**Suggested Citation:**"Closing Remarks." National Research Council. 2000.

*Mathematics Education in the Middle Grades: Teaching to Meet the Needs of Middle Grades Learners and to Maintain High Expectations: Proceedings of a National Convocation and Action Conferences*. Washington, DC: The National Academies Press. doi: 10.17226/9764.

**Suggested Citation:**"Closing Remarks." National Research Council. 2000.

*Mathematics Education in the Middle Grades: Teaching to Meet the Needs of Middle Grades Learners and to Maintain High Expectations: Proceedings of a National Convocation and Action Conferences*. Washington, DC: The National Academies Press. doi: 10.17226/9764.

**Suggested Citation:**"Closing Remarks." National Research Council. 2000.

*Mathematics Education in the Middle Grades: Teaching to Meet the Needs of Middle Grades Learners and to Maintain High Expectations: Proceedings of a National Convocation and Action Conferences*. Washington, DC: The National Academies Press. doi: 10.17226/9764.

**Suggested Citation:**"Closing Remarks." National Research Council. 2000.

**Suggested Citation:**"Closing Remarks." National Research Council. 2000.

**Suggested Citation:**"Closing Remarks." National Research Council. 2000.

**Suggested Citation:**"Closing Remarks." National Research Council. 2000.

**Suggested Citation:**"Closing Remarks." National Research Council. 2000.

**Suggested Citation:**"Closing Remarks." National Research Council. 2000.

**Suggested Citation:**"Closing Remarks." National Research Council. 2000.

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_~ :r;w~ REFLECTIONS ON THE CONVOCATION Edward Silver, Chair, Program Steering Committee for the Convocation, Learning Research and Development Center, University of Pittsburgh

Adaptedfrom the transcript of remarks by Edward Silver Chair, Program Steering Committee for the Convocation Learning Research and Development Center, University of Pittsburgh Most of us came to this Convocation because we have a serious commitment to enhancing the quality and quantity of mathematics learning in the middle grades, such as the development of important algebra and geometry con- cepts. Most of us also came here because we have a serious commitment to ad- dressing other needs of young adoles- cents such as their healthy social and emotional development. The Convoca- tion took as a premise that this dual commitment to both the development of mathematical ideas and the development of children actually could, in many ways, mask differences in perspective regard- ing these two emphases; that is, individu- als and groups might differ with respect to the relative emphasis of these two commitments in their work. For most of us, although both are important, one looms larger in our thinking than the other. This (lifference in relative empha- sis has been evident in the discussions. A second premise was that the educa tion of young a(lolescents would be enhanced if we took off the mask, expose(1 these (1ifferences in perspec- tive, seriously examined them in order to identify the convergences and diver- gences, then trie(1 to crystallize some issues, concerns, and questions that would benefit either from some kind of concerted action or from further serious (leliberation. A third premise was that it would be productive in the search for these issues, concerns, anti questions, to look for them along three different (limensions curricular, pe(lagogical, and contextual and we organized this conference along these dimensions. So now the question is: What have we learne(l? Several (lifferent kin(ls of learning might have occurred. We've i(lentifie(1 some things we (lo agree on anti some things we (lon't agree on. We have gaine(1 an enriched un(ler- standing of the issues and deeper insights into questions. We now understand some things a little deeper,

a little differently than when we walked in. And we have identified some poten- tial areas for concerted action. Finally, we have also identified a set of issues about which we need to think much harder and much longer. In the remainder of my remarks, I'm going to give you examples of a few areas in which ~ think we made some progress during the meeting. Let me turn to the first one. INTEREST AND RELEVANCE We have broad agreement that the study of mathematics is important for young adolescents. There is also broad agreement that we want mathematics to be interesting to students. The question is, then, how do we make it interesting. We have had a set of examples pre- sented to us that ~ think challenge a view (which is very prevalent these days) some of us might have held walking into this Convocation that in order for students to be interested in mathematics, the mathematics has to be relevant. That is, good mathematics for middle grades students has to be tied to some important application or some- thing related in some important way to students' lives. Glenda Lappan told us on the first night that we need to connect students to things that are generally interesting Ml DDLE GRADES CONVOCATION to them, and Tom Dickinson described students who were gathering (lata in experiments about questions that were genuinely of interest to them and then displaying the data. Students were ([riving the investigations anti making (recisions about how to (lisplay the (lata in the best way to answer their ques- tions. This is a good example of what it might mean to have interesting math- ematics for students. But then, on the other hand, we saw different examples in the videos. One almost by inference in Linda Foreman's case and the other more directly in the video that Nannette Seago showed of Cin(ly's teaching, in which students were engage(1 in the investigation of mathematical ideas in problems that you could hardly call applied. They were problems that (li(ln't come from a meaningful context. They lacked the connection to thematic or application oriented settings that many of us might take to be boun(1 up inextricably with this notion of what's interesting to kills. There is no question that in order for young a(lolescents to learn mathemat- ics, they're going to have to fin(1 it interesting. The question is what is it that makes the mathematics interest- ing. The videos and the discussion about the videos help to remind us that students can find mathematics inher- ently interesting. They can find "ap- plied, real-world" tasks interesting, but

