graphing technology to enhance, not to replace, the students' mathematical skills."Geared toward helping students learn and apply mathematics by incorporating the hallmark Bittinger Five-Step problem-solving process, this text integrates real world applications with proven pedagogy and an accessible writing style. This text is graphing calculator required but retains a strong balance of skill and drill |
CAMMath Classes (Computer Assisted Modular Math)
The School of Academic Advancement is offering a different delivery model for developmental mathematics—CAMMath (Computer Assisted Modular Math). It has been piloted and revised since Summer 2013. We are excited about this alternative option for students and believe that it can make the developmental math experience a better fit for each individual student. Note: There are still an equal amount of traditional sections being offered as well.
The School of Academic Advancement is offering a new delivery model for developmental mathematics. We are excited about this alternative option for students and believe that it can make the developmental math experience a better fit for each individual student.
What is different about the CAMMath classes (Math Concepts and Basic Algebra)? Why are the classes scheduled at the same time, with the same instructor? • Students that place into Math Concepts and Basic Algebra meet at the same time in the same computer classroom/lab. • These classes use MyMathLab for web-based computer-assisted instruction that allows students to proceed at their own pace through a set of content based modules. • MyMathLab tracks student progress and allows teachers to deliver just-in-time instruction. • Motivated students who place into Math Concepts may be able to complete requirements for both Math Concepts and Basic Algebra in a single semester.
How will the class be organized/what will instruction look like? • Each student's learning experience will be customized to their individual needs – students who come to class with mastery of introductory material will be able to demonstrate their mastery via pre-tests, and homework is customized for each student based on performance on module pre-tests. • The course is scheduled in a computer classroom/lab, where students work online in a specialized math environment that allows students to watch lecture videos, read e-texts and receive guided instruction through complex problems. • The students receive just-in-time instruction as needed and also participate in group activities/instruction with students who are working on similar skills and concepts. • Students are expected to attend the classes as scheduled. • Students are expected to work an additional 4-8 hours per week outside of class online (similar to the outside of class study time and homework expectation in traditional courses). • All assessments (pre- and post-tests) are done online, but within the scheduled class time. This provides a proctored environment.
Who should take the class? • This class is intended for students whose COMPASS score placed them into Math Concepts or Basic Algebra. • Students should be technology friendly, very self-motivated, and able to work independently in and out of class.
Could someone complete both Math Concepts and Basic Algebra in one semester? Should they sign up for both? • Since the modular system is self-paced, students successfully completing the Math Concepts modules may continue into the Basic Algebra modules and complete all 16 modules in a single semester. • Students should enroll only into the course identified by COMPASS. • If a Math Concepts student successfully completes both Math Concepts and Basic Algebra coursework and exams, they will be provided permission to enroll into their next math course.
What about students who struggle and do not complete all of the modules? • One of the advantages of the modular system is that students may re-enroll for the next semester and be able to continue from the point at which they ended the initial semester, unlike the traditional classroom. • For example, a student needs to complete 16 modules to demonstrate mastery of the content in Basic Algebra. If the student is able to complete modules 1 thru 12 in the first semester, the student would begin with module 13 in the subsequent semester, and would only need to complete 4 modules to complete the course.
What books are needed for this course? (versus face-to-face) • Instead of a textbook, students will purchase access to MyMathLab software via a course materials fee.
