Pam Harris  00:00

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

Kim Montague  00:07

And I'm Kim. 

Pam Harris  00:08

And we make the case that mathematizing is not about mimicking steps or rote memorizing facts. Y'all, it's about thinking and reasoning, about creating and using mental relationships. We take a 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  00:37

So last week, we started a conversation about what makes a rich task rich. And of course, we have plenty of things to say about all the things so we decided that we would pick up where we left off and talk some more about rich tasks this week.

Pam Harris  00:51

Yeah, so if you haven't listened to the last episode, check it out. Because we go through some lists and describe some things and ways of thinking about what makes a rich task rich. Today, we'd like to give maybe some examples and further exemplify kind of what we talked about with those lists. And maybe also talk a little bit about what we think rich tasks or not. So I'm going to start by talking about, I had a conversation once with Glenda Lappan, who I give a lot of credit to her and Betty Phillips and the whole Connected Math project team for creating, I think, some wonderful rich tasks. And one of the things that she and I talked about was that they really tried for contexts that made sense to students, things that students could relate to. They actually gave students surveys, to find out what students were interested in, what they thought about, what things that could help them sort of realize the math. And then they tried to stick those contexts where they made sense as they created rich tasks, not in a fake kind of weird way. But I think they did a fantastic job of doing things with pizza and video game reaction times. I love that and bicycles, shoes, clothes, amusement park rides, Olympic times and scores like just to name a few of some of the wonderful contexts that they used that could help make those rich tasks, where they help make the math realizable. Not that it has to be real world. But just things that kids were interested in that would sort of perks of interest. And I honored the fact that they were actually thinking about not just real world, but things that would be sort of interesting to kids. And then let's be clear, they did not make them too hokey, not too, like try to force the math into the context. Often they sometimes had contexts that were more whimsical or more, sort of fantastical. Sometimes they had them purely mathematical. So it's not about making everything real world. But I can honor the fact that they thought about if there were ways they could use things that were that that interest teenagers, like young middle school kids. I thought that was pretty cool.

Kim Montague  03:01

Well, not only did they think about it, but they actually surveyed kids, right? That's really cool. And it's not, you know, you're talking about that, it makes me think about how not too terribly long ago, one of my sons came home and his teacher tried really hard, right, to make the assignment be like something that kids can connect to, by putting her students' names in the worksheet. And so it was like, you know, my son's teacher, it would have his name and one of the problems. But, you know, you run into that risk of, like, what happened with my son is that he was like, "This is a problem about me, but I totally do not like the thing that I'm supposed to be talking about me liking." I was like, "Oh, sorry, buddy. Sorry."

