Leslie P. Steffe

• University of Georgia

This is an earlier version of Chapter 9 of the now published book “Children’s Fractional Knowledge” by Steffe & Olive (2010). References are made to earlier chapters in the published book. This version is a more detailed account and contains material that was eventually cut because of space limitations.

> Context . Ernst von Glasersfeld introduced radical constructivism in 1974 as a new interpretation of Jean Piaget's constructivism to give new meanings to the notions of knowledge, communication, and reality. He also claimed that RC would affect traditional theories of education. > Problem . After 40 years it has become necessary to review and eva...

Through my work in mathematics education, I have come to the realization that constituting mathematics education as an academic field entails constructing models of mathematical minds that are constructed by students in the context of mathematics teaching beginning in early childhood and proceeding onward throughout the years of schooling. In this...

Leslie Steffe, among the foremost mathematics education researchers in the world, has had a profound influence on three generations of researchers. In 2006, he received the first-ever Senior Scholar Award from the AERA Special Interest Group: Research in Mathematics for the excellence and seminal nature of his work. Steffe shares his thoughts about...

Purpose: My goals in this paper are to comment on some of the roles that Ernst von Glasersfeld played in our work in IRON (Interdisciplinary Research on Number) from circa 1975 up until the time of his death, and to relate certain events that revealed his character in very human terms. Method: Among my recollections of Ernst, I have chosen those th...

The constructivist teaching experiment is used in formulating explanations of children’s mathematical behavior. Essentially, a teaching experiment consists of a series of teaching episodes and individual interviews that covers an extended period of time—anywhere from 6 weeks to 2 years. The explanations we formulate consist of models—constellations...

〉 Purpose • One of my goals in the paper is to investigate why realists reject radical constructivism (RC) as well as social constructivism (SC) out of hand. I shall do this by means of commenting on Peter Slezak's critical paper, Radical Constructivism: Epistemology, Education and Dynamite. My other goal is to explore why realists condemn the use...

During her fourth grade, Melissa had been paired with another child who had constructed only the tacitly nested number sequence. The teacher of the two children geared her activities to the other child and Melissa essentially served as the other child’s interlocutor. The other child did not construct any fraction schemes during her fourth grade and...

The reorganization hypothesis – children’s fraction schemes can emerge as accommodations in their numerical counting schemes – is untenable if counting is regarded only as activity. Focusing only on the activity of counting, however, does not begin to provide a full account. When using the phrase “the explicitly nested number sequence,” I am referr...

When fractions are introduced in school mathematics, they are usually introduced in the context of continuous quantity. Number sequences are essentially excluded because, as quantitative schemes, they are thought to be relevant only in discrete quantitative situations. Even though I developed number sequences in Chap. 3 in the context of discrete q...

In this chapter, we trace the construction of the fraction schemes of two of the children in our teaching experiment, Nathan and Arthur, who apparently had already constructed a generalized number sequence before we began working with them. We interacted with these two children as we did with the other children in the sense that our history of the...

The separation of the study of whole numbers and fractions is historical and contributes to the legendary difficulty children experience in the learning of fractions that inspired Davis et al. (1993) to comment that “the learning of fractions is not only very hard, it is, in the broader scheme of things, a dismal failure” (p. 63). I cite Davis et a...

By the end of his fourth grade year, Jason had constructed the partitive fraction scheme and Laura had constructed the part-whole fraction scheme, and the children had used these schemes to produce fractional parts of a fractional whole. But the children could not use them to produce fractional amounts that exceeded the fractional whole. Nor could...

The primary goal of this chapter is to present a model of important steps in children’s construction of their numerical counting schemes because the basic hypothesis that guides our work is that children’s fraction schemes can emerge as accommodations in their numerical counting schemes. I consider a number sequence to be the recognition template o...

Children's Fractional Knowledge elegantly tracks the construction of knowledge, both by children learning new methods of reasoning and by the researchers studying their methods. The book challenges the widely held belief that children's whole number knowledge is a distraction from their learning of fractions by positing that their fractional learni...

As stated at the beginning of the first chapter, the basic hypothesis that guided our work is that children’s fraction schemes can emerge as accommodations in their numerical counting schemes. We explained our way of understanding this hypothesis as follows. The child constructs the new schemes by operating on novel material in situations that are...