Are students required to have a computer (with internet access) at home? • A home computer and broadband internet access are not required, but are strongly recommended. Students are expected to work an additional 4-8 hours per week outside of class online. To do so, they will need to plan for where this work can be done. • The software is web-based so that students can do the online homework (viewing videos and doing problems) anywhere there is internet access. The libraries and Student Achievement Centers are places on campus that students can do this online work. • Our modular system, in its initial form, is NOT intended to be an online class – our research has shown that for the majority of students, interaction with qualified and caring instructors is vital to success in mathematics – much like in a structured course setting. Our expectation is that students will attend class consistently and interact with instructors. |
The best Collection of Math cheat sheetsMany Cheat Sheets in your mobile to have the formulas wherever and whenever you want.With this application you can use your travel time to study, or just have it as a quick reference when needed.NOTE: App can be moved to the SD Card!!Contents:=========Algebra. Elementary techniques for factoring binomials and trinomialsAlgebra. Exponent laws and factoring tipsAlgebra. Solving quadratic equations by completing the squareAlgebra. College Algebra quick referenceAlgebra. Solution of the 3rd degree polynomial equationAlgebra. Solution of the 4th degreee polynomial equationTrig. Basic trig identitiesTrig. Law of sines cosines etc and other triangle formulasTrig. Graphs of the trig functionsTrig. Inverse trig functionsTrig. Power reducing formulas for powers of sines and cosinesTrig. Graph paper for plotting in polar coordinatesTrig. Two unit circles with trig funcion valuesTrig. Single unit circle with trig function valuesCalculus. Basic differentiation formulas and some useful trig identitiesCalculus. Basic differentiation and integration formulasCalculus. Definitions and theorems pertaining to Riemann sums and definite integralsCalculus. A quick reference sheet on Taylor polynomials and seriesCalculus. A summary of convergence testsCalculus. Guidelines for evaluating integrals involving powers of sines and cosinesCalculus. Guidelines for evaluating integrals involving powers of secants and tangentsCalculus. Standard forms for conic sectionsCalculus. Common infinite seriesCalculus. Trigonometric substitutionCalculus. Cylindrical coordinatesCalculus. Spherical coordinatesCalculus. Hyperbolic functionsCalculus. Applications of integralsCalculus. Applications of integralsCalculus. Common ordinary differential equationsCalculus. Common ordinary differential equationsCalculus. Common ordinary differential equationsCalculus. Undetermined coefficients and variation of parametersCalculus. Vector formulasCalculus. Simple summary of cylindrical and spherical coordinatesMisc. Some prime and composite numbersMisc. Sets. Functions lines and sequencesStatistics formula sheet Page 1Statistics Page 2Statistics Page 3----------------------------------------SD Installation supportPERMISSIONS:============Why READ_PHONE_STATE / CHECK_LICENSE permissions?The application verifies it has been been purchased using google licensing library (CHECK_LICENSE). Google recommends to use the telephone identifier (READ_PHONE_STATE) to avoid piracy. Not used for anything else |
Shopping & Services
Best Parenting Classes Denver 2003 - Parents' Mathematics Institute Community College of Aurora
Show of hands: How many parents have had palpitations when a graphing calculator showed up on the seventh-grade back-to-school supply list? Was it just the cost, or was it also the fact that you had no earthly clue what a graphing calculator does or why anyone would need to know such a thing? The good news is that the folks at CCA realize there are major mathematical concepts that have been discovered since you left junior high. They've put together a two-week, mini-crash course, the Parents' Mathematics Institute, for moms and dads of kids in seventh grade and up that covers elementary math concepts, including writing, solving and graphing inequalities, estimating answers and beginning algebra. The bad news is that the course, complete with an introduction to that pesky graphing calculator and separate workshops for kids, took place earlier this month. |
4 Answers
On top of what you've got there already, I would recommend signal processing or some similar course that covers Fourier transforms and the like. Besides being useful as a foundation for wavelets, Fourier theory will give you a new way of looking at data that is often useful. Wavelets will probably be part of the curriculum for more advanced signal processing courses.
If you want to do research then I'd recommend learning both discrete and continuous. Most people only know one or the other in detail and it is seriously stifling research. There is a lot of opportunity for cross pollination here.
It sounds to me like you should just start learning about wavelet transforms and then figure out gaps along the way. They're not that involved. Fourier transforms etc are just an example of an orthogonal basis that is part of linear algebra. |
Linear Algebra
Linear Algebra (LA) is the study of systems of equations. Specifically LA teaches an approach to solve complex, multivariable equations, starting with an approach called Gaussian elimination. There is a more abstract way of thinking about systems of equations embodied in the notion of vector spaces, which can be thought of as an extension of the euclidean plane into multi-dimensional spaces. LA has a lot of practical applications including statistics, computer science and probability, so a deep knowledge of LA can go a long way. Understanding LA requires some basic mathematical thinking (solving systems is nothing more than doing simple arithmetic and algebra) - past this there is a lot of abstract mathematics in thinking about how systems of equations take on a spatial meaning.
I've studied Linear Algebra formally (I took it in college my senior year and received an A in the course) and have been exposed to different applications of Linear Algebra in my MA studies at Columbia in various statistics and computerscience courses. As a biochemistry student in college I used LA extensively to solve problems in quantum/physical chemistry |
Precalculus : Functions and Graphs - 3rd edition
Summary: Making the transition to calculus means being prepared to grasp bigger and more complex mathematical concepts.Precalculus: Functions andGraphs is designed to make this transition seamless, by focusing now on all the skills that you will need in the future. The foundation for success begins with preparation andPrecalculus: Functions and Graphswill help you succeed in this course and beyond.
