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~ +.~ - ~ [~5 ~1 ~ Fir. arm_ As an initial step to address national, state, and local issues of teaching and learning mathematics in the middle grades, the Mathematical Sciences Education Board (MSEB) of the Na- tional Research Council's Center for Science, Mathematics, and Engineering Education (CSMEE), the American Educational Research Association (AERA), the National Council of Teach- ers of Mathematics (NCTM), and the National Middle School Association (NMSA) co-sponsored a National Convocation on Middle Grades Math- ematics. The Convocation was held at the National Academy of Sciences in Washington DC, on 25-27 September 1998 with support from the U.S. Depart- ment of Education and the AERA. The goals of the Convocation were to . challenge the nation's mathematical sciences community to focus its energy and resources on the improve- ment of middle grades mathematics education and begin an ongoing national dialogue on middle grades mathematics education, bringing together those with different perspectives and responsibilities to jointly consider the issues. The Convocation consisted of plenary sessions attended by all of the partici- pants and small focused discussion groups. Over 400 participants inclu(ling mathematicians, mathematics teacher educators, state and district mathemat- ics education policy makers, national policy makers, mathematics education researchers, classroom teachers, curriculum developers, and school boar(1 members atten(le(1 the Convoca- tion. Some of the attendees came as individuals. Manyothersweremem- bers of the more than 50 (1istrict teams that addressed the issues in terms of their own communities anti needs. Prior to the convocation, attendees reviewe(1 the following backgroun materials: a paper commissione(1 for the Convo- cation, ' What is Sth Grade Mathemat- ics: A Look from NAEP" by John Dossey, the abri(lge(1 version of Turning Points from the Carnegie Council on A(lolescent Development,

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This We Believe from the National Middle School Association, an article from the Kappa n Articles online, "Speaking with One Voice, a Manifesto for Middle-Grades Reform" by Joan I~ipsitz, Hayes Mizell, An- thony Jackson, an(1 Leah Meyer Austin, and "Middle Grades Mathematics Educa- tion: Questions and Answers" a paper prepared by Andrew Zucker for the U.S. Department of Education On the first evening of the Convoca- tion, Dr. Bruce Alberts, President of the National Academy of Sciences, wel- comed the group and spoke of the need to invest ourselves in improving educa- tion and to be open to new ways to meet the challenges of the task. He was followed by the Honorable C. Kent McGuire, Assistant Secretary, Office of Educational Research and Improve- ment, who brought greetings from the U.S. Department of Education. The Convocation Program Steering Commit- tee Chair, Edward Silver, University of Pittsburgh, offered brief remarks about the goals of the Convocation and set the stage for the rest of the activities. Two presentations followed, each represent- ing a different perspective. Glenda Lappan, Michigan State University, discussed the importance of laying a firm foundation for un(lerstan(ling mathematics during the middle years. She note(1 the (1ual goals of respecting the developing capabilities of middle school students while engaging these energetic adolescents in learning mathematics for their own future. Rather than listing a set of topics, she proposed a strand approach to the mathematics students should learn that would be central to further study of mathematics or to being a good and productive citizen. Thomas Dickinson, In(liana State University, then use(1 "small stories" as a way to characterize successful middle schools. He dis- cussed development in the context of the child as well as development and its connection to teaching. He gave several examples of teachers teaching individu- als, anti teachers teaching mathematics where development was placed within a context of individuals and the individu- als in a context of learning. CONTENT AND LEARNING ISSUES After greetings from Luther Williams, Assistant Director, Directorate for Educa- tion and Human Resources, National Science Foundation, the first focus of the second day of the Convocation was on content and learning mathematics at the middle grades. Nancy Doda, from National-Louis University, spoke of a crossroads in middle school reform. She noted the need for a reexamination of the MATHEMATICS EDUCATION IN THE MIDDLE GRADES

