Can we really be efficient in solving multiplication problems without algorithms? What about the really cranky ones? In this episode Pam and Kim discuss the quarter strategy- a clean, efficient, figureoutable way of solving traditionally difficult multiplication problems.
Talking Points:
Click here for more information on "Using Quarters" https://www.mathisfigureoutable.com/quarters
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Can we really be efficient in solving multiplication problems without algorithms? What about the really cranky ones? In this episode Pam and Kim discuss the quarter strategy- a clean, efficient, figureoutable way of solving traditionally difficult multiplication problems.
Talking Points:
Click here for more information on "Using Quarters" https://www.mathisfigureoutable.com/quarters
Check out our social media
Twitter: @PWHarris
Instagram: Pam Harris_math
Facebook: Pam Harris, author, mathematics education
Linkedin: Pam Harris Consulting LLC
Kim 00:01
I'm telling on you.
Pam 00:03
Hey, fellow mathematicians! Welcome to the podcast where Math is Figure-Out-Able!
Kim 00:07
I...
Pam 00:08
I'm Pam Harris. No!
Kim 00:11
Already today I'm a mess.
Pam 00:13
Whoo!
Kim 00:14
I totally just found the intro, and I was like, "Oh, find the intro! Find the intro!"
Pam 00:20
And you're going to say, "I'm Pam."
Kim 00:21
Yes. Well, I read... I... It's fine.
Pam 00:24
Okay, who are you?
Kim 00:25
Hey, Pam. I'm Kim.
Pam 00:27
Hey, Kim.
Kim 00:28
Hi, Pam.
Pam 00:28
And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. Ya'll, we can mentor students to think and reason like mathematicians do. Not only are algorithms not particularly helpful in teaching mathematics, but rotely repeating steps actually keep students from being the mathematicians they can be.
Kim 00:53
You know, I'm blaming the fact that I got these nice new headphones, and I'm still thinking about how cushy they are on my ears compared to the old ones I had.
Pam 01:03
I'm so glad we treat you well here at Math is Figure-Out-Able that you have cushy. I will tell everybody. With her cushy earphone, she just sent me a picture of them, so I can get the cushy earphones. Because one of my least favorite parts about recording podcasts is that my ears always bug me by the end of (unclear).
Kim 01:21
I'm also going to tell you. People can start looking at the clock. That we said 20 minutes. We're going to aim for 20 minute. It will not happen. But I'm looking at the time right now.
Pam 01:30
Kim and I are cranking out the Problem String books, and so we have to limit the amount of time we spend recording podcasts today, so.
Kim 01:37
Yeah.
Pam 01:38
This is going to... Alright, so, Kim, let's dive in. Go, go, go.
Kim 01:40
Okay. Alright, alright. So, I am going to take a second for review. FBOHannon said, "I happened across Pam and Kim two years ago after 20 years of teaching math in grades two through eight, and they were looking for a better way. Since then, I've listened to every single podcast, participated in every challenge, and try every MathStratChat. I've learned so much these past couple years. My 13-year-old son is usually in the car with me and we'll listen to the podcast together. I have him hit pause when Pam throws out a question, and we work through it together. His thinking has improved as well. Just recently, I had to take the math practices for grad school, and I found myself using ratio tables and reasoning through questions. Ya'll, I aced that test. Perfect score." That's exciting! Oh, wow! Nice! So, many fun things. Hey, and actually, that is really cool. When I took the GRE it was the same kind of thing. Like, I was like, "I have no idea." Yeah. I'm just, "Let's take it cold. Let's see what happens." And yeah, it went quite well because I was able to use what I knew and not rely on memorizing something from high school. So Math is Figure-Out-Able!
Pam 02:49
Oh, that's amazing. (unclear).
Kim 02:50
Thanks for listening.
Pam 02:51
Yeah, thanks FBOHannan That's, yeah. Nice. Thanks, Kim, for sharing that...
Kim 02:57
Yeah.
Pam 02:58
...review.
Kim 02:58
I love finding those.
Pam 02:59
And, ya'll, thank you for sharing your reviews and giving us... What do you call the five star thing? What's that? (unclear)
Kim 03:08
Rating?
Pam 03:08
A rating! There we go.
Kim 03:10
The words are fun, though. It makes my day when I get to go look for something like that.
Pam 03:13
And both of those help other people find the podcast because that's how people... Well, you know, when they're looking for a podcast, if you have rated it, given us a review, then more people will get a chance to see it, and we can spread the word more and more. Yeah, so thanks a lot. Okay, so Kim.
