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This study aimed to examine the quality of the GeoGebra applets for fraction division by attending to the conceptualizations, representations, and cognitive actions prevalent in these digital resources. The results reveal that a majority of the existing applets conceptualize fraction division as measurement while other conceptualizations of fraction division are underrepresented. Among the various representations used, the length model in the form of fraction bar and area model are the predominant choices, with the number line representation being notably less prominent. The results from this study also show that although most applets attempt to visualize the process of fraction division, there are limited opportunities in most applets for users to enact mental actions associated with fraction division, such as partitioning, unitizing, iterating, and disembedding. These results not only increase our understanding of the affordances and limitations of existing applets for fraction division so that we can become more intentional in our choice of them but also inform the design of new applets that support students' development of a rich and robust understanding of fraction division. [ABSTRACT FROM AUTHOR]

In this preliminary theoretical paper, we describe our use of conjecture mapping (Sandoval, 2014) to guide the design of a digital task sequence to support 6th graders’ meanings for points as a simultaneous representation of the amount-ness of two quantities. The conjecture map ultimately serves as a theoretical framework with testable conjectures about how the design of the digital task sequence might promote the intended learning outcomes. [ABSTRACT FROM AUTHOR]

This paper presents an initiative aimed at integrating tailor-made micro-programming environments (MPEs) into middle school mathematics education to foster student learning and enhance computational thinking skills. We examine the effectiveness of MPEs in engaging students in computational thinking, aligning with mathematical practice standards, and usability in promoting interdisciplinary connections between mathematics and computing in the middle school setting. Findings suggest MPEs can advance computational thinking skills and enrich lesson alignment with educational standards. Most participants indicated approval of integrating MPEs into middle school classrooms. [ABSTRACT FROM AUTHOR]

This study explores how inclusive and differentiated instructional practices, specifically focused on designing and implementing mathematical tasks, influence student learning outcomes in mathematics education across Nigeria, Kenya, Ghana, and the USA. By integrating insights from prominent research in each context, the research aims to uncover the transformative potential inherent in mathematical tasks. Drawing from established concepts like higher-order thinking, problem-solving, scaffolding, and differentiated instruction, the study navigates diverse educational contexts, emphasizing the impact of socioeconomic contexts, inclusivity, and collaborative learning environments. The findings aim to offer practical guidance for educators and policymakers, contributing valuable insights to enhance mathematics learning environments. [ABSTRACT FROM AUTHOR]

Numeracy has played an increasingly important role in schooling in recent years, with the recognition that individuals need to be numerate in order to cope with the mathematical demands of everyday life. One societal issue involving a great deal of mathematical information is climate change. We conducted a study to understand the Australian general public’s (n = 144) climate change views, practices, and numeracy capabilities. Based on our analysis of questionnaire data, we found that many participants lacked understanding of climate change information presented in stacked bar graphs. Furthermore, several participants seemed unable to focus on the mathematical content of the numeracy questions; instead, they responded in emotional, non-mathematical ways. Consequently, we argue that numeracy education needs to include a focus on the dispositional aspects of numeracy, per the Goos et al. (2014) model. [ABSTRACT FROM AUTHOR]

How preservice teachers (PSTs) learn to visually represent fraction multiplication with partitioning is an ongoing area of research. Our study investigated what strategies 15 elementary PSTs used to solve and model two non-unit fraction multiplication tasks before instructional guidance. We found the majority of PSTs used algorithmic approaches to provide solutions and often did not provide an explanation or a visual representation to support their reasoning. Two special cases provided insights into challenges PSTs may face when depicting and partitioning the unit within area models. These findings have implications for purposeful sequencing of fraction tasks and intentional instruction around representing and partitioning units. [ABSTRACT FROM AUTHOR]

Copyright of Conference Papers -- Psychology of Mathematics & Education of North America is the property of Psychology of Mathematics & Education of North America and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Vague or imprecise language has been shown to hinder students’ mathematical understanding. To investigate how imprecise language contributes to student errors, researchers analyzed the work of college freshmen on a trigonometry problem. Findings showed that students struggled with conceptualizing radians and moving from understanding trigonometric functions in the context of triangles to the context of the unit circle. Imprecise diagrams and formulas evidenced students’ lack of full understanding. Researchers recommend the use of precise language that prioritizes coherence, consistency, and conceptual clarity. [ABSTRACT FROM AUTHOR]