The fourth-grade teaching experiment with Joe and Patricia constituted a “replication” of the teaching experiment with Jason and Laura while they were in their fourth and fifth grades. We use scare quotes to indicate that the intent is not to repeat the experiment with Jason and Laura under the exact same conditions. Rather, the intent is to genera...

In a year-long teaching experiment with two 7th grade students, this study investigated how those students dealt with enumerative combinatorial problems based on their additive and multiplicative reasoning. The results show that three distinctive levels of enumeration appeared in the students' mathematical behavior: additive enumeration, multiplica...

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 su...

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 m...

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 diffe...

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...

Our guiding principle when designing the TIMA was to create computer tools that we could use to achieve our goals when teaching children. The design of the TIMA took place in the context of a constructivist teaching experiment with 12 children that extended over a three-year period. Three different TIMA were designed and used in the teaching experi...

In his paper on A New Hypothesis Concerning Children’s Fractional Knowledge, Steffe (2002) demonstrated through the case study of Jason and Laura how children might construct their fractional knowledge through reorganization of their number sequences. He described the construction of a new kind of number sequence that we refer to as a connected num...

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 us...

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 ofVygotsky's work, nor does it reflect Vygotsky's opinion of Piaget's work. We question L...

Discusses the development of multiplicative and divisional schemes within a constructivist framework. Illustrates child thinking and child methods relative to the meanings of these operations using interviews with children. Compares the constructivist perspective on the development of meanings of multiplication and division to what is found in text...

In the design of computer microworlds as media for children's mathematical action, our basic and guiding principle was to create possible actions children could use to enact their mental operations. These possible actions open pathways for children's mathematical activity that coemerge in the activity. We illustrate this coemergence through a const...

The caption that I have chosen for my discussion of the papers by Peter Renshaw and Paul Cobb is particularly relevant because it encapsulates much of what I want to say. Both authors emphasize that socio-cultural theory, whatever its form, has come to the fore in early childhood mathematics education. Renshaw provides an excellent overview of Vygo...

We start this discussion of the article by Martin A. Simon with two major conjectures that we will try to substantiate. The first is that there is a kind of teaching that can legitimately be called “constructivist teaching.” The second is that Simon's model of teaching his prospective elementary school teachers, if modified, would fit our understan...

It is fitting to call this short feature on Henry Van Engen “An Analysis of Meaning in Arithmetic” because his famous articles in the Elementary School Journal (Van Engen, l949) go by that title. In Van Engen's operational school of meaning in arithmetic, the meaning of a symbol is an intention to act.

Introduces the idea of using computer microworlds--interactive software for exploration of specific concepts--for mathematics education, the theme of this issue's articles. Discusses their development and the advantages of their use. Suggests that they can be successfully used for interactive teaching or for interpreting mathematics textbooks, whic...

In this article we propose the beginnings of a constructivist model of mathematical learning that supersedes Piaget's and Vygotsky's views on learning. First, we analyze aspects of Piaget's and Vygotsky's grand theories of learning and development. Then, we formulate our superseding model, which is based on the interrelations between two types of i...

Our intention in this article 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 epi...

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 cogniti...

The construction of schemes of action and operation involving composite units is crucial in the mathematical development of children. In the article, six different composite units are identified along with two general categories of schemes involving composite units—units-coordinating schemes and unit segmenting schemes. The schemes within these two...

Based on principles of constructivism, an analysis is made of how practice in mathematical education might be reformed towards a professional practice. In addition to the widespread recommendations that mathematical teaching be based on interactive communication and that mathematical learning be active, we argue that conventional school mathematics...

Segmenting sensory experience into units is advanced as the fundamental operation that produces object concepts. How such unitizing activity might be produced by children through interactions with elements in their environments is suggested by an interpretation of several recent works on the construction of objects by infants. The arithmetical unit...

Researchers in Finland have warned for some years that children learn mathematics too mechanically in our comprehensive school. They learn rules and tricks. but not mathematical thinking. It is rote learning without meaning. I think this is what happens often with the fraction concept.

