Pooja G. Sidney

Making Connections in Math: Activating a Prior Knowledge Analogue Matters for Learning.

This study investigated analogical transfer of conceptual structure from a prior-knowledge domain to support learning in a new domain of mathematics: division by fractions. Before a procedural lesson on division by fractions, fifth and sixth graders practiced with a surface analogue (other operations on fractions) or a structural analogue (whole number division). During the lesson, half of the children were also asked to link the prior-knowledge analogue they had practiced to fraction division. As expected, participants learned the taught procedure for fraction division equally well, regardless of condition. However, among those who were not asked to link during the lesson, participants who practiced with the structurally similar analogue gained more conceptual knowledge of fraction division than did those who practiced with the surface-similar analogue. There was no difference in conceptual learning between the two groups of participants who were asked to link; both groups performed less well than did participants who practiced with the structural analogue and were not asked to link. These findings suggest that learning is supported by activating a conceptually relevant prior-knowledge analogue. However, unguided linking to previously learned problems may result in negative transfer and misconceptions about the structure of the target domain. This experiment has practical implications for mathematics instruction and curricular sequencing.

Linking Teacher Instruction and Student Achievement in Mathematics: The Role of Teacher Language

Although high-quality early educational environments are thought to be related to the growth of children’s skills in mathematics, relatively little is known about specific aspects of classroom instruction that may promote these abilities. Data from a longitudinal investigation were used to investigate associations between teachers’ language while teaching mathematics and their students’ growth in mathematical skill during the 2nd grade. Specifically, the extent to which mathematics lessons included cognitive-processing language (CPL)—instruction that is rich in references to cognitive processes, metacognition, and requests for remembering—was related to changes in students’ math achievement. Demonstrating the role of the language of instruction, the findings indicated that children whose 2nd-grade teachers included greater amounts of CPL during instruction evidenced greater growth in math fluency and calculation than did their peers whose teachers employed lower levels of CPL.

Does Comparing Informal and Formal Procedures Promote Mathematics Learning? The Benefits of Bridging Depend on Attitudes toward Mathematics.

1 Temple University, 2 University of Wisconsin—Madison Students benefit from learning multiple procedures for solving the same or related problems. However, past research on comparison instruction has focused on comparing multiple formal procedures. This study investigated whether the benefits of comparing procedures extend to comparisons that involve informal and formal procedures. We also examined how learner characteristics, including prior knowledge and attitudes toward mathematics, affect learning from comparing procedures. We addressed these issues in college students’ learning procedures for solving systems of equations problems in algebra. Learners who liked mathematics learned equally well whether they received comparison or sequential instruction. However, among learners who did not like mathematics, instruction that included support for comparisons between the formal and informal procedures led to greater gains in conceptual knowledge than did sequential instruction of the procedures. Correspondence: Correspondence concerning this article should be addressed to Shanta Hattikudur, via email to shanta.hattikudur@temple.edu.

From continuous magnitudes to symbolic numbers: The centrality of ratio

Leibovich et al.s theory neither accounts for the deep connections between whole numbers and other classes of number nor provides a potential mechanism for mapping continuous magnitudes to symbolic numbers. We argue that focusing on non-symbolic ratio processing abilities can furnish a more expansive account of numerical cognition that remedies these shortcomings.

Who uses more strategies? Linking mathematics anxiety to adults’ strategy variability and performance on fraction magnitude tasks

ABSTRACT Adults use a variety of strategies to reason about fraction magnitudes, and this variability is adaptive. In two studies, we examined the relationships between mathematics anxiety, working memory, strategy variability and performance on two fraction tasks: fraction magnitude comparison and estimation. Adults with higher mathematics anxiety had lower accuracy on the comparison task and greater percentage absolute error (PAE) on the estimation task. Unexpectedly, mathematics anxiety was not related to variable strategy use. However, variable strategy use was linked to more accurate magnitude comparisons, especially among adults with lower working memory performance or those who use mathematics less frequently, as well as lower PAE on the estimation task. These findings shed light on the role of strategy variability in fraction problem solving and demonstrate a link between mathematics anxiety and fraction magnitude reasoning, a key predictor of general mathematics achievement.

Creating a Context for Learning: Activating Children's Whole Number Knowledge Prepares Them to Understand Fraction Division

When children learn about fractions, their prior knowledge of whole numbers often interferes, resulting in a whole number bias. However, many fraction concepts are generalizations of analogous whole number concepts; for example, fraction division and whole number division share a similar conceptual structure. Drawing on past studies of analogical transfer, we hypothesize that children’s whole number division knowledge will support their understanding of fraction division when their relevant prior knowledge is activated immediately before engaging with fraction division. Children in 5th and 6th grade modeled fraction division with physical objects after modeling a series of addition, subtraction, multiplication, and division problems with whole number operands and fraction operands. In one condition, problems were blocked by operation, such that children modeled fraction problems immediately after analogous whole number problems (e.g., fraction division problems followed whole number division problems). In another condition, problems were blocked by number type, such that children modeled all four arithmetic operations with whole numbers in the first block, and then operations with fractions in the second block. Children who solved whole number division problems immediately before fraction division problems were significantly better at modeling the conceptual structure of fraction division than those who solved all of the fraction problems together. Thus, implicit analogies across shared concepts can affect children’s mathematical thinking. Moreover, specific analogies between whole number and fraction concepts can yield a positive, rather than a negative, whole number bias.

Are Books Like Number Lines? Children Spontaneously Encode Spatial-Numeric Relationships in a Novel Spatial Estimation Task

Do children spontaneously represent spatial-numeric features of a task, even when it does not include printed numbers (Mix et al., 2016)? Sixty first grade students completed a novel spatial estimation task by seeking and finding pages in a 100-page book without printed page numbers. Children were shown pages 1 through 6 and 100, and then were asked, “Can you find page X?” Children’s precision of estimates on the page finder task and a 0-100 number line estimation task was calculated with the Percent Absolute Error (PAE) formula (Siegler and Booth, 2004), in which lower PAE indicated more precise estimates. Children’s numerical knowledge was further assessed with: (1) numeral identification (e.g., What number is this: 57?), (2) magnitude comparison (e.g., Which is larger: 54 or 57?), and (3) counting on (e.g., Start counting from 84 and count up 5 more). Children’s accuracy on these tasks was correlated with their number line PAE. Children’s number line estimation PAE predicted their page finder PAE, even after controlling for age and accuracy on the other numerical tasks. Children’s estimates on the page finder and number line tasks appear to tap a general magnitude representation. However, the page finder task did not correlate with numeral identification and counting-on performance, likely because these tasks do not measure children’s magnitude knowledge. Our results suggest that the novel page finder task is a useful measure of children’s magnitude knowledge, and that books have similar spatial-numeric affordances as number lines and numeric board games.