Ep 86: Are Algorithms Necessary?

Pam Harris  00:01

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

Kim Montague  00:08

And I'm Kim.

Pam Harris  00:09

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. We take the strong stance that 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  00:40

So if you've been listening to the podcast for a while, or you've heard Pam speak, then you know, we have some strong feelings or opinions or things. 

Pam Harris  00:49

I mean.

Kim Montague  00:50

So in today's episode, we're going to spend some time here talking about algorithms and discuss are algorithms necessary.

Pam Harris  00:58

Yeah, or even helpful or perhaps hurtful. Constance Kamii is a researcher who said the teaching of algorithms is actually hurting students. And I remember when I read that, early in my dive into research, and I thought, "Whoa, that is a strong statement to make. You better back that up." And she does. And we have carried on in that work. Not only asking are they particularly helpful, but necessary? Hurtful? Like, where do algorithms fit? To start that off, Kim, tell us about your son, Luke, the other day. This is amazing. Like, tell us about that.

Kim Montague  01:37

Yeah. So actually, the other day really close to here. Luke was working on something in geometry. And to be honest, I can't remember exactly what it was. It was like, arc length or something that I'm not sure. And he called me over because he was really excited because he had found some relationship on his own, whatever he was supposed to be doing. And he said, "Hey, look, I figured something out." And what he had figured out, after he called Pam and talked to her, was he had figured out a formula. He had solved the problem using relationship and later figured out, "Oh, it was the formula that was in his book." It is so interesting, right? 

Pam Harris  02:16

So cool.

Kim Montague  02:17

And what the conversation between me and Luke was that he had been told a lot of formulas and algorithms before, and then figured out why they had worked in the past. But this time, he messed with it and figured out before he even knew that that was a thing that he could get a formula for, right? And what he said I thought was really interesting. He said, "Why do they always tell us first and then we practice it, instead of helping us figure it out?" And he literally said, "It seems like they take all that learning away." And of course, I called you right away, because it was so interesting.

Pam Harris  02:55

It's so fascinating, that this high school kid, okay, he's kind of still middle school taking high school courses, like is clear that he learns more, that learning is happening when he's using relationships, and figuring out how those relationships connect together. And how we can use those to solve problems. He's clear that that's learning. 

Kim Montague  03:16

Yeah. 

