Cobb P. (1987) Information-processing psychology and mathematics education: A constructivist perspective. Journal of Mathematical Behavior 6(1): 3–40. https://cepa.info/2968

Discusses the implications of information processing psychology for mathematics education, with a focus on the works of schema theorists such as D. E. Rumelhart and D. A. Norman and R. Glaser and production system theorists such as J. H. Larkin, J. G. Greeno, and J. R. Anderson. Learning is considered in terms of the actor’s and the observer’s perspective and the distinction between declarative and procedural knowledge. Comprehension and meaning in mathematics also are considered. The role of abstraction and generalization in the acquisition of mathematical knowledge is discussed, and the difference between helping children to “see, ” as opposed to construct abstract relationships is elucidated. The goal of teaching is to help students modify or restructure their existing schema in predetermined ways by finding instructional representations that enable students to construct their own expert representations.

Elby A. (2000) What students’ learning of representations tells us about constructivism. Journal of Mathematical Behavior 19(4): 481–502. https://cepa.info/4657

This article pulls into the empirical realm a longstanding theoretical debate about the prior knowledge students bring to bear when learning scientific concepts and representations. Misconceptions constructivists view the prior knowledge as stable alternate conceptions that apply robustly across multiple contexts. By contrast, fine-grained constructivists believe that much of students’ intuitive knowledge consists of unarticulated, loosely connected knowledge elements, the activation of which depends sensitively on context. By focusing on students’ intuitive knowledge about representations, and by fleshing out the two constructivist frameworks, I show that they lead to empirically different sets of predictions. Pilot studies demonstrate the feasibility of a full-fledged experimental program to decide which flavor of constructivism describes students more adequately.

This article presents a case study of a seven-year-old girl named Amanda who participated in an eighteen-week teaching experiment I conducted in order to model the development of her intuitive and informal topological ideas. I designed a new dynamic geometry environment that I used in each of the episodes of the teaching experiment to elicit these conceptions and further support their development. As the study progressed, I found that Amanda developed significant and authentic forms of geometric reasoning. It is these newly identified forms of reasoning, which I refer to as “qualitative geometry,” that have implications for the teaching and learning of geometry and for research into students’ mathematical reasoning. Relevance: This article describes a study for which I used Steffe & Thompson’s teaching experiment methodology to produce a learning trajectory (Steffe 2003, 2004) resulting from the actual teaching of children. In order to perform the conceptual analysis, the theoretical framework draws on von Glasersfeld’s scheme theory, which is an interpretation of Piaget’s theory of cognitive development.

Hackenberg A. J. (2007) Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. Journal of Mathematical Behavior 26: 27–47. https://cepa.info/764

This article communicates findings from a year-long constructivist teaching experiment about the relationship between four sixth grade students’ multiplicative structures and their construction of improper fractions. Students’ multiplicative structures are the units coordinations that they can take as given prior to activity – i.e., the units coordinations that they have interiorized. This research indicates that the construction of improper fractions requires having interiorized three levels of units. Students who have interiorized only two levels of units may operate with fractions greater than one, but they don’t produce improper fractions. These findings call for a revision in Steffe’s hypothesis (Steffe L. P. (2002). A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior 20: 267–307) that upon the construction of the splitting operation, students’ fractional schemes can be regarded as essentially including improper fractions. While the splitting operation seems crucial in the construction of improper fractions, it is not necessarily accompanied by the interiorization of three levels of units. Relevance: This article takes a radical constructivist approach to mathematical learning and develops local theory about how students’ units coordinations are related to the fraction schemes they can construct.

Hackenberg A. J. (2013) The fractional knowledge and algebraic reasoning of students with the first multiplicative concept. Journal of Mathematical Behavior 33: 1. https://cepa.info/992

To understand relationships between students’ quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. Six students, with each of three different multiplicative concepts, participated. This paper reports on the fractional knowledge and algebraic reasoning of six students with the most basic multiplicative concept. The fractional knowledge of these students was found to be consistent with prior research, in that the students had constructed partitioning and iteration operations but not disembedding operations, and that the students conceived of fractions as parts within wholes. The students’ iterating operations facilitated their work on algebra problems, but the lack of disembedding operations was a significant constraint in writing algebraic equations and expressions, as well as in generalizing relationships. Implications for teaching these students are discussed. Relevance: In this paper the author uses second-order models of students’ multiplicative concepts and fractional knowledge built from radical constructivism to explore relationships between students’ fractional knowledge and algebraic reasoning. The paper is therefore one contribution to the construction of second-order models of students’ algebraic reasoning, which is sorely needed by the field of mathematics education, particularly for students who struggle to learn algebra.