In this paper we suggest that the use of conceptual models, as developed in cybernetics and used in cognitive psychology, may help to bridge the gap between educational research and teachers. The notion of model is discussed in the context of explanatory theories, teachers' understanding of students, and students' understanding of mathematics. Piag...

In the summer of 1985, Carl Bereiter published an article in the Review of Educational Research titled Toward a Solution of the Learning Paradox. Ever since, it has been my intention to provide a counterexample to the paradox. Fodor (1980b), who is credited by Bereiter as clearly stating the learning paradox, views learning as being necessarily ind...

Discussed are inconsistencies and cognitive conflict with respect to current mathematical knowledge of students and how that knowledge might be modified is discussed. The inconsistencies that students generate for themselves and those produced by the teacher are described. (KR)

The purpose of this article is to discuss how children give meaning to situations which, from an adult's perspective, could be solved by multiplying or dividing (Steffe and von Glasersfeld 1985). We have found in our research that a child's understanding of these situations may be significantly different not only from those of the teacher but also...

In this book, we present the results of a teaching experiment that was carried out during the academic years 1980-1981 and 1981-1982. . . . Because it also involves experimentation with the ways and means of influencing children's knowledge, the teaching experiment is more than a clinical interview. Whereas the clinical interview is aimed at establ...

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 c...

The integration operation emerged in the context of patterns for Tyrone, Scenetra, and Jason in contrast to its emergence for Brenda, Tarus, and James. We are therefore obliged to add numerical concepts to those explained in Chapter V. When the elements of a spatial pattern serve as material of an integration operation, we call the resulting patter...

From our point of view, strategies for finding sums and differences involve the coordination of arithmetic symbols that signify systems of integration operations and their products. Moreover, because mental operations (including integration operations) can be expressed in terms of action, the coordination of symbols implies a corresponding coordina...

We now turn to the three children who entered the teaching experiment as counters of motor unit items: Tyrone, Scenetra, and Jason. These three children had already reached the figurative stage of their counting scheme at the beginning of the teaching experiment, in that they could count their motor acts as substitutes for countable perceptual or f...

In Chapters II and III, we focused on changes in the unit items that Brenda, Tarus, and James created and counted over the two-year duration of the teaching experiment. Knowledge of the periods in the development of the three children’s counting schemes, however, only partially specifies the possible lexical and syntactical meanings they gave to nu...

In this chapter, we present the results of the teaching experiment as they pertain to the construction of motor unit items by the three children who began the teaching experiment as counters of perceptual unit items. We start with the hypothesis that adaptations of the counting scheme involve changes in the assimilatory structures of the scheme whi...

Observations of children using thinking strategies to find sums and differences are reported in the literature (Brownell, 1928, 1935; Brownell & Chazal, 1935; Carpenter, 1980; Carpenter & Moser, 1982; Ginsburg, 1977; Hiebert, 1982; Hiebert, Carpenter, & Moser, 1982; Ilg & Ames, 1951: Rathmell, 1978; Smith, 1921; Steffe, Hirstein, & Spikes, 1976; St...

We have argued that the shift in counting type from perceptual to motor unit items constituted a stage shift in the counting scheme, and we have called these stages the perceptual and the figurative stages. Re-presentation was the vital mechanism in the children’s progress; however, the material they re-presented was predominantly restricted to fin...

Arithmetical knowledge is viewed as simply the coordinated schemes of actions and operations the child has constructed at a particular point in time. A fundamental distinction is drawn between operative and figurative schemes in the context of additive schemes. Numerical extension, an operative scheme, is distinguished from intuitive extension, a f...

Mathematics educators have for some time been interested in psychological bases for the teaching of mathematics in the schools (Buswell, 1951). They have naturally turned to cognitive theorists in their quest to understand such terms as knowledge, meaning, concepts, mental operations, problem solving, and insight . Gagné's (1983) paper is bur one e...

The constructivist teaching experiment is used in formulating explanations of children's mathematical behavior. Essentially, a teaching experiment consists of a series of teaching episodes and individual interviews that covers an extended period of time—anywhere from 6 weeks to 2 years. The explanations we formulate consist of models—constellations...

One primary reason for writing this critique of the report by Hiebert, Carpenter, and Moser (1982) is to emphasize the intricate relationship between theory and data. First, we question the classical empiricist assumption that observation is free from the prejudices of theory. We then consider the role that explanatory theoretical constructs played...

