Geometry Instruction [MTBoS #2]

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:

  1. 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
  2. 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?

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8 comments

  1. Your comment on frustrating struggle: I like to let the students draw the images that relate to the particular theorems, draw some conclusions about how the images are ‘all doing the same things, have similar issues, etc. and then we talk about the theorem (s) that get applied. As I am a great fan of similarity and difference, I then let them try to create situations to prove the rule wrong (great group work). I find this helps them with the ‘rules’ and then we can talk about proof where we need to – the students can then explain ‘why’ they now know something is true. But to begin with, I have the theorems in front of them, so nobody has to remember something they have not yet experienced.
    Next time I teach geometry, I’d like to add challenges that let them tell me which ‘rules’ apply to given situations, or maybe why a rule doesn’t apply, as part of the everyday conversations in class.

    1. Thanks for the feedback! I haven’t done a lot of activities where we try to prove something wrong, so that would definitely be a great concept to incorporate. I’m curious- how invested are students in trying to prove a theorem wrong? Do you find that students are willing to accept that something is true simply because you (the teacher/theorem) says it is?

      1. My students are brainwashed that the teacher knows it all. At the first of the year, I would say, are you sure-they would immediately think they were wrong because I didn’t say they were right. They got involved with trying to disprove theorems by accident- one of our students questioned everything ( a real mathematician!) I also offered candy to anyone that could prove the theorem wrong. This involved them understanding what the proof was saying. I wish I had kept some of their papers! It was a culture of competition and it took some time to foster that!

  2. I don’t teach geometry, but students in any class often want to default to the comfortable “you show me how, I practice and memorize, you test, I regurgitate and promptly forget”. It is hard to get students to realize that all meaningful learning comes through struggle and frustration.

  3. I teach Geometry too, and I think you and I probably share similar struggles (frustrating and meaningful, though different than what our students face!) as well as teaching view points. I agree that your gut is probably right–when students say more examples, they may think they want more direct examples, but what they really need is to see more of a reason WHY we use something. You can do 1000 textbook problems, and still not be able to recall a formula or concept when an application problem appears. Of course, your gut is the more challenging of the two approaches. I think Dan Meyer has a lot of other great things to say about it. I read one of his posts on abstract vs. concrete tasks (http://blog.mrmeyer.com/?p=15420) that guides my thinking a lot when trying to come up with these types of “examples” where students actually understand the purpose behind “why what I’m doing makes sense.” It doesn’t have to be a “real world” context, but the application has to make sense. (Of course, this is easier said than done!)

    In an ideal setting, students could discover a lot of Geometry through exploration and manipulation. But this type of freedom does not go over well with my students. One strategy I’ve been using this year has been a sort of hybrid of “traditional instruction” and “discovery.” I will teach them a new term or rule (like linear pairs and vertical angles), but I will let them discover the relationships and what they mean by having them measure angles and figure out the patterns (this, of course, is inductive reasoning). I’ve liked this because they have an introduction to an idea so they have a term for what they are working with and have a general idea of what they are looking for, but they are a little more empowered in the “discovery” of the relationship, and it sticks better. Perhaps this “hybrid” method could help you bridge some of the gaps between where you are and ultimately where you are trying to go?

    Anyway, I look forward to following your blog, and good luck!

    1. Thanks for your feedback! The “hybrid” sounds like it’s that ideal balance- although probably harder to achieve for every lesson than it seems. I think I need to pull pack a little bit from discovery in order to set my students up better. Welcome to the MTBoS! 🙂

  4. Heya. Phil Daro repurposed the structure as “You Do, We Do, I Do” at a conference last year. I’d say this is about right. The “I Do” isn’t the problem. At all. The problem is that sometimes students haven’t done anything themselves, or done anything together, to generate the need for the teacher to do anything.

Learning is having new questions to ask (thanks, Chris Danielson).

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