# Ep 99: If Not Algorithms, Then What? Finale!

May 10, 2022 Pam Harris Episode 99
Math is Figure-Out-Able with Pam Harris
Ep 99: If Not Algorithms, Then What? Finale!

In the last 5 episodes we've laid out the most important strategies for students to be able to solve for all four operations. So what? In this episode Pam and Kim solve a gnarly problem to demonstrate how mathematizing can empower you and your students, and open up to the world of Real Math.
Talking Points:

• A gnarly problem solved using Real Math
• It's all about efficiency and developing more sophisticated reasoning
• Partial strategies are a great starting point but not the ending place
• Kudos to Pam's research and thoughtfulness
• Why does Pam seem to focus on computation?
• What believers are saying
• Other ways to learn with Pam
• We can make higher math more accessible
• BONUS: Kim's failed attempt at the standard algorithm

Pam Harris  00:01

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

Kim Montague  00:07

And I'm Kim Montague.

Pam Harris  00:08

And we make the case that mathematizing is not about mimicking (unclear). Let me try that again. And we make the case that mathematizing is not about mimicking steps, or rote memorizing facts, but it's about thinking and reasoning about creating and using mental relationships. People don't realize we record this every time. We take a strong stance, and not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keep students from being the mathematicians they can be. We answer the question: If not algorithms and step by step procedures, then what?

Kim Montague  00:48

Okay, so today, we're wrapping up our series on the major important strategies for each of the four operations. And, listen, we have been giving you this amazing download, this free ebook that has all kinds of goodies in it. And we want to take just a minute to mention how we anticipate, how we think that you will use that or not use that. So what we've said before, add that we want to reiterate is we don't think that you're going to run off a copy of each of these for your students. We don't think -

Pam Harris  01:23

Kim Montague  01:25

Pam Harris  01:29

In fact, Kim, before you go on, sometimes when we have a teacher handout at a workshop, I will kind of jokingly say, "So right up on the top of this write: Not for student use."

Kim Montague  01:38

Yeah.

Pam Harris  01:39

And it always cracks me up a little bit. Because people will dutifully 'write not for student' - I just say that every time you know, like, here's your hint, this is not for student use. This is for teacher use. This is teacher learning, teacher understanding, and then you're going to use it as you teach students.

Kim Montague  01:54

Yeah, so absolutely. This is really about your learning. And it's really an opportunity for you to study and to say, "Man, which of these strategies do I find myself regularly using? Do I understand each of the strategies? Could I find problems where I would use each of the strategies?" I really feel like a way I would use it is I would study it. I'd put it away. And I would say to myself, "I'm going to dive into MathStratChat, I'm going to dive into looking for problems in the world." And then after a while, I think I'd pull that booklet back out. And I would say, "Oh, I use 5 is Half of 10 a lot. I use the Over strategy a lot. But I'm not really finding myself using Equivalent Ratio a lot. Let me re-familiarize myself with that. Let me put it away. Let me focus on that particular strategy." Go do some strings.  Yeah

Pam Harris  02:44

Go do a lot of Problems Strings with that with your students, so that you get used to numbers that it makes sense to do that particular strategy. You build those relationships in your brain, you're building in your students' brain. Ah, all of a sudden, that strategy is going to ping for you because those relationships are created now mentally.

Kim Montague  02:59

Yeah. Another note is that for teachers and leaders, we want you to build yourselves alongside students. this is not the thing where you have to own every single strategy. And then you can teach strategies through Problem Strings. You are -

Pam Harris  03:13

Not that. Not that.

Kim Montague  03:14

Not that. Right? It's not be perfect, know all the things before you can actually get started. But there is a little bit of some relief maybe about knowing just a bit more, at least owning the fact that there are alternative strategies so that you can be open to listening to your students, and figuring out which relationships are worth chasing down.

Pam Harris  03:35

Yeah, that's really well said, because when you know that these are the major ones, then you have sort of a sense as you're listening to students, Ah, yep, I have I, you're gonna be able to listen, you're gonna be able to hear what they're doing the relationships they're using a little bit easier because you have created those relationships. And let's say that you don't have them all, yet, as you listen to students, you're going to create them better in yourself. But you're starting with this notion that there are these major relationships. And like Kim said, then you'll have a better sense and you'll learn more as you go which strategies you want to chase down, which ones are going to be more fruitful for everybody to compare to and listen to and construct in class.

Kim Montague  04:17

Yep. And if you're listening for the first time, and you're like, "Wait, I don't know what you're talking about." You are gonna want to go right now to download that ebook, that resource, that fabulous download at mathisFigureOutAble.com/big because it is incredible.

Pam Harris  04:31

Kim Montague  06:20

Okay. Say all the things.

Pam Harris  06:24

Say all the things. Yeah, like get our, I have the other meaning of explicit in my head right now. What am I?