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fundamental philosophy of the middle school concept to understand what is needed to ensure academic success for all children. Doda contended that in many cases, middle schools have been more successful in altering aspects of climate and structure than of the curriculum and instruction. There is now a pressing demand to refine the elements of the middle school concept to build an explor- atory curriculum that is also intellectually demanding. Kathleen Hart, the University of Nottingham, based her remarks on projects from the United Kingdom. Middle grades are recognized as the time students should move from the concrete to the abstract, but the transi- tion must be carefully developed so that understanding emerges. She gave examples where the children she studied did not connect concept devel- opment with formal work with aIgo- rithms, often due to the fact that teach- ers did not make strong connections between the two. She noted how important it is for teachers to under- stand where mathematical ideas are leading and to be ready to build on the different pieces of knowledge individual children take away from a lesson. Following the plenary session, partici- pants engaged in a mathematical task in small (liscussion groups (Marcy's (lots, see page 581. They were asked to renect on the mathematics involve(1 anti EXEC UTIVE SUMMARY to analyze student responses as a backdrop for their discussion about content and learning mathematics in the mi(l(lle gra(les. TEACHING ISSUES The afternoon sessions were focused on teaching in the middle grades. In the plenary session, participants viewe(1 two video clips. Nanette Seago, California Mathematics Renaissance Project, showed a videotape of an eighth grade class during a lesson on patterns in algebra. The viewers consi(lere(1 how listening to conversations among students enables teachers to learn about student understanding. Groups dis- cussed the decisions the teacher made as she pursue(1 the lesson anti renecte on the impact of these (recisions on the outcome of the lesson. Line Foreman, Portian(1 State University, showed a videotape of middle grades students making a presentation at the 1998 National Coun- ci} of Teachers of Mathematics Annual Meeting in Washington, DC. The students spoke about what they hall learned in mathematics and what their teacher had done to enable that learning to happen. In particular, the students recognized and supported the notion that learning (li(1 not take place without some "(lisequilibrium" anti that strug

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gling with learning was a natural part of the process. These students had been taught by Foreman for four years, and she suggested that structuring learning so students and teachers stay together over time is one way to create a success- fu} community of learners. A pane} consisting of Hyman Bass, a mathematician from Columbia University, Deborah Ball, mathematics educator from the University of Michigan, and Sam Chathn, a middle grades teacher from William H. English Middle School in Scottsberg, Indiana, reacted to each videotape. Bass observed the blend of algebra and geometry in both videos and noted that use of video might be an appropriate too} to help bridge the gap between mathematics as content and mathematics in practice. ChatUn's com- ments related to the environment estab- lished by the teacher in each case as evidenced by the kinds of questions and answers and by the confidence students displayed about their work. Ball framed her remarks around the interplay between the mathematics to be learned and the role of discussion where the teacher's decision about how to frame a question and how to respond drives what students learn. In the small group discussion session that followed, the participants were asked to resect on teaching issues raised by the videotapes as wed as on the use of videotape as a means for stimulat- ing resection and discussion. ORGANIZATIONAL ISSUES The last theme of the Convocation was the organization of schools for middle grades students and its relation to teaching and learning mathematics. A pane} presentation framed the issues for (liscussion. Craig Spilman,a principal from Canton Middle School in Baltimore, MD, emphasized the need to make intelligent use of data to promote articu- lation among elementary and middle grades teachers to understand where students are in their learning. The design and implementation of programs should be flexible enough to accommo- date students as they grow. He spoke of the nee(1 for principals to communicate with anti counsel teachers to ensure that their mathematics instruction is centered around student learning. Mary Kay Stein, a research scientist from the University of Pittsburgh, hypothesized that the developmental approach teaching mathematics with a focus on the whole child an(1 the subject matter approach teaching mathematics with a focus on the con- tent each have news. She propose(1 a middle school organization and structure that is jointly informed by subject matter anti (developmental concerns. Stein argued that the mathematics for middle school students should take into account the (levelopmental needs of adolescents. For this to happen, professional develop MATHEMATICS EDUCATION IN THE MIDDLE GRADES

