Sunday, May 29, 2016

Memorial Day Miscellania

Pedagogy Riff

I've been musing about this conversation:



I see a lot of "answer seeking" as well in my kids from time to time which is rich since Math club is just an optional after school activity and clearly has no grades.  The behavior I most want to alter is a student asking "Is X the answer?" and then if you tell them that's its not correct, they immediately and blindly guess a nearby number rather than analyzing their work for errors.  My first instinct is that this is not an issue with being advanced. There's this false narrative that only being quick to find the result is advanced but there's a reverse one that advanced students are only good at calculation and somehow lack conceptual understanding or appreciation for Math. Focusing on finding the correct answer is a natural instinct and built into the type of activities Mathematics is founded around.  In fact, I believe that concrete structure is what draws many initially to Mathematics: Here's a powerful tool for discovering truths.  I still personally derive much pleasure from carefully solving something that's difficult. However, as noted in the original conversation this can easily go awry.  On reflection, I tend to often pick puzzle type problems which lend themselves to bad habits if you're not careful.  Its the end of the year so I won't probably pick up on this fully until next time but;

  • I want to emphasize this more in the initial charter discussion.
  • I need to remember to also talk about answer seeking on a day to day basis My standard response is always "I'm interested in your thinking not the answer per. se."
  • Should I thread more open-ended exercise in to try to break down this habit?
  • I think part of the antidote here is to appreciate the math itself in a problem. What I find most satisfying for this is to riff on an already completed problem.  Now I need some good examples to use in a group setting.


On a different note, I've also been thinking about this post: https://samjshah.com/2016/05/25/merblions/which had an interesting approach to experimenting with inscribed arcs using geogebra. What I like here is the sense of  free play this encourages with the geometrical constructions while heading towards an important property.


I'd love to do something similar with other geometry topics like similar triangles or scaling but I don't have access to large numbers of computers. My thought experiment for now: could we do this as a group with a single laptop or go quickly enough using graph paper to approximate the same sense of experimentation?

Planning

There are only 3 session left to plan. The last will definitely be a party/game day. So on my remaining idea list so far:

I've seen a fun worksheet from nrich with shapes times shapes: https://nrich.maths.org/5714 that  I'm fairly sure I'll be using as a warmup.

Geometry Binge

Maybe it was because of the holiday, or seeing the twitter feed for gogeometry but I ended up doing a few proofs in a row.






This one is a good example of the power of  common inscribed arcs from above. After finding all the congruent angles and similar triangles. I went with a straightforward approach of overlaying the line segments that actually worked on the first try.




This one is actually a very similar variant to the prev. one and has a similar solution as  well. First we  divide the line into 2 pieces one of which is equal to the first segment and then prove the remaining piece equals the second one. This  version has a particularly pleasing tiling in the proof. Where it divides into an equilateral triangle and parallelogram.















Like most ratio proofs this one starts from similar triangle ratios. After identifying all 3 in the picture it was then a process of working backwards to get the desired ratio to look like one of 2 sides of one of the triangles. 





This was a quick extension on the previous problem. I saw a version using only area addition (no similar triangles)  from @five_triangles later on. Basically you extend the picture to be a full rectangle and since the the 2 triangle halves are the same area and the sub triangle also are, the remaining two little rectangles (s^2 - bs and ab - bs) must be equal.

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