Teaching

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; Understanding Math, Part 5: Multiplying Fractions; and Understanding Math, Part 6: Algebraic Multiplication.

Earlier in this blog post series, I gave you three middle-school math rules. But by exploring the concept of rectangular area as a model of multiplication, we discovered that in a way they were all the same.

The rules are not arbitrary, handed down from a mathematical Mount Olympus. They are three expressions of a single basic question: what does it mean to measure area?

There was only one rule, one foundational pattern that tied all these topics together in a mathematical web.

What Is Your Worldview?

Many children want to learn math instrumentally, as a tool for getting answers. They prefer the simplicity of memorizing rules to the more difficult task of making sense of new ideas. Being young, they are by nature short-term thinkers. They beg, “Just tell me what to do.”

But if we want our children to truly understand mathematics, we need to resist such shortcuts. We must take time to explore mathematics as a world of ideas that connect and relate to each other in many ways. And we need to show children how to reason about these interconnected concepts, so they can use them to think their way through an ever-expanding variety of problems.

Our kids can only see the short term. If we adults hope to help them learn math, our primary challenge is to guard against viewing the mastery of facts and procedures as an end in itself. We must never fall into thinking that the point of studying something is just to get the right answers.

We understand this in other school subjects. Nobody imagines that the point of reading is to answer comprehension questions. We know that there is more to learning history than winning a game of Trivial Pursuit. But when it comes to math, too many parents (and far too many politicians) act as though the goal of our children’s education is to produce high scores on a standardized test.

What If I Don’t Understand Math?

If you grew up (as I did) thinking of math as a tool, the instrumental approach may feel natural to you. The idea of math as a cohesive system may feel intimidating. How can we parents help our children learn math, if we never understood it this way ourselves?

Don’t panic. Changing our worldview is never easy, yet even parents who suffer from math anxiety can learn to enjoy math with their children. All it takes is a bit of self-discipline and the willingness to try.

You don’t have to know all the answers. In fact, many people have found the same thing that Christopher Danielson described in his blog post “Let the children play” — the more we adults tell about a topic, the less our children learn. With the best of intentions we provide information, but we unwittingly kill their curiosity.

• If you’re afraid of math, be careful to never let a discouraging word pass your lips. Try calling upon your acting skills to pretend that math is the most exciting topic in the world.
• Encourage your children to notice the math all around them.
• Search out opportunities to discuss numbers, shapes, symmetry, and patterns with your kids.
• Investigate, experiment, estimate, explore, measure — and talk about it all.

The Science of Patterns

Patterns are so important that American mathematician Lynn Arthur Steen defined mathematics as the science of patterns.

As biology is the science of life and physics the science of energy and matter, so mathematics is the science of patterns. We live in an environment steeped in patterns — patterns of numbers and space, of science and art, of computation and imagination. Patterns permeate the learning of mathematics, beginning when children learn the rhythm of counting and continuing through times tables all the way to fractals and binomial coefficients.

— Lynn Arthur Steen, 1998
Reflections on Mathematical Patterns, Relationships, and Functions

If you are intimidated by numbers, you can look for patterns of shape and color. Pay attention to how they grow, and talk about what your children notice. For example, some patterns repeat exactly, while other patterns change as they go (small, smaller, smallest, or loud, louder, loudest).

Nature often forms fractal-like patterns: the puffy round-upon-roundness of cumulus clouds or broccoli, or the branch-upon-branchiness of a shrub or river delta. Children can learn to recognize these, not as a homework exercise but because they are interesting.

Math the Mathematician’s Way

Here is the secret solution to the crisis of math education: we adults need to learn how to think like mathematicians.

For more on what it means to think about math the mathematician’s way, check out my Homeschooling with Math Anxiety blog post series:

• Mathematicians avoid busywork as if it were an infectious disease.
• Mathematicians always ask questions.
• And most of all, mathematicians love to play.

As we cultivate these characteristics, we will help our children to recognize and learn true mathematics.

