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• #### Appendix E: A Framework for Constructing a Vision of Algebra: A Discussion Document 145-190

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Synthesis of Day One ~ Henry Poling Henry Pollak offered a summary of the issues raised on Day One of the symposium. He remarked, "One of the first things I noticed is a very healthy sense of rebellion among the group here. When you didn't think a question was the right question, you said so." He then provided a synopsis of responses to the following questions posed to the group on Day One. Question 1: To develop coherence and depth in the algebra curriculum across K-14, what transitions do students need to make and over what period of time? Also, how long must students be engaged with an important mathematical idea so that understanding is achieved ? Pollak noted that "the first transition is the one from the concrete to pictures of the concrete to symbolic representation. You go from play with things to a pattern to perhaps a numerical expression of that then to a symbolic expression of that, and all those together allow you to form a conjecture. With that conjecture, you can make predictions. Predictions give you control and that, finally, allows you to do verification or proof, depending on the level you are in. It's a very interesting sequence of events, and a number of you talked about it. You said, when you teach, don't try to go too fast. Students have to have a chance to prepare for these things. If, for example, you as a teacher want to use algebra tiles, students need to have played with things like that in the earlier years. You cannot suddenly hit students with algebra tiles, and say, 'See.' " Pollak also noted the participants' interest in emphasizing links to the other parts of mathematics: "One of the important transitions, for example, is geometry and its connections to algebra. Explore an area, look at it in many different ways, persevere in what you and the students are doing, such as cooperative learning, use of English, and emphasis on reading." In moving to the second part of Question 1, "How long must students be engaged with an important mathematical idea so that understanding is achieved? Pollak noted the participants' sense that this is the wrong question and that the straightforward response from many was simply, "As long as it takes." Question 2: To develop criteria or lenses with which to select curriculum, appropriate pedagogy, and assessment, how should algebraic reasoning be derned? Also, how should state, national, or college placement exams be taken into account? * The following remarks by Henry Pollak and Gail Burrill were edited from verbatim transcriptions and serve as a transition from Day One to Day Two of the symposium. 49

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so THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM Pollak said that "the answer I heard to the first part of Question 2 was that the transitions more than anything else help." He then discussed examinations state exams, national exams, and college placement exams. He noted that "Everybody agreed that assessment drives instruction. They wish it were the other way around, but it isn't. As Hugh Burkhardt from England likes to say, 'What you test is what you get.' Something I hadn't realized is how important college placement exams are. People are more interested and more worried about these than the other exams. As someone here commented, if the college placement exam is deemed so important, what are we going to do about that at the K-12 level? Connections with the colleges are not good. The alignment between what goes on in the schools and what the college exams test is a real problem. There is a desire for teachers to be able to influence these tests in a serious way, including aligning the college placement tests with the Standards. It was noted that the SAT has changed, so why don't other tests change? At the same time that we need to know why each exam exists and what its goal is, others make the comment that students with real mathematical power should be successful with most tests, so quit worrying. Certainly this has been an important point of view in many of the curriculum revisions that I have seen in recent years." Pollak also summarized reaction to the proposed national eighth-grade mathematics exam: "Most of you who talked about it were afraid of it. It was something that you were quite worried about. It may not go in the right direction. It may be at too low a level." Question 3: To develop long-term support for teachers and Standards-based algebra, how much help do teachers need? Also, how do we develop public supportfor teachers and Standards-based algebra? Pollak said, "A major idea repeatedly presented was that teachers need time to talk to each other and to understand a district's scope and sequence. Elementary-, middle-, and secondary-level teachers in the system need time to work with each other and to get a holistic view of the curriculum. This is a very strong and useful idea. The general feeling was that high-school teachers need inservice primarily on pedagogy and elementary-school teachers need inservice primarily on content. Teachers for grades K to 5 particularly need materials that develop algebraic thinking. Coming from Teachers College as I do, I need to think about all that, too. We all need to integrate our efforts and to get support from families, administrators, and boards for what we are doing. But, specifically what do we do to develop support for teachers and standards-based algebra? Working with PTAs and business roundtables, writing newspaper columns, and gaining