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Developing Essential Understanding of Proof and Proving for Teaching Mathematics in Grades 9–12

Amy Ellis, Kristen Bieda, and Eric Knuth
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Woong Lim
, on

Do you enjoy teaching content courses for mathematics education majors? Has your chair just asked you to teach a course for education students on writing proofs? I suspect you might have the same dilemma as mine: finding a course text that is not intimidating and still enables students to reason with definitions, axioms and theorems. I think I have found one that works for my students and recommend that you consider adopting it as your course reading. The book is written by Amy Ellis, Kristen Bieda, and Eric Knuth. Ellis and Knuth teach at University of Wisconsin-Madison, and Bieda teaches at Michigan State University. All are highly respected names in the field of mathematics education. The book is loaded with the language of reformed mathematics education as supported by the NCTM standards and research-based perspectives of practice.

The book engages the reader with fundamental ideas about proof and proving within mathematics in grades 9–12. It presents rigorous mathematical contexts in which mathematical concepts related to conjecturing and generalizing are elaborated; pedagogy (e.g. strategies, vignettes, student thinking, student-teacher dialogues, research findings) is delineated; and most notably, the voice of the narratives situates the reader as the teacher and challenges him or her to make connections from one mathematical idea to another, from pedagogy to mathematics, or vice versa. When I asked my students to evaluate the book in the course evaluation, I was pleasantly surprised when they recognized the benefits of the book: “I enjoyed their discussion about evolving forms of proof and the various roles of proof since I’ve always thought about two-column proofs only and didn’t appreciate the variety,” “I felt like the book talks to me as if I were the teacher, and it helped me learn to talk like teachers,” and “the text is really well written and made me keep reading… it was one of the first math textbooks ever I actually read from cover to cover.”

My class usually has a good mix of those who want to be certified to teach middle grades and secondary mathematics and those who are math majors but have some interest in teaching careers. My struggle has been to offer a math course in which prospective teachers learn to write proofs, use the formal language of mathematics to make mathematical arguments, and have opportunities to experience the fundamental ideas that serve as a bridge to abstract mathematics. It worked quite well with my students to utilize this book as a supplementary reading. First I assigned reading selected portions of the book as homework. Then in class we together did the mathematics drawn from the examples in the book or extended the book’s discussion. I assigned proofs to read and asked them to produce a formal proof and write narratives that were similar to the examples in the book. My students appeared to be doing the homework reading, thus tended to be prepared for these class activities, perhaps more so than with other reading assignments.

I believe that mathematics education majors must learn mathematics and pedagogy together. This book definitely will dispel naïve notions of deductive reasoning while fostering thinking related to probing, testing, and refining conjectures through verbal or written arguments, diagrams, symbolic notations and narrative articulations. What comes along with these for mathematics education majors are the opportunities to think about the challenges of supporting their future students when they get to teach proof and proving. Such opportunities in content courses are priceless!

Woong Lim ( is an Assistant Professor of Mathematics Education at Kennesaw State University, GA. His research interests include interrelations between language and mathematics, content knowledge for teaching, and social justice issues in mathematics education.