Steffe L. P. (1991) The constructivist teaching experiment: Illustrations and implications. In: Glasersfeld E. von (ed.) Radical constructivism in mathematics education. Kluwer, Dordrecht: 177–194. https://cepa.info/2098

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 and how 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. (1994) Children’s multiplying schemes. In: Harel G. & Confrey J. (eds.) The development of multiplicative reasoning in the learning of mathematics. SUNY Press, New York: 3–39.

The work on children’s multiplying and dividing schemes in which I am engaged is based on two theoretical assumptions, both of which I have substantiated. The initial assumption is that children, when faced with their first arithmetical problems, use their current mathematical schemes to attempt to solve them. They seem to persist in doing this in spite of their teacher’s explanation of accepted methods. Moreover, if the children’s attempts are successful in producing an answer that is seen as ‘‘correct” by the teacher, the teacher often remains unaware of the fact that the children use their own methods. These methods, however, eventually become dysfunctional in current school mathematics programs because the children are discouraged from using them in favor of standard methods, imposed upon them by their teachers.

Steffe L. P. (2002) A new hypothesis concerning children’s fractional knowledge. Journal of Mathematical Behavior 20: 267–307. https://cepa.info/1056

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.

Steffe L. P. (2004) On the construction of learning trajectories of children: The case of commensurate fractions. Mathematical Thinking and Learning 6(2): 129–162. https://cepa.info/2113

Learning trajectories are presented of 2 fifth-grade children, Jason and Laura, who participated in the teaching experiment, Children’s Construction of the Rational Numbers of Arithmetic. 5 teaching episodes were held with the 2 children, October 15 and November 1, 8, 15, and 22. During the fourth grade, the 2 children demonstrated distinctly different partitioning schemes-the equi-partitioning scheme (Jason) and the simultaneous partitioning scheme (Laura). At the outset of the children’s fifth grade, it was hypothesized that the differences in the 2 schemes would be manifest in the children’s production of fractions commensurate with a given fraction. During the October 15 teaching episode, Jason independently produced how much 3/4 of 1/4 of a stick was of the whole stick as a novelty, and it was inferred that he engaged in recursive partitioning operations. An analogous inference could not be made for Laura. The primary difference in the 2 children during the teaching episodes was Laura’s dependency on Jason’s independent explanations or actions to engage in the actions that were needed for her to be successful in explaining why a fraction such as 1/3 was commensurate to, say, 4/12.

Steffe L. P. (2007) Radical Constructivism: A Scientific Research Program. Constructivist Foundations 2(2-3): 41–49. https://cepa.info/28

Purpose: In the paper, I discuss how Ernst Glasersfeld worked as a scientist on the project, Interdisciplinary Research on Number (IRON), and explain how his scientific activity fueled his development of radical constructivism. I also present IRON as a progressive research program in radical constructivism and suggest the essential components of such programs. Findings: The basic problem of Glasersfeld’s radical constructivism is to explore the operations by means of which we assemble our experiential reality. Conceptual analysis is Glasersfeld’s way of doing science and he used it in IRON to analyze the units that young children create and count in the activity of counting. In his work in IRON, Glasersfeld first conducted a first-order conceptual analysis of his own operations that produce units and number, and then participated in a second-order analysis of the language and actions of children and inferred the mental operations that they use to produce units and number. Further, Glasersfeld used Piaget’s concept of equilibration in the context of scheme theory in a second-order analysis of children’s construction of number sequences and of more advanced ways and means of operating in the traffic of numbers. Research Implications: The scientific method of first- and second-order conceptual analysis transcends our work in IRON and it is applicable in any radical constructivist research program whose problem is to explore the operations by means of which we construct our conceptions. Because of the difficulties involved with introspection, conducting second-order conceptual analyses is essential in exploring these operations and it involves analyzing the language and actions of the observed. But conceptual analysis is only a part of the research process because the researchers are by necessity already involved in creating occasions of observation. The “experimenter” and the “analyst” can be the same person or they can be different people. Either case involves intensive and sustained interdisciplinary thinking and ways of working if the research program is to be maintained over a substantial period of time as a progressive research program.

Open peer commentary on the target article “Who Conceives of Society?” by Ernst von Glasersfeld. Excerpt: My goal in this commentary is to say enough to suggest that the meanings children impute to the language and actions of other children are based on their current conceptual schemes and that, if the schemes are at different levels of the constructive process, it is no easy feat for children to use their schemes in interactive mathematical communication.

Steffe L. P. & Glasersfeld E. von (1985) Helping children to conceive of number. Recherches en Didactique des Mathématiques 6: 269–303.

Number is presented as a uniting operation that can have collections of sensory items as material and a unit of units as the result. One thesis of the paper is that children who have constructed this uniting operation have not necessarily constructed number sequences. Problem situations are suggested for such children that might encourage the internalization of counting and the concomitant construction of the iterable unit of one. Situations are then suggested in which the child can use the number sequence that is based on the iterable unit of one to construct other iterable units and their corresponding number sequences. A second thesis of the paper is that for children who are yet to construct the uniting operation, counting is a sensory-motor scheme that should be coordinated with spatial, finger, and auditory patterns. Problem situations are suggested for these coordinations. While a teacher cannot give a child the uniting operation, the suggested situations can encourage its construction by the child.

Steffe L. P. & Glasersfeld E. von (1988) On the construction of the counting scheme. In: Steffe L. P. & Cobb P. (eds.) Construction of arithmetical meaning and strategies. Springer, New York: 1–19. https://cepa.info/1399

In an earlier publication (Steffe, von Glasersfeld, Richards, & Cobb, 1983), we presented a model of the development of children’s counting schemes. This model specifies five distinct counting types, according to the most advanced type of unit items that the child counts at a given point in his or her development. The counting types indicate what children’s initial, informal numerical knowledge might be like, and reflect our contention that children see numerical situations in a variety of qualitatively different ways. These constructs constituted the initial theoretical basis of the teaching experiment and served as a catalyst for the first years work. Consequently, we provide an explanation of the counting types as we defined them in 1980.

Steffe L. P. & Tzur R. (1994) Interaction and children’s mathematics. In: Ernest P. (ed.) Constructing mathematical knowledge: Epistemology and mathematics education. Falmer Press, London: 8–32. https://cepa.info/2103

In recent years, the social interaction involved in the construction of the mathematics of children has been brought into the foreground in order to sptcify its constructive aspects (Bauersfeld, 1988; Yackel, etc., 1990). One of the basic goals in our current teaching experiment is to analyse such social interaction in the context of children working on fractions in computer microworlds. In our analyses, however, we have found that social interaction does not provide a full account of children’s mathematical interaction. Children’s mathematical interaction also includes enactment or potential enactment of their operative mathematical schc:mes. Therefore, we conduct our analyses of children’s social interaction in the context of their mathematical interaction in our computer microworlds. We interpret and contrast the children’s mathematical interaction from the points of view of radical constructivism and of Soviet activity theory. We challenge what we believe is a common interpretation of learning in radical constructivism by those who approach learning from a social-cultural point of view. Renshaw (1992), for example, states that ‘In promulgating an active. constructive and creative view of learning,… the constructivists painted the learner in close-up as a solo-pia yer, a lone scientist, a solitary observer, a me:ming maker in a vacuum.’ In Renshaw’s interpretation, learning is viewed as being synonymous with construction in the absence of social interaction with other human beings. To those in mathematics education who use the teaching experiment methodology, this view of learning has always seemed strange because we emphasize social interaction as a primary means of engendering learning and of building models of children’s mathematical knowledge (Cobb and Steffe, 1983; Steffe, 1993).