> > I very much agree that the tension between language, thought, and > mathematics is a key issue [see my word problems book excerpt to what > I view as a way to overcome it]. > > I would (counterfactually? ;-)] be careful about using > counterfactuals > with students. The reason is exactly because of the problem I believe > many have with it (illustrated in the example below). The reason is > that counterfactuals require more mental processes than factual > language. One has to hold both states in his mind when one is using > counterfactuals. It's more accurate mathematically (logic wise) but > more complex thinking wise. > > Thus, I prefer them to simplify the language and then translate to > mathematics. >
But is it? All I'm saying is that after seeing students (including when I was a student) struggle and struggle with trying to make sense of the conditional, and teachers trying and trying to explain it, I kept asking myself, "Why doesn't the teacher just change the tense of the verb?" Is is really less simple language to use a tense other than the present tense? We have children pretend things all the time, using the counterfactual tense. Example: As a creative writing exercise, we could have them fill in the blank in "If I were a bird, then _____." I don't see that it's necessarily more simple and more appropriate language to use the present tense, and ask, "If I am a bird, then _____."
Another example, in the math realm: Which question would more likely cause less of a problem for students, the first one using the counterfactual tense, or the second one using the present tense? (1) If Earth's radius were the same as Jupiter's, then what would be Earth's circumference? (1) If Earth's radius is the same as Jupiter's, then what is Earth's circumference?