Pam Harris  03:17

And he's able to look back on these other experiences, where they've handed him a formula or a rule and step by step procedures, an algorithm, and then said, go do that. And he's clear, then he's like he wants to understand. So he figures it out, then, you know, like, then he figures out the relationships. And it feels better to him the other way, it feels better. That's fascinating. So y'all, people ask me about algorithms all the time. And I'm finding it fascinating that we're actually differing on the definition of an algorithm. And so I want to sort of pause it. I want to, like tell you my position, that I think we're messing up when we aren't using the correct definition of an algorithm. And you might be like, "Pam, like, what? You're correct, no one else is correct?" I mean, it's the definition in the dictionary. And it's the definition that computer scientists who are sort of the algorithm gurus, it's the one that they use. So I'm going to fall on that side of the definition, most of the definitions I hear that are not what I mean, for sure, and it doesn't agree with kind of the computer science definition. Most of them are coming from math educators that are kind of using it in this loose term, this loose way, this loose sort of, you know, like student algorithms. You know, when students generate their own algorithms. When students create the, you know, they find patterns and they make some algorithms. They're using it in a way that I would say means strategy. So to be really clear, what we're talking about today is not student generated strategies where students are using relationships, to pull things together and solve problems. That's a strategy that's like how I'm thinking about what's happening and using that to solve the relationship. That's different than a generalized procedure that works for any problem of its type. So that a computer, let's be clear is dumb, because all it can do is what we tell it to do. It cannot intuit. It can't use intuition to make decisions, it only can follow steps. Therefore, we have to give the computer all the steps every time, because the computer can't choose when to Give and Take or when to use difference or removal, or when to find an equivalent ratio so that the problem is easier to solve. Or when to use the distributive property instead of the associate part. Like a computer cannot intuit that. We have to give it the steps to solve the problem every time. So algorithms aren't bad. Algorithms are brilliant. To come up with a good generalized procedure is hard. And so my hat is off to the eighth century, seventh century mathematicians that came up with these step by step procedures that freed up the common man to be able to compute. No longer were the only computers, the people who were the aristocracy, who were able to go to school to learn how to use an abacus. Now, anybody could learn the step by step procedures and therefore could be the bookkeepers. And could sort of do the computation that was necessary to do. That's like seventh and eighth century, y'all like early man, that was important. But that is not important anymore. Now, anybody can compute because we have technology. But even more important, we now know ways to help more people mathematize, not just mimic step by step procedures, but actually use relationships and connections do what mathematicians do. So I'm not dogging algorithms. What I'm suggesting is the teaching of math is not giving students a step by step procedure that will work every time for any problem of its type. That's what computers need. What students need is to develop intuition and use that intuition. So that the numbers, the structure influences how they solve the problem. So some of you have heard about my kids. I have a son, who is right now a junior, you might be almost a senior at Brigham Young University. He's a computer science major. He is specializing in artificial intelligence, and virtual reality. I love it. It's like, fun when my kids go do these crazy things. And so he's in school, and he's literally taking a machine learning class right now. And next semester, he has to take an artificial intelligence class. So like, it's all of this, this AI stuff that's happening. He's learning. He's already a programmer, he's doing all this stuff. And in this machine learning class where they're like figuring out how to help machines sort of learn. It's like, how do we program, what kind of algorithm can we put into a computer, so that it can kind of get better at doing things? But notice, you have to write the program, you have to write the algorithm in order to help the computer do things more efficiently. So he sent me this quote, just yesterday, he said, "My machine learning Professor calls the algorithm part of the machine, the monkey pushing a button." So if you talk about machine learning, the algorithm part of machine learning is the monkey pushing a button, unquote. And then my son continues. "Today's class was about recognizing all of the business/statistics domain knowledge that makes monkey pushing a button useful." Now, that was a little bit weird to understand. But basically, like the whole class was about if we're going to be writing algorithms, or excuse me, if we're going to be doing machine learning, in business and statistics, in order to have all that working, the monkey pushing a button part, we have to let's talk about making that useful. So y'all, what we don't want in our math classrooms is to create monkeys that push buttons. That's what Kim and I are talking about when we advocate not demanding students use algorithms and classrooms. And we're even going to go further and say, so don't even give them algorithms and suggest that it's a good way to solve the problem. Not for humans, it might be a great way for someone who can't intuit like a computer. But it's not such a great way for humans. There are better ways to solve problems than especially just mimicking a step by step procedure. And people always say, "But Pam, what if they understand the steps of the algorithm?" Y'all, it's not about understanding the steps of an algorithm. It's about mathematizing. It's about connect, making more mental connections, neural connections, between what you know, and some new sort of relationships so that then I can continue to use those and create more relationships. That's what mathematicians do. That's how they solve problems.

Kim Montague  10:14

So what I'm hearing you say that's applicable for teachers right now is that what we're against is handing kids an algorithm, practicing a few times and calling it math? That's not what learning math is.

Pam Harris  10:26

That's not what learning math is. That's not what mathematizing is.

Kim Montague  10:30

Yeah. 

Pam Harris  10:30

Yeah. Thank you for capitalizing that. Like to be helpful, we're not saying algorithms are evil or bad. We're saying there's a place for them. And it's not in the mathematics classroom. It doesn't help your students mathematize.

Kim Montague  10:42

Yeah. So we have another colleague, right? Who's got a spunky, spunky son.  Spunky's good. Yep. Yep.  And he shared a story with us that I would love for you to share.