In this paper, we present part of a larger study investigating the development of multiplicative ideas in young children before their introduction to multiplication. Arrays were displayed as Quick Images (briefly displayed for 5-10 seconds), including those with hidden and missing amounts, over nine days across 3 weeks. Second-grade students (n=20) were asked to determine the total amount, followed by whole-class sharing and discussions. Using a constant comparative method, we analyzed students’ problem-solving strategies reported here. Results indicated an initial reliance on subitizing and counting strategies, with neglect of row/column size in determining the total amount. Students exhibited a shift in attending to rows/columns as composite units, indicating early multiplicative ideas, particularly when making sense of arrays with partially missing amounts. Implications for teaching and research are also discussed. [ABSTRACT FROM AUTHOR]

We examined students’ thinking of graphs around a graphing task from 14 individual interviews, in terms of three layers—frames of reference, coordinate systems, and graphs—and explored their productive and intuitive strategies. As a result, we present a framework that offers a characterization of students’ graphing activities. We then discuss implications of the framework. [ABSTRACT FROM AUTHOR]

As graph literacy continues to be necessary to communicate in STEM fields, conventions around such graphs have developed for students to work and reason with. We describe a fifth grader’s, Emma’s, thinking through non-conventional graphical representations of a linear relationship. We argue that Emma relied on mathematical reasoning when faced with conflict in conventions and was able to make sense of unconventional graphs by using quantitative strategies. Although Emma acknowledged her known conventions of graphing, she was not bound by these conventions but rather leaned on her reasoning about quantities and flexible use of reference frames. We use Emma’s activity to argue possible implications for research and teaching regarding graphing conventions. [ABSTRACT FROM AUTHOR]

Situated within efforts to understand the complex interplay among learners, teachers, and tasks in mathematical modeling education, we examine how contingent scaffolding moves influence the modeling process. Using mixed methods, we coordinated qualitative frameworks for scaffolding and modeling competencies through their application to task-based cognitive interviews with undergraduate STEM majors. A mixed logistic regression model with participant random effect analyzed the temporally-linked frequencies of codes. The model sustains claims about the compatibility of the frameworks and predicts moves eliciting competencies. [ABSTRACT FROM AUTHOR]

The purpose of this study was to investigate 236 U.S. in-service elementary and middle grades teachers’ responses to a measurement fraction division problem including their strategies and types of errors. Using an inductive content analysis approach, one main result was that relatively high number of teachers (218 out of 236 teachers; 92.4%) responded to the problem correctly. A second main result was that a greater percentage of teachers had the quantitative meanings of measurement division, indicating flexibility with referent units, in comparison to those teachers and preservice teachers in past research. Regarding informal strategies, teachers relied on the common denominator and decimal strategies; however, strategies such as repeated subtraction, unit rate, or dividing numerators and denominators were not present. We discuss implications for improving teachers’ knowledge of fraction division. [ABSTRACT FROM AUTHOR]

In this report, we share analysis of 55 students’ interview responses to a “suspension of sensemaking” story problem involving how far two people live from one another. The prompt occasioned different ways of reasoning. Some students reasoned about lines; others about triangles; and still others about circles. Students’ differing responses indicate across-student variation in spatial organization and the quantitative roles of distance in those spatial organizations. We describe students’ mental operations of reasoning about points as varying and static and relate it to research on students’ meanings for a circle as representing all points an equal distance away from a center point. Overall, this study provides a quantitative reasoning lens to better understand how students make sense of a well-studied problem. [ABSTRACT FROM AUTHOR]

Graphical representations are commonly used in everyday life and are important in STEM fields. Interpreting graphs entails understanding the underlying structures of graphs, including coordinate systems and reference frames. In this report, we characterize one student’s constructions of coordinate systems. These constructions indicate two distinct types of reference frames not currently distinguished in the literature: (a) continuous and (b) ordered-discrete. Using data from a 10-session teaching experiment, we discuss the interplay of a student’s perception of tasks, the reference frames she reasoned with, and differences in those reference frames. We consider how the interplay of the aforementioned items may have influenced the quantities she considered as well as the coordinate systems she constructed. We conclude with suggestions for research and teaching that support students’ productive graphing activity. [ABSTRACT FROM AUTHOR]

This case study investigates the thinking a student utilizes while exploring a task involving a linear function application. The study tracks the level of covariational reasoning and uses of slope a student applies while making sense of the situation contextually, on a Cartesian graph, and on a dynagraph. Although the thinking revealed aligns with past research findings that students often have conceptions of slope that are isolated from covariational thinking, the student in this case study demonstrates strong covariational thinking across representations that enables him to use slope for different purposes while never referencing the term “slope” and never displaying any signs of shape thinking. [ABSTRACT FROM AUTHOR]