they can also find tasks that aren't applied in the real world interesting- because these are tasks that arise within a classroom community of students who are investigating math- ematical ideas about something that they value. Yesterday, Deborah Ball talked about the notion that students' interest can be cultivated. That is, students bring their interests to the classroom, but teachers have the capacity to cultivate new areas of interest, as well. In many of the things that I've been involved with in the Quasar Project, and in the work that we're doing related to the new NCTM stan(lar(ls, we've been trying to grapple with this idea of how you cultivate students' interest and thoughtful en- gagement in classrooms. It is very clear that we have examples, some of which you saw yesterday an(1 many of which you can see in other locations, demon- strating that students can be challenged an(1 supporte(1 in engaging with math- ematical ideas and find them quite interesting in a variety of ways. What we should take from this is not that students should never see context nor that everything has to be embedded in context. Rather, our thinking about this issue is enriched when we resect back on the set of examples that we saw anti the way in which they represent mathematical possibilities. They show us that students are engaged when they CLOSING REMARKS have interesting things to think about, and mathematics is filled with interest- ing things to think about. We need to give students the chance to see math- ematics as being interesting and some- thing to think about. As ~ reflected on this matter of rel- evance and engagement, ~ recalled something from my first year of teach- ing. ~ taught seventh grade in the South Bronxin New YorkCity. Oneofthe students in my class was named Jeffrey. ~ want to tell you about Jeffrey. He was very pleasant, and he had learned that the way you get through school is to smile at the teacher and be polite. Jeffrey was a won(lerfu} little boy, but academic school was not a priority for him. Nevertheless, during the year, Jeffrey, for some reason, became very, very interested in palindromes. For those of you who (lon't know about palin(lromes, a number like i,331 is a palin(lrome because if you write the number forwar(1 or backwards, it's the same number. Jeffrey got very inter- este(1 in palin(lromes not because he could apply them to his every (lay life, but because they struck his curiosity. Jeffrey spent most of the seventh gra(le in an in(lepen(lent exploration of palin- dromes. And it turns out that you can learn a lot of algebra by exploring palindromes and looking at the struc- tures of these numbers and what hap pens if you multiply them by certain

numbers, what happens if you combine them in certain ways, how many possi- bilities there are for certain forms, and so on. Jeffrey had his own independent study course going on because that's really what engaged his interest. To bring closure about what makes it interesting for kids to grapple with mathematics, the examples we've seen at this Convocation suggest that the "interest" can be in the tasks themselves that we give students, or it can be in challenges raised by the tasks and in the process of grappling. Students can find it very interesting and can learn a lot from struggling with challenging tasks. They can derive a tremendous amount of well-deserved self-esteem from this. Mathematics is not easy. It is not always fun. It is something that's worth strug- gling with and worth doing well. This struggle can be a very rewarding experience, and it can be meaningful. Now ~ want to draw attention to a second area that ~ heard as a popular topic at this Convocation algebra. ALGEBRA There were many mentions of algebra in the plenary sessions, and from looking at the records from the discus- sion groups, it is clear that algebra came up quite frequently there. It is quite possible that some of us came to this Ml DDLE GRADES CONVOCATION meeting with a view of algebra in the middle grades as a course very much like the first year course in high school. That is, algebra for middle school students would mean that students in some gra(le before high school would take this course, whether it's eighth grade or seventh grade or sixth grade. This conception of algebra as the only notion of algebra in the middle grades was called into question by much of the discussion and many of the examples that we saw. That is not to say that one cannot have a one-year course in aIge- bra. But even a one-year course that focuses on algebra can be (lifferent than what we might expect. If you think about the video involving Cindy, she was teaching an algebra course, but the way she was teaching that algebra course strikes me as somewhat (lifferent than our caricature of the way in which the first year of high school algebra is typically taught. If we think about the set of i(leas that Glen(la talked about on the first night, the set of ideas that you might have read about in the first discussion session on algebra in the mi(l(lle gra(les section (lrawn from the Principles and Standards for School Mathematics: Discussion Draft, or in other materials for this Convocation, you get a (lifferent view. This view suggests it might be possible to think about algebra and the develop- ment of algebraic ideas over grades six