Pam Harris  03:47

It happens. Yeah. So it's a little less of that maybe and a little more of like, if we could put something in a context that might make it relatable for kids, that that might be a thing to try if we could. An example that I'll use from my work, I've said before, I'm a curriculum designer, I think that that's a very interesting place to be in education because I have to think about a lot of things and think about the mathematical terrain and how to make a rich task rich. And I was seeking for, so we mentioned maybe a little bit before, that we're creating right now, a Building Powerful Linear Functions high school workshop. It'll be an online workshop. It'll launch in the fall of 2022. Very excited about it. I think it's some of my best work. And as we were writing and piloting and testing and rewriting and revamping, I became really clear that a lot of work that I had been doing and working with people was having kids write functions, for discrete situations. I just got kind of mathematical idea there. So hold tight. A function is by nature often continuous, so it should ideally represent contiguous data, not broken up data, or not discrete data. Discrete data is like individual points of data that doesn't tend to, like time is continuous, right? Distance and measurement is continuous. But we could have other data like numbers of dogs, numbers of pets that you have that would be more discrete data. And so often we were doing things with discrete data, that we were having kids write continuous functions to model that data. And that could be an okay thing, if there are trends that we want to look at and some things. But I really wanted the first time that I was going to have a group of Algebra 1 students write a function, I wanted it to be for data that was at least felt continuous. So I was seeking for a measurement context. Like I really wanted to have a measurement because measurement is continuous, right? Depends on how precise you want to get with the measurement. And so I thought really hard about that. And y'all I worked for quite a while looking at what context would be somewhat intriguing to kids. I had Glend Lappan on my mind. And what could be something that kids could relate to, but the math could be realizable through. And then I realized, "Oh, I could do it with frozen yogurt." That was a thing that was popping up around. We had one just in our little town where you could go. You take the cup, put it underneath the spigot thing, and you could put as much of that yogurt in, and then you could go to the next and put in as much. You could choose different kinds of yogurt. Then there was this whole topping bar. And then you paid by weight. And so if you've ever had an experience with that, there was a, it was a really nice context. Where we could talk about the price could be dependent on the cone that you put it in, whether it was a waffle cone, or a dipped, waffle bowl or not dipped waffle bowl. Those cost different amounts, or if you just put it in a cup. And then how much yogurt, that was sort of the fixed cost. And then how much yogurt that you put in the cup would depend on how much your yogurt would cost. And so that was a scenario situation, that I was like, "Oh, this could work." But then I had to think about equity. I had to think about the fact that we would have students conceivably, who didn't live anywhere near a frozen yogurt shop. And what could we do to make sure that that scenario, that context was relatable enough for students. So we shot this really cool video, where down at our local frozen yogurt shop where the owner talked was all about all the math that she had to consider is she priced different toppings and the way that they were priced. And zeroing out the scale when they measured the, or when they weighed the yogurt. It was awesome. It was a great conversation. So we shot this video. And so one of the rich tasks that I use is this idea of writing a function for the cost of frozen yogurt, depending on how much frozen yogurt I put in, depending on whether it's a cup or a bowl or a dipped waffle bowl. And so then we thought if we show this video, we give kids this experience where they get to feel that, then we thought we had written what could be an example of a good, rich task. And I'll leave maybe the end of that, the way that we got them to write that function is really kind of clever. But that's an example of some things that I thought about while I was creating a rich task. What could be a context that could be intriguing for kids and make the math realizable so it fit the mathematics. But also what's a way that then we could give kids experience so that we had equity, that we had everybody could dive in, everybody could be able to make sense of what was happening in the scenario.

Kim Montague  08:39

And it sounds like you spent a lot of time thinking about that, right? It's not like a super quick, just like whip something up?

Pam Harris  08:47

It was no, definitely not. In fact, I tried several different things. And then pilot tested several different things. And then once we sort of decided frozen yogurt was the thing, then we pilot tested different ways of giving kids experience and also different ways of asking the questions to make sure that what we got was kids actually thinking about, "Hmm, well, I could solve that. But I don't know how to, I don't know how to solve that one, because you're like, you haven't told me how... Oh, I could represent all of them." And that's becoming an increasingly important conversation for me in high school math, is how can I represent all of the scenarios? Or how can I represent all of the possibilities, then becomes that linear function. A linear function represents for any x, what would the outputs be for any x in this scenario? So that became a really important question that we found after researching and pilot testing different ways of writing that rich task. So Kim, if that's a way of thinking about what rich tasks are, what are they not? What are some things that we agree which tasks don't have? 

Kim Montague  09:54

So you know, I was thinking about this not too long ago, actually, because we like to think about problems right? And so if they're kind of fun or interesting, but they don't lead somewhere, you talked about the fact that you love that they lead somewhere in last week's episode. But if they're fun and interesting, but they don't kind of mathematically go anywhere, that's not going to be a rich task, right? They don't have the extensions that I love, then it's also probably not a rich task. And the reason I was thinking about this is because we have a teammate who loves to throw out some problems in our communication, we slack, but our communication. And they're fun to do and think about and talk about, but we don't consider them a rich task, because they're not really connected. They're kind of in isolation. They don't really go anywhere. It's just a single problem that you might tinker with a little bit. And they're just kind of lame. That's where it is.

Pam Harris  10:47

So to be clear, we're not saying don't ever do those, don't ever throw them out, don't work with students. It's just that when we talk about curriculum design, and we talk about how you sort of set up a unit of study, and we say, "Oh, like it should be centered around rich tasks." What we don't mean are those things that you just said. Things that are aren't extendable, that they don't have, they don't lead somewhere. They're fun in themselves. But when we want to kind of focus or center our sequence of tasks around a rich task, that's not what we mean. 

