John Olive

Children's fractional knowledge

Chapter I. A New Hypothesis Concerning Childrens Fractional Knowledge The Interference Hypothesis The Separation Hypothesis Next Steps Chapter II. Perspectives on Childrens Fraction Knowledge On Opening the Trap Fractions as Schemes Mathematics of Living Rather Than Being Chapter III. Operations that Produce Numerical Counting Schemes Complexes of Discrete Units Recognition Templates of Perceptual Counting Schemes Recognition Templates of Figurative Counting Schemes Numerical Patterns and the Initial Number Sequence The Tacitly Nested Number Sequence The Explicitly Nested Number Sequence An Awareness of Numerosity: A Quantitative Property The Generalized Number Sequence An Overview of the Principal Operations of the Numerical Counting Schemes Final Comments Chapter IV: Articulation of the Reorganization Hypothesis Perceptual and Figurative Length Piagets Gross, Intensive, and Extensive Quantity Composite Structures as Templates for Fragmenting Partitioning and Iterating Final Comments Chapter V: The Partitive and the Part-Whole Schemes The Equi-Partitioning Scheme Segmenting to Produce a Connected Number Making a Connected Number Sequence An Attempt to Use Multiplying Schemes in the Construction of Composite Unit Fractions Lauras Simultaneous Partitioning Scheme Jasons Partitive and Lauras Part-Whole Fraction Schemes Establishing Fractional Meaning for Multiple Parts of a Stick Continued Absence of Fractional Numbers An Attempt to Use Units-Coordinating to Produce Improper Fractions Discussion of the Case Study Chapter VI. The Unit Composition and the Commensurate Schemes The Unit Fraction Composition Scheme Producing Composite Unit Fractions Producing Fractions Commensurate with One-Half ProducingFractions Commensurate With One-Third Producing Fractions Commensurate With Two-Thirds An Attempt to Engage Laura in the Construction of the Unit Fraction Composition Scheme Discussion of the Case Study Chapter VII. The Partitive, the Iterative, and the Unit Composition Schemes Joes Attempts to Construct Composite Unit Fractions Attempts to Construct a Unit Fraction of a Connected Number Partitioning and Disembedding Operations Joes Construction of a Partitive Fraction Scheme Joes Production of an Improper Fraction Patricias Recursive Partitioning Operations The Splitting Operation: Corroboration in Joe and Contraindication in Patricia A Lack of Distributive Reasoning Emergence of the Splitting Operation in Patricia Emergence of Joes Unit Fraction Composition Scheme Joes Reversible Partitive Fraction Scheme Joes Construction of the Iterative Fraction Scheme A Constraint in the Childrens Unit Fraction Composition Scheme Fractional Connected Number Sequences Establishing Commensurate Fractions Discussion of the Case Study Chapter VIII. Equi-Partitioning Operations for Connected Numbers: Their Use and Interiorization Melissas Initial Fraction Schemes A Reorganization in Melissas Units-Coordinating Scheme Melissas Construction of a Fractional Connected Number Sequence Testing the Hypothesis that Melissa Could Construct a Commensurate Fractional Scheme Melissas Use of the Operations that Produce Three Levels of Units in Re-presentation A Child-Generated Fraction Adding Scheme An Attempt to Bring Forth a Unit Fraction Adding Scheme Discussion of the Case Study Chapter IX. The Construction of Fraction Schemes Using the Generalized Number Sequence The Case of Nathan During his 3rd Grade Multiplication of Fractions and Nested Fractions Equal

