Mathematical Thinking 101

Disciplinary thinking asserts that students need to spend more time behaving like mathematicians, artists, scientists, poets, etc. than they do simply trying to memorize the results of the work of experts in the field.

In Stage 2 of our framework we advocate against “telling” and “showing” students as primary modes of instruction. Instead, we hope teachers will ensure students are actively processing what they are learning by making predictions, categorizing, comparing similarities and differences, explaining in their own words, etc.

If you have not yet experimented with this shift, here is what will likely happen if you jump right into Stage 4, Disciplinary Thinking:

You will “show and tell” kids how to think like a mathematician. They will write down what you told them and practice exactly what you showed them.  You will pat yourself on the back but the students will not be any closer to thinking like a mathematician. They have just demonstrated their ability to follow directions and mimic your actions.

Here are steps to help us get started.

Step 1: Think deeply about what math is and isn’t.

Choose the best description of mathematics:

a. Manipulating symbols by carefully following prescribed and memorized rules and procedures

b. The art of pattern making with careful explanation of ideas

Choice “a” is what the vast majority of math classrooms look like, especially in the United States. Choice “b” is probably what a real live mathematician might say. What would math classrooms look like if choice “b” were at the forefront of teachers’ minds every single day?

Step 2: Plan for students to struggle through figuring out the mathematics for themselves.

There are scores of resources to help you do this. Here are a few:

Teaching Student-Centered Mathematics (practical tools and explanations that are useful for all grade levels)

Problem-Based Learning

Illuminations from NCTM

Dan Myer’s blog

Step 3: Require a justification and explanation of every single answer. The way most math classes are structured communicate to students that the goal of math is to “find the correct answer.”  Let’s say the answer to an equation is 345. The assumption we make is that all students who reached 345 understand the mathematical concepts and relationships to reach it. The trouble is there is no way to know if they understood or not without hearing or reading how they got it. And if we don’t ask for it we reinforce the notion that math is a collection of rules to be memorized and applied mindlessly.

Try out these three steps and let us know how it goes!

Tomorrow will feature Scientific Thinking 101 and compare thinking between the humanities and math/science.

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Categories: Stage 4: Disciplinary Thinking

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