Ep 59: Why Not Manipulatives?

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 mathematical relationships; that math class can be less like it has been for so many of us, and more like mathematicians working together. We take the strong stance that not only are algorithms not particularly helpful in teaching, but that mimicking algorithms actually keeps students from being the mathematicians they can be. We answer the question, if not algorithms and step by step procedures, then what?

Kim Montague  00:49

In this episode, we are continuing the conversation about models and modeling, the idea that we use those words to mean lots of things, but that we need to parse out and be more particular about words, what we're saying, so that we can understand each other and therefore make more progress and making more math more Figure-Out-Able. So Pam, last week, we talked about a variety of meanings for the word model.

Pam Harris  01:15

Yeah, and the most important takeaway is this important sequence of modeling. First, it's sort of a model of the situation, as kids are in a context, we want to be able to model that situation and really understand what's happening. But it's an important model of that situation, because then that model of the situation then sort of morphs into where as the students are solving problems, then the teacher represents their thinking. We make a model of their thinking, make the thinking visible. And as the teacher makes that thinking visible as students say, "Oh, when my brain does that, it could look like that?" Then we transition to have students then use that model to represent their thinking. Then we ask students, "Hey, when your brain does that, how could you make that visible? How could you like, show me your thinking?" But that has to come after we've sort of used a model to represent their thinking, giving them ideas of how they could represent that. And then lastly, after lots of experience, and really using that model a lot to represent their thinking, then that becomes a tool for thinking. Then they can actually use that model in order to think. So that's kind of this really important sequence or progression of modeling that we like to think about when we work with teachers. Now, Kim, there are some popular progression videos out there. And even when I just said progression of modeling, you're like, "Oh, yeah, I've seen some progression videos. Those are cool." And they're very well made, and they sort of show a progression of manipulatives through the grade levels. And they end with, wallah, that's why the algorithms work. They're really well done. And they're entertaining. And they do explain well the algorithms and  maybe many people understand the algorithms for the first time because of these progression videos. Which might be noteworthy about the usefulness of understanding the algorithms. Just saying. So you might find it interesting that we don't promote these progressions, because remember, the algorithm is not our goal. Nope, nope, not our goal. So today, I'd like to discuss another reason that we don't promote those progressions. And it's about the way those videos use manipulatives. So today, let's focus more on one of those things we said in Episode 58 that we don't mean, we do not mean when we use the word model. When we talk about how important it is to use models in math class, in math teaching, we don't mean manipulatives, at least carte blanche.

Kim Montague  03:53

Manipulatives. So I was on Facebook recently. And, you know, I keep seeing these posts, lots of posts in different teacher groups about people changing grade levels or starting teaching. And they're asking things like, "What's the best manipulatives that I need for this particular grade level? Y'all give me some ideas?" Or, you know, "I got a couple hundred dollars as a new teacher, and I'm trying to hear from others what manipulatives I should get? What do you recommend?" And I find that so fascinating that that's the question that they're asking.

Pam Harris  04:26

Yeah, like Kim, let's talk about manipulatives. Give us some examples of what we mean - we have high school teachers on here - what do we mean by manipulatives? 

Kim Montague  04:35

Oh, there's so many. Right? So there's base 10 blocks. We've got counters. We've got two sided counters, little colored teddy bears maybe in younger grades.

Pam Harris  04:46

 Cute, cute.

Kim Montague  04:47

Yeah, there's unifix or centimeter cubes, dice, little clocks, tiles, algebra tiles, hundreds charts, students size hundreds charts, all the things, right, to get students holding, moving, manipulating.

Pam Harris  05:05

Manipulating, that's why they're called manipulatives. So I accept that they can sort of move. And all of that is in the name of helping students feel the math, do something with the math, experience the math, but ironically, not in the way that we mean, at least much of the time. Alright, so a quick story. When I was a new teacher, brand new teacher, I think it was in my like third year of teaching. I read a really well written, well said, President's Message from NCTM. President Gail Burrill. Y'all, she is one of those people who has really impacted the way I think about teaching and learning mathematics. And not just from this really well written President's Message but others. In fact, I'll give a shout out to Gail Burrill. I used to tell people, when I would talk about going to conferences, I would say, "Hey, when you go to conferences, don't look at the session titles. Go to the speaker index and look at who's speaking. Choose who you go listen to by who's speaking." And I would flat out say, there are five people that it doesn't matter what they're talking about. I will go listen to those people, even if I've heard them before. I learned something new from those five people. And Gail Burrill was one of those five. I have learned so much from her and y'all it's 5. It's like a shortlist. Now, that list has changed over time. You might be interested someday - hey, maybe we'll do a podcast episode where you can ask me my current list of five people are. Not today. So in that President's Message, this specific one that I'm thinking of, so this is a few years ago. When was Gail president? This might be in the 1990s, I think? She wrote this President's Message - y'all do you remember the old Wendy's commercial? So I'm dating myself a little bit. But there was a Wendy's commercial that was, "Where's the beef?". It was this little old granny and her little spiel was, "Where's the beef?" Because like Wendy's had more beef in their hamburgers, whatever. And so a lot of things at that point in time kind of took off on that slogan, and it was this, 'where's the beef' kind of thing? Well, Gail, as the NCTM president said that she'd been traveling around the country in her role as being NCTM President. And she was in all sorts of math discussions and classrooms. And she was seeing many, many manipulatives. But she worried if the mathematics were missing. And ready, the title of her President's Message was, 'Where's the Math?' And I remember like, 'Where's the math?' And I remember thinking, but isn't that what we're supposed to be doing? Like isn't it all about - because I was big into like CRA - and aren't we supposed to be making it concrete? Right? Because manipulatives, they embody the math, right? Or do they?

