Ep 2: Math Does Not Equal Algorithms!

Pam Harris  00:02

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

Kim Montague  00:08

and I'm Kim Montague. 

Pam Harris  00:09

And we're here to suggest that mathematizing is about thinking and reasoning about creating and using mental relationships. We answer the question, if not algorithms, then what? 

Kim Montague  00:22

In today's podcasts, we're going to talk about how Pam's work differs from others that you might find out there. There are a handful of people who are doing really good work around numeracy around student thinking and alternate strategies. Math looks different than it did when we were in school, and that can be confusing for parents and teachers. Pam, can you talk to us a little bit about how your work is unique? 

Pam Harris  00:43

You bet. So Kim, I have a story I'd love to tell you. I was at a large national conference. And this was a few years ago when the math focal points had just come out. NCTM had put out these math focal points. They were kind of short lived. They weren't really used. I think maybe towards assessment which is they didn't have a very long shelf life. But, when the focal points came out, one of the authors was a colleague of mine who I knew and they were doing a session at a large national conference about the math focal points. And at the end of the session, they had a large Q & A. So it's big ballroom, lots of people. And they had a question answer session at the end. And I raised my hand and an honest question I had was, "Why did the math focal points put so much emphasis on algorithms?" Now algorithms, to be really clear, to define an algorithm: an algorithm is all the steps all the time, that's kind of my informal definition. It's a series of steps. It's a procedure to solve every problem in the class. So if I have a, not a class of students, but a class of problems, so if I have a set of problems, a certain kind of problems, then we create algorithms that computers can use to solve any problem like it. And computers need to have algorithms because they're not smart enough to let the numbers dictate the strategy they're going to use. They just have to crunch the numbers. And so we have to give the computers all the steps, all the times because computers can't choose what to do based on the numbers. So at this point, the math focal points had put what I felt was too much emphasis on students using algorithms. At that point in my career, I had learned that there were other ways to think and reason. And I knew that mathematicians don't typically use algorithms when they solve problems. They don't mimic steps. They don't use all the steps all the time. They try to take shortcuts. They use relationships to solve problems. And so I knew that and I was really curious about why the authors would do that. Well, when I asked the question in that large Q & A with all those people in the room, this friend of mine up there kind of gave me the "thanks a lot look", and then gave sort of a politically correct answer about you know, algorithms are important in math and blah, blah, blah. And I wasn't satisfied. I'm still not satisfied with that answer. And so at the end of the session, I walked up to this colleague of mine and I said, "Hey, are we still friends?" And she said, "Yeah, yeah, but why did you do that?" Like, "Pam, you're secondary." Now at this point, we've started to walk through the conference. You can picture it's a big hallway, there's all the people and we're starting to walk through the hallway to the next session. All the people are walking, you know, it's a lot of people were walking.And she said, "Pam, your're secondary, you appreciate the fact that often we just need kids to solve the problem. We just need kids to get the answer in the most efficient way. We just need the kids to use the algorithm, you know that." And I said, "Wow, hang on a minute, the most efficient way? Like, what's 99 plus anything?" She kind of looked at me and I said, "What's 99 times anything?" She stopped in the hallway. Right then and there; all these people *crashing noises* kind of like plow into us from behind. And she looked at me and she said, "Is that what you people are talking about?" Now this is really noteworthy because this is a mathematician. This is a person who's written a book for elementary educators to become math teachers. She's obviously, she's a person that well received in the education community, knowing what she's talking about with math education, but didn't really understand what we meant when we say, solve a problem more efficiently. She actually thought that the algorithms were efficient. And until I said, "What's 99 plus anything?" She again, she just like, stopped in the hallway and looked at me. "That's what you mean?" Well, so let's talk about 99 times anything. Because this story illustrates to me what it means to be a math teacher out there today, and not really understand. Maybe we're not communicating. When people like me say it's not all about the algorithm. There are other things we can do in math. So I think my work is unique, because I have this bent on what mathematizing, what mathematicians really do, and it's not mimicking the steps that we learned as students. It's not just repeating a bunch of algorithmic steps to get an answer.

Kim Montague  05:06

So it's not about mimicking. Could you give us a little summary, if you could share with us a few bullet points. Tell us about your work in math education about teaching mathematics.

