Mathematical Sciences

We describe the practices of a team of U.S. university teacher/researchers who were invited to attempt to alleviate students’ proving difficulties in an undergraduate real analysis course by offering a voluntary “proving skills supplement.” We analyze what happened in the supplement and why it happened in terms of our theoretical perspective concerning actions in the proving process. This perspective includes that the proving process is a sequence of actions, some of which are not visible or are difficult to recall, and that understanding the justification for an action differs from a tendency to execute it autonomously. Also, the real analysis course and that teacher’s somewhat traditional style of teaching are briefly described, and a comparison is made between proofs co-constructed in the supplement and proofs assigned in the real analysis course. Finally, some student difficulties, views of the supplement given by three students, and the effect of the supplement are briefly discussed.

This paper points out a number of features of the distinctive style in which proofs are written and links them to minimizing validation errors due to working memory overload, rather than the enhancement of insight or conceptual understanding.  Although a theoretical paper, it suggests several empirical questions concerning the mathematics research literature and the role of working memory in validation errors.

This was done by a student on the blackboard. The levels were indicated by her after the proof had been completed. She wanted to remind herself which was the Level 1 and which was the Level 2 proof framework. She indicated on the bottom that the Level 2 proof framework was gotten by unpacking the conclusion, which appears at the bottom of the Level 1 proof framework.

This draft version of our theoretical paper suggests a perspective for understanding university students' proof construction based on the ideas of conceptual and procedural knowledge, explicit and implicit learning, behavioral schemas, automaticity, working memory, consciousness, and System 1 and System 2 cognition. In particular, we will discuss proving actions, such as the construction of proof frameworks that could be automated, thereby reducing the burden on working memory and enabling university students to devote more resources to the truly hard parts of proofs.

First, we discuss a whole class teaching experiment for helping advanced undergraduate and beginning graduate mathematics students construct proofs. This course has been taught at least eight times since Fall 2007 and each time we are learning something more about students’ proving capabilities and difficulties. For example, there are certain aspects of proving that mathematicians do automatically, but that students are often unaware of. We define the formal-rhetorical part of a proof to be those aspects of a proof that can be written by examining the logical structure of the statement of a theorem and by unpacking associated definitions. This is something we have called a proof framework. Examples include writing the first and last lines, “unpacking” the meaning of the last line, and considering what that means for the structure of a proof. Writing the formal-rhetorical part of a proof can expose "the real problem(s)" to be solved. We call the remainder of the proof the p...

This paper presents the results of an empirical study of the proof validation behaviors of sixteen undergraduates after taking a transition-to-proof course that emphasized proof construction. Students were interviewed individually towards the end of the course using the same protocol used by Selden and Selden (2003) at the beginning of a similar course. Results include a description of the students’ observed validation behaviors, a description of their proffered evaluative comments, and the suggestion that taking a transition-to-proof course does not seem to enhance students’ validation abilities. We also discuss distinctions between proof validation, proof comprehension, proof construction and proof evaluation and point out the need for future research on how these concepts are related.

This case study elucidates the difficulty that university students' may have in unpacking an informally worded theorem statement into its formal equivalent in order to understand its logical structure and facilitate constructing a proof. This situation is illustrated with the case of Dori who encountered just such a difficulty with a hidden double negative. She was taking a transition-to-proof course that began by having students first prove formally worded " if-then " theorem statements that enabled them to construct proof frameworks, and thereby make initial progress on constructing proofs. However, later, students, such as Dori, were presented with more informally worded theorem statements to prove. We discuss what additional linguistic difficulties students might have when interpreting informally worded theorem statements and structuring their proofs. This paper sits at the border between linguistics and mathematics education. It considers linguistic obstacles that university students often have when unpacking informally worded mathematical statements into their formal equivalents. This can become especially apparent when students are attempting to prove such statements. We illustrate this with an example from Dori, who was taking a transition-to-proof course that began by having students construct proofs for formally worded " if, then " theorem statements. Early on, she was introduced to the idea of constructing proof frameworks (Selden & Selden, 1995, 2015) and was successful. Later, she encountered difficulty when attempting to interpret and prove an informally worded statement with a hidden double negative. Theoretical Perspective We adopt the theoretical perspective of Selden and Selden (2015) and consider a proof construction to be a sequence of mental or physical actions, some of which do not appear in the final written proof text. Each action is driven by a situation in the partly completed proof construction and its interpretation. For example, suppose that in a partly completed proof, there is an " or " in the hypothesis of a statement yet to be proved: If A or B, then C. Here, the situation is having to prove this statement. The interpretation is realizing that C can be proved by cases. The action is constructing two independent sub-proofs; one in which one supposes A and proves C, the other in which one supposes B and proves C. A proof can also be divided into a formal-rhetorical part and a problem-centered part. The formal-rhetorical part is the part of a proof that depends only on unpacking and using the logical structure of the statement of the theorem, associated definitions, and earlier results. In general, this part does not depend on a deep understanding of, or intuition about, the concepts involved or on genuine problem solving in the sense of Schoenfeld (1985, p. 74). The remaining part of a proof has been called the problem-centered part. It is the part that does depend on genuine problem solving, intuition, heuristics, and a deeper understanding of the concepts involved (Selden & Selden, 2013).

ABSTRACT. We describe the practices of a team of US university teacher/researchers who were invited to attempt to alleviate students' proving difficulties in an undergraduate real analysis course by offering a voluntary "proving skills supplement."

We describe a perspective and a framework for understanding the role of nonemotional cognitive feelings in proving theorems. We begin with a brief discussion of the nature of affect, emotions, and nonemotional cognitive feelings. We see kinds of situations as mentally linked to kinds of feelings that then participate in enacting behavioral schemas to yield actions.

In this largely theoretical paper, we discuss the relation between a kind of affect, behavioral schemas, and aspects of the proving process. We begin with affect as described in the mathematics education literature, but soon narrow our focus to a particular kind of affect – nonemotional cognitive feelings. We then mention the position of feelings in consciousness because that bears on the kind of data about feelings that students can be expected to be able to express. Next we introduce the idea of behavioral schemas as enduring mental structures that link situations to actions, in short, habits of mind, that appear to drive many mental actions in the proving process. This leads to a discussion of the way feelings can both help cause mental actions and also arise from them. Then we briefly describe a design experiment – a course intended to help advanced undergraduate and beginning graduate students to improve their proving abilities.