through eight in some way that doesn't require a separate course for the year we decide to teach algebra. This kind of algebra instruction would be integrated algebra, integrated with the stu(ly of numbers, with the study of geometry, and so on. This is, in fact, the approach that is represented in most of the curriculum materials that have been produced and released in recent years. Many of those materials differ in the way that they go about doing this, but they all share a commitment to trying to develop algebraic knowledge or fluency in a more integrated way throughout the middle grades rather than concentrated in a single year. This view of algebra is really quite different than the view that some of us might have had coming into this Convocation. And it is a view that challenges us to think harder about what it would mean to learn algebra in the middle grades. Now ~ want to connect that to a last point which ~ think came through very clearly in the pane} session, much more strongly than it had in earlier sessions. Good curriculum and good intentions and good practices and pedagogy may not be enough. There is a range of policy and political matters that need to be consid- ered. As we heard this morning, other kin(ls of support from parents, a(lministra- tors, and organizational context matter, and they matter a great deal. There were a number of people who CLOSING REMARKS talked about the politics that surround reform ideas, whether they were middle school reform ideas or mathematics reform ideas. And many kinds of politics have been mentioned in this Convocation community politics, district politics, school politics, personal politics, professional politics, and so on ~ remember from my first year of teaching when we were trying to create "open classrooms," which was then the avant-garde reform idea. But we didn't have the kind of physical space needed for open classrooms. We had a very old building with lots of walls. In response, we rearrange(1 space anti use(1 the hallways. We arranged students in groups rather than having them sitting in straight rows of desks. But we didn't have tables in fact, we didn't have any of the things that now are standard practice. Instead, we had individual student desks, and so the desks were organized into small groups to allow students to work together. Some students would work in different loca- tions in the room, some out in the hallway, and so on. Every night Tom, the janitor, would come to my room, take all of the (leeks, whether they were in the hallway or in (lifferent corners of the room, wherever they were anti arrange them into straight rows. Tom had (lone this for 22 years in this school An(1 the fact that a new teacher thought that the furniture was going to be

arranged in some other way had no impact whatsoever on Tom. Every night Tom would come and move the desks. And it quickly became a joke. The first thing the students would do in the morning was rearrange the furniture. No intervention by the principal, no (liscussion with Tom, nothing would have an impact on this. We did this all year long, and mine wasn't a unique case in the building. There were other teachers doing the same thing. So there is politics even at that level, let alone dealing with commu- nities and parents and so on. How does this relate to algebra in the middle grades? At this Convocation, we've heard that there is a pretty w~de- spread view of algebra, what it means and how it looks. Essentially, this is the view ~ described as the one that many of us might have had walking into this Convocation algebra that looks just like the first year of a high school course, all happening in one year. An(1 if that is, in fact, the common perception of what algebra is, and if you're trying to implement some other way of teaching algebra, there's likely to be a problem because people won't understand it, whether it's Tom the janitor or whether it's Tom's great grandson who is now in your class or his grandson who's a parent in the community. There is a serious set of issues that have to be addressed in terms of unpacking for Ml DDLE GRADES CONVOCATION ourselves and for the whole community what it means to say that students are learning algebra, and what it would mean for all students to learn algebra. Can we design programs so that students succeed in learning algebra in the middle grades? Bob FeIner's com- ment about "no acceptable casualties" is a very important one. We don't have a very good conception of what this means. Our programs in mathematics have not always been designed so that everybody could be successful with them. Mathematics education has generally been organize(1 to fin(1 the few students who could be successful, so they could get on to the next course. Some folks are working har(1 to change this way of thinking, anti it is now a goal for many that all students should be more successful in mathematics. But we need to recognize that there is a huge education and political job to be done in "unpacking" what it means for "all students" to learn algebra, if we want something that's (lifferent from just taking that high school course one or two or three years early. We need to (levelop a broa(1 un(lerstan(ling of this notion of algebra with others, including administrators anti parents anti other members of the public. We nee(1 to systematically examine different in- structional anti curricular arrangements that are (lesigne(1 to have all students learn algebra. We have a lot of hunches

