*Click to read the earlier posts: Understanding Math, Part 1: A Cultural Problem; Understanding Math, Part 2: What Is Your Worldview?*

From the outside, it’s impossible to tell how a person is thinking. A boy with the instrumental perspective and a girl who reasons relationally may both get the same answers on a test. Yet under the surface, in their thoughts and how they view the world, they could not be more different.

“Mathematical thinking is more than being able to do arithmetic or solve algebra problems,” says Stanford University mathematician and popular author Keith Devlin. “Mathematical thinking is a whole way of looking at things, of stripping them down to their numerical, structural, or logical essentials, and of analyzing the underlying patterns.”

And our own mathematical worldview will influence the way we present math topics to our kids. Consider, for example, the following three rules that most of us learned in middle school.

- Area of a rectangle = length × width.

- To multiply fractions, multiply the tops (
*numerators*) to make the top of your answer, and multiply the bottoms (*denominators*) to make the bottom of your answer.

- When you need to multiply algebra expressions, remember to FOIL: multiply the First terms in each parenthesis, and then the Outer, Inner, and Last pairs, and finally add all those answers together.

While the times symbol or the word multiply is used in each of these situations, the procedures are completely different. How can we help our children understand and remember these rules?

Over the next three posts in this series, we’ll dig deeper into each of these math rules as we examine what it means to develop relational understanding.

Many people misunderstand the distinction between Instrumental and Relational Understanding as having to do with surface-level, visible differences in instructional approach, but it’s not that at all. It has nothing to do with our parenting or teaching style, or whether our kids are learning with a traditional textbook or through hands-on projects. It’s not about using “real world” problems, except to the degree that the world around us feeds our imagination and gives us the ability to think about math concepts.

This dichotomy is all about the vision we have for our children — what we imagine mathematical success to look like. That vision may sit below the level of conscious thought, yet it shapes everything we do with math. And our children’s vision for themselves shapes what they pay attention to, care about, and remember.

*Click to continue reading Understanding Math, Part 4: Area of a Rectangle.*

**CREDITS:** “Math Workshop Portland” photo (top) by US Department of Education via Flicker (CC BY 2.0). This is the third post in my Understanding Math series, adapted from my book *Let’s Play Math: How Families Can Learn Math Together—and Enjoy It,* available at your favorite online book dealer.