0:00 Hello, and thank you for listening to the mathematics teacher educator journal podcast. The mathematics teacher educator journal is co sponsored by the Association of mathematics teacher educators and the National Council of Teachers of Mathematics. My name is Eva Anheuser. I'm talking with Andrew ejack, who is a professor at Tufts University in the School of Arts and Sciences in the Department of Education. We will be discussing the article using coordinated measurement with future teachers to connect multiplication, division and proportional relationships, published with co authors Tori colo at Portland State University. sybilla Backman at the University of Georgia Dean Stevenson, at Prince William County Schools and Barack Obama's at the University of Georgia in their September 2019, issue of the mathematics teacher educator journal, we will begin by summarizing the main points of the article and discuss in more depth the lessons they shared in the article their successes and challenges and how these lessons relate to their work. Andrew, thank you so much for joining us. 1:04 Thank you for having me. 1:05 So I'll jump right into the first question. Can you give us a brief summary of the article including the results? Yes, the 1:13 article presents a way of thinking about multiplication that we have been developing in the context of pre service Teacher Education at the University of Georgia. And sort of the main thing that we hope readers will get is a vivid existence proof of future teachers being able to construct on their own sound explanations for division by fractions and solving proportional relationships topics, that research article after research article reports, teachers have difficulties with 1:53 Yeah, this is actually something I was thinking about. Because your article isn't like a one time innovation, it's kind of an overview of several activities within a course. So it's like you just said it's an existence proof that this can be done with some examples. And I'm just going to go ahead and say that there is a lot of stuff in the appendix of the article. If people do want to try out some of the activities, they're all included there. So let's jump into the next question. Who do you think should read this article or could use these activities? So of course, we 2:27 think everybody should read the article. I think we really have sort of two kind of overlapping audiences in mind, folks who identify themselves as a mathematics teacher, educators who are working with future teachers around topics related to multiplication, particularly factions, proportional relationships, linear function, and also mathematics, education, researchers more broadly. And of course, there are many, many people who sort of live at the intersection of those identities. 3:01 Yeah, so anybody who is teaching a course, where this is a topic in the course, could benefit from pulling some of those activities. What important problem or issue does your article address, the main 3:17 sort of problem that we're addressing is sort of this thing I alluded to earlier, which is that in the sort of literature on on teacher education, there's been a sort of several decades of research in which numerous reports say that teachers have difficulty understanding sort of how to explain division by fractions, how to just explained other topics related to multiplication. And we were interested in developing approaches to multiplication, which would provide sort of scaffolding contexts in which teachers could be able to achieve sort of success stories where teachers are able to sort of make sense out of these topics. 3:59 I was very intrigued when I was reading this article, because this, you know, this falls squarely into what I do regularly, which is teach content courses. And there was just so much in there, right? Like, it's kind of like a whole course, but also pieces that can be pulled out. So as you based your work on a lot of prior work in the field, right, like you said, multiple times, you want to pull out a few studies that you were building on, 4:29 of course, are these bodies of empirical studies, you know, the Deborah ball eating law stuff around, you know, teachers having difficulty construct us teachers, not Chinese, but us teachers having trouble constructing word problems that would illustrate something like one in three quarters divided by a half, we have studies, and I think that's sort of circa 1990, who I think was when Deborah ball was first publishing sort of stuff out of her dissertation about that. So we're sort of thinking about results as far back as that and also results There's a much smaller body, but a growing body of research on teachers difficulty with proportional relationships, they sort of know to cross multiply, but they have, you know, other kinds of troubles distinguishing things that are linear, but not proportional additive relationships, from multiplicative relationships, we sort of have all of that in the back of our minds. But the other thing that we have in that we are also thinking hard about are are sort of how we as sort of researchers in the field and teacher educators in the field conceptualize multiplication. So there's a lot of, you know, you can read a lot of studies in which someone will say, you know, multiplication is not repeated addition. And I think sort of within the research community, people understand sort of the limitations of thinking about multiplication, as repeated addition, and maybe the primary audience for those kinds of statements might be parents, might be school administrators might be some teachers who have, because this kind of meaning of multiplication is sort of in the bit embedded in the culture. Another kind of meaning is that fractions