and a lot of opinions about which way is best or which way will work. Many people believe that if we just did it the old way, it would be fine. But, if we could be more precise about what it is that we're trying to get students to be able to do, what it is that we want them to know, and how it is that we would like them to be able to perform, we could then ask whether or not taking that high school course one or two or three years early really meets these goals. We would also have to ask ourselves whether putting students through a well-taught version of an innovative middle school mathematics program does this. We have to push ourselves to ask this evidence question. What is the evidence that we can, in fact, produce the kinds of competence we want in our students through these different in- structional approaches? So for me, this issue has gotten no less complex. But it strikes me that this is a place where we have an opportunity to begin to work together, because ~ think the middle school community and the mathematics education community are both very interested in finding ways to increase the competence and confi- dence of students with respect to mathematics. We want all students to have the opportunities that mathemati- cal competency and mathematical proficiency affords them. Some people call it "mathematical power." Some CLOSING REMARKS people don't like that term, but that's what it's about. It's about having math- ematics, owning it, having it be your own and being able to open the doors that mathematics can open. We want this to happen for students, and we want this to happen in ways that are sensitive to their needs. This Convocation has crystallized some of this for us, sharp- ened some of the issues, and left us with a number of other issues about which we have to continue the conversation. The issue of"mathematics for all stu- dents" is one in which we might be really to begin to act on. OTHER ISSUES There are also a few other issues of note that were raised in this morning's conversation an(1 in the (1iscussion groups. The issue of teacher preparation was not (liscusse(1 explicitly in any of the sessions but was certainly a running theme along with teacher professional (levelopment. An(1 those two coalesce around questions about teacher short- ages and turnover. Occasionally, there was mention of the special needs of students anti teachers in high poverty communities. This is a very important issue that is (lifferent in rural communi- ties and in urban communities. This plays out in the mi(l(lle gra(les in quite (lifferent ways. The organization of

schools is often different in those com- munities, and the ways of thinking about specialist teachers and generalists is quite different also. More conversation is needed about these issues of diversity. Racial and ethnic diversity, linguistic diversity, special needs students, and mainstreaming all need to be considered as we move forward. Technology was strangely absent in most of the conversations, although it came up in this morning's discussions about the home-school interface in reference to students who have access to technology of a certain kind in the home, but not in school. Technology is very important for students of this age, but technology also has an impact on what's taught, how it's taught, and what the possibilities are for teacher profes- sional development and teacher assis- tance. Consider, for example, the amount of help that could be provided to teachers to do a better job through the uses of technology. Some people ~ know in this audience are thinking very hard about that. And we need to be looking at that. And then lastly, ~ want to mention the notion of identity because Mary Kay Stein talked a little bit about it this morning, and it came up very strongly in the discussion group sessions. Maybe not everybody would attach the word identity to this notion, but some participants are asking questions about how a teacher should balance attention Ml DDLE GRADES CONVOCATION to competing demands for a group of students. How do you balance your attention to the student with your attention to the discipline or the subject matter? How do we deal with the generalist/specialist potion? How do you balance affiliation with fellow teachers of mathematics versus mem bership on a cross-disciplinary team of teachers for a group of students? There are ways to frame the question that set up false dichotomies as if it has to be one or the other. Those of you who live in classrooms every (lay know it isn't that simple. But it is clear that how you think of yourself has an influence on what happens in classrooms. When you think of yourself as a mathematics teacher, you have a particular set of resources and colleagues as well as a set of constraints on what you do. When you think of yourself primarily as a middle school teacher, then you have a different set of resources and colleagues anti so on. We have to think about ways of forming a community that has a joint identity and that helps to move the agenda of this Convocation forward. And ~ just want to close by reiterating something that Steve Gibson ma(le a point of saying this morning that we nee(1 to keep in min(l. Engaging in (liscussions anti (1ialogues such as we have experience(1 at this Convocation is the way in which we're likely to make progress. Thank you for being part of this very productive first step.