Preparing Boys for Life.
The Art of Building Proofs
The Art of Building Proofs
The Haverford School

Geometry teacher Sam Walters recently finished a textbook (co-written with former faculty member Zack Murtha), based upon a research project he completed with the University of Delaware to examine how students best learn the art of building proofs. Here, we interview Walters about the experience and what he's learned.

1. What was your inspiration for the textbook?

My colleague and I embarked on a research project with the University of Delaware in 2010 to examine how proof is taught and how to effectively build students' understanding. The Haverford School was one of two schools selected to participate. In the initial phase, which lasted two years, we developed a number of scaffolding tools to help students understand the pieces of proof and how proof fits together like a puzzle. We presented our findings at the annual national conference of the National Council of Teachers in Mathematics, demonstrating how teachers can take the skills needed to write a proof and break them down into smaller digestible chunks.

From there, we wrote a cohesive curriculum based upon the research. The textbook focused on building the knowledge in a way to effectively reason with it. The material helps students to make and interpret drawings, draw conclusions, and substantiate those conclusions.

2. How has the textbook been received by students?

Well! Unlike traditional textbooks, we have the flexibility to make continual edits and refinements. Our boys like the workbook format, which reinforces the material in the form of in-text questions and provides space for problem solving. Using a single text, rather than jumping between textbooks and handouts and worksheets, makes the learning experience more fluid and also more focused.

After each lesson, I reflected on my experience teaching from the textbook and gained feedback from the students to better understand what was missing and what needed to be improved. There has been a significant increase in the number of students who are capable of writing a well-justified proof.

3. How has the textbook evolved over time?

After each lesson, I reflected on my experience teaching from the textbook and gained feedback from the students to better understand what was missing and what needed to be improved. I incorporated additional problems to reinforce the materials, as well as to assist with quiz and test preparation. I also added honors supplements into the text, allowing all students to see the more advanced topics and attempt the accompanying problems. Finally, I have aligned the textbook with the Haverford math department's horizontal and vertical goals -- continuing to use and reinforce the algebra skills learned in previous years, while emphasizing the logic and reasoning component of geometry. To ensure continuity in student learning from one grade to the next, it was important to emphasize the connection between geometry and algebra.

4. What differences have you seen in the classroom as a result of using this textbook?

There has been a significant increase in the number of students who are capable of writing a well-justified proof. They are illustrating a logical progression using theorems, definitions, and postulates to develop and substantiate a proof. The boys can apply this same use of solid logic and reasoning to understand any problem more fully and to make better decisions. The overall way they approach their thinking has changed, and that's the real advantage.

Sam Walters teaches Geometry and Honors Geometry in Haverford's Upper School and also coaches squash and ultimate frisbee. Before joining Haverford in 2009, Walters taught at Avon Old Farms School and The Hill School. He began his career as a government services consultant, working with the Navy, Air Force, and IRS. Walters holds a B.S. in manufacturing engineering from Northwestern University.

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