4. Exponential and Logarithmic Functions. Exponential Functions and Their Applications. Logarithmic Functions and Their Applications. Rules of Logarithms. More Equations and Applications.
5. The Trigonometric Functions. Angles and Their Measurements. The Sine and Cosine Functions. The Graphs of the Sine and Cosine Functions. The Other Trigonometric Functions and Their Graphs. The Inverse Trigonometric Functions. Right Triangle Trigonometry.
03215011146 |
Mathway is a mathematics problem solving tool where students can select their math course - Basic Math, Pre-Algebra, Algebra,...
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Mathway is a mathematics problem solving tool where students can select their math course - Basic Math, Pre-Algebra, Algebra, Trigonometry, PreCalculus, Calculus or Statistics and enter a problem. The computer solves the problem and shows the steps for the solution. It also has a worksheet generator.
math911 contains step by step tutorials in Introductory Algebra, Intermediate Algebra, PreCalculus and Introductory...
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math911 contains step by step tutorials in Introductory Algebra, Intermediate Algebra, PreCalculus and Introductory Statistics. The given link is to the setup file. After setup, you should see a math911 icon on the desktop. Click on the icon to run math911. Vista, Windows 7, Windows 8 users might need to right click and in the drop down list select 'Run As Administrator.'To set up a grade report file you will be asked to enter your name, birth date (or any other date) and select a course.A grade report file will be created (FLmmdd.mdb where F=first initial, L=last initial, mmdd= month and date). Grades are saved AUTOMATICALLY to this file..Select a chapter, then a lesson and a problem will appear.For a tutorial: You can step through the solution (click See Solution, See All Steps, See Next Step, etc.)To answer: Type the answer and press ENTER Only correct answers count. Wrong answers are ignored !If your answer is incorrect: Click on See All Steps, See Next Step, etc.Use the up/down arrows to enter exponents and move between the numerator and denominator of a fraction. There are up to 8 levels for each lesson and all the types of problems for you to master AlgebraIMPORTANT. math911 is activated for Introductory Algebra. For full activation to other courses, RIGHT click on the ABOUT button and enter the code: essex.Once fully activated you can switch back and forth to the other courses.Math911 is continuously updated. For technical help please call Professor Martin Weissman 347-528-7837
Old Egyptian Fractions at MathCats(Web and android version)Old Egyptian Math Cats knew fractions like 1/2 or 1/4 (one piece...
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Old Egyptian Fractions at MathCats(Web and android version)Old Egyptian Math Cats knew fractions like 1/2 or 1/4 (one piece of a pie).But to make fractions like 3/4, they had to add pieces of pies like 1/2 + 1/4 = 3/4.Old Egyptian Math cats never repeated the same fraction when adding.They never wrote: 1/4 + 1/4 + 1/4 = 3/4How it works:Choose puzzles from the list on the top. ( * = easier, **** = very hard.)Add 2 or 3 fraction pieces below.After you find one solution, the puzzle is marked "Solved." Can you find more solutions? (Click to see them listed on the bottom.)There's the android version in the mirror site link
'Brain Teaser measures the ability to differentiate, to analyze, to reason out, and to apply knowledge. It is one of the best...
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'Brain Teaser measures the ability to differentiate, to analyze, to reason out, and to apply knowledge. It is one of the best ways to engage yourself as well as to check your intelligence.This version fixes all the issues reported by some users for previous version. We have tested each question / answer set for accuracy.HOW TO USE THIS APP ?=====================- To start, select the number of questions you want in the current session. - All the questions are in simple language. In each case 4 answers are given, and what you have to do is simply choose the right answer. - There is only one correct answer to each question. - Answer the current question to proceed to next question.SCORING / HIGH SCORE====================- The more you answer correctly more you will gain points.- You need to score at least 50% to get a chance in the High Score charts.- Keep track of your progress or compete with your friendsBENEFITS OF BRAIN TEASER=========================- Children can take this test for preparing aptitude test.- giving your brain a good exercise while you spend time in commute or a boring lecture ;)- fun to challenge your friends to beat your high score.- NO blinking ads means you can focus on the questions better.★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★ ★This is not just a game but a POWERFUL tool to practice, learn and develop MATHS and LOGIC skills.While anyone can attempt to play this game BUT if you are looking for a real tough set of questions which are actually asked in the Competitive examinations, then this app is a must have for you.'This app costs $0.99 |
CK-12 Middle School Math Grade 6, Volume 2 Of 2
CK-12 Middle School Math Grade 6, Volume 2 Of 2 Middle School Math Grade 6 covers the fundamentals of fractions, decimals, and geometry. Also explored are units of measurement, graphing concepts, and strategies for utilizing the book's content in practical situations. Volume 2 includes...