are, you know, so many pieces out of so many pieces, three out of four slices. So I think there's a kind of a well established sort of stands among mathematics, education, researchers and teacher educators about sort of trying to shift other adult points of view about these topics, we are actually trying to shift sort of the research fields perspective on these topics as well. So we've done a bunch of work kind of revisiting and kind of critiquing work by Bernie Oh, by Pat Thompson, various other people have sort of written extensively about multiplication Davidov, and have sort of developed an approach to to these topics, based in what we call coordinated measurement. And much this is sort of laid out and briefly in the article, and we're kind of using that sort of reconceptualization of the meaning of multiplication in order to sort of as the foundation for driving everything that we do, 7:04 can you explain that a little bit, this reconceptualized version of multiplication, what distinguishes 7:09 multiplication from addition is that you have some quantity, which we refer to as a product amount, and you are simultaneously measuring that quantity in two different units. one unit, we call a base unit, the other unit we call a group. So you might have some length, and you might measure that in inches. And you might measure that in feet. And so this is our idea of taking a given quantity and measuring it simultaneously in two different units. That's what is characteristic of multiplication. And if you adopt that point of view, it actually in other articles that we've published, like in JME, and Educational Studies in mathematics, it actually sort of propagates out and has lots of consequences for how you think about multiplication, across different things like area situations, equal group situations, rate situations, how you think about division, how you think about these things, is laying a foundation for proportional relationships, 8:11 I don't know you started with the feet and inches, if you could give an example, that would highlight how you describe the relationship between multiplication and division in this article. 8:24 So we think of multiplication, as far as division is multiplication with an unknown factor of pre service teachers often come sort of thinking that division is about partitioning. And that's also kind of an idea that sort of out there kind of in the culture, when you take something and you separate it out into equal sized groups, and people use the word division to refer to that activity frequently. But it actually causes all kinds of problems. In our work with you to teachers, we refer to that as partitioning that activity of separating things into equal sized groups. And division for us instead is is sort of the answer to one of two different kinds of questions. One kind of question is you have your product amount, and you want to know how many groups you have. So that would be measurement division, or quarter division, in sort of terminology that's often used in the field. And another kind of situation is you might ask, how many units do you have in one group? So that would be like that would be part of it or sharing division. But these are dealing with another and we're asking for different kinds of measurement questions when it's sort of a quote unquote, measurement or quoted of division type situation from questions that we're asking when it's a measurement, or sharing or part of division situation. But all the questions are about measurement. 9:45 One of the big unifying pieces of your work, I think, is this notion that multiplication and division are the same thing, but they're part and parcel of this sort of same structure that helps you kind of see a lot of mathematics connected that Maybe previously you didn't see as connected? 10:02 Yes. Could you 10:04 just for people who maybe haven't spent that much time thinking of it, just give us a specific example that you know of one of these equations and how it's multiplication and division, 10:14 we could do a, you know, an example with whole numbers, is that sort of what you have in the mind? Yeah, we can go back to the length example, I have some length, I have some group that I'm using to measure the length. So let's say, I know the length is and see if I get this right. So I know that let's say I know the length is like 60 inches. And I know that I'm measuring it in this groups. And there are five groups that also make that same length, then the part of oversharing Division question and the situation is how many units make just one of those groups 1212 inches make the group? So that's how you would think about sort of sharing a part of the division in this context or coordinated measurement? For the other kind of division question, you would know that you that you've got 12 inches in a foot, and you're again, measuring the same length, which in the end, we're going to know is 60 inches, when you're asking sort of how many of these groups fit into that larger length? That's measurement or qualitative division. 11:19 So one of the examples that you listed in the article that was powerful for me was this notion of, I think, teaspoons and cups, is that an example you guys use? And how you if you're in a recipe, and you have like a lot of teaspoons, you could say, well, this many make a cup, and then you could like think in cups versus in teaspoons. I don't know if that was exactly what you had. But I don't know why. But that really helped me think about this coordination idea of units. What research questions did you study? In this paper that you published? 