Lately I’ve been reflecting on feedback from my students- it’s the end of quarter 1 so a natural opportunity arose. Many asked for me to have “more examples”.
I’m not sure if “more examples” means:
- More lessons that are traditional “I do one example while students furiously scribble into notebooks, we do one example where I cold call from the class, you do 20 of these silently and independently” OR
- More examples/experiences on how/where to apply concepts we are learning (this one’s a bit more nebulous for me)
If I’m not doing enough of 2, then I’m failing as a teacher.
I lean towards pretty much flat out refusing to do much more of #1 because I never want my students to think that math is about mimicry and that I can somehow magically model every geometry problem they would/could ever encounter. Also, geometry in TN is not a state-tested subject, so there is not a lot of pressure towards an end-of-course performance test. This allows me with a lot of flexibility to make my curriculum a little more… shall we say progressive? I resonate with and align my core teaching values with many of the values that I’ve observed in the rest of the MTBoS re: quality math tasks.
Enter Dan Meyer.
No one I respect thinks students should discover all of geometry deductively. … To motivate a proof, students need to experience that “Wait. What?!” moment of perplexity, the moment where the seemingly-true has revealed itself, a perturbing moment experienced by so many mathematicians before them.
Which has forced me to reevaluate and define the difference between meaningful struggle (what Dan would refer to as perplexity) and frustrating struggle [it sounds so obvious written out- I guess that’s why I should blog more often?]. If I have to present my students with something in order to allow us to engage in more meaningful struggle, I am not robbing my students of the opportunities to make deductive conclusions. I am supporting them by preventing frustrating struggle that leads to disengagement and shutdown.
Enter Chris Danielson and the Van Hiele levels of Geometry Instruction. These are not something that I’ve paid any attention to when planning my lessons, which
should probably will definitely change. Like Chris said, high school students come in, or maybe should come in?, around a level 1.
Frustrating struggle [current]: My questions/examples are aimed towards getting students to remind themselves of properties (which maybe 1 person in a class can currently provide for us).
Meaningful struggle [goal]: I need to purposefully and consistently provide the opportunity for students to make deductions about properties and shape relationships based on experiences (data collection -> observations) AND for students to learn how to defend those relationships (whether presented or deduced) through quality math talk.
Now, to bring it back to Dan’s comment. How do you decide what is appropriate for students to discover deductively? Is a traditional “I do, we do, you do” lesson ever appropriate? If so, when?