Constructivist Teaching Experiment Research Articles

Unitizing Predicates and Reasoning About the Logic of Proofs

This article offers the construct unitizing predicates to name mental actions important for students’ reasoning about logic. To unitize a predicate is to conceptualize (possibly complex or multipart) conditions as a single property that every example has or does not have, thereby partitioning a universal set into examples and nonexamples. This explains the cognitive work that supports students to unify various statements with the same logical form, which is conventionally represented by replacing parts of statements with logical variables p or P(x). Using data from a constructivist teaching experiment with two undergraduate students, we document barriers to unitizing predicates and demonstrate how this activity influences students’ ability to render mathematical statements and proofs as having the same logical structure.

Diego’s Number Understanding Development through His Subitizing and Counting

ABSTRACT Subitizing, a quick apprehension of the numerosity of a small set of items, has been found to explain students’ number understanding when counting. We utilized a constructivist teaching experiment methodology to investigate how the counting and subitizing activity of one student, Diego, related to his number understanding (described by his pre-numerical units construction) when solving early number tasks. Subitizing and counting tasks were designed to assess and leverage Diego’s pre-numerical units development. Findings indicated that as Diego’s pre-numerical units included more than one way to construct five, he still relied on external material. Implications of this study suggest that task design should be grounded in unique aspects of student reasoning and should be designed to promote students’ partitioning as well as use of finger patterns, pointing, and number words.

Linking Teacher–Learner Discourse With Mathematical Reasoning of Students With Learning Disabilities: An Exploratory Study

Informed by a constructivist-based, student-adaptive pedagogical approach, this study explores the benefits of teacher–learner discourse moves for the mathematics learning of students with learning disabilities (LD). During a constructivist teaching experiment for nurturing the multiplicative reasoning and problem solving of five third-grade students with school-identified LD, we analyzed the global trends and detailed dynamics of teacher–student interactions with statistical discourse analysis. We found that the teacher’s discourse moves to support each individual student’s problem solving helped engage them in mathematical reasoning, which improved their problem-solving performance. Thus, this study contributes to knowledge in the field of special education by (a) specifying ways in which discourse-oriented mathematics instruction can help each student with LD and (b) showcasing a novel statistical analysis of teacher–student discourse.

Maddie's units coordinating across contexts

Units coordinating has emerged as an important construct for understanding students’ mathematical thinking, particularly their concepts of multiplication and fractions. We conducted an eleven-session constructivist teaching experiment with a pair of sixth-grade students to investigate how children coordinate whole number and fractional units across quantitative contexts. In this paper, we focus on one student, Maddie, who was assessed as reasoning with three levels of whole number units at the onset of the teaching experiment. We illustrate Maddie's units coordinating with fractional units over the course of the teaching experiment: it began with incorrect responses that indicated she had yet to coordinate three levels of fractional units; it progressed through her coordination of three levels of fractional units in situations for which she had a tangible referent for partitioning. We describe how students’ ways of experiencing quantitative situations are important considerations in their development of units coordinating structures.

Subitising activity relative to units construction: a case study

ABSTRACTSubitising, a quick apprehension of the numerosity of a small set of items, has been found to change from an individual's reliance on perceptual to conceptual processes. In this study, we utilised a constructivist teaching experiment methodology to investigate how the subitising activity of one preschool student, Amy, related to her construction of prenumerical units. Subitising and counting tasks were designed to assess and perturb Amy's thinking relative to her construction of units, and to observe changes in Amy's activity associated with the different tasks. Findings indicate that as Amy's subitising activity changed from perceptual to conceptual, she constructed subitised motor units and subitised figurative units. Implications of this study suggest that the construction of subitised units may support young children's later development of arithmetic units.

An emerging methodology for studying mathematics concept learning and instructional design

Our microgenetic research methodology was developed for two purposes: 1) to develop greater understanding of conceptual learning and instructional design that promotes conceptual learning and 2) to develop empirically-based hypothetical learning trajectories for particular mathematical concepts. The challenge in developing the methodology was to afford analysis of the learning process (e.g., the transition), not just identification of conceptual steps through which learners progress. The methodology, developed by the Learning Through Activity research program, was based on constructivist teaching experiment methodology. Modifications were made to data collection for the purposes outlined and a multi-level retrospective analysis was developed.

