Math is Figure-Out-Able!
Math teacher educator Pam Harris and her cohost Kim Montague answer the question: If not algorithms, then what? Join them for ~15-30 minutes every Tuesday as they cast their vision for mathematics education and give actionable items to help teachers teach math that is Figure-Out-Able. See www.MathisFigureOutAble.com for more great resources!
Math is Figure-Out-Able!
Ep 246: Avoid the Trap of Using Less Sophisticated Reasoning
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If a student can perform an algorithm, have they developed the mathematical reasoning that operation calls for? Or are they answer-getting? In this episode Pam and Kim discuss a major trap students fall into when we teach mathematics as a series of steps to rotely memorize.
Talking Points:
- A closer look at the multiplication algorithm and a major trap.
- Problem String that helps give students a high does of patterns for a multiplicative strategy.
- Algorithms disregard reasonableness.
- Memorizing an algorithm is forgetable but every strategy learned helps build mental maps to the make subsequent strategies easier to develop.
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Pam 00:01
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able. I'm Pam, a former mimicker turned mather.
Kim 00:09
And I'm Kim, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.
Pam 00:17
Because we know that algorithms are amazing human achievements. But, ya'll, they are not good teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop.
Kim 00:31
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.
Pam 00:37
Thanks for joining us to make math more figure-out-able! (unclear).
Kim 00:39
Hi there! Yeah.
Pam 00:44
Let's dive in today.
Kim 00:46
Yeah.
Pam 00:47
Because I'm super excited. So, you know that my brand new book is out. Developing Mathematical Reasoning: Avoiding the Trap of Algorithms.
Kim 00:52
Mmhm! Yes, ma'am.
Pam 00:56
So, Kim, today I am hoping that you're okay. Let's... If you're alright with it, let's talk about one of the major traps of the algorithms.
Kim 01:07
Yes!
Pam 01:08
And let's be honest. So... Let's be honest. That's not the what I meant to say.
Kim 01:12
We're always honest, Pam.
Pam 01:13
Yeah. Let's be... Let's fine tune that? What I'm trying to say is, in the book, I outline major traps of many of the traditional algorithms that we teach in schools.
Kim 01:27
Yep.
Pam 01:28
But one of the things that I did was specifically outline that there are three... No. Yeah. Oh, heavens.
Kim 01:37
Are you tired?
Pam 01:38
I'm going to take a breath. I have so many things happening in my head right now. It's like trying to parse them out.
Kim 01:42
I know.
Pam 01:43
I outlined three traps that fit, I think, all algorithms.
Kim 01:47
Yeah.
Pam 01:47
So, pick a specific algorithm, and there are a lot of traps in that specific algorithm.
Kim 01:51
Yep.
Pam 01:52
But I think there's three main traps that kind of fit all the algorithms. So, I'm super curious. Ya'll, if you could... I'm going to... What I propose today, Kim, is that we go through one of those traps, kind of in detail.
Kim 02:03
Okay.
Pam 02:04
And I really want to invite our listening audience to think about any algorithm that is taught typically at your grade level and your subject, and think to yourself, do these... This one. We're going to go over one today. Does this trap, do I find that in my particular algorithm?
Kim 02:22
Yeah.
Pam 02:23
I've just thought about as many of them as I can generate, but I'm really curious if you could think of one that maybe this trap doesn't fit for. I'm going to bet not. This one's so important. Alright. So, Kim, the trap that I inviting us to talk about today, we actually mentioned in the intro.
Kim 02:30
Okay. Okay.
Pam 02:42
And we mentioned in the intro because I say that mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop.
Kim 02:53
Mmhm.
Pam 02:53
So, I call this the "less sophisticated reasoning" trap.
Kim 02:57
Yep.
Pam 02:58
What do I mean by that? What I mean is kids can get correct answers using that particular algorithm, but they can get away with using less sophisticated reasoning than we should be developing with them if those are the kinds of problems kids are learning to solve.
