Musings: If Not Methods, Then What?

Last week, I quoted Pam Harris calling out a foundational myth of math education, the idea that we need to teach kids the methods that work on even the most difficult math problems.

“We have a misconception in math education that we think we need to teach methods so that kids can answer the craziest kind of a particular problem.

“We would be far better served to teach kids to think about the most common kinds of questions WELL, and let technology handle the crankiest.”

—Pam Harris

Since many of us grew up in schools that taught these methods, they may feel like the only sensible approach to math. Without the standard procedures, how will our kids learn to do math?

If we don’t teach subtraction with borrowing/renaming, how can students figure out calculations like 431 − 86? If we don’t teach fraction rules, how will they handle problems like 1 ¹/₂ ÷ ³/₈?

Levels of Mathematical Reasoning

For many students, multi-digit subtraction opens a gateway to the world of math-as-rote-nonsense, where the only thing that matters is following someone else’s rules. The rules don’t have to make sense as long as they produce correct answers.

To adults, the idea of borrowing/renaming numbers makes sense. It’s like shopping: If we don’t have enough dollar bills to pay for a purchase, then we may need to break a twenty (or whatever larger bill is in our wallet).

That’s additive thinking, where we imagine numbers as chunks that can be put together or taken apart in different ways.

Children begin their experience with numbers by counting. As they grow, they develop more sophisticated concepts, such as additive thinking, multiplicative thinking, reasoning about proportional relationships, and finally, working with algebraic functions.

Additive reasoners can think of $13.27 as:

• 13 dollars and 27 cents
• (5 + 5 + 3) dollars and (25 + 2) cents
• a bit less than $20
• or many other combinations

But math standards introduce the subtraction algorithm at an age when many children are still in the counting stage. They haven’t developed an additive-thinking mindset, so the procedure doesn’t make sense.

Even though we try to explain how our place value number system works — and even if they can parrot back what we say — children who aren’t ready to think this way only “hear” the garbled blah-blah-blah of an adult in a Charlie Brown cartoon.

What’s wrong with the Standard Algorithms?

The standard algorithms, the traditional pen-and-paper methods for arithmetic calculations, were amazing inventions for their time. They allowed mathematicians, scientists, merchants, and bankers to forget about the difficulty of working with numbers and focus their attention on other things.

These algorithms work by using simple math facts over and over. We don’t have to think about what the numbers in our calculation really mean, just pay attention to one place-value digit at a time.

The power of these methods inspired mathematical philosopher Alfred North Whitehead to write:

“It is a profoundly erroneous truism, repeated by all copy books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing.

“The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.”

— Alfred North Whitehead,
An Introduction to Mathematics

Children Learn What They Think About

But this same power that made the algorithms valuable to history’s Scientific and Industrial Revolutions, the ability to produce correct answers with minimal thought, makes them terrible for teaching math.

Human minds are not computers. We can’t give our children a program to store and expect them to recall it on demand. Human minds only remember what they think about. And the deeper we think, the stronger we learn.

Which means, for educational purposes, the standard algorithms are the enemy, because they were designed to eliminate the need for thought.

If we want children to understand math, to build true fluency, then we need to get them thinking and making sense of the subject. We need to start where they are and help them reason about the numbers.

And this process of working through the idea will lead to growth, building those thinking mindsets that serve as a foundation for future learning.

Talk Math with Your Kids

There may be only one right answer to a math problem, but there’s never only one right way to get that answer.

What matters in math is the journey. How do your children make sense of the problem and reason their way to that answer?

In my next few posts, I’ll look in more detail at some of the common stumbling blocks on the road to mathematical mastery, the places we adults are most tempted to offer our children a crutch.

We’ll look at how you can use these tough spots to develop true understanding by pulling out and building on your children’s own ideas.

With this approach, you may only get through a few problems in each lesson. But by going into depth and making sense of the math, your students will learn much more than they’d ever get from rotely following a memorized procedure.

As always, real math is not about the answers but the thinking.