it, not just how much time is allocated for mathematics but how that time is spent. They need to investigate not just whether calculators or other resources are used, but how they are used.70 Research that looks across countries can provide a sharper picture of what matters in instruction aimed at developing proficiency.

A second set of issues concerns instruction over time. Although learning is fundamentally temporal, too little research has addressed the ways in which instruction develops over time. Many studies are restricted to isolated fragments of teaching and learning, providing little understanding of how the interactions of teachers, students, and content emerge over time, and how earlier interactions shape later ones. How do ideas developed in class affect later work, and what affects teachers’ and students’ ability and inclination to make such links, as well as their use of such connections over time? How is time used, and how does its use by teachers and students affect the quality of instruction?

A third arena concerns students and how their diversity affects instruction. Too little research offers insight into the experience of students and how the instruction offered, together with their responses to it, affects their learning. Still more important, there are too few well-designed studies that would offer insight into how instruction might be developed to work effectively for all students. Too often, research on classroom teaching and learning either studies faceless, colorless students and teachers out of context, or it is situated in particular contexts but lacks a design that permits analyses that could provide the knowledge needed for effective instruction in mathematics.

Fourth, too little research has addressed what it takes for students to learn mathematics in class. What do students need to do, and know how to do, in order to profit from the instruction offered by each of our four teachers? A cursory glance at any mathematics class makes plain that the skills, abilities, knowledge, and dispositions displayed by students are not the same, and yet teachers and researchers rarely attend to what students need to know and be able to do in order to use instruction effectively. People seem to assume implicitly that instruction acts on students and that opportunities to learn are actually moments of learning. Research that examined both what students have to know and do in mathematics instruction and what teachers can do to enable all students to make use of that instruction would add significantly to the knowledge base on teaching and learning mathematics.

A fifth set of issues has to do with reconnecting research on teacher knowledge with instructional effectiveness. Although most people believe that teachers’ knowledge of mathematics and of students makes a difference for



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