What we say/What they hear

An important paper is about to appear in the Journal for Research in Mathematics Education, exploring why lecture is so ineffective for so many students: “Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey” by Kristen Lew, Tim Fukawa-Connelly, Juan Pablo Mejia-Ramos, and Keith Weber. The authors video-taped a portion of a lecture given in a junior-level real analysis course and performed a detailed analysis of the differences between what both the professor and his peers thought had been conveyed and what the students were able to take from it.

The study used a class by a professor at a large public university who is widely recognized as an excellent lecturer. It focused on a 10-minute stretch in which a proof was presented. The theorem in question is, “If a sequence {x_n} has the property that there exists a constant r with 0 < r < 1 such that |x_n–x_n–1| < rⁿ for any two consecutive terms in the sequence, then {x_n}is convergent.” The four authors of this paper and an additional instructor who teaches real analysis each observed the video and noted the messages that they saw the professor conveying. They then interviewed the professor who identified five messages that he was trying to convey during this lecture. These are listed below. All except the first had been noted by all of the other peer observers. A full transcript of what transpired during these 10 minutes is included in the appendix to the paper. You may want to check whether you can see these points.

1. Cauchy sequences can be thought of as sequences that “bunch up”
2. One can prove a sequence with an unknown limit converges by showing it is Cauchy
3. This shows how one sets up a proof that a sequence is Cauchy
4. The triangle inequality is useful in proving series in absolute value formulae are small
5. The geometric series formula is part of the mathematical toolbox that can be used to keep some desired quantities small

Six students from this class agreed to participate in the extensive interviews required for the study. They were put into three pairs in order to encourage discussion that would help draw out and verbalize what they remembered.

About two or three weeks after the class in question, students were asked to review their notes about this proof and identify the points that the professor had made. These were compared with the professor’s five points. None of the pairs brought up any of the instructors messages. This is not particularly surprising. Students tend to restrict what they write in their notes to what is being written on the board, and all five of the professor’s points had only been made orally.

As a second pass, each of the students was given a transcript of all that had been written on the blackboard during this proof and then watched the 10-minute lecture, with the hope that they could now focus on what was being said rather than what had been written. They were again asked to identify the points that had been made. One pair did note the emphasis on the importance of the triangle inequality. Another pair noted the third point, that this was about how to set up a proof that a sequence is Cauchy. Nothing else from the list was mentioned.

At a third pass, the students were shown just the five short clips where these five points had been made. Two of the pairs now picked up the first message, two picked up the second, and two picked up the fourth. No one new picked up the third point, that the professor had been illustrating a general approach to proving that a particular sequence is Cauchy.

Finally, the students were told that these five messages might have been contained in the lecture and were asked whether, in fact, these points had been made. Now most of the students were able to see most of these messages, but one pair never acknowledged the second point, that one way to prove that a sequence converges is to show that it is Cauchy, and, even after seeing the clip in which this point was made, none of them acknowledged that the professor had made the fifth point: that the geometric series is part of the toolbox for approaching such proofs.

What I find particularly interesting is the sharp distinction between what was seen in this lecture by those who are familiar with the material and what was seen by those who are still struggling to build an understanding. This echoes much of the work of John and Annie Selden who have shown how difficult it is for undergraduate students to extract the significant features of a proof. This paper shows that it is not enough to accompany what is written on the board with oral indications of what is important and how to think about it. It is not even enough when these indications are repeatedly emphasized.

In the introduction, this paper presents the example of the Feynman Lectures, widely considered to be some of the finest scientific expositions ever made. Yet, the fact is that when they were given at Cal Tech, “Many of the students dreaded the course, and as the course wore on, attendance by the registered students dropped alarmingly.” (Goodstein and Negebauer, 1995, p. xxii–xxiii). There is no doubt that lectures have an important role to play in conveying information for which the recipients have a well-structured understanding in which to place it. However, as this study strongly suggests, lectures are not very helpful for students who are trying to find their way into a new area of mathematics and who still need to build such a structure of understanding.

References

Goodstein, G. & Negebauer, G. 1995. Preface to R. Feynman’s Six Easy Pieces. pp. xix–xxii. New York: Basic Books.

Lew, K., Fukawa-Connelly, T., Mejia-Ramos, J.P., and Weber, K. 2016. Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey. Journal for Research in Mathematics Education. Preprint retrieved from pcrg.gse.rutgers.edu on January 24, 2016.