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ment should be part of the day-to-day administrative functioning of the school. Robert FeIner, National Center on Public Education, University of Rhode Island, spoke about project work that focused on high performing learning communities. A key finding was that if educational improvement efforts do not attend to the full ecology of the setting, they will ultimately fail. Although some schools have success in raising student scores for a time, the success is not sustainable, and, in fact, according to this research, raising student achievement over time is related to the degree of implementation of key structural changes in the school. He also men- tioned that parent involvement that correlates with gains in student achieve- ment is "sending home information about how to work with and talk to your children." In response to questions, panelists pointed out that raising expecta- tions is a critical part of raising student achievement, that structural changes should be accompanied by thoughtful support for teachers, and that even though we continue to improve, the task keeps changing, masking the gains. CLOSING REMARKS In his closing remarks, Edward Silver noted the dual commitment of the Convocation participants: to enhancing the quality and quantity of mathematics EXEC UTIVE SUMMARY learning in the middle grades and to a(l(lressing other nee(ls of young a(loles- cents. He suggested a major concern is how to make mathematics interesting and important to young adolescents. Silver pointed out that the examples presented during the Convocation indicated students can be interested in the mathematical tasks we give them, in the mathematics itself, or in the process of struggling with the tasks, and that the role of the teacher is to cultivate this interest. He challenged the audience to contrast their own view of algebra with the algebraic ideas that were presented in the Convocation sessions and to renect on what it means to say that students are learning algebra and what it would mean for ah students to learn algebra. He advocated a systematic examination of different instructional and curricular arrangements designed to have ah students learn algebra. His closing comments addressed the issue of using the generalist/specialist notion as a way to set up a false dichotomy and caped for thinking about ways to form a commu- nity with a joint identity that moves the Convocation agenda forward. ACTION CONFERENCE ON THE NATURE AND IMPACT OF ALGEBRA AT THE MIDDLE GRADES The agenda for the Action Conference was (lesigne(1 to bring attention to

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different possibilities for algebra in the middle grades and to the issues in- volve(1 in implementing any of these possibilities. Discussion was framed by six questions presented by the Confer- ence organizer, Hyman Bass, a math- ematician from Columbia University. The questions covered the following topics: attention to subject matter vs. attention to students; algebra as the language of mathematics; real world contexts vs. generalization and abstrac- tion; covering mathematics vs. uncover- ing mathematics; situating algebra in the mathematics curriculum; materials, design, selection criteria for mathemat- ics curriculum. km Fey from the University of Maryland suggested that the important aspects of algebra are the concepts and techniques for reasoning about quantitative conditions and relationships. With this as a theme, Fey claimed that moving high school aIge- bra into the middle grades wait not be sufficient. Al Cuoco, Director of the Center for Mathematics Education at EDC, presented a view of algebra that relied more on symbols and problems from the world of mathematics, with an emphasis on the ways of thinking that can emerge from reasoning about calculations and about operations. Bass suggested that Fey's and Cuoco's approaches were two (lifferent aspects of the same thin". OritZaslavsky,a mathematics education researcher from Israel, postulated that learning is about constructing meaning that can change over time, across learners, anti across contexts. Learning algebra in the mi(l(lle gra(les is just a beginning, where examples play a critical role and the issue of representation is inherent. The second day of the conference featured approaches to a middle grades algebra curriculum from the Connected Mathematics Project, University of Chicago School Mathematics Project, Mathematics in Context, and Saxon Mathematics in which the presenters described the nature of their algebra stran(1 anti what works in practice. As part of a pane} on general implementa- tion issues, Anne Bartel, from the Minnesota project SciMath, pointed out that many of the issues are tied to people's belief systems about whether algebra is focused on skills or thinking and about what "algebra for all" really means. She closed with a discussion of the characteristics of effective profes- signal development, including the need to make the algebra content an(1 corre- spon(ling instructional strategies ex- plicit. Vern Williams, Gifted anti Tal- ente(1 Coordinator from a Virginia mi(l(lle school, emphasize(1 that some children nee(1 more than the norm with an emphasis on theory, structure, an problem solving. These students nee(1 to be challenge(1 every (lay, anti a gifted and talented course in algebra opens the MATHEMATICS EDUCATION IN THE MIDDLE GRADES

