Search for: All records

The Reading and Appreciating Mathematical Proofs (RAMP) project seeks to provide novel resources for teaching undergraduate introduction to proof courses centered around reading activities. These reading activities include (1) reading rich proofs to learn new mathematics through proofs as well as to learn how to read proofs for understanding and (2) reading mathematician stories to humanize proving and to legitimize challenge and struggle. One of the guiding analogies of the project is thinking about learning proof-based mathematics like learning a genre of literature. We want students to read interesting proofs so they can appreciate what is exciting about the genre and how they can engage with it. Proofs were selected by eight professors in mathematics who as curriculum co-authors collected intriguing mathematical results and added stories of their experience becoming mathematicians. As mathematicians of colour and/or women mathematicians, these co-authors speak to the challenges they faced in their mathematical history, how they overcame these challenges, and the key role mentors and community have played in that process. These novel opportunities to learn to read and read to learn in the proof-based context hold promise for supporting student learning in new ways. In this commentary, we share how we have sought to humanize proof-based mathematics both in the reading materials and in our classroom implementation thereof. 

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. 

This theoretical paper sets forth two aspects of predication, which describe how students perceive the relationship between a property and an object. We argue these are consequential for how students make sense of discrete mathematics proofs related to the properties and how they construct a logical structure. These aspects of predication are (1) populating the way students generate sets of examples of the property, and (2) testing membership how one tests whether or not a given object has a specific property. Using data from two teaching experiments in which undergraduate students read proofs of theorems about the discrete concept of multiple relations, we illustrate the nature of these aspects of predication and demonstrate how they help explain student interpretations of the proofs. We argue that these particular properties from number theory likely have correlates in many other discrete mathematics topics because of the role of computation/algorithms for defining and testing properties as well as the role of iteration and recursion in populating examples. We anticipate that these constructs will be useful to teachers and researchers of discrete mathematics to foster and assess student understanding of various mathematical properties. They provide tools for thinking about what it means to understand properties in a rich and coherent way that supports understanding complex lines of inference and generalizations. 

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. 

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.