Sequences and Series - or - Collaboration in the MTBoS

Like everyone else I've been watching the conversations on the State of the MTBoS with a balance of interest and trepidation.  I don't have a good sense of what it's like to break into this community because I learned about it while at PCMI.  I spent 3 weeks with Ashli, Kate and Sam, getting to know them as Ashli, Kate and Sam, before I learned about their existence as @mythagon, @k8nowak and @samjshah.  There were many other awesome people at PCMI who are both online and not, but the point is, I started out interacting with some really active members of the MTBoS.  I try to be welcoming to newbies and have enjoyed mentoring new teachers and hopefully have helped include new members of the community regardless of their number of years in teaching, but I don't feel that I can offer a good perspective on that aspect of this conversation.

However, I have heard concerns that we aren't collaborating.  And I know that's not true.  Collaborating takes many different forms, and since it crosses platforms it may be hard to follow, but it is absolutely happening.  I'd be hard pressed to find a post on this blog that isn't inspired by someone else and the majority of the lessons that I share are adapted from someone else.  Today I'd like to tell you how Sam and I collaborated on our Sequences and Series Unit.

Last summer Sam and I found out that we would both be teaching PreCalculus this year, a new prep for each of us.  Sam decided he wanted to write a unit on Sequences and Series before school started.  He wrote, he shared with me, I provided some feedback (no recollection if it was helpful or not, that was a year ago!).  In August he wrote a post, I saved the documents knowing I'd want them later.  Sometime mid-year I mentioned that my Sequences and Series unit was coming up, Sam sent me the updated version of the document.  I saved that too.  I didn't end up teaching the unit until the end of May - at that point I knew my students really well and could determine if his approach would work for me.  We also had a hiccup- the school was nearly out of toner for the copiers and totally out of funding.  I needed to find a way to do this without printing the packet - enter the dry erase template!

Sequence Template

All my students have a plastic sheet protector in their binder and I have a basket of dry erase markers available for their use.  Slide this page into the sleeve and project the images or problems from the packet - voila!  One sheet of paper per kid.  More than saving paper, this template also helps kids form a set of questions they can ask themselves.  Some students didn't need to find the 100th term to know the nth term while others needed help finding the 10th term, but all of them could look at the table and recognize a pattern.

The grid works for drawing as well as graphing.
Some students don't need the table - and that's fine!

Another change I made for the first section: instead of specifying what patterns students should describe, I had them pick - squares? perimeter? toothpicks? go ahead and define your variables and explore!  This was motivated by a need to give the students who finished quickly another task before the whole class was ready to discuss and move on, but I also think it's important for students to see that math is about describing what they notice.  I was able to really highlight this idea when we shared different ways of writing a formula:

We were studying patterns this morning and as I walked around the room I took note of who wrote their formula in a different way.  After having a couple students share their methods and then having the class prove their equivalence, I asked a third student if she was the one who had “n+(n-1)” (because I couldn’t remember whose paper it was).  The other students yelled out, “obviously that’s the same as 2n-1″ but I shushed them and had her share.  After the presentation, one of the students who had said it was obviously the same announced “that’s a totally different approach! It’s the same thing to say n+n-1 and n+(n-1) but the parentheses mean something really different here”  I was thrilled, because that was exactly the point I was trying to make and he couldn’t have said it better if I’d given him a script.  This is the same student who 5 minutes before was blowing on his phone (who invented an app that involves blowing on the screen??).  He may not be 100% on task, but I’ll take an easily distracted student who shares insights with enthusiasm over an obedient one any day.

(Reposted from One Good Thing: n+n-1 vs. n+(n-1)) 

So now, Sam gets to decide "This template is awesome, I want to use it!" or "The template would be better if..." or "My school has plenty of paper and ink and my way worked."  And over the course of a year we will have worked together to make a great unit that we will continue working on and discussing as we teach it again and as others try it or leave feedback.  Collaboration looks different when the collaborators are running on totally different time lines, but it's happening, I promise.