Pam Harris  05:17

Yeah, thanks. So math teaching is about listening to students. You've got to know your content and know your kids. Math is also not about mimicking a series of steps. It's not about repeating an algorithm. Mathematics is about letting the numbers and the structure dictate the strategy that you choose.

Kim Montague  05:38

Those are some really important ideas. Can we back up a little bit and I heard you say know your content, know your kids. Can you say more about what you mean by that?

Pam Harris  05:47

Yeah, sure. So there's a lot of really good people out there. And I think most of us when I talk to colleagues at the university, when I go to conferences, most of us agree on that sort of first point that I made that math teacher is about knowing your content, knowing your kids that we need to listen to students. We need to notice what students are thinking, we need to notice the way that they're attacking problems, we need to notice the strategies that they're using, we need to notice the way that they are thinking about relationships. There's a whole thing out there that the Drexel Institute came up with the notice and wonder that's wonderful. A lot of people have picked up on it. And so there's people out there that are doing rich tasks and three act tasks. But for the most part, all of that, all of those really good people are agreeing that we need to listen carefully to kids' thinking. But their end goal is still the algorithm. So they do a lot of really good work towards getting students to be able to estimate and use reasonableness and, get some number sense, and all that stuff. Like, of course we should do that. But their end goal is the algorithm, is getting kids good at repeating these steps in order to get an answer. Their goal is to get kids good at mimicking all the steps all the time, they're headed towards the algorithm. Not me. I'm not headed towards having kids just do this one thing all the time to solve every problem in that class.

Kim Montague  07:12

That's a pretty bold statement you're making that algorithms are not your goal.

Pam Harris  07:16

Yeah. And it's where I differ. So it's one of the reasons that we're talking about this on this podcast so early is to sort of say, my work isn't just about getting kids kind of good, at you know, like some number sense and some being able to, you know, kind of reason a little; nah like, that's actually my goal. My goal is number sense. Number sense isn't just this little thing about being able to estimate or being able to look at your answer and make sure that it's reasonable. Number since is a huge part of what it means to be a mathematician. That I use relationships and connections I know. That instead of looking at a problem and saying, "Oh, this is an addition problem, I must now do these steps" or, "Oh, this is solving a proportion, I must now cross multiply and divide." Instead, it's looking at the problem and saying, "Ooo, how are these numbers affecting me? What relationships do I see here? And how do I want to then let those relationships influence how I'm going to solve the problem?" I let the numbers and the structure dictate the strategy. It's not about one set of solution steps all the time. It's about ooo, based on what I see in this multiplication problem, ooo, I see these numbers, this relationship, I'm going to use those relationships to solve this problem. So, Kim, I happen to know you well enough that if I say, "Hey, give me an example of what you do." So I mentioned earlier 99 plus anything. So let's just pick an ugly number, like 47. Give me an example of what you do. If I said, "Hey, Kim, what's 99 plus 47?" How do you let those numbers influence how you solve that problem? 99 plus 47.

Kim Montague  08:48

Sure. So 99 to me is really close to 100. So I'm gonna think 100 plus 47, which is 147. And then since I only needed to add 99, I'm going to backup one, you get 146. 

