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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 Sennheiser, and I'm talking today with bourbon, Oberlin and Gil seren. A car from bogazici University, we will discuss the article exploring real numbers as rational number sequences with prospective mathematics teachers that was published in the September 2020 issues of the mathematics teacher educator, 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 the other work. Thank you so much for joining us. Thank you for taking us today. And I just wanna like point out that we are talking from Portland, Oregon, which is on the west coast of the United States, to Turkey. So this is gonna be our first international mte podcast recording. Can you just get started by giving us a brief summary of the article? Okay. In our article, we described an innovative instruction on real numbers, especially be constructed real numbers through rational number sequences by quantitative reasoning. That was the main point of our article. And we worked with prospective mathematics teachers, then we provide a classroom teaching experiment in our article with data from our prospective mathematics teachers. That was the main point of our article, what did you find, just really briefly, we'll get into it in detail later. Okay, the instruction was a four phase instruction. In the first phase of instruction, prospective teachers were able to define fractions as a quantity. And based on this idea, they further reasonable equivalent fractions through diagrams as a quantitative as quantities. And then, especially in phase one, we focused on what prospective teachers know about the basic mathematics concepts, such as rational numbers, fractions, and equivalent fractions, but our focus was on their reasoning quantitatively about fractions and equivalent fractions. In the second phase of the instruction. Prospective teachers work on two examples. One is seven over two, and the other one is 11. Over three, we selected these examples by purposefully, seven over two is our example for nonterminating expansion, decimal expansion, and a little over three is four, seven over two is for terminating expansion, and 11 over three is for non terminating expansion. In the first example, seven over two teachers, prospective teachers used three representations. One is diagram, the other is division algorithm, and the other one is long division. By using these three representations and shifting between these representations. They constructed this small representation of seven over two, using quantitative reasoning on diagrams. And then we continue with 11 over three, the same process of cubes, they again use three representations for these diagrams. The other one is division algorithm. And the other one is long division. They again, using reasoning to quantitatively they constructed the decimal representation of 1103. By using these three representations, they found that the using these two representations refer to the same number, actually, they get that idea. And then in phase three of the instruction, we continued with 11 over three example. And we try to construct the sequences that we can see equals the 11 over three number, we get the nested intervals by sequencing 11 over three, but again, by referring to the phase two, that in phase two, that the teachers get to decimal representation of 11 over three. Again, in phase three, they considered any positive rational number x, and they got a generalization that they can represent in a real positive rational number. True rational number sequences, one is increasing and the other is decreasing. In the last phase of the instruction, we focus on irrational numbers we use the example of square root of three. In this case, they use numbers

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Line, they placed the skirt of tree on number line by using nested intervals, these nested intervals are the terms of the sequences one is increasing, and the other one is decreasing rational number sequences. By increasing the terms of the sequence, they get the idea that the two sequences one is increasing, and the other one is increasing approach to the irrational number which is square to three. And then we ask them what they get at the end, they work for a positive rational number, they work for a positive irrational number. So, we can say that all positive real numbers can be represented through rational number sequences. That was the idea that they got at the end of the instruction. So, if I am trying to rephrase what you said that you developed a four step instructional sequence, to get your participants to develop the idea that all real numbers can be represented as rational number sequences. Yes. Okay. So I have tons of follow up questions before I jump back. But first, I wanted to take like a little time because I'm not super familiar with how things work in Turkey. So you said prospective teachers, can you give us a little background of what Teacher Education looks like and what grade levels are in those kinds of things, a teacher education in Turkey at universities actually occurs and it happens and any university having Faculty of Education Can I will say produce teachers, future teachers, and especially in our university, this happens in just a few universities in Turkey actually, what happens, there are two different types of branches that prospective teachers need to take. One is that they take courses science and mathematics courses from the science department and an attack it is how it happens in our university and also mathematics education or science education courses, classes are taken from the Faculty of Education. So, so resembles education mostly in the United States from that perspective, and in our actually department Mathematics and Science Education Department, when students come to their third or third grade, when they are juniors and they what they start taking a module I will say, consisting of several courses, they start with mathematics and science education methods course one, and mostly all prospective teachers, I will say to you know, your teachers, prospective teachers, what from science and mathematics, they take that course together. And then in the fourth year, they they start taking, say mathematics education matters courses, from the mathematics educators, and science education matters course from science educators. And also they start taking internship courses during the last year, again, has two modules in it. During the first semester, they observe teachers and teaching in the schools and during the second semester, actually happens in springs, generally, they start also teaching, they do their internships, and also they take seminar courses. Of course, beside these courses, they also take assessment and evaluation courses or, you know, classroom management courses, etc. So, and there's a package that they have to take to be able to become teachers in the future. That's how things are in our university and the sequence that we're talking about which course was that situated in this was actually illustrated in math mathematics course. But it was actually an extra hour course an extra hour. And David I will say, because we teach math courses or our civic per week and then during the day we asked participants if they will participate in the study after the you know, after those four hours, is this like secondary teachers? And by secondary I mean high school I don't know what the system is there. Yes, yes, it's secondary actually. There are two programs in Turkey in terms of in general but especially in our you know, university two, one is majoring in elementary mathematics education, elementary meaning middle school mathematics education, and in the equality elementary actually in Turkey. Actually, those you know, Paxman suffer from secondary school mathematics measure and secondary is what age children 15 to I will say ninth grade, last grade, okay. All right. Thank you so much that helps such so it is not that different in terms of like what we would say secondary teachers.

