Keith Weber has written an article that was recently published in The Journal of Mathematical Behavior. The article is entitled How syntactic reasoners can develop understanding, evaluate conjectures, and generate counterexamples in advanced mathematics. Here is the abstract of Weber’s article:
This paper presents a case study of a highly successful student whose exploration of an advanced mathematical concept relies predominantly on syntactic reasoning, such as developing formal representations of mathematical ideas and making logical deductions. This student is observed as he learns a new mathematical concept and then completes exercises about it. The paper focuses on how Isaac developed an understanding of this concept, how he evaluated whether a mathematical assertion is true or false, how he generated counterexamples to disprove a statement, and the general role examples play for him in concept development and understanding.