By David Bressoud

On January 31, I published a piece in The Conversation, “Why Colleges Must Change How They Teach Calculus.” The following is one of the statements that I made in this article:

Active learning does not mean ban all lectures. A lecture is still the most effective means for conveying a great deal of information in a short amount of time. But the most useful lectures come in short bursts when students are primed with a need and desire to know the information.

There is no simple binary choice between an active learning classroom and straight lecture. Furthermore, making a class an effective locus for student learning requires more than just active learning.

An article by Campbell et al. (2017), “From Comprehensive to Singular: A latent class analysis of college teaching practices,” reports on an interesting study of what happens in college classes (not just STEM classes), adding a few layers of complexity that are useful for anyone thinking about how to be a more effective teacher. The authors observed 587 courses in nine colleges and universities, ranging from Research 1 (public and private) to comprehensive state schools to liberal arts colleges at a range of levels of selectivity. They looked for seven types of activities in the classroom.

One of these is lecture, defined as “A presentation or recitation of course content by the faculty member to all students in the class.”

They split active learning into three sub-categories:

●      Class discussion. Back and forth conversation between instructor and students or among students about the course content.

●      Class activities. A structured activity where students engaged with the course content (e.g., case studies, clickers, group work).

●      Student questions. Students asking individual questions of the instructor about the course content.

They also picked up the three practices laid out in Neumann’s (2014) description of cognitively response teaching. Active teaching should be cognitively responsive. Unfortunately, as their observations showed, it often is not. These three practices are:

1. Core subject matter ideas. The instructor introduced in depth one or more concepts that are central to the subject matter of the course, the instructor created multiple representations of “core ideas,” or the instructor introduced students to how ideas play out in the field.

2. Connections to prior knowledge. The instructor surfaced students’ prior knowledge about the subject “core ideas,” or the instructor worked to understand students’ prior knowledge about the subject matter “core ideas.”

3. Support of changing views. The instructor provided a space for students to encounter dissonance between prior knowledge and new course material, or the instructor helped students to realize the difference similarities and sometimes conflict between prior knowledge and new subject matter ideas.

Developing over the past few decades and now accelerating thanks to the work of the community engaged in research in undergraduate mathematics education, there have been remarkable strides in understanding the misconceptions that are barriers to student learning. To cite just two examples that I have discussed elsewhere, students often have difficulty making the transition from trigonometric functions in terms of triangles to the circle definition, and they tend to interpret functions as static objects, impeding an understanding of them as descriptions of the linkage between variables that vary. I discussed this issue of the disconnection between what we say and what students hear in two columns in 2016: What we say/what they hear and What we say/what they hear II. The instructor who does not try to understand the prior conceptions and knowledge that students bring into the classroom is setting a large proportion of the students up for failure.

For the last practice, support of changing views, the physics education community knows how important this is. With their Force Concept Inventory (FCI), Halloun, Hestenes, and Wells (see Hestenes et al., 1992) demonstrated that prior concepts are powerful. Students are reluctant to release them, even in the face of what instructors consider to be clear exposition of the actual state of affairs. Getting students to recognize cognitive dissonance requires skill.

Campbell et al. observed that traditional lecture—what the Progress through Calculus study (Apkarian and Kirin, 2017) has revealed to be standard practice in 72% of all Calculus I classes in university mathematics departments with PhD programs—did a pretty good job on core subject matter ideas, but almost nothing with connections to prior knowledge or support of changing views. And, of course, traditional lecture involved none of the first two active learning sub-categories. Less obvious but not surprising, student questions were seldom observed in traditional lecture.

Active lecture is the second most common form of calculus instruction, found in about 14% of the PhD-granting mathematics departments we surveyed in progress through Calculus (3% of departments relied mainly on active learning practices in the classroom and the remaining departments reported too much variation by instructor to classify their course as one type). These introduced class activities and did not decrease core subject matter ideas. Campbell et al. found that they noticeably increase student questions, but do nothing in and of themselves to improve connections to prior knowledge or support of changing views.

These last two practices were almost never observed in either traditional or active lecture classes. The only classes that were observed to improve these aspects of cognitively responsive teaching were those that made a point of employing all seven behaviors, including lecture. In other words, connections to prior knowledge and support of changing views do not come for free once one is using active learning. They have to be intentionally incorporated, and they rely heavily on carefully guided class discussion.

The lesson is that lecture has its place, and active learning is only one piece of what is needed for a truly effective class. David Hestenes (1998) summed it up nicely in “Who needs physics education research!?”:

Managing the quality of classroom discourse is the single most important factor in teaching with interactive engagement methods. This factor accounts for wide differences in class FCI score among teachers using the same curriculum materials and purportedly the same teaching methods. Effective discourse management requires careful planning and preparation as well as skill and experience … Effective teaching requires complex skills which take years to develop. Technical knowledge about teaching and learning is as essential as subject content knowledge.

Read the Bressoud’s Launchings archive.

References

Apkarian, N. and Kirin, D. 2017. Progress through Calculus: Census Survey Report. https://www.maa.org/sites/default/files/PtC Technical Report_Final.pdf

Bressoud, D. 2016. What we say/What they hear. Launchings. http://launchings.blogspot.com/2016/02/

Bressoud, D. 2016. What we say/What they hear. II. Launchings. http://launchings.blogspot.com/2016/03/

Bressoud, D. 2018. Why colleges must change how they teach calculus. TheConversation. January 31, 2018. https://theconversation.com/why-colleges-must-change-how-they-teach-calculus-90679

Campbell, C.M., Cabrera, A.F., Michel, J.O., and Patel, S. 2017. From Comprehensive to Singular: A Latent Class Analysis of College Teaching Practices. Research in Higher Education. 58: 581–604. https://link.springer.com/article/10.1007/s11162-016-9440-0

Hestenes D., Wells M., Swackhamer G. 1992. Force concept inventory. The Physics Teacher 30: 141-166. http://aapt.scitation.org/doi/10.1119/1.2343497

Hestenes D. 1998. Who needs physics education research!?. Am. J. Phys. 66:46.5. http://aapt.scitation.org/doi/abs/10.1119/1.18898

Mathematical Association of America. 2017. Instructional Practices Guide. https://www.maa.org/programs-and-communities/curriculum resources/instructional-practices-guide

Neumann, A. 2014. Staking a claim on learning: What we should know about learning in higher education and why. The Review of higher Education. 37:249–267. http://www.ashe.ws/files/Past Presidents/37.2.neumann.pdf