Understanding Math: What Is Your Worldview?

Humphreys High School FootballClick 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

Math-WorldviewEvery 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.

performing in middle school math class

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.

Math Practice 3

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

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


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

LPM-ebook-300This 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.


howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and sign up to hear about new books, revisions, and sales or other promotions.


Understanding Math: A Cultural Problem

Thinking

All parents and teachers have one thing in common: we want our children to understand and be able to use math. Counting, multiplication, fractions, geometry — these topics are older than the pyramids.

So why is mathematical mastery so elusive?

The root problem is that we’re all graduates of the same system. The vast majority of us, including those with the power to shape reform, believe that if we can compute the answer, then we understand the concept; and if we can solve routine problems, then we have developed problem-solving skills.

Burt Furuta

The culture we grew up in, with all of its strengths and faults, shaped our experience and understanding of math, as we in turn shape the experience of our children.

Six Decades of Math Education

math on slateLike any human endeavor, American math education — the system I grew up in — suffers from a series of fads:

  • In the last part of the twentieth century, Reform Math focused on problem solving, discovery learning, and student-centered methods.
  • But Reform Math brought calculators into elementary classrooms and de-emphasized pencil-and-paper arithmetic, setting off a “Math War” with those who argued for a more traditional approach.
  • Now, policymakers in the U.S. are debating the Common Core State Standards initiative. These guidelines attempt to blend the best parts of reform and traditional mathematics, balancing emphasis on conceptual knowledge with development of procedural fluency.

Model Math Problems

The “Standards for Mathematical Practice” encourage us to make sense of math problems and persevere in solving them, to give explanations for our answers, and to listen to the reasoning of others‌—‌all of which are important aspects of mathematical understanding.

But the rigid way in which the Common Core standards have been imposed and the ever-increasing emphasis on standardized tests seem likely to sabotage any hope of peace in the Math Wars.

What Does It Mean to “Understand Math”?

Math-HomeworkThrough all the math education fads, however, one thing remains consistent: even before they reach the schoolhouse door, students are convinced that math is all about memorizing and following arbitrary rules.

Understanding math, according to popular culture‌—‌according to movie actors, TV comedians, politicians pushing “accountability,” and the aunt who quizzes you on your times tables at a family gathering‌—‌means knowing which procedures to apply so you can get the correct answers.

But when mathematicians talk about understanding math, they have something different in mind. To them, mathematics is all about ideas and the relationships between them, and understanding math means seeing the patterns in these relationships: how things are connected, how they work together, and how a single change can send ripples through the system.

Mathematics is the science of patterns. The mathematician seeks patterns in number, in space, in science, in computers, and in imagination. Theories emerge as patterns of patterns, and significance is measured by the degree to which patterns in one area link to patterns in other areas.

Lynn Arthur Steen

Click here to read Understanding Math, Part 2: What Is Your Worldview?


CREDITS: “Thinking” photo (top) by Klearchos Kapoutsis via Flicker (CC BY 2.0). “Math on a Slate” (middle) by Pranav via Flicker (CC BY 2.0). “I Can Model Problems” poster by Nicole Ricca via Teachers Pay Teachers. “Math Homework” photo (bottom) by tracy the astonishing via Flickr (CC BY-SA 2.0).

LPM-ebook-300This is the first 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.


howtosolveproblemsWant to help your kids learn math? Claim your free 24-page problem-solving booklet, and sign up to hear about new books, revisions, and sales or other promotions.


PUFM 1.5 Multiplication, Part 2

Poster by Maria Droujkova of NaturalMath.com. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

Multiplication is taught and explained using three models. Again, it is important for understanding that students see all three models early and often, and learn to use them when solving word problems.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

I hope you are playing the Tell Me a (Math) Story game often, making up word problems for your children and encouraging them to make up some for you. As you play, don’t fall into a rut: Keep the multiplication models from our lesson in mind and use them all. For even greater variety, use the Multiplication Models at NaturalMath.com (or buy the poster) to create your word problems.

Continue reading PUFM 1.5 Multiplication, Part 2

PUFM 1.5 Multiplication, Part 1

Photo by Song_sing via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

My apologies to those of you who dislike conflict. This week’s topic inevitably draws us into a simmering Internet controversy.

Thinking my way through such disputes helps me to grow as a teacher, to re-think on a deeper level concepts that I thought I understood. This is why I loved Liping Ma’s book when I first read it, and it’s why I thoroughly enjoyed Terezina Nunes and Peter Bryant’s book Children Doing Mathematics.

repeatedaddition

Multiplication of whole numbers is defined as repeated addition…

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not… Adding numbers tells you how many things (or parts of things) you have when you combine collections. Multiplication is useful if you want to know the result of scaling some quantity.

— Keith Devlin
It Ain’t No Repeated Addition

Continue reading PUFM 1.5 Multiplication, Part 1

PUFM 1.4 Subtraction

Photo by Martin Thomas via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

When adding, we combine two addends to get a sum. For subtraction we are given the sum and one addend and must find the “missing addend”.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

Notice that subtraction is not defined independently of addition. It must be taught along with addition, as an inverse (or mirror-image) operation. The basic question of subtraction is, “What would I have to add to this number, to get that number?”

Inverse operations are a very fundamental idea in mathematics. The inverse of any math operation is whatever will get you back to where you started. In order to fully understand a math operation, you must understand its inverse.

Continue reading PUFM 1.4 Subtraction

PUFM 1.3 Addition

Photo by Luis Argerich via flickr. In this Homeschooling Math with Profound Understanding (PUFM) Series, we are studying Elementary Mathematics for Teachers and applying its lessons to home education.

The basic idea of addition is that we are combining similar things. Once again, we meet the counting models from lesson 1.1: sets, measurement, and the numberline. As homeschooling parents, we need to keep our eyes open for a chance to use all of these models — to point them out in the “real world” or to weave them into oral story problems — so our children gain a well-rounded understanding of math.

Addition arises in the set model when we combine two sets, and in the measurement model when we combine objects and measure their total length, weight, etc.

One can also model addition as “steps on the number line”. In this number line model the two summands play different roles: the first specifies our starting point and the second specifies how many steps to take.

— Thomas H. Parker & Scott J. Baldridge
Elementary Mathematics for Teachers

Continue reading PUFM 1.3 Addition