11:57 The main thing? And it's stated somewhere? I think so it was not until maybe the middle of the paper was always good to bury your research question. In the middle of the paper, the main thing that we were asking you know, is if you provide this sort of instructional context for future teachers, in which you're using this explicit, meaning for multiplication, grounded and coordinated measurement, you start with whole number examples. You also provide certain we also use particular kinds of drawings, double number lines and strip diagrams that we use across all these different topics in the course. So there's sort of like a coherent backbone, we're always referring to the same meaning of multiplication, we're always using the same kinds of drawings, even if the question shear for multiplication questions to division questions, even if the kinds of numbers that we're working with from whole numbers to fractions will always have the sort of same core pieces that are showing up across, you know, week after week after week. And of course, if you provided that kind of support, could future teachers just figure out how to divide fractions by themselves? Could they figure out you know how to solve proportional relationships, again, sort of just generate sort of sensible solutions to these kinds of problems. 13:10 If we go back the example I mentioned earlier, with 60 inches and the feet, and we imagine this number line, so we would have to use a double number line, right, so we would have two number lines, and one would be cut into feet, and one would be cut into inches, 13:26 how you allocate the number lines depends on whether or not you're using doing measurement division or part of division. Okay, so if you're at least that's how we did probably other ways to do it, at least that's sort of how we do it in this course. So if you're our thinking measurement division, so you have this length, that 60 inches, and you want to know, sort of how many groups of 12 inches fit into that, then you would assign both number lines would be in terms of inches, okay, so one number line would be showing you the sort of the original 16 inch length. And the second number line would be shorter showing or showing you the sort of 12 inch segments that you're iterating or using to build up and trying to figure out how many of those are going to fit Okay, that kind of reasoning works if you didn't ask this both to sort of make the point here this kind of sort of approach to thinking about a problem works equally well with whole numbers and with fractions 14:19 Okay. Then for part of division, you have the the assignments for the number lines are different. One number line is the unit is inches. And the other number line the unit is groups. So in this case, sort of what the one hole refers to and each number line is different. You started alluding to this, but I mean, a follow up is so if we work with this equation that you illustrated earlier for relating multiplication and division, and these double number lines, how does this help students create meaning for division of fractions? 14:56 First of all, some familiarity with sort of how to think about double number lines as a resource for solving problems. We don't think that reasoning about number lines is self evident, self evident, or more easy. We think it's what it takes, sort of practice, and time to sort of build up familiarity with how to do that. And that's one of the reasons that's that's the main reason why we really, in terms of drawn models in our courses, we just have two that we use consistently, the double number line distributed strip diagrams, we use them for everything, because we sort of think that we're investing time and learning how to reason with those representations in the first place. And then once you've made that investment, of course, you want to reap the benefits of learning how to reason with them. And then the meaning of multiplication. The question is the same, you're still asking sort of how many groups do you have, if we're thinking about a measurement division situation. And in the first example, with the 60, in the 12, and the five, everything was in a hole number, so that you can ask the same kind of question with fractions instead, in terms of solving this problem with drawings, the extra twist is that you have to learn how to partition, you have to learn how to sort of take lengths, and break them up into equal sized pieces that allow you to sort of coordinate the partitioning of one thing with another. So coordinating partitioning, say, a cup in thirds with a cup and fifths. So we spend a lot of spend, we explicitly develop partitioning skills and talk about how you can use sort of factor product combinations. to partition wheat in the context of fractions, we, we spend quite a bit of time talking about how, in some situations, you're looking for the common multiple of the denominators. But for other kinds of situations, you're looking for common multiples of the numerators. There's a whole kind of facility with partitioning, that we think is really critical for being successful in this domain that really is not well developed, certainly in in school curriculum that we've seen. And we suspect in a lot of teacher education as well. So we invest quite a bit of time and sort of learning how to partition and that sort of helps you then extend measurement division double number lines with whole numbers into fractions, 17:20 as I'm listening to you and reflecting on reading the paper. And thinking about my situation where the whole numbers course is a different course from the fractions course, it seems like it would be really important to have some consistency across the courses with what models you're using to be able to leverage those in. When you hit the fractions course, that seems to be what you're saying, right? You read you just using two models, but you're using the models that you think are going to be most useful for the students throughout 17:53 links are very sort of fast. So you can kind of use some standard things that are letting so we don't have experience with pre service teachers, for instance, having trouble using lengths to represent volume. So you can have it you know, this problem about cups and teaspoons, and you could use lengths about that we've never sort of detected that using sort of a length quantity to represent a volume quantity has caused problems for teachers. And the nice thing about using legs is that you know, down the road, you want to sort of connect things to Cartesian graphs and slopes of lines and unit links for that. So we might as well just start there. 