How Do Gestures Influence Thinking and Speaking? The Gesture-for-Conceptualization Hypothesis.

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

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\u27 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

Middle School Students’ Conceptual Understanding of Equations: Evidence from Writing Story Problems

This study investigated middle school students’ conceptual understanding of algebraic equations. 257 sixth- and seventh-grade students solved algebraic equations and generated story problems to correspond with given equations. Aspects of the equations’ structures, including number of operations and position of the unknown, influenced students’ performance on both tasks. On the story-writing task, students’ performance on two-operator equations was poorer than would be expected on the basis of their performance on one-operator equations. Students made a wide variety of errors on the story-writing task, including (1) generating story contexts that reflect operations different from the operations in the given equations, (2) failing to provide a story context for some element of the given equations, (3) failing to include mathematical content from the given equations in their stories, and (4) including mathematical content in their stories that was not present in the given equations. The nature of students’ story-writing errors suggests two main gaps in students’ conceptual understanding. First, students lacked a robust understanding of the connection between the operation of multiplication and its symbolic representation. Second, students demonstrated difficulty combining multiple mathematical operations into coherent stories. The findings highlight the importance of fostering connections between symbols and their referents

Embodied truths: How dynamic gestures and speech contribute to mathematical proof practices

Grounded and embodied theories of cognition suggest that both language and the body play crucial roles in grounding higher-order thought. This paper investigates how particular forms of speech and gesture function together to support abstract thought in mathematical proof construction. We use computerized text analysis software to evaluate how speech patterns support valid proof construction for two different tasks, and we use gesture analysis to investigate how dynamic gestures—those gestures that depict and transform mathematical objects—further support proof practices above and beyond speech patterns. We also evaluate the degree to which speech and gesture convey distinct information about mathematical reasoning during proving. Dynamic gestures and speech indicating logical inference support valid proof construction, and both dynamic gestures and speech uniquely predict variance in valid proof construction. Thus, dynamic gestures and speech each make separate and important contributions to the formulation of mathematical arguments, and both modalities can convey elements of students’ understanding to teachers and researchers

Evidence for overt visual attention to hand gestures as a function of redundancyand speech disfluency

We investigated the effect of gesture redundancy and speechdisfluency on listeners’ fixations to gestures. Participantswatched a speaker producing a redundant or non-redundantgesture, while producing fluent or disfluent speech. Eyemovements were recorded. Participants spent little time on aspeaker’s gestures regardless of condition. Gestureredundancy and speech disfluency did not affect listeners’percentage dwell time to a speaker’s gestures. However,listeners were more likely to fixate to a speaker’s gestureswhen they expected the gesture to be non-redundant.Listeners were also more likely to fixate to a speaker’sgestures when the speaker was disfluent. Thus, listenersallocate overt visual attention based on the expectedusefulness of a speaker’s gestures, although evidence does notsuggest that they spend more time fixating on these gestures.Furthermore, listeners are sensitive to disfluency in aspeaker’s utterance and change how they attend to gesturesbased on qualities of the speech

The Role of Comparison in Mathematics Learning

To better understand how comparison can be effectively used in mathematics instruction, we reviewed research in psychology and education, with the aim of identifying types of comparison that take place in mathematics learning, and considering the effects of comparison on procedural and conceptual understanding. We identified three types of comparison that are commonly utilized in mathematics instruction and learning: (1) problem-to-problem comparisons, (2) step-to-step comparisons, and (3) item-toabstraction comparisons. Of these three types, only the effects of problem-to-problem comparisons on learning have been well documented. This paper therefore highlights the need for further research to elucidate the unique contributions of different types of comparison in mathematics learning

Interpreting Data Tables: Can Variable Symmetry Scaffold Performance?

Data interpretation is crucial in modern society. One common data structure that people frequently encounter is 2 x 2 tables. Past work suggests that the nature of the variables affects how people interpret 2 x 2 tables. Specifically, people interpret tables with symmetric variables (present/present; e.g., treatment A vs. treatment B) more accurately than tables with asymmetric variables (present/absent; e.g., treatment vs. no treatment). This study tested whether interpreting tables with symmetric variables could scaffold later interpretation of tables with asymmetric variables. Undergraduates interpreted tables and rated the importance of each cell to their interpretations. Some participants interpreted tables with symmetric variables before tables with asymmetric variables; others interpreted only tables with asymmetric variables. Participants who first interpreted tables with symmetric variables later judged cells in the bottom row of asymmetric tables to be more important. Thus, experience with symmetric variables shifted participants’ views of tables with asymmetric variables

The mental representation of integers: Further evidence for the negative number line as a reflection of the natural number line

Humans are able to make sense of extraordinarily abstract concepts in mathematics (e.g., negative numbers). What is the underlying representation of these concepts? Integers extend natural numbers by including zero and negative numbers. To study the mental representation of integers, we employed a number comparison task in an online context. We replicated the previously-reported distance effect, in that far comparisons were faster than near comparisons. Namely, we observed reliable distance effects for positive and negative comparisons, and critically, an inverse distance effect for mixed comparisons. We conclude that the mental representation of integers may align with a hypothesis proposing the mental number line for negative numbers mirrors the natural number line. Moreover, we conclude that web-based data collection is a promising tool for future numerical cognition research