Answers to “Common Concerns about Active Learning” from the American Mathematical Society

The February 2017 issue of the Notices of the AMS carried an article about active learning strategies.  Here is the piece of that article I enjoyed the most.  Thanks to Sam Huckaba for sharing.

As I’ve said previously, I am only qualified to comment on teaching and learning in physics.  But this article was written by experts in the teaching and learning of math.  I guess active learning is good for math as well.  

Common Concerns about Active Learning

While active learning has many advocates among mathematicians,  there are also responsible teachers who have reasonable concerns about active learning methods. We address four of them here.

How will students learn the mathematics if we do not clearly tell them everything about it?

The historical dominance of the lecture format rests on the belief that learning occurs as a result of transmission of information from instructor to student and that students learn by a process of taking in bits of information that their instructors say or write. Further, because of our passion and love for mathematics, a natural human impulse is for mathematics faculty to tell students about the ways we have come to understand our discipline, to shed light on the subtleties that surround most mathematical ideas, and to explain the fundamental insights of our field. Our common experience, supported by research, demonstrates that learning is not this simple. For example, almost every teacher has experienced telling a student a certain mathematical fact—such as (a+b)^2
does not equal a^2 + b^2—only to have them demonstrate on a test that they have not learned it. Such experiences suggest that it is not enough for students simply to be told information if we want to produce deep and meaningful learning. Thus, the key is to find an effective balance between direct instruction and active learning, wherein instructors provide guidance through a combination of explanations and active learning tasks.

What if I can’t cover the same amount of material?

Direct instruction alone can be an efficient way of getting through material. However, the example of students not knowing that (a+b)^2 does not equal a^2 + b^2 should not be far from our minds: lecturing in order to cover more material is not always effective for students. By exclusively considering course content coverage and responding to content coverage with telling, we risk forgetting the many other elements of student learning that active learning addresses, such as the cognitive goals for students outlined in the 2015 MAA CUPM Curriculum Guide, including:

  • recognize and make mathematically rigorous arguments,
  • communicate mathematical ideas clearly and coherently both verbally and in writing,
  • work creatively and self-sufficiently,
  • assess the correctness of solutions,
  • create and explore examples,
  • carry out mathematical experiments, and
  • devise and test conjectures.

In addition to the recognition that content topics are not the exclusive subject of coverage, recent research suggests that coverage of material is less important for student persistence and achievement in mathematics than the use of teaching techniques that address these other types of learning goals.

How do I know if I’m doing a good job with my teaching?

The crafting of rich lectures contributes to mathematicians’ feelings of efficacy in their discipline.  There are, however, a number of other ways in which teachers may gain efficacy while balancing traditional lecture with active learning in their classrooms. These include activities such as choosing problems, predicting student reasoning, generating and directing discussion, pushing students for high-quality explanations, asking for questions that extend student knowledge, and obtaining immediate feedback from students regarding what they just learned. Reflecting in this manner shifts the way we measure our own teaching away from the quality of our presentations and toward the quality of the tasks we provide students. Further, many mathematicians who implement active learning report that they have a deeper understanding of student progress and can observe changes in students more clearly than in their previous courses.

I didn’t need active learning, why do my students?

Although this is changing, many mathematicians have not personally experienced undergraduate teaching environments that include active learning components. Thus, for many mathematicians and graduate students, their first experience with active learning techniques will be as teachers rather than as students. However, we should be careful when comparing our own experiences with those of our students; as Carl Lee has written:

I often engaged in math classes at a high cognitive level merely as a result of a teacher’s direct instruction (“lecture”). As a teacher I quickly learned that I engaged few of my students by this process. Not all developed their “mathematical habits of mind” or “mathematical practices” through my in-class lectures and out-of-class homework (often worked on individually).I now better appreciate the significant role of personal context and informal education in the development of students’ capacity.

Research suggests that active learning has a strong positive impact on a wide range of students, not only those who enter our courses ready to independently engage with math at a high cognitive level. That research also suggests that active learning does not harm, and may further benefit, already high-achieving students. Reflecting on our own educations, the authors agree that we would likely have built a firmer mathematical foundation had we experienced more active learning environments and that active learning would have prompted in us an earlier understanding of mathematics as an inquiry-based discipline.

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