| Mathematical problem posing as a link between algorithmic thinking and conceptual knowledge | 45$-$60 |
Abstract
The paper proposes an idea of using problem posing as a link between conceptual and procedural mathematical knowledge. Two levels of conceptual understanding---basic and advanced---have been considered. Examples of the interplay between the two types of knowledge are presented. The paper is informed by the author's work with teacher candidates in different technology-enhanced mathematics education courses.
Keywords: problem solving; problem posing; technology; conceptual understanding; algorithmic thinking; reflective inquiry.
MSC Subject Classification: 97C30, 97D50
MathEduc Subject Classification: C33, D53
| From differentiation in affine spaces to connections | 61$-$80 |
Abstract
Connections and covariant derivatives are usually taught as a basic concept of differential geometry, or more precisely, of differential calculus on smooth manifolds. In this article we show that the need for covariant derivatives may arise, or at lest be motivated, even in a linear situation. We show how a generalization of the notion of a derivative of a function to a derivative of a map between affine spaces naturally leads to the notion of a connection. Covariant derivative is defined in the framework of vector bundles and connections in a way which preserves standard properties of derivatives. A special attention is paid on the role played by zero--sets of a first derivative in several contexts.
Keywords: Affine space; second derivative; connection; vector bundle.
MSC Subject Classification: 97I99, 97G99, 53--01
MathEduc Subject Classification: I95, G95
| Cousin's theorem and two other basic properties equivalent to it | 81$-$83 |
Abstract
We present direct proofs of the equivalence between both the compactness and the connectedness of the interval $[a,b]$ and the Cousin's theorem in ways that allow their beauty to go through.
Keywords: Cousin's theorem; compactness; connectedness.
MSC Subject Classification: 97I20
MathEduc Subject Classification: I25
| Division---a systematic search for true digits, II | 84$-$92 |
Abstract
This paper is an improvement and a continuation of the author's previous paper published under the same title (The Teaching of Mathematics, 2005, vol. VIII, 2, 89--101).
Keywords: long division; canonical division; ``increase-by-one'' methods.
MSC Subject Classification: 97F30
MathEduc Subject Classification: F32