I'm planning to capitalize on the notions that Judy and others have made explicit, i.e., that kids often have good intuitions about things but are unable, or unwilling, to express those intuitions in terms of the conventions that mathematicians and experienced doers of mathematics use.
Next week-ish I'll be starting a unit on statistics via a problem setting that I hope will have meaning for my kids. Their task will be to develop a presentation for a buyer for a local department store chain that will give that buyer an accurate representation of what seventh graders would buy and wear. In the course of the unit we'll cover data collection (surveys, random samples, etc.); data representation (different types of graphs, charts, tables, etc.); and data analysis (measures of central tendency, interpreting pre-existing data, etc.).
I'm fully counting on my assumption that my kids, much like Judy's, will have fairly strong and fairly reasonable ideas about what we're doing even before we start. One of the biggest parts of my job, then, will be not to present new concepts but to help my kids make their existing understandings coherent to themselves and, in this case, to an outsider. In the course of this work I fully expect that some of what we do will be foreign to some kids initially. When such situations arise I'll do some direct instruction if necessary or, more likely, will ask for those who do understand to explain it in their terms. Again, my task will be to help guide the kids toward speaking, writing, and (eventually) thinking in ways that fit with conventional mathematical representations.
In all this I'm assuming some notions of fairness (for survey purposes) and magnitudes of numbers (for representation and analysis purposes) on the parts of my students. These are hard things even for adults to express, so I expect that there will be many situations in which the students will say that they "just know," without being able to articulate their own understandings. Hence this is one of my explicit objectives: to help my students develop the ability to know and to say what they're thinking.
Kreg A. Sherbine | To doubt everything or to believe Apollo Middle School | everything are two equally convenient Nashville, Tennessee | solutions; both dispense with the email@example.com | necessity of reflection. -H. Poincare