Pam Harris  09:01

Cool, yeah, we call that the Over strategy. So you're going to add a little bit too much, and then you kind of have to adjust. So you might find it interesting in my university classes early in the semester, well all semester long I have my students do short interviews with people for part of their homework. And so near the beginning of the semester, I give them the specific interview assignment that they need to go find a couple people and need to ask them, "What is 99 plus anything?" And now I'm trying to help them become better questioners. And so when I suggest to them, when the person goes, "What? What do you mean by that?" Then they say, "Well pick an ugly number," kinda like I did with you pick 47. So what's 99 plus 47? If the person then still goes, "What are you talking about? What do you mean?" Or they take the 99 plus 47. And they line it up and they start to do the small digits first, and they add just those single digits together, and then they carry the one and then they add that this. Those are the steps of the algorithm. That's what a kid would do if they were going to do all the steps all the time. But for 99 plus 47, why would you do all those steps? And for heaven's sakes, why would you think about the tiny numbers first? In fact, research has proven that if I just give kids a problem like 99 plus 47, before we have taught them an algorithm, or without teaching them those steps, kids will always look to the big numbers first. It's natural for them to think about the big numbers first. It's not natural for them to think about the tiny numbers first. And so, if a person when they in this interview when they say, "Hey, so what's nine plus 47?" If the person begins to do all the steps, all the time. Or if they are kind of stymied, then I suggest to them that they kind of try to trigger some prior knowledge in that person. So I suggest to them that they go. "Ninety-niiiiiiiine." Because as soon as you do that, "Ninety-niiiiiiiine plus 47," then it sort of triggers in all of us when we have kind of done that counting thing, where we're like 97, 98, ninety-niiiiiiiiine, 100. And so, "Well what's ninety-niiiiine plus 47?" Often people will go, "Oh, well 99 plus one is 100. I was supposed to add 47. So now ninety-niiiiiine 100, I just still need to add the 46 left over. So what's 100 and 46, ah it's just 146." That's a great other strategy that you might use for that problem. That's an example of how Kim might use the Over strategy, I might use the sort of idea of getting to that next friendly number and adding what's left over. Either of those work really well and are far more efficient for a problem like 99 plus 47. So this friend of mine, when we were walking down the hall when I said, "What's 99 plus anything?" She could picture in her mind. Oh, those two strategies, and she could be like, "That's what you mean." Like that's an example of where we can be far more efficient than all of those steps. Now, you might be thinking right now, well, what about other numbers? We're going to hit other numbers. Today, we're going to do some kind of specific problems that are easier to think about how other strategies might be used instead of the algorithms. Kim? What if I asked you 99 times anything?

Kim Montague  11:57

So I was just thinking about the Over strategy for Addition and I thinking that I would also do the same for this problem. If it was a hundred 47s, then I would have 4700. But I don't want a hundred 47s, I only want ninety-nine 47s. So then I could just subtract 47 from 4700.

Pam Harris  12:15

Bam. And if you could do that subtraction, which will work on the subtraction, sure enough, I know you can, then instantly with just a couple steps, you've got ninety-nine 47s. You've done what with the traditional algorithm would be several steps like I'm suggesting in the teen steps, it will be a ton of steps to actually do all of them for 99 times 47. But you're suggesting you can just think about a hundred 47s, and get rid of just one of those 47s, bam. So y'all, I have a challenge for you. Go out and ask people, like literally go up to people. Take your significant other, your kids, anybody you run into and ask them, "What's 99 plus anything?" And then if they are like, "What do you mean?" Then you could say. "What's 99 plus random 47? What's 99 plus 47?" And if they pause, you know, like what or they start lining them up, then you can say, "Ninety-niiiiiine." And just see. Now pause along that way, because you might not need to even give them any hint. They might go, "Well, 99 plus anything, that's just 100 plus that thing minus one." They might have already some thoughts about how they could use relationships to solve 99 plus anything. And if they don't, I bet you could help trigger that prior understanding, that prior knowledge in them. And then they do. I want you to be interested in how other people are thinking about numbers. And this is a great one that almost anybody that you meet on the street can think about. And then we can become better listeners, and better understanders of what it really means to think and reason using mathematics. So, to recap about what makes my work unique. Math teaching is about listening to students. You got to know your content, but you really have to know your kids. You have to be willing to listen to what how they are thinking and reasoning about numbers, and use that to help them progress to become more sophisticated mathematicians. Mathematics is not about mimicking an algorithm. It's not about doing all the steps all the time. And what is math teaching? Math teaching is about letting the numbers or the structure in the problem dictate the strategy that you use to solve that problem. Alright you guys, I hope you've enjoyed joining us today. If you're go on to your favorite podcast site, and like the podcast, give us a review, we'd appreciate it so more people can see it. Check out the website mathisfigureoutable.com for more information. We'd love for you to join us on Wednesdays at Math-Strat-Chat on Twitter, Instagram, and Facebook. So if you are interested to learn more math and you want to help students become the mathematicians they can be. Then the Math is Figure-Out-Able Podcast is for you. Because math is Figure-Out-Able.