I am just back from a splendid couple of weeks in France, working with Guy Brousseau. Steep learning curve, as ever, and most of it not in particularly newsletterable. Two things struck me as being worth a screen or two, though.
The first was a workshop I went to in Montpellier. I rode in cheerfully on Guy's coattails - he was the guest lecturer for both days of the event. Since the structure of the French educational system has me totally befuddled, I didn't really ask what we were doing. I just tooled along, sat myself down in the classroom and whipped out my ball point. But as the time went along and the conversations developed, I was overwhelmed as I have never been before by the similarities between this gathering and some of our CCML workshops for middle and high school teachers. Part of it was the banter and the in-jokes - they were clearly very much a community, despite long gaps between their gatherings. Part of it was the level of congeniality. By the end of lunch (cous-cous and rose wine under a very Mediterranean sky - not something we offer our CCML participants, I fear!) I was so comfortable with them I could ask "So tell me, just who are you?" It turned out they were a mixture of teachers taking a day off for professional development, teachers who have switched (at least for the moment) into doing professional development and faculty members from the closest approximation to a college of education. That knowledge gave me the perspective to spot more similarities: some left early to get to an after-school commitment (sports, even!), one missed half of an afternoon because after her strenuous efforts to persuade some teachers to spend class time on statistics succeeded, she discovered that their statistics knowledge was thin and fragile, and she needed to go support them. So what were they doing there? Well, the first day, which had the broader spectrum of attendees, was devoted to the study of how to analyze a class, using a transcript. One among them (brave soul!) had had his class videotaped, and then someone had put in the labor required to produce a written version of the videotape. The day was devoted to discussing how this transcript might be chunked up in ways that make it possible to think clearly about how different parts of the class hang together, and how the students' thinking develops (or doesn't!). The second day the core of the group continued, using the same transcript to address the issue of "What does it mean for students to be doing mathematics in class (in contradistinction to simply carrying out instructions or doing exercises)? How do we recognize it, and how do we optimize the chances that it will occur?" [That's a free translation if ever there was one!] Very familiar issue - and no easier to solve there than here!
On a very different note, one of the issues that Guy and I tackled for the umpteenth time was one which arose at the end of the Brown Bag for which Guy was our guest a couple of years ago. On this visit for the first time I got enough of a handle on it to attempt to write about it. I shall copy in the resulting passage from the introduction to the article he and his wife and I are working on : "Rational and Decimal Numbers in the Required Curriculum". My hope is that you will be A) intrigued and B) maybe even enlightened. If, on the other hand, you are C) bored, you know where to find the delete button!
Our language, although generally rich in synonyms, fails to supply us with a pair of words to correspond to the French near-synonyms "savoir" and "conna”tre". Both are translated as "to know". Likewise the nouns associated with them, "les savoirs" and "les connaissances" are generally both translated as "knowledge". At times this is fine, at other times it is a problem. The latter is notably the case in dealing with Didactique, where the words are frequently and highly intentionally distinguished. In the following paragraph we will attempt to make the distinction clear.
At a first pass, the verbs can be thought of as "to be familiar with" (conna”tre) and "to know for a fact" (savoir). For some examples the distinction is clear and useful: "Conna”tre" a theorem means to have bumped into it sufficiently often to have an idea of its context and uses and of more or less how it is stated; "Savoir" a theorem means to know its statement precisely, how to apply it, and probably also its proof. On the other hand, when it comes to an entire theory, with a collection of theorems and motivations and connections, what is required is to conna”tre it. Savoir at that level is not an available option - but on the other hand, no real connaissance is possible without the savoir of some, in fact of many, of the theory's constituent parts. The corresponding distinction exists between the two words for "knowledge", with the additional complication that each of the French words has both a singular and a plural form.
Before offering a solution, I propose to give an example of a way in which having the two words is both thought-provoking and a material aid in analyzing what's going on. Currently in American mathematics education there is considerable debate about the status of certain kinds of knowledge. One side is accused of interesting itself solely in "skill-drill" and computation, the other of interesting itself solely in "fuzzy math", where anything goes as long as it is in the right general vicinity. Consider instead the following description: all school learning is an alternation of savoirs and connaissances. Isolated parts are acquired as savoirs connected by connaissances. Without the connaissances, the savoirs have no context and are swiftly mixed or lost. Without the savoirs, the connaissances are more touristic than useful. Imbedded in connaissances, savoirs can develop gradually into a solidly connected chunk - in fact, a savoir, which is then available to be set into a wider connaissance. Thinking this way then provides a tool for contemplating another of the current hazards of mathematics education: assessment. It is a clear need, but a thorny issue. And one of the causes of its thorniness is that all that can be assessed on a standardized test is savoirs. The state of a student's connaissances is visible to the teacher if enough time in the classroom can be devoted to the kind of activity where connaissances are built and used. But an over-emphasis on visible, "testable" knowledge leads to attempting to teach the savoirs without the connaissances to hold them together and carries with it the danger of damaging the entire fabric of the learning. --