MoreCK-12's Middle School Math Grade 6 covers the fundamentals of fractions, decimals, and geometry. Also explored are units of measurement, graphing concepts, and strategies for utilizing the book's content in practical situations. Volume 2 includes the last 6 chapters.' |
A typical Algebra 1 course reinforces the very basics of solving, graphing, and writing linear equations and inequalities. The next step introduces powers and exponents, quadratic equations along with polynomials and factoring. I like to emphasize WHY a particular concept is being learned along with the real-life CONNECTION |
Algebra2 and Trigonometry is a new text for a course in intermediate algebra and trigonometry that continues the approach that has made Amsco a leader in pre-senting mathematics in a modern, integrated manner. ... Algebra2 and Trigonometry Author: AMSCO
† Complete AnswerKey with solutions. With Amsco's Preparing for the ACT: English and Reading, ... Curriculum for Algebra2 and Trigonometry as covered in Amsco's Algebra2 and Trigonometry and Preparing for the Regents Examination Algebra2 and Trigonometry.
ANSWERKEY Preparing for the REGENTS EXAMINATION GEOMETRY AMSCO N 81 CD. ... 12 Algebra is the study of operations on sets of numbers. 13 Trigonometry is the study of triangles. 14 Geometry is the study of shapes and sizes.
expanded as a bridge to Algebra2 and Trigonometry. iii ... Boolean algebra is key in the development of com-puter science and circuit design. ... Answer (2) The hypothesis pis "2 is a prime number," which is true.
ANSWERKEY. Preparing for the. REGENTS ... One possible answer is: Every square has . 17 This PDF book include amsco workbook latin four years answers guide. To download free ... This PDF book include algebra2trigonometry test one answers information. To download free scoring key ...
answerkey includes information regarding the competency to which ... Investigating college algebra and trigonometry with technology. Emeryville, CA: Key College Publishing. ... New York: AMSCO School Publications. |
Understanding the concepts of algebra 2 and trigonometry are paramount in getting through this course and moving on to calculus. Precalculus takes you through an in-depth look at functions, graphs, and the relationships between geometric shapes (mainly triangles |
this highly dependable book retains its best features–accuracy, precision, depth, and abundant exercise sets–while substantially updating its content and pedagogy. Striving to teach mathematics as a way of life, Sullivan provides understandable, realistic applications that are consistent with the abilities of most readers. Chapter topics include Graphs; Trigonometric Functions; Exponential and Logarithmic Functions; Analytic Geometry; Analytic Trigonometry; Counting and Probability; and more. For individuals with an interest in learning algebra and trigonometry as it applies to their everyday lives. |
groups of managers, clients, customers, and the general public. Able to apply concepts of basic algebra and geometry such as fractions, percentages, ratios, and proportions to practical situations. Able to calculate figures and amounts... |
As a person who has a healthy interest in mathematics and has taken many classes, this is definatley one of the best! Professor Valenza taught it (he has been teaching this Linear Algebra class at CMC for ten years) and his book is essentially an excellent compilation of the lecture notes from his class. It takes a very different tack from most linear algebra texts: Usually, a linear algebra text begins by inroducing matrices and solving simultaneous equations, teaching computational methods. Prof. Valenza starts with the structure BEHIND all of that math however: Sets, Groups, and Vector Space properties. This structure is absolutely essential to knowing what's going on: My father took a (less superior) linear algebra class many years ago, and he never understood the concepts behind the mathematical manipulations; I actually sat down with him and taught him the things that I learned in Prof. Valenza's class. I really think that the knowledge in this book is invaluable to someone who wants to know what Linear Algebra is really about. Just a few examples of the truly deep knowledge that this book communicates follows. For instance (this will ring a bell for those who have taken calculus) the "constant of integration" that must be added when doing an antiderivative is actually a property of group homomorphisms. The "absolute value" that must be introduced when taking square roots is structurally THE SAME property of group homomorphisms. Also, we all know that you can't divide by zero; it's just not allowed. But, the reason for that is ultimatley rooted in group theory; namely, the real numbers are NOT a group under multiplication. This type understanding has EVERYTHING to do with matrices and systems of equations! For instance, the fact that only square matrices can be inverted is a trivial consequence of a property of function mappings called "bijectivity." (a mapping from three- to two- dimensional space can't be bijective, for example) Many seemingly complex linear system problems can be simplified to a trivial questions by, for example, investigating the "span" of the column vectors of a matrix. There are countless problems that simply can't be understood without the kind of structural knowledge that Prof. Valenza's book gives. Understanding the basic properties that underlie so many mathematical objects has been a true delight for me, and anyone who wants to know what is really going on "behind the scenes" with linear equations would be wise to investigate Prof. Valenza's book. It's no accident that he also wrote a book on Fourier Analysis; understanding structure is simply the key to higher math.Read more ›
This book treats the basic principles of abstract algebra. It is targeted to graduate students that need a more theoretical approach to mathematics (instead of the usual calculus courses) This book is the best introduction to abstract algebra for the following reasons -its style : good introduction in each chapter, making the reader curious to read further. -its rigor : everything is well explained in full details with proof. -its elegance : This book treats the abstract structural aspects of algebra and then suddenly shows how more concrecte applications follow from these abstract results. This is the kind of elegance and style that makes mathematics an art : build a very abstract theory and then see how more concrete stuff follows immediately as special case of this abstract framework. This way, new things can be discovered and most of the time (as in this book), you can explain practical calculation rules in a short and rigorous way. Definitely the finest there is ....
