Three parts on fostering Mathematical Knowledge Knowledge for Teaching mathematics

- All three parts: (1) The Didactic Triad, (2) Supporting teachers' internalizing the Triad, (3) Phases in Developing a Key Pedagogical Understanding
- Last two parts: (2) Supporting teachers' internalizing the Triad, (3) Phases in developing a Key Pedagogical Understanding
- Last part: (3) Phases in developing a Key Pedagogical Understanding

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.

Keynote presentation given at *The Conference on Epistemic Algebra Students*, Athens, GA. I present a framework for thinking about students' development of the idea of magnitude, starting from perceptual magnitude and ending with relative magnitude as the foundation for understanding physical quantities in science.

Presentation given to the Chandler, Arizona Parents of Academically Talented Children. Theme of presentation is, since WW II, the U.S. has had its best ideas in many areas (including mathematics education) picked up by other countries while ignored in the U.S. The U.S. has created a culture of low expectations of student learning. Other countries have built cultures of high expectations for student reasoning with curricula and instruction to support them.

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.

The 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.

Animations developed as a companion to an article in *Oncore*, the online journal for the Arizona Association of Mathematics Teachers. The article and animations constitute an example of how animations can be used as didactic objects in mathematics instruction.