Working Definition of Conceptual Understanding:A student who achieves conceptual understanding in our course is understanding the meaning of a concept well enough (or deeply enough) to be able to "adapt, modify and expand"[1] that concept in order to apply it in novel situations or novel ways (within an application or within a purely mathematical context).
Assessment of Conceptual Understanding:
As instructors, we focus a huge amount of energy on providing the course activities and classroom environment that we believe will help our students develop conceptual understanding. However, for us to truly know the effectiveness of this work we must be equally invested in the assessment of this conceptual understanding. We quickly discover that it is not easy to assess conceptual understanding, as students could correctly answer many questions intended to test a concept by using a method that they have developed or memorized while working on a similar question in previous course work. Therefore, to truly assess conceptual understanding we must examine a concept in a setting that is novel enough to require that the student is indeed reasoning from an understanding of the concepts in order to correctly answer the questions.
This is easier said than done. Many students enter our class with the opinion that being given a problem which they haven't seen before on a homework or an exam is not fair. In a traditional classroom, they might even be justified as they have probably not had an opportunity to develop the ability to reason from a concept to solve an unfamiliar problem. In order to prepare students for these novel problems, we must first carefully build both their confidence to attack problems that are not of a "type" they recognize and their ability to use their understanding of concepts to answer these questions. We do this by engaging students in this type of reasoning in every class activity and in every homework set. We also discuss regularly with our students how important it will be in their future careers to be able to approach an unfamiliar problem and form a strategy for answering it. After this work in class and on homework, we can ask a question on an exam that involves a novel context or use of a concept without the reasoning experience itself being novel or unexpected.
For more information on how we write such questions please see the attached paper below pages 10 - 14. This paper has been published in a slightly different form in the journal Transformative Dialogues and is available here.
[1] One characteristic of understanding concepts is the ability to adapt, modify and expand on a concept according to "Analysis of the Learner Characteristics of Students Implied by the Perry Scheme" by Cornfeld, J.L. and Knefelkamp, L.L. Copyright (c) 1979 by L. Lee Knefelkamp. According to Perry and Knefelkamp this level of understanding enables a student "to investigate and compare things and to make judgements about adequacy or inadequacy, appropriateness or inappropriateness.”
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