torquetum

Selden, J. & Selden, A. (1995). Unpacking the logic of mathematical statements. Educational Studies in Mathematics, 29(2), 123-151.

From the abstract: “This study focuses on undergraduate students' ability to unpack informally written mathematical statements into the language of predicate calculus... We discuss this data from a perspective that extends the notion of concept image to that of statement image and introduces the notion of proof framework to indicate that part of a theorem's image which corresponds to the top-level logical structure of a proof... We infer that these students would be unable to reliably relate informally stated theorems with the top-level logical structure of their proofs and hence could not be expected to construct proofs or validate them, i.e., determine their correctness” (p. 123).

Students can be confused between proof and evidence, also, some accept proofs as valid, but still like to see empirical evidence. There are some nice references to different ways to teach proofs to students. The authors look at students’ abilities to clarify the logical structure of mathematical statements in the context of constructing and validating proofs themselves. They found that this ability was not successfully represented by most of the students in the study. When judging the correctness of a proof, there are questions which need to be asked and answered by the person reading the proof. Their study showed that students could neither ask those questions nor answer them. The paper goes on to define and discuss terms like “proof”, “unpack” and “formal and informal statements”. Also a proof framework is defined: “By a proof framework we mean a representation of the "top-level" logical structure of a proof, which does not depend on detailed knowledge of the relevant mathematical concepts, but which is rich enough to allow the reconstruction of the statement being proved or one equivalent to it. A written representation of a proof framework might be a sequence of statements, interspersed with blank spaces, with the potential for being expanded into a proof by additional arguments” (p. 129).

The authors argue that students “who cannot reliably unpack the logical structure of informally stated theorems, also cannot reliably validate their proofs” (p. 130) and support their argument with discussion of proof frameworks. There is a lovely discussion of “statement images” which is a collection and network of all knowledge items associated with a particular statement and which can be very rich and detailed. There is reference to a Radford 1990 paper which I’d like to dig out. The data is presented in detail and finally three suggestions for teachers are made (summarised from p. 142):

First, since many undergraduate students have difficulty unpacking informally written statements, it might be useful to present theorems and definitions both in a more informal way and in a more formal way.

Second, it might be useful to offer university students some explicit instruction or advice on validation, an area currently more or less neglected. There is no reason for such instruction to be restricted to proofs.

Finally, because university students who still have weak validation skills may not be able to distinguish proofs from supplementary or explanatory comments, it might be good to present material in a way that makes this distinction clear.

Do not treat this blog entry as a replacement for reading the paper. This blog post represents the understanding and opinions of Torquetum only and could contain errors, misunderstandings and subjective views.