Pat begins by describing the work of Dewey, Piaget and others who explained that the communication of meaning lies at the heart of effective teaching. However, communicating meaning is extremely difficult. As Pat says,

Figure [1] shows Persons A and B attempting to have a meaningful conversation. Person A intends to convey something to Person B. The intention is constituted by a thought that A holds that he wishes B to hold as well. The figure shows A not just considering how to express his thought, but considering how B might interpret A’s utterances and actions. It is worthwhile noting that A’s action towards B is not really towards B. A’s action towards B is towards A’s image of B. In a sophisticated conversation A’s action towards B is not just towards B, but it’s towards B with some understanding of how B might hear A. Likewise, B is doing the same thing. He assimilates A’s utterances, imbuing them with meanings that he would have were he to say the same thing. But B then colors those understandings with what he knows about A’s meanings and according to the extent to which A said something differently than B would have said it to mean what B thinks A means. B then formulates a response to A with the intent of conveying to A what B now has in mind, but B colors his intention with his model of how he thinks A might hear him, where the model is updated by anything he has just learned from attempting to understand A’s utterance. And so on. (Thompson, 2013, p. 63)

Figure 1. Summary of intersubjective operations involved in the communication of meaning. (Thompson, 2013, p. 64)

The problem is that just because each party has a mental image of the other as understanding their meaning is no guarantee that there is such a mutual understanding:

In Piaget’s and Glasersfeld’s usage, A’s and B’s conversation enters a state of intersubjectivity when neither of them has a reason to believe that he has misunderstood the other. They may in fact have completely misunderstood each other, but they have not discerned any evidence of such. (Thompson, 2013, p. 64)

I’d like to offer my own interpretation of what was happening in the class that Lew et al. observed. This is pure speculation, but it is based on more than forty years of teaching. I believe that the instructor and the students had attributed very specific and very different meanings to the proof that was presented in class.

To the instructor, this proof was an opportunity to showcase general approaches. The fact is that the theorem that was proven, “If a sequence {

*x*} has the property that there exists a constant

_{n}*r*with 0<

*r*<1 such that |

*x*

_{n}–x_{n–1}| <

*r*for any two consecutive terms in the sequence, then {

^{n}*x*} is convergent,” is not particularly important to the study of convergence. What is clear from the five points that instructor believed he had made was that this provided an opportunity to showcase the usefulness of the Cauchy criterion, the triangle inequality, and the geometric series. This was his meaning. The ease with which peers identified the majority of these messages signifies that they shared his image of the meaning of this example.

_{n}The student inability to recognize the points that the instructor had intended to convey suggests that their meaning for this proof was very different. They probably understood the instructor’s intention as one of communicating that this is a valid result worthy of being noted and remembered. Just laying out a formal proof immediately communicates this message to most students in real analysis. The fact that the only things written on the board were the steps in the proof of this result almost certainly reinforced their belief that it was the validity and significance of this statement that was the instructor’s meaning.

I suspect that, had the instructor written his five points on the board, that might have succeeded in shifting the understanding by some of the students of the instructor’s meaning for this proof. But I would be willing to wager that not all of them, probably not even a majority of them, would have seen these as being as important as the actual statement of the theorem. Their reluctance to even recognize that the instructor had made particular points when these were singled out from the lecture suggests that just writing them on the board would not have been sufficient.

Anyone who has probed student understanding has seen this miscommunication. This raises the obvious question: How do we manage to establish a shared understanding? Certainly, a lecture format with only occasional interaction between instructor and students is fertile ground for intersubjectivity that has nothing to do with mutually shared meanings. This is where clickers can help, especially in large format classes. But their effective use relies on a thorough understanding of the range of possible student understandings of the meanings of the lesson. And this understanding must be accompanied by careful construction of questions that can both identify miscommunication and create the cognitive dissonance that moves students toward understanding the instructor’s meaning.

Flipped classes can be even more effective in establishing common meanings, but they also are not easy to run effectively. The work that is done in class must be carefully tailored to identify student misinterpretations of the intent of the lesson, complete with leverage points for addressing these misunderstandings. It is too easy for a flipped class to degenerate into supervised practice. How much more instructive it would have been for the instructor to ask the students to work on a proof of the stated result, emphasizing the usefulness of each of the tools needed for the proof as students discovered—or were led to discover—them. And, of course, you do not just do this once. You need the students to encounter multiple instances where these tools are useful before they fully grasp their versatility and importance. This approach is not easy. It requires an instructor who is finely attuned to the knowledge and the ability to draw on that knowledge of each of the students. And it requires a considerable investment of time. Such an approach takes far more than the ten minutes the instructor actually spent on this theorem.

However, as the work in this study revealed, that ten minutes was largely wasted. The intended messages were never heard. If these were important messages, and I think that most of us who teach real analysis would acknowledge that they are, then they are worth the effort to communicate this importance. Inevitably, that will require “covering” less material. It forces the instructor not only to prioritize the understandings she intends that students carry away from this course, but also to prioritize her efforts to determine what students think she is saying.

**References**

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.

*Vital directions for research in mathematics education*(pp. 57-93). New York, NY: Springer.

Thompson, P. W. (2015). Researching mathematical meanings for teaching. In English, L., & Kirshner, D. (Eds.),

*Third Handbook of International Research in Mathematics Education*(pp. 435-461). London: Taylor and Francis.

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