Great book ! If you read this book, you will not only gain knowledge of abstract algebra, but also understand clearly why mathematics is art. It was a real fun reading this book. The topics are presented in such a way that the author leads you to a climax, making you curious to read further, and help you to explore the beauty in all the ideas of abstract mathematics. This book is the best book I ever read on abstract algebra.
The emphasis in on rigor and abstract structural concepts. It is nice to see that the more practical applications follow as a special result from the abstract structural concepts. This is a very elegant approach !!
This is a beautiful book though you have to belong to a certain reader segment to appreciate it.
The readers that will like this book probabely are beginning undergraduate students that want to build a mathematical career and want a first and quick introduction to abstract mathematics. The reader is not overwhelmed by exotic topics that are rarely used, but is introduced to abstract basic principles needed to understand other courses like for instance quantum mechanics,more advanced graduate courses in algebra or functional analysis.
The power of this book is that it covers just enough material to have a solid foundation of algebra for other abstract courses like functional analysis,
When I compare it for instance with the book of Shilov, I strongly prefer this book since it is better organised, covers less topics, but enough to know the basics. This book succeeds in providing shorter proofs compared to Shilov without sacrifying rigor and clarity. How is this possible ?? Ah my friend, this is a reward coming from abstract reasoning as illustrated by this book. |
"It is fun to figure out the puzzle of how children go about making sense of mathematics and then how to help teachers help kids.â€Â John A. Van de Walle, Late of Virginia Commonwealth University This is the philosophy behind Elementar y and Middle School Mathematics: Teaching Developmentally. John A. Van de Walle wrote this book to help students understand mathematics and become confident in their ability to teach the subject to children in kindergarten through eighth grade. Although he could not have foreseen the changes in mathematics teaching over the last three decades, he was at the forefront of the movement towards a constructivist view of teaching, or teaching developmentally. Constructivism says that children construct their own knowledge. They are not...
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Nabu Press 9781172264452 Vector Analysis; A Text-Book for the Use of Students of Mathematics and Physics, Founded Upon the Lectures of J. Willard Gibbsi Rudolf Steiner - Steiner, Rudolf THIS IS A BRAND NEW UNOPENED ITEM. Description Collected here in this 5-in-1 omnibus e
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This Teaching Notes and Solutions Manual accompanies Singapore Math's sold-separately Discovering Mathematics Textbook 7B Softcover.
Free Delivery Worldwide : Essential Mathematics for Cambridge IGCSE Student Book : Paperback : Oxford University Press : 9780199128747 : 019912874X : 02 Feb 2012 : Written by an examiner to exactly match the 0580 syllabus, this text gives you a sequential and logical teaching path through the full syllabus, addressing the four curriculum areas. A full set of solutions, exam practice and worked examples encourage active learning, and a CD offers dynamic material for every section of the book.
This Teaching Notes and Solutions Manual accompanies Singapore Math's sold-separately Discovering Mathematics Textbook 7A 174 pages, softcover.
