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Vision in Elementary Mathematics (Dover Books on Mathematics) Paperback – November 2, 2011
Scientific Teaching Series
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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.Read more ›
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.Read more ›
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!!!!!