Table representations of functions allow students to compare rows as well as values in the same row.
Erik D. Jacobson
This study (n = 1,044) used data from the Teacher Education and Development Study in Mathematics (TEDS-M) to examine the relationship between field experience focus (instruction- or exploration-focused), duration, and timing (early or not) and prospective elementary teachers' intertwined knowledge and beliefs about mathematics and mathematics learning. Early instruction-focused field experience (i.e., leading directly to classroom instruction) was positively related to the study outcomes in programs with such field experience of median or shorter duration. Moreover, the duration of instruction-focused field experience was positively related to study outcomes in programs without early instruction-focused field experience. By contrast, the duration of exploration-focused field experience (e.g., observation) was not related to the study outcomes. These findings suggest that field experience has important but largely overlooked relationships with prospective teachers' mathematical knowledge and beliefs. Implications for future research are discussed.
Erik D. Jacobson
Mathematicians and university math educators insist that multiplication is not repeated addition; however, the interpretation works when solving most elementary school mathematical problems. Sure, mathematicians might make a distinction, but does such hair splitting matter to third graders? Should it matter to their teachers?
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, 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 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.