Kim Montague  06:29

Metacognitive.

Pam Harris  06:29

What do you say?

Kim Montague  06:30

Metacognitive? And then like, say it out loud.

Pam Harris  06:32

Yes. Be very metacognitive. Think about our thinking. Alright, so, Kim, here's your question. How would you solve this lovely, because it is a lovely, a lovely problem. It took me a minute to come up with this problem, because I really wanted one that the people are going to listen to and go blah. 49 times 67. Tadaa.

Kim Montague  06:55

Okay, I just just wrote it down. Oh, yeah, 49 times 67.

Pam Harris  06:59

Oh, yeah. Because Einstein said paper is for the things you need to remember. And your brain is for thinking.

Kim Montague  07:05

I'm thinking. Like, I'm looking at it for a second.

Pam Harris  07:08

Yeah.

Kim Montague  07:08

So um, this is me saying that. Okay.

Pam Harris  07:10

And that's legal, right? Let's be really clear, it is legal. I remember, while you're thinking, when I was the math student, I knew in my heart of hearts that what I was supposed to do was see a problem and instantly know what am I supposed to do? And we're saying: not true. In fact, what mathematicians do as mathematicians look at a problem and they consider, they play around with things. So Kim, what were you just thinking?

Kim Montague  07:33

I was looking at 49 and 67. And I was wondering which one I want to mess with?

Pam Harris  07:39

Say more.

Kim Montague  07:39

So I don't love 67.

Pam Harris  07:42

When you say 'mess with', I'm going to be really explicit. You were wondering, do you want to find forty-nine 67s or -

Kim Montague  07:48

Or sixty-seven 49s.

Pam Harris  07:50

Thank you.

Kim Montague  07:51

And 49 is near some things that I like. So I'm thinking that I want to mess with forty-nine 67s rather than sixty-seven 49s because I can alter that 49 to my liking. Like I can mess with it a little bit more readily.

Pam Harris  08:11

Okay.

Kim Montague  08:12

So I'm gonna go with forty-nine 67s. Okay?

Pam Harris  08:15

We shall let you.

Kim Montague  08:16

Okay, thank you. So I'm actually thinking, I want to do a hundred 67s. And I'm going to tell you why in just a second if you'll let me. So I know a hundred 67s because I know how to scale up by 10, and then by 10, again. I'm gonna say that a hundred 67s is 6700. So is that clear enough? Good on that one?

Pam Harris  08:40

Yeah, that's good.

Kim Montague  08:41

So then the reason I went to 100, is because I wanted to eventually get to fifty 67s. And so I went way over, so that I could do 5 is Half a 10. I get fifty 67s.

Pam Harris  08:54

That makes sense.

Kim Montague  08:55

So then I know that I need to halve the 6700.

Pam Harris  09:00

Because you halved 100 to get to 50.

Kim Montague  09:02

And I actually went from left to right there. And I halved 6000 to get 3000. And I halved 700 to get 350.

Pam Harris  09:12

So half of 6700 is -

Kim Montague  09:15

3350.

Pam Harris  09:16

So if I can pause you for just a second, because I wonder if listeners might find it noteworthy that as you're talking, I have written down a ratio table. So I have one to 67.

Kim Montague  09:27

Okay.

Pam Harris  09:27

100 to 6750 and 50 to 3350.

Kim Montague  09:33

Interesting.

Pam Harris  09:34

What do you have on your paper?

Kim Montague  09:35

I wrote down equations actually.

Pam Harris  09:37

I can see that. I've seen you do that before. So you have like 100 times 67 equals 6700.

Kim Montague  09:42

And then 50 times 6. Yeah. Okay. And as I'm saying what I'm saying I'm underlining to make sure that I've included the words.

Pam Harris  09:52

Alright, so you're now at fifty 67s.

Kim Montague  09:54

Yeah. And so then I know that I have fifty 67s and I'm going to just go down a 67, I put down on my paper.

Pam Harris  10:02

Why?

Kim Montague  10:02

I'm going to subtract because I need forty-nine 67s. And I already have 50 of them. So I just need 49 of them so I can subtract one 67. Yeah.

Pam Harris  10:17

From?

Kim Montague  10:17

From 3350.

Pam Harris  10:20

Cool. And how are you going to subtract 67 from 3350.

Kim Montague  10:23

Yeah, at that point, I subtracted 50. To Get to a Friendly Number. So I started with 3350. And I subtracted the 50. I actually wrote it on a number line, subtract 50 to get to 3300. But I still need to subtract, my total to subtract was 67. And I did 50. So I still need to subtract 17. And so I'm at 3300, and then subtract 17. And at that point, I just knew it was 3283.

Pam Harris  10:54

And how do you know?