Kim 03:14
Yeah, absolutely.
Pam 03:15
Alright, so here's an example. If I were to ask you to solve a multiplication problem, and you use the traditional algorithm that's out there.
Kim 03:26
Mmhm.
Pam 03:26
Then you could... Well, so it's multiplication.
Kim 03:30
Mmhm.
Pam 03:30
So, if you think about developing mathematical reasoning, that graphic that I've created. And we want to develop from counting strategies to additive thinking to multiplicative thinking. If we're talking about the multiplication algorithm, we should be developing multiplicative thinking.
Kim 03:44
Mmhm.
Pam 03:45
That's what we're doing, right? You're learning multiplication. You should be reasoning multiplicatively. We need to develop multiplicative reasoning because we build on that to build proportional reasoning. And we need to build multiplicative and proportional reasoning because we build on that to build functional reasoning and other things.
Kim 04:01
Mmhm.
Pam 04:02
So, if we're doing multiplication, then we need to be building that kind of reasoning. So, Kim, if I were to ask you to solve the problem 14 times 21, and you wrote that down with a traditional algorithm...
Kim 04:16
Mmhm.
Pam 04:16
...then you might put the 14 on top and the 21 on the bottom. That doesn't really matter. Oh, this is a terrible problem to use as an example. You know what? I actually meant to give you 14 times 19. That would be a much better example. Sorry. This is going to be a great podcast, I can tell. Okay, so...
Kim 04:34
Are you asking me to write these steps down?
Pam 04:38
You know what? I'm going to give you actually a separate, one more problem.
Kim 04:41
Okay. (unclear).
Pam 04:42
You're just totally laughing at me.
Kim 04:44
No.
Pam 04:45
Oh, do you the other day when you tried to do the algorithm, you did it wrong. That was hilarious.
Kim 04:50
Yes. (unclear).
Pam 04:50
Yeah, because it's been so long since you've actually done it. Alright, 29 times 13 is the one that I'm going to give you. Yeah. Okay, so if we were... I'll help you, Kim. It's alright. If we were doing the traditional algorithm, then we might...
Kim 05:01
Pause for just a second. I can't not think of the problem I want it to be.
Pam 05:05
Haha!
Kim 05:06
Like, I'm so ingrained in actually thinking.
Pam 05:09
You're mathematical brain is just...
Kim 05:11
(unclear) I want it to be something different. Yeah, okay.
Pam 05:12
Your mathematical brain is mathematizing as we go.
Kim 05:14
Mmhm, yeah.
Pam 05:14
And you're like, "Really? You're going to make me do these dumb steps?" Yeah, yes. I am.
Kim 05:19
Am I writing down?
Pam 05:21
Yeah, write down 29. And then underneath it, write 13.
Kim 05:23
Okay.
Pam 05:24
And then put the little x next to it. Draw the line.
Kim 05:26
Yep, I got ya.
Pam 05:27
Okay, so the very first step that we would have students do is do 9... Or, excuse me, 3 times 9. Or 9 times 3. Either one. If I am a student doing the algorithm, do you agree with me that I could now say I need three 9s.
Kim 05:41
Mmhm.
Pam 05:41
Or nine 3s.
Kim 05:42
Mmhm.
Pam 05:43
And then I could say, "Hmm. Ready begin." 9 plus 9 is 18 plus another 9 would be 27. Like, you could actually skip count to find three 9s. Or you could think about nine 3s and you could skip count. 3, 6, 9... You could literally find nine 3s.
Kim 06:00
Mmhm.
Pam 06:00
So, you could get the correct 27, and you could write down the 7 below and put the 2 above, but the whole time you are thinking additively because you're skip counting.
Kim 06:11
Mmhm.
Pam 06:12
Not multiplicatively. You could agree that could be true?
Kim 06:14
For sure.