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Education here. And it was situated in a methods course. Yeah, I think you say in the paper that it's like, technically it's outside the course because they stayed extra. But it's kind of that population. Yes. Okay. So let's jump into the question, what is the important problem that you're solving? Like? Why did you embark on this study? When we read the literature on real numbers, we faced with some difficulties that alters the mathematics educators pointed to that was attracted us for that kind of innovation and thinking, how to, how can we teach real numbers conceptually, is the main problem for us? And how can we develop prospective mathematics teachers knowledge of real numbers, conceptually, was the main point for us. And in the other site, we read many articles on quantitative reasoning. And as we understood, quantitative reasoning is considered as a tool for leading to conceptual understanding of mathematics on the part of students. So we thought that we may,

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in order to teach real numbers, conceptually, we may use quantitative reasoning, but how can we use quantitative reasoning for real for teaching real numbers was the main issue for us. So the idea was for students to really understand conceptually, what real numbers are. Yeah. And I can maybe just add just some points. Some other point, the issue, mostly, I mean, since I have come back from the States, you know, after my graduation hit there, like for the last 12 years, I think, or so I have been teaching mostly maths courses, and especially with high school mathematics measures, I will say that there is this missing point on the part of the learner. The missing point is that they, I mean, most prospective teachers think that they will be teaching high school mathematics. So other mathematics as if it is not related, you know, to their profession, and they don't even maybe have to think about that part of mathematics. So, for them to be able to realize that mathematics is connected, no matter what I mean, starting from I mean, fractions, you can build up on it, and then evolve into real numbers, or even maybe complex numbers later on, you know, so that's an issue for them to understand or maybe to reconceptualize when they start thinking about how to teach mathematics. So that was one of the, again, major issues that we would like them to establish. Especially we do that mostly in matters courses, or maybe some other content courses. Again, that's another branch that I maybe need to mention, like in education of teachers, what we try to do here at our university, especially, and I see that also in other universities, too, that there's this like teaching geometry, teaching statistics, teaching, algebra, teaching, calculus, you know, precalculus idea. So we also offer those courses, on top of everything that they need to learn about methods and other you know, issues of assessment and classroom management, you know, that we try to establish on their part that mathematics is connected, then it's one unified, I will say phenomena. Yes, that was another reason. I'm glad you mentioned that. Because as I was reading the article, I was remembering how the different number systems seem to be very disjointed to myself when I was growing up, right, there's like, so distractions and there's decimals, and those are different things. And you know, now it's hard for me to see how you could even see them as not being the same, but I distinctly remember, like, seeing everything is different. And so having your instructional sequence really does pull all the various pieces together.

