the answer "that question is ridiculous" must be accepted

My friends who work in education of the young (the 'fuzz included) like to quote studies that show students answering without comment or concern questions like "Farmer Jake has 13 sheep and walks them 21 miles. How old is Farmer Jake?" There are many mixed-up reasons for this problem; some relate to rote learning; some relate to the artificial dichotomy set up between reading and math; some relate to the decontextualized ways we teach math; some relate to the testing environment that saturates schools; and so on. I feel all these things!

Imagine we want to see students using their common sense and their judgement with every question they consider and answer. I think that would be good. How do we foster this kind of thinking and exercise of common sense? I think we have to let the students call "bullshit".

Here's an example: "Johnny has twelve toy cars. He gives eight to Frances. How many does he have left?" Obviously we should accept the answer "four". But we should also accept the answer "No way! Who would give more than half of his toy cars to someone else?" If we don't accept that answer, we are saying to the students "calculate without thinking". That might be okay for quantum physicists (though I disagree), but it isn't okay for the rest of us.

what's the right answer?

In her K/1 class (ages five to seven), the 'fuzz was doing "the weather" when an argument broke out between those who thought the clouds "move the Sun around" and those who thought the clouds "block out the Sun". She let the discussion proceed, encouraging contributions. In the end, the "move the Sun around" group got the consensus. What to do? You can correct them all, and then they learn that scientific truths are handed down by more knowledgeable authorities ("How do you know the Universe is expanding?" "I read it in a book."). Or you can let it lie, in which case they go home thinking they know something that in fact is wrong. Or (best, but extremely time-consuming), you can go through the process of having them turn their pseudo-scientific explanations into predictions about other phenomena, or have them extrapolate their model into other domains, and then see why or where it breaks. That's (to my mind) the only solution you could possibly call science, but it would require an absolutely radical replacement of the current curriculum and structure of school. The 'fuzz didn't have the right (it was someone else's classroom) to blow the schedule, and she didn't want to be a priestess, so she let it lie and moved on to the next activity.

don't make students hate math!

In my previous post I implied, inadvertently, that we should only teach useful math. That was not my point. My point was that we should not teach math if the effect of that teaching is to cause most students to hate it. Of course if we could teach it such that the effect was to cause most students to love it, I would be all for teaching it!

don't teach math!

In a nice conversation about writing for education, Adam Gidwitz (the author of A Tale Dark and Grimm) pointed me to The Mathematician's Lament. The book makes (much more clearly than I) a point I have been making informally for years: If you want students to know and love math, you definitely should not teach it in school! (Same for literature.) The mathematics requirements in school empty the subject of its meaning and point, and are useless to boot. How many non-scientists use the quadratic formula, ever? Discovering the formula would be fun, using it is a drag (and exceedingly rare).