Pam 06:15
Cool. Then I'm going to do the next one, and now I'm just supposed to do 3 times 2. So, I could go 2, 4, 6... So, I could skip count three 2s. Add the 2. Write that down correctly. Then the magic 0, or the turtle lays an egg, or whatever it is. And then I could do the next row. And I could think about all of those single-digit multiplications. I could actually do them additively. So, if you can picture the development of mathematical reasoning graphic where I've got counting to additive to multiplicative. I'm supposed to be building multiplicative reasoning, but I'm getting away with using additive reasoning. And I could do that through the entire problem. I could use additive reasoning, and then add all the columns of things at the end, and I could get a correct answer. And my teacher could say, "Yay! You got a correct answer to a multiplication problem! You must be reasoning multiplicatively!" When, in reality, you are actually reasoning additively the whole time.
Kim 07:11
Yes.
Pam 07:11
And that is the less sophisticated reasoning trap.
Kim 07:16
Mmhm.
Pam 07:16
That I'm getting correct answers, looking like I'm reasoning in a certain way, but I'm actually getting away with using less sophisticated reasoning. And to be clear, when I do those columns of additions at the end of the problem, I could actually also do those counting by ones. I could add all of those columns up because it's just single-digit plus a single-digit. I could count by ones to add those up. So, even in the additive part, I could be using counting strategies, less sophisticated reasoning. So, I look like I'm an additive, multiplicative reasoner because I'm getting this multiplication problem correct, but I'm actually using reasoning in the level before. I've been trapped. My brain has not been encouraged to grapple with the multiplicative relationships, to actually think and reason multiplicatively. I've been allowed to just reason additively, get correct answers. And then when I move on to the next topic, the next year, the next whatever, my brain hasn't grown. I haven't been given the opportunity, especially I haven't been encouraged and nudged in the opportunity to reason multiplicatively. That is the sophisticated reasoning trap. Alright, what are you thinking
Kim 07:25
Mmhm.
Pam 07:26
right now?
Kim 07:29
I think that many, many students in many, many grade levels are having this exact trap on a daily basis.
Pam 08:37
Yeah.
Kim 08:37
I mean, I'm picturing being a third, fourth, fifth grade teacher, and how many times, you know, I see in these classrooms, fingers, counting as students are adding columns of digits, and skip counting. You know, you said...
Pam 08:39
(unclear). Oh, go ahead. Keep going.
Kim 08:54
When you said 9 times 3 and 3 times 9, I'm remembering like my first year teaching, hoping that kids would skip count by 9s instead of skip counting by 3s and calling that a win.
Pam 09:06
Yeah, for sure.
Kim 09:07
Mmhm. You know?
Pam 09:07
Which is a fine win in third grade when we're very starting to develop multiplication as repeated addition. But we have got to get more sophisticated. We can't leave kids there. Hey, Kim, when you just said that you can picture a classroom where kids are kind of counting on their fingers or maybe they're skip counting, I can also picture classrooms, well intended classrooms. Ya'll, it's no blame, right? We're just all helping each other move forward. I can picture classrooms where students are looking at a multiplication table. For those very same facts. I would submit that that's using not even additive reasoning. That's using looking up on a table reasoning.
Kim 09:45
Mmhm.
Pam 09:45
Which is not the reasoning. What are we supposed? If we're doing multiplication, we're supposed to be developing multiplicative reasoning.
Kim 09:51
Mmhm.
Pam 09:51
Now, we're just having students just like look them up on a table. Or a student who types in those single-digit facts in the calculator.
Kim 09:59
Yeah, that's not a less sophisticated reasoning trap. That's a no reasoning trap. (unclear).
Pam 10:05
I would agree completely. Yeah. Let me give you one more no reasoning trap. Where I have a kid go, "Oh, let's see. 3 times 9 is like the diamond. If it rolls, then it's 20." I'm trying to come up with a rhyme, and I don't have a rhyme for 3 times 9.
Kim 10:20
You don't have a poster?