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Dear Valued Customers...- Brand New,US version - Written in English - Picture shown exactly the book we sell Please read the following description carefully before buying because all sales are FINAL.Terms:Only ship to Paypal registered residential or office address NO P.O. BOX, APO, FPO, Puerto Rico, Guam address We ship WORLDWIDE Telephone number must be provided in Paypal when you checkout for delivery contact purpose It takes 7-10 working days by standard mail or 3-5 working days by expedite
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Nabu Press 9781178338676 Higher Mathematics for Students of Chemistry and Physics with Special Reference to Practical WorkFree Delivery Worldwide : About Teaching Mathematics : Paperback : Math Solutions Publications : 9780941355766 : 0941355764 : 04 Sep 2007 : Marilyn Burns' signature resource continues to be at the forefront of current trends and practices for improving the teaching and learning of mathematics. Supporting a student- centered, problem-solving approach proven by research to best develop students' thinking and reasoning of mathematics, Marilyn presents invaluable insights in a standards- driven, practical, and accessible manner. This essential resource includes more than 240 classroom-tested activities and presents the content and pedagogy...
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Nabu Press 9781178076660 A Text-Book of Mathematics and Mechanics Specially Arranged for the Use of Students Qualifying for Science and Technical Examinations: With Numerous |
Description of Item
Does your child understand how formulas and ratios work? The LIFEPAC Pre-algebra & Pre-geometry I Unit 8 Worktext is full of step-by-step instruction in these critical math skills. This eighth worktext in a series of ten has a large variety of student-directed lessons covering topics such as writing formulas, function machines, equations, and ratios and proportions.
But how will you know your student is learning what he needs to? To encourage mastery of important math skills, this student-paced Alpha Omega curriculum includes text-based instruction, easy-to-understand examples and models of math concepts, colorful illustrations, self tests, and ample opportunity for practice of lesson content. A teacher-administered unit test provides opportunity for assessment of student progress. In addition, your student will feel a sense of accomplishment at the completion of each slim worktext—usually in just three to four weeks. See for yourself what makes LIFEPAC such a popular homeschool math curriculum—get the LIFEPAC Pre-algebra & Pre-geometry I Unit 8 Worktext today!
Product:
Lifepac Math Grade 7 Formulas & Ratios Unit 8
Edition Number:
2014
Media Type:
Book
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7th Grade
Maximum Grade:
7th Grade
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0.29 pounds
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0.125 inches
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MAT 0708
Subject:
Math
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Intermediate Algebra - With 2 CDs - 10th edition
Summary: This concise and cumulative guide shows students the art of technical writing for a variety of contexts and institutions. Using examples from the business and non-corporate world, the book emphasizes transactional writing through practical explanations, real-world examples, and a variety of ''role-playing'' exercises. Each section builds on the next as readers learn a variety of models of style and format. This edition features a stronger emphasis on electronic commu...show morenication, integrated coverage of ethics, and more explanation of how to create technical documents that produce concrete results. ...show less
3Chapter 4: Systems of Linear Equations
4.1 Systems of Linear Equations in Two Variables 4.2 Systems of Linear Equations in Three Variables 4.3 Applications of Systems of Linear Equations 4.4 Solving Systems of Linear Equations by Matrix Methods
9.1 The Square Root Property and Completing the Square 9.2 The Quadratic Formula 9.3 Equations Quadratic in Form Summary Exercises on Solving Quadratic Equations 9.4 Formulas and Further Applications 9.5 Graphs of Quadratic Functions 9.6 More about Parabolas and Their Applications 9.7 Quadratic and Rational Inequalities
11.1 Additional Graphs of Functions 11.2 The Circle and the Ellipse 11.3 The Hyperbola and Functions Defined by Radicals 11.4 Nonlinear Systems of Equations 11.5 Second-Degree Inequalities and Systems of InequalitiesAcceptable
AllAmericanTextbooks_com Ypsilanti, MI
0321443624 Cover worn but text is still usable. Text is in great shape. ISBN|0321443624 Intermediate Algebra, 10th Edition (PJ) WQ
$4.83 |
Math Competition LinksAmerican Mathematics Contest 8 (Middle School) The AMC 8 is a 25 question, 40 minute multiple choice examination in junior high school (middle school) mathematics designed to promote the development and enhancement of problem solving skills. The examination provides an opportunity to apply the concepts taught at the junior high level to problems that not only range from easy to difficult but also cover a wide range of applications.
American Mathematics Contest 10 (Secondary Grades) The AMC 10 is a 25-question, 75-minute multiple choice examination in secondary school mathematics containing problems which can be understood and solved with pre-calculus concepts. Calculators are allowed. The main purpose of the AMC 10 is to spur interest in mathematics and to develop talent through the excitement of solving challenging problems in a timed multiple-choice format. The problems range from the very easy to the extremely difficult.
