by Cathy Kessel and Liping Ma
October 2000
What we think that elementary teachers need is a solid and substantial school mathematics that forms a web of connections—a road system connecting students' understandings with the mathematics and statistics that we want students to know. Teachers, education researchers, curriculum developers, and mathematicians all know some locations and pathways. Developmental psychologists and teachers of early grades know about preschoolers' and kindergarteners' understandings. Teachers and mathematicseducation researchers know about the kinds of understandings that students receive from particular curricula. Mathematicians and statisticians know their disciplines well. But only rarely do members of these different groups compare their pieces of knowledge and think about paths between them or requirements for constructing such paths.
What is needed to create this school mathematics is substantive collaboration that maps the routes from early perceptions of change, pattern, shape, number, and space to sophisticated understandings of measurement, functions, geometry, algebra, probability, and statistics. Each of these early categories contains rudiments of subsequent categories—but does not become any one of those subsequent categories. Instead, each provides the beginnings of later learning. Pattern contributes to geometry as well as to function and to algebra. Shape has obvious geometric entailments, but it also contributes to measurement and to calculus. Number contributes not only to algebra but also to function (think of number patterns) and to geometry (think of figurate numbers). Space is a rudiment of geometry, as well as of measurement and of topology.
Solving problems, proving conjectures or finding counterexamples, making mathematical connections, and using representations to do and communicate these activities occur in rudimentary form in early years and in more sophisticated form in later years. A solid school mathematics program must provide pathways that lead, for example, to understanding that certain algorithms (or strategies, concepts, proofs, or representations) that once looked different are really the same, to the propensity to look for these connections, and to the ability to explain them.
Prospective teachers need a solid basis on which to build their understanding of such a school mathematics—a basis that includes not only mathematical knowledge and mathematical attitudes but also a sense of how students learn. When they begin teaching, they need welldesigned curriculum materials that are based on that school mathematics. They need time (and energy!) to read, reflect on, and discuss those materials. And they need assessments that are based on that school mathematics and that support them in helping students learn.
Others must understand these needs well enough to make wise decisions concerning teachers and students. First, mathematicians and teacher educators must help preservice teachers gain a solid mathematical basis. District and school officials must foster working conditions that allow teachers' knowledge to grow. Curriculum and assessment developers must create appropriate instructional materials—moreover, publishers must understand that developing such materials takes time and must be willing to publish them. Finally, when considering curricula, members of state and local boards must know routes to mathematical understanding, as well as the nature of this understanding, when selecting tests for students and setting criteria for teacher certification.
Elementary mathematics is not superficial. Neither is preparing and supporting elementary teachers.
Cathy Kessel, kessel@soe.berkeley.edu, is an independent scholar. She is the managing editor for Research in Collegiate Mathematics Education and was an additional writer for NCTM's Principles and Standards for School Mathematics (2000).
Liping Ma, lipingma@leland.stanford.edu, is an independent scholar and the author of Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States. She and Cathy Kessel are currently working on an intervention for elementary mathematics students.