Kim Montague  10:56

Well, yeah, so I know my combinations of 100 really well, because I play I Have, You Need a lot. But I could have subtracted 10 and then subtracted seven if I wasn't sure.

Pam Harris  11:06

So 3300 subtract 10 would be 3290. Subtract 3 more would be 3283. And you also could have subtracted from that 3300. You could have subtracted 20. What is that?

Kim Montague  11:21

You want me to do that? So 3300 minus 20, will be 3280. And then because I've subtracted too much, I did 3 too much, then I could add 3 to the 3280 and get 3283.

Pam Harris  11:35

So you can use sort of an Over subtraction strategy to do that?

Kim Montague  11:39

Yeah.

Pam Harris  11:40

So you're saying that forty-nine 67s is -

Kim Montague  11:46

3283.

Pam Harris  11:48

And in solving that you used the 5 is Half a 10 multiplication strategy.

Kim Montague  11:55

Over.

Pam Harris  11:55

Over multiplication strategy. And then you use partners of 100 to do the subtraction. But you could have done an Over subtraction strategy.

Kim Montague  12:03

Well, first I did Remove to a Friendly Number.

Pam Harris  12:05

True.

Kim Montague  12:06

Well, I actually, interestingly, because of what I had, I Removed a Friendly Number that also Got me to a Friendly Number.

Pam Harris  12:13

Sure enough, yep. Yep.

Kim Montague  12:15

And then we could have done Over subtract, or I knew partners of 100. Yeah.

Pam Harris  12:21

Yeah. So I don't know if anybody finds that interesting that we just kind of wanted to do a problem where we sort of parsed out all the things that we're thinking about. But Kim, let's be clear, like if you hadn't had to say all that stuff?

Kim Montague  12:35

Oh, yeah.

Pam Harris  12:35

Do you think? How many?

Kim Montague  12:37

It would be -

Pam Harris  12:38

Kim Montague  12:38

It would have been much quicker. Not that speed is everything. But you slowing me down to make me say all the things might feel to our listeners like, "Wow, that took a really long time."

Pam Harris  12:51

You really had to slog through all that. Wow, that was a lot of effort?

Kim Montague  12:54

But no, it would not have been.

Pam Harris  12:56

So if I may, in actuality, you looked at a problem and you said, "Forty-nine 67s. I'd rather do that than sixty-seven 49s." Because you knew 49 was almost 50. And bam, 50 is half of 100. So you found 100 of them, halved it to get 50 of them, took one of them away. And you're done.

Kim Montague  13:14

Yeah, yeah.

Pam Harris  13:16

And in reality, that was fairly, it was very efficient, and far fewer steps than if you would have done all of the single digit multiplication that would have been needed to do the traditional algorithm, with all those opportunities for errors. And also the opportunity that you could have been using Additive Reasoning, or at best Multiplicative Reasoning with single digits. And then instead, you're using Multiplicative and Additive Reasoning as you quickly or efficiently and using sophisticated strategies, were able to solve what I think is a pretty gnarly problem.

Kim Montague  13:49

Yeah.

Pam Harris  13:50

Nicely done.

Kim Montague  13:50

Can I tell you that you know you just said less steps? I think that if you had said, "You have to do the traditional algorithm." I would have been very uncomfortable, not because I don't love it. But because it has been so long since I have, that I would not have known if my answer was correct. Like, I feel so sure about the fact that because I'm thinking about the entire amount that I'm dealing with.

Pam Harris  14:16

The magnitudes involved.

Kim Montague  14:17

Yes, I would be very unsure whether my answer was correct using an algorithm.

Pam Harris  14:23

So it's not like you couldn't have done it. You could have done the steps and done all the things or whatever. But at the end of it, you're like, "If I would have made just one little error in any of that." The magnitude of your answer could be completely crazy off.

Kim Montague  14:35

Right.

Pam Harris  14:36

But instead, you're like, "No, I'm really clear. 6700, 3350, just one less 67. We're done.

Kim Montague  14:43

Yeah.

Pam Harris  14:44

Yeah. And that's fascinating to me that sometimes teachers get so upset about how their kids' answers aren't reasonable. And my suggestion is we've taken them out of reasoning land to begin with. If we stay in reasoning land, we're so much more confident that the answer is reasonable. Because we're reasoning the entire way along. Nicely done. Anything else before we do this next part?

Kim Montague  15:06

Nope, I think I'm good.

