Click to read the earlier posts in this series: Understanding Math, Part 1: A Cultural Problem; Understanding Math, Part 2: What Is Your Worldview?; Understanding Math, Part 3: Is There Really a Difference?; Understanding Math, Part 4: Area of a Rectangle; and Understanding Math, Part 5: Multiplying Fractions.
We’ve examined how our vision of mathematical success shapes our children’s learning. Do we think math is primarily a tool for solving problems? Or do we see math as a web of interrelated concepts?
Instrumental understanding views math as a tool. Relational understanding views math as an interconnected system of ideas. Our worldview influences the way we present math topics to our kids. And our children’s worldview determines what they remember.
In the past two posts, we looked at different ways to understand and teach rectangular area and fraction multiplication. But how about algebra? Many children (and adults) believe “math with letters” is a jumble of abstract nonsense, with too many formulas and rules that have to be memorized if you want to pass a test.
Which of the following sounds the most like your experience of school math? And which type of math are your children learning?
Instrumental Understanding: FOIL
Every mathematical procedure we learn is an instrument or tool for solving a certain kind of problem. To understand math means to know which tool we are supposed to use for each type of problem and how to use that tool — how to categorize the problem, remember the formula, plug in the numbers, and do the calculation.
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.
Relational Understanding: The Area Model
Each mathematical concept is part of a web of interrelated ideas. To understand mathematics means to see at least some of this web and to use the connections we see to make sense of new ideas.
The concept of rectangular area has helped us understand fractions. Let’s extend it even farther. In the connected system of mathematics, almost any type of multiplication can be imagined as a rectangular area. We don’t even have to know the size of our rectangle. It could be anything, such as subdividing a plot of land or designing a section of crisscrossed colors on plaid fabric.
We can imagine a rectangle with each side made up of two unknown lengths. One side has some length a attached to another length b. The other side is x units long, with an extra amount y stuck to its end.
We don’t know which side is the “length” and which is the “width” because we don’t know which numbers the letters represent. But multiplication works in any order, so it doesn’t matter which side is longer. Using the rectangle model of multiplication, we can see that this whole shape represents the area 
But since the sides are measured in pieces, we can also imagine cutting up the big rectangle. The large, original rectangle covers the same amount of area as the four smaller rectangular pieces added together, and thus we can show that 
With the FOIL formula mentioned earlier, our students may get a correct answer quickly, but it’s a dead end. FOIL doesn’t connect to any other math concepts, not even other forms of algebraic multiplication. But the rectangular area model will help our kids multiply more complicated algebraic expressions such as 
Not only that, but the rectangle model gives students a tool for making sense of later topics such as polynomial division. And it is fundamental to understanding integral calculus.
To be continued. Next up, Understanding Math Part 7: The Conclusion…
CREDITS: “Math Workshop Portland” photo (top) by US Department of Education via Flicker (CC BY 2.0). 
This is the sixth 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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I love this area model. Thanks for sharing a good visual representation of it.
Yes, the area model is flexible and powerful. A great tool for students and teachers alike!