# Selected Presentations

## (Page in Progress)

### 2008, U. Michigan

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

### 2012, Variation in Proportionality

An argument that proportionality as equivalent to linearity entails images of smooth continuous variation.

### 2012, Integers

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.

### 2013, Schemes for Thinking with Magnitudes

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.

### 2014, Houston, We have a problem: Math Education in the U.S. and in the World

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.

### 2016, Kaput Lecture, University of Massachusetts at Dartmouth

Under the Radar: High School Mathematics Teachers' Mathematical Meanings and Lessons for the Mathematical Preparation of Teachers

Results and implications of the NSF project Mathematical Meanings for Teaching secondary mathematics (MMTsm) for the preparation and professional development of high school mathematics teachers. Video at YouTube. Slides here.

### 2016, University of Adger, Norway

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.

### 2018, A Rigorous Approach to Conceptual Calculus

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.

### 2018, Imagery and the Reconstitution of Meaning in Advanced Settings

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.

### 2019, Unpacking Simultaneous Graphs

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.

### 2019, Making the Fundamental Theorem of Calculus fundamental to student's calculus

Plenary presentation given at the Conference on Calculus in High School and College, University of Adger, Kristiansaand, Norway, August 6, 2019. Click here for the paper that accompanied this presentation.

### 2019, Israel

I gave four lectures in Israel during November of 2019 as Fellow of the Israel Academy for Humanities and Social Sciences

• Structure, like beauty, is in the eye of the beholder. (Weizmann Institute, Nov. 4)
I present data from an international study of U.S. and South Korean high school mathematics teachers’ schemes for seeing structure in mathematical expressions. I embed the presentation within a theory of meaning and conveyance of meaning particularly suited for thinking about classroom interactions between teachers and students.
• Taking Quantitative Reasoning Seriously in Calculus. (University of Haifa, Nov. 13)
Many people think quantitative reasoning is fundamentally about using concrete settings. I clarify that the goal of having students reason quantitatively entails the goal that they develop symbolic reasoning as a means of expression. I also give examples of difficulties instructors have with envisioning instruction to support students reasoning quantitatively and what it looks like when they do.
• Mathematical Meanings for Teaching Mathematics. (Tel Aviv University, Nov. 19)
I distinguish between mathematical meanings and mathematical knowledge. I will explain, with illustrations, why this distinction is important for theorizing about teachers’ mathematics teaching and about students’ mathematical learning in the context of instruction.
• US High School Students’ Preparation for Calculus. (Ben Gurion University, Nov. 20)
I report on a recent study of U.S. high school students' preparation for calculus in terms of their understandings of foundational ideas like variable, function, and rate of change.

### 2020, University of Sonora, Mexico

This is an invited plenary at the 30th National Conference of Research and Teaching in Mathematics. In it I highlighted the ways in which DIRACC Calculus synthesizes ideas of Newton, Leibniz, Robinson, and Glasersfeld.

I and Fabio Milner also presented a workshop introducing ideas of DIRACC to U. Sonora faculty and graduate students and high school mathematics teachers in the State of Sonora.