The carnival is short this month, but full of treats. Enjoy!

“So here we are on December 23, 2015, the 93rd edition of Math Teachers At Play! As per tradition, what’s so fascinating about the number 93?

“First, it’s a prime number! No. Wait, that’s clearly false. So 93 is not a prime number. But that’s not very fascinating. Aha! But 93 is a semi-prime! since 93=31×3. Even more interesting is that 94 and 95 are semi-primes. So a question is, is there another triplet of positive integers that are also semi-primes? It’s a good question to ask your students! …”

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

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.

Instrumental Understanding: Math as a Tool

Fractions confuse almost everybody. In fact, fractions probably cause more math phobia among children (and adults) than any other topic before algebra.

Children begin learning fractions by coloring or cutting up paper shapes, and their intuition is shaped by experiences with food like sandwiches or pizza. But before long, the abstraction of written calculations looms up to swallow intuitive understanding.

Upper elementary and middle school classrooms devote many hours to working with fractions, and still students flounder. In desperation, parents and teachers resort to nonsensical mnemonic rhymes that just might stick in a child’s mind long enough to pass the test.

Relational Understanding: Math as a Connected System

Now let’s zoom in on our rectangle. Imagine magnifying our virtual grid to show a close-up of a single square unit, such as the pan of brownies on our table. And we can imagine subdividing this square into smaller, fractional pieces. In this way, we can see that five-eighths of a square unit looks something like a pan of brownies cut into strips, with a few strips missing:

But what if we don’t even have that whole five-eighths of the pan? What if the kids came through the kitchen and snatched a few pieces, and now all we have is three-fourths of the five-eighths?

How much of the original pan of brownies do we have now? There are three rows with five pieces in each row, for a total of 3 × 5 = 15 pieces left — which is the numerator of our answer. And with pieces that size, it would take four rows with eight in each row (4 × 8 = 32) to fill the whole pan — which is our denominator, the number of pieces in the whole batch of brownies. So three-fourths of five-eighths is a small rectangle of single-serving pieces.

Notice that there was nothing special about the fractions 3/4 and 5/8, except that the numbers were small enough for easy illustration. We could imagine a similar pan-of-brownies approach to any fraction multiplication problem, though the final pieces might turn out to be crumbs.

Of course, children will not draw brownie-pan pictures for every fraction multiplication problem the rest of their lives. But they need to spend plenty of time thinking about what it means to take a fraction of a fraction and how that meaning controls the numbers in their calculation. They need to ask questions and to put things in their own words and wrestle with the concept until it makes sense to them. Only then will their understanding be strong enough to support future learning.

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.

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.

Once again, Nrich.Maths.org and Plus.Maths.org are offering a selection of activities to encourage your students’ mathematical creativity, one for each day in the run-up to Christmas. Click one of the images below to visit the appropriate Advent Math Calendar page.

“Wild Maths” puzzles and articles for teens and up. When you get to the +Plus Magazine website, you can tell which links are live by the drop shadow under the picture. One link becomes live each day — so come back tomorrow and discover something new!

My November “Let’s Play Math” newsletter went out early Monday morning to everyone who signed up for Tabletop Academy Press math updates. This month’s issue focused on figurate numbers, and it also included the latest updates on the Let’s Play Math paperback edition (coming in early 2016).

If you didn’t see it, check your Updates or Promotions tab (in Gmail) or your Spam folder. And to make sure you get all the future newsletters, add “Denise at Tabletop Academy Press” [Tabletop Academy Press @ gmail.com, without spaces] to your contacts or address book.

And if you missed this month’s edition, no worries—there will be more playful math snacks in the next issue. Click the link below to sign up today, and we’ll send you our free math and writing booklets, too!