Kim Montague  07:54

Yeah, that's so good. So let's describe a little bit of what she might have been seeing. Right? Let's say it's time to be working with negative numbers. And so we'll see teachers take out integer chips, which are counters that are red on one side and yellow on the other. The CRA approach would have kids build the numbers, and talk about zero pairs. And then the kids start moving tiles around. And then the teacher has kids draw the same things. And then the abstract, right, then they write the numerals.

Pam Harris  08:26

Yeah, CRA. Concrete and then the representational and the abstract. Well said. That is a way, not so much what we suggest. So then what are we suggesting? What does it mean to do kind of more of what we are talking about? Let's do that with these integers, with positive and negative numbers. So we suggest starting from important contexts, because the first step in what we recommend is a model of the situation. And so we want to help kids really think about temperature, and elevation, and debt, and even American football. So I know we have lots of international listeners. And so American football, not soccer, because we have this line of scrimmage thing. So if you're not a big football person, you can probably leave out football, but that one works, as long as your zero is the line of scrimmage. We don't want zero to be the zero yard line on the football field. So what do we do? We talk about temperatures that rise and fall and get above zero and below zero. We talked about elevation above sea level and below sea level. We talk about debt, and we're sort of in the black and in the red and what that all means. And we ask kids really good questions and we let an open number line be a representation of the situation, of the context. So sort of our kind of beginning modeling sequence is that we use an open number line to represent the sequence. And then we ask kids really good questions. And as they think and reason about falling below, like I was 10 feet above sea level. And then I fell down in a cave 20 feet and so where am I now? And as kids think about that. Or hey, we're in crazy Texas where this last winter, it was 32 degrees, which is cold for us. And then all of a sudden it was like seven degrees. And so how far did it drop? And as kids reason about that, or like when I lived in Michigan, and it was two degrees above zero, and then it was 10 degrees below zero. Ah! Kidding? No, I'm not kidding. So as we have kids reason about those situations, we model their thinking. We represent it on an open number line. So as they sort of think about, oh, well, that would mean that I dropped, oh, then we write that as minus 10. Or we have an addition of a debt that we write that as plus a negative, a negative $50, if that was our debt. And so we represent those negative numbers and those movements on that open number line. We represent those things, and we represent their thinking. And then we ask them, "Hey, now that you've seen me represent your thinking, can you represent your thinking using this number line?" And then with lots of experience and lots of nudging, we help kids then move to where that open number line becomes a tool for them to reason with about negative and positive numbers, about integers. So I'm going to quote Cathy Fosnot, who said, "The mathematics is not in a model to be seen. The mathematics is in a child's mind. You can't just hold up a visual picture and expect that children see the mathematics in that model" Unquote. Y'all you might listen to that, again. It's like really important. We think that the manipulative embodies the mathematics, but it's only because we understood the mathematics enough to create the manipulative. We need students to build that mathematics in their minds. It's not just embodied in that physical thing. We can't just hand them the physical thing, and expect the mathematics to magically happen. Another quote by Koeno Gravemeijer is "the materials cannot transmit knowledge, the learner must construct the relationships.". 

Kim Montague  12:14

So good.

Pam Harris  12:16

And we agree. So let's talk about contexts. You might notice that I use some very specific context for integers. They need to be contexts that translate, in this case, to number lines. So it's not just about any random context to get kids interested. Sometimes people hear, "Oh, you've got to make it real so kids are interested." It's not about that. It's not about real world. It's about context that make the mathematics realizable. And then we can translate to those really important models. So again, the sort of modeling that we think about is that it's first a model of the situation. Then the teacher models student thinking so that then students can model their own thinking. And then we transition, and this takes time and experience, to a model for thinking or in order to think.

Kim Montague  13:12

So listeners, if none of this makes sense, think of an analog clock. The math is clearly in that clock. You and I know that, right? What the numbers represent and what's going on. We've already messed with those relationships in time. But do kids understand that analog clock by looking at it or even messing with one where the hands move? Or do they need to have experience with the relationships? Absolutely. Just having that manipulative does not ensure - we can't you just hand it to them - and ensure that they understand time.

Pam Harris  13:45

Especially if we just hand it to them and say, "Now do this. Now do that. And you got it down? Okay, got those little steps? Alright. Now you understand time." What? Ah!

Kim Montague  13:53

Yeah, so are we suggesting that manipulatives are bad? Absolutely not. But we so want to be particular about which ones and how and when we use them. And in the next few episodes, we're going to take a bit of a deeper look into what we do recommend.

Pam Harris  14:10

So if you want to learn more math and refine your mathematics teaching so that you and your students 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!