Unpacking the logic of mathematical statements

Abstract

ABSTRACT. This study focuses on undergraduate students' ability to unpack informally written mathematical statements into the language of predicate calculus. Data were collected between 1989 and 1993 from 61 students in six small sections of a "bridge" course designed to introduce proofs and mathematical reasoning. 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. For simplified informal calculus statements, just 8.5% of unpacking attempts were successful; for actual statements from calculus texts, this dropped to 5%. 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.

Relevance of Learning Logical Analysis of Mathematical Statements

“…-The absence of the existential quantifier produces statements congruent to the initial statement that do not express this statement. -We find again in the students' productions the same phenomena spotted by Selden and Selden (1995), namely, the poor capacity of students in making explicit the logical structure of informal statements.…”

“…Regarding the construction of a proof, Selden and Selden (1995) argue that when students cannot easily make the structure of a logical informal 2 statement explicit, they cannot easily determine the structure of the proof of these statements. Indeed, the logical structure of statements provides indications of how the proof can be undertaken.…”

Mathematicians’ perspectives on the teaching and learning of proof

“…One needs to instantiate in order to have objects to think creatively about and in order to subject inferences to counterexample checks. Instantiation is not strictly necessary for structural thinking, but I would argue that it can help to clarify the structure of a statement, especially in cases involving suppressed quantifiers (Selden & Selden, 1995). For example, in a recent real analysis class, I had a student argue that the statement "Every bounded sequence contains a convergent subsequence" is false because the sequence 1, 0, 1, 0, 1, 0, 1, .…”

“…. ," (Dubinsky & Yiparaki, 2000;Epp, 2003;Zepp, Monin, & Lei, 1987) and that this can interfere with students' abilities to make use of logical structures in constructing and validating proofs (Selden & Selden, 1995, 1999. It is worth noting that effective structural thinking is "syntactic" in the sense in which we use the term in Weber and Alcock (2004) -to refer to correct reasoning based on form and logic -and not as this term is sometimes used to refer to the meaningless manipulation of symbols or ritual use of a certain format.…”

Proof Validation and Modification by Example Generation: A Classroom-Based Intervention in Secondary School Geometry

“…The study reported in this paper addresses proof validation (Selden & Selden, 1995, 2003, which is defined as reading and checking an argument constructed as a proof, to determine whether the argument really constitutes a legitimate proof, that is, whether the argument can establish the truth of a statement. Proof validation is a crucial activity, at not only the undergraduate level, but also primary and secondary school levels, since the Common Core State Standards Initiative (2010) emphasises the ability to critique the reasoning of others as one of the standards for mathematical practice.…”