publicity for student work all were mentioned as ideas." Pollak observed further: "One important thought about mathematics education is to involve employers. The system is too nearly a closed system. You need to get employers to think about the future rather than the past; what employees will need, not what they have needed in the past. You need to get them to think about how mathematics will be used rather than how they, themselves, learned it. That could be of great help. Please remember that standards and traditions in schools are not the same thing." DISCUSSION RELATED TO JERK CONFREY'S PRESENTATION In discussing Jere Confrey's presentation, Pollak observed: "Evervbodv agreed that there is no single encroach to algebra that will serve the needs of all children. Participants expressed an interesting worry about the implications of saying that, and whether this means that we are going to have tracking. People wanted to say, 'No,' to that, but they were a little afraid to. What are the varieties of approaches? Cultures, expectations, assessments, families, abilities, pedagogics, lengths of class periods these are all things that differ and are all things that affect how we learn and teach. What approaches should you use? Every approach you can think of was discussed today- the constructivist approach, the structural approach, and the application or context-driven approach, an approach through functions, an approach through connections to other parts of mathematics, an approach through group work, a traditional approach, a technology and math-lab driven approach, an approach through historical perspec- tives. These all have their place. The feeling was that you need a mixture of them." Further, Pollak said, "Others noted that parents expect their children to have what they had. Remember the worry that if there should be more than one approach, that might automatically imply there is a need for tracking? Some people said, 'Hey, I don't like that word 'approach.' What does it mean? Is this a trap?' And they perhaps take care not to teach in a way that closes doors for any of their students." "Another discussion question from Day One," Pollak noted, "was, What happens to the curriculum beyond the first course in algebra? The reaction was that this is the wrong question. If we are talking about the grades K

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SYNTHESIS OF DAY ONE 51 through 14 curriculum, why is anyone even mentioning a first course in algebra as if it were gospel? Could we talk not about a first course but rather integration, as in the 'Interactive Mathematics Project' ?" Also, "another question was, What is the role and responsibility of teachers in changing success rates in algebra? And the reaction was to ask, what do you mean by success? Who defined it? Again, the answer seems to be, consider the transitions, consider discussions across grade levels, have the whole district work together, and so on. In staff development, people need to buy into the vision, and there needs to be involvement by teachers in the planning of the curriculum. School-university partnerships also are important for success, including those horrible placement exams. A major comment was that we all know what needs to be done. We should stop talking end dolt." AREAS NEEDING FURTHER DISCUSSION "Last," Pollak observed that "the notion of algebra as a way of thinking versus algebra as an object of thought came up yesterday. I want to just ask you something. How much can you use algebra as a way of thinking before it has been an object of thought? How do you do that? What is the way in which the two of them can go together? I would like to hear some more discussion about that. "I [also] want to mention that in three of the talks yesterday, piece-wise defined curves were mentioned. We had discussion of piece-wise defined curves on the CBL. We had discussion about renting videotapes and the different fees involved. We had discussion of the costs of the different ways of getting Internet services. Each of these is a piece-wise defined function. We do so little that is extremely practical in algebra and in mathematics in general. The practical includes piece-wise defined functions. I think we ought to be paying a lot more attention to them." COMMENTS ON HYMAN BASS'S PRESENTATION Pollak closed his observations about Day One of the symposium by "making some comments about some of the things that were said yesterday, from Hyman Bass's talk in particular. Hy emphasized the many different interpretations of a mathematical operation. Take something like multiplication. You can interpret it as repeated addition. You can interpret it as counting arrays. You can interpret it as area. You can do all of these. The important thing is that some of those interpretations generalize, and some of them don't. Repeated addition is not easily generalized to fractions. Counting arrays is not easily generalized to fractions and real numbers. Area is. What you so often do in mathematics is have many contexts for an operation, and you look at them all. Then you generalize from the ones that generalize." Pollak also noted that, "in the current rethinking of the Standards, one of the issues that people are working on is what is meant by algorithmic thinking. Hy gave a beautiful example of that in connection with the greatest common divisor. Yes, you can talk in terms of factoring the numbers and making a pool of all the factors that are in common. The trouble is that in the real world that is impractical. What is practical in the real world is, for example, the Euclidean algorithm. There are differences in the ability to carry out various algorithms. That is at the heart