NCTM Adventures, Day 1

This week was the regional NCTM conference for the East Coast.  Since it was in Hartford, one town away from where I grew up and where my parents still live, it seemed like an excellent opportunity to finally attend.  Lately I have more big ideas, strategies and philosophies to try than time to implement them, so I chose sessions that were hands on and related to Geometry or PreCalculus specifically.  I came away with a lot of great activities that I'm excited to share with you, my department and my students.

Standards Based Instruction by Suzanne Mitchell, NCSM President

She showed us some cool sample problems which included a ferris wheel problem that I thought would be a nice addition to the one I currently use, but a search of mathedleadership.org leads me to believe such things are only available to members.  Maybe I will expand my investigation of the ferris wheel beyond the problem we currently do though (which I might blog about someday).

Got Something to Prove? by Ralph Pantozzi

First of all, Ralph was an awesome presenter.  He was funny, jumped up on chairs to reach people with the microphone and spoke passionately about his students.  Beyond that, I really like what he had to say.  His central theme was "Let them wonder about it first."  Not only did he use this phrase several times, but he modeled it - giving us plenty of wonder time - and shared that he types this sentence across the top of any page he will use when working with students, including the notes for his presentation that day.  The goal is to convince students (and remind yourself) that all of mathematics is something that someone had a question about.  We should ask those same questions, and use proof because we genuinely wonder what works, why and how.  Proof doesn't need to be formal (no mathematician will be coming in the door to say "That's not a real proof!") but it needs to convince the audience - in formal mathematics, theories are accepted by people talking to each other, the same should hold true in the classroom.  Ralph likes to use the phrase "convince me" with his students, I wonder if that's better than the phrase I use, "defend your answer."  My intention isn't to have students be defensive...  Sprinkled throughout the session were some fascinating puzzles:

1 + 2 =3
4 + 5 + 6 = 7 + 8
(no questions, just two equations that he put up on a slide while waiting for us to wonder)

Divide a square up into squares.  (No shapes other than squares are allowed.)

P points are connected with segments in pairs.

The segments can’t intersect.

Draw a closed doodle.  Place a point at each intersection, each arc/squiggle/segment(s) between two points is called a section.  Count points, sections, regions (include the space outside the doodle).

The original presentation is now available as a pdf.

Origamics by Michael Serra

I was feeling really spiffy because the first thing we folded was the project I just completed in class!  Then we got to the second construction and I was stumped.  Luckily the woman I sat next to was stumped as well, and we were able to work through part of it together before he gave us an overview of the million similar triangles we had missed.  I spoke to Peg Cagle about the session later and she shared that she has students write "always, sometimes, never" statements about these types of things, especially the 'homework' which heads students towards sophisticated generalizations.  The constructions we did (NCTM Hartford), along with other sessions, are available on Michael's website.

Hands on Activities for Geometry by Carol J. Bell

The focus of this session was on proof without words.  I really liked the proofs we did, but they lacked scaffolding which made them difficult to approach.  Some simply needed the addition of a diagram since we couldn't see from the back which angles she referred to, while others jumped directly to abstraction without any numerical exploration.  That said here are some cool proofs that need lessons built around them:

Hands on Geometry

Customized Web Homework by Stephen Kuhn, Sandy Watson

I attended this session out of curiosity but with no real intention of using web homework.  Internet access still isn't quite universal enough among my students for this to be fair, plus it's something else to do and I promised myself to only take away activities (at least for implementation in the near future).  That said, I think the system they're using is promising (multiple types of answers, unique problem sets for each student with multiple attempts allowed on auto-generated similar problems and built in math type communication with instructors).  But for right now, I'm really excited about one aspect- shared problem sets among teachers!  I haven't had a chance to figure things out yet, but at some point I will be searching for PreCalculus problem sets.  Googling "whs web homework system" gets you to the website, several articles on the research study and more. 

And that concludes Day 1, read about Day 2.