Pam Harris  10:55

Yeah, so this colleague of ours was talking to us about the spunky kid of his. And they were going back and forth. I think it was about division, maybe I don't know, they were going back and forth about something. And the son has heard the colleague or colleagues talk about, you know, kind of what mathematizing is and stuff. And so the son has kind of a clue. This is I think, a fourth grade or fifth grade, or sixth grade or somewhere in there, somewhere sort of middle grades. So divisions like a thing, right, fourth, fifth, sixth grade. But this kid is clear, what it means to learn and what it means to just sort of mimic things, or at least is becoming more clear. And so we got this quote, this email the other day, who said, "Why is this person even teaching?" So talking about this kids teacher. "Why is my teacher even a teacher? Because she doesn't listen to what we're thinking? Like, it's just like, she wants to just tell us everything. And act." Oh, you know what? I'm not quoting it right? He actually said 'they' sorry. So, "Why don't they listen to what we're thinking? They just want to tell us everything and act like we don't have things already in our head." 

Kim Montague  12:00

Hmm. 

Pam Harris  12:01

I think that's such a fascinating quote. 

Kim Montague  12:03

Yeah, spunky alright. 

Pam Harris  12:05

I mean, spunky, like, they're acting like we don't even have things in our head. Just like, like. Now, that's a different sort of pedagogical bent. Like, I might be, pedagogy is not the right word there. What do I mean? It's like there are different theories about learning. And so you might be the person who says, "I can unzip a kid's head and pour knowledge in. They are a blank slate." We don't agree with that. We agree with the sort of bent that we start with stuff. Like kids have, just like this kid says, "I have things in my head. Why aren't they asking me about what I know, and helping me get from where I am, what I know, to that thing that they're trying to teach?" I feel like you want to say something, a minute ago, sorry.

Kim Montague  12:47

Well, I was just going to say we've had this conversation with a colleague. And we know that this kid has had a few experiences in previous grades where they had teachers who had given experiences where they were used to talking about math. 

Pam Harris  13:01

Mathematizing. Relationship. 

Kim Montague  13:03

Sorry, yeah. And sharing like what they were thinking about. And so it's, I think there's just an experience this year for this kid who the teacher was a little more traditional, maybe. And was, is very in the school of thought, where like, I have to tell you all these things, and like, get through all this material. 

Pam Harris  13:20

I do, we do. And then you go do. 

Kim Montague  13:23

Yeah, yeah. And the kid has some feelings, right about that?

Pam Harris  13:26

I think it's fascinating that the kids clear. Yeah, like, "Why are they treating me like I don't have anything in my head?" They're just like, "Why is this person even teaching?" It's a little harsh. 

Kim Montague  13:38

Yeah. 

Pam Harris  13:38

"Why are this person even teaching?" Because this kid has had really good teaching and learning experiences and isn't this year. That's interesting.

Kim Montague  13:47

Well, and it brought us to the conversation that like if this with his colleague, like their parent, during the parenting situation. And so we were talking about if this resonates with listeners, as a parent, if your students have a teacher who is pretty traditional, that they could still talk about rich problems, and talk and play about real math at home.

Pam Harris  14:08

Yeah, so if you are that parent, you have sort of some influence over home, like you can still create that rich, richer experience with your students at home. And we would encourage you to do that. And, you know, it's kind of fun. We actually are gaining quite a homeschool following. So shout out to all of our homeschool parents that are listening in. Way to go. Nice. We appreciate you. We appreciate all of the teachers that are parents. And we appreciate the teachers that are listening to the podcast that are trying to make math more and more Figure-Out-Able. 

Kim Montague  14:40

Yeah, if you want to know how to teach so you are mentoring your students to be mathematicians, and you would like some help doing it, we have online workshops. And they actually start running today. 

Pam Harris  14:51

Today.

Kim Montague  14:53

Whenever you're listening, check out our asynchronous online workshops at mathisFigureOutAble.com/workshops.

Pam Harris  15:00

And all of you listeners that are in the workshops. I am thrilled to be able to interact with you in the message boards. And we're gonna have some live Q&A's. It's going to be a blast. Thanks for taking the workshop. So if you want to learn more mathematics and refine your math teaching so that you and students are mathematizing more and more, then join the Math is Figure-Out-Able movement and help us spread the word that Math is Figure-Out-Able.