of algorithmic thinking. Hy also talked a lot about extending numbers in operations, so I want to add a thought to that. Say you start out with positive integers. You go from there to counting numbers. You go from there to all the integers. You go to the rationale. You go to the reals. You go to the complexes. If you include exponentiation, there are all sorts of stages in between. Each time you generalize, you increase the repertoire of operations, and you have to extend all the previous operations to the newly defined numbers." Pollak added that, "the interesting fact is that every time you do this you lose something as well as gain something. It's like the economists say, 'There is no free lunch.' You push in here, and it comes out there. Every time you make an extension, you lose something as well as gain something. What do you lose when you go from the positive integers to all the integers? Well, the positive integers have the property that any non-empty set of positive integers has, i.e. a smallest member. Once you go to all the integers, that is not true. There may not be a smallest one in a subset of all the integers. What do you lose when you go from the integers to the rational numbers? When you take one integer divided by another, you no longer have a unique representation. Two over six and three over nine mean the same thing. Now you have lost uniqueness. As Hy pointed out, when you go to the reals, you have lost finite representation. You have, in fact, lost the ability to write down the number competely. As he said, if you then try to do multiplication by going from right to left like you are supposed to, the question arises of how you can

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52 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM start if there is no right end. So, you have lost something. When you go from the real numbers to the complex numbers, what do you lose? Order. There is no such thing as one complex number being less than or greater than another. This cannot be defined. In every one of the extensions that we talk about, there is a loss as well as a gain." In a final comment, Pollak said that, in "addressing the differences between arithmetic and algebra something we haven't talked about as much as I had expected let me mention one aspect that has always bothered me. In much of arithmetic, all of the numbers are the same. There is no difference between 15 and 16 and 17 and 18. All the operations are carried out in the same way. In algebra, though, there is a big difference, because you have to look at things not only additively but multiplicatively. Interesting facts then are that 15 appears in the multiplication table once, 16 appears twice, and 17 doesn't appear at all. In arithmetic, the numbers are all the same, but in algebra, this knowledge of the biographies of the numbers is very valuable. When you start thinking about things like ratio and proportion, particularly in middle school, you are beginning to make the transition from thinking about just the additive structure into thinking about the multiplicative structure. That is part of the road from arithmetic to algebra." Gail Burr`R OBSERVATIONS STIMULATED BY JOHN DOSSEY'S PRESENTATION Gail Burrill noted that she had been thinking about what John Dossey said and about the notion "that we have lost a dynamic, growing view of what algebra is. I also am thinking about the observation that we are survivors and the notion that technology allows us to go back to a dynamic view. In my classes, using some of the technology has opened the students to thinking and developing and growing in their understanding of what algebra is all about, what solving equations is, how many different ways this can be done. So, it seems to me we can use technology as a way to get back to a dynamic, growing view. One topic that didn't get addressed in this technology conversation is momentum for change and the direction we take now. And how we have to be very careful that we are not just doing the old things in new ways. Why do I say that? Because I could already graph a function without technology. So, if I just keep graphing a function and graphing a function and just doing it faster using technology, I have not actually made the technology help me understand. I think we can use the technology to do old things in new ways, but we have to allow for the expansion of, for instance, the piece-wise functions that we were looking at earlier. That is something that was not actually in most of our curriculums at least not until calculus but it is now accessible to students. When they have access to these kinds of technology, students can think about and do things that were not part of the curriculum before." TECHNOLOGY AND ALGEBRAIC REASONING QUESTIONS Burrill said, "Let us look at algebraic reasoning and the question about what it is and how it differs from other lands of reasoning. I do not know if this was healthy rebellion or confusion, but when I read the algebraic reasoning papers, the coverage included patterns and generalizations and problem-solving and symbols and graphing and variables. There seemed to be a huge set of things that people think of as algebraic reasoning. In some of the discussion, I heard the questions, How do we know if we are seeing algebraic reasoning? Do we know it when we see it? There was a really intense conversation in one of the groups about just what does it look like when students are doing algebraic reasoning. If one is applying a definition, is that doing algebraic reasoning? I think we need to pay some attention to this." Further, "When the question was asked, Does technology promote or inhibit the development of algebraic reasoning and does the answer depend on the kind of technology employed you all nearly unanimously said, 'Of course, sometimes technology inhibits, and sometimes it promotes. It depends entirely on the situation that you are