Pam 10:21
I mean, I'm sure there's one out there somewhere with a story, whatever. So, if it's a story, or a rap, or a rhyme, if it's a mnemonic, if it's a memory retrieving device, then that's not even reasoning at all. At least skip counting is additive reasoning. Well, I say that. Skip counting is additive reasoning if you're adding the numbers, if you are singing the skip counting song from... What's the PBS multiplication? Schoolhouse Rock!
Kim 10:51
Oh, yeah.
Pam 10:52
Now, I'm not knocking Schoolhouse Rock, but I am knocking it if you're pretending that singing 3, 6, 9, 12, 15, 18... Yeah, I can't even see the song anymore. But if that's what you're singing, then you're singing a song. You're not even using additive thinking. So, it's like you said. It's not reasoning at all. So, not only is it the less sophisticated... Well. Less sophisticated. No sophisticated reasoning trap?
Kim 11:17
Yeah.
Pam 11:18
Yeah. What were you going to Maybe.
Kim 11:18
Yeah.
Pam 11:19
say?
Kim 11:20
I was going to say that you and I are firmly, firmly grounded in the belief that that is the job. The job is to build reasoners, not just get answers.
Pam 11:29
Well, because that's where mathers are. Yeah.
Kim 11:31
Right.
Pam 11:31
Mathers actually reason mathematically. They don't just mimic. Yeah.
Kim 11:35
Right. So, I think for some people who are hearing us say like poster, song. Like, if your belief is the job is to just get answers, then you might be wondering like why is that a problem?
Pam 11:50
Sure. Yeah. And, ya'll, again, this is not about shaming anybody. This is about like lifting everybody up into real math. I am the poster child of we sang songs, and raps, and rhymes to memorize stuff because I was convinced that mathing was about squiggling and that there was no shark. It was just like let's memorize the squiggles in the... If you don't know what I'm talking about with the shark, go listen to last week's episode.
Kim 12:19
Yeah.
Pam 12:20
So, yeah. What we're trying to do is help everybody realize that the purpose of math class is to build mathematical reasoning. And we can! The good news is we can! Everybody can do more real math than fake mathing, than fake math mimicking. Absolutely.
Kim 12:34
Yeah.
Pam 12:34
Alright. So, Kim, how could we do a better job with the problem that I gave you earlier? Let's do a quick Problem String.
Kim 12:41
Okay.
Pam 12:42
Alright. So, first problem. What is 14 times 10?
Kim 12:46
140.
Pam 12:47
Alright, and I'm going to record that in a ratio table. So, I've just written down 1 to 14, and then I've written down 10 to 140. So, if we're starting, if we're thinking about fourteens, you told me that 10 of them was 140. So, 1 to 14, and then 10 to 140. What if I only wanted 9 of those 14s?
Kim 13:07
That's going to be one less group than we just had.
Pam 13:10
Okay.
Kim 13:10
So, I'm going to take 140 and subtract 14.
Pam 13:13
Mmhm.
Kim 13:13
Which is 126.
Pam 13:16
Alright, so 1 less group of 14. 140 minus 14. You got 126. Cool.
Kim 13:20
Mmhm.
Pam 13:21
What if we wanted twenty 14s. 20 times 14 or 14 times 20?
Kim 13:27
I'm going to double the ten 14s.
Pam 13:29
Okay, alright.
Kim 13:31
Which is 140, and so now it's 280.
Pam 13:33
Nice. I'm kind of curious. Totally makes sense that you did that. You got anything else for that one?
Kim 13:39
For 20?
Pam 13:40
Yeah.
Kim 13:41
Yeah. We don't have 2 on the ratio table.
Pam 13:45
Sure enough.
Kim 13:45
But you could do 2 is 28, and then times 10.
Pam 13:50
get the 20? To
Kim 13:51
Mmhm.
