Newsletter #61     Review of Liping Ma

This is an historic first occasion, because basically the entire newsletter is written by somebody else. I do need to introduce it, though. Back in March, I went to a conference in DC on the preparation of elementary school teachers (I even wrote about it--that was Newsletter #52.) Part of the preparatory reading was a couple of chapters from a new book by Liping Ma studying differences between some Chinese elementary teachers and some American ones. My first reaction was "gosh!" and my next was "yes, but" and eventually I settled into a resounding "ummmm". As it turned out, Liping Ma herself was at the conference, and I had a chance to hear her say some things, which produced a cyclic repetition of the same three reactions. Since the conference I have heard occasional reverberations, but tending to be highly laudatory remarks by mathematicians specializing in highly non-laudatory remarks about our current state of education, so it didn't elucidate much. Then some brilliant person suggested that I ask Judy Roitman of the University of Kansas to write a review of the book for the education column of the Association for Women in Mathematics. This week I got the review, and suddenly a) it all makes much more sense to me and b) I really want to read the whole book (those of you in the Seattle area will have a chance to borrow it--but not until I finish it!)

The column will not appear for a couple of months, but Judy kindly gave me permission to give youall a sneak preview, so herewith her review: Review of Liping Ma, Knowing and Teaching Elementary Mathematics, Lawrence Erlbaum Associates, Mahwah, NJ, 1999 Reviewer: Judith Roitman Liping Ma found herself, as a very young teenager during the cultural revolution, sent to a town far away from the center. She was, like millions of other patriotic Chinese teenagers, supposed to learn the good earthy values of the peasants. Instead, the peasants wanted their kids to know what she knew, and she found herself (shades of the U.S. frontier) teaching elementary school, all grades at once. She had found her calling. The vagaries of life eventually brought her to graduate school at Michigan State University, where she worked with Deborah Ball's group at the National Center for Research on Teacher Education. More vagaries of life brought her to Stanford to finish her dissertation under Lee Shulman. This book is essentially her dissertation. It should be noted, as Ma herself more modestly notes, that the uncommon elegance of the writing is the result of a partnership between Ma and Cathy Kessel who, as a member of Alan Shoenfeld's mathematics education research group at Berkeley, took on the editing of this book, which appears in a series edited by Schoenfeld. Thus, this work is firmly situated in the education research community, especially in the work of Ball and her colleagues. That mathematicians have discovered this book --- it was reviewed by Roger Howe in the AMS Notices, for example, and was circulated widely in manuscript form by Dick Askey --- is quite wonderful, but the book is often discussed in the mathematics community as if it is an isolated, revolutionary work. This gives the unfortunate impression that Ma's is the only work on education that mathematicians need to read, rather than opening up to mathematicians a literature and set of questions which are a focus of continuing activity in the mathematics education research community. Ma herself is quite clear about her intellectual influences, and discussions of her work should firmly root it in the larger research agenda that produced it. When Ma was at Michigan State, Ball and her colleagues were using a series of mathematical questions to study the mathematical understanding of a group of American elementary teachers. These questions are exquisitely designed to gain insight into teacher's content knowledge by posing hypothetical classroom situations and, essentially, asking ``what would you do?" These questions aim for the jugular of elementary mathematics, and Ma used four of them in her work. The questions she used involve regrouping, place value, meaning of operations, and area and perimeter. Ma found herself, as she politely puts it on p. xix, ``intrigued" by American teachers. They had so much formal education! And they knew so little! So she compared the data on 27 subjects from the study of Ball and her colleagues, with her own data on 72 Chinese teachers. "Data" here does not mean simply performance on the four questions, but extensive interview responses, many of which she quotes at some length. It is important to emphasize that she did not administer an instrument to a large group. Ma's is a qualitative study. Her purpose is not to draw conclusions about Chinese teachers vs. American ones. It is, rather to gain an understanding about what it is that the teachers successfully answering these questions (who tend to be Chinese) know that the teachers who answer unsuccessfully (who tend to be American) do not and then trying to find the conditions that encourage that knowledge. The easy answer for what many of the Chinese teachers know is ``content knowledge." But this is too easy an answer, and it tells us nothing. What kind of content knowledge? Not differential equations, that's for sure. Another easy answer would be "pedagogical content knowledge," for example (at the secondary level) knowing that many students overgeneralize linearity, thinking, e.g., that (a+b)^2 = a^2 + b^2. But that is not it either. Ma's notion is something she calls profound understanding of elementary mathematics, or PUFM, and it is not what you might think. That the integers form a ring, and the rationals a field, under the usual operations, is not PUFM but simply irrelevant. On the other hand, a strong understanding of how algorithms and procedures involving place value are grounded in the distributive law is relevant. Also relevant are things that are barely, or frequently not, in the mathematician's vocabulary: grouping/regrouping and composing/decomposing of numbers, for example; or notions of meaning, such as the partitive interpretation of division. This sort of knowledge is a major part of PUFM. Ma is reminding us that the basic mathematics taught in the elementary classroom is intellectually respectable and worthy of careful attention. But she is also careful to point out that this is not the same understanding that a mathematician might emphasize. I would add that it is not the same understanding that a mathematician might even have, or might have so implicitly she might not even notice it, much less realize how important it is for teachers. Ma's exploration of how teachers gain PUFM is in part an exploration of what is already widely known through various international studies: teachers in east Asia often collaborate, have much more time to plan, work in an atmosphere of on-going professional development, and are generally treated as professionals. She explores how these factors affect how and what teachers learn mathematically. But she also raises other issues: attitudes towards the children in their classes; the sorts of supplemental curriculum material available to teachers; access to materials that encourage on-going learning of fundamental mathematics; teachers' attitudes towards their own learning of mathematics; and, most intriguing (at least, to me), the differences in attitude and absorption between people preparing to be teachers and people who are already experienced teachers. Apparently there is a sharp learning curve that takes place in China after teachers start teaching. The wide-spread assumption that content is essentially learned before teachers begin teaching appears to be essentially flawed. Ma's work poses serious challenges for American education. Because of its subtlety it can easily be misinterpreted (Lee Shulman's introduction delineates some of ways it can be misunderstood: as primarily a cross-cultural study, for instance), especially if the reader is too quick to try to fit it into comfortable pigeonholes. It is suggestive, both building on previous research and pointing to further research. While one may wish that it provide easy answers, it has too much integrity to do so. It is both a graceful introduction (for mathematicians and other neophytes) to an important area of mathematics education, and an interesting theoretical work in its own right. I recommend it highly.

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