The Role of Reasoning-and-Proving
We hear a lot of debate about ‘fake news’ in our politics these days – what it is, what it is not, and how we can get honest information and guard ourselves against being deceived. What does fake news have to do with math class, though? Everything!
Particularly in middle and high school, we as mathematics teachers need to equip our students with the ability to determine what is mathematically true, what needs to be justified, and what is not mathematically true. This work of reasoning-and-proving is the topic of our recent volume, We Reason & We Prove for ALL mathematics: Building Students’ Critical Thinking, Grades 6-12.
As teachers, we can’t just skip or lightly treat the work on reasoning-and-proving. Reasoning-and-proving is a practice that’s fundamental to establishing mathematical truth and helping students ensure that the mathematics they are doing is valid and generalizable. Reasoning-and-proving involves more than proof – it includes identifying patterns, making conjectures, and making both proof and non-proof arguments. Often, the biggest hurdle in teaching reasoning-and-proving in our classrooms is to convince students that there’s a need to engage in the practice. Why does reasoning-and-proving matter in mathematics? Why can’t we just use the formula, tool, theorem, or rule that you as the teacher told us as students?
Reasoning-and-proving matters a great deal, and it should be more than telling students to take our word for it that what we’re saying is true. This process is what establishes truth in mathematics. As teachers, our role is to facilitate students’ learning, not to be the arbiter of what is and isn’t true in mathematics. As such, we need to support students in developing their own ability to determine what is mathematically true, as well as to make and justify claims of their own. In addition, students often enter secondary school with established mathematical identities and senses of mathematical authority. For many students, they don’t enjoy math and look to the teacher to evaluate what is correct and what is not in mathematics. If we want students to take ownership of the discipline and to see themselves as capable of knowing and doing mathematics, we must shift the narrative about who can do mathematics and who determines mathematical truth. Engaging your class in reasoning-and-proving will help your students develop positive and productive mathematical identities and to see themselves as mathematical authorities.
I close this post with a few quick dos and don’ts to support your students engaging in reasoning-and-proving throughout the middle and high school math curriculum:
…have students consider examples as a gateway to generalizing and justifying
While it’s important to note that examples don’t serve as proof or a generalization for a claim, tinkering with examples can help students identify patterns and see the structure in repeated reasoning that can lead to general justifications.
…show how thinking and reasoning evolves through the process of reasoning-and-proving
We should model for students how mathematical thinking happens, from making claims to determining and argument to justifying them. Make sure you show students that authentic mathematical thinking can be messy and take twists and turns before arriving at a destination.
…press past the what to the why
When students see a pattern in a task that we give them, like a general formula for a visual pattern, it’s natural for them to jump right to writing a rule. Press students to describe why the components of their rule model the situation, and why their generalization of the claim means that it will work every time.
…ask students to use theorems and rules extensively before they work to justify them
When we give students rules to use, but don’t engage them in justifying them to understand why they work (and under what conditions), we undermine our efforts to get students to see why reasoning-and-proving matters in mathematics. Work early with your students to get them in the habit of not using rules until they’ve justified them. They can handle it!
…overemphasize efficiency or rely solely on one form or proof
Efficiency matters in mathematics, but save that discussion for after providing a proof argument for a new claim. If students focus only on efficiency, they won’t explore alternative paths to justify!
…value the product over the process
Often times, particularly in high school, we focus students on the proof as the goal. It’s the journey, not the destination, that helps students develop the reasoning skills. Find ways to value the process of reasoning-and-proving, including in how we assess students.
Take a sneak peek of We Reason & We Prove for ALL Mathematics to learn more about how to support students in engaging in reasoning-and-proving in all secondary mathematics topics.