This is mostly a preview of an AWM column with ruminations inspired by
the sessions I attended at the Joint Meetings in Atlanta. I see no benefit
to producing a general chronicle of what I did and heard, but one tidbit
that didn't fit the column nonetheless deserves relating. It was part of a
report made to the Committee on Mathematical Education of Teachers. By this
time, I imagine anyone reading these newsletters is pretty familiar with the
NCTM Principles and Standards (or at least with its predecessor, the NCTM
Standards). What may be less well known is that one result of the way that
the Standards burst upon the world was that in almost every state (48, as I
recall) they spawned a set of State Standards -- sometimes called a
Framework, or Benchmarks, or in the case of our own state, the Essential
Academic Learning Requirements (known as the EALRs, pronounced "eelers").
Sometime in 2004, a large group containing mathematicians, mathematics
educators, supervisors, administrators and more gathered to compare and
contrast these documents. They worked extremely hard, and I believe that a
report of some sort is in the works. What I really liked, though, was the
result of an informal survey done right at the end amongst the participants.
They were asked to rate which state's Standards. Of the 48 under
consideration, 24 appeared on one response or another. The they were asked
to rate which state's Standards were the least helpful, and the number that
appeared was a mere 14. The intersection of the two lists contained 8
On to the AWM Education column. I gave it the working title of "Explicitification", though I'm not sure that will be seeing the light of day.
I have just been ambushed by an idea. It's not a new one – I have met it from time to time for a number of years – but at the Joint Meetings in January it seemed to be lurking behind every third talk that I went to, ready to pounce on me in a new guise. Needless to say, I have been thinking a lot about it since.
The idea first came into my consciousness in the mid-eighties in an article that put words to a vague worry I had been carrying with me since I first ran into math "manipulatives" – physical objects designed to enable a child to learn a variety of concepts. Many of these objects are wonderful, and to a mathematically inclined adult they clearly produce physical manifestations with deep connections to significant mathematical concepts. The question is: are those connections going to be made by the child? Answer: not necessarily. Seeing children engaged and successfully carrying out the relevant tasks can produce a great sense of satisfaction, and with it the temptation to assume that the task of the teacher is done. That assumption is false – if the connections are not made explicit, then by and large they will not happen, and the learning produced by the physical manipulations, not being part of any intellectual network, will swiftly fade out.
One of my early bits of bonding with Didactique occurred when I found this same idea in Guy Brousseau's writing. He was discussing the impact of Zoltan Diénes's "blue blocks", and maintained that the ideas they produced, while sometimes pretty sophisticated, rarely fed into children's overall learning. Not only did this give me a bond, it gave me a verb that English lacks: expliciter. I envy the French that one, so much more emphatic than "to make explicit." That's what needs to be done about the connections between the manipulations and the concepts.
After that I began making some connections on my own. For instance, there is an issue that can produce serious difficulties in doing professional development for teachers. A mathematician planning to teach a workshop or institute wants to offer an intellectually exciting experience which he or she knows will enrich the teachers' understanding of some mathematical concepts, thereby strengthening their capacity to teach those concepts. When the teachers fail to get excited, the mathematician tends to decry their lack of curiosity and/or ability and finish up in an (intellectual) huff. It's very easy to say "All they want is something to take into the classroom next Tuesday." Sometimes (let's be honest!) it's even true. On the other hand, a workshop that begins with an explicit discussion of how the topics undertaken will directly strengthen what the teachers can do for their students stands an excellent chance of being thoroughly successful without a single make-it-and-take-it item.
On to the January meetings. The first session I got to was one sponsored jointly by MER (Mathematicians and Educational Reform) and the MAA's COMET (Committee on the Mathematical Education of Teachers). It had a number of excellent talks, but the moment that grabbed me came in a talk by Cameron Sawyer. She was describing a capstone course for future high school teachers that she has developed at Southwestern University with COMET support. The course focus is well represented by the title of its textbook, Solomon Usiskin's High School Algebra from an Advanced Standpoint. And the capstone project for this capstone course is to find three sets of problems, each set consisting of one from the course itself, one from a different college level course and one from a high school text book, and to show how each element of the set connects to the other two. Explicit? Yes, indeed!
Avoiding a laundry list format, I shall skip to the last and most dramatic example. That one came in a pair of sessions entitled Using mathematically rich activities to develop K-12 curricula. The curriculum under discussion, which is still a work in progress, developed out of the work of Bob Moses, author of Radical Equations: Math Literacy and Civil Rights. Moses' contention is that mathematical literacy is the major gatekeeper shutting people off from escaping poverty. In support of that theory he has worked for many years in a school in Mississippi. First, with indefatigable patience and energy (not to mention good teaching) he earned the trust of the school administration. Then he began to introduce and work on a curriculum that would enable many more students from that school, with a largely African-American, economically distressed population, to learn enough to handle college admission tests and college itself. The idea, of course, is to make that curriculum available ultimately to many more such students. The design of the curriculum has now become a project involving a number of people besides Moses, and it was to that design that we were introduced.
We started off taking part in one of the actual activities. A simple set-up where we moved between "buildings" (pieces of paper labeled 1,2,3,4) taped to the floor according to a couple of specific rules led into a puzzle that my group definitely had to think about. Other groups did, too, and came up with solutions not at all similar to ours. A great activity, in short. And then the presenter said "And you can all see that this is a perfect set-up for understanding composition of functions," thereby setting all manner of alarm bells jangling in my head. They continued their jangling until the next session, when we found out just how the lessons progressed. Allowing for a little reconstruction from slightly sketchy notes it looks like this:
"Set up a chart that shows what happens when I require you to make move A. Now do a chart for move B. Now see what happens if you do A and then B. Confused? Go back to the buildings on the floor and walk it through. Now see if you can come up with a notation for what the charts are telling us so that we can describe things more compactly. Let's look at everybody's notations and see how they work. Here, practice the one this group came up with. Now practice the one from that group. OK, mathematicians call this a function and here's the notation they generally use. Let's practice that one, too. Why don't you walk through the movements that this function over here would indicate? Now what happens if you do this function and then that function? Walk it!"
Now that's what I call explicitification raised to a high art!