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So let's talk a little bit about that. You already talked a little bit about the literature, but that's give you a chance to kind of talk a little bit more about what you build on. Especially, there are several difficulties on real numbers that researchers point to to one of them is for instance, when a student is asked whether 3.9999 equals to four, and the student says it's not equal to these numbers are not equal to each other. But when we come back to our study, we aim to be able to discuss with students the idea that two rational number sequences may represent the same real number. So we point to that issue in in our instruction. And another difficulty is that, especially students prefer to deal with decimal representations more than fractions. So they

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may not see the connection between them, they need to use the multiple representations. And they need to understand these multiple representations refer to the same thing mathematically, that was another cool tip. And also, there are some suggestions that researchers pointed to regarding teaching. And they refer to the use of multiple representations and focusing on both terminating and non terminating decimal expansions. And also use of the limits of sequences. In this way, they concluded that they suggested that there is a conceptual understanding about the number system and students may construct how rational numbers are connected to fractions, fractions are connected to decimal representations, and how all of these are connected to real numbers, they may have an idea about this issue if multiple representations are used, and the connection between them are emphasize. This was the literature that used in our study. That's all that I want to say. Okay, so I'm jumping into the next question, which asks about innovation. And we already discussed that at the beginning of the article. But let me try to summarize. So you have an instructional sequence that pretty much anybody could implement in their own secondary methods or content course, potentially, even with secondary students that has for four distinct phases that you run a call, and it's like packaged in the article if anybody's interested in using it. So you highlighted the difficulties, I'm assuming that this is also built on your own experience teaching and you struggle teaching and try to develop something that works better. Is that fair? Yes. It's fair. Yes. Fair enough. Yes. Yeah. Especially with quantitative reasoning. I would like to say, I think quantitative reasoning, I just love quantitative reasoning, I will say, because, first, it helped me understand mathematics more, personally. And whenever I asked a question about mathematics, how ideas are related with each other? Whenever I start thinking about asking the questions and thinking about those questions, I find myself thinking on quantities, I will say, I mean, depending on the concept, of course, but then I realized that I am able to construct, reconstruct, actually some ideas on my own. So as a person, I say that if I can do that, then anyone can, then with the you know, this being this mathematics educator, having this mathematics educator part of us in ourselves, then we start thinking about how can we induce such understanding on the part of others. So as you said, that's one part of us, putting into work, whatever we realized, we whatever we learned in terms of the frameworks, I mean, quantitative reasoning, it's, it's a way of thinking mathematical thinking that we actually was going to ask you to define what you mean by quantitative reasoning. It's thinking about I will say, mathematical ideas in a quantitative structure. What I mean by that is, is that mathematical ideas can be I will say, concepts can be quant can be taught through quantities, but they mean by quantity, and quantities and quantitative relationships. So quantitative reasoning is a way of thinking, that enables the person to work with quantities and the relationship between quantities, so that the ideas are in a structure related with each other. And they make sense to her to the learner. And so what we mean by quantity is the quality measurable quality of an object, say that I have a pencil. And if I start thinking about its weight, or it's maybe linked, then I start thinking about quantities, I don't have to measure it physically, or I mean, I don't have to actually measure the length or maybe the muscle mass of the object, and it's an external object. But then when I start thinking about those qualities, measurable qualities, then it becomes internal. So quantities are always conceptual entities like that, we think in our minds, and then with them, we can then start thinking about how those quantities relate with each other. So say, real numbers here, like racial numbers, how much of a quantity I mean, in terms of positive racial numbers, we can think of it how much different to how much and then I don't have to even think about one half tutors. Of course, we re re re present those you know ideas in our minds when when somebody says tutors, right? But what I mean is that I don't have to eat

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Put them on a nice number line externally, if I can think about them in my mind, then I start thinking about them quantitative, those ideas. So real numbers are distances, when you think about in terms of quantities. So any real number is can be taught as a distance from a reference point and reference point can be zero. So, that number line.

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Actually, this reminds me of another follow up question I wanted to ask earlier, when you talked about the three representations. You said division and long division? What's the difference? I thought the division algorithm and long division, can you tell us what the difference is between those two? By division algorithm, I mean, the following, we have a number that we divide with some number, the number equals two divided times quotient plus remainder. divided.

