The National Council of Teachers of Mathematics' Curriculum and Evaluation Standards in 1989 was pivotal in mathematics reform. The National Science Foundation funded several curriculum projects to address the vision described in the Standards. This study investigates students' learning in one of these Standards‐based curricula, the Connected Mathematics Project (CMP). The authors of CMP believe that the teaching and learning of algebra is an ongoing activity woven through the entire curriculum, rather than being parceled into a single grade level. The content of the study investigates students' ability to symbolically generalize functions. The data regards the solutions of four performance tasks dealing with three different types of relationships—linear, quadratic, and exponential situations—completed by five pairs of eighth‐grade students. The major finding claims that middle to high achieving students who had 3 years in the CMP curriculum demonstrated achievement in five strands of mathematical proficiency of a significant piece of algebra.
When worthwhile mathematical tasks are used in classrooms, they should also become a crucial element of assessment. For teachers, using these tasks in classrooms requires a different way to analyze student thinking than the traditional assessment model. Looking carefully at students' written work on worthwhile mathematical tasks and listening carefully while students explore these worthwhile tasks can contribute to a teacher's professional development. This paper reports on a professional development activity in which teachers analyzed mathematical tasks, predicted students' achievement on tasks, evaluated students' written work, listened to students' reasoning, and assessed students' understanding. Teachers' engagement in this way can help them develop flexibility and proficiency in the evaluation of their own students' work. These experiences allow teachers the opportunity to recognize students' potential, strengthen their own mathematical understanding, and engage in conversations with peers about assessment and instruction.
How do Your Students do When you Ask them to generalize? Do they reason numerically or geometrically? Can they make connections among representations? Algebraic reasoning and representations are foundational for middle-grades students as they prepare to enter high school and pursue future studies. Many policymakers suggest an “algebra for all” approach (Silver 1997). Although there is no consensus on what this algebra might look like, many agree that understanding patterns, relations, and functions and representing and analyzing mathematical situations with variables are important components of algebra (NCTM 2000). Studying patterns in middle school builds a foundation for more formal investigations with functions in high school.
In recent years, elementary school preservice teachers often have a fieldwork experience before student teaching. However, the quality of these experiences varies greatly (Wilson, Floden, and Ferrini-Mundy 2002). What supports a good fieldwork experience? Certainly we want students to be taught in classrooms in which they are asked to reason, represent, and communicate. At the University of Michigan–Dearborn, we strive to find these sorts of placements—first, by working with districts using reform-based materials and, second, by asking local district leaders to identify exemplary teachers. Moreover, our future teachers have experienced inquiry-based lessons in the mathematics and science courses they take as university students. Even with this careful design, when we observe classrooms in the field we find that future teachers focus on surface aspects rather than on mathematical thinking. This experience concurs with the findings of Moore (2003) and Putnam and Borko (2000), who found that novices focus on management and procedures, not on learning. So we asked ourselves, How do we sharpen the future teachers' focus?
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