An official website of the United States government
This article offers the construct unitizing predicates to name mental actions important for students’ reasoning about logic. To unitize a predicate is to conceptualize (possibly complex or multipart) conditions as a single property that every example has or does not have, thereby partitioning a universal set into examples and nonexamples. This explains the cognitive work that supports students to unify various statements with the same logical form, which is conventionally represented by replacing parts of statements with logical variables p or P(x). Using data from a constructivist teaching experiment with two undergraduate students, we document barriers to unitizing predicates and demonstrate how this activity influences students’ ability to render mathematical statements and proofs as having the same logical structure.
more »
« less
Epistemological obstacles related to treating logical implications as actions: The case of Mary
Understanding how students reason with logical implication is essential for supporting students’ construction of increasingly powerful ways of reasoning in proofs-based mathematics courses. We report on the results of an NSF-funded case study with a mathematics major enrolled in an introductory proofs course. We investigate the epistemological obstacles that she experienced and how they might relate to her treatment of logical implications as actions. Evidence shows that an action conception may pose challenges when students transform or quantify implications and may contribute to erroneous assumptions of biconditionality. Our report on available ways of operating with logical implications as actions is a first step in designing instructional tasks that leverage students’ existing reasoning skills to support their continued development.
more »
« less
- Award ID(s):
- 2141626
- PAR ID:
- 10547677
- Publisher / Repository:
- Proceedings of the Forty-fifth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Reno, NV.
- Date Published:
- Format(s):
- Medium: X
- Location:
- https://files.eric.ed.gov/fulltext/ED658354.pdf
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Cook, S.; Katz, B.; Moore-Russo, D. (Ed.)This study explores how instructional interventions and teacher moves might support students’ learning of logic in mathematical contexts. We conducted an exploratory teaching experiment with a pair of undergraduate students to leverage set-based reasoning for proofs of conditional statements. The students initially displayed a lack of knowledge of contrapositive equivalence and converse independence in validating if a given proof-text proves a given theorem. However, they came to conceive of these logical principles as the teaching experiment progressed. We will discuss how our instructional interventions played a critical role in facilitating students’ joint reflection and modification of their reasoning about contrapositive equivalence and converse independence in reading proofs.more » « less
-
Spatial reasoning comprises a set of skills used to mentally visualize, orient, and transform objects or spaces. These skills, which develop in humans through interaction with our physical world and direct instruction, are strongly associated with mathematics achievement but are often neglected in early grades mathematics teaching. To conceptualize ways to increase the representation of spatial reasoning skills in the classroom, we examined the outcomes of cognitive interviews with kin- dergarten through grade two students in which they engaged with one spatial reasoning task. Qualitative analyses of students’ work samples and verbal reasoning responses on a single shape de/composition task revealed evidence of a continuum of sophistication in their responses that supports a previously articulated hypothetical learning progression. Results suggest that teachers may be able to efficiently infer students’ skills in spatial reasoning using a single task and use the results to make instructional decisions that would support students’ mathematical development. The practical implications of this work indicate that additional classroom-based research could support the adoption of such practices that could help teachers efficiently teach spatial reasoning skills through mathematics instruction.more » « less
-
Olanoff, D. (Ed.)This report documents how one undergraduate student used set-based reasoning to reinvent logical principles related to conditional statements and their proofs. This learning occurred in a teaching experiment intended to foster abstraction of these logical relationships by comparing the predicate and inference structures among various proofs (in number theory and geometry). We document the progression of Theo’s emergent set-based model from a model-of the truth of statements to a model-for logical relationships. This constitutes some of the first evidence for how such logical concepts can be abstracted in this way and provides evidence for the viability of the learning progression that guided the instructional design.more » « less
-
Based on data from a teaching experiment with two undergraduate students, we propose the unitizing of predicates as a construct to describe how students render various mathematical conditions as predicates such that various theorems have the same logical structure. This may be a challenge when conditions are conjunctions, negative, involve auxiliary objects, or are quantified. We observe that unitizing predicates in theorems and proofs seemed necessary for students in our study to see various theorems as having the same structure. Once they had done so, they reiterated an argument for why contrapositive proofs proved their associated theorems, showing the emergence of logical structure.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