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SYNTHESIS OF DAY ONE 53 working in and the goal that you have and how the whole thing is handled, what kind of training the teachers have, what kind of access.' " In addition, Burrill noted that, "when the question was, How important is it to develop algebraic reasoning, everyone said, 'Well, that is a really silly question because of course it is very important to develop algebraic reasoning.' There was that healthy rebellion. When exploring the question of what reasoning and content skills should be in an algebra course or strand that promotes algebraic reasoning again, the responses were varied." HIDING BEHIND EXCUSES Burrill observed that, "one issue that came up in our discussions is that there is a tendency to hide behind excuses. I think that is what Bea Moore-Harris told us, and I think that is really important because it was reflected in the comments that I read. We can stack up a whole lot of excuses for why we cannot make things happen, then we can sit behind them and hope that someone else is going to solve them. We heard about the issue of articulation with the community colleges and the universities. Evidently this is a critical problem. We cannot hide behind it. We must do something about it. I am not exactly clear about what each of us can do, but I suspect that collectively in your states or in your communities you can start addressing this issue." Burrill observed further that, "access to technology is an excuse that we can hide behind very easily, but there are states and there are cities that have made a difference in terms of giving every student access to technology. Every student that I know at my school has their own basketball boys and girls. They can have their own calculator, too." NATIONAL TEST "Another question that came up involved the national voluntary test in mathematics," Burrill noted. "There were people who were very concerned that this assessment would set a minimum level, when, individually, each of us wants to be higher than the minimum. I would hope that, as a community, we can make sure that if we do have this kind of assessment or if you have an assessment for your state that it is not at a minimum level. There are two points here. One is what we may be after is a consistent vision, that there may be uniform kinds of things that students ought to be able to do at a certain point in time. The other point is that we need to be very careful about how we think about assessment: minimum level indicates that we are ranking kids, and I think that we do not want that to be the point of our assessment. I think we want more out of assessment than sorting students into piles. If we think about a minimum, we are not thinking about looking for what students know and are able to do. So, we need to push on that." ADULT LEARNERS Burrill observed that, "the following question came up once or twice, and it is really important: Do adult learners in community colleges or universities need a different kind of developmental structure to understand algebra? Or can you do the same things with them as with younger learners? This is critical because, as we learned yesterday, we have many students learning algebra when they come back to school as adults or even when they go right into college and have to take it over again." PROFESSIONAL DEVELOPMENT "The next-to-last topic I want to mention is professional development," Burrill said. "We need to have professional development, but we cannot have it for today's technology. I am sorry, given one of the comments I heard, but we should not teach every teacher how to use the CBL and how to use a TI85 or 86 because we do not know what is going to happen tomorrow. We cannot anchor the development of teachers to a particular piece of equipment at a particular point in time. This puts a really big burden on thinking about what is important in terms of content for teacher preparation because that has to transcend changing technologies. That is why this whole notion of symbols and technology is critical. We need to sort out what the technology is allowing us to do from what the important mathematics is going to be."

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54 THE NATURE AND ROLE OF ALGEBRA IN THE K-14 CURRICULUM EQUITY Finally, Burrill observed that, "equity issues were mentioned just once or twice, but it is important to recognize that teachers who tend to avoid technology are going to disenfranchise their students. This is a different kind of equity than we usually discuss. So, if there is a colleague who is not letting students use graphing calculators or computers, at some point, those students are going to suffer." In addition, "another point I picked up from Bea Moore-Harris's talk is that it is really important for us to go beyond our own understanding. I had done the borders and blues problem that Betty Phillips shared with us. I had thought really hard about different ways to do that problem, yet I was amazed at all of the representations that came up. They never, ever would have occurred to me. I have to be able to move beyond my own understanding of how to fill in the holes so that I can let students have their choices of ways to think about things. When we have this notion of equity and the notion of trying to move forward, we need to put together technology, equity issues, and professional development in ways that will help us to help students understand algebra."