Sunday, November 23, 2014

New Plan for Teaching Proofs

I know it's been a long time between posts.  In addition to the normal teaching stuff, I've been co-advising our senior class, which has been  a ton of work lately.   So far this year, I have taught three-dimensional measurement, coordinate geometry, lines and angles (without proof), and transformations.  Basically this whole time, I've been putting a lot of thought into how to teach proofs differently this year. 

Year one, I used the textbook to teach proofs.  No success.

Year two, I used past Regents questions to teach proofs.  Minimal success.  (Here students really learned to memorize proofs because they were constantly seeing similar set-ups.  There's only so much you can do with 2 triangles.)  I was given advice from a retired math teacher to skip algebraic and line and angle proofs entirely, and to just start with congruent triangle proofs.  I tried to use an activity with straws and pre-cut angle measurements to help students determine the postulate/theorems used to prove two triangles are congruent, but it took 3 days (42 minute periods) and the students did not retain anything they learned from the activity.

Year three, I need something new.

Here's the plan so far:
  • Justify teaching proofs to the kiddos.  (Read:  attempt to justify teaching proofs to the kiddos.)  At this time I can introduce them to writing paragraph proofs and two-column proofs.  I feel like I always hear complaints from Geometry teachers about two-column proof, but the one thing I notice is that when using paragraph proof, students routinely forget to justify their statements with reasons.  When doing two-column proofs, even when they have trouble coming up with the reasons, they at least know that they need to have a reason.
  • Begin with algebraic proofs.  Students will know exactly what do to for the statements, they will have a list of reasons.  They just have to put some thought into matching them up.  I'll do this for two class days.  (I am beginning my unit on proofs tomorrow, and we only have school tomorrow and Tuesday.  Plus I don't want to get super heavy into a topic right before a break.) 
  • Move from simple algebraic proof to line and angle proofs.  Begin doing this by solving algebraic problems based on line and angle relationships.  This will hopefully get students comfortable with using line and angle relationships as reasons.  From here we'll move on to proving theorems about lines and angles without using algebra.  (G-CO.9)  
  • From there we can start congruent triangle proofs.  I still want to incorporate some kind of activity for students to understand WHY SSS, ASA, SAS, AAS, and HL work.  (Side note:  What happened to HL?  I'm seeing so many resources not listing it as an option.)  I'm thinking of progressing through congruent triangle proofs differently than I have in the past.  I'm thinking that I will start with an activity introducing all 5 postulate/theorems, spend time analyzing diagrams to decide to use which postulate/theorem to use, then begin doing actual proofs.  At this point, I want to introduce students to flow proofs.  I understand how it works, but I haven't used it in the past because I don't know any of the details of how to use it.  This year I am making a point of it to learn and then teach flow proofs.  Then students will have three choices of proof method. 
  • One of the most important things that I need to do is allow students to talk to each other about proof.  Students need to articulate, compare, contrast, exchange ideas, share feedback, and challenge each other.  This is the most important change I need to make.  Rather than use time as an "excuse" for why this doesn't happen, I have to make the time to do this.  
Here are links to the websites that helped inform this plan the most:

Most Influential - http://evanweinberg.com/2011/11/05/teaching-proofs-in-geometry-what-i-do/, http://evanweinberg.com/2012/09/27/first-day-of-geometry-proofs-refining-my-process/

The student explanations of proof are awesome - http://mathforum.org/sanders/exploringandwritinggeometry/proof.htm

Good website with tons of information - http://www.mathedpage.org/proof/math-2.html

Possible activity idea - http://www.learnnc.org/lp/pages/7483

Does anyone out there have any advice?  Am I missing anything?



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