Originally posted on The Learning Exchange by Marian Small | October 5, 2015

I sometimes reflect on what it is about math that makes me love it. I think that what attracts me is the opportunity to think and reason. But for lots of folks, math is not really about the thinking as much as it about getting answers to problems (or exercises).

I’ve been interacting with teachers around the world about the difference between “doing” math and “thinking” math. I believe that a change of teacher mindset in terms of viewing the study of mathematics as an opportunity to think, not just perform, could be the key to changing teacher practice as well as student attitudes and performance. And I think this is equally valid in the early primary years as it is in secondary school.

## So what does all that mean?

Consider these alternative questions.

In each case, there is what I call a Type I (doing) question and a Type II (thinking) question. The Type I questions are legitimate mathematical questions that we want students to have answers to, but I propose that it’s not enough to only, or even mostly, ask Type I questions. I believe a large proportion of Type II questions is essential.

Notice that embedded in each Type II question is a requirement that students use mathematical knowledge (knowledge that could be demonstrated by responding to Type I questions), but they use that knowledge in interesting ways- to wrestle with mathematical ideas, not just show what they recall or can do.

**Example 1:**

Type 1: How many dots are there?

Type II: How would you arrange eight dots to make it easy to tell it’s eight?

Example 2:

Type I: What is the name of this shape?

Type II: What shapes are a lot like rectangles, but not quite rectangles?

Example 3:

Type I: What is the value of the represented number?

Type II:

- A number is represented with some hundred flats, twice as many ten rods and three times as many ones as rods.
- What could the ones digit be? Why?

Example 4:

Type I: Read this number: 4023

Type II: What numbers take exactly four words to say?

Example 5:

Type I: Measure this angle.

Type II: An angle’s measure is really easy to estimate. What might it look like?

Example 6:

Type I: What is the perimeter of this rectangle?

Type II: A rectangle has a perimeter seven times its width. What could its dimensions be?

Example 7:

Type I: What is the 20th term in 3, 5, 7, 9, …?

Type II: The 20th term in a growing pattern is 41. What are possibilities for the pattern?

Example 8:

Type I: What is 10% of 300?

Type II: Is 10% a lot or not?

Example 9:

Type I: What is the solution to

3/4 x – 2 = 5/8 x + 9?

Type II: Do equations with fractions in them usually have whole number solutions or fraction solutions?

Consider how your classroom would change if your focus was engaging students in Type II questions instead of Type I questions.

I envision much more rich mathematical discussion, much more engagement, much more out-of-the-box thinking and a much greater sense of accomplishment.

Although there are many ways to engage students, for example, by appealing to contexts of interest, one profound way to engage students, as proposed here, is to let them just play with mathematical thinking through Type II questions. Not only do we engage them, but we really share the joy of considering and discovering ideas.

**Marian Small **is an Author and International Professional Development Consultant for Improved Mathematics Education in Ontario, Canada.