The question is, "Where does math teaching reform need to go from here?" Our answer is to get the mathematical community working together in the quest.

The summer of 1998 was a time of debate, challenge, attack and defense. It wasn’t a very good year for changing people’s minds, though.

Why not? Debate and parallel presentation of positions simply isn’t meant to change minds. It is rhetorical jousting in which people take positions, challenge opponents’ positions, defend positions, and try to win a tournament. At best, the kind of contest we witnessed between George Andrews and Jerry Uhl in Toronto cheers supporters of positions, and angers opponents of speakers’ stances.

What is a position? Here are two examples, one from what might be considered the conservative element in these discussions, and one that might be defended by people who support change and reform:

Either of those positions would serve well as the premise for a debate. It could be fun to participate. It would be irritating to watch. The result would probably be an audience full of people with the same positions they had at the beginning, only holding to those positions even more strongly than they did upon entering the hall.

What’s that George Andrews quote about? After the Toronto meeting, Bill Davis and George Andrews exchanged e-mail about the style and content of such encounters. In response to a suggestion that we try to mobilize the math community toward improving the situation in math teaching, Professor Andrews started a note with that sentence.

George Andrews is absolutely right. We aren’t ready to act. As a community, we don’t even have a common understanding of the problems we face. We don’t understand each other’s beliefs. We won’t understand each other as long as we continue the debates, the sermons, and the defense of the righteous positions we take. We could put together a very long list of items we have never discussed: items whose meaning we can only assume.

Each of us looks at statements of purpose or content through our own eyes. In fact, it is possible to read the same document in many ways. I, for example, have read what I believe to be the lead document at the Mathematically Correct web site in different ways at different times. I can read it making my own assumptions about the meanings of statements about content and concept, and find a lot to agree with. I can then read it from the perspective of a person who believes that it represents a thin defense of a reactionary approach to teaching mathematics and come away angry. What I can’t really know is which ideas and issues are shared by the Mathematically Correct and me, and which aren’t. I can guess, and I do, but I can’t know.

Here’s an instance. A common complaint of faculty is that students come poorly prepared, and lack much of the basic knowledge we should expect them to bring to our courses. You’ll hear that from both camps as a defense for their positions. What I don’t think we have is any clear statement of just what it is we want students to bring with them. Even if we were to agree, for example, that every student entering a calculus course should bring along ‘completing the square’, it’s not clear what we would mean. In fact, I think we would mean different things. Do we mean we want our students to be able to take a quadratic expression, x² + A x + B, and find the coefficients a and be which make it (x + a)² + b, or do we mean that we want them to know that it is possible to write it as (x + a)² + b, to know why one might want to do that, and to have the basic skills to find those constants even if they can’t immediately recall the algorithm for finding them? I don’t believe they are the same, and I don’t believe the second follows from the first.

Clarity is important. It determines curricula and syllabi. It lies underpins testing. It is a key element of any statement of standards. It shouldn’t be left to the individual teacher any longer. How can we clarify what we mean if we only debate? It is surprising to me that we mathematicians are willing to proceed into lively argument and debate without our axioms and definitions.

George Andrews ended that note to Bill Davis with a statement that he thinks we don’t have enough common to proceed to reasonable understanding of shared goals. I’m an optimist. I think we have an enormous common set of goals and desires. I believe we can get together and sort that part out, and to decide on strategies on how to proceed toward improving the situation we think we see.

There are things we can do. There are ways for people to get together to find common ground and then decide whether or not to proceed toward change. Here’s a brief summary of a process for change. I suggest that we get together and use one such process. Here’s a description of one the authors of this paper know well.

1. Identify and describe the components.
2. Analyze the current status.
3. Share and engage in Dialogue.
4. Identify the issues.
5. Discuss and make decisions and recommendations for change.
6. Describe the impact of the changes.
7. Create a working plan.

This process must be moderated by trained professionals. Each step in the process requires that certain skills be exercised by the participants. The moderators teach the skills as the process proceeds.

You will notice that items 3 and 5 center on dialogue and discussion. A little while ago I condemned debate. Now I’m asking for dialogue and discussion. What’s the difference? These are the terms used in this process for the conversations which must occur.

Dialogue is conversation whose only purpose is understanding each others’ points. In this context, dialogue has nothing to do with reaching agreement. The point is to allow for assigning meaning to words and ideas, create common understanding of the terms and issues among the participants, sharing information, and learning how everyone views the issues involved. We aren’t used to participating in dialogue. The facilitator is, and knows how to get participants to practice the necessary skills.

Discussion follows dialogue and identification of the issues. The purpose is to take what has been learned and make decisions about what can be done. In other words, discussion is needed for making decisions, solving problems, creating options, increasing mutual respect, serving values, and reaching stated goals. That’s tough stuff.

We propose that at least one of our professional societies adopt the process and endorse a sequence of workshops aimed at finding common ground and determining how to exploit their findings. Each of the workshops must be run by moderators skilled in all aspects of the process, and skilled in their implementation.

There should be a kickoff meeting whose participants represent the most respected leaders in the current debates. It needs to fun for several days, and the goal should be at least a clear statement of good targets for change.

Next, there needs to be a sequence of training workshops to increase the number of people who have the skills to run workshops employing this process for change. At the same time, a sequence of regional sessions should be scheduled which will concentrate on making specific recommendations for action to the mathematical community.

This procedure should begin as soon as participants and sponsors can be identified.

One warning: Whoever decides on the lists of participants for each of these activities must be very careful to select people who do want us to work together, whether or not they believe that is possible. There are people participating in the debates and list-serv discussions who are very happy to have the current confrontations continue. They simply enjoy the conflict and the debate. They shouldn’t be invited to participate.