The Exponential Curve, Phase II

I haven't posted anything for a couple of years now, and I think I'm finally able to start up again.  I'm at a very different place in my life now, compared to when I first started this blog, and, thankfully, compared to when I stopped writing.

I am really excited by the move toward Common Core, and the abandonment of the CST tests.  I've felt my spirit and teaching practice withering and dying with each new modification I made to try and cram more standards in faster and more efficiently.  When students would ask why they need to learn math, I would give a standard reply about problem solving abilities and critical thinking in all areas of life, and this felt more and more cynical as I compared what I was saying to the actual content I was delivering.

This year, in my intensive algebra 1 class (for students who are well below grade level), I've quit worrying about skills lists.  I'm focusing on using high-quality problems and resources.  I'm trying to actually do what I say, and engage them with activities that require thinking, explaining, justification, problem solving, and persistence.  And since the new Smarter Balanced tests, according to David Foster, will be only 31% material at levels 1 and 2 of Webb's DOK, and 69% at levels 3 and 4, I feel like I can justify my new approach to any skeptic.

Just yesterday, I spent nearly 30 minutes playing the Game of 21 with them, a quick and easy misere game.  (The first person says 1, 2, or 3.  The next person increases by 1, 2, or 3.  Alternate turns.  You can't go past 21, and the first person to say 21 loses).  It was great to watch them struggle with it.  Of course, I made them start, and so I won game after game.  But then some students started realizing that I was going to win as soon as I said 16, and after a while longer, they realized I would win when I said 12.  Some students were frustrated, some continued to challenge me blindly, and some were clearly paying close attention and trying to develop a strategy.  I love the moment when the first kid tells me that, no, *I* need to go first.  And then when they finally beat me, it's a great moment!  At this point, I stopped the game without discussing the strategy, and told them to play it against a friend or family member for homework.  We'll pick it up again on Monday and see if more can beat me.

We've been working on finding out the rule for patterns, using only visual examples, building them with toothpicks, etc.  So I decided to start them on the non-linear Growing Staircases POM from SVMI for the second half of the class (A pattern where you start with one square, then add two, then three, etc).  First, I spoke with them about what perseverance means, and why it's so important.  I told them that we were going to all get through level C of the POM, but we were not going to finish today, or even by the next class.  That real math problems take longer than an hour to solve (or the 15 seconds that they are accustomed to).  They mostly completed level B (figuring out how many squares it would take to build a 10-step staircase), and they used counting and other patterns to do so.  Level C (find a rule for the number of squares in an n-step staircase) is going to be a lot harder, and I'm looking forward to seeing what they come up with.

I decided to do a lot of patterning work with them before we even talk about solving equations, using their visual experience with developing a rule as leverage for understanding.  When they see something like 4n + 3 = 51, I want them to think something like "That's 4 groups of n-blocks, and then 3 blocks more, giving a total of 51 blocks.  So the 4 n's would have to make 48 blocks.  So each one would be 12!".   Right now, their pattern work ends with coming up with a rule.  Soon, I'll start asking them to figure out what step number it would have to be, given a specific total.  I'll treat this as a problem-solving exercise now, and eventually formalize the steps into the standard solving equations algorithm.

I've also been drawing heavily upon the excellent work found in the Formative Assessment Lessons at the Mathematics Assessment Project .  These are really meaty lessons that incorporate excellent group tasks, and really help push students to use the CCSS Math Practice standards.  So far, in this algebra class, I have used Positive and Negative Numbers in Context and Interpreting Algebraic Expressions.  The one caveat I have is that the amount of time it takes to run one of these lessons is well underestimated in their notes.  The expressions FAL took my intensive class four lessons to complete, not the optimistic 1 lesson with a 20 minute follow-up!  But of course this will vary from class to class.

Aside from this, I am also teaching Algebra 2, and I am using CPM instead of whatever NCLB book we have sitting in our library.  This will be the subject of other posts, as my students and I wrestle with a completely different way of teaching and learning math.

And, as if this weren't enough, I am teaching the first year of IB Math SL (which is roughly equivalent to the first year in a pre-calculus/calculus sequence).  This class is filled with really positive, engaged, motivated students, and it's a joy to teach, so I shouldn't complain too much about the extra prep!

So hopefully this post will be the first of many more to come.   Being part of the mathtwitterblogosphere is invaluable, and I want to start adding my voice to the mix again.