Completely stuck on this problem, so I decide to look at a solution.
I look at an answer. I find this:
I find the art of adding auxiliary lines entire mysterious. There are a few moves I have, there are a few things that I notice help pretty often (adding altitudes to diagrams; finding right triangles; adding lines that create congruent angles and possibly congruent/similar triangles) but I have no sense of when to use which move. My scrap work on each of these problems involves lots of random line-drawing. It’s entirely guess-work, random for me.
I don’t know this area of math very well. Am I having fun?
The short answer is, yes, I am.
Do I want a teacher to show up and make this mess more systematic?
This is a tougher question. I’m having fun with this, I’m in no rush to get better at this. It is clear to me at this stage that if I wanted to get better at these problems in a jiffy I should spend a lot more time trying to make sense of the answers that other people have written than I currently am. It’s unclear to me how much time spent working on the problem was beneficial for my learning.
I was also hit hard by this paragraph from Max’s post:
I also believe that students’ beliefs about things like who does math, what it means to do math, how math is learned, etc. are so important that pedagogy has to take them into account — even if that means sacrificing clarity or efficiency to support the belief that math can make sense to students, and good math ideas can come from anyone.
I have no idea how these experiences are shaping my perceptions of myself in relation to math. I think my mathematical personality is pretty stable — stuff doesn’t come so easily to me, but I know if I work at it eventually I can get it — but I’m not at all confident that I can see myself clearly.
I’m also working on these problems by myself. I feel no social pressure to “keep up” with anyone else. So little is at stake with my work here. I think that we often underestimate the feeling of bla that comes from seeing everyone around you engage with math while feeling lost. In such situations, I think arguably Priority #1 for fostering a productive mathematical disposition is getting the kid doing the same mathematical task as everyone else. (This is why I get a bit queasy when people recommend giving kids a special new task as a form of feedback.)
Another sort of stray observation: it’s damn near impossible to generalize from one problem. If I want to get any better, I need to go back and try to find some connections between the problems that I’ve seen so far. This is really tough math work — trying to cull some generalizable principles from different-seeming problems.
Some questions, to wrap this up:
- If I keep at these geometry problems, will I eventually start to see some connections and patterns and get auxiliary lines down? Probably, right?
- I wonder if I can get better at sensing and responding to when my students are feeling the way I felt — like I had gotten everything that I could out of the problem.