By Benjamin Braun, Editor-in-Chief, University of Kentucky; Priscilla Bremser, Contributing Editor, Middlebury College; Art Duval, Contributing Editor, University of Texas at El Paso; Elise Lockwood, Contributing Editor, Oregon State University; and Diana White, Contributing Editor, University of Colorado Denver.
Editor’s note: This is the second article in a series devoted to active learning in mathematics courses. The first article in the series is here.
Mathematics faculty are well-aware that students face challenges when encountering difficult problems, and it is common to hear instructors remark that successful students have high levels of “mathematical maturity,” or are particularly “creative,” or write “elegant” solutions to problems. To appreciate research results regarding active learning, it is useful to make these ideas more precise. Motivated by research in education, psychology, and sociology, language has been developed that can help mathematicians clarify what we mean when we talk about difficulty levels of problems, and the types of difficulty levels problems can have. This expanded vocabulary is in large part motivated by…
…the “cognitive revolution” [of the 1970’s and 1980’s]… [which] produced a significant reconceptualization of what it means to understand subject matter in different domains. There was a fundamental shift from an exclusive emphasis on knowledge — what does the student know? — to a focus on what students know and can do with their knowledge. The idea was not that knowledge is unimportant. Clearly, the more one knows, the greater the potential for that knowledge to be used. Rather, the idea was that having the knowledge was not enough; being able to use it in the appropriate circumstances is an essential component of proficiency.
— Alan Schoenfeld, Assessing Mathematical Proficiency 
In this article, we will explore the concept and language of “level of cognitive demand” for tasks that students encounter. A primary motivation for our discussion is the important observation in the 2014 Proceedings of the National Academy of Science (PNAS) article “Active learning increases student performance in science, engineering, and mathematics” by Freeman, et al. , that active learning has a greater impact on student performance on concept inventories than on instructor-written examinations. Concept inventories are “tests of the most basic conceptual comprehension of foundations of a subject and not of computation skill” and are “quite different from final exams and make no pretense of testing everything in a course” . The Calculus Concept Inventory is the most well-known inventory in mathematics, though compared to disciplines such as physics these inventories are less robust since they are in relatively early stages of development. Freeman et al. state:
Although student achievement was higher under active learning for both [instructor-written course examinations and concept inventories], we hypothesize that the difference in gains for examinations versus concept inventories may be due to the two types of assessments testing qualitatively different cognitive skills. This is consistent with previous research indicating that active learning has a greater impact on student mastery of higher- versus lower-level cognitive skills…
After introducing levels of cognitive demand in this article, our next article in this series will directly connect this topic to active learning techniques that are frequently used and promoted for postsecondary mathematics courses.