Instructional designs that include two or more artifacts (digital manipulatives, tables, graphs) have shown to support students’ development of reasoning about covarying quantities. However, research often neglects how this development occurs from the student point of view during the interactions with these artifacts. An analysis from this lens could significantly justify claims about what designs really support students’ covariational reasoning. Our study makes this contribution by examining the “messiness” of students’ transitions as they interact with various artifacts that represent the same covariational situation. We present data from a design experiment with a pair of sixth-grade students who engaged with the set of artifacts we designed (simulation, table, and graph) to explore quantities that covary. An instrumental genesis perspective is followed to analyze students’ transitions from one artifact to the next. We utilize the distinction between static and emergent shape thinking to make inferences about their reorganizations of reasoning as they (re-)form a system of
instruments that integrates previously developed instruments. Our findings provide an insight into the nature of the synergy of artifacts that offers a constructive space for students to shape and reorganize their meanings about covarying quantities. Specifically, we propose different subcategories of complementarities and antagonisms between artifacts that have the potential to make this synergy productive.
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STUDENTS’ REORGANIZATIONS OF VARIATIONAL, COVARIATIONAL, AND MULTIVARIATIONAL REASONING
In this paper we examine sixth grade students’ constructions and reorganizations of variational, covariational, and multivariational reasoning as they engaged in dynamic digital tasks exploring the science phenomenon of weather. We present case studies of two students from a larger whole-class design experiment to illustrate students’ forms of reasoning and the type of design that supported those constructions and reorganizations. We argue that students constructed multivariational relationships by bridging, transforming, and reforming their reasoning and that the nature of the multivariational relationship being constructed affected this process.
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- Award ID(s):
- 1742125
- NSF-PAR ID:
- 10383421
- Editor(s):
- Olanoff, D.; Johnson, K.; Spitzer, S.
- Date Published:
- Journal Name:
- Proceedings of the forty-third annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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We describe a functional programming approach to the design of outlines of eighteenth-century string instruments. The approach is based on the research described in François Denis's book, Traité de lutherie . The programming vernacular for Denis's instructions, which we call functional geometry , is meant to reiterate the historically justified language and techniques of this musical instrument design. The programming metaphor is entirely Euclidean, involving straightedge and compass constructions, with few (if any) numbers, and no Cartesian equations or grid. As such, it is also an interesting approach to teaching programming and mathematics without numerical calculation or equational reasoning. The advantage of this language-based, functional approach to lutherie is founded in the abstract characterization of common patterns in instrument design. These patterns include not only the abstraction of common straightedge and compass constructions, but of higher-order conceptualization of the instrument design process. We also discuss the role of arithmetic, geometric, harmonic, and subharmonic proportions, and the use of their rational approximants.more » « less
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Abstract Background In scientific inquiry learning, students often have difficulties conducting hypothetical reasoning with multiple intertwined variables. Concept maps have a potential to facilitate complex thinking and reasoning. However, there is little investigation into the content of student‐constructed concept maps and its association with inquiry task performance.
Objectives This study explored students' concept mapping style and its association with task performance in computer‐based inquiry learning.
Methods An exploratory study was conducted with 80 Grade 11 students, who collaboratively constructed concept maps in a free style to support inquiry learning with a virtual ecosystem. Student‐constructed concept maps was analysed by firstly identifying different types of propositions formed in the maps and then determining the style of each concept map based on the dominant type of propositions in the map. Finally, the association between the concept map style and inquiry task performance was explored.
Results and Conclusions Two major concept map styles were identified: (1) knowledge‐oriented concept maps (KCMs) mainly representing problem‐related subject knowledge as a set of concepts and their relationships, and (2) problem‐oriented concept maps (PCMs) mainly representing problem situation as a sequence of changes and their causal relationships. Compared with those constructing KCMs, the students constructing PCMs formed higher‐quality propositions in their maps and performed better in hypothesising, reasoning, and drawing conclusions in the inquiry task.
Implications Besides KCMs, students in inquiry learning can be encouraged to construct PCMs to foster effective thinking and reasoning; that is, constructing a concept map to represent the problem situation as a sequence of changes and the causal relationships between the changes.
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