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 are here to suggest that mathematizing is not about mimicking or rote memorizing, but it's about thinking and reasoning; it's about creating and using mental, mathematical relationships. That mathematics class could be less like it has been for so many of us and more like mathematicians working together. We answer the question: If you're not teaching steps in algorithms in your class, then what are you teaching?

Kim Montague  00:38

Awesome. So for the last few weeks, we've been asking you to submit questions that you want answered on the podcast. And today is that day. 

Pam Harris  00:45

Doo-ta-doo.

Kim Montague  00:46

So we decided to just grab a couple of those questions and spend some time on them to let Pam give you her take over a couple of episodes. All right. Are you ready?

Pam Harris  00:56

I'm ready. So -

Kim Montague  00:57

Okay,

Pam Harris  00:58

Bring them on! What questions are we answering today?

Kim Montague  01:00

So today, we got an email from Drew Purdue. And this is what Drew said, "Hi, this is probably not quite the kind of question that you're looking for. But I keep wondering - "

Pam Harris  01:11

It's exactly the right kind! 

Kim Montague  01:12

Right, if you are wondering. "So as adults, we often think about coins to help us solve math problems. Obviously, money is a great way to visualize decimal amounts. My question is, do you think that today's children will grow up using coins often enough for that to continue to be a relevant strategy for them? As our lives become more and more digital, should we imagine today's children using coins in their adult lives? Which leads to a larger question about strategies changing with the times? Thanks for all your work, the podcast is very inspiring to me."

Pam Harris  01:46

Well, Drew, thanks, we appreciate that. We're so glad that the podcast is expiring - I just said expiring -  inspiring to you. And we really appreciate you reaching out and a super great question. Let's get right at it. So I wanna talk about a couple things in your question. First of all, I want to point out: nicely done in recognizing how important it is for students or people to have experience with relationships. 

Kim Montague  02:11

Right. 

Pam Harris  02:12

In other words, for you to ask the question, we're not dealing with coins so much anymore, dimes, nickels, pennies, quarters. And so it's been a great model in the past, is it going to be a good strategy for students to use? And I'm going to say it's more of a model for students to use as they grow up, because they're not using coins as much in their everyday lives. So recognizing that it is the experience that we've had that has really helped us as adults, to be able to use that, it helps us think about decimals. That brilliant pick up on that that it is very important for us to have experience with them. This is one of the reasons - I'm gonna land here for a second - this is one of the reasons why I stress so much that I do not think it is the place of a mathematics classroom to drill the steps of an algorithm. 

Kim Montague  03:02

Right.

