Geometry CP students' Favorite Topics:
Ch 1
Ch 5
Ch 10
Ch 12
Triangles
Triangles
quads
quads
quads
Solving Triangles
Similar triangles
Ratio/proportion
Proportion
Area
Area
Area/Volume
Trig (interesting)
Pythagorean thm
Pythagorean thm
Translation
Polygons
Polygons
SAS etc. Theorems
Geometry CP students' Hard Topics:
Nothing
Nothing (if I paid attention)
I don't know
I don't remember
A lot
Everything
Everything
Everything
Everything
Everything
Everything
Triangles
Finding lengths and angles
Ratios/proportions
Trig
Trig
Trig
Trig
Trig
Trig
Trig
Trig
Trig
Trig
Trig
Trig
Trig
Proofs
Proofs
Proofs
Circles
Area formulas
What's the most striking thing? The million Trigs!
That pattern isn't surprising to me at all. I haven't figured out yet what I will do next year (suggestions??), but Trigonometry was the only section where I had students asking "When will I ever need to know this?" We studied plenty of topics that were difficult (at best) for students to see the applications of, but they never stopped to ask that question because they were interested, involved, curious and they understood enough to be able to work toward the problem. In trig, that didn't happen. I saw confusion, frustration and kids giving up. It may have started when the first investigation we did gave data that was too far off to see real patterns (perhaps technology would be better than measuring by hand for this?) or perhaps when we started synthesizing too many ideas at once. When 'solving a right triangle' (finding all the side and angle measures given a few) we applied angle sum rule, pythagorean theorem, trig ratios and a lot of algebraic manipulation. Next year I'd like to do more problems throughout the year that synthesize topics so hopefully that won't be so overwhelming. Last, but certainly not least, trig was the first topic we did after MCAS (the state exam, required for graduation) and students feel like they should be done when they've finished that test (even though it happens mid-May and we didn't finish until June 29 this year). Overall, trigonometry got a bad deal last year. I'll try to do it more justice in the future.
Proofs are another challenge, and I believe that a lot of that is related to how they are presented. Student in my classes are accustomed to "defending their answers" and usually can do so well. However, when it comes to writing a proof they get caught up in format, and formal language. Precision is key and I certainly want my students to be able to write concise and carefully worded explanations, but I wish that they were more willing to just write something to start with. Does anyone have a method of draft proofs or easy entry formats?
Otherwise, I appreciate the overlap of favorite topics and hard topics. Different students had different preferences, and geometry has plenty of variety so most students get to experience a balance of topics they enjoy and others that they struggle with.
Take aways:
Do multi-step, synthesizing problems (before Trig)
Make proofs more 'low threshold'
I like to start proofs with logic puzzles instead of anything involving geometry. That way students can focus on the ideas of good proving methods without simultaneously learning new concepts.
ReplyDeleteThis is the format I used last year. First they looked at several different types of proofs to see what made them work and what made them not work. Then I had them write proofs of their solutions, then we did a proof writer's workshop where they gave one another feedback on their work and then they revised & rewrote the proof in small groups.
There is a sample of one of the problems I used for this activity in my post "Rosie and the Round Table".
Thanks! That looks like a great way to start. I have a bunch of logic puzzles in the files somewhere, I'll have to pull them out and see which ones lend themselves to proofs.
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