American Mathematics Contest 12 (Secondary Grades) The AMC 12 is a 25-question, 75-minute multiple choice examination in secondary school mathematics containing problems which can be understood and solved with pre-calculus concepts. Calculators are allowed. The main purpose of the AMC 12 is to spur interest in mathematics and to develop talent through solving challenging problems in a timed multiple-choice format. Because the AMC 12 covers such a broad spectrum of knowledge and ability there is a wide range of scores. The National Honor Roll cutoff score, 100 out of 150 possible points, is typically attained or surpassed by fewer than 3% of all participants. The AMC 12 is one in a series of examinations (followed in the United States by the American Invitational Examination and the USA Mathematical Olympiad) that culminate in participation in the International Mathematical Olympiad, the most prestigious and difficult secondary mathematics examination in the world.
The Mandelbrot Competition (Secondary Grades) In those ten years the contest has grown to two divisions encompassing students from across the United States as well as from several foreign countries. Nearly half of the competitors in the USA Math Olympiad in the last couple of years have been Mandelbrot competitors. The Mandelbrot Competition is split into two divisions: Division A for more advanced problem solvers and Division B for less experienced students.
Mathcounts (Grades 7-8) Each year, more than 500,000 students participate in MATHCOUNTS at the school level. Those who do tell us that their experience as a "mathlete" is often one of the most memorable and fun experiences of their middle school years.
Math Problems of the Week (Grades K-12) The Problem of the Week is an educational web site that originates at the University of Mississippi. All the prizes are generously donated by CASIO electronics. All contest winners are chosen randomly from the pool of contestants that successfully solve that week's problem. |
worry about starting algebra. Pre-Algebra Essentials For Dummies provides an overview of critical pre-algebra concepts to help new algebra students (and their parents) take the next step without fear. Free of ramp-up material, Pre-Algebra Essentials For Dummies contains content focused on key topics only. It provides discrete explanations of critical concepts taught in a typical pre-algebra course, from fractions, decimals, and percents to scientific notation and simple variable equations. This guide is also a perfect reference for parents who need to review critical pre-algebra concepts as they help students with homework assignments, as well as for adult learners headed back into the classroom who just need to a refresher of the core concepts. |
One of the most effective ways to stimulate students to enjoy intellectual efforts is the scientific competition. In 1894, the Hungarian Mathematical and Physical Society introduced a mathematical competition for high school students. Among the winners were Lipot Fejer, Alfred Haar, Todor Karman, Marcel Riesz, Gabor Szego, and many others who became world-famous scientists. The success of high school competitions led the Mathematical Society to found a college-level contest, named after Miklos Schweitzer. The problems of the Schweitzer contests are proposed and selected by the most prominent Hungarian mathematicians. This book collects the problems posed in contests between 1962 and 1991, which range from algebra, combinatorics, theory of functions, geometry, measure theory, number theory, operator theory, probability theory, and topology to set theory. Solutions are included. The Schweitzer competition is one of the most unique in the world. Experience shows that this competition helps identify research talents. This collection of problems and solutions in several fields in mathematics can serve as a guide for many undergraduates and young mathematicians. The large variety of research-level problems should interest more mature mathematicians and historians of mathematics as well.
Book Description:Springer. Hardcover. Book Condition: New. 038794588110245
387945880
Book Description:Hard35
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Book Description:19957518901
Book Description:Springer. Hardcover. Book Condition: New. 038794588143 |
When you don't have the time...but you still need the grade! If your life is too busy to spend hours ploughing through weighty textbooks, and you need every study minute to count, this Schaum's Easy Outline is perfect for you! This super-condensed, high-torque study guide gives you what you need to know in a fraction of the time. Super-Impact Built for quick, effective study, this Easy Outline packs exciting new learning tools that make mastering biology fast, funand almost automatic. Speedy Quick...
Passing grades in two years of algebra courses are required for high school graduation. Algebra II Essentials For Dummies covers key ideas from typical second-year Algebra coursework to help students get up to speed.
Based on a NATO ASI, held at Leuven, Belgium, August 1992, this title gives an account of research advances in numerical techniques used in large scale and real-time computations and their implementation on high performance computers.
Suitable for both Junior and Senior High School use, the videos and workbooks offer a quick and interactive way for students to revise and improve in their understanding and competency in any part of the high school maths courses.