Kim Montague  11:22

Yeah. And it's probably not something, a rich task might not be something that you do every single day. Because they're not the easiest, like you mentioned earlier, they're not the easiest to write or to find, right? It's not like you're going to do a fresh, rich task every single day. But you might have a sequence where a rich task is the center. And you might do some problem strings leading up to it. You might back out of it with a problem string or some other kind of routine.

Pam Harris  11:47

And then we really like those rich tests that you can build from, right? That have like a second step to them. Like,"Ooh, what if we..." and I think we'll mention a couple examples of those in just a minute.

Kim Montague  11:58

Yeah, so I know, you know this. But Cathy Fosnot knows 10 day units are some of my very favorite rich tasks. When you and I were writing it was a lot easier to spend the time to come up with really rich tasks. And so I don't, I never recommend to teachers, like, "Oh, just go whip some out real fast." What I recommend is that they use and take kind of their discernment to parse out, is this that I'm finding a rich task? Is it going to be the best use of my time with my students, rather than to create their own. Although I have seen some people come up with some really good ones. But Fosnot's 10 day units are definitely at the top of my list. Rich.

Pam Harris  12:38

Cathy Fosnot is a master at writing incredibly, what's a good word? Just well written, well planned, well developed, extendable, that go somewhere, they have, they are replete with patterns. I think I have learned so much from doing her rich tasks and thinking about the sequencing that she's done with rich tasks. And really, my hat is off, I think she's a master curriculum designer. And so some of our favorites, we really like her sub sandwich investigation. It's so well done. I've tried to do kind of a version where I tweaked the numbers and the numbers she chose are so well chosen that there's just some really cool things that can come out of it. And the more that you do some examples of, I'll tell you a few more examples of hers. But the more that you do, rich tasks like that, the more you realize, oh, like it's, this is how you can pack lots of different patterns. Because you have a meaty enough like the sub sandwich, which we've done in episode, we'll put that in the show notes. We talked about her sub sandwich rich task in one of our episodes early on. So we'll put that in the show notes if you want to listen to that one. Also, Measuring for the Art Show, I think is super brilliant about helping kids conceive of and kind of create the open number line as a model of themselves. Also, there's we call it the 'Turkey Investigations'. But I think her ten name is called 'The Big Dinner'. This is one that has a, it's a really good example of kind of a follow up, where in The Big Dinner, the teacher buys a turkey. And so it's all about finding the cost of a turkey. And I think it's 24 pounds and the turkeys $1.25 per pound, and all the wonderful conversation that can come out of that. And then after you've had that conversation, and maybe followed with some Problem Strings and routines. Then after that, then there's this, "Hey, we bought the turkey. And we have this really great Congress about how you guys were talking about how you could figure out how much the turkeys are going to cost. Well, now we've got to bake the turkey. So now we're gonna bake that 24 pound turkey and you're gonna bake it at 15 minutes per pound." And now you have a quarter hour whereas before we had sort of $1 and a quarter in money and now you have a quarter hour. And all like, all of those relationships. So we've got money and we've got time and we've got quarters in both of them and. And you have a number of 24 that is so rich and, so many factors and ways of partitioning. Anyway, there's, it's just brilliant. That's an excellent example. So we really like that one. Also, her Exploring Parks and Playgrounds is another one that has some fantastic risk tasks in it. Really, I think, I don't know that I've ever met a ten day unit that Cathy Fosnot has written that I didn't like at least lots of it. So highly recommend that. Well just give a couple other examples, as you're looking. We think Bridges and Mathematics has some decent examples of rich tasks at the elementary level. For middle school level, we like to Connected Math Project. We also like Math in Context, which is out of print now, I believe. But if you ever get a hold of the Math in Context book, it has an excellent rich tasks, also some really good modeling in it. For the highschool level Discovering Mathematics, Discovering Algebra, Discovering Geometry, Discovering Advanced Algebra has some excellent and they call them investigations, rich tasks. Desmos has an excellent teacher tasks that I would call rich tasks. Open Up Resources at the highschool level that was created from the NVP Project. So the high school level Open Up Resources, I think has some excellent rich tasks in it. And maybe end this episode, talking about one more rich task that I've worked on, and I got the inspiration from the Discovering Math series, where they use a CBR. It's called a Calculator Based Ranger, but that's not important. What's really important is it's a motion detector. Where they use motion