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The many opportunities for students to make conjectures.  
Squaring binomials.  
Adding and subtracting like terms.  
Subtracting a negative from a negative.  
Multiplying a negative by a negative.  
Simplifying expressions such as 50  (20  x).  
An approach that allows students to reason their way toward the algorithm for long division. (I confess that what I was taught in grade four was meaningless, and that as a default option I just memorized the process.)  
A novel method that illustrates on graph paper which points satisfy an equation.  
A discussion regarding the language that we use when we say times for numbers greater than one, of for numbers less than one, and the use of the single symbol x for both words. This discussion includes a sequence of whole numbers with alternating blanks where the reader is asked to fill in the blanks. This pattern recognition activity suggests how times and of are closely related. 
Vision in Elementary Mathematics deals mainly with the beginnings of algebra, but it nonetheless recognizes that understanding arithmetic is important for students to realize that "the pattern of the algebra is exactly like the pattern of the arithmetic" (p. 332). Sawyer adds that if a student "is not clear about algebra, it will only increase his mystification if we try to teach him arithmetic through algebra" (p. 333). For example, the student must first understand how to work with fractions in arithmetic in order to appreciate their connection with fractions in algebra.
As mathematics teachers, there are mathematical concepts that we have all wrung our hands over when determining how we can most meaningfully teach them to our students. It is tempting to merely look up how these concepts were handled in Vision in Elementary Mathematics. However, as demonstrated in the book, students might not have the prerequisite understanding in order to appreciate what is being explained. This has implications for not only the teacher seeking teaching ideas, but also for the approach to teaching mathematics by K12 schools. Sawyer's work subtly implies the need for more consistency and communication by mathematics teachers in how they teach mathematics — which is especially true with the need to build upon students' understanding.
Also, students beginning to study algebra need to understand the practical benefits of the subject. As recommended by Sawyer, this can be accomplished by having students begin by solving problems that they can solve, and then later progress to problems that they cannot solve due to the inadequacy of their mathematical background. Students will then have the knowledge and motivation for a more systematic study of mathematics.
I am aware that this book is almost 40 years old. Nonetheless, I am impressed with how Sawyer has skillfully used problems that build upon what students know and what he has already developed while incorporating multiple representations and opportunities for students to make conjectures. As a result, it is recommended that the chapters be read in order initially to appreciate Sawyer's fine work. I enjoyed reading Vision in Elementary Mathematics and I found that a lot of the issues then are still relevant today. I am sure that all readers will find that there is at least one useful idea in this book that would prove beneficial for their teaching and/or learning mathematics.
Publication Data: Vision in Elementary Mathematics (second edition), by W. W. Sawyer. Dover, 2003 (original edition 1964). Softcover, 346pp, $14.95. ISBN: 048642555X.
Rick Seaman (Rick.Seaman@uregina.ca) is Associate Professor of Mathematics Education at the University of Regina in Regina, SK, Canada.
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Read This! is the MAA Online book review column. Contributions are welcome; contact the editor if you'd like to be one of our reviewers. Books for review should be sent to the editor: Fernando Gouvêa, Dept. of Mathematics, Colby College, Waterville, ME 04901. Publishers, please check our reviews information page. 