Book Review by David Palmer-Stone

The Art of Problem Posing
1983 by Stephen I. Brown and Marion I. Walter
The Franklin Institute Press

People tell horror stories about math teachers, and about math in general. Math dysfunctions have new, popular labels -- "math anxiety" and "innumeracy" (from the books by Sheila Tobias and John Allen Paulos). The standard curriculum has failed, according to this view, because of its authoritarian emphasis on the "right" way to get the answers, and its demoralizing judgments of those who make mistakes.

Adequate mathematicians can solve problems. But for some, math is intrinsically rewarding, like art. They don't just tolerate it: they like it, and choose to do it.

This book's authors suggest that good mathematicians have unconsciously developed strategies for fostering mathematical insight (in spite of the school curriculum), and that we can help students learn to consciously adopt these strategies.

This "non-standard curriculum" focuses on process and questioning, not on getting the right answer. With the threat of "judgment" and "failure" removed, self-esteem is no longer on the line, and the student becomes interested in exploring math through the following four interrelated stages:

1. List attributes of the problem, statement, phenomenon, etc., that you are starting with. This list must be as highly detailed as possible. (This first stage of intense observation is essential to any creative or critical-thinking activity).
   
2. Ask "What-if-not?" for the attributes. This is really the core of this process. You are asking, "What if this attribute were different?" There is a virtual infinity of ways that something can be different from what it is. Different people will notice different attributes, and replace them differently, so this and the previous step open the door for individual ideas.
   
3. Explore, ask questions, come up with examples, in the form of data, answers, ideas, etc.
   
4. After the problem is solved, ask "What does this mean or imply?; What have I actually done?; How far can it be extended?; Why did it work?" and so on. Analyze the problem for its significance.

Good mathematicians will look for regularities, generalities, limits and exceptions to rules, new viewpoints, implications, etc. The above steps provide a fertile hunting ground for these.

The authors suggest broad implications of this work:

1. Listing attributes – and then challenging them – can show us our unconscious assumptions that limit us -- our beliefs, our "givens";
   
2. Generating problems is vital for understanding and creating. You only understand something as you change it; just solving problems does not change the "given" that the problem provides. One should question the question itself.
   
3. Observing one's own learning ("metacognition") helps one learn how to learn. The exercises in the book provide an opportunity to watch and explore your own learning and thinking strategies, and to experiment with new ones.

I highly recommend this book. It has interesting ways to get started, and examples, including student work. The exercises (most are aimed at a high-school level) are generally easy enough to get into, yet hard enough to be challenging. The technique is readily transferrable to other, including creative and even practical, activities. It is good-natured, playful, fun, and at the same time profound. And I agree with its philosophy.

Thomas Homer-Dixon lecture review

Review of Thomas Homer-Dixon Lecture

David Palmer-Stone

Thank you, everybody, for the great participation in our LSABC get-together this spring! I think that this blog (we will be eternally grateful to you, Tlell and Cindy, for setting this up) has great potential for us! I received a nice affirmation of this opinion shortly after our meeting. Arleigh sent me a note informing me that Thomas Homer-Dixon was speaking at UVic (for free!) I attended the lecture since I had read his book, The Ingenuity Gap, some years ago, and found it very thought provoking. He argued in that book that we are creating problems that are so complex that we don't have the wherewithal to solve them. Of course this piqued my interest, as a learning specialist, and as a person interested in the ways we address the challenges presented to us by intractable problems. (In fact, I have been pondering this issue over the past few years, off and on, and some of my ideas informed the ways in which I approached my work on developing our recent workshop.)

So it was with interest that I attended Homer-Dixon's lecture. And I was not disappointed -- in fact it struck me as fortuitous to have had the opportunity to hear what he had to say, in the aftermath of our LSABC conference. For those of you who are unfamiliar with him, he is from The Island, and received his undergraduate degree from UVic. He has a Ph.D. from MIT, and seems to be interested in quantifying global problems, somewhat along the lines of Jared Diamond (Guns, germs, and steel: the fates of human societies, and Collapse: how societies choose to fail or succeed; both engaging books to read); although Homer-Dixon says that he is not as pessimistic as Diamond.