detectors hooked up to a graphing calculator, and you walk or move in front of that motion detector and the resulting graph of the distance versus time, the how far you are from the motion detector over time, that graph gets put into the, the data gets put to a graphing calculator, and it displays a graph. And when I first encountered that, there were some fine sort of activities that I did kind of as part of T cube, the TI outreach to Teach with Technology. And then we've developed over time a series of rich tasks that really take advantage of that experience. So think equity, if we put a CBR in kids hands, if we put that motion detector in kids hands, and we actually ask them to walk in front of it, and how your motion affects the graph, how you're actually making that graph change based on your motion. And then we have a series of rich tasks throughout high school, where we continue to then lean back on that experience, and continue to build richer and richer connections. And so as an example of that, a one of those rich tasks that was an extension of the sort of beginning task, where we asked kids how you walk affects the graph. And then later on, we then use that as an extension where we actually have kids write a function for their own motion. And they write sort of their first continuous function for their own motion and it's really cool. As I was working with that task, there was a follow up Problem String that I was writing. And as I was writing that follow up Problem String, I was kind of fussing around, we'd already pilot tested it a bit. And the results weren't quite what I was hoping for. And so I was kind of tweaking some things. And so, like, what I often will do, I just called my kids. I was like, "Hey, hey, you come here, you're, you're in geometry, you've had algebra one." And it was an Algebra 1, it was designed to be an Algebra 1 task. And say, "Hey, kid come over here. How would you solve this?" And that kid who had already had Algebra 1, he was like, "Well, I can think about it this way." And Kim, I don't know if you remember this, he actually used your strategy, which I'd never even thought of before. I was like, "Oh, okay, this is a real strategy. Kim use it. Now Craig is using it, okay. It's a real strategy." I was really thinking about it and fussing with it. Well, then I had a kid who was a senior. And he was taking Physics and Calculus. And so I was like, "Hey, how are you? How would you solve this problem?" And he paused for a second, Matthew is really thoughtful. And he looked at me and he goes, "Well, do you want to Calculus solution or Physics solution?" I was like, "Yes. Okay. That means this is a really rich problem." Because not only can we have an Algebra 1 solution, which Craig had used. But we can also have a Calculus and Physics solution that Matt was thinking about. Meanwhile, my sixth grade daughter was tugging on me. And she was like, "Mom, I want to try it." And I was like, "Whatever, whatever. Your sixth grade, this is an Algebra problem. Like walk away." She's like, "Mom, like I want to try." I was like, so in my head, I thought, "Fine. I'll give you the problem. You'll realize you can't do it. Then I can move on." Right? You'll quit buggin me. Good mom move, huh. Yeah. So I was like, "Okay, fine, fine. Here it is." And my sixth grade daughter was like, well, and then thought through the solution. I was flabbergasted. I was like, that is brilliant. Y'all. It was a transformation solution. It was a really like spatial solution that would lead to transformations and a way of thinking about transformations with functions. And I was like, "Yes, that is amazing." And I realized how I could tweak the task, so that more kids would would sort of be inclined to think the way she was. And that could be one of the strategies that we brought out. And in that moment, I was like, "This is a rich test." So if I just may, the Hallmark, a hallmark of a rich task is that you can use different thinking, that I could use more of a spatial approach to come at it. I could use more of a numeric approach to come out. I could use more of, if I was to describe, the other one is kind of, one of them was more of a step by step, numeric, more of a small step, numeric approach. And another one was more of a, you'd almost have to have more experience, but more of a global numerical approach. And that I can have a Calculus and Physics approach. Like the fact that you could come at it with different thinking, that can be a hallmark of a rich task. Because then we want to bring all those together so that we can then use different strengths of learners to come together and build all of us into more rich problem solvers. And if that intrigues you at all, y'all that series, that sequence of tasks is what I am putting out to the world, in the Building Powerful Linear Functions workshop, which will be debuting this fall. So fall of 2022. If you haven't checked it out yet. Check out the sequence of Problems Strings and rich tasks and structural routines that will be coming out to teach linear functions, Building Powerful Linear Functions fall of 2022. So excited again. We're creating it right now. And it's just like we're creating the workshop right now. We have the tasks already. So excited to put it out to the world. Y'all if you want to learn more math 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.