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
The role of unitizing predicates in the construction of logic
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
- Award ID(s):
- 1954613
- PAR ID:
- 10355729
- Date Published:
- Journal Name:
- Proceedings of the 12th Congress of the European Society for Research in Mathematics Education
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
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
-
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
-
In many advanced mathematics courses, comprehending theorems and proofs is an essential activity for both students and mathematicians. Such activity requires readers to draw on relevant meanings for the concepts involved; however, the ways that concept meaning may shape comprehension activity is currently undertheorized. In this paper, we share a study of student activity as they work to comprehend the First Isomorphism Theorem and its proof. We analyze, using an onto-semiotic lens, the ways that students’ meanings for quotient group both support and constrain their comprehension activity. Furthermore, we suggest that the relationship between understanding concepts and proof comprehension can be reflexive: understanding of concepts not only influences comprehension activity, but engaging with theorems and proofs can serve to support students in generating more sophisticated understanding of the concepts involved.more » « less
-
Cook, S. ; Katz, B. ; Moore-Russo, D. (Ed.)This paper presents six categories of undergraduate student explanations and justifications regarding the question of whether a converse proof proves a conditional theorem. Two categories of explanation led students to judge that converse proofs cannot so prove, which is the normative interpretation. These judgments depended upon students spontaneously seeking uniform rules of proving across various theorems or assigning a direction to the theorems and proof. The other four categories of explanation led students to affirm that converse proofs prove. We emphasize the rationality of these non-normative explanations to suggest the need for further work to understand how we can help students understand the normative rules of logic.more » « less