Pam Harris  03:03

In part because that takes time. In order for students to get good at the steps of an algorithm, we have to spend time drilling them, practicing them, doing problem over and over and over, the steps over and over, looking at their steps, tweaking their steps, making sure they remember the order of the steps, if the steps are really making sense for them. We need that time to be giving students experience with the relationships that are important. We need that time not to be drilling those steps, where I would submit we're wasting that time. We are not using that time to our benefit, if all we're doing is drilling steps of an algorithm. Because then what do we get? We get steps of an algorithm where kids are getting answers where they're not really thinking and reasoning. They're not building their reasoning for sure. They're just doing the same thing over and over again. Maybe they're getting correct answers. But y'all, if that's what you want, are correct answers without building reasoning, just use a calculator. We're spending way too much time at that. So we need to use that time instead to help students gain experience with the important relationships, therefore then those important relationships can ping. Ping, I'm saying ping like, ping, like in a submarine. When you have, "Just one ping." I'm trying to remember the movie Hunt for Red October where he goes, "Just one ping." When you see numbers, when you see something, what do you want to ping for you? In fact I was just telling Kim the other day, I was messing around with MathStratChat coming up with a new problem. By the way, I think carefully about those problems. And as I was thinking about the new problem, it was really important for me to know what 99 times 12 was. Well, Kim knows we deal with a problem 1188 divided by 12 a lot. It's a great problem. We give it to students all the time, we have video of kids solving it. And so because of that, I know that 99 times 12 is 1188. I know that relationship, I've done it so much, I have experience with it. Well having it ping for me while I was solving this other problems was really helpful. And I noticed that, like I was aware, "Oh, that was so helpful that that pinged." Now I'm not saying that everybody has to have that one. But there are some important relationships that we want to build in kids and experiences where it's at. We need time to build that experience. One other thing I want to mention about your question is, you said: "This leads to a larger question about strategies changing with the times", I want to actually parse out the word strategy just a little bit. In a prior episode, Episode Seven I think, we talked about models versus strategies. Very important. If you haven't heard that, go check out that episode. It's a very important distinction. However, today, I want to parse out the difference between different strategies, meaning, there are teaching strategies, how you teach the way you teach, the way you approach teaching, the way you approach students' thinking. And there are also mathematical strategies, the way you solve problems, the way you use relationships, the way you attack, the way you're going to think about solving a problem or attack using relationships to get further in what you understand or how you're thinking about a phenomena. Those are different. And sometimes I think we might be talking past each other when we use the word 'strategy', because we're not sure. So I gotta be honest with you. Drew, in your question, you said, "This leads to a larger question about strategies changing with the times," that could mean are we going to want to change our teaching strategies with the times, to which I'm going to say maybe. Maybe we're going to be able to learn from each other some new ways of tweaking Problem Strings to make them even better. New ways or new instructional routines, like Which One Doesn't Belong is a fairly recent addition to our repertoire of instructional routines, and I love it. I love especially when we do Which One Doesn't Belong. We'll put that in the show notes, where we, when we play it, we don't just ask kids to come up with one thing, one of the choices that doesn't belong. But where we ask them to think about a reason for each one of the four choices to not belong, that's when it becomes really powerful, right? Not that we demand that. But we challenged them to do that. And then we compare and there's like always pushing kids, encouraging them, challenging them to come up with more things. So will we come up with more teaching strategies? Yeah, maybe. I think another example would be Peter Liljedahl, in his new Thinking Classroom book, relatively new, where he talks about some teaching strategies about vertical non permanent surfaces and about visibly helping students see that we are choosing their groups randomly. Those are relatively new to me. And I like them, I think there's some really powerful things about those teaching strategies. Let's separate that then from the mathematical problem solving strategies. I think that the relationships that are important, aren't going to necessarily change. So for example, if we think about solving addition problems with whole numbers, or even whole numbers and decimals, I think the relationships of the friendly landmark numbers, and then either getting to a friendly landmark number or adding a friendly landmark number, like those aren't going to change. That those are sort of major relationships that we need to build in kids, for them to then be able to solve any addition problem that's reasonable solve without a calculator. Those relationships, I don't know that those are gonna change. I think the major mathematical relationships that are sort of inherent in the math - so for example, if I'm solving quadratic equations. The idea of knowing that those are connected to the x intercepts on a graph, the idea that knowing that those are connected to the zeros of the function, the solutions to that quadratic equation, that those mathematical relationships, those aren't going to change over time. Now, maybe we'll get a little deeper or better, or maybe we'll learn more connections. But I think the major relationships in the major phenomena, I don't think those are changing. We might disagree when I say, "I think I know the five major important multiplication strategies, that if you know these five multiplication strategies, you could solve any problems reasonable to solve without a calculator." We could maybe have a conversation about that. You might say, "Well, I have a different five, or I agree with four of yours. But this last one, I would tweak or maybe we call it different things." But the relationships themselves, those are inherent in the mathematics, those are inherent in what we're doing. I don't think those are going to change. So what I think you're asking more is now that kids have less experience with coins, and I agree with cash, like they just don't count back change like we used to have to do. I think that because kids have less experience with that, I think you're asking, Is it still relevant? Is it still helpful to ask them to think about money to help them to think about decimal relationships? And as in a couple of past episodes that we just did using money as a model to help them think about fractions and denominators that have factors of 100. To which I say, I think you're absolutely correct that we have less experience with cash and coins and counting back change. So what I'd like to do is give kids that experience, I think we need to understand money. I think it's really important that we understand money, maybe look at our national debt today and go, "Wow, we need to understand money better." Look at the rate of consumer debt. And again, maybe we need to understand money a whole lot better. So what I would advocate is doing work with money in mathematics, to help promote students being able to use those relationships mathematically, but also to help promote them understanding money. So to me, I totally agree with you, they have less experience, let's give them the experience and then use it for our benefit. I think those can have happen in tandem. I don't think we have to do one first and then the other. I think that can happen at the same time. Let me mention one other sort of thing that maybe it might give you a vision to what I'm talking about, a glimpse, a better visualization of what I mean. I used to be a graphing calculator expert. I kind of could still do that. I've turned my attention to other things. I used to do whole workshops on how to use graphing calculators to teach math better. I still think we don't take advantage of the power of visualization like we could. So I would highly recommend that if you get a chance to take a good workshop on that, by all means do. As soon as I come out with my high school workshops, which we are working on right now, then you'll see some examples of that. But it'll be embedded in teaching the math. It'll be less of like, here's how to just use technology, it'll be like, here's how to teach linear functions. And of course, we're going to use technology as a tool to do that. The point I'm bringing up here is early in my career, when I was sort of just doing more of how to help teachers learn how to use graphing calculators to teach high school math, I was having a conversation with another colleague of mine. We were both T3 instructors for TI. T3 stands for teachers teaching with technology. And a new calculator had just come out. And we were talking about did we feel comfortable doing a workshop on that new calculator? You know, what, how did we feel about that? And I found it interesting that he said, "Whoa, I'm not ready, I'm not ready at all." And the more that we talked, it became kind of about the button pushing. It became about like, where would I find that thing? And I found it noteworthy, like he was already better at that specific new calculator and where the buttons were, and how to kind of access things. But I was actually pretty okay with doing the workshop that I was going to do with that new calculator, because I knew I was going to do the same kinds of things, aka teaching strategies that we brought up a little bit earlier. So the teaching strategies weren't going to change. How I exactly did them with that technology, I might need to like figure out where the button was or, you know, like, what's the keystroke sequence. But to me, that's not important. Honestly, kids these days are going to be able to help me figure that out. They're so intuitive. Right? Get the kids involved. Kim, how many times lately have you said to your kid, "Hey, help me with this on my - "