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Yes, divisor times quotient plus remainder is the division algorithm for us. By long division, I mean that when we have a remainder, we continue to do division by putting a zero next remainder. And the coma or.it depends on the depends on the country insertive use comma and I think in us you use that when we put zero next to the remainder and como dot next caution and continue to the division. This is the long division process for us. So the difference being that one is division with remainder and the other is where you work, continue and don't have the remainder, the division. Albertson represents the relationship between remainder quotient divided and divisor. This okay for us. Okay, yeah, that makes sense. By the way, where I grew up also uses different commas and dots than where I live in. It still confuses me to this day, I have to pause when I make deposits at the bank. I'm like, Okay, what country Am I in, I want to make sure I'm doing this correctly. I don't know why. But that's just it's so engrained how you grow up with numbers, or the other thing we just recently looked at, right, with all these big numbers being in the news. what is called a billion here is not called a billion, where I come from in Germany. And so the differences are kind of funny. All right, so let's jump into I think we have a handle on the innovation that you created. And you said it worked. So let's kind of be more specific in what was the research question that you asked to show that it worked? And then what evidence do you have that it actually worked? Our research question was how do prospective mathematics teachers reason quantitatively, while representing real numbers as rational number sequences? In the process of teaching experiment, custom teaching experiment, we try to ask questions to reveal students thinking processes during the intervention, we basically focused on how students present quantitatively. For instance, in phase one, while working on fractions, we ask prospective teachers to give examples of fractions and show in diagrams, and then we asked them how they reasoned on this process, how they construct two diagrams, and what do they represent for us quantitatively. And then, when we work on equivalent fractions, in the phase one of the instruction, we again, ask prospective teachers to give some examples of equivalent fractions and show them on the diagrams. Most of the teachers initially used two different diagrams to show that the examples that you provided for equivalent fractions are equivalent fractions in diagrams, for instance, they gave one over two and two over four equivalent fractions, but they use different diagrams, and show that they think that they show that they are the equivalent fractions, but then we asked them, if you can show them in one diagram, they then talk about, think about, okay, I draw one over two. And then how can I get to our four from this diagram, and then they think about partitioning and this partitioning process is for us a quantitative process. This represents the quantitative operation for us. This is the indicator for us that students think quantitative in the process of our intervention. For instance, in the US

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second phase of the intervention. As I mentioned previously, we work on two examples seven over two and 11 over three. Again, when they use diagrams, the first row 747, over three, three hosts, and another diagram for representing one through one or two, three holes and, and the one number two. And then we asked them to continue with the long division also, they write seven over two, they found three is the quotient, and one as the remainder. And then we asked them, How can we continue to division, they said that we may add zero to the next to the remainder and come on next to the como dot next to the caution. When we asked them what that means. In the diagram, they again, try to think about equal partitioning and grouping. What they did, when in the long division, they put a comma next to the remainder they get 10, right. And then they put a comma next to the quotient, which was to a three sorry, when we asked them, what's the meaning of that tan that you get after putting zero next to the remainder, they refer to the diagram that they partition each piece of the diagram into 10 equal parts, it then is a whole day yet 17 equal parts. And by grouping two of them, they can't put one over tense into diagram. In one piece of the diagram, they have 10 pieces after they read partitioned by 10. by grouping One, two groups of the vanover 20s, they got five vanover 10s. In the diagram, they continued that process. And we refer to the country to operation theories partitioning and grouping. We focused on these ideas. While prospective teachers were explaining their reasoning in the process. This was the indicator for us that they reasoned quantitative throughout the process. Also, when we came to the 11, over three, the same process occurred, they when they continue to do long division, they put a zero next to the remainder. And they come back to the diagram and explained what happens in the diagram. And they connected these processes throughout the intervention. Even in the third phase of the instruction, we go to the sequences, rational number sequences, then they obtain terms of the sequences, they also refer to the refer to the ideas that they got in phase two, they focus on they again explain re partitioning and grouping that they did in the phase two, by reasoning on the sequences and how they obtained the sequences in phase three. That was the indicator for us. And in phase four, why dealing with irrational numbers, they use the same similar process, we can say, in this phase, they used number line and the quantity dating the numbers approximate distance to the reference point, which was zero, that was the quantity for us. And that was the reason process that we expect the prospective teachers might go through.