Dylan’s units coordinating across contexts

Units coordination has emerged as an important construct for understanding students’ mathematical thinking, particularly their concepts of multiplication and fractions. To explore students’ units coordination development, we conducted an eleven-session constructivist teaching experiment with a pair of sixth-grade students, investigating how they coordinated whole number and fractional units in discrete and continuous settings. In this paper we focus on one student, Dylan, who reasoned with whole number units but not fractional units at the beginning of the teaching experiment. We describe Dylan’s development of units coordination as he continued to reason with whole number units in fractional situations, and we discuss implications for instruction.

The tacitly nested number sequence in sixth grade: The case of Adam

Student learning in the middle grades is afforded and constrained by basic numerical operations characterized in the number sequences framework (Steffe, 2010a), originally developed with younger children. I investigated its use with four sixth-grade students through a constructivist teaching experiment. I found that one student, Adam, was able to strategically operate on composite units, which is a strong indication that the he had constructed an explicitly nested number sequence (ENS). However, he was not able to reversibly disembed subsequences or construct iterable units of 1, both principal operations of an ENS. I conclude that Adam was constrained to the construction of the less sophisticated tacitly nested number sequence (TNS), and that his advanced behavior is a natural outgrowth of his increased experience operating with a TNS. I suggest a refinement of the number sequences framework to support research and teaching at the middle grades.

Ben’s perception of space and subitizing activity: a constructivist teaching experiment

This 22-session constructivist teaching experiment set out to investigate a preschool student’s number understanding relative to his subitizing activity. Subitizing, a quick apprehension of the numerosity of a small set of items, has been found to characterize perceptual and conceptual processes students rely on as their understanding of number develops. The purpose for this study is to investigate how a preschool student’s, Ben, perceptual subitizing activity changed relative to the density of items and the development of his number understanding. Findings indicated that early on in the teaching experiment, Ben’s perceptual subitizing activity was influenced by his primary reliance upon the perceived amount of space between items. Shifts in reasoning when perceptually subitizing indicated physiological and experiential development in Ben’s number understanding, as Ben described the number of items increasing when the perceived amount of space between items decreased. Number conservation was considered as relevant to these findings because Ben’s explanation for why a number could increase or decrease mirrored similar logic when unable to conserve number. Implications of this study suggest nuances in number understanding development which can explain preschool students’ reliance upon a more refined set of perceptual subitizing.

Units coordination and the construction of improper fractions: A revision of the splitting hypothesis

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.

From fractions to proportional reasoning: a cognitive schemes of operation approach

Four seventh grade students participated in a constructivist teaching experiment in which manipulatives within a computer microworld were used to solve fractional reasoning tasks followed by tasks that involve concepts of rate, ratio and proportion. Through a retrospective analysis of video tapes, their thinking processes were analyzed from the perspective of the types of cognitive schemes of operation used as they engaged in the given problem situations. One result of the study indicates that the modifications of the students’ available schemes of operation when solving the fractional reasoning tasks formed a basis for the cognitive schemes of operation used in their solutions of tasks involving proportionality.

Design and Use of Computer Tools for Interactive Mathematical Activity (TIMA)

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 experiment: Toys, Sticks, and Bars. These tools were designed to provide children contexts in which they could enact their mathematical operations of unitizing, uniting, fragmenting, segmenting, partitioning, disembeding, iterating and measuring. As such, they are very different from the drill and practice or tutorial software that are prevalent in many elementary schools. We provide examples of how the TIMA were used by children to engage in cognitive play and, through interactions with a teacher/researcher and other children, transform that play into independent mathematical activity with a playful orientation. The role of the teacher in provoking perturbations that could lead eventually to accommodations in the children's mathematical schemes was critical in the use of the TIMA as learning tools.

Computer Tools for Interactive Mathematical Activity in the Elementary School

The computer tools for interactive mathematical activity (TIMA) were designed to provide children a medium in which they could enact their mathematical operations of unitizing, uniting, fragmenting, segmenting, partitioning, replicating, iterating and measuring. As such, they are very different from the drill and practice or tutorial software that are prevalent in many elementary schools. The TIMA were developed in the context of a constructivist teaching experiment focused on children's construction of fractions. They were used to promote cognitive play that could be transformed into independent mathematical activity. Teaching interventions were often critical in bringing forth mathematical activity. Students' interactions were also important provocations for mathematical reasoning with the TIMA. The TIMA do not, by themselves define a microworld. Rather it is the children's activity and their interpretations of the results of that activity, while interacting with others, that bring forth a microworld of mathematical operations. Designers of computational environments for children need to take into account the contributions children need to make in order to build their own mathematical structures. For teachers to make effective use of software such as the TIMA they need to understand (and share) the views of learning that shaped the development of the software.

