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.

Key words: education, mathematics

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.

Key words: cognition, mathematics, education

intention in this mticle is to provide an interpretation of the influence of constructivist thought on mathematics educators starting around 1960 and proceeding on up to the present time. First, we indicate how the initial influence of constructivist thought stemmed mainly from Piaget’s cognitive development psychology rather than from his epistemology. In this, we point to what in retrospect appears to be ineTitable distmtions in the interpretations of Piaget’s psychology due primarily to its interpretation in the framework of Cartesian epistemology. Second) we identify a preconstructivist revolution in research in mathematics education beginning in 1970 and proceeding on up to 1980. There were two subperiods in this decade separated by Ernst von Glasersfeld’s presentation of radical constructiTism to the Jean Piaget Society in Philadelphia in 1975. Third, we rnark the beginning of the constructivist revolution in mathematics education research by the publication of two important papers in the JRME (Richards & van Glasersfeld, 1980; van Glasersfeld, 1981). Fomth, we indicate how the constructivist revolution in mathematics education research served as a period of preparation for the reform movement that is currently underway in school mathematics.

Lerman, in his challenge to radical constructivism, presented Vygotsky as an irreconcilable opponent to Piaget’s genetic epistemology and thus to von Glasersfeld’s radical constructivism. We argue that Lerman’s stance does not reflect von Glasersfeld’s opinion of Vygotsky’s work, nor does it reflect Vygotsky’s opinion of Piaget’s work. We question Lerman’s interpretation of radical constructivism and explain how the ideas of interaction, intersubjectivity, and social goals make sense in it. We then establish compatibility between the analytic units in Vygotsky’s and von Glasersfeld’s models and contrast them with Lerman’s analytic unit. Consequently, we question Lerman’s interpretation of Vygotsky. Finally, we question Lerman’s use of Vygotsky’s work in mathematics education, and we contrast that use with how we use von Glasersfeld’s radical constructivism.

A primary purpose for using teaching experiment methodology is for researchers to experience, firsthand, students’2 mathematical learning and reasoning. Without the experiences afforded by teaching, there would be no basis for coming to understand the powerful mathematical concepts and operations students construct or even for suspecting that these concepts and operations may be distinctly different from those of researchers. The constraints that researchers experience in teaching constitute a basis for understanding students’ mathematical constructions. As we, the authors, use it, “constraint” has a dual meaning. Researchers’ imputations to students of mathematical understandings and operations are constrained by the language and actions they are able to bring forth in students. They also are constrained by students’ mistakes, especially those mistakes that are essential; that is, mistakes that persist despite researchers’ best efforts to eliminate them. Sources of essential mistakes reside in students’ current mathematical knowledge. To experience constraints in these two senses is our primary reason for doing teaching experiments. The first type of constraint serves in building up a “mathematics of students” and the second type serves in circumscribing such a mathematics within conceptual boundaries.

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).

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.

Key words: Constructivism, Methodology, Teaching experiment, Instrumental understanding

Excerpt: The constructivist teaching experiment emerged in the United States circa 1975 (Steffe et al. 1976) 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 (e.g., Piaget and Szeminska 1952; McLellan and Dewey 1895; Brownell 1928). 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 (Wirszup and Kilpatrick 1975–1978). 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.

Key words: constructivism, methodology, teaching experiment, instrumental understanding.

Based on the constructivist principle of active learning, we focus on children’s transformation of their cognitive play activity into what we regard as independent mathematical activity. We analyze how, in the process of this transformation, children modify their cognitive play activities. For such a modification to occur, we argue that the cognitive play activity has to involve operations of intelligence which yield situations of mathematical schemes.