Pam Harris  15:08

So one of the things I wanted to emphasize as we sort of wrap up our series on this idea that there are these major relationships that if a person has developed the relationship so that these strategies are, they own these major strategies, then they can solve any problem that's reasonable to solve without a calculator. That, as we do that, I just want to bring out the point that too many places, what we see are people stop with the partials. We have programs out there and whole whole sort of fields of thought where, "Okay, give kids some conceptual understanding that we're going to do partial sums, partial differences, partial products, partial quotients." And once we've done those, "Okay, now our next step, bam, is the algorithm." And we either then see kids doing these really inefficient partials. Partials are good, and they're unnecessary starting place, but they can't be the ending place. But also, the ending place is not the traditional algorithm. The ending place is, really I should even say 'the ending place', because we just keep getting more and more sophisticated, that after we've got the partials, then we continue to build kids sophistication. We continue to help them think about bigger and bigger jumps, bigger and fewer jumps, bigger and fewer chunks. And as they do that their brains literally have more and more neural connections that then we can build on as we want to think even more multiplicatively as we build Proportional Reasoning.

Kim Montague  16:39

Yeah.

Pam Harris  16:40

So we cannot stop with just the partials. Go ahead.

Kim Montague  16:42

Yeah, and I want to give you a lot of props, a lot of credit to my thinking, and to a lot of people's thinking in that. Because I know that you spent a lot of time looking that at all that's, the kind of put out in the world. And that you have been very clear for a very long time that either the work was conceptually based and just kind of stopped at partials, or that it was all about the algorithm in the end, right? And I remember you, I mean, you've come to me and asked me how I've solved stuff. And you know, I've given you kind of my gut response. But then you have taken that and you've asked, like all the people. And you've known that we need to generalize and create a system and like this cohesive plan, so that there's no holes. Because your expertise, you know, started with high school. And so you knew and have known that, like all this work that we do, and the elementary has to be carried through. It's gotta make sense in higher math. And, you know, I appreciate what you've given to me, but also to the world in that arena.

Pam Harris  17:41

Kim Montague  19:58

Yeah.

Pam Harris  19:58

Kim Montague  24:12

Totally free. In fact, it is at mathisFigureOutAble.com/freeworkshop. It's so fantastic. You'll love it. And if you've already done DMR then you should join Journey, which is your oh gosh, what do I say.

Pam Harris  24:27

Our signature implementation support product. That is where we join with teachers and leaders all around the world as we are all working together to teach more and more Real Math.

Kim Montague  24:37

Yeah, you can find that at mathisFigureOutAble.com/journey.

Pam Harris  24:42

Kim Montague  26:30

This is like a bonus that we get.

Pam Harris  26:34

We shut off the episode and Kim keeps talking. She says to me, go ahead, are you, can you talk? You are crying laughing.

Kim Montague  26:41

I am. I said, "Do you want to know what I was doing while you were talking there at the end? I wrote down 49 times 67 and did the traditional algorithm and -

Pam Harris  26:51

You just decided to have fun. You're just gonna play with the algorithm for a little bit, because it is not really fun.

Kim Montague  26:55

Because to be fair, it has been - (pauses) - Holy cow - 15 years?

Pam Harris  27:03

A few years.

Kim Montague  27:04

I mean, maybe longer since I literally wrote a traditional algorithm for multiplication. So I'm out of practice. And I wrote 49 times 67 and got it wrong.

Pam Harris  27:19

Then I said, "We got to record this." So I hit the record button. Because I'm like, Kim, I want to hear. What did you do wrong?

Kim Montague  27:27

Here's what I did wrong.

Pam Harris  27:28

You don't remember a fact correctly? You wrote the zero in the wrong place?

Kim Montague  27:31

I did every algorithm, well, two things that I did wrong. One: when I said 9 times 7 is 63. In my head, I was thinking of the value 63. And you know what I wrote first? The six. I was thinking 63. And so I put six and I carried the three and in my mind right now I'm thinking, I wonder if that's why kids do that. I wonder if they're thinking 63. And they write 60, carry the three. I've never considered that before.

Pam Harris  28:02

That's fascinating. Yeah, absolutely. What was the other thing you did wrong? Um, so then when I wrote the zero, whatever, moved on to the next step, I carried a five. And you know what I did? I counted. I tapped it five times. Back to Counting strategies. So you didn't really do that wrong, but you found yourself doing something less sophisticated.

Kim Montague  28:23

I found myself like reverting to, I'm putting like, thought behind. 'Did I do that correctly?' and I went back to counting strategies. Absolutely.

Pam Harris  28:34

I think that's fascinating.

Kim Montague  28:35

Yeah. So I got 3256 for my answer, and that would have been a big fat red X.

Pam Harris  28:42

And no credit for all that you did right.

Kim Montague  28:45

And no understanding of what I did right or incorrectly.

Pam Harris  28:49

Yeah. No understanding that. Yeah. Your teacher would have just been like, "Try, do it again."

Kim Montague  28:53

Yeah.

Pam Harris  28:54

Yeah. And also, without any relationships, you wouldn't have had an idea if you were reasonable or not.  Right. Alright. Well, I don't know if you guys were listening to the end of this episode, because we kept talking. There's your bonus to the end of that episode. Standard algorithms stink, because they don't build mathematicians.