Pam 13:51
280? Okay, cool. Just curious. What if we only wanted nineteen 14s. 19 times 14. So, so far we have twenty 14s because I know a lot of people are listening to this in their car. So, we have twenty 14s is 280. But now we only want 19 times 14. Okay.
Kim 14:06
Yeah, so I'm just going to subtract a group of 14 from the 20 that I have there. So, 20 groups was 280, so 19 would be 266.
Pam 14:15
266, nice. And we could talk about that subtraction. But today, we're not. Alright, cool. What if, instead, I wanted twenty-one 14s.
Kim 14:25
Mmm, that's a lot of 14s. I'm going to add 14 to the twenty 14s we already figured out. So, 280 plus 14 is 294.
Pam 14:36
Cool. So, so far, we've kind of solved a lot of problems.
Kim 14:39
Mmhm.
Pam 14:40
We've gotten ten 14s, nine 14s, twenty 14s, nineteen 14s, twenty-one 14s.
Kim 14:45
Mmhm.
Pam 14:46
So, at this point in class, I might have a conversation about like what kind of pattern do you guys feel like we're playing with here? Can you put some words to kind of? Are you feeling like a we're playing with something here?
Kim 14:59
Yeah, I think we're thinking about groups of 14s that we know. Like, the 10 of them, and the 20 of them. And then we're going like a little under or a little over.
Pam 15:08
19, 21. Once we found the twenty 14s, we could back. Or once we found the ten 14s, then we could ask you for 9. Little over, a little under.
Kim 15:17
Mmhm.
Pam 15:18
Okay, cool. Next problem. How about... We're going to change the... I'm no longer in a ratio table with fourteens. We're going to think about 13s. And this time, I'm looking for thirty 13s.
Kim 15:28
Mmhm. So, I'm going to say I know three 13s is 39.
Pam 15:33
Okay.
Kim 15:34
So, thirty 13s is 390.
Pam 15:37
So, I've just written down 1 to 13.
Kim 15:40
Mmhm.
Pam 15:40
And then 3 to 39 because three 13s, like you said, is 39. And then 30 to 390.
Kim 15:45
Mmhm.
Pam 15:46
Okay, cool. So, we got 30 times 13. And then, but, Kim. Just kind of with the same pattern that we've kind of just been talking about, little over, a little under. I actually only want twenty-nine 13s.
Kim 15:58
So, I'm going to say thirty 13s was 390, so twenty-nine 13s is subtract a 13, so 377.
Pam 16:00
So, you think that 377 for twenty-nine 13s. And we just sort of went a little over and under. And I don't know if you remember, but that is the problem I finally landed on when we were doing the algorithm earlier.
Kim 16:22
Mmhm, yeah.
Pam 16:22
29 times 13.
Kim 16:23
Yeah.
Pam 16:24
Where we could skip count our way through that problem. Or use no reasoning if we're just singing the song or using a multiplication table. Whatever.
Kim 16:32
Mmhm.
Pam 16:33
Or we could kind of develop this Over Under strategy, and we could help kids. We can high dose kids with the pattern of... How did you say it? Using what we know, like a multiple of 10, like you found. Well, if I can find thirty 13s, bam, I can use that to help me find 29.
Kim 16:50
So, I just... You know, when I said I can't not like think about something...
Pam 16:55
Yeah, what were you going to?
Kim 16:56
...that makes sense to me. I was going to do 30 because 29 so close to 30. I was instantly thinking to myself thirty 13s is 390, so it's going to be around there. And I just want to point out that when we wrote the 29 times 13, like did the 9 times 3 stuff, the first row has 87 in it on my paper. Which is a partial, but it's not even like the biggest partial. It's nowhere near the 390, so when I...
Pam 17:22
Mmhm. Go ahead.
Kim 17:32
Go ahead. When I wanted to go with 30 times 13, I was like within the ballpark. I was in the hundreds that I was going to be. It's just the idea of magnitude and the idea of reasonableness. I knew that when I landed at 377 that it was near my estimation. And you can't do that when you're starting off with an algorithm and thinking about the one.