Kim 03:27
Yeah.
Pam 03:29
Today we're going to talk a little bit about a particular strategy. But I wanted to give a little bit of history. So, one day... Well, so many of you know that Kim was a third through fifth grade teacher who actually taught my personal kids. So, my sons had Kim. And at one point... You know, we learned. Like, we've told everybody that I did all this research, and then I dove into my kids classrooms, and we experimented, and learn, and, you know, collected data. And we've learned so much. Well, in the midst of that, one of the things that we learned is that we had to get the kids more sophisticated, that we had to help them become more efficient. If we weren't going to teach the steps and drill the steps of an algorithm, and we were teaching the kids to think and reason, we couldn't leave them being too inefficient, too unsophisticated. So, in the midst of that, while kids were really probably a little bit still less sophisticated than we now know that we would wanted them to get, you had my fifth grade kid. And I'm not sure actually if it was you or Het. Monica Hettenhausen.
Kim 04:35
Oh, my fav. She did science and social studies, and I did math for both of our classes.
Pam 04:40
Okay, so Craig would have had you for math in fifth grade.
Kim 04:43
Yep.
Pam 04:43
Okay, cool. So, at that point, a lot of the students that were Craig...and Craig's my son...and his counterparts, were kind of doing multiplication with partial products.
Kim 04:53
Sure.
Pam 04:53
Maybe a few other strategies. But we weren't stressing the more sophisticated strategies yet. They were using Investigations in Data, Number, and Space. Which really kind of didn't have a hierarchy. It was kind of like, "Whatever kids did, goes," lot of emphasis on partial products. And it was kind of this sense that all strategies were equal. Whatever kids did, as long as they were thinking, it was okay. So, then, Craig moved to sixth grade. I was not working with the district at this point. And there was a brand new to district teacher who was teaching Craig. That's who Craig had in sixth grade. And she was brand new to the district. And so, since I was no longer working with the district, and the district made some, I would say, poor choices to not replace me with someone who could train the teachers, this poor teacher got students in her class who were all using reasoning to solve problems, and she didn't recognize it. She had never been trained. The only way she knew how to multiply was the traditional algorithm. And so, it kind of freaked her out.
Kim 05:59
Yeah.
Pam 06:00
Now, unfortunately, my kids district tracked. And so, at this point, Craig is in sort of the upper track. And this teacher had that class, and then a bunch of general, regular tracks. And so, in the regular tracks, she said, "Whoa, Whoa, Whoa. I don't even know what's going on here. And enough of you are taking too long to do these problems. And enough of you are not getting right answers. I'm going to teach you quote unquote, the way to do it." She forced the traditional algorithm on them. With Craig's class, she said, "Ya'll are getting right answers enough, and you're quick enough, that I'm not going to worry about it. I have other things." Because the upper track had to get a lot more math in. She said, "I'm not going to about it now because you're getting enough right answers. I'm just going to dive into this rest of the math that I have to. I have to shove all this math in." Because they had to do a year and a half of the standards. But after they took the test, Craig came home. And he said, So, "Mom, you're actually going to have to teach me the traditional algorithm for multiplication." And I said, "Uh, why?" And he said, "Well, my teacher said that we can't use our baby strategies anymore. And so, she created this worksheet. And she put these problems on here. And she taught us the algorithm. And we now have to do the algorithm. We can't do baby strategies." (unclear).
Kim 07:10
I remember you telling me about "baby strategies".
Pam 07:13
Mmhm, mmhm.
Kim 07:13
And a being a little...
Pam 07:15
Yeah., Yeah, well, hang on. Because he said, "We're too inefficient, Mom, with our baby strategies. We need to be more efficient." Kim, the first problem, handwritten. This teacher created a handwritten page of problems that are, "If you do your baby strategies, it will take too long. It will be too inefficient." Now, I'll give her some credit that she chose some problems that if you do partial products, you're going to have to do all four partial products. And when you do the addition, it's going to be kind of cranky. And the first problem on that page was 75 times 36. In that moment, I said to Craig, I said, "Hey, Craig. You know what? Before I... Okay, you're telling me I got to teach you the traditional algorithm. You're going to have to do it on this page. Before you do that, I'm just really curious." So, listeners, maybe think about 75 times 36 just for a second. I said to Craig, "I'm really curious. What do you think about when you see 75?" And he goes, "Quarters." And I said, "I'm really curious. If you're thinking about quarters, could you think about one-quarter of 36?" And he goes, "Well, yeah. A quarter of 36. It's like 36 divided by 4, so that would be 9." And I said, "If a quarter of 36 is 9, what is three-quarters of 36? And he goes, "Well, if one-quarter of 36 is 9, and I need 3 of them to get three-quarters, then that's 27." And I can kind of... Yeah, three-quarters of 36 is 27. And I said, "So, then, what is 0.75 times 36?" Now, in that moment, ya'll, he paused.