18:31 So you just hinted at, you didn't see difficulties with that. Did you run into any difficulties with the materials that you developed? Were there any challenges that you want to share? 18:43 Are you asking about sort of over the years of developing this approach, or you're asking about sort of pedagogical things, we've learned sort of typical stumbling blocks that future teachers encounter? 18:55 Yeah. Now I want to ask both of those. 18:58 So let's just answer with respect to development, with course, we spent a lot of time I guess tinkering would probably be a fair term tinkering with the meaning of multiplication. So in earlier iterations, this sort of sense of coordinated measurement wasn't developed as fully, but we read the daveed of work on multiplication and measurements, and a lot of that is restricted to whole numbers. Then we started really sort of thinking about sort of a measurement sense of fractions with respect to the multiple canned so you know, if you have four recipes, and each one requires three fourths of a cup of sugar, so we're sort of thinking about sort of the measurement sense of three fourths of a fraction there. So how many cups make the smaller quantities, the cup, the one cup is the measurement unit for the smaller quantity, and then we thought, Well, you know, why not just use that meaning for the multiplier as well. So that that was something that that got added later. And now we're kind of in this place where we're sort of all numbers in the multiplication equation are interpreted from this measurement point of view that has in the paper, we sort of explained that has sort of a lot of consequences that sort of propagate out from that. That's one thing that and of course, we tinkered with activities. Now we try it activities and sort of saw how future teachers engaged with them and sort of thought, Well, you know, we'd like to make various changes to this the next time around, the ones that are in the appendix with the article are ones that we have used several times and feel pretty good about in the sense that at least for us, we've been able to generate the conversation with future teachers using those activities in the context of our of our courses, things that teachers have difficulty with this measurement sense of fractions. So thinking about a fraction from a measurement point of view that takes a while to get used to. In this conversation today, we haven't talked about that too, too much yet. So with whole numbers, you have a sort of a sense of measurements, so you know, how many feet you know, make up that 60 inches, while there's five copies of that one foot. In order to extend the sort of measurements and stuff fractions, you have to keep that one foot as your measurement unit. And now you're asking how many of those make something small, it's a like an inch. So we're using one foot the longer piece to measure the smaller piece. And that's how you think about 112? How many of the big piece make the small piece so that that's a measurement sense of unit fraction. From there, you can iterate the unit fractions to get a measurement sense of 512 or 1512, proper fraction improper fractions would have you building that sort of measurement sense of a fraction of measurement sense of a number in general, that's not easy for future teachers necessarily. 21:51 Yeah. And you lay that out in the paper with some pictorial examples. So we started with this in let's kind of think about closing the article out by a giving you a chance to add things that we haven't gotten a chance to talk about yet, but you would like to add and be revisiting what the contributions are that you made to the field with this paper in particular, but potentially beyond that with your approach, and how other people might use this innovation. And I'm realizing I'm asking you three questions in one. But so let's just kind of think about, maybe start with re summarizing the contributions that you made. And then add in other things that we haven't gotten a chance to talk about yet that you would like to include, can I do those in the other worker? Yes, you may 22:46 one other pieces really critical about the contribution is that we have two different ways of sort of thinking about proportional relationships, one we refer to as multiple batches. And this is a perspective on proportional relationships, which is very well represented in the research literature on sort of children's thinking. So if you think of stuff like Joanna Bono's work on composed unit reasoning, we sort of form some sort of a batch and you make copies of it. So this is often a sort of a very consequential moment in children's thinking about fractions, because the sort of the first it is a way in which they can initially experience success, coordinating variation in two different quantities at the same time. So I might be sort of thinking that a third of a cup is a half of a recipe. And I sort of think of that as sort of, you know, a single thing, and I made copies of that. So I can ask how many