> Regarding pictures... > > My son was working through a set of problems > converting improper fractions to mixed numbers. His > speed was fine and his answers correct. I asked him > to explain one of his answers and he took a long time > before he even began to respond and then started to > draw something. I stopped him and asked "What are you > doing?" He replied "You asked me to explain what I > was doing and I need to draw a picture." I said "You > took less than 5 seconds to answer some of those > questions, there is no way you were drawing pictures > in your head. Tell me what you were doing when you > were answering those questions." It wasn't easy to > drag it out of him but finally he said "I was finding > the biggest (multiple) of the bottom number that > wasn't bigger than the top number and then > subtracting it from the top number to get the > fraction (part)." I said "Thank you, that is exactly > what I expected you to say. Why didn't you just say > that?" > > You have to be very careful with aids becoming > illegitimate proxies for what it is you are actually > trying to teach. Unfortunately, his book uses > pictures EVERYWHERE. It is rather sad that my son was > on the verge of fraction mathematics but had no > example in his curriculum to relate to. SOMETHING I > WILL NOT LET GET BY ME AGAIN. Nowhere was there a > discussion of dividing the numerator by the > denominator and taking the remainder, even though all > of these pieces were talked to earlier, with pictures > of course. I am certain that his teacher knows the > mathematical sense of "fraction" (I have met him). I > mean, I could ask him if 23/64 is greater than or > less than 3/8 and he would answer swiftly. But as > with most teachers, the job of teaching seems to > eventually push aside the act of teaching. How can > someone that knows fractions accept the notion that > pictures are sufficient? How many of his students > were on the verge of fractions but didn't have a > parent with the odd hobby of pedagogy? I have > accepted that this must come with the job. Like the > way police officers act a certain way that we find > hard to relate to, but put one of us in their shoes > and after some time, we would probably act the same > way. > > Bob Hansen > Bob, maybe this is just the teacher in me, but I found myself curious what picture he was going to draw. Are you sure that the answer he gave you demonstrated that he understood it or might he have just memorized an algorithm?
In the classroom, I usually try to accept multiple explanations - at least initially - even if it what I am getting might be inefficient.
Please don't take this as criticism. Your son is very fortunate to have you as his most important math teacher.