Kim 08:57
Mmhm.
Pam 08:58
So, on the paper right now I have three-quarters of 36. I literally wrote the word "of" 36 is 27. And then, I said "What's 0.75 times 36?" And he paused. And he looked at me. And he thought. And he goes, "Seriously?" And I said, "What are you thinking about right now?" And he goes, "Well..." I'm not actually sure the correct order of exactly what he said. But he's hot at this point. He's thought. He's kind of sad. And he's mad. And he looks at me, and he goes, "Baby strategies?" And I said, "What are you doing? And he goes, "Well, 0.75 times 36 is the same as three-fourths of 36. That's 27. And so, I can just scale up from there to find 75 times 36. That's 2,700." And I said, "Tell me what you're thinking." And he goes, "Baby strategies?" Kim, I got to admit, the initial problem was actually 76 times 36. Sorry, I'm telling this story not very well today. The initial problem was 76 times 36, and then everything follows from there. I said, "What do you think about when you hear 75?" And he said, "Quarters." And I said, "The quarter of..." Blah, blah, blah. So, at this point, he said, "If 0.75 times 36 is 27, then 75 times 36 is 2,700." And I said, "Here's the hard part, Craig, then what's 76 times 36?" And he kind of laughed at me because he knew I was joking. And he goes, "I just need one more 36. Good heavens, Mom! That's just 2,736!" And then he said, "Baby strategies?" And I said, "Yeah, tell me more." And he goes, "Mom, she said to us, "Don't use your baby strategies. Let me teach you the traditional algorithm. So, she wrote down..." He's telling me this story. She wrote down 76, and then 36 underneath it, and then wrote a little times sign and drew the line. And then, she drew this kind of loopy thing around the 77, and then down to the six in the 36, and then back up. And he said, "That's supposed to be a turtle." And she even drew an eye on the turtle. Now, listeners, if you can't picture the turtle, if you really want to, you could go Google turtle multiplication, and you'll see.
Kim 09:08
Mmhm. But please don't.
Pam 09:42
Yeah, please don't. But it's basically this attempt to make the steps something that you can memorize with this kind of picture of this turtle. People will say to me, "That doesn't look like a turtle." I know. I know it doesn't look like a turtle. But it's a good faith attempt to help kids memorize the steps. And he said, "So, she drew the turtle, and you do this multiplication, and then you do that multiplication. And then, Mom, are you ready? And then, the turtle lays an egg. And that's where the magic 0 comes in." And then, you do the rest of it, and all the blah, blah, blah, And he goes, "Baby strategies? Baby!" I mean, this kid. You have to... Part of what's interesting about this story is Craig is so even tempered. He's so...
Kim 11:50
Yeah, he is.
Pam 11:51
Like, he's just this chill, nothing gets him upset. He has a newborn, and I have seen him take that screaming newborn, and go, "Dude, it's okay! Life's alright. Life's good." Like, "He just smiles at him, and he goes "Hello!" And this kid's like screaming at him. And he can just calm him down because he's just this even tempered. Kim in that moment, he was hot, "Baby strategies!" And then, he said to me, "All year long, we have been rote memorizing stuff about fractions, decimals, and percents, and doing all these rules and procedures. We could have been doing work like this. This is the kind of work we should have been doing. Baby strategies!" He threw the pencil down and stalked off. It was so unlike him. He was so fired up about it. Kim, that story is burned into my head.
Kim 12:43
Yeah. Well, and you're right. Craig is super even tempered and super, super kind. But it probably like shook him that he spent all that time thinking that math as he got older was going to be different. You know (unclear)
Pam 12:58
Less Figure-Out-Able. Mmhm.
Kim 13:00
Mmhm. Yeah, I mean, that would be, you know, a quite a big awakening to realize that.
Pam 13:07
He kind of wasted his time, I think, is what he was feeling like. You know?
Kim 13:10
Sure.
