This was an exciting moment in my day. I was stuck on this problem for weeks, and yesterday I decided it was time to check out the solutions. Reading solutions was very satisfying, but there was something else that struck me today.
The problem I was stuck on was this:
After reading the first solution I found (it involved auxiliary lines and jumps of intuition that struck me as prophetic) I read a second solution that hit me hard.
Let E be the intersection of AC and BD. For triangle AED, angle(AEB) = x + 60. For triangle EBC, angle(ECB) = angle(AEB) – angle(EBC) = x + 60 – (x + 30) = 30.
So angle(ADB) = 2.angle(ACB), points A, B and C belong to a same circle with center D, AD = DC, triangle ADC is isosceles and x = 10.
In my own work, I had figured out that angle BCA is 30 degrees, but this is new for me. Since that angle is 30, and it connects up with angle BDA, you can see BDA as a central angle and BCA as an inscribed angle. Do D is the center of this added circle, and that means DC is a radius and that means x is 10.
This feels so new to me. I immediately felt the urge to find more problems that I could tackle with it. I started looking back on other geometry problems I had worked on in recent weeks and months. I soon spotted GoGeometry Problem #3.
If D is the center of a circle, DC and AD would be radii, and then x would be the inscribed angle that goes with 45, and therefore be 22.5. That is a cool new way to see this problem! Is it powerful?
Now, to be sure, there’s more work to do in Problem #3. In particular, you have to worry about whether B actually lies on the edge of the circle. If it does lie on the edge of the circle, then a theorem that I teach my kids says that it has to be 90 degrees, so the work becomes trying to show that angle B is a right angle. And if you knew that angle B is a right angle, well, you’d be done for reasons that have nothing to do with this circle perspective.
But the circle perspective tells me something else very quickly. It tells me that, ultimately, we’re dealing with two isosceles triangles glued together along BD, since BD has to be a radius. This is a new “way of seeing,” and it’s one that is going to help me see possible ways forward in other problems.
What I’m most excited about, though, it the promise of this perspective for helping me make sense of when and how to add auxiliary lines. Consider Go Geometry Problem #1:
I looked at someone’s solution. They added auxiliary lines. It was mysterious to me. Then, I looked at this scenario again from the circle perspective, and things opened up quickly.
Oh, wow. So that’s how you could know to extend into a right triangle!
This month, I’m reading Making Sense of Algebra, Lesh & Zawojewski (2007), I’m working on these geometry problems (and a few complex numbers problems), I’m looking carefully at visual pattern puzzles and I think I’m starting to see one way that these disparate sources come together: we need an expanded notion of what students learn in math.
These “ways of seeing” aren’t any more central to mathematical learning than content knowledge, mathematical practices, strategies or metacognition, but they are as crucial to learning as any these things. Likely, for any given topic or sub-domain of math (e.g. linear equations, auxiliary line geometry, visual patterns) these aspects of mathematical knowledge grow together in a way that makes causality hard to distinguish. (Is my “circle perspective” a development of my habit of looking for new perspectives? Am I using finding a new way to use the solve a simpler problem strategy? Am I learning a theorem?) This tangling doesn’t bother me very much, but maybe it should.
We talk about pedagogical content knowledge. Maybe I’m just arriving back there. Truly being aware of all aspects of a specific content area that are especially important and challenging is the sort of sensitivity to content that teachers need to be armed with. Sometimes those sticking points will be ways of seeing (e.g. visual patterns, auxiliary lines). Other times they’ll be self-conscious use of a strategy (e.g. adding decimals, solving 1-variable equations). Maybe other times the sticking point for kids will tend to be something dispositional, like a lack of perseverance (e.g. debugging software).
So, maybe, that’s the project. To focus in on narrow areas of content. To smoke out the different ways that kids think about these, and to make explicit the whatever that makes this specific area of learning challenging. To then share ideas for moving kids past those stuck-points so they have more great ideas.
Maybe there are no shortcuts. Maybe the only way to talk about what feedback is best to give, or how to craft learning goals or to give good hints is to deeply ground this sort of talk in specific content. This would be a big change in how I do things.