Kim Montague  13:08

Oh, yes, more than I'd like to admit.

Pam Harris  13:12

Because you got kids that are messing with it. And they're really intuitive. And they play with it. And so then you do your strengths, which is helping me help teachers, and you pull your kids in when you need a little help with technology. That's not important to you, right? So you're spending your time doing that. So in the same way, I'm still an expert on helping teachers realize how to teach using visualization, using the power of technology to do things quickly, and to notice patterns. I don't need to be, I don't think, the expert on exactly the buttons to push and what the keystroke order is. Kids can help us with that. Teachers, especially if you're my age or older, maybe even a little bit younger than me, I think we need to be comfortable kind of being in front of kids going, "Hey, let's learn together. You guys are better at tech maybe and I have the math knowledge. But maybe we can learn both of those together. You're going to teach me some things about math. And I'm going to teach you some things about tech. But we're all in this together. And we're all going to continue to learn." So hopefully, that helps a little bit about whether we could use coins and money as we go and whether teaching strategies are going to change over time. Regardless, I hope the podcast is still a little bit inspiring to you.

Kim Montague  14:27

Excellent. Fabulous. So we just spent enough time for one whole podcast with one question. So we have several other questions that we're going to tackle in future episodes. Thanks so much for all the questions that you have sent in. Remember to join us on MathStratChat on Facebook, Twitter, or Instagram on Wednesday evenings where we explore problems with the world.

Pam Harris  14:49

Yeah, if you find the podcast helpful, please rate it and give us a review. That way more people can find it wherever they get podcasts and we could spread the word even farther that Math is Figure-Out-Able.

Kim Montague  15:00

Keep sending us your comments and suggestions. We love hearing from you.

Pam Harris  15:04

So if you're interested to learn more math and you want to help yourself and students develop as mathematicians then don't miss the Math is Figure-Out-Able Podcast because Math is Figure-Out-Able!