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Day 38: Discovering Diagonals

The goal seemed so simple: task the class with creating three hypotheses related to the diagonals of a square. Problem #1: Students ignored that bit about diagonals and just regurgitated previously known facts like the sides are congruent. Problem #2: Students don't make any distinction between an obvious observation and a hypothesis that will require investigation.

The idea was that with each student having their own unique square, any hypothesis they created could immediately be checked, albeit informally, with other squares for validation. The three conclusions I was looking for were: pairs of diagonals are always the same length; the point of intersection cuts the diagonals into equal pieces (haven't defined the word midpoint yet); and that the diagonals are perpendicular.

The first two are fairly easy for students to "discover" if they're willing to put forth the effort to measure the segments. The last one is often thought up, but students have no idea how to prove it. I tried to lead students to the connection between "90 degrees" and "perpendicular" and hope they make the connection back to Unit 1 and see slope as a method (these squares are on graphs for a reason).

For each class I made the brief point about why we spent these 3 days the way we did - I could have simply told everyone those formulas and conclusions on Day 1, but in the long run they wouldn't have a deep understanding of what they were actually doing. Some kids bought my explanation, but others (generally the regular Geo class) would just complain "this is stupid" and "just tell us the answers already."

I don't know how to reach kids that have such a combative attitude toward learning. All the literature I've read claims that if students can take ownership of their education, they'll change their attitude, but I have NEVER seen that work in practice. I still think this is a better way to teach, but I'm stymied about how to handle students who actively fight against the class methods.

Hi Barry,

ReplyDeleteI've been enjoying reading about your Geometry class this year. Thank you for detailing it on this blog!

About "taking ownership of their own education": You asked them to make conjectures of their own about the diagonals of a square. Yay, conjecturing! However, though this prompt seems open-ended, there really isn't much exploring for them to do. They are perhaps right in feeling that you are fishing for some particular, pre-ordained conjectures. Also, there's a strong sense in which there is only one square--so there's not so much to look into or explore.

Here's a different prompt: "Create a quadrilateral whose diagonals are related in some interesting way. Then find two more quadrilaterals that have this same property. Describe the property and make a conjecture about it."

Prompted in this way, a student might find something that none of her peers do--might find something that you weren't expecting at all. Then it becomes "Jaime's property" or "a Jaime quadrilateral" and then ownership happens. The problem space is big, and while perhaps they won't come up with all of the diagonals-relationships you want to end up discussing, you can always tack the remainder on later.

How do those thoughts strike you?

Thanks again!

Wow, my first comment!

ReplyDeleteYour critique is absolutely valid - I essentially even told the students that there were 3 "discoveries" that I was looking for. Some students actually came up with different ones (interior triangles are congruent for example), but most involved concepts we hadn't discussed as a class yet.

And I love your idea - I've done as much in physics letting students make up a name for the units for momentum. That was something they really enjoyed. As for this activity though, we haven't even defined the word quadrilateral yet. I'm trying to keep things as ridiculously basic as long as can so we can build a strong foundation before moving forward.

A final thought: the biggest obstacle I face with my students is the motivation to do *anything* that's not directly step by step. Heck, I struggled with this activity as limited as it was. If I were to task them with creating something as vague as "quadrilateral with related diagonals," I would be coaxing 35 students to even draw a 4 sided figure :(

I really do appreciate the feedback and the comments - keep 'em coming!