Pam 17:59
The smallest, least consequential numbers first. Absolutely.
Kim 18:02
Yeah.
Pam 18:03
Yeah. So, it was interesting. We just had Jo Boaler on our challenge, and she talked about her book Math-ish. And when you said that instantly you thought about thirty 13s, and so you knew it was going to be around 390, but you were too high, so it's going to be a little bit lower, you were finding an "ish" number.
Kim 18:20
Mmhm.
Pam 18:20
You found an "ish" number. Jo Boaler says, 'Find your ish number first." It's not about this whole other thing to teach kids about approximating. Or what's the other word I want? Maybe just approximating. Estimating, yeah. It really is getting kids to kind of like think bigger picture.
Kim 18:39
And it's also not find the answer of 377, and then round 400 or 380 or whatever else to fill in the blank.
Pam 18:46
Which is totally what I did as a student.
Kim 18:48
Oh, so many do, yeah.
Pam 18:49
Yeah. Well, and if it's all about answers, if the purpose of math class is all about answers, then why not do what I did?
Kim 18:56
Yeah.
Pam 18:56
Because I got the correct. I knew I was going to have to get the exact answer anyway, so I might as well just get it and round off.
Kim 19:02
Right.
Pam 19:02
But if the purpose is to develop mathematical reasoning, then all of a sudden, finding an "ish" answer is is part of what you're doing. I love that you point out that the very first steps of the algorithm immediately sends you to the least consequential part of it, and it doesn't help you "ish" the answer. It doesn't help you get a sense and a feel kind of for where you're going to end up. But reasoning does. And I'll also say that it's not enough to just tell kids to ish. I think we really do... Most kids need a higher dose of patterns to be able to have something to ish with. I think there's a lot of kids that will look at 29 times 13 and not be sure how to start to estimate. And so, then teachers come up with more squiggles to memorize about how to estimate (unclear).
Kim 19:48
Round this one to the tens. Round this one to the hundreds. Wait, how long? Yeah.
Pam 19:52
Yeah. And we're not about more steps to memorize and mimic. We're really about what are the major relationships that we can high dose.
Kim 20:00
Yeah, it's so good! Yeah.
Pam 20:00
High dose kids with, so that they can be thinking and reasoning the way, doing the mental actions that mathematicians actually do. Alright. So, Kim, the first trap that I think hits all algorithms is that kids could mimic those steps of the algorithm and get correct answers, but actually being trapped in less sophisticated reasoning than the problems are there for to help develop. Hey, one other quick thing that I want to mention is that a thing to think about with these algorithms is that every algorithm learned, if the goal is to get answers and I'm learning these steps of algorithms to mimic them, every time I learn, I rote memorize the steps of another algorithm, it's another thing for me to get confused about. It's another thing for kids to go, "Oh, my gosh. I have to memorize another one! I don't know if my brain is going to start leaking out! I only have so much. You know like, I need to quirk this ear because it's going to start. There's too much! I'm filling my brain with all this nonsensical shark dots that aren't connected. And when are we ever going to use this?" It's one more thing to get confused about. But every strategy learned makes the next subsequent strategy easier to learn because every strategy, not memorized but strategy developed, every set of major relationships where we actually build those mental maps in kids heads becomes a more robust map upon which we can build the next strategy, and it makes... Right?! It's so good! Strategy learning, relationship learning is synergistic. And algorithm learning is really just a dead end that traps kids. And less and less kids move on and succeed. Yeah.
Kim 21:51
Yeah. Alright, ya'll, get the Developing Mathematical Reasoning: Avoiding the Trap of Algorithms. Out now.
Pam 21:57
Bam.
Kim 21:57
Get yourself a copy.
Pam 21:58
And, ya'll, thanks for tuning in and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. And keep spreading the word that Math is Figure-Out-Able!