Just How Helpful is Knowing Your Multiplication Facts?

A thought experiment

Here’s a weird idea I sometimes like to toss around, as I think it clarifies why interventions in math are so hard:

What if there was a magic pill that instantly let you know every single-digit multiplication fact by heart?

Maybe a brain implant, not a pill. Or a little nanobot that encodes it directly via your synapses. Or maybe it is a pill, I mean why not? We’re playing pretend.

No matter the mechanism, that’s all it does—you get the neural implant, inhale the bots, take the pill, and suddenly you forever know the answers to 7 x 8, 3 x 6, 9 x 6, and all the rest.

How much would this help? And how fast would we see results? My sense is that some of us out there think that the answer is that it would help a ton, very quickly. Some think that knowing multiplication facts would be an instant boost for struggling math students, no matter the age. Let’s take a close look at all that.

Question #1: After taking this magic pill, do high school students suddenly get better at math?

We should probably start by noting that, as far as I can tell, this hasn’t been studied. I don’t just mean because of the sci-fi aspect. When I reviewed the multiplication literature, I wasn’t able to find any multiplication interventions specifically involving adolescents. It’s possible I missed a paper, but that’s not the point—if it was widely studied, I would have found something.

Probably the reason nobody has studied this is because…it definitely wouldn’t help?

To be fair, there are a few places in high school were students are asked to multiply. I don’t know if any of you have tried to teach kids to multiply algebraic binomials when they can’t handle the usual, non-algebraic stuff. It’s painful and depressing.

If you can’t multiply 9 x 6, you can’t multiply this either.

The pill would help with this—kids could learn to multiply binomials. Maybe they could even learn to factor them. Hooray!

Except…

Kids who don’t know multiplication facts well enough to get by in high school are typically struggling across the board. They don’t know how to make sense of negatives. They don’t understand ratios or proportions. Fractions freak them out. The list goes on and on and on for these poor students.

It’s true that every little bit helps. The pill would magically help with factoring problems. And besides that? I honestly am not sure.

Conclusion: the pill wouldn’t help much.

So let’s set the dial back a few more years and try again.

Question #2: After taking this magic pill, do middle school students suddenly get better at math?

This year I teach 7th Grade. Multiplication is everywhere. I think the pill has a better shot of helping here! But it’s still a bit confusing to figure out where.

Middle school math has one foot in arithmetic, the other in algebra. The algebra involves the Distributive Property, things like 5(13 + 2a). I think the pill would help with that.

But there’s a lot more to algebra in middle school. Kids have to solve equations, combine like terms, graph two-variable equations. Would knowing 5 x 9 help a kid avoid saying that 5a + 9 is 14a? Would our sci-fi wizardly make a kid know how to solve 5.3 - 2x = 10.7? How to graph y = 3x? Or y = (2/3)x?

And will the kids know whether to divide or subtract something from both sides for 3a = 10? Oh god I’m so sick of seeing a = 7. Multiplication/division undo each other…would these kids know that?

Once again, the problem is that multiplication facts not nearly enough. These kids typically can’t handle fractions, decimals, negatives. The other problem is that there’s a lot of multiplication knowledge that goes beyond the facts. There’s no reason why the kids who magically get the facts would also automatically get the understanding.

Conclusion: the pill would help a bit, but not much.

Now, set the clock back one more time.

Question #3: After taking this magic pill, do elementary school students suddenly get better at math?

Yes, they would! I don’t know how much, but a bit.

First, they’d get better at all the multiplication. Like, instantly. Multiplying 37 x 4 is much easier if you know 3 x 4 and 7 x 4. This is a huge win. Ditto with division.

So, there are some quick wins. But everything else will take time.

These kids might have more opportunities to learn the sophisticated multiplication/division stuff that will help them with proportional reasoning in middle school. The thing that will help them realize that, yes, you should divide to solve 3a = 10. I don’t know about “suddenly get better at math.” It depends on when they’re taught this stuff. But knowing the facts means they’ll better be able to hang with the lesson.

The big, big, big thing though is: fractions. To learn fractions, kids need to know how to multiply. You can’t add fractions if you can’t multiply single-digit numbers easily. You can’t simplify fractions like 15/36 if you don’t recognize 3 as a common factor. You can’t find equivalent fractions without multiplying. You’re just stuck, stuck, stuck.

This is something you know if you’ve taught, and you could also know it from research. Multiplication fluency is a “key predictor” of fraction procedural knowledge.

Hansen, N., Jordan, N. C., Fernandez, E., Siegler, R. S., Fuchs, L., Gersten, R., & Micklos, D. (2015). General and math-specific predictors of sixth-graders’ knowledge of fractions. Cognitive Development, 35, 34-49.

For fraction procedures, attention and whole number computational abilities (multiplication fluency and division) emerge as key predictors, as has been noted in previous research (e.g., Hecht & Vagi, 2010)… In contrast, multiplication fluency was uniquely predictive fraction procedures but not concepts, which makes sense given the importance of multiplication for finding common denominators (e.g., 3/4 + 2/3).

OK, but again: when is too late?

If you give the pill in 5th Grade, you’re going to have to also remediate a lot of missing fraction skills. Sure, there are a lot of opportunities to catch up in 5th Grade itself, but learning is not magic—except for the one bit of magic we’ve already helped ourselves to. Kids will need to be taught the procedures they missed in 3rd and 4th Grade, back when they didn’t know what 3 x 4 too well.

Ideally, then, you’d do what researchers have done and combine this magic pill with some sort of systematic fractions instruction. Absent something like that, you’d be relying on luck that kids will connect these dots on their own.

All of this is to say: yes, it will help. Some of this will be sudden. Some of it will take time and ideally be paired with fractions (and I guess multiplication/division) instruction.

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What all this clarifies for me is that even this magic pill is no magic solution. In general, and especially as kids get older, it becomes less and less necessary to directly remember multiplication facts. (Maybe this is why some adults are confused as to why knowing them is helpful.)

Meanwhile, they are directly necessary for learning fraction procedures. They help us learn all sorts of multiplication/division concepts. And knowing all these things helps us learn the next layer of stuff—multiplicative thinking, proportional reasoning, ratios, etc.

But if you could give kids a magic pill, you’d want to give it early. Maybe you’d even want to give it before there’s trouble. Because by the time kids are a little bit older, it’s almost too late for even magic to help.