This article reports on students' learning through conjecturing, by drawing on a semester-long teaching experiment with 6 sixth-grade students. It focuses on 1 of the students, Josh, who developed especially powerful ways of operating over the course of the teaching experiment. Through a fine-grained analysis of Josh's actions, this article integrates Piaget's scheme theory (1950/2001) and Peirce's logic of abduction (1998) into a new theory about conjecturing that explains Josh's learning. Results indicate the power of Josh's *operational conjectures* in resolving problematic situations and constructing new schemes. Because of the context in which the teaching experiment and Josh's conjecturing occurred, results hold implications for research on fractions and on a particular operation called *splitting* (Confrey, 1994; Empson, 1999; Sáenz-Ludlow, 1994; Steffe, 2003). The theoretical integration of scheme theory and abduction also holds implications for resolving the learning paradox (Fodor, 1980; Glasersfeld, 2001).

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## Josh's Operational Conjectures: Abductions of a Splitting Operation and the Construction of New Fractional Schemes

### Anderson Norton

## What's on Your Nation's Report Card?

### Anderson H. Norton III

A report on a sample of fourth graders' responses to the 2003 NAEP exam. Readers are invited to analyze the responses to assess students' concepts of fractions. The article has four examples of student work.

## Mathematical Letter Writing

### Anderson Norton, Zachary Rutledge, Kareston Hall, and Rebecca Norton

Mathematical letter writing can be a mutually beneficial partnership between high schools and universities.

## Unit Transformation Graphs: Modeling Students’ Mathematics in Meeting the Cognitive Demands of Fractions Multiplication Tasks

### Anderson Norton, Catherine Ulrich, and Sarah Kerrigan

This article introduces unit transformation graphs (UTGs) as a tool for diagramming the ways students use sequences of mental actions to solve mathematical tasks. We report findings from a study in which we identified patterns in the ways preservice elementary school teachers relied on working memory to coordinate mental actions when operating in fraction multiplication settings. UTGs account for the constraint of working memory in sequencing mental actions to solve mathematical tasks. They also explain the power of units coordinating structures in offloading demands on working memory. At the end of the article, we consider some of the research implications for these findings—specifically, ways that UTGs can lend explanatory and illustrative power to analyses of students’ mathematics.

## Using Steffe's Advanced Fraction Schemes

### Andrea V. McCloskey and Anderson H. Norton

Recognizing schemes, which are different from strategies, can help teachers understand their students' thinking about fractions.

## Coordinating Units at the CandyDepot

### Anderson Norton, Steven Boyce, and Jennifer Hatch

A new app helps students learn to construct fractions and develop algebra readiness, to take some of the challenge out of both areas.

## ZPC and ZPD: Zones of Teaching and Learning

### Anderson Norton and Beatriz S. D'Ambrosio

The goal of this article is to examine students' mathematical development that occurs as a teacher works within each of 2 zones of learning: students' zones of proximal development (ZPD) and students' zones of potential construction (ZPC). ZPD, proposed by Vygotsky, is grounded in a social constructivist perspective on learning, whereas ZPC, proposed by Steffe, is grounded in a radical constructivist perspective on learning. In this article, we consider potential pragmatic differences between ZPD and ZPC as instantiated during a semester-long teaching experiment with 2 Grade 6 students. In particular, we examine the constructions that a teacher fostered by working with these students in each zone of learning. The data suggest that operating in their ZPD but outside of their ZPC impacts the learning opportunities and resulting constructions of the students. Finally, we characterize aspects of ZPD and teacher assistance that foster the development of mathematical concepts.

## The Splitting Loope

### Jesse L. M. Wilkins and Anderson Norton

Teaching experiments have generated several hypotheses concerning the construction of fraction schemes and operations and relationships among them. In particular, researchers have hypothesized that children's construction of splitting operations is crucial to their construction of more advanced fractions concepts (Steffe, 2002). The authors propose that splitting constitutes a psychological structure similar to that of a mathematical group (Piaget, 1970b): a structure that introduces mutual reversibility of students' partitioning and iterating operations that the authors refer to as the splitting loope. Data consisted of 66 sixth–grade students' written performance on 20 tasks designed to provoke responses that would indicate particular fractions schemes and operations. Findings are consistent with hypotheses from related teaching experiments. In particular, they demonstrate–consistent with the notion of the splitting loope—that equipartitioning and the partitive unit fraction scheme mediate the construction of splitting from partitioning and iterating operations.

## The Splitting Group

### Anderson Norton and Jesse L. M. Wilkins

Piagetian theory describes mathematical development as the construction and organization of mental operations within psychological structures. Research on student learning has identified the vital roles of two particular operations–splitting and units coordination–play in students' development of advanced fractions knowledge. Whereas Steffe and colleagues (e.g., Steffe, 2001; Steffe & Olive, 2010) describe these knowledge structures in terms of fractions schemes, Piaget introduced the possibility of modeling students' psychological structures with formal mathematical structures, such as algebraic groups. This paper demonstrates the utility of modeling students' development with a structure that is isomorphic to the positive rational numbers under multiplication–the splitting group. We use a quantitative analysis of written assessments from 58 eighth grade students to test hypotheses related to this development. Results affirm and refine an existing hypothetical learning trajectory for students' constructions of advanced fractions schemes by demonstrating that splitting is a necessary precursor to students' constructions of 3 levels of units coordination.

## Modeling Students' Mathematics Using Steffe's Fraction Schemes

### Anderson H. Norton and Andrea V. McCloskey

Each year, more teachers learn about the successful intervention program known as Math Recovery (USMRC 2008; Wright 2003). The program uses Steffe's whole-number schemes to model, understand, and support children's development of whole-number reasoning. Readers are probably less familiar with Steffe's fraction schemes, which have proven similarly useful in supporting children's development of fractional reasoning. The purpose of this article is to introduce some of these schemes. We provide examples of student work accompanied by discussions of how fraction schemes can be used as tools for insight into student reasoning. We hope that teachers will find the schemes useful in understanding their students as mathematicians.