One of the topics that is guaranteed to stir up conversation among mathematicians dealing with secondary and post-secondary students is proofs - whether students understand them, why students don't understand them, when they should be taught, how they should be taught, by whom they should be taught, etc. At the high school level, discussion often revolves around the "two-column proof" and its effectiveness and/or necessity. At the college level, once past the moaning phase, the conversation broadens out to optimal timing for a course addressing proof directly, best form for such a course, expected clientele and what is being done at other colleges and universities. In fact, a number of interesting things are indeed being done - but that's another column. This particular column deals instead with a much younger bunch of learners.
First let me give some context. I recently had the good fortune to be one of the presenters of a workshop on Didactique at the seventh Symposium on Elementary Mathematics Teaching at Charles University in Prague. Didactique is a French research program in mathematics education which is well into its fifth decade, but is only now getting to be known to the English-speaking world. Since the official language of the symposium was English, this seemed to Guy Brousseau, founder of Didactique, to be a good opportunity to spread the word. By way of rounding out the information imparted in two relatively general introductory sessions, he opted to present in detail one single set of lessons. His choice was the topic of counterexamples. I was so taken by the sequence and the depth of its possibilities that I decided to present it here.
Before I do that, though, I need to give a little more context, as well as a disclaimer about my intentions: this is not a panacea to be hurled at the heads of innocent primary teachers! At the core of Didactique is the Theory of Situations, which looks at teaching and learning not by focusing on one or two elements of the teacher-student-subject triangle, but by studying in all its complexity the set of relationships of the learners, the mathematics and the milieu. The milieu includes the other students, the concepts being learned, the concepts available to be used in constructing an understanding of the concept at hand, and other things of similar nature. The Situations in question are not recipes on offer, but elements of a study of what understanding it is possible for students to develop under a given set of circumstances. Many of the Situations were first tried out at an observation school (a public school in a blue-collar district) where students were taught all other topics in the standard way, and mathematics in a special way, often using such situations, and often under very careful observation. A rich collection of conclusions can be drawn from these studies - conclusions about the growth of knowledge, and the mathematical capacities of students who are given the opportunity to take responsibility for some of their own mathematical understanding. Furthermore, these conclusions and the study that goes into them provide ideas and perspectives which have a lot to offer towards deepening the understanding of mathematics education. The only danger lies, as I indicated above, in regarding the Situations as recipes.
Onward to specifics: the Situation I am about to describe was studied with a class of fourth graders and is entitled the Biggest Number. The objective was to give students the opportunity to engage in the activity of a mathematician, rather than simply making use of a process mathematicians have produced. One very fundamental day-to-day activity in mathematical research is testing for validity by looking for counterexamples. The Biggest Number Situation was developed around the idea of counterexamples. I will first describe the bare bones of the Situation, then point out how it is used and what it achieves. Phase one: the teacher announces, "I am going to give you five numbers. Your job is to use each of those numbers exactly one time and apply the regular operations (+, -, x, and ÷) to produce the biggest number you can. If you can say exactly what you did, carry out the operations accurately, and get the biggest number anyone gets, you get a point." She then gives them the numbers 3,8,7,5,4. Essentially all of them stick with multiplication. A few computational errors arise and get sorted out, but most of the discussion comes in attempting to describe the method - how to verbalize it and whether order matters. Once that is done, they practice on a few more and begin to think it's all a ruse for getting them to practice multiplication.
Then comes the next phase: "This time you are going to describe a method before you have the numbers. It has to be a description that someone else can use, and it has to apply no matter what numbers I give you. Work with three other students, and when your team has agreed to a method and a description of it, make a poster of it. Then all of you will look at the posters and each team will choose one produced by a different team and will apply that to our next set of numbers. If your team's poster is chosen, your team gets a point, but if it turns out not to work, you have to return the point plus a penalty. Are you ready?" And they get to work producing and choosing posters.
Once every team has made a choice and committed itself to following the method chosen, the teacher presents the numbers 4,6,2,0,5. After a brief period of chaos and rebellion, they settle back down to reformulate their methods, requiring that any zeros be added and everything else be multiplied. The teacher follows up with 3,4,6,1,2, and once the dust settles from that, with 5,1,4,1,7. The rest of the sequence you can deduce. So what, besides fun and games and a highly engaged class, goes on during this sequence? For a start, there is the tremendous challenge of articulating a process instead of just following one - a particularly acute challenge when one considers that these are fourth graders, as yet unfamiliar with algebraic notation (the teacher suggests using letters to represent the unknown numbers, but in several repetitions of the Situation not one student took the suggestion). Then there is the shift in perspective from regarding a mathematical statement not as an edict from God (or the teacher, which is roughly equivalent) but as something about whose validity they need to be concerned. Then finally there is the repeated realization that all it takes is one counterexample to destroy a conjecture's validity (a realization so often repeated, in fact, that one student finished up with a plaintive query whether there were any mathematical theorems that didn't have counterexamples!) There is even the opportunity, given a skilled teacher, to discuss the fact the no collection of examples can constitute a proof, while one counterexample constitutes a disproof.
Wouldn't you love to have students coming to you - whether you are teaching them in middle school, high school or college -- with those ideas firmly inculcated? --