Many students continue to struggle in high school math courses because they failed to master the basic mathematical skills. REA's new Ready, Set, Go! Workbook series takes the confusion out of math, helping students raise their grades and score higher on important exams—including the NJ HSPA. What makes REA's workbooks different? For starters, students will actually like using them. Here's why: * Math is explained in simple language, in an easy-to-follow style * The workbooks allow students to learn...
Based on presentations given at the NordForsk Network Closing Conference "Operator Algebra and Dynamics," held in Gjáargarður, Faroe Islands, in May 2012, this book features high quality research contributions and review articles by researchers associated with the NordForsk network and leading experts that explore the fundamental role of operator algebras and dynamical systems in mathematics with possible applications to physics, engineering and computer science. It covers the following topics: von...
REA's Algebra & Functions Workbook Perfect for students struggling with math! This book will help high school math students at all learning levels understand basic algebra. Students will develop the skills, confidence, and knowledge they need to succeed on high school math exams with emphasis on passing high school graduation exams. More than 20 easy-to-follow lessons break down the material into the basics. In-depth, step-by-step examples and solutions reinforce student learning, while the "Math...
Designed with New York State high school students in mind.CliffsTestPrep is the only hands-on workbook that lets you study, review, and answer practice Regents exam questions on the topics you're learning as you go....
REA's Algebra and Trigonometry Super ReviewGet all you need to know with Super Reviews!2nd EditionREA's Algebra and Trigonometry Super Review contains an in-depth review that explains everything high school and college students need to know about the subject. Written in an easy-to-read format, this study guide is an excellent refresher and helps students grasp the important elements quickly and effectively.Our Algebra and Trigonometry Super Review can be used as a companion to high school and college...
Algebra for the Urban Student offers an algebra textbook for the typical math student. In many cases, such textbooks are written for people who love mathematics and understand the jargon. Teacher Canaa Lee has incorporated her personal experiences as a high school mathematics teacher into a textbook that is specially geared toward students? needs. Most students dislike mathematics because the subject has always been difficult for them to master. With this in mind, Algebra for the Urban Student builds...
This book is a handbook on topics in High School Algebra. We have covered the topics like - Ratio, Proportion, Variation, Progressions, Surds, Imaginary Quantities, Quadratics, Permutations, Combinations, Mathematical Induction, Binomial Theorem, Logarithms, Inequalities, Probabilities and Determinants. Each chapter is follows a simple structure. A definition of terms and detailed explanation of concepts; followed by derivation of useful formulas through first principles. A few problems are solved...
Until geometric programming...
This eBook reviews some advanced topics in algebra, including exploring the nature of polynomials, functions, equations and identity's, examining the mathematical nomenclature used in multiplication and division. We consider multiplying out brackets, taking out common factors, manipulating algebraic fractions and simplifying expressions. Further, we include an extensive selection of questions for the student to enjoy
This eBook introduces the subject of algebra to the student encompassing, inverse operators, equations, the order of precedence, algebraic conventions, BODMAS, expressions, formulae, factorising, rearranging and solving linear, quadratic and simultaneous equations as well as inequalities.This eBook is part of our range of Grades 6, 7 & 8 math eBooks that are aligned with the North American maths curriculum.Our Grades 6, 7 & 8 math eBooks comprise three principle sections. These are, notably:*Number...
Using Graphing Calculators and Computers Can Help Make Learning Algebra More Interesting. Here's a book of engaging blackline-master activities for algebra students to use with their graphing calculators or graphing software technology commonplace in the high school maths classroom. Creating graphs is no longer a time-consuming task for students, which leaves them more time to use graphs to study the properties of functions...
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High level linear algebra book that blends both computational and theoretical aspects, using each to enhance the other. Explains the key points of the Gaussian elimination algorithm.Discusses vector spaces and linear transformations using matrix computations.Takes advantage of software packages such as MATLAB, Mathematica, and Maple.
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Mark Dugopolski was born and raised in Menominee, Michigan. He received a degree in mathematics education from Michigan State University and then taught high school mathematics in the Chicago area. While teaching high school, he received a master's degree in mathematics from Northern Illinois University. He then entered a doctoral program in mathematics at the University of Illinois in Champaign, where he earned his doctorate in topology in 1977. He was then appointed to the faculty at Southeastern...
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The advent of quantum chromodynamics (QCD) in the early 1970s was one of the most important events in twentieth-century science. This book examines the conceptual steps that were crucial to the rise of QCD, placing them in historical context against the background of debates that were ongoing between the bootstrap approach and composite modeling, and between mathematical and realistic conceptions of quarks. It explains the origins of QCD in current algebra and its development through high-energy... |
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