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?



Sunday, November 2, 2014

Geometry Scope and Sequence Realignment Reflection

This year I made the brave decision to begin Geometry with our unit on three-dimensional measurement.  I've considered this for the past two years.  Since last year moving the Coordinate Geometry unit to the beginning of the year was a success, I decided to try moving the measurement unit to the very beginning of the course.

My reasons for this are:
  • This unit encompasses the topics that students expect when beginning Geometry.  The definitions at the beginning, the lines and angles that are not parts of actual shapes, etc. seem to really bother students.  It's all the foundational knowledge of Geometry, but students don't see it as Geometry right away.  They expect to see complete shapes, and expect to calculate perimeter, area, and volume of shapes because this is what they've experienced in the past.
  • This unit relies on basic, foundational Algebra which serves to reinforce and review the key concepts of applying formulas and solving equations.  This was the key to my decision.  My students had multiple Algebra teachers the previous year, and their Algebra skills that I assessed the first days of school proved that their skills are weak.
For my reasons, the move was successful.  Students were able to practice basic problem solving, and they've shown improvement there.  I heard less complaints that "we're not doing Geometry."  However, there were some issues that arose from moving this unit to the beginning.  The main issue is that I was limited in the practice problems that I could assign.  Our textbook has awesome practice problems that include review from previous topics, but we have yet to learn those topics, so they were unusable.  (For now.  I have to remember to include these problems when I go back to review later in the year.)

Overall, I noticed that students struggled with this unit the same way the students did when I taught it last year.  I need to find more hands-on activities for this unit.  I feel that I really missed out on incorporating great learning experiences in this unit.  I'll have to make finding activities for this unit one of my goals for this summer.

For this unit, my Interactive Notebook pages were not much different from the Measurement unit pages last year.  For now, I am not planning to post about these pages.

Made4Math: Feather Pencils

I've been trying to come up with an idea all summer long to maintain my stock of pencils.  I know it's inevitable that my pencils are going to just get up and walk out of the classroom, but anything I can do to minimize the loss will be a great help.  Last year I tried wrapping colorful mini duct tape around the top of the pencil to identify my pencils.  Within two weeks, I was finding the tape adhered to my desks, chairs, and adorning my classroom walls.

I've seen all over pinterest teachers that taped flowers to their pencils.  It will definitely remind students that the pencil does not belong to them, but I can only imagine how annoying it is to write with.  The real problem though is that a large flower renders the eraser unusable.  These teachers have stated that they just hand out erasers for students to use instead.  This would be a wonderful solution, but my students will take one eraser and magically turn it into several mini projectiles.  (In case you haven't picked up on this fact so far in this post, or any of my previous posts, my students are very destructive.)

I was stuck on the idea of attaching something to the pencil though.  Something small, flexible, colorful, and easily identifiable.  Finally it dawned on me.  I went shopping at Hobby Lobby and came across these:

I know.  1998 called and it wants its pens back.  They were only $0.99, and couldn't resist the fun nostalgia. 

So I decided that since feather pens are so much fun, why not try feather pencils?  I bought a pack of bright, colorful feathers, and attached them to the pencils with washi tape.  My only hope is that students don't take these apart.  And of course they do.  (The most annoying thing is when they have scissors for Interactive Notebooks, some of them will cut the feathers off.)  They complain that the feathers annoy them.  I have two things to say when they tell me this:  (1)  Just be glad I didn't put a large flower on it; (2)  If you don't like my pencils, bring your own pencil.

One of the things I love about this method is that there is no need to sign up to borrow a pencil.  Just take one and put it back when you're done.  (Although I've had to remind students that this is what we will have to resort to if they keep taking/mutilating my pencils.)  I will be able to scan the room before the end of class and notice who has one of my pencils and remind them to return it.  Another bonus is when students do find my pencils in another classroom, they bring them back because they know the pencil is mine.  In the past two months, I have lost approximately 30 pencils, which is a HUGE improvement from the pencils I lost by this time last year.

Thank you!

Thank you for being patient as I got my new blog set up.  I am slowly releasing updated versions of old posts while I add new content.  Plea...