Catherine Ulrich is an assistant professor of mathematics education in the School of Education at Virginia Tech. She studies how students’ reasoning with integers is constrained and enabled by their current conceptions of addition and subtraction and their ways of operating with counting numbers. She is also studying the interplay between the development of additive and multiplicative reasoning at the middle school level. She teaches a doctoral course on children’s construction of number and works with pre-service teachers at both the elementary and secondary levels in developing effective methods of mathematics instruction. Prior to being a faculty member, she taught secondary mathematics for 6 years.

Steffe L. P. & Ulrich C. (2013) Constructivist teaching experiment. In: Lerman S. (ed.) Encyclopedia of mathematics education. Springer, Berlin: 102–109. Fulltext at https://cepa.info/2959

In an epistemology where mathematics teaching is viewed as goal-directed interactive communication in a consensual domain of experience, mathematics learning is viewed as reflective abstraction in the context of scheme theory. In this view, mathematical knowledge is understood as coordinated schemes of action and operation. Consequently, research methodology has to be designed as a flexible, investigative tool. The constructivist teaching experiment is a technique that was designed to investigate children’s mathematical knowledge it might be learned in the context of mathematics teaching (Cobb & Steffe, 1983; Hunting, 1983; Steffe, 1984). In a teaching experiment, the role of the researcher changes from an observer who intends to establish objective scientific facts to an actor who intends to construct models that are relative to his or her own actions.

Steffe L. P. & Ulrich C. (2014) The constructivist teaching experiment. In: Lerman S. (ed.) Encyclopedia of mathematics education. Springer, Berlin: 102–109. Fulltext at https://cepa.info/6312

Excerpt: The constructivist teaching experiment emerged in the United States circa 1975 in an attempt to understand children’s numerical thinking and how that thinking might change rather than to rely on models that were developed outside of mathematics education for purposes other than educating children. The use of the constructivist teaching experiment in the United State was buttressed by versions of the teaching experiment methodology that were being used already by researchers in the Academy of Pedagogical Sciences in the then Union of Soviet Socialist Republics. The work at the Academy of Pedagogical Sciences provided academic respectability for what was then a major departure in the practice of research in mathematics education in the United States, not only in terms of research methods but more crucially in terms of the research orientation of the methodology.

Tillema E. S., Hackenberg A. J., Ulrich C. & Norton A. (2014) Authors’ Response: Interaction: A Core Hypothesis of Radical Constructivist Epistemology. Constructivist Foundations 9(3): 354–359. Fulltext at https://cepa.info/1092

Upshot: In reading the commentaries, we were struck by the fact that all of them were in some capacity related to what we consider a core principle of radical constructivism - interaction. We characterize interaction from a radical constructivist perspective, and then discuss how the authors of the commentaries address one kind of interaction.

Ulrich C. (2014) Issues Around Reflective Abstraction in Mathematics Education. Constructivist Foundations 9(3): 370–371. Fulltext at https://cepa.info/1094

Open peer commentary on the article “Examining the Role of Re-Presentation in Mathematical Problem Solving: An Application of Ernst von Glasersfeld’s Conceptual Analysis” by Victor V. Cifarelli & Volkan Sevim. Upshot: Cifarelli and Sevim’s analysis of Marie’s problem solving activity raises two questions for me. The first regards what Marie is reflectively abstracting: the use of the generic phrase her solution activity finesses a largely unarticulated disagreement in the mathematics education community about what the nature of actions are in Piaget’s theory. The second question involves the implications that radical constructivist research in mathematics has for day-to-day mathematics instruction.

Ulrich C., Tillema E. S., Hackenberg A. J. & Norton A. (2014) Constructivist Model Building: Empirical Examples From Mathematics Education. Constructivist Foundations 9(3): 328–339. Fulltext at https://cepa.info/1084

Context: This paper outlines how radical constructivist theory has led to a particular methodological technique, developing second-order models of student thinking, that has helped mathematics educators to be more effective teachers of their students. Problem: The paper addresses the problem of how radical constructivist theory has been used to explain and engender more viable adaptations to the complexities of teaching and learning. Method: The paper presents empirical data from teaching experiments that illustrate the process of second-order model building. Results: The result of the paper is an illustration of how second-order models are developed and how this process, as it progresses, supports teachers to be more effective. Implications: This paper has the implication that radical constructivism has the potential to impact practice.