Ep 34: Multiplication Facts: The Good, the Bad, and the Ugly Pt 1

Pam Harris  00:02

Hey fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.

Kim Montague  00:09

And I'm Kim.

Pam Harris  00:10

And we're here to suggest that mathematizing is not about rote memorizing or mimicking. But it's about thinking and reasoning; about creating and using mental relationships, that mathematics class could be less like it was for so many of us, and more like mathematicians working together. We answer the question: if you're not teaching students to do algorithms, then what?

Kim Montague  00:36

So today, we're going to begin a topic in a three part series that is huge. So many people have pretty fond feelings one way or the other about this topic, and we decided it was a great time to dive in and give our thoughts to the world about multiplication facts.

Pam Harris  00:54

This is a hot topic, everybody. And so let's address it and address it well today. The multiplication facts are so important, so misunderstood, so stressful, and unnecessarily stressful, in our opinion. Are the facts important? Yes! 

Kim Montague  01:12

Absolutely. And it's not really the facts themselves that are the issue. It's the pressure surrounding them and the ways of practicing that people argue over. So let's spend a minute defining and clarifying.

Pam Harris  01:24

Yeah, so let's do alright. So what do we mean when we say 'multiplication facts' or 'multiplications tables'? What does that mean? I don't actually use the multiplication tables very often. But the more I work internationally, the more I hear people talking about multiplication tables. So we want students to know single digit multiplication facts. Sometimes we want, some standards call for multiplication facts, through 10 by 10, sometimes 12 by 12. So that's like, sometimes. I even just heard somebody say that their kids have to memorize through their 15. 

Kim Montague  01:57

Right. 

Pam Harris  01:58

Ah, that's like maybe a little bit a lot. So for example, when I say 10 by 10, or 12 or 12, or through the 15s. If we were to pick like seven, so 7x1, 7x2, 7x3, all the way to if we're doing 10 by 10, all the way to 7x10. If we're doing the 12, all the way through 7x12. Or even for that one teacher, 7x15. We're having kids sort of know those multiplication facts or multiplication tables.

Kim Montague  02:24

Yes. Yeah. And most of the time that someone talks about those facts they hear "recall from rote memory." And Pam, and sometimes you get misheard. People have mistakenly thought in the past that you think that kids don't need to know their facts. Can you tell me more about what you really mean here? Because here in the States, the standards say students need to know their facts. In fact, this common core standards in grade three, I'm going to quote this, it says, "Fluently multiply and divide within 100 using strategies such as the relationship between multiplication and division, (and it gives an example) or properties of operations." And it says, "By the end of third grade, know from memory, all products of two one-digit numbers." And here in the state of Texas, where we are our standards say, "Recall facts to multiply up to 10 by 10 with automaticity." Yeah, so this gets a little sticky. So let's be clear. Does Pam Harris think students need to know their facts? Yes. But let's define 'know'. What does it mean to know their facts? And I use the word 'know' on purpose. What I didn't say was does Pam Harris think students need to memorize facts? Because the word 'memorize' is tricky. Because it means different things to different people. It's a tricky word. And I think that when we use the word 'memorize', and even when we use the word 'know', and some other things, that we kind of talk past each other, because we assume the other person knows what we mean. So let's define 'knowing', versus 'rote memorizing'.  Okay. 

Pam Harris  03:53

Because the wording of these standards is problematic. They try to walk the line, they try to please everyone in the way that they word it and because of that it gets muddy. So let's clear it up. What's the difference between rote memorizing and knowing? So I don't even use memorizing anymore because it's so so confusing. And so whatever. So rote memorizing and knowing. So defining some terms. I'm going to define rote memorizing is recall without meaning, and knowing is to have at your fingertips. I believe that that's what the standards mean when they say automaticity, and when they say know all products of two one-digit numbers. They mean, know to have at your fingertips to be able to use, to be able to not haltingly spend forever on, or not know at all. Like if as soon as I don't know, then obviously, they're not being able to pull it off. But it also means the student who doesn't take five minutes to figure out the fact. 