Mathematical equality and equivalence are critical concepts that span K-12 and tertiary education. Few studies use sociocultural approaches to examine thinking about equivalence in postsecondary education. This case study utilizes a discursive approach to explore how one student graduating from a developmental mathematics program at a community college thinks about equality and equivalence. We analyze this student’s word use, visual representations, routines, and narratives as components of her discourse to explore the patterns in her thinking. The results provide the student’s rich and nuanced thinking while revealing when and how consistencies and inconsistencies occur in her thinking. Questions allowing students to generate their own examples and definitions can be useful in eliciting their thinking and a discursive approach can have implications in enhancing communication in postsecondary classrooms. [ABSTRACT FROM AUTHOR]

In this study, I present how eight U.S. college calculus instructors with different patterns of inquiry practices used instructional situations to frame instructional tasks for introducing derivatives graphically to students. During four interviews, the instructors proposed up to eight tasks for introducing derivatives physically, graphically, verbally, and symbolically (Zandieh, 2000). The findings focus on the two tasks proposed by each instructor that centered the graphical representation of the derivative: derivative at a point as the slope of a tangent line, and derivative as a function presented by its graph. While no two instructors proposed the same tasks with the same instructional situations for teaching these concepts, they relied on graphing, conjecturing, and calculating situations to frame their tasks. [ABSTRACT FROM AUTHOR]

Students with learning disabilities/difficulties in mathematics often apply ineffective procedures to solve word problems. Given that current mathematics curriculum standards emphasize conceptual understanding in problem solving as well as higher-order thinking and reasoning, the purpose of this study was to evaluate the impact of a model-based problem-solving (MBPS) intervention program on elementary students' word problem-solving performance through analyzing the error patterns. Results indicate that after the MBPS intervention, participants significantly improved their problem-solving performance and made less errors on solving problems across a range of problem situations. Implications of the study will be discussed in the context of National Council of Teachers of Mathematics' calling for teaching big ideas to help students develop a deep understanding of mathematics knowledge. [ABSTRACT FROM AUTHOR]

A new classification for semantic structures of one-step word problems is proposed in this paper. The classification is based on illustrations of word problem situations in Common Core State Standards (CCSSM, 2010) and related historical studies (e.g. Weaver, 1973, 1979, 1982), as well as conceptual elaborations of embodied and grounded nature in Lakoff et al. (2000). The classification identifies two main classes: action on/change of an initial quantity and coordination/comparison of two quantities, providing a unifying characteristic of basic operations of quantities. This classification is more comprehensive and differentiated than the classification of CCSSM (2010) and Polotskaia et al. (2021), as it emphasizes conceptual demands of children's mathematics, coherence and continuity of progressions, and consistency with thinking modes and/or problem-solving strategies. [ABSTRACT FROM AUTHOR]

This paper provides a case study account of a preservice secondary mathematics teacher's thinking while engaging in slope tasks using dynagraphs. The data included audio recordings and screen captures of a small group of preservice teachers engaging with these tasks, with our analysis focusing on the case of Robin. Despite familiarity using slope to measure steepness and determine relationships in a Cartesian plane setting, results indicate Robin struggled to reason about these same uses of slope when presented with a dynagraph. Using the APOS-Slope framework as a lens, the analysis suggests that Robin is limited to thinking of slope as an Action when using slope to measure steepness and determine relationships, relying heavily on shape thinking in the Cartesian plane. Implications for further research are provided. [ABSTRACT FROM AUTHOR]

The construct of static and emergent shape thinking (Moore & Thompson, 2015) characterizes differences in students' reasoning about graphs. In our previous work with middle school students, we found that this construct may also be useful in characterizing students' reasoning about other representations such as simulations and tables. In this paper, we present data from six students' reasoning to initiate a discussion around the possible nature of static and emergent shape thinking in the context of simulations and tables that would contribute to an expansion of the current framework to include these representations. [ABSTRACT FROM AUTHOR]

There are strong motivations to implement integrated STEAM activities that engage with key ideas in mathematics. In integrating mathematics with other STEM disciplines, however, epistemic tensions can emerge. Rather than attempting to suppress, avoid, or adjudicate these tensions, we propose a strategy of "epistemic rekeying," in which epistemic tensions are offered as provocations for students to create playful and artistic responses. This approach takes epistemic tensions seriously and makes them accessible to young learners. We give the rationale for this approach and describe settings where students' creativity suggested its potential to us. [ABSTRACT FROM AUTHOR]

This study addresses the need to better describe instructional strategies used by middle grades mathematics teachers. After coding 177 videos of grades 6-8 mathematics instruction for indications of effective instructional practices, we further analyzed 8 of the highly scored videos with specific attention to teachers' implementation of strategies associated with Explicit Attention to Concepts (EAC) (Champion et al., 2020). We found that these effective teachers of mathematics tended to enact EAC by using a preferred strategy more predominantly than others, though all teachers used at least two EAC strategies during a lesson. Additionally, most participants in our study used an Initiate, Response, Evaluate (IRE) format (Mehan, 1979) when enacting EAC. We illustrate examples of their instruction with vignettes. [ABSTRACT FROM AUTHOR]