CREDITS: “Frabjous 01” photo (top, text added) by Windell Oskay and “Back to School” photo (middle) by Phil Roeder via Flicker (CC BY 2.0). “I Can Solve Problems” poster by Nicole Ricca via Teachers Pay Teachers. This is the final 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.

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</a (CC BY 2.0, text added)>. 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.

Click here to read Part 1: Understanding Math: A Cultural Problem.

Educational psychologist Richard Skemp popularized the terms instrumental understanding and relational understanding to describe these two ways of looking at mathematics. It is almost as if there were two unrelated subjects, both called “math” but as different from each other as American football is from the game the rest of the world calls football.

Which of the following sounds the most like your experience of school math? And which type of math are your children learning?

Instrumental Understanding: Math as a Tool

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. To be fluent in math means we can produce correct answers with minimal effort.

Primary goal: to get the right answer. In math, answers are either right or wrong, and wrong answers are useless.
Key question: “What?” What do we know? What can we do? What is the answer?
Values: speed and accuracy.
Method: memorization. Memorize math facts. Memorize definitions and rules. Memorize procedures and when to use them. Use manipulatives and mnemonics to aid memorization.
Benefit: testability.

Instrumental instruction focuses on the standard algorithms (the pencil-and-paper steps for doing a calculation) or other step-by-step procedures. This produces quick results because students can follow the teacher’s directions and crank out a page of correct answers. Students like completing their assignments with minimal struggle, parents are pleased by their children’s high grades, and the teacher is happy to make steady progress through the curriculum.

Unfortunately, the focus on rules can lead children to conclude that math is arbitrary and authoritarian. Also, rote knowledge tends to be fragile, and the steps are easy to confuse or forget. Thus those who see math instrumentally must include continual review of old topics and provide frequent, repetitive practice.

Relational Understanding: Math as a Connected System

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. Giving a correct answer without justification (explaining how we know it is right) is mere accounting, not mathematics. To be fluent in math means we can think of more than one way to solve a problem.

Primary goal: to see the building blocks of each topic and how that topic relates to other concepts.
Key questions: “How?” and “Why?” How can we figure that out? Why do we think this is true?
Values: logic and justification.
Method: conversation. Talk about the links between ideas, definitions, and rules. Explain why you used a certain procedure, and explore alternative approaches. Use manipulatives to investigate the logic behind a technique.
Benefit: flexibility.

Relational instruction focuses on children’s thinking and expands on their ideas. This builds the students’ ability to reason logically and to approach new problems with confidence. Mistakes are not a mark of failure, but a sign that points out something we haven’t yet mastered, a chance to reexamine the mathematical web. Students look forward to the “Aha!” feeling when they figure out a new concept. Such an attitude establishes a secure foundation for future learning.

Unfortunately, this approach takes time and requires extensive personal interaction: discussing problems, comparing thoughts, searching for alternate solutions, and hashing out ideas. Those who see math relationally must plan on covering fewer new topics each year, so they can spend the necessary time to draw out and explore these connections. Relational understanding is also much more difficult to assess with a standardized test.

What constitutes mathematics is not the subject matter, but a particular kind of knowledge about it.

The subject matter of relational and instrumental mathematics may be the same: cars travelling at uniform speeds between two towns, towers whose heights are to be found, bodies falling freely under gravity, etc. etc.

But the two kinds of knowledge are so different that I think that there is a strong case for regarding them as different kinds of mathematics.

—Richard R. Skemp

For Further Reading

• The difference between instrumental and relational understanding
• Richard Skemp’s Relational Understanding and Instrumental Understanding
• Instrumental vs. Relational: Student Response

Click to read Part 3: Is There Really a Difference?

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CREDITS: “Humphreys High School Football” photo (top) by USAG- Humphreys via Flicker, and “Performing in middle school math class” (middle) by woodleywonderworks via Flicker all (CC BY 2.0). “I Can Explain My Thinking” poster by Nicole Ricca via Teachers Pay Teachers. “Snow globe” photo by Robert Couse-Baker via Flickr (CC BY 2.0, text added).

This is the second 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.