Newsletter #73     On NCTM "Principles and Standards"

The bulk of this is going to be another preview -- I've written a column for the AWM (=Association for Women in Mathematics) Newsletter which I think (hope--what have you) will be of interest. Before I launch it, though, I have three small things to say. Two of them have to do with the "Newsbit" which I put out a couple of weeks ago. The content of that one was that one of our Brown Bag guests, one Greg Miller from Civil Engineering, had volunteered to help set up cross-campus connections such as to get Engineering faculty to drop in on some of the applied or applicable undergraduate courses and chat briefly about just how the material in the course turns up in his/her field. It seemed a splendid idea in the abstract, and I can now report on a tiny chunk of it in the concrete. By something closely resembling pure chance, I was in a 308 class when Greg himself dropped in, and it was great. His examples were very convincing and the students came up with some excellent questions, which is a pretty good tip-off that they were well tuned in. Here's to more of that scheme!

Also in the Newsbit, by way of introduction, I mentioned that I was going ahead and putting it out because nothing was brewing that would make a full newsletter worthwhile. Little did I know that even as I spoke Jim Morrow was putting the final touches on a Community College Symposium. I don't know many details, but I do know that what he spotted was that faculty members at community colleges teach the same material, usually with the same course numbers, as we do, and could benefit greatly from finding out just what goes on in our version. In particular, they need to know what our expectations are of students in the sequels. So he set up a visiting day, and arranged for a whole bunch of community college guests to visit the courses of their choice and then have a good conversation over lunch with that course's instructor. Brilliant, I call it!

We've had another Brown Bag since then, which I for one thoroughly enjoyed. I can't do it justice in reporting, though, because a lot of what was nice about it was that it was a conversation among a batch of people who have Lived Linear Algebra one way or another, and that's not a course I teach these days. The three graduate students who led off had just (thanks to the PFF) been at a meeting of the Washington Mathematics consortium focused on the teaching of linear algebra, and most of the rest of those present are teaching or have recently taught it. Good conversation, as I said. From my perspective, the striking thing was the parallel with issues we look at constantly in working with high schools: technology (courses run the gamut from completely computer lab based to completely classroom based) and applicability. On the latter front, one point of strong unanimity on the part of those who have had students do a project of their choice was that students are absolutely delighted to be able to work on something in which they can recognize their field of interest. Right precisely in line with some of the most dearly held beliefs of those who are now working on K-12 education -- which provides a neat segue into the column I promised:

In the course of the nineties I took part in many conversations and discussions about "the Standards" (the NCTM's Curriculum and Evaluation Standards, published in 1989). Out of all of commentary, one line resounds most clearly in my memory. It was uttered by a speaker at the first of the MER (= Mathematicians and Educational Reform) fora to which mathematicians and mathematics educators were invited together. "The Standards," he said plaintively, "are being regarded as Holy Scripture when they were meant to be a subject for hot debate." Certainly there was plenty of evidence to support his statement. From the moment the Standards hit the press, no self-respecting publisher would put out a K-12 text-book without claiming that it was Standards-based, even if the statement of that claim was the only thing that distinguished the text from its previous edition. Eventually, though, I came to the realization that the lack of debate should not simply be blamed on those whose reaction to the Standards was to genuflect. A lot of the blame belonged to those whose reaction was to yawn, and that comprised almost the entire of the community of mathematicians, with the exception of a few in the sparsely populated domain where the mathematics and mathematics education communities intersect.

I was not alone in this perception. When the time came for a new edition of the Standards, an entire process was set up for involving the mathematical community and eliciting the maximum amount of feedback possible. A number of mathematics organizations, including the AWM, were asked to set up ARGs (Association Review Groups). A bunch of interesting, thorny and provocative questions were then hurled at the ARGs, and the responses taken very seriously. A dedicated collection of writers representing a broad range of backgrounds spent huge amounts of time and effort sorting through the ARGs' ideas and each other's and arriving at a coherent document. The document was printed up in draft form and widely circulated through many communities with repeated requests for feedback. And finally, out of all this came, last April, the Principles and Standards for School Mathematics.

The story doesn't end there, though. All the communities involved, and in particular the mathematical community, are very much invited to continue the conversation thus begun. And although most of us are probably more experienced than we like to admit at sounding intelligent on the strength of very small knowledge (one of the more dubious of the skills honed by academia), we do even better when we actually know something. So here is a thumbnail (or possibly pinky-nail) sketch of the document and a set of pointers to further information and to some of the related articles that I have especially enjoyed. Following those leads will lead you, if you like, to more articles. Many, many more articles.

For a start, observe the change in title. This is not a chance event. One entire chapter is devoted to the statement of a set of six principles which underlie the recommendations in the whole document. Probably the one which would receive the loudest "Yes!!" from AWM members is the leading one, which states that excellence in mathematics education requires equity - high expectations and strong support for all students. The next four, on curriculum, teaching, learning and assessment would also be hard to argue with (and their content generally applies far beyond mathematics itself.) The last one will undoubtedly stir up wrath in some quarters, since it states that technology is essential in teaching and learning mathematics. It was not casually arrived at, and a good deal of effort is made in the rest of the book to support and clarify the statement.

Beyond the principles come the standards, clearly derived from the 1989 standards but with some notable modifications. One that leaps to the eye is an attempt to repair an extremely prevalent misinterpretation of the earlier version: from the statement that brute computation should be de-emphasized many leapt to the conclusion that it would be better yet to omit it altogether. This was never the intention of the original writers, but with that interpretation loose in the community, the writers felt the need to make quite explicit certain skills which should unambiguously be mastered by certain stages. It was also observed that the original division into K-4, 5-8 and 9-12 lumped far too many developmental stages into the first set, so the new standards are divided in four levels: pre-K-2, 3-5, 6-8 and 9-12.  Each of ten standards is carried through each of the levels, with indications of reasonable expectations for a student at that level and types of problems and activities that support the learning of the standard. Five of the standards are "process standards," dealing with problem-solving, reasoning and proof, communication, connections, and representation, and five are "content standards," dealing with numbers and operations, algebra, geometry, measurement, and data analysis and probability. As the introduction states firmly and the examples make clear, these are decidedly non-discrete topics - overlapping and integration abound.

One other comment before I launch into the list of interesting articles on the subject: something that should not be lost to sight in the midst of the significance of the book's impact and the seriousness of the process of writing it is that the book is fun to read. It is full of mathematical situations that are intriguing to think about and activities one would love to try out with kids, as well as classroom vignettes that give flashes of insight into how children think and learn. Not that it should be read from cover to cover at a sitting, but reading from it should certainly not be regarded as an onerous task. Now about the references: for a start, the document itself and a lot of surrounding information are available at the NCTM web site: Also on the web is a National Education Association article, found at And available either on the web or in hard copy, two that I particularly enjoyed are in the AMS Notices. One by Joan Ferrini-Mundy entitled Principles and Standards for School Mathematics: a Guide for Mathematicians appeared in volume 47, number 8 (September, 2000) and another entitled Four Reactions to Principles and Standards for School Mathematics appeared in the following issue (October, 2000). Both can be found online at

So please go and find out some more about this. It is unlikely that you will agree with all of it (how boring would that be, anyway?) but you will be much better attuned to the discussions of a profoundly important issue: the mathematical education of all of our children, present and future.

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