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. 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.
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. When they begin teaching, they need well-designed curriculum materials that are based on that school mathematics. 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.