FAQ: The Necessity of Math Facts

Ah, math facts — the topic that just won’t stop giving grief to students and anxiety to their parents. So it happened that I got another question, but this one leaned in a more philosophical direction…

“I enjoyed your podcast interview on Cultivating Math Curiosity and Reasoning in Kids. I love the idea that we don’t have to make our children memorize everything in math. We can give them freedom to make mental connections for themselves.

“But on the other hand, we don’t have unlimited time for them to figure things out on their own, do we? What about children who can’t make these connections for themselves?

“For example, what about the math facts? If my kids aren’t picking them up, don’t they just have to memorize them?”

Learning Is Making Connections

A wise teacher once pointed out that children are born persons, with the ability to learn built into their minds from the beginning. Just as they come into the world ready and able to digest physical food, so they also come ready and able to digest mental food.

We don’t teach them how to learn, how to make mental connections, because they’ve been doing it from birth.

It’s not that some kids are able to make mental connections, but others need us to do it for them. Children must make their own connections. No one else can do it for them.

The challenge of education is to determine how best to support and encourage children so they can make strong, deep, and lasting connections in their own minds.

A Necessary Distinction

There are some types of knowledge that we need to tell children and they need to memorize. We call such facts arbitrary or social knowledge. For example, this thing with four legs is a “chair,” while that other thing with four legs is a “dog.”

Math symbols are arbitrary knowledge, and children do have to memorize them. There’s no way by logic to deduce the meaning of “2” or “4” or “+” or “=.”

But math concepts are not arbitrary. “2+2=4” is a necessary fact, no matter how we say the names for those symbols. As soon as children understand what we mean by the words “two” and “four,” they will be able to tell you that two cookies on their plate and two cookies in their hands is four cookies in all. They do not need to memorize the math fact, because it is inherent in the meaning of the words.

All of the math facts are necessary knowledge. The answers follow logically, as soon as we agree on the meaning of the symbols. Doubling seven leads inexorably to fourteen; there is no other possibility.

The Real Question Is…

So the important question for math teachers is, “What’s the best way to help children master necessary knowledge?”

Arbitrary knowledge must be taught and memorized. Should we handle necessary knowledge the same way?

I believe memorization is the worst possible way to teach necessary knowledge, because the more we overload memory, the less we can rely on it. Memory is a weak mental connection, so we should reserve it for situations in which nothing else will work.

But that doesn’t mean we leave our students alone to flounder, expecting them to develop all of mathematics on their own.

Rather, it means we give them plenty of opportunities to think about math concepts and experience math relationships. We help them put their thoughts into words by using tools like the Notice-Wonder-Create cycle. And we share our own thinking, not because we want them to “do it my way” but because learning is a conversation.

How Shall We Teach?

For example, in the podcast interview, we talked about the fraction equation:

2/3 + 3/4 = ?

…and how we might help children who had no idea how to solve it.

[Did you miss the podcast? Find it here.]

Our goal with this problem is to teach a necessary truth: When we add or subtract fractions, we need common (shared) denominators.

We could treat this concept as if it were arbitrary knowledge, like the words for “chair” and “dog.” We could give our students a rule to memorize and practice. A few of our students would intuit the sense of the rule and add it to their working knowledge of math concepts. But most would add it to their mental file cabinet of things to remember by rote.

Or we could take the time to have a real conversation about the concept, like the Notice-Wonder-Create discussion in the podcast.

• Notice means to focus your attention on the situation. Open your eyes, take time to see the details.

• Wonder is to bring your curiosity to the fore. Ask questions. Think about how the situation connects to things you already know or how it might extend beyond them.

• Create means shaping these things you’ve noticed and wondered about into something new, some new understanding, some new explanation of the puzzle. Consolidate the ideas in your mind, making them fully your own.

We have less control over this conversation than over a traditional classroom lesson. But a teacher who understands math herself can draw out from children the idea of getting all their fractions into same-size pieces, which is the key insight behind that common-denominators rule.

As children work through the whole process, it builds much stronger mental connections than a memorized rule, and the next problem we do will reinforce those connections, requiring significantly less talk before reaching a solution.

After several problems of this type, we may even be ready to ask the kids themselves to give us a rule for solving this sort of calculation. A few of our students may still end up forgetting everything else and rote-memorizing the final rule. But most of them will have made sense of the math and truly mastered it.

Where Do We Find the Time?

The process does take time. But it also saves time because all this deep thinking about fractions builds a strong foundation that will make future concepts feel natural, even inevitable. Students who comprehend the purpose behind common denominators will easily make the jump to adding mixed numbers or algebraic fractions.

We spend time now to save it in the future, because making deep connections enables future learning in a way that the crutch of a teacher-provided rule could never do.

Going back to the topic of math facts, we reap the same time-saving benefit by delaying memorization as long as possible, pushing students to think more deeply about number relationships and patterns. The math facts are limited to numbers that fit on our fingers, but the patterns and relationships apply to all numbers and form the foundations of algebra.

We want students to own the math facts. But that will not happen if we teach those facts as if they were arbitrary, like the word “dog.” True ownership comes from plenty of thinking, reasoning, figuring-out, noticing, wondering, and playing with numbers.

For More Information

Annie Fetter’s NCTM Ignite talk “Ever Wonder What They’d Notice?”:

• https://www.youtube.com/watch?v=a-Fth6sOaRA

Dan Meyer’s TEDxNYED talk “Math class needs a makeover”:

• https://www.ted.com/talks/dan_meyer_math_class_needs_a_makeover

At a more academic level, there’s Dave Hewitt’s “Arbitrary and Necessary” series of articles in the journal For the Learning of Mathematics:

• https://www.lboro.ac.uk/media/media/services/lumen/Dr%20Hewitt%20-%20Arbitrary%20and%20Necessary%20part%201.pdf
• https://www.lboro.ac.uk/media/media/services/lumen/Dr%20Hewitt%20-%20Arbitrary%20and%20Necessary%20part%202.pdf
• https://www.lboro.ac.uk/media/media/services/lumen/Dr%20Hewitt%20-%20Arbitrary%20and%20Necessary%20part%203.pdf