Kim Montague  04:55

Right. 

Pam Harris  04:56

Either one of those to know means that we sort of have it at your fingertips. It's in your psyche. But rote memorizing to recall without meaning, is not what I would suggest, it's not the intent of the standards.

Kim Montague  05:10

And so you said automaticity. So let's define that. What does automaticity mean? We believe it means what people mean when they say know your facts. They mean have at your fingertips be able to use them. And really, they don't have to fuss or struggle or reach without success. Yeah, and I believe that math people came up with the term automaticity and fluency, because they see some students fluently using facts, and other students not. And they wanted to be able to help us all think about the difference between rote memorizing, that recall without meaning, and knowing which is owning down deep, like you just said that automaticity. Yeah, I think it's a great time for you to share the quote that you wanted to talk about with Cathy Fosnot and her young mathematicians at work series.

Pam Harris  05:57

So she has a section in our books called memorization or automaticity. Question mark. And she says, "Memorization of basic facts usually refers to committing the results of operations to memory, so that thinking is unnecessary. Isolated, multiplications and divisions are practiced one after another. The emphasis is on recalling the answers. teaching facts for automaticity, in contrast, relies on thinking. Answers to facts must be automatic, produced only few seconds, counting isn't sufficient. But thinking about the relationships among the facts is critical. A child who thinks of nine times six as 10 times six minus one six, produces the answer 54 quickly. But thinking not memorizing, (and I'm going to put the word in here rote memorizing) is at the core, although over time these facts are remembered. The issue here is not whether facts should eventually be memorized, (I would put in parentheses known) but how this memorization, (knowing that's my word knowing) is achieved by rote drill and practice, or by focusing on relationships." And then she talks about ways to develop relationships and says, "In this way, the facts become automatic, but the relationships, the heart of mathematics are not sacrificed." Brilliant, right? The relationship, which is the heart of mathematics, are not sacrificed.

Kim Montague  07:25

So the standard say that students need to know the facts. And we're not debating that.

Pam Harris  07:29

Nope, not debating that. 

Kim Montague  07:30

What about students who are slow to rote memorize or when memory fails for them? We have to give students relationships and connections to be able to derive a fact, because let's be honest, memory fails for every student, everyone's memory fails at some point.

Pam Harris  07:45

Everybody's fails at some point. We don't want to leave kids without something in that moment. We need them to have the relationships that they can then get that fact quickly, easy relationships. Yeah.

Kim Montague  07:54

And we do a lot of work on campuses, right. And I have been on campuses where kids do the exact same thing in third grade, then in fourth grade, then the same rote memory practice and time tests and fifth grade over and over and over and get the same results. And at some point, for me, that's the definition of insanity. It just is.

Pam Harris  08:14

Right? Doing the same thing over and over and expecting different results. That is the definition of insanity. You might find it interesting that I often work with high school math teachers, calculus teachers, and when I ask them if there's a fact they don't know, but they refigure it in a mathy sophisticated way so it's doesn't bog them down in what they're doing. Every single workshop, at least one teacher, usually more, admits it. 

Kim Montague  08:38

Yeah.

Pam Harris  08:39

Y'all, this calculus teachers. Every single one admits there are facts, they don't know. But it's not like they're like, "Woe is me. I should have known it. Maybe fire me for my job." No, no, no, no, they confidently tell me. And it's almost like, "Well, of course, there are some facts that I refigure." Their attitude is very much like, "I'm busy thinking about all these higher level math things. Every mathematician knows that you don't have to fill your brain with that stuff. Just find it when you need it." 

Kim Montague  09:04

Yeah.

