College Students’ Difficulties in Learning to Do Mathematical Proofs

<Doctoral dissertation> 

Robert Crumley, Moore, (1990)

To explore the part of language in Moore’s study , I read the sections of language in his dissertation which examines difficulties of college students when learning mathematical proofs. 

  

In terms of language and notation, one of his finding of the students’ challenge in learning mathematical proofs is that “the students were unable to understand and to use mathematical language and notation.” For most of the proofs during the course he observed, definitions gave the students both language and notation, and the symbolic form of them showed whole structures of given proofs. Although some students need doing examples to grasp or develop concepts of definitions and theorem they learned, notation sometimes stops those students doing examples in addition to preventing them from understanding definitions and theorem. Additionally, some terminologies were defined in different notations (deliberately by the professor to make the students understand them), and the students sometimes decided the easier one of them based on their experience, concept images, and context. However, the students believe rigid a “mono” definition or notation while the professor explained them flexibly. In the interview with the professor in the study, he said his students seem to translate logical statements from the symbols to the English words without thinking the meaning of them, so “they translated by rote from symbols to words” (p. 89). The author argues that definition provided practice of formal mathematical language for both reading and writing with right words, grammar, and symbols, but informal explanations of concept including examples, diagram were often necessary for the students to understand the notation.

Comment 

One of his finding that the students needed informal expressions of definition to understand the concepts and the professor provided multiple ways whereas some of the students understand only the easier one regardless formal or informal, is considerably interesting to me. Most teachers might attempt to use simple expressions before leading the students to understand the difficult idea. For the students, however, ironically, they cannot easily distinguish which words, grammars, phrases are “formal” or “informal” in the mathematics context. Within mathematical discussions/dialogues in math classes, informal mathematical expressions could be accepted by others unless it does not make sense. However, in terms of mathematical proof, only formal mathematical explanations are accepted as a correct answer. This point should be tough for language learners or bilingual students, and this may be one of the reasons why many students have relied on rote when doing proofs.

In the Japanese national curriculum, proofs in mathematics can be only seen in the unit of geometry in middle school (as far as I remember, but I need to double check). Moore’s study targeted college students in equivalence class, so it should be different between the Japanese students and his students. Currently, I am not sure how mathematical proofs in geometry can be difficult for the bilingual students in secondary school, but it should be one of their challenges in mathematics learning. I think it will be important to know and reveal how they cope with those problems to help other language learners to do mathematical proofs.