*Click to read the earlier posts in this series: Understanding Math, Part 1: A Cultural Problem; Understanding Math, Part 2: What Is Your Worldview?; and Understanding Math, Part 3: Is There Really a Difference?*

In this post, we consider the first of three math rules that most of us learned in middle school.

- Area of a rectangle = length × width

### Instrumental Understanding: Math as a Tool

The *instrumental approach* to explaining such rules is for the adult to work through a few sample problems and then give the students several more for practice.

In a traditional lecture-and-workbook style curriculum, students apply the formula to drawings on paper. Under a more progressive reform-style program, the students may try to invent their own methods before the teacher provides the standard rule, or they may measure and calculate real-world areas such as the surface of their desks or the floor of their room.

Either way, the ultimate goal is to define terms and master the formula as a tool to calculate answers.

Richard Skemp describes a typical lesson:

Suppose that a teacher reminds a class that the area of a rectangle is given by A=L×B. A pupil who has been away says he does not understand, so the teacher gives him an explanation along these lines. “The formula tells you that to get the area of a rectangle, you multiply the length by the breadth.”

“Oh, I see,” says the child, and gets on with the exercise.

If we were now to say to him (in effect) “You may think you understand, but you don’t really,” he would not agree. “Of course I do. Look; I’ve got all these answers right.”

Nor would he be pleased at our devaluing of his achievement. And with his meaning of the word, he does understand.

As the lesson moves along, students will learn additional rules.

For instance, if a rectangle’s length is given in meters and the width in centimeters, we must convert them both to the same units before we calculate the area. Also, our answer will not have the same units as our original lengths, but that unit with a little, floating “2” after it, which we call “squared.”

Each lesson may be followed by a section on word problems, so the students can apply their newly learned rules to real-life situations.

### Relational Understanding: Math as a Connected System

In contrast, a *relational approach* to area must begin long before the lesson on rectangles.

Again, this can happen in a traditional, teacher-focused classroom or in a progressive, student-oriented, hands-on environment. Either way, the emphasis is on uncovering and investigating the conceptual connections that lie under the surface and support the rules.

We start by exploring the concept of measurement: our children measure a path along the floor, sidewalk, or anywhere we could imagine moving in a straight line. We learn to add and subtract such distances. Even if our path turns a corner or if we first walk forward and then double back, it’s easy to figure out how far we have gone.

But something strange happens when we consider distances in two different directions at the same time — measuring the length and width of the dining table automatically creates an invisible grid.

In measuring the length of a rectangular table, we do not find just one point at any given distance. There is a whole line of points that are one foot, two feet, or three feet from the left side of the table.

And measuring the width shows us all the points that are one, two, or three feet from the near edge. Now our rectangular table is covered by virtual graph paper with squares the size of our measuring unit.

The length of the rectangle tells us how many squares we have in each row, and the width tells us how many rows there are. As we imagine this invisible grid, we can see why multiplying those two numbers will tell us how many squares there are in all.

That is what the word *area *means: the area of a tabletop is the number of virtual-graph-paper squares it takes to cover it up, which is why our answer will be measured in square units.

### Making Sense of Mixed Units

What if we measured the length in meters and the width in centimeters?

With a relational understanding of area, even a strange combination of units can make sense. Our invisible grid would no longer consist of squares but of long, thin, rectangular centimeter-meters. But we could still find the area of the tabletop by counting how many of these units it takes to cover it.

Square units aren’t magic — they’re just easier, that’s all.

*Click to continue reading Understanding Math, Part 5: Multiplying Fractions…*

**CREDITS:** “Framed” photo (top) by d_pham via Flicker (CC BY 2.0). This is the fourth 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.

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Is maths really taught in the manner of that ‘typical lesson’?! I was frustrated recently at the number of books I came across TELLING us the formula for the area of a triangle, and really had to scour the web for a way to show my kids how to figure it out for themselves. We then embarked on a several-week-long geometry adventure playing with rectangles, parallelograms, trapeziums etc (even circles) with right-angled and non-right angled triangles along the way. I know I was taught in that ‘typical’ way at school but it just seems unbelievable now!

Well, as Skemp told it, the “typical lesson” was a review. So the teacher had probably given demonstrations and explanations of some sort to the class in previous sessions.

But whatever had been the lead-in to the formula didn’t really matter. The key thing, for both the teacher and the student, was to know what the letters stood for so that one could follow the formula and get the right answer.

What you describe sounds like more fun, and also more likely to lead to long-term retention!

Unfortunately, too many teachers (1) don’t understand math themselves or (2) feel too rushed to “cover” a mile-wide list of topics-that-will-be-on-the-test so they don’t have time to examine things in depth.

I am loving this series of posts! You are really making that Skemp article accessible for more people.

Thank you, Sue.

This topic was the most important thing missing from my book and the main reason it has taken so long to finish the paperback version. I needed to figure out how to express Skemp’s categories in words and examples that I hope will communicate to people who don’t already think of math this way.

Of course, being a tinkerer, I’ve also been adding more resources and quotes throughout. Just yesterday, I found this gem from Paul Halmos for my high school chapter.