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At the beginning of the Convocation, the participants were invited to think about areas of agreement and continuing challenges as they took part in the Convocation activities. Given that the Convocation involved over 400 partici- pants with diverse backgrounds and perspectives, the conversations were far reaching. Mathematics education is a discipline in which it is difficult to find absolutes. Many factors contribute to successful mathematics teaching and learning an(1 what is perceive(1 as con- vincing evidence from one perspective is not at all convincing from another. The fohow~ng observations renect comments made in the reports from the final discussion session during which each group was asked to make statements about areas of agreement and challenges or issues needing more work. The observations are a sampling of the many interesting and important ideas raised. AREAS OF AGREEMENT WITHIN DISCUSSION GROUPS The areas of agreements offered by the discussion groups ranged from specific statements about important content to agreeing about a set of challenges faced by the education community as they worked to improve mathematics education at the mi(l(lle grades. The set below renect some of the thinking. It is important to enhance the quality anti (1epth of mathematics learning in the middle grades and to sensitively a(l(lress the needs of young a(loles- cents as in(livi(luals as well as learn- ers. Participants observed that it is possible to honor these commitments to students and to content simulta- neously in mi(l(lle gra(les mathemat- ics classrooms, but the intent to do so should be clearly highlighted in the school structure anti organization anti (lelivery of learning. Participants commented on a clear anti competing need to have higher expectations for all mi(l(lle gra(les students. Teachers, schools, an(1 parents should have high expectations for student achievement in mathemat- ics and recognize that to realize these expectations will take renewed effort and commitment from everyone. .

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. . . To deliver the kind of mathematics content in ways that respects middle grades students as learners demands a well prepared and motivated teacher. Few existing teacher prepa- ration programs meet this need, and certification requirements do not support adequate content and peda- gogical preparation. To achieve the kind of instruction that wall maximize learning for all students in middle grades, the issue of on-going professional development activities in which teachers focus on their practice and become part of a community with shared goals must be addressed. Teachers in such a community {eke time to renect on their teaching and its relation to what students learn and work together to improve what they do in their classrooms. The organizational structure of a school system is important but equally as important is the support and commitment for mathematical excellence and for equal opportunities for all. Teachers need adequate time to plan their lessons and work through their curriculum as well as support for staff development. CHALLENGES RAISED BY DISCUSSION GROUPS The improvement of mathematics education in the middle grades faces PARTICIPANT OBSERVATIONS curricular, pedagogical, and contextual challenges. The following are some of the challenges and issues highlighted by the discussion groups in their final reports. Good curriculum and pedagogy may be insufficient if policy and political issues are not taken into consider- ation. Because public perception of a quality mathematics program at the mi(l(lle gra(les may be in connict with the goals of mathematics educators, success in implementing new pro- grams (lepen(ls on buil(ling public understanding of the changes. Fami- lies should be informe(1 about the content expectations of the overall program and specific grade levels within the program. Districtleve} policy makers should understand the nature of the program and provide support within the system to make changes. Mathematics education researchers should be convinced to investigate questions around reform issues anti to produce (1ata that will help the public understand that schools and districts are making informe(1 (1ecisions about teaching anti learning mathematics. Both internal articulation within school systems anti external articula- tion within states are critical. Man- (lates concerning assessment, stan- dards, and curriculum often signal different messages about what is

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important in mathematics at the middle grades. When the mathemati- cal content for the middle grades is not clearly delineated, there is overlap and confusion about content, particu- larly in the areas of number and algebra. In teaching students about number and operations, the curricu- lum can become repetitive contribut- ing to the "mile wide, inch deep" characterization of the middle grades curriculum by TIMSS. On the other hand, the emphasis on algebra in the middle grades has raised articulation issues about high school credits in middle grades, or students repeating content at the high school level with no recognition of their work in the middle grades. The practice of grouping students by perceived ability can appear to be in direct connict with the goal of provid- ing the same mathematics for all students. While many teachers support heterogeneous grouping in theory, in practice they find it difficult to implement. The need to simulta- neously provide remediation and acceleration in large classes can prove overwhelming, yet separating stu- (lents for (lifferent academic pro- grams raised questions for the teach- ers about equitable mathematics opportunities for all students. The nature anti role of algebra at the mi(l(lle gra(les raises curricular MATHEMATICS EDUCATION IN THE MIDDLE GRADES questions as well as articulation issues. Too much emphasis on algebra comes at the expense of other important mathematical topics. For teachers to be successful with algebra embe(l(le(1 throughout the curricu- lum, they should have a high level of mathematical knowledge, for which many have not been prepared. Imple- menting a program that teaches algebra only to a select set of students inhibits organizing students into houses, a typical middle grades approach, or forces an arbitrary ability grouping layer on the housing structure. Acceleration for some can lower expectations for others. It is important for mi(l(lle gra(les students to see the connections among mathematical topics, their lives, anti what they learn. The challenge, however, is to achieve a balance between student needs and content; integration should not happen at the expense of mathematics. An overem- phasis on developing students as in(livi(luals can result in a loss of instructional focus on content. Stu- (lents should come to un(lerstan(1 that while mathematics is not taught in a vacuum, it is a (liscipline of its own. Unless carefully constructed where the mathematics is not just a(l(le(1 on when seen as useful, thematic units can be a (letriment to a coherent anti complete mi(l(lle gra(les mathematical

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. curriculum. Many middle grades teachers do not have mathematical backgrounds and find it difficult to make links with mathematics. Communication among the diverse audiences involved in middle grades mathematics programs is problem- atic. The term "developmentally appropriate" is not well un(lerstoo(1 by many in the broad education commu- nity. Some questioned what it meant to characterize a mathematical con- cept as (levelopmentally appropriate PARTICIPANT OBSERVATIONS for a student. Integrated mathematics means different things at the high school level where it indicates blend- ing of mathematical topics with no clear demarcation between algebra and geometry and at the middle grades where it usually refers to blending of content from different disciplines, mathematics with science and literature, for example. To have productive conversations there has to be an attempt to develop a common language.

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