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Vision in Elementary Mathematics

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For those not familiar with Sawyer's other books (*Prelude to Mathematics* -- also by Dover Press, and *Mathematician's Delight* -- by Penguin, but out-of-print), W. W. Sawyer was a Brit who excelled in making even the most opaque topic in mathematics understandable and clear to his many students and readers.

Sawyer's *Vision in Elementary Mathematics* adds to his sterling reputation. On the surface, *Vision* focuses on various topics in 'elementary' mathematics including: algebra, arithmetic, and geometry. A deeper look at this book reveals, however, that Sawyer's book goes well beyond rehashing basic concepts in math.

The focus in this book -- as the title *Vision* suggests -- is on helping both students and teachers have a stronger inuitive grasp of basic concepts of mathematics that many teachers tend to gloss over. As many people have experienced in their own education, mathematics is often treated as a mysterious, 'black box'-like subject. That kind of mindless and unthinking approach to mathematics teaching and learning tend to turn many people off to the subject. This ongoing tragedy in American education has been well researched by another excellent book, *Knowing and Teaching Elementary Mathematics* by Liping Ma.

Sawyer wrote this book in order to combat the unthinking approach to math education. The remedy he offers to that kind of approach is to encourage a deeper understanding of mathematics from relatively simple concepts like fractions, arithmetic, and number properties all the way up to polynomial equations. My favorite quote from this book, which is repeated in different forms, is: "We first try to make sure that we can see what the problem means - for if we do not understand the question, we have no hope of finding the answer." As this quote suggests, *Vision* focuses on helping people to truly UNDERSTAND what mathematical problems really mean.

*Vision* utilizes various techniques to help readers to have a deeper level of understanding of mathematics. Sawyer encourages people to develop some geometrical heuristics to help them grasp a problem. A great example of this is how Sawyer explains how to easily multiply and divide anything from numbers all the way up to polynomial algebraic equations using 'matrix'-like geometric concepts -- no mean feat!

Another great thing about *Vision* is that Sawyer encourages experimentation by students in this book. As many math-lovers know, you can't really learn mathematics without doing it. Sawyer not only encourages people to 'do' math but to experiment with it. By encouraging students to think about math problems in creative and novel ways -- and to not be afraid of making mistakes and, hopefully, learning from them -- *Vision* will help many math students to get a greater appreciation for the subject and will encourage them to be much more creative about the subject. Furthermore, by encouraging student to find out for themselves the whys and hows of a correct solution, *Vision* will encourage the kind of mentality one has to have to become a good mathematician or scientist.

In this regard, it should also be noted that Sawyer includes several exercises at the end of each chapter and includes ANSWERS at the end of the book for each and every question (very helpful for self-learning).

A surprising -- given its focus on 'elementary' mathematics -- benefit of this book is that it provides solid foundations for more advance topics in mathematics. *Vision* provides excellent foundations for number theory, linear algebra, calculus, trigonometry, and combinatorics. Another plus for this book is that Sawyer deliberately tries to emphasize the fact that mathematics -- at any level -- can be useful to real-life problems because mathematics underlies so much of what we see in the natural world.

Who should read this book? The obvious audiences are teachers and students of mathematics in anywhere from elementary school up to high school. This book would be especially helpful to home-schoolers who are interested in providing their kids with a solid grounding in mathematics that can be built upon when they attend university.

But those aren't the only people who should read this book. This book should be read by people who feel that they had an 'incomplete' education. People, like myself, who felt a bit cheated by the poor state of the educational system should definitely read this book no matter how confident (or un-confident) about their mathematical abilities.
This book can and should be read by advance students and teachers of mathematics in fields ranging from engineers to scientists (including professional mathematicians). Those advanced users of mathematics can use this book both for review and to get some additional insights into their chosen subject. I have no doubt that Sawyer has insights and creative approaches in *Vision* that will prove valuable to mathematicians at any level.

Finally, ANYONE interested in getting a 'profound understanding of fundamental mathematics' (often abbreviated PUFM in educational policy circles) for whatever reason MUST read this wonderful book. *Vision* perfectly fulfills Sawyer's vision of writing a book that helps people to look at mathematics in a novel way that will help them to have a deeper understanding and appreciation for mathematics.

Sawyer's text describes visual methods for making concepts from elementary algebra and those areas of arithmetic that tend to pose difficulties for students, including manipulating negative numbers and fractions, understandable. This text was written in response to Sawyer's experiences teaching algebra to a class of mathematically advanced elementary school students, but the methods he discusses here are more widely applicable. Teachers of pre-algebra and elementary algebra will benefit from reading this text.

Sawyer discusses concepts from arithmetic and algebra, including even and odd numbers, divisibility tests, negative numbers, fractions, operations on polynomials, graphs, and the Pythagorean Theorem. As Sawyer discusses each topic, he introduces visual models that help students understand the concept that they are learning. Throughout the text, Sawyer warns the reader about the type of mistakes students are prone to make, why they make them, and how to address them.

Sawyer objects to the practice of teaching algebra by having the students learn a series of skills that allow them to solve increasingly complicated problems without first placing what they are learning in context. He feels that method of teaching discourages students from wanting to learn algebra. Instead, he advocates posing problems that can be represented visually so that students can see what is happening rather than memorizing a rule. Sawyer has his students explore variations on a problem, including their own versions, and search for patterns. This process, which mimics what mathematicians do, helps students understand the concept, which is preferable to having them memorize a rule without understanding the reason for it. He also stresses the importance of having the students master a concept before building to the next concept, arguing that teaching new concepts to students with a poor foundation does them a disservice since they won't be able to understand the new concept until they have that foundation.

The text includes exercises, most of which are routine calculations. However, there are some more interesting problems in which you are asked to search for a pattern. Answers to all of the exercises are provided in the back of the text.

I have some caveats. American readers should be warned that this is a British text, so there are differences in terminology and notation. As Sawyer notes, Americans refer to indices as exponents. Also, the roles of the dot used as a decimal point and the dot used for multiplication are interchanged. Readers outside the United States should be aware that the text is old enough to refer to old English units rather than the metric system. It is also old enough to refer to sixpences, half crowns, and shillings, coins that the British have replaced.

Sawyer's writing is clear, but it is also dry and pedantic. Consequently, I found the text to be a slow read. Therefore, while I can recommend it to teachers of algebra, I do not recommend it to lay readers.

Clear instruction on guiding students through the the most frequently stumbled upon topics in elementary math and algebra: fractions, negatives, combining vs. multiplying terms, distributing negative signs, and a wonderful introduction to simultaneous equations.
His methods leave the "rules" out letting students SEE why a rule may be made. Sawyer is wonderful. I recommend this along with all of his other books. Anyone struggling with Abstract Algebra should find an old copy of his concrete approach book, it's simply the best.

IF YOU TEACH MIDDLE SCHOOL MATH OR HIGH SCHOOL- YOU HAVE TO READ THIS!!!!!