Our concern in this study was to examine the relationship between problem-solving performance and the quality of the organization of students' knowledge. We report findings on the extent to which content and connectedness indicators differentiated between groups of high-achieving (HA) and low-achieving (LA) Year 10 students undertaking geometry tasks. The HA students' performance on the indicators of knowledge connectedness showed that, compared with the LA group, they could retrieve more knowledge spontaneously and could activate more links among given knowledge schemas and related information. Connectedness indicators were more influential than content indicators in differentiating the groups on the basis of their success in problem solving. The tasks used in the study provide straightforward ways for teachers to gain information about the organizational quality of students' knowledge.

# Search Results

## Knowledge Connectedness in Geometry Problem Solving

### Michael J. Lawson and Mohan Chinnappan

## “We Want a Statement That Is Always True”: Criteria for Good Algebraic Representations and the Development of Modeling Knowledge

### Andrew Izsák

This article presents a case study in which two eighth-grade students developed knowledge for modeling a physical device called a *winch*. In particular, the students learned (a) to distinguish equations that are true for any value of the independent variable from equations that constrain the independent variable to a unique value and (b) to solve the latter type of equation to determine when specific physical events occur. The analysis of how these understandings emerged led to two results. First, the analysis demonstrated that students have and can use criteria for evaluating algebraic representations. Second, the analysis led to a theoretical frame that explains how students can develop modeling knowledge by coordinating such criteria with knowledge for generating and using algebraic representations. The frame extends research on students' algebraic modeling, cognitive processes and structures for using mathematical representations, and the development of mathematical knowledge.

## Measuring the Effects of Professional Development on Teacher Knowledge: The Case of Developing Mathematical Ideas

### Courtney A. Bell, Suzanne Wilson, Traci Higgins, and D. Betsy McCoach

This study examines the impact of a nationally disseminated professional development program, Developing Mathematical Ideas (DMI), on teachers' specialized knowledge for teaching mathematics and illustrates how such research could be conducted. Participants completing 2 DMI modules were compared with similar colleagues who had not taken DMI. Teacher knowledge was measured with multiple-choice items developed by the Learning Mathematics for Teaching project and open-ended items based on problems initially developed by DMI experts. After controlling for pretest scores, a hierarchical linear model identified statistically significant differences: The DMI group outperformed the comparison group on both assessments. Gains in teachers' scores on the more closely aligned measure were related to the degree of facilitator experience with DMI. This study adds to our understanding of the ways in which professional development program features, facilitators, and issues of scale interact in the development of teachers' mathematical knowledge for teaching. Study limitations and challenges are discussed.

## Reconceptualizing Procedural Knowledge

### Jon R. Star

In this article, I argue for a renewed focus in mathematics education research on procedural knowledge. I make three main points: (1) The development of students' procedural knowledge has not received a great deal of attention in recent research; (2) one possible explanation for this deficiency is that current characterizations of conceptual and procedural knowledge reflect limiting assumptions about how procedures are known; and (3) reconceptualizing procedural knowledge to remedy these assumptions would have important implications for both research and practice.

## Investigating Measurement Knowledge

### Jenni K. McCool and Carol Holland

Collaborating with a researcher, this teacher uses two fifth graders' assessment results to inform her whole-class instruction and gain insight into all her students' conceptual knowledge.

## Enriching Number Knowledge

### Nancy K. Mack

Exploring number systems of other cultures helps students deepen mental computation fluency, knowledge of place value, and equivalent representations for numbers.

## The Tree of Knowledge

### W. D. Reeve

In the Hall of Science of the Century of Progress Exhibition last year in Chicago there was represented on one of the walls the “Tree of Knowledge,” a photograph of which we present (by permission) on page four of this issue of *The Mathematics Teacher*. As can be seen from this picture mathematics furnishes the central root and vitalizing energy for the basic sciences such as astronomy, botany, chemistry, geology, and physics and, together with them, furnishes strong support for the applied sciences on the higher branches such as social studies like economics and sociology and engineering of various kinds.

## Fostering Middle School Teachers’ Mathematical Knowledge for Teaching via Analysis of Tasks and Student Work

### S. Asli Özgün-Koca, Jennifer M. Lewis,, and Thomas Edwards

Introduction Lee Shulman’s groundbreaking notion of “pedagogical content knowledge” ( 1986 ) suggested that beyond content knowledge, teachers possess forms of knowledge that are specialized for teaching that content. Shulman called for ways

## New Learning and Subverbal Knowledge

### Roy D. Hajek

Every student has a hidden asset. Each has a large store of subverbal knowledge and ability that can be tapped to contribute to his progress in mathematics. This knowledge, accumulated through the sum of past experience in the surrounding world, is seldom displayed, but certainly does contribute to attitudes, conceptions, and reasoning processes. Though the knowledge is not neatly catalogued nor readily available for specific recall, it can form an important basis for progress in new mathematical situations. It guides, outlines, and either limits or extends the amount of insight each person is able to exhibit in new situations.

## A Framework for Investigating Qualities of Procedural and Conceptual Knowledge in Mathematics—An Inferentialist Perspective

### Per Nilsson

Over the years, research has resulted in several conceptual frameworks for describing central and generic components of mathematical knowledge ( Haapasalo & Kadijevich, 2000 ; Star, 2005 ), including relational and instrumental understanding