A teacher's role in helping students understand mathematics and engage in sense-making is crucial. Therefore, we investigated how teachers viewed instructional gestures after they participated in a content and pedagogy based professional development project. The findings revealed that teachers thought about gestures as a means of communication and cognition. Implications for supporting teachers' use of instructional gestures to help students to make mathematical connections through integrated content-based professional development are discussed. [ABSTRACT FROM AUTHOR]

Copyright of Conference Papers -- Psychology of Mathematics & Education of North America is the property of Psychology of Mathematics & Education of North America and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This project seeks to understand the emergence of mathematical meanings mediated by learners' interactions with multiple artifacts. Extending our prior work which took an enactivist approach and revealed the dynamics of embodied interactions fundamental to understanding fraction division, we now employ a semiotic lens to illuminate how learners make personal meanings from their engagement with multiple artifacts and translate them into more generalized mathematical meanings. We are doing so by taking a semiotic approach to tracking the emergent phenomenon of two learners' meaning making as it arises from the complex interplay of signs. We rely on our findings to argue that semiotic theory can be used as a resource to complement and enhance an enactive analysis of the unfolding of sense making with multiple artifacts. Implications for the design of learning experiences with multiple artifacts are proposed. [ABSTRACT FROM AUTHOR]

Recently, abstracted quantitative structures (AQS), a construct from quantitative reasoning, has been offered as a means to conceptualize and study mathematization during mathematical modeling. Extending this theoretical work, we provide empirical evidence that an intervention targeting participants' AQS can assist in aligning modelers' models with normatively correct models. We report on a pre/post intervention study designed to elicit alignment between symbolic forms and AQS and alignment between AQS and modeling scenarios. We used the Sørenson-Dice coefficient and cluster analysis to identify shifts in student associations of symbolic forms with modeling scenarios. [ABSTRACT FROM AUTHOR]

The purpose of this study was to examine the relationship between in-service teachers (ISTs') reported use of manipulatives and their pedagogical content knowledge for teaching fractions (PCK-Fractions). The study's results indicated no significant relationship between ISTs' reported use of visual representations and their PCK fractions. However, trends were observed across ISTs' education, taught grade levels, PCK, and use of visual representation. The implications and future needs for the study are discussed in the paper. [ABSTRACT FROM AUTHOR]

Many studies use instructional designs that include two or more artifacts (digital manipulatives, tables, graphs) to support students' development of reasoning about covarying quantities. While students' forms of covariational reasoning and the designs are often the focus of these studies, the way students' interactions and transitions between artifacts shape their actions and thinking is often neglected. By examining the transitions that students make between artifacts as they construct and reorganize their reasoning, our study aimed to justify claims made by various studies about the nature of the synergy of artifacts. In this paper, we present data from a design experiment with a pair of sixth-grade students to discuss how their transitions between artifacts provided a constructive space for them to reason about covarying quantities in graphs. [ABSTRACT FROM AUTHOR]

Hundreds charts are an essential tool for teachers in the mathematics classroom. The present study investigated the effects of spatial configuration in hundreds charts on children's placevalue knowledge and their strategies for solving arithmetic word problems. Kindergarten and first-grade students were assigned to three instructional conditions in which they worked with a researcher to solve arithmetic problems. The charts in the conditions differed by spatial configuration. Results showed that the children improved their place-value knowledge and were more likely to use strategies based on place-value concepts when the numbers in the chart increased from top to bottom than when the numbers increased from bottom to top. [ABSTRACT FROM AUTHOR]

The primary aim of the present study was to examine whether the studs on LEGO® bricks act as irrelevant details when solving fraction division problems. We also investigated whether prior knowledge plays a role in the irrelevant details effect and in children's problem-solving strategies. Fifth- and sixth-grade students (N = 38) first completed a fractions test to assess their prior knowledge. A video-recorded lesson then was delivered to all students on how to solve fraction division problems with LEGO® bricks. Most of the participants were not distracted by the studs on the LEGO® bricks when solving fraction division problems on a transfer task. Prior knowledge was related to accuracy on learning and transfer performance, with low prior knowledge students generating less accurate solutions than to high prior knowledge students. [ABSTRACT FROM AUTHOR]

The constitution of a mental object in a student implies their competence to perform an adequate reading and execute processes in the different contexts in which it is possible to find that related concept. As part of the constitution of the mental object system of linear equations, we consider that pre-university students must be competent in the use of a resolution method that allows recognizing a system of equations as a mathematical object and identifying the nature of its solution set. In addition, we consider necessary the competence to identify the elements of systems of linear equations in different mathematical sign systems. [ABSTRACT FROM AUTHOR]