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universe of mathematics for them. Nancy Doda, National-Louis University, disagreed with Williams and advocated that all students needed to be chal- lenged and raised issues of equity. In closing, participants discussed algebra in relation to mathematics content, curricular design, and use of research in the context of a search for guidance on how to scale up promising programs to realize improved mathematics learning for more students. ACTION CONFERENCE ON RESEARCH IN THE TEACHING AND LEARNING OF MATHEMATICS IN THE MIDDLE GRADES The focus of the conference was to help define research required to better understand and articulate the assump- tions that underpin activities aimed at improving the mathematics curriculum, teaching, and learning in the middle gra(les. The group was charged with making suggestions to those in the field about what further research is needed and advising the Department of Educa- tion and National Science Foundation on where strategic investments in research might be made. Invited talks were organized to move from broad theoreti- cal and practical issues on how to address research to specific research EXEC UTIVE SUMMARY efforts to the applications of research knowledge to recent curriculum devel opment projects. From the first perspective, lames Hiebert, University of Delaware, described a tension between solving practical problems and doing good research and offered a framework to better understand and resolve these tensions. Alan Schoenfeld, University of California-Berkeley, responded to Hiebert by arguing for a stronger theoretical base for and through research. Mary Kay Stein illustrated how research can grow from attempts to solve problems of practice. Richar Mesh, In(liana University-Pur(lue, commented on how thinking can be change(1 over time anti the nee(1 to be explicit about the big ideas in the middle grades curriculum. lames Fey, University 0 f Mary}an(1, and Koeno Gravemeijer, Freudenthal Institute, the Netherlands, described ways that the extent research on rational number anti proportional reasoning shape(1 (1esign (1ecisions in their respective curriculum projects. Ju(ly Sow(ler, University of California- San Diego, (liscusse(1 ways to (1eepen teachers' knowledge of mathematics. The participants ma(le recommen(la- tions in three areas: teaching anti teacher learning; student learning; anti communicating with a variety of interested constituencies.

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ACTION CONFERENCE ON THE PROFESSIONAL DEVELOPMENT OF TEACHERS OF MATHEMATICS IN THE MIDDLE GRADES The Action Conference was designed to afford an opportunity for participants to examine promising approaches to professional development by creating an analytic and practical conversation about the sorts of opportunities in professional development most likely to lead to teachers' learning and improvements in their practice. Deborah Ball, University of Michigan, conference organizer, framed the discussion from the vantage point of teacher educators considering sites through which teachers might most profitably learn mathematics content needed in teaching, based on tasks which teachers regularly do as pert of their leaching. Participants discussed what is known about profes- sional (levelopment, teacher learning, anti the improvement of practice. As an initial example of how teachers use knowledge of content to shape their teaching, Ball together with Joan Ferrini-Mun(ly, Center for Science, Mathematics, and Engineering Educa- tion engage(1 participants in reformulat- ing a mathematical task and considering the mathematical knowledge used to create anti evaluate these new tasks. Margaret Smith, Pennsylvania State University, involved the participants in a case stu(ly of student work anti (liscusse(1 how analyzing student responses led to a discussion of the importance of the mathematical knowI- edge of the teacher. Karen Economopoulos, TERC, posed two questions for reflection and discussion: How might curriculum materials offer professional (levelopment opportuni- ties for teachers and how can these materials influence or support teach- ers' daily decisions? Nanette Seago, Mathematics Renaissance Project, using a video of an eighth grade math- ematics lesson, facilitated a discussion of the use of videotape as an instruc- tional medium in professional (levelop- ment. The closing session feature(1 a pane} that presented their reflections on the improvement of professional (levelopment. Iris Weiss, from Horizon Research, argue(1 for the nee(1 to help teachers (levelop some way to filter and make (recisions, anti raise(1 a concern about how to scale up professional development models. John Moyer from Marquette University, reflected on professional (levelopment with urban, large city middle grades teach- ers using teacher responses to an observer's comments to promote teachers' reflection on their practice. Stephanie Williamson, Louisiana Systemic Initiative, described the work MATHEMATICS EDUCATION IN THE MIDDLE GRADES

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done to build collaboration among school systems and universities in Louisiana that led to the development of a document used to guide decisions about professional development pro- grams. Participant's comments at EXEC UTIVE SUMMARY the end indicated that the Conference did take seriously professional devel- opment as a field and attempted to create a frame for thinking about theoretical, research, and practice- based learning.