Pam 13:10
Yeah. And he was kind of like, "Wait, what?" Like, "Oh, man. I could have..." Yeah. And he was very polite to the teacher.
Kim 13:18
Oh, yeah, sure.
Pam 13:19
And he then proceeded to never learn any traditional algorithms from then on out. And he just continued to reason and think. And he is the kid who graduated a year ago from Brigham Young University with a BS in computer science. He works as a computer coder today. He's phenomenal. His interest is in artificial intelligence. Never rote memorizing traditional algorithms at all.
Both Pam and Kim 13:44
Yeah.
Kim 13:45
You actually had a couple of problems not too long ago. Well, it's been several months now. Where it was like 16 times 75.
Pam 13:55
Okay.
Kim 13:56
And I have a sixth grader... No, I'm a liar. He's seventh grade now. (unclear). He's a seventh grader. And I was super excited because, you know, he has done a lot of work with me with different strategies, but really isn't getting that at school. And so, because he's not doing a lot of multiplication stuff right now, I still, you know, will give him some MathStratChat stuff. And he's super interested and reminds me most of the time. And you had a problem that was like 16 times 75. And I just wondered. You know, it had been a while since we talked about that. And I was really excited to see that he also thought about a fourth of 16, to get three-fourths of 16, to get 75 times 16.
Pam 14:41
That's awesome.
Kim 14:42
Yeah. And I think what's really important about your story and, you know, how Cooper applied it is that they they weren't just following a bunch of problems that you asked, you know? Cooper's like applying the idea that Craig, I'm sure, is applying also, that you you kind of awakened in him. "Oh, 75. I can figure it in this way." Of any 75 or any 76. It's not just like I have the answer to one specific problem at the end of the string.
Pam 15:12
If quarters ping for me, then I can think and reason using quarters. Cool. So, Kim, let's take just a brief moment here to develop. We probably have listeners who are like, "Wait, wait. What? I kind of followed what you did." Let's build that strategy a little bit using my favorite Instructional Routine and Problem String. Here we go. First problem, Kim. What is 20... Oh, actually. Sorry. What I'd like to do is... I'm remembering my plan here is I'd like to tell the listeners
Kim 15:39
(unclear) 20 minutes.
Pam 15:40
Well, and I want to stay in 20 minutes as well. Yes, yes, yes, yes. I want to tell the listeners kind of how I usually do this Problem String.
Kim 15:46
Okay.
Pam 15:47
So, I usually say what is 25 times 24? And then, I let people use whatever strategies they've got. I've usually worked with teachers or students enough at this point that some people will... Well, so listeners. Shut off the thing. Figure out 25 times 24, however you want to. And then, I will share a couple of strategies. Usually we'll share something with Doubling and Halving. Maybe a smart Partial product. Since it's 25, we could do something with Five is Half of Ten? And I'll share a couple of strategies, and then I'll say, "Hey, next problem. Totally unrelated..." And I'm not actually going to say what that answer is right now, Kim, if that's okay.
Kim 16:21
Sure.
Pam 16:21
I'll say "Totally unrelated. A new problem. I know Problem Strings are usually related, but this one's not one-fourth of 24." And people will say, "Oh, well fourth 24 is..." Go ahead, Kim.
Kim 16:33
Six.
Pam 16:33
Six. And then, sometimes people will say, "Well, I thought of half of 24 is 12, so half of 12 is 6." You can also use Craig's strategy to divide by 4. Kim, I don't know. What do you do when I say a fourth of 24?
Kim 16:47
I...
Pam 16:48
(unclear).
Kim 16:48
No.
Pam 16:49
It's just sort of there.
Kim 16:50
Yeah.
Pam 16:51
Yeah, okay. You divide it by 4 maybe. Okay. And then, I'll say what's 0.25 times 24. And then, I'll pause. And people will pause just like Craig did. And then, they'll kind of crack a grin. Or they'll look like, "Hmm." And then, I'll say "What is 0.25 times 24." And they'll kind of smile, and they will say, "Well, it's also 6." And we'll talk just a little bit about the connection between a fourth and 0.25 and "of" and times. Just a little bit. And then, I'll say, "Hey, is there any connection to that very first problem?" Because the very first problem's answer was 600. So, they've got 25 times 24 is 600. A fourth of 24 is 6. 0.25 times 24 is 6. And then, we'll kind of talk. "Oh, you could just scale. You can just sort of scale." So, I'll say, "Could you have solved 25 times 24 using quarters? Could you have thought about a quarter of 24 is 6, and then since that's 0.25 times 24 is 6, then you could just scale both of those up and get 25 times 24 being 600. Huh, interesting." Next problem. Could I ask you something like 0.24 times 24? Now, depending on the audience we might pause a bit longer. But, Kim, how are you thinking about 0.24 or 24/100 times 24?
