Abstract
Although proof comprehension is fundamental in advanced undergraduate mathematics courses, there has been limited research on what it means to understand a mathematical proof at this level and how such understanding can be assessed. In this paper, we address these issues by presenting a multidimensional model for assessing proof comprehension in undergraduate mathematics. Building on Yang and Lin’s (Educational Studies in Mathematics 67:59–76, 2008) model of reading comprehension of proofs in high school geometry, we contend that in undergraduate mathematics a proof is not only understood in terms of the meaning, logical status, and logical chaining of its statements but also in terms of the proof’s high-level ideas, its main components or modules, the methods it employs, and how it relates to specific examples. We illustrate how each of these types of understanding can be assessed in the context of a proof in number theory.
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Notes
We note that there were two related types of proof comprehension mentioned by mathematicians that were not discussed in the mathematics education research literature—recognizing where a proof becomes non-routine or difficult, and recognizing why the obvious approach to proving a particular theorem fails.
The proof used in this paper introduces new terminology such as “triadic,” but as the following question illustrates, one need not only ask about new terminology.
In this sense, this aspect of proof comprehension sometimes differs from the “surface level” described by Yang and Lin. This difference is due, in part, to the content of proofs in high school geometry and advanced mathematics. It is uncommon to create new terminology for a specific proof in high school geometry.
A proof might have several methods. However, we anticipate that many proofs for which one would test students’ comprehension have a fairly well-delineated method.
To avoid students answering this question by matching symbols rather than using a deep understanding, one could also ask the student to justify why this choice of M was appropriate or to present them with a poor choice of M and ask why the proof would not work in that case.
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This material is based upon work supported by the National Science Foundation under Grant No. DRL0643734.
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Mejia-Ramos, J.P., Fuller, E., Weber, K. et al. An assessment model for proof comprehension in undergraduate mathematics. Educ Stud Math 79, 3–18 (2012). https://doi.org/10.1007/s10649-011-9349-7
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DOI: https://doi.org/10.1007/s10649-011-9349-7