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.

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### Andrew Izsák

### Sybilla Beckmann and Andrew Izsák

In this article, we present a mathematical analysis that distinguishes two distinct quantitative perspectives on ratios and proportional relationships: variable number of fixed quantities and fixed numbers of variable parts. This parallels the distinction between measurement and partitive meanings for division and between two meanings for multiplication—one rooted in counting equal-sized groups, the other in scaling the size of the groups. We argue that (a) the distinction in perspectives is independent from other distinctions in the literature on proportional relationships, including the within measure space versus between measure space ratio distinction; (b) the psychological roots for multiplication suggest the accessibility of the two perspectives to learners; and (c) the fixed numbers of variable parts perspective, though largely overlooked in past research, may provide an important foundation for central topics that build on proportional relationships. We also suggest directions for future empirical research.

### Andrew Izsák and Erik Jacobson

Past studies have documented students' and teachers' persistent difficulties in determining whether 2 quantities covary in a direct proportion, especially when presented missing-value word problems. In the current study, we combine a mathematical analysis with a psychological perspective to offer a new explanation for such difficulties. The mathematical analysis highlights numbers of equal-sized groups and places reasoning about proportional relationships in the context of reasoning about multiplicative relationships more generally. The psychological perspective is rooted in diSessa's (diSessa, 1993, 2006; diSessa, Sherin, & Levin, 2016) knowledge-in-pieces epistemology and highlights diverse, fine-grained knowledge resources that can support inferring and reasoning with equal-sized groups. We illustrate how the combination of mathematical analysis and psychological perspective may be applied to data using empirical examples drawn from interviews during which preservice middlegrades teachers reasoned with varying degrees of success about relationships presented in word problems that were and were not proportional.

### Andrew Izsák, Sybilla Beckmann, and Torrey Kulow

This article explores teaching practices described in NCTM's *Principles to Actions: Ensuring Mathematical Success for All*. Common factors, common multiples, strip diagrams, and double number lines are discussed in this, the third installment in the series.

### Andrew Izsák, Erik Jacobson, and Laine Bradshaw

We report a novel survey that narrows the gap between information about teachers' knowledge of fraction arithmetic provided, on the one hand, by measures practical to administer at scale and, on the other, by close analysis of moment-to-moment cognition. In particular, the survey measured components that would support reasoning directly with measured quantities, not by executing computational algorithms, to solve problems. These components—each of which was grounded in past research—were attention to referent units, partitioning and iterating, appropriateness, and reversibility. A second part of the survey asked about teachers' professional preparation and history. We administered the survey to a national sample of in-service middle-grades mathematics teachers in the United States and received responses from 990 of those teachers. We analyzed responses to items in the first part of the survey using the log-linear diagnostic classification model to estimate each teacher's profile of strengths and weaknesses with respect to the four components of reasoning. We report on the diversity of profiles that we found and on relationships between those profiles and various aspects of teachers' professional preparation and history. Our results provide insight into teachers' knowledge resources for enacting standards-based instruction in fraction arithmetic and an example of new possibilities for mathematics education research afforded by recent advances in psychometric modeling.

### Andrew Izsák;, Erik Tillema, and Zelha Tunç-Pekkan

We present a case study of teaching and learning fraction addition on number lines in one 6th-grade classroom that used the Connected Mathematics Project *Bits and Pieces II* materials. Our main research questions were (1) What were the primary cognitive structures through which the teacher and students interpreted the lessons? and (2) Were the teacher's and her students' interpretations similar or different, and why? The data afforded particularly detailed analyses of cognitive structures used by the teacher and one student to interpret fractions and their representation on number lines. Our results demonstrate that subtle differences in methods for partitioning unit intervals did not seem important to the teacher but had significant consequences for this student's opportunities to learn. Our closing discussion addresses knowledge for teaching with drawn representations and methods for examining interactions between teachers' and students' interpretations of lessons in which they participate together.

### Andrew Izsák, Erik Jacobson, Zandra de Araujo, and Chandra Hawley Orrill

Researchers have recently used traditional item response theory (IRT) models to measure mathematical knowledge for teaching (MKT). Some studies (e.g., Hill, 2007; Izsák, Orrill, Cohen, & Brown, 2010), however, have reported subgroups when measuring middle-grades teachers' MKT, and such groups violate a key assumption of IRT models. This study investigated the utility of an alternative called the *mixture Rasch model* that allows for subgroups. The model was applied to middle-grades teachers' performance on pretests and posttests bracketing a 42-hour professional development course focused on drawn models for fraction arithmetic. Results from psychometric modeling and evidence from video-recorded interviews and professional development sessions suggested that there were 2 subgroups of middle-grades teachers, 1 better able to reason with 3-level unit structures and 1 constrained to 2-level unit structures. Some teachers, however, were easier to classify than others.

### Andrew Izsák, Torrey Kulow, Sybilla Beckmann, Dean Stevenson, and Ibrahim Burak Ölmez

We report results from a mathematics content course intended to help future teachers form a coherent perspective on topics related to multiplication, including whole-number multiplication and division, fraction arithmetic, proportional relationships, and linear functions. We used one meaning of multiplication, based in measurement and expressed as an equation, to support future teachers' understanding of these topics. We also used 2 types of length-based math drawings–double number lines and strip diagrams–as media with which to represent relationships among quantities and solve problems. To illustrate the promise of this approach, we share data in which future secondary mathematics teachers generated and explained *without direct instruction* sound methods for dividing by fractions and solving proportional relationships. The results are noteworthy, because these and other topics related to multiplication pose perennial challenges for many teachers.