cups I need for a whole recipe for two recipes for you know, five halves a recipe that we refer to that as multiple batches. This coordinated measurement structural sense of multiplication also reveals that there's another way to think about proportional relationships, which we refer to as variable parts. But which has essentially been completely overlooked by the research field, we've been writing about it, presenting about it at a variety of conferences, a mte, and ctmp, na p International. And this way of thinking about proportional relationships is supported with the strip diagrams. And so the idea, if you imagine sort of a three part strip and a five part strip, maybe the three part strip is red paint, and the five part strip is blue paint, all the convention is all the parts have to have the same number of gallons and them. And the way that you would get covariation is not by making extra copies of these strips, but by just putting more and more units inside of each part. So you sort of think about it almost like the numbers of parts are fixed, but you allow them to dilate. And this was his way of thinking about proportional relationships is novel for the fields. It's a very important contribution of the work and we sort of show examples in the paper of future teachers who are able to sort of Pick quite successfully, one of the places that this is really important is sort of a little bit more locally, it provides a very elegant and accessible way of thinking about why slope of the line is invariant. And more generally, it provides a very elegant way of sort of thinking about invariants for, for any kind of geometric similarity, anything that can be sort of conceived of as sort of geometric similarity, this variable parts perspective on proportional relationships, that sort of its natural habitat provides, we think, sort of new and really, potentially promising avenues and just thinking about central content. So that's another really sort of critical piece of the contribution. The first question you wanted to ask was what I've drafted by now. I answered your second question. First, 25:47 I think that it was related, like what would you like to say that we haven't gotten to yet? And what is your contribution to the field, which I think you answered both of those with your response. It seems like another contribution, at least I took away, I don't know if that was an intended contribution of this paper, it was this really unifies multiplication and division, this approach. That's something that's sometimes difficult for pre service teachers to see that those are not two separate things. 26:20 I'm glad that they came to clearly we, you know, one of the things that we really care about in this approach, and we think that sort of developing a coherent approach to all these different topics related to multiplication is critical for as a kind of a foundation for designing environments in which future teachers are able to sort of really make sense of this content for themselves. However, we don't, in our experience, sort of interacting with reviewers, I don't think so much for the mtps. But sort of for other kinds of plate experiences where we've been disseminating our work, we have a sense that it's not sort of a given that the field as a whole things that coherence across, you know, topics, waves multiplication is important or even a good idea. I think we're a little bit surprised by this. But the sort of idea that coherence is a desirable sort of thing to work towards, may actually be some sort of a conversation for the field to have, 27:20 that'll be a good conversation to have. All right, let's close out by just hearing you maybe talk a little bit about what you think the mte listener or reader can take away from this paper into their own practice. 27:38 And my sort of appeal to what I think is sort of a final paragraph in the paper. I think it's important when people read the examples of the future teachers, reasoning that we present in the paper along with the written work that they're producing, and which they're giving sort of very lovely explanations for things like seven divided by three fifths is 35 thirds and offering explanations for proportional relationships. Those accomplishments are sort of embedded in a pretty complicated sort, of course design, many of the features, you know, we've talked about attention to coherence using an explicit meaning of fixed, explicit meaning of multiplication across topics to support that coherence using the same kinds of math drawing the double number lines in the strip diagrams to support that coherence. It's a quite sustained sort of environment or complicated environment that we have created, in which these teachers are experiencing these kinds of successes. It's entirely possible that a reader might take one or two of our activities that are in the appendix and try them and they might achieve some sort of success. But whether or not they would achieve the same kind of success that we report in the article with those examples, is unclear to me. There really is sort of a push to sort of thing, you know, really systemically about sort of approaches to multiplication. 29:03 So maybe another message that your article sense is really about coherence, whether within this topic or other topics that as we think about developing courses, we don't think in terms of activities, but we think in terms of the whole course, and housings or maybe even across courses, in my case, for example. All right. Well, thank you so much for joining us. For further information on this topic. You can find the article on the math teacher educator website. This has been your host, Eva Sennheiser. Thanks for listening. And goodbye. 29:41 Thank you very much for having me.