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So I'm just really curious, I'm gonna go off script a little bit. What about the sequence that you developed? made them be able to reason this way that they weren't able to reason before? Was it like this targeted selection of numbers? Was it the questions like What about this allowed them to reason we try not to direct them through the solution? We just asked how and why questions depending on their previous answers to our questions. Depending on their questions V shaped instruction, we actually I have a we have a learning trajectory that we have in our mind. But in the throughout the instruction, we try to decentralize ourselves from the instruction because I need to focus on what students think during the instruction. That was the main point I think. So really building on where they currently are and then asking questions, but I would like to maybe add something else like together with this hypothetical learning trajectory, as I said like earlier, as researchers or teachers, we first like to we actually do these those thought experiments and think about how but mentally

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Actions one might go through to be able to develop some idea. And then we try to come up with those possible paths, I will say towards that target that, you know, that that target, right. And then we have an agenda actually before starting Of course, but then their difficulties or maybe their reasoning at the moment, it shapes also how the lesson will go. In the flow, like novena was saying that they had some difficulties, and we had to deal with how to, without even like, I mean, I, I know, Marvin was a teacher, and I was also watching the videos and you know, and then thinking about how and also there was, of course, piloting and you know, so many things before then that how to without interfering, how to accomplish I mean, how to overcome those difficulties such as whenever you ask them to come up with an equal production to anything, any fraction, then they start using even not even two different say, rectangular diagrams, but one is rectangular, and the other is, you know, maybe circle and then you then start thinking talking about how come they are different, how come they are similar, you know, that that is also you know, happen. But at the same time, like, especially with seven over to like long division, the relationship between long division and a division algorithm how that's coming, okay, putting zero near the remainder, or in the long division? How does what does that mean in the diagram or in the division algorithm, then, even just without telling them? How can we establish on their part, the thinking, so that they will focus on in those pieces, but at the same time, they will be the ones who is thinking? Yes, yeah. So all right, so let's wrap up our podcasts by talking a little bit about how you see other people using your intervention. So anybody who's listening, what do you want to tell them wanting, maybe I can start Marriner, then maybe we can continue also, one thing that I think of is especially that makes educators in the field, we can use the I think they can, because I as I said, if one person can do this, and if several people can compatibly talk about the issues and come to some generalizations on their own, going through some tasks sequence of tasks. And then it's possible that others also might get benefit of it. The issue is the experience the essence of experience, depending on their background knowledge, if they are similar, if they have similar background knowledge, then they can also build up those ideas on their own. So maybe in the future, but I'm thinking in terms of teaching prospective teachers education in this innovation or the sequence of tasks might be used, again, depending on I mean, the discussions probably will differ might differ. And some other ideas will come up maybe, you know, depending on the audience, I mean, the learner participants, but still, I think it will have some I mean, the metamath, educators will have some benefits, in terms of using it in their content or maybe matters courses. And I would like to quote platform, he says that everything number is the beginning of everything. So maybe we can talk about number. This is one way. And yes, no matter what you say, just listening to both of you talk today, the way you described connecting the representations is very powerful. So I think if somebody were just to take the sequence in I personally teach elementary teacher education. But I was even thinking as we were talking about how I can adapt some of those things in my courses, because like when you define quantitative reasoning, right, connecting across representations is a big part of that. And that just seems, seems really powerful. So I want to thank you guys, did you want to add something my winner to how people might use this in the instruction actually, when the readers read our article, they may see that we connected the different mathematical ideas together to obtain a general idea. We started for fractions and go through the real numbers. So they will see the span of mathematical ideas and how they are connected to each other. In that sense, they may they may benefit also, I think this is an example for that, I think. Well thank you so much. This is really fun. Thank you. Thank you everything that we are really thankful that you listen to us. And you know you had asked today. Thank you so much. It was great fun. I am so lucky that I get to do this because I learned from every conversation so it makes my like I read your papers, and then I talk

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To you and I feel like the conversation always adds to the understanding of the paper. Plus, it's so cool to hear the author's voices. I just think there's a that's a nice addition. Yeah, so let me round out this podcast for further information on this topic. You can find this article on the mathematics teacher educator website. This has been your host, Ava Sennheiser. Thank you for listening and goodbye.