Three parts on fostering Mathematical Knowledge Knowledge for Teaching mathematics
An argument that proportionality as equivalent to linearity entails images of smooth continuous variation.
An illustration developed for students in MTE 320 Conceptual Foundations of Curriculum and Assessment Grades 7-12. It reflects the mathematical construction of integers from whole numbers as equivalence classes of differences. It also reflects Vergnaud's (1983) analysis of students' confusions of whole number amounts and changes in whole number amounts, Thompson and Dreyfus (1988) investigation of two students' construction of integers as transformations, and Thompson's (1993) analysis of students' difficulties in thinking of differences (results of additive comparisons) as quantities in themselves.
Data on U.S. and South Korean teachers meanings for function and function notation given at ICME-16 (Germany) and published in Thompson & Milner. The presentation also contains data on U.S. and South Korean teachers' structure sense — ways of thinking about the structure of symbolic expressioins.
Role of differentials in establishing links between rate of change and accumulation functions in DIRACC Calculus 1 and 2.Presented to ASU's First Year Mathematics faculty.
A presentation given at the Collective Contributions of Constructivist Research Programs held at NYU September 28-30, 2018. I illustrate the idea of differential as a basic idea and how it must be reconstituted in meaning when the idea is used in more advanced contexts. Understanding the role of imagery in the development of schemes is key.