Pam Harris  09:04

That's huge! Like that speaks to this fact that when we're thinking about higher math, we're really clear, we don't necessarily have to have rote memorize all of these facts. What we want are relationships and connections. And then we want to deal with those facts a lot. We want to do things where it demands that we have those facts that we use them, that we play with them, that we are thinking about the relationships between the facts. And then if one of them doesn't pop out when we need it, we can refigure it quickly in a multiplicative, mathy kind of way. There's no detriment to that. We're not at a loss because we had to refigure it quickly, because we had these relationships in our fingertips so that we can refigure them. So these calculus teachers are really clear that mathematizing is about using relationships. What they didn't do was fret about the fact that they didn't know some of the single digit facts. That's not a, they're not stressed about. Oh, if we could give that unstressed that de-stressing, I don't know if unstressed as a word, de-stressing to our students, so many of them would be able to relax into just using the facts, enough that they become then automatic. Of course, we want students to know their facts, of course, we want them at their fingertips. But we don't want all of the stress that comes when we ask kids to rote memorize them without meaning. In fact, Kim, I want to share a quick story with you. We were at the table, I was talking about the fact that we were going to do a series pretty soon. And my husband, bless his heart who struggled in math all of his life, said to me, "If my teachers would have helped me understand that there were relationships," he said, "I literally thought I was memorizing a bunch of disconnected facts that they had nothing to do with each other. It was like if you would have said, 'apple plus oranges, banana,' and then I was supposed to remember the apple plus, or times whatever times pineapple was a mango, that it was that disconnected, that it had one there was no rhyme or reason, there was no sense." He's like, "I just couldn't make sense of it. And I was horrible at the rote memorizing and so I fail all that stuff, and then over, you know, just getting escalated." So let's be clear. If your perspective has been that the goal of your math class is for students to get to say, the traditional long multiplication algorithm, then we get why it would make sense that you would think, okay, in order for students to be successful in this multiplication algorithm, they must be able to recall the single digit facts. Because the only thing you do in that multiplication algorithm is do single digit multiplication facts over and over and over, there's a magic zero that shows up. And then you do a bunch of addition at the end. So if you're gonna be successful at it, yeah, you're gonna be able to have, you want those single digit facts at your fingertips. So I get that. I understand why there would be this emphasis on knowing the facts. However, even if that was your goal, what if you could get to having kids have those multiplication of acts at their fingertips, but without all the stress? If we do it without all the stress, then we can get kids that don't get further further behind, because the stress is impeding them. We get kids who can actually use those single digit multiplication facts in that algorithm. But we're suggesting that's not the goal. It's not the goal to just have kids get answers by not thinking, recalling from rote memory, a bunch of single digit facts, and then the magic zero and then they add the stuff up at the end. That's not our goal. Our goal is mathematizing. Our goal is helping students develop the way they think and reason and use relationships. Therefore, we recommend helping students learn the facts through relationships. And we're going to get into this more as our series progresses about how to do that. Today, we just want to lay the background. But if our goal is on helping students develop multiplicative reasoning, then we want to actually use the facts to help develop their multiplicative reasoning, not just rote memorize them, so we can use them. Actually, let's know them. Let's learn them. Let's own them down deep by investigating the relationships. Now we've done it so much, we're using the relationships and and using the facts a lot that then they sort of become part of us. They become at our fingertips, we know them down deep and we can use them. So for more on that the goal being mathematizing and developing mathematical reasoning, and specifically multiplicative reasoning, check out podcast Episodes five and six, where we really dive deep into what the goal of mathematics education should be, that it's not just kids being able to get answers, recalling steps from rote memorizing an algorithm that requires rote memorizing a bunch of facts.

Kim Montague  13:44

So today, we're just starting the series right by trying to give you a little bit of a background and some things to think about in relation to multiplication facts. But we are gonna be back so stay tuned more about this hot topic in mathematics next week with some more. Remember to join us on MathStratChat on Facebook, Twitter, and Instagram on Wednesday evenings where we explore problems with the world.

Pam Harris  14:08

And please share the podcast with your friends and colleagues, parents, and if you don't mind giving us a rating and a review so that more people could find the podcast. We'd really appreciate it. So if you're interested to learn more mathematics and you want to help yourself and your students or your personal kids develop as mathematicians, then don't miss the Math is Figure-Out-Able podcast because Math is Figure-Out-Able.