Hunt J. & Tzur R. (2017) Where is difference? Processes of mathematical remediation through a constructivist lens. The Journal of Mathematical Behavior 48: 62–76.

In this study, we challenge the deficit perspective on mathematical knowing and learning for children labeled as LD, focusing on their struggles not as a within student attribute, but rather as within teacher-learner interactions. We present two cases of fifth-grade students labeled LD as they interacted with a researcher-teacher during two constructivist-oriented teaching experiments designed to foster a concept of unit fraction. Data analysis revealed three main types of interactions, and how they changed over time, which seemed to support the students’ learning: Assess, Cause and Effect Reflection, and Comparison/Prediction Reflection. We thus argue for an intervention in interaction that occurs in the instructional process for students with LD, which should replace attempts to “fix” ‘deficiencies’ that we claim to contribute to disabling such students.

The basic hypothesis of the teaching experiment, The Child’s Construction of the Rational Numbers of Arithmetic (Steffe & Olive 1990), was that children’s fractional schemes can emerge as accommodations in their numerical counting schemes. This hypothesis is referred to as the reorganization hypothesis because when a new scheme is established by using another scheme in a novel way, the new scheme can be regarded as a reorganization of the prior scheme. In that case where children’s fractional schemes do emerge as accommodations in their numerical counting schemes, I regard the fractional schemes as superseding their earlier numerical counting schemes. If one scheme supersedes another, that does not mean the earlier scheme is replaced by the superseding scheme. Rather, it means that the superseding scheme solves the problems the earlier scheme solved but solves them better, and it solves new problems the earlier scheme did not solve. It is in this sense that we hypothesized children’s fractional schemes can supersede their numerical counting schemes and it is the sense in which we regarded numerical schemes as constructive mechanisms in the production of fractional schemes (Kieren, 1980). Relevance: This paper relates to Ernst von Glasersfeld’s reformulation of Piaget’s concept of scheme.

Steffe L. P. (2003) The fractional composition, commensurate fractional, and the common partitioning schemes of Jason and Laura: Grade 5. Journal of Mathematical Behavior 22(3): 237–295.

A case study of two 5th-Grade children, Jason and Laura, is presented who participated in the teaching experiment, Children’s Construction of the Rational Numbers of Arithmetic. The case study begins on the 29th of November of their 5th-Grade in school and ends on the 5th of April of the same school year. Two basic problems were of interest in the case study. The first was to provide an analysis of the concepts and operations that are involved in the construction of three fractional schemes: a commensurate fractional scheme, a fractional composition scheme, and a fractional adding scheme. The second was to provide an analysis of the contribution of interactive mathematical activity in the construction of these schemes. The phrase, “commensurate factional scheme” refers to the concepts and operations that are involved in transforming a given fraction into another fraction that are both measures of an identical quantity. Likewise, “fractional composition scheme” refers to the concepts and operations that are involved in finding how much, say, 1/3 of 1/4 of a quantity is of the whole quantity, and “fractional adding scheme” refers to the concepts and operations involved in finding how much, say, 1/3 of a quantity joined to 1/4 of a quantity is of the whole quantity. Critical protocols were abstracted from the teaching episodes with the two children that illustrate what is meant by the schemes, changes in the children’s concepts and operations, and the interactive mathematical activity that was involved. The body of the case study consists of an on-going analysis of the children’s interactive mathematical activity and changes in that activity. The last section of the case study consists of an analysis of the constitutive aspects of the children’s constructive activity, including the role of social interaction and nonverbal interactions of the children with each other and with the computer software we used in teaching the children.

Thompson P. W. (1982) Were lions to speak, we wouldn’t understand. The Journal of Mathematical Behavior 3(2): 147–165. https://cepa.info/3047

Discusses criteria for evaluating clinical research in mathematics education. It is suggested that any set of criteria must be consistent with the underlying epistemology from which that research is carried out. The implications of environmentalist vs constructionist world views of mathematics education for establishing criteria are examined, and a set of criteria applicable to constuctivist mathematical-education research is presented.