Building A Thinking ClassRoom: Covering A Huge AMount of COntent In A Single Lesson

I (might have) had a momentous week in my Building A Thinking Classroom journey this week.

In the Using Hints and Extensions, chapter, Liljedahl talks about how to sequence curricular tasks using variation theory, giving this example regarding factoring polynomials:

For the non-math teachers out there reading, any where you see red in the sequence above would typically be a new lesson when teaching factoring polynomials, as each new red spot represents a new level of complexity.  So according to the sequence above, teaching factoring might take about 5-6 days, which feels about right based on my long-ago experience teaching 8th grade and doing so.

Liljedahl tells that, in his experience, when things get humming in a Thinking Classroom and a curricular task like the one above is sequenced and varied (I also read and hear this called "thin slicing" among other Building Thinking Classrooms gurus) just right according to variation theory, that "we get through the entire [5-6 day] sequence in 40-60 minutes.  Balancing ability and challenge," he goes on to say, "allows you to cover a huge amount of content in a single lesson" (p.151).

I may have just had my first experience with this.

And I didn't even do that good of a job with my "thin slicing."

This week I started teaching ratio problems.  A basic example would be a problem like "if 2 cups of flour are needed to make 5 batches of cookies, how many cups of flour are needed to make 30 batches of cookies."

According to my notes from last year, I spent twelve days teaching about ratio situations:

I don't have to teach quite so many strategies this year because of a state curriculum change in Georgia this year, but suffice it to say it still would have been an 8-10 day sequence (and I planned it as such for this year, too).  

I used a 3-act task as the impetus for a "thin sliced" curricular task every day last week (except for Wednesday, when I made up my own) and asked similar questions every day.  This is Tuesday's, which I'll refer to as an example below.

Act I Arcade Basketball Insanity from Brian Lack on Vimeo.

The "thin slicing" was intended to follow this sequence:
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• A few questions about ratios that occur in exact multiples of the given one (i.e, "how many shots would he make in the full 60 seconds?" or "how long would it take him to make 75 shots?"​

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• A few questions that would require reducing the ratio ("how many shots would he make in 2 seconds?" or "how long does it take him to make 10 shots?")
• A few questions that are not exact multiples, but are exactly half way in between two multiples ("how many shots in 9 seconds?  How about 33 seconds?")
• A few questions that are not exact multiples and that are not halfway between multiples ("how many shots would he make in the remaining 34 seconds?" or "how long would it take him to make 100 shots).

Each day was intended to follow the same sequence, but for a different engaging ratio situation.  Were a group to get through this entire sequence, it really would have been basically the full 8-10 days worth of learning.

And it worked.

I was seeing certain students and groups make incredible progress on these tasks, getting all the way to the end some days.  It wasn't everybody and it wasn't every day, but it definitely happened for certain students on certain days.

I'll be honest, too, I didn't even do that good of a job with the tasks​.  Here's a list of missteps I made:

Having watched one of my students (not pictured) eat all the marshmallows in her Lucky Charms school breakfast and throw the rest of the cereal away, I decided to make my own task about the ratio of marshmallows to cereal pieces in Lucky Charms for Wednesday.

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• I didn't think of all of the thin slices until Wednesday
• On Tuesday, I rushed my own math and had some of the wrong questions in the wrong sections
• On Thursday, the initial task was more difficult than anticipated, and just getting most groups to the answer of the main question was a challenge, with very few groups making it much further
• On Friday, I again miscalculated some of the questions and had very not-thin slices
• I ran out of time for check-your-understanding questions Monday-Thursday

Nonetheless, here are some positive outcomes:

• On my "what should you be forgetting today?" practice questions to kick of class each day, the overwhelming majority of the kids were able to transfer their ratio skills from the tasks to new problems.
• On Friday, during an extended and extensive check-your-understanding block, I got very few questions from the kids and the ones I did get were on reasonably high level questions
• Quite a few kids have asked me why we've spent so long on ratios since they're so easy and seem like they should be in a younger grade level
• Most of the kids have already figured out how to calculate, understand, and use unit rates to their advantage, and that wasn't even a goal for this set of content (it's up next).

I really wish I had had the assessment planned for this topic ready earlier.  I might have given it to some kids on Wednesday to see if they really did learn two weeks of content in two days.  I suspect at least a quarter of them did.  I wish I had some evidence, but I wasn't on my formative assessment game.  I would love to see just how extensive this accelerated learning was.

Pretty exciting.

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