Kim 18:05
Well, I'm thinking about if the problem you just asked was $0.25, 24 times, then this is $0.24, 24 times
Pam 18:15
Nice.
Kim 18:16
So, I need $0.01 less 24 times, so that's $0.24. So, I went with $6.00 minus $0.24, which is 576. $5.76.
Pam 18:30
Or 5.76. Nice. Often people will say, "Well, that's just like, 0.25 times 24 is is 1 penny or 0.01 off of 0.24 times 24. So, they'll say, "Well, what is 0.01 times 24?" I love how you thought about it as pennies. Often people will sort of kind of think about what's a hundredth of 24, and they'll say, "Okay, there's your 0.24," and then subtract it like you did. So, 0.24 times 24 is 5.76. Nice. And then, I might say, "Could you think about 0.25 times 24? And go ahead, Kim.
Kim 19:10
So, it's that same $0.24 difference. So, a quarter times 24 was $6.00. So, another penny 24 times is $0.24, so I'm adding the $6.00 and the $0.24 to get 6.24.
Pam 19:26
Nice, excellent. And then, the next problem I might say how about 0.75 times 24?
Kim 19:34
Mmhm. (unclear).
Pam 19:35
(unclear) A little bit. Yeah, go ahead.
Kim 19:38
0.25 times 24 was 6. So, now I have 3 times as much because I have 0.75 instead of 0.25. So, 3 times 6 is 18.
Pam 19:50
So, three-fourths of 24 is 18. 0.75 times 24. Nice. And then the piece de resistance. I don't speak French. I don't know how you (unclear).
Kim 19:59
I don't either.
Pam 20:00
Is 0.76 times 24. 0.76 times 24. Mmhm.
Kim 20:05
So, that difference between 75 and 76, 0.75 and 0.76 is still... 0.24, got 1 penny more. So, I'm adding the $18.00 from three-fourths of 24 plus 0.24, so I got 18.24.
Pam 20:26
So, you just did double digit decimal multiplication, but thinking about quarters. Then, if we had to, we could also say what's 76 times 24?
Kim 20:35
And that would be 1,824. 1,824.
Pam 20:39
You scale it by 100, and there you go. One of the reasons that this strategy, I call it the quarter strategy, is one of the major strategies that I advocate students need to have experience with, so they build a relationship, so that becomes a natural outcome for them. One of the reasons that I advocate this strategy is because we need to be able to multiply 25 times stuff and 75 times stuff. 25 times stuff isn't usually too bad because we can double and half and pretty quickly get to 100 times things Doubling and Halving. But 76 times stuff, 74 times stuff, that's trickier. But if we've got this quarter strategy down, A, we're dealing with quarters, and we're making connections between decimals, percents, and fractions. But we also gain access to multiplying by certain numbers. And that's one of the things I did was looking at if we are not going to force kids to learn the traditional algorithm for multiplication, what are the ways... We got to make sure that we can hit all the numbers. We need to be able to multiply any number, any problem that's reasonable to solve that a calculator. Well, things like 74 times stuff and 76 times stuff, bam, we can use this quarter strategy to help us get there. And I would submit, more efficiently than the traditional algorithm. And building all that proportional reasoning at the same time. Double whammy, triple whammy. It's exciting.
Kim 22:01
Yeah.
Pam 22:02
Kim, one of the things that we've done is in our Problem String books, we created a shared vision about how we could develop the major relationships that kids need to be able to solve any problems reasonably will solve a calculator, and we thought hard about what the Problem String structures needed to be, how long to spend on a strategy. All of that kind of stuff is in our upcoming Problem String books. They are coming out soon. You guys are going to love them!
Kim 22:31
Yay!
Pam 22:32
Stay tuned. We'll announce on the podcast when you can get. They're gonna come out kind of piecemeal. But they'll all be out soon. So exciting.
Kim 22:39
Yeah.
Pam 22:40
Thank you for tuning in, everybody, and teaching more and more real math. To find out more about the Math is Figure-Out-Able movement, visit mathisfigureoutable.com. And thank you for spreading the word that Math is Figure-Out-Able!