Problem Solving Classroom

Tips from a Master Teacher (This is a work in progress.)

Proving Outside the Box

by Mark Illingworth
Proving Outside the Box
Expand Your Definition

I invite you to expand your definition of proofs and bring less formal but equally important thinking into the mix. I know that for some of you this statement is sacrilege. I don’t mean for it to be. I understand the importance of—and even appreciate the beauty of—a well constructed, lock tight proof as much as the next math teacher. It’s just that we lose opportunities to give students the skills they need to thrive if we tell them that proving things is more about specific syntax than it is about sound reasoning. There's room for a wide range of proofs.

Proofs in the Real World

Proving things (in the real world) involves presenting rational explanations and arguments to justify decisions, choices, or viewpoints, and to recommend courses of action. As a former mechanical engineer at Pratt & Whitney, I know that you’d get laughed out of the meeting if you submitted a two-column proof using the Equality Property of Multiplication. I also know that it will not go over well with your colleagues if you support an important engineering choice by telling them that it just felt right to you. It is obviously important that we train our students to be good at creating arguments and explanations based on sound reasoning. We have, however, many more opportunities to do this beyond solely the formal proof.

When students solve they types of rich problems I hope you’re giving them, they should develop the habit (in class and in their small groups) of justifying their steps with postulates, theorems, properties and definitions.

Come back to this one later, David. I'm still writing it.

Article Topics

In a Nutshell


Proofs can take many forms. They always include logical reasoning, but they don't always involve traditional formal syntax.


Real-world proofs require clear communication using arguments and explanations based on evidence and sound reasoning.


You can develop sound reasoning by changing the way you ask questions.

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