One primary reason for writing this critique of the report by Hiebert, Carpenter, and Moser (1982) is to emphasize the intricate relationship between theory and data. First, we question the classical empiricist assumption that observation is free from the prejudices of theory. We then consider the role that explanatory theoretical constructs played...

this question historically depended on whether one was a drill theorist or a meaning theorist. The drill theorist would answer in the negative, and the meaning theorist would answer in the affirmative. Obviously, the question has no answer solely in empirical research because the empirical research always stands to be interpreted. A prime example i...

This report presents the results of a teaching experiment which investigated (1) the role of mathematical experiences on the development of counting, addition, subtraction, mental arithmetic, classification, and other arithmetical topics and (2) the role of quantitative comparisons and class inclusion as readiness variables for learning the content...

This volume includes reports of six studies of the thought processes of children aged four through eight. In the first paper Steffe and Smock outline a model for learning and teaching mathematics. Six reports on empirical studies are then presented in five areas of mathematics learning: (1) equivalence and order relations; (2) classification and se...

A sample of 42 kindergarten children was partitioned into 27 conservers and 15 nonconservers of matching relations “as many as,” “more than,” and “fewer than,” based on an 18-item conservation of matching relations test (CMRT), 6 items for each relation. An 18-item transitivity of matching relations test (TMRT) was then administered to each subject...

The purpose of this study was to investigate interrelationships among 4- and 5-year-old children's ability to conserve length relations not involving the asymmetric property and consequences, conserve length relations involving the asymmetric property and consequences, use the reflexive and nonreflexive properties of length relations, and use the t...

The purpose of this study was to investigate interrelationships among 4- and 5-year-old children's ability to conserve length relations not involving the asymmetric property and consequences, conserve length relations involving the asymmetric property and consequences, use the reflexive and nonreflexive properties of length relations, and use the t...

The goal of this study was to investigate improvement in the usage of equivalence and order relations under specified instructional conditions. Forty-eight upper middle class kindergarten children were given a 14-session pretreatment designed to define two equivalence and four order relations of length and matching, followed by tests of conservatio...

The major objective of this study was to investigate differential performances among 5-year-old children when using transitivity of matching relations. Instruments were constructed to measure the subjects' (1) knowledge of matching relations, (2) ability to conserve the relations, and (3) proficiency in making inferences using the transitive proper...

One major purpose of this investigation was to determine and compare the effects of three different perceptual situations (screening, neutral, and conflict) on kindergarten and first grade children's ability to use the transitive property of the equivalence relation "same length as" and the two order relations "longer than" and "shorter than". A to...

A sample of 111 first-grade children was partitioned into 4 categories, where the categorization was determined by an ability to make quantitative comparisons and IQ. 6 problems of each of 8 problem types were presented to each child in a randomized sequence. Approximately one-half of the children in each category were randomly assigned to 1 of 2 l...

A sample of 111 first-grade children was partitioned into 4 categories, where the categorization was determined by an ability to make quantitative comparisons and IQ. 6 problems of each of 8 problem types were presented to each child in a randomized sequence. Approximately one-half of the children in each category were randomly assigned to 1 of 2 l...

4 levels of Quantitative Comparisons (Q) and 3 IQ Intervals (I) were used to partition a population of 341 first-grade children. Each child in an ordered random sample, taken in such a way that 33 children were at each level of Q and 44 children were in each IQ interval, was individually administered a set of 18 addition problems and a set of 6 add...

This study is one of a series which attempts to arrive at generalizations about the learning of mathematics and the use of its terminology in the context of mathematical structure by young children. The first half of the document describes an experimental training program designed to integrate mathematical concepts of metric space, arc length, and...

Six tests were constructed, four on a pictorial level and two on a symbolic level, to measure the performance of fourth-, fifth-, and sixth-grade children, in three different ability groups, on problems concerning ratios or fractions. Two variables were of interest in the four tests on a pictorial level: (a) "equal" ratio situations vs. "equal" fra...

Typescript. Thesis (Ph. D.)--University of Wisconsin--Madison, 1966. Vita. Includes bibliographical references.