My intentions have changed over the years, I started a bit towards the notation side and have since swung to free-spirit. There is evidence to suggest I've gotten better at it as the years add up, I can better predict struggles and pinpoint problem areas in advance. But it dawned on me that maybe I'm just better at teaching to the test, and not really getting better at teaching proof, logic, argument and reasoning.
So I set out to find a concrete reason we teach proof, and I saw it in a dirty subway station in NY city.
A subway map.
In our proofs we have a starting point (given) and a destination (prove). I liken it to needing to get form point A to point B in NY. I like proofs and in the same way I admire the transit system in NY. It is a puzzle, you know where you are and you know where you want to be, the challenge is deciding what route to take.
If you live and ride in the city every day you are probably good at the subway puzzle. Much like me completing my 7th year of teaching proofs, I see a start and an end and I know how to get there, math teachers know which routes to take.
If you live and ride in the city every day you feel safe on the subway (at least about where you are going). You are on the red line and you don't need to look at the map, you don't need to check the sign at every stop and count down in your head how many stops till yours, you just know when to get off and transfer. Math teachers are the same way with proofs, we know when the segment addition property is coming up, we don't need no stinkin' maps.
If you live and ride in the city every day you know there are a couple of routes you could take, but you know the best and fastest ones. "Oh you didn't know this was an express and missed your stop" doesn't happen to you. Math teachers don't go down wrong avenues either, we don't make mistakes and get on the wrong subway.
Our students are tourists lost in NY with a crappy map and little desire to get anywhere.
So how do we take these lost little souls and turn them into veteran subway riders.
- Break down the train lines (aka properties) get them comfortable with what each train line does. They should know that when they need to get to JFK they need to take the blue line. When they see midpoint use midpoint formula.
- Start with shorter routes and work to longer ones. Confidence is half the battle, they starting feeling good on the trains they will take that next route right or wrong, instead of sitting at the station afraid to ride any trains (blank paper).
- Do mix in long routes. Shorts are good but they usually aren't great proofs. They leave you feeling like "Why the hell did we just prove that, it was obvious and pointless." Don't let that become your first 3 days of proofs. You wouldn't go to NY and take the red line one stop for the first 3 days.
- Remind them the lines don't change and they go both directions. Green will take you to the Bronx every time. That goddamn 'definition of congruent' ain't gonna do nothing but change from congruent to equal and back, don't use it for anything else.
- If a student gets on all the right trains and makes it to point B celebrate that. Don't worry too much about the names of the trains, in fact don't worry at all about the names. If they can just describe the train and what it does be happy.
- "Segment Addition Property" -good
- "The property that says you can take part + part = whole" -THIS IS FINE
- Students aren't going to use segment addition property after your class, they will use logic and reasoning though, so let them reason it out for themselves instead of jamming specific words, definitions, and theorems in their proofs.
So we have the metaphor,we just need a way to carry it out.
Stay tuned for part two of proofs.
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