Robert Speiser

How far can you go with block towers

Abstract This report focuses on the development of combinatorial reasoning of a 14-year-old child, Stephanie, who is investigating binomial coefficients and combinations in relationship to the binomial expansion and the mapping of the binomial expansion to Pascals triangle. This research reports on Stephanies examination of patterns and symbolic representations of the coefficients in the binomial expansion using ideas from earlier explorations with towers in grades 3–5 to examine recursive processes and to explain the addition rule in Pascals triangle. This early work enabled her to build particular organization and classification schemes that she draws upon to explain her more abstract ideas.

Modeling Perspectives in Math Education Research

Too often powerful and beautiful mathematical ideas are learned (and taught) in a procedural manner, thus depriving students of an experience in which they create and refine ideas for themselves. As a first step toward improving the current undesirable situation in undergraduate mathematics education, this chapter describes several different modeling perspectives and their implications for teaching and learning.

Block Towers: From Concrete Objects to Conceptual Imagination

In previous chapters, we looked at the development of various forms of reasoning in students working in a classroom in small group settings. In this chapter, we focus on an individual student – we examine Stephanie’s development of combinatorial reasoning. In previous chapters, we saw how Stephanie, working with others and on her own, made sense of the towers and pizza problems. In this chapter we see how Stephanie extended that work. In her examination of patterns and symbolic representations of the coefficients in the binomial expansion, using ideas from earlier explorations with towers in grades 3–5, she examined several fundamental recursive processes, including the addition rule in Pascal’s Triangle.

Formal meromorphic functions and cohomology on an algebraic variety

Let X be a projective Gorenstein variety, Y ⊂ X a proper closed subscheme such that X is smooth at all points of Y , so that the formal completion of X along Y is regular.