An Integrated Study of Children's Construction of Improper Fractions and the Teacher's Role in Promoting That Learning

In this constructivist teaching experiment with 2 fourth graders I studied the coemergence of teaching and children's construction of a specific conception that supports the generation of improper fractions. The children's posing and solving tasks in a computer microworld promoted a modification in their fraction schemes. They advanced from thinking about a unit fraction as a part of a whole to thinking about it as standing in a multiplicative relationship with a reference whole (the iterative fraction scheme). In this article I report an intertwined analysis of the children's construction of this multiplicative relationship and an examination of the teacher's adaptation of learning situations (tasks) and teacher-learner interactions to fit within the constraints of the children's mathematical activity.

Third Graders' Interpretations of Equality and the Equal Symbol

This paper analyzes the interpretations of equality and the equal symbol of the third-grade children who participated in a year long whole-class socio constructivist teaching experiment. These children initially interpreted the equal symbol as a command to perform an arithmetical operation; it was less natural to them to interpret it as a relational symbol to compare two quantities. By the end of the school year, children were able to conceptualize the ‘quantitative sameness’ of two numerical expressions and describe it by using the phrase ‘is the same as’, the words ‘equal’ or ‘equals’, or the symbols ‘=’ or ‘=s’. These children expanded their conceptualizations of equality due to their active role in class discussions, the arithmetical tasks that took into account children's difficulties, and the teacher's intellectual sensitivity to ‘strike a delicate balance between the force of teaching and the freedom of learning’ (Freudenthal, 1991, p. 55).

Justification in the mathematics classroom: A study of prospective elementary teachers

This study focuses on mathematical justification in the context of a mathematics course for prospective elementary teachers. The course was run as a whole-class constructivist teaching experiment. The analysis of the data provides a detailed look at how classroom norms for mathematical justification were established given the prospective teachers' traditional expectations of school mathematics and the teacher's reform-oriented notions. Three episodes from the class transcripts are analyzed in this article. They reveal the interactive process of negotiating new classroom norms for doing mathematics, the range of student responses to situations for justifications, and the complex interrelationship between the investigation of mathematical validity and development of mathematical understanding in the classroom community.

Reconstructing Mathematics Pedagogy from a Constructivist Perspective

Constructivist theory has been prominent in recent research on mathematics learning and has provided a basis for recent mathematics education reform efforts. Although constructivism has the potential to inform changes in mathematics teaching, it offers no particular vision of how mathematics should be taught; models of teaching based on constructivism are needed. Data are presented from a whole-class, constructivist teaching experiment in which problems of teaching practice required the teacher/researcher to explore the pedagogical implications of his theoretical (constructivist) perspectives. The analysis of the data led to the development of a model of teacher decision making with respect to mathematical tasks. Central to this model is the creative tension between the teacher's goals with regard to student learning and his responsibility to be sensitive and responsive to the mathematical thinking of the students.

Reconstructing Mathematics Pedagogy from a Constructivist Perspective

Constructivist theory has been prominent in recent research on mathematics learning and has provided a basis for recent mathematics education reform efforts. Although constructivism has the potential to inform changes in mathematics teaching, it offers no particular vision of how mathematics should be taught; models of teaching based on constructivism are needed. Data are presented from a whole-class, constructivist teaching experiment in which problems of teaching practice required the teacher/researcher to explore the pedagogical implications of his theoretical (constructivist) perspectives. The analysis of the data led to the development of a model of teacher decision making with respect to mathematical tasks. Central to this model is the creative tension between the teacher's goals with regard to student learning and his responsibility to be sensitive and responsive to the mathematical thinking of the students.

Ann's fraction schemes

A longitudinal constructivist teaching experiment that lasted approximately one academic year was conducted with six third graders. The purpose of the teaching experiment was to analyze the itinerary of children's ways of operating while solving fraction tasks. Ann was one of the third graders who participated in the teaching experiment, and her case study presents the author's interpretation of the generation and evolution of Ann's fraction schemes.

The Constructivist Researcher as Teacher and Model Builder

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 of theoretical constructs--that represent our understanding of children's mathematical realities. However, the models must be distinguished from what might go on in children's heads. They are formulated in the context of intensive interactions with children. Our emphasis on the researcher as teacher stems from our view that children's construction of mathematical knowledge is greatly influenced by the experience they gain through interaction with their teacher. Although some of the researchers might not teach, all must act as model builders to ensure that the models reflect the teacher's understanding of the children.