Homer-Dixon has written a new book -- Upside of down: creativity and regeneration in the face of collapse -- and his talk was based in the concerns that he addresses in this book. I will not discuss the problems that he presented. You can check out the book on your own if you want to (I intend to!). What I was more interested in was a nice instance of synchronicity. He was very enthusiastic about the potentials of such technologies as "wikis" and "blogs" as tools for addressing intractable problems, and it struck me as a nice coincidence that we, as learning specialists, have just now taken the plunge to use these technologies ourselves.

I will summarize what I can remember of his comments here -- perhaps adding a few of my own -- because I think they are pertinent to our work with our students, our profession, and our position as professionals within the academic community. (Don't rely on these comments as a source of factual information about the lecture -- I didn't take notes during the lecture and I am basing this on mind maps that I made the next day from memory to review the lecture -- I figure this learning skills strategy works fine for me -- the gain in critical thinking is worth the cost of loss of detail.)

Homer-Dixon had what I thought was an interesting comment in response to a question from a member of the audience. He said that there are two paradigms in our society for addressing problems: 1. The top-down hierarchical approach, in which experts solve problems and apply their solutions in a hierarchical power system; 2. The bottom-up, democratic, "grassroots" approach in which people act responsibly and autonomously, with the designated "leaders" eventually following their initiatives. Homer-Dixon suggests that both approaches are necessary and that Internet systems that use "open architecture" provide a venue within which such work may be engaged. He was enthusiastic about such applications as Wikipedia and noted that in spite of nay-sayers, that system has done very well in developing -- through diligent self reflection -- systems for fostering accurate, unbiased information.

He mentions four serious problems, however, that such systems must address, to be able to work effectively; and I thought that it would be relevant for me to pass these along: 1. Being hijacked. Wikipedia recognizes the need to moderate content, and the people who run the service reserve the right to bar topics or individuals. He refers to such individuals as "extremists," which he defines as people who will not listen or engage in discussion. (I recall reading an article about instances of abuse, for example, in topics of an emotionally volatile political nature.); 2. The role of experts. These systems need to be set up so they will not stifle democratic discussion by deferring to experts, as a matter of course; but at the same time, they will not work properly if "ranting" gains the upper hand, and if expertise is disdained; 3. "Winnowing." This is to do with information overload and is a problem addressed by all information systems now. He suggests that out of 10,000 submissions, only 10 may be worth considering; but how does one narrow down the selection in an effective way? 4. Cooperation and synergy. After the number of items has been winnowed down, they must be enabled to amalgamate into something more intricate and better than even "the sum of the parts."

It seems to me that there are a number of areas of crossover between Homer-Dixon's thesis and our area of work as learning specialists: 1. If anyone is interested in solving intractable problems, not so much as far as content is concerned, but in a generic, process-oriented way, it is us; 2. We delight in collaborative work. If there's one thing I value about our group, it is that there is a complete lack of competition -- we share freely, we are open-minded, we are curious about our own and one another's learning and working processes, we enjoy one another's company when we work together on tasks related to learning, and we are happy to share our learning experiences with one another; 3. We value "high-level" learning in which emergent meanings are fostered. If humanity is to stand a hope of addressing intractable problems, then we must turn our attention to developing ways of developing increasingly complex, adaptable, transferable, applicable, elegant, formulations of understanding; 4. We value, and engage in, self reflection about learning, such that we may discover what works so that we can do more of it. And we often do this as learners, ourselves, "from the learner's point of view."

To my mind, Homer-Dixon's lecture lent great support to our vocation as learning specialists. In fact, if he had known about us, I think he would have said that we are doing cutting-edge work. That is why I am excited about our move into this new medium of blogging, and perhaps developing a "learning wiki," and about our intention of connecting with people involved in the scholarship of teaching and learning. As Homer-Dixon suggested, catastrophes can be the birthplace of insight. I think that the emergence of insight is something we all, in this area of work, value passionately. And the world has never needed it more than right now.