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

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.     

A quantitative analysis of children's splitting operations and fraction schemes

Teaching experiments with pairs of children have generated several hypotheses about students’ construction of fractions. For example, Steffe (2004) hypothesized that robust conceptions of improper fractions depends on the development of a splitting operation. Results from teaching experiments that rely on scheme theory and Steffe's hierarchy of fraction schemes imply additional hypotheses, such as the idea that the schemes do indeed form a hierarchy. Our study constitutes the first attempt to test these hypotheses and substantiate Steffe's claims using quantitative methods. We analyze data from 84 students’ performances on written tests, in order to measure students’ development of the splitting operation and construction of fraction schemes. Our findings align with many of the hypotheses implied by teaching experiments and, additionally, suggest that students’ construction of a partitive fraction scheme facilitates the development of splitting.     

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.     

A collective chain of signification in conceptualizing fractions: a case of a fourth-grade class

Fourth graders who participated in a yearlong teaching experiment constructed their own bridge between natural number knowledge and initial conceptualizations of fractions in discrete wholes. This bridge was established by a chain of signs individually and collectively generated. Students came to re-conceptualize natural number as “manifold of units” mediated by linguistic arithmetical expressions, double counting, and numerical diagrams. Linguistic arithmetical expressions, double counting, and numerical diagrams were powerful “representamens” dialectically determining and representing the inverse relations between iteration and decomposition and between part and whole in order to mediate the construction of quantitative fractional relations. These fourth graders’ transition from decomposition and iteration of natural number units to conceptualization of fractions in discrete wholes is analyzed using the Peircean notions of sign, chain of signification, and semiosis.     

Learning trajectory of a student with learning disabilities for the concept of length: A teaching experiment

This study aims to map the learning trajectory (LT) of a student with learning disabilities (LDs) regarding the unit concept in length measurement and the usage of rulers. The article draws on data from a teaching experiment with a 10-year-old student with LDs in Turkey. Data were analyzed in two stages, including microanalysis, where each successive teaching session was separately analyzed, and macroanalysis, where the teaching sessions regarding interrelated instructional goals were analyzed to construct the LT. The main findings of the study illustrate that this student with LDs eliminated her misconceptions about the unit concept and using a ruler, accomplished the determined instructional goals to a large extent, and reached a higher level of thinking with a 4-month teaching experiment designed based on her specific developmental capacity.     

Quantifying the value of buyer–vendor coordination: Analytical and numerical results under different replenishment cost structures

Despite a growing interest in channel coordination, no detailed analytical or numerical results measuring its impact on system performance have been reported in the literature. Hence, this paper aims to develop analytical and numerical results documenting the system-wide cost improvement rates that are due to coordination. To this end, we revisit the classical buyer–vendor coordination problem introduced by Goyal [S.K. Goyal, An integrated inventory model for a single-supplier single-customer problem. International Journal of Production Research 15 (1976) 107–111] and extended by Toptal et al. [A. Toptal, S. Çetinkaya, C.-Y. Lee, The buyer–vendor coordination problem: modeling inbound and outbound cargo capacity and costs, IIE Transactions on Logistics and Scheduling 35 (2003) 987–1002] to consider generalized replenishment costs under centralized decision making. We analyze (i) how the counterpart centralized and decentralized solutions differ from each other, (ii) under what circumstances their implications are similar, and (iii) the effect of generalized replenishment costs on the system-wide cost improvement rates that are due to coordination. First, considering Goyal’s basic setting, we show that the improvement rate depends on cost parameters. We characterize this dependency analytically, develop analytical bounds on the improvement rate, and identify the problem instances in which considerable savings can be achieved through coordination. Next, we analyze Toptal et al.’s [A. Toptal, S. Çetinkaya, C.-Y. Lee, The buyer–vendor coordination problem: modeling inbound and outbound cargo capacity and costs, IIE Transactions on Logistics and Scheduling 35 (2003) 987–1002] extended setting that considers generalized replenishment costs representing inbound and outbound transportation considerations, and we present detailed numerical results quantifying the value of coordination. We report significant improvement rates with and without explicit transportation considerations, and we present numerical evidence which suggests that larger rates are more likely in the former case.     

A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution

In this paper, we establish a version of the Feynman–Kac formula for multidimensional stochastic heat equation driven by a general semimartingale. This Feynman–Kac formula is then applied to study some nonlinear stochastic heat equations driven by nonhomogeneous Gaussian noise: first, an explicit expression for the Malliavin derivatives of the solutions is obtained. Based on the representation we obtain the smooth property of the density of the law of the solution. On the other hand, we also obtain the Hölder continuity of the solutions.     

Fostering construction of a meaning for multiplication that subsumes whole-number and fraction multiplication: A study of the Learning Through Activity research program

We report on a teaching experiment intended to foster a concept of multiplication that would both subsume students’ multiple-groups concept of whole number multiplication and provide a conceptual basis for understanding multiplication of fractions. The teaching experiment, which used a rigorous single-subject methodology, began with an attempt to build on students’ multiple-groups concept by promoting generalizing assimilation. This was not totally successful and led to a redesign aimed at promoting reflective abstraction. Analysis of this latter phase led to several significant conclusions, which in turn led to a revised hypothetical learning trajectory. The revised trajectory aims to foster a concept of multiplication as a change in units.