The construction of an iterative fractional scheme: the case of Joe

Abstract 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 number sequence (CNS). A CNS can result from the application of a child’s explicitly nested number sequence, ENS (Steffe, L. P. (1992). Learning and Individual Differences, 4 (3), 259–309; Steffe, L. P. (1994). Children’s multiplying schemes. In: G. Harel, & J. Confrey (Eds.), (pp. 3–40); Steffe, L. P. (2002). Journal of Mathematical Behavior, 102 , 1–41) in the context of continuous quantities. It requires the child to incorporate a notion of unit length into the abstract unit items of their ENS. Connected numbers were instantiated by the children within the context of making number-sticks using the computer tool TIMA: sticks. Steffe conjectured that children who had constructed a CNS might be able to use their multiplying schemes to construct composite unit fractions. (In the context of number-sticks a composite unit fraction could be a 3-stick as 1/8 of a 24-stick.) In the case of Jason and Laura, his conjecture was not confirmed. Steffe attributed the constraints that Jason and Laura experienced as possibly stemming from their lack of a splitting operation for composite units . In this paper we shall demonstrate, using the case study of Joe, how a child might construct the splitting operation for composite units, and how such a child was able to not only confirm Steffe’s conjecture concerning composite unit fractions, but also give support to our reorganization hypothesis by constructing an iterative fractional scheme (and consequently, a fractional connected number sequence ( FCNS )) as a reorganization of his ENS.

Symbolizing as a Constructive Activity in a Computer Microworld

In the design of computer microworlds as media for childrens 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 childrens mathematical activity that coemerge in the activity. We illustrate this coemergence through a constructivist teaching episode with two children working with the computer microworld TIMA: Bars. During this episode, in which the children took turns to partition a bar into fourths and thirds recursively, the symbolic nature of their partitioning operations became apparent. The children developed their own drawings and numeral systems to further symbolize their symbolic mental operations. The symbolic nature of the childrens partitioning operations was crucial in their establishment of more conventional mathematical symbols.

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 childrens mathematical schemes was critical in the use of the TIMA as learning tools.

Building a New Model of Mathematics Learning

Abstract In this summary article the process of building a new model of mathematics learning is illustrated through a synthesis of the Fractions Project articles and commentary on the two reaction articles. Crucial aspects of the Project and critiques are brought together to focus on work that has been accomplished in the Project and work that needs to be considered. Of particular concern is maintaining the constructivist view of research and teaching.

Introduction to Section 2

This introduction sets the scene for the volume section 2 on the theme of learning and assessing mathematics with and through digital technologies. It first describes the section’s points of departure. Then each of the chapters of the section is briefly addressed. The introduction ends with a short reflection on the section as a whole, noting that the major content emphases are on algebra and geometry, with only limited attention to calculus, statistical reasoning, and proof. In closing, we call for a closer relationship between mathematics education research and educational science in general.

Chapter 7 Technology and school mathematics

Abstract Many educators believe that technology can be a catalyst for the reform of school mathematics. That promise is largely unfulfilled as over the past decade computers have been used mainly for drill-and-practice type tutorials. Technologies that provide teachers and students with tools for doing mathematics, dynamic environments for exploration, experimentation and conjecturing, and powerful languages for creating algorithmic solutions to problems have transformed the teaching and learning of mathematics. Multi-media environments that are under the joint control of teacher and students also show promise for changing the meaning of teaching and learning. The teachers are critical factors in how technology is used in their classrooms. My thesis is that a technology-driven reform effort will have little effect on the classroom curriculum until teachers are given the opportunities, support and training to take charge of the technology and use it to transform their own teaching environments.

The Partitive, the Iterative, and the Unit Composition Schemes

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 generate observations that can be used not only to corroborate the previous observations, but to refine, extend, and modify them as well. In a teaching experiment, the teacher does not establish a hypothetical learning trajectory at the outset of the experiment for the entire experiment. Rather, the teacher/researchers hypothesize what the children might learn in the next, or even in the next few, teaching episodes based on their current interactions with the children and their interpretations of it, and it is the testing of these hypotheses in the teaching episodes that, in part, constitutes the experimental aspect of the teaching experiment. Both the possibilities that are opened by the particular children and the constraints that the researchers’ experience that emanate from within the children provide new observations that can be retrospectively analyzed to generate a “replicate” case study.

The Construction of Fraction Schemes Using the Generalized Number Sequence

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 children along with their current mathematical activity served in creating possibilities and hypotheses that we continually explored in teaching episodes. In our interactions, we found that their construction of the operations that produce the generalized number sequence opened possibilities for their constructive activity that we did not experience with the other children. We did not decide a priori to use higher-order tasks in our interactions with these two children than we used with the other children. Rather, their ways of operating served as the basis for our constructions of tasks that we used. Nathan participated in the teaching experiment during his third, fourth, and fifth grades, whereas Arthur participated only during his fourth and fifth grades.