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

February 09, 2021Pam HarrisEpisode 34

Math is Figure-Out-Able with Pam Harris

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

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Math is Figure-Out-Able with Pam Harris

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

Feb 09, 2021Episode 34

Pam Harris

In this episode Pam and Kim begin the first of another series: Multiplication Facts! What role can mathematizing and real math play in helping kids master their multiplication facts? Listen in as Pam and Kim outline the differences between 'knowing' and 'memorizing' multiplication facts. Talking Points:

Why multiplication facts are important

The difference between knowing and memorizing facts

How to get rid of all the stress around multiplication facts

Is the goal to get through memorizing the facts, or to develop multiplicative reasoning?

In this episode Pam and Kim begin the first of another series: Multiplication Facts! What role can mathematizing and real math play in helping kids master their multiplication facts? Listen in as Pam and Kim outline the differences between 'knowing' and 'memorizing' multiplication facts. Talking Points:

Why multiplication facts are important

The difference between knowing and memorizing facts

How to get rid of all the stress around multiplication facts

Is the goal to get through memorizing the facts, or to develop multiplicative reasoning?

Hey, fellow mathematicians, welcome to the podcast where math is figure-out-able!

I'm Pam Kim:

And I'm Kim.

Pam:

And we're here to suggest that mathematizing is not about 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:

So today we're gonna began, a topic in a three-part series that is huge. So many people have pretty firm 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:

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

Yes! Kim:

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:

Yeah. So let's do. Alright. So what do we mean when we say multiplication facts or multiplication tables? What does that mean? I don't actually use 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 standards call for multiplication facts through 10 by 10, sometimes 12 by 12. That's like maybe a little bit, if we were to pick like sevens, when I say 10 by 10 or 12 or 12, So for example, I even just heard somebody say that their kids have to memorize it through their 15s. a lot. or through the fifteens, so seven times one, seven times two, seven times three, all the way to, if we're doing 10 by 10, all the way to seven times 10, if we're doing to 12 then all the way through seven times 12, or even for that one teacher seven times 15, we're having kids sort of know those multiplication facts.

Kim:

Yeah. And most of the time that someone talks about those facts, they hear recall from rote memory. And, Pam, 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 us 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 a hundred 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.

Pam:

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 this:

Pam Harris thinks students need to memorize their facts. Because the word memorize is tricky because it means different things to different people. 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.

Pam:

Because of the wording of these standards is problematic.

Kim:

Okay. 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 wrote memorizing and knowing. So I don't even use memorizing anymore because it's so, so confusing. So wrote memorizing and knowing. So defining some terms I'm going to define wrote memorizing as 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 2 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 a student says, I don't know, then obviously they're not being able to pull, but it also means the student who doesn't take five minutes to figure out the fact, 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. 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 them at your fingertips, be able to use them. And really they don't have to fuss or struggle or reach without success.

Pam:

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.

Kim:

Yeah, I think it's a great time for you to share the quote that you wanted to talk about with Kathy Fosnot and her young mathematicians at work series.

Pam:

So she has a section In her books called 'memorization or automaticity?'. 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 in only a 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 of 54 quickly, but thinking not memorizing -". And I'm going to put the word in here wrote 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.". That would be a brilliant, right? The relationship, which is the heart of mathematics are not sacrificed.

Kim:

So the standards say that students need to know the facts and we're not debating that.

Pam:

Nope, not debating that.

Kim:

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

Pam:

We don't want to leave kids without something in that And we do a lot of work on campuses,

Kim:

Yeah. then get that fact quickly using relationships. moment. We need them to have the relationships that they can right? And I have been on campuses where kids do the exact same thing in third grade then in fourth grade, then the same wrote-memory practice, and time tests in fifth grade over and over and over and get the same results. And at some point for me, that's the definition of insanity. Do the same thing over and over and expecting different results.

Pam:

Right. 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 doesn't bog them down and what they're doing - every single workshop, at least one teacher usually more admits it. Yeah. Y'all these are calculus teachers. Every single one admits there are facts they don't know, but it's not like they're like, Whoa is me. I should've known it. Oh, maybe fire me from my job. No, 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, 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. Yeah. That's huge. That speaks to this fact that when we're thinking about higher math we're really clear we don't necessarily have to have wrote-memorized 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 at 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 out there. They're not stressed about it. Oh, if we could give that unstressed that 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, I'm going to share a quick story with you. We were at the table. I was talking about the fact that we were going to do this 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 orange is banana. And that I was supposed to remember that Apple plus or times whatever times pineapple was a mango, that it was that disconnected, that there was no rhyme or reason. There was no sense. And he's like, I just couldn't make sense of it. And I was horrible at the memorizing. And so I failed all that stuff. And then over the years, you know, it just escalated. So let's be clear if your perspective has been that the goal of your math classes 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 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 having those multiplication facts at their fingertips, but without all the stress. If we do it without all the stress, then we could get kids that don't get further, and 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 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:

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 going to be back. So stay tuned more about this hot topic in mathematics next week. Remember to join us on #MathStratChat on Facebook, Twitter and Instagram, on Wednesday evenings, where we explore problems with the world.

Pam:

And please share the podcast with your friends and colleagues, and 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!