During my second volunteer block, the fellow I was assisting would put me on the spot during the prep period and ask me the occasional philosophical question. One of those questions still haunts me. I'd just helped out with his 11 applied math class and a conversation similar to the following occurred:
- 'So Roy, why do we teach them trigonometry and quadratic equations?'
- (naively I replied) 'Well, you've got to know those things, you need them in real life.'
- 'Roy, you already know based on your assessments and interactions that not all of these students will be going to university or college - and many that do won't be in a math based course. When are they going to need trigonometry? When have you used trigonometry?'
- (digging myself deeper) 'This past summer I was cutting baseboards and measured the width of the board, knew it was a 45 degree cut, and I calculated the amount to add to the measurement using trig.'
- 'Did you do this for every cut?'
- (eyes just peeking out of the hole) 'no, I went back to cutting the boards a bit longer and marking them to cut them to length.'
- 'So Roy, why do the students need to know trig and quadratics? You've just confirmed that most of these students won't need trig for future studies and unless they're conducting a thought experiment, they'll never use them in life. The time you need to figure this out is now, and not when you're sitting across from an irate parent.'
Answer #2 is my 'I am smarter than my calculator' response. I sincerely hope that I'll be able to instill enough adaptability into my students so that a broken calculator won't be the end of math until another calculator is found. I want them to be able to say to me that they know the answer is wrong, and ask me to help them figure out why. Similarly, I will cringe whenever I hear a version of "that's the answer that the calculator gave ... it must be wrong." Just as I will never forget the endings for Latin first declension nouns, I'm hoping that I can find a way to nurture some exceptionally strong estimation and verification skills among my future students. And the only way that they will be able to estimate well is to know what their calculator should be doing ... ie know the underlying trig skills. Oh, and I should add that when my first year engineering prof tried to teach us estimation, my response was 'what a waste of time ... why not just calculate it?'
My final response will be the 'I am not Nostradamus' reply. Dennis Dyack used the old Norse he learned during his undergrad within one of the video games he developed. I returned to studying history 20 years after I had last taken a history class, and used the the same answer template for identify and significance questions that I learned in grade 10 history. I should probably add that I also hated creative writing with a passion and avoided English classes like the plague. Yup, you'll never know where you'll be in 10, 20, 30 years and what skills you might need at that time. So why limit your options; learn the skills now and they'll be waiting for you if you ever need them later.
So, given Zoe's comment (and thank you), I have to ask why we are not teaching high students logic? If my response #1 is correct, that mathematics is an attempt to translate instinct to awareness, then why are we not going to the next step? Formula manipulation is just a variation of the law of identity (A=A therefore A-3=A-3). When we ask students to check their work, we're asking them to verify their premises (steps) in order to verify their conclusion (answer). Given that students are already applying symbolic logic within math class, why not show them what they're doing instinctively?
I can be nudged off of my soapbox when it comes to math, but I have real troubles leaving informal logic out of high school humanities classes. We're asking students to critically analyze sources, but we're not reinforcing the tools to do so. The students recognize racist arguments are wrong, but is this because they recognize that ad hominems are fallacies or based solely on their moral compasses; what happens if the moral compass shifts? Should they be able to recognize straw man fallacies when they come across them - as a mature student I get away with them all the time. Finally, some of these are so slippery that even knowing about the fallacy only results in a slight sense that something is amiss (if ... then statements are the worst for this). And in a world where we're moving away from absolutes, I'd much rather that we equip students with all the tools necessary to avoid generalizations while ensuring that they have well-constructed arguments for moving away from existing myths.