Bifurcations – mathematical and cognitive –
via the epistemic actions model for abstraction in context

Tommy Dreyfus, Tel Aviv University, Israel

The emergence of a new mathematical knowledge construct may be considered as a process of abstraction. In order to help researchers analyze such processes of abstraction, as they occur in specific learning contexts, Hershkowitz, Schwarz and Dreyfus (2001) have proposed the RBC-model for abstraction in context. The model is based on the idea of epistemic action, and specifically on the three epistemic actions of Recognizing, Building-with, and Constructing (whence its name: RBC-model).

Examples of knowledge constructs include fraction as a number, the distributive law, algebra as a tool for proof, 2-dimensional sample spaces in probability, rate of change as a function, the equivalence of infinite sets, and bifurcation in dynamic systems. Contexts include historical (students’ personal and common learning experiences), social (interaction with others such as another student or group of students, a tutor, or a teacher) and material (computer tools, books, worksheets) components.

In the talk, I plan to present the basics of the RBC-model, using the example of the distributive law as constructed by an interacting pair of junior high school students. I will then concentrate on a number of modifications of the model that were found relevant when using it for analyzing processes of abstraction in advanced mathematics, specifically for the topic of bifurcations in dynamic processes. For example, I will show how constructing actions may go on in parallel and interact, and more specifically how they may bifurcate, thus giving a double meaning to the idea of bifurcation.

Hershkowitz, R., Schwarz, B., & Dreyfus, T. (2001). Abstraction in Context: Epistemic Actions.  Journal for Research in Mathematics Education, 32 (2), 195-222.

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