A Small Accomplishment

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 AlgebraLesh & 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.

9 thoughts on “A Small Accomplishment

  1. Hi. Michael. I don’t mean to stalk you, but my google+ feed opened to this latest post of yours, which I find riveting, even if it’s been too long since my last geometry class for me to completely get my head around the specific problems presented.

    Of course, that’s less important than what you think you’re discovering about feedback. And what you’ve suggested there leads me back to Carl Anderson’s work on writing conferences. I think he is suggesting the same thing you’re offering here: specific content requires specific behaviors and understandings of students, places certain predictable demands on them.

    Carl’s subject is writing. He’s studied the writing process, both by observing his own and also by observing the habits of other writers, and can anticipate the demands different aspects of that process are likely to place on students. In addition, he’s studied how writing develops in children over time. The feedback that is appropriate for a child in a given moment is best delivered from a teacher’s awareness of the demands writing places on the writer, in general, as well as an awareness of where the specific child is developmentally and, hence, what may help her/him develop increasing agency as a writer in that particular moment.

    I may be off here, of course. Math and writing are not the same thing. But I couldn’t help making the observation.

    Looking forward to seeing you and the other Fellows soon!


    1. It hardly counts as stalking to leave a thoughtful comment on my public blog, Vicki!

      Anderson is on the extended reading list, but your comment immediately resonates with me. Feedback is a huge part of a teacher’s work, and it’s such a broad concept that it’s hard to meaningfully describe “effective feedback” at the top level. We need some sort of analysis.

      I had been trying, so far, to break down analysis by some sort of description of content-agnostic classroom situations. (e.g. “You give back a quiz and the class has consistently gotten one question wrong”; “A kid is a high-flier, but has a blind spot”; “Your kids are stuck on a problem, but a certain strategy could help them”).

      Maybe, though, this is wrong-headed. Maybe the real work needs to be analyzing specific mathematical content, describing the way kids think about it, and then helping provide feedback for those scenarios. And any generalizations would be organized around generalizations of content (e.g. “If your kids are trying to find an equation, function, pattern or formula, you’ll tend to see these ways of thinking….and these are the appropriate sorts of questions/suggestions to offer students to nudge them to one of these new ways of thinking.”).

      This is, basically, what CGI did. Right?


  2. You’re reminding me I haven’t dug into Childrens Mathematics, the second edition yet. And it’s been about 12 years since I took a good look at the first edition, but, yes, now that you mention it, I believe this is exactly what they propose. So you have the new edition that just came out?


    1. Yeah, and the 2nd Edition is great. Though I was a little bit sad that they moved away from that austere thing of just describing how kids learn, and adding a bunch about how teachers should teach. There was something beautiful about the restraint in that first edition.


  3. That was meant to be DO you have the latest edition of Childrens Mathematics.
    I can set you up with that if you’ve not seen it yet. The latest edition includes I think more than 90 video clips via QR code, which shows CGI in action.


  4. it looks like I must have been writing my last post as you were posting yours. Now you’ve got me curious about how the 2nd edition of CM differs from the 1st. And you’ve got me asking this question from a publishing perspective : are you suggesting the previous edition invited Ts to make their own observations and discoveries about the particular children in their classrooms and how to respond to them, while the second edition is a bit
    more–searching for the right word here–prescriptive? Didactic?

    Now adding CM to the summer reading list with this question in mind.


    1. I wrote about the differences between the 1st and 2nd edition of Children’s Mathematics over on my old blog.



      The authors are far less cautious about offering classroom recommendations in this new edition than they were in the first. There are two new chapters detailing their advice for teaching through problem solving in elementary classrooms. I’m sure that many will find these chapters helpful, but there’s something austere and lovely that I’ll miss in the restraint of the first edition.

      I’ve had my theories about what’s behind this change (first edition is closer to the math wars? second edition is more useful as a text for teaching methods classes?) but I should probably just ask you or KB?


  5. This post is a great example of why one of my favorite feedbacks starts with “do you want to know what I would try?” Which I probably picked up from David Coffey.


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