Making the Transition to Formal Proof

Robert Crumley, Moore (1994)

The author investigates the cognitive challenges of university students when learning mathematical proofs. His analysis of the mathematics course participated by students majoring mathematics or mathematics education, reveals that there are three types of difficulties to do mathematical proof for the students: mathematical language and notation, concept understanding, and getting started on a proof. Furthermore, the author analyzes concept understanding based on the scheme and classifies it into students’ five difficulties as Figure 1 shown.

In addition to the idea of Vinner and others (Vinner, 1983; Tall and Vinner, 1981, etc.), the author adds the conception usage as another aspect in concept understanding, so his concept understanding scheme is structured by three aspects: definition, image, and usage.

• Concept definition: defined as “a formal verbal definition that accurately explains the concept on a noncircular way.”
• Concept image: defined as “the set of all mental pictures that one associates with the concept.” 
• Concept usage: defined as “the ways one operates with the concept in generating or using examples or in doing proofs.”

The author claims that this scheme can be helpful for other learning situations in mathematics. For instances, several students who were required proofs in upper-level course said they depended on memorizing the proofs instead of understanding it and how to write it. The model of this schema help for both teacher and students to understand what the students are lacking and confused.

 
Fig 1. Model of the major sources of the students’ difficulties in doing proofs (Moore, 1994, p. 253)

Comment

The model of students’ difficulties in doing a mathematical proof is impressive to me. As I mentioned in another post, I thought I have to memorize the whole procedures of mathematical proofs when I learned mathematical proofs (especially, high school and undergraduate). For example, when I learned mathematical induction in high school mathematics, honestly, I did not know the meaning of “induction” and necessity of it, but I knew this way could get marked as a right answer. Proofing requires students to correctly utilize the concepts they learned with high accurate language whereas students try to cope with them by memorizing like the author mentioned in this paper.

 My curious point here is D6 (Mathematical Language and Notation). What kind of challenges do students face to understand and use mathematical language and notations? How about language learners? (It might be different from native speakers) It would be different between understanding and using language/notations.