Six Ways I Use The Two Types Of Tasks (Building THinking Classrooms IN Mathematics)

In Building Thinking Classrooms in Mathematics, author Peter Liljedahl differentiates between two types of thinking task in which to engage students - curricular tasks and non-curricular tasks.  Non-curricular tasks, as the name implies, are tasks that, while qualitative and mathematical in nature, don't directly address specific learning goals for students.  They are intended, rather, to capture students' attention, to broadly grow their interest in mathematics, and to serve as tasks through which we can teach them how to think and how to engage in a thinking classroom productively.  Liljedahl recommends card tricks, numeracy tasks, and "good problems" for these types of tasks.  I've had good luck with all of these, as well as with Ted Ed riddles.  My students love all of these!

Curricular tasks, on the other hand, directly address standards or learning goals set for the specific course in which students are enrolled.  These types of tasks are designed to guide students to learn the content of third grade, of pre-algebra, of calculus, and so on.  They are the core work of a thinking classroom, and if you were to observe such a classroom, they would be the star of the show.  Witnessing students working collaboratively at standing whiteboards, teaching themselves math with nothing more than guidance, hints, and extensions provided by the teacher or by other groups, is truly a sight to behold!
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"I'm checking the math, and it looks like he uses thinking tasks in six different ways, not two."

As I keep experimenting as I try to build my own Thinking Classroom in Mathematics this year, I've found that mastering thinking tasks is the highest leverage practice in the program; the gap between getting these right and getting these wrong is much bigger than it is for any other practice.  While I've made breakthrough improvements with random grouping, note-making, and consolidation this year, those improvements have been more icing than cake.  There were times I paused doing some of those altogether while I tried to master other practices, and we still made good progress.

Thinking tasks however, make or break the whole class.  Every day that they are on point feels like a breakthrough, to the point that there are days when we cover several days of content in a single lesson.  On the other hand, when they miss, they really miss.  It can be a total disaster.  If you're anything like me, there was even a point when a series of bad curricular tasks made me want to abandon the Thinking Classrooms concept entirely.

Getting the right task for the right goal - that's the game.
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"I think all he really does is try to match up these instructional goals with these task options."

In my half year of learning and experimenting, I've found myself using the two types of thinking tasks in six different ways.  In the space below, I'll explain what they are, when I use them, and how I create them.
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1. Culture-Setting Non-Curricular Tasks

​What are they?  Culture setting non-curricular tasks are non-curricular tasks just as Liljedahl speaks about them in the book.  They are highly-engaging tasks used in the first 3-5 days of a new term with a new class.  These tasks capture kids' attention and interest, and they send the message that what we do in this class will not be what you do in other classes.  They set a culture of thinking from the very get-go, endear kids to the process, and give them early opportunities to get used to thinking and collaborating.

"I guess figuring out this card trick counts as thinking about math?"

When do I use them? I used these for the first four days of school, right in line with the advice from the book.  On day one (literally the first day of middle school for my kids), I did a card trick.  The kids liked it and worked decently well at it, but expected me to just give them the answer before class ended, which of course I didn't in order to establish the importance of the thinking we would be doing, so we also worked on this card trick on day 2.  On day 3, I did one of the numeracy tasks on Liljedah's website.  It was a blast!  On day 4, another card trick, and this time, if the groups didn't figure it out, they just didn't get an answer and I told them to either keep working on it or to find another group to explain it to them.  They were four great days!

How do I plan them?  I don't.  They all come from Liljedah's website catalog.
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2. Problem-Solving Curricular Tasks

What are they?  Problem-solving curricular tasks involve some kind of engaging prompt or situation that can be solved using on-grade-level math.  They often come from 3-act tasks, YouCubed, and other such sources that were making these types of "math worth doing" tasks before Building Thinking Classrooms came into the picture.  In podcasts and webinars, I often here Liljedahl reference a task like this involving stacking books on a desk to teach linear equations, which sounds like a terrific example.

The key features of these tasks are that 1) they involve some sort of graphic, video, live prompt, or other tangible, engaging "hook," and 2) they have a solution.

When do I use them?  I think it is important to use these at the beginning of a topic of study to "launch" the topic.  For example, I used several 3-act tasks on ratios (this one, this one, and this one) to introduce the concept of a ratio before students had learned anything about ratios.  Because of that timing - giving these tasks before any content-specific learning has taken place - these tasks feel like non-curricular tasks to my students.
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Why schedule them this way?  These types of tasks only require thinking if students don't have the tools necessary to solve them.  Once they do have the tools to solve them, they don't require thinking anymore - they're just really engaging practice problems.  For example, if you don't know what a ratio is or how to solve a problem involving one, figuring out how many points the Arcade Basketball Prodigy will score is quite challenge and requires a lot of thinking.  If you do know how to solve problems involving ratios, figuring out his score requires no thinking at all - it's just a more fun version than asking the same question in a textbook.

How do I plan them?  I was using 3-act tasks and other similar tasks to launch topics of study long before I was building a Thinking Classroom, so I have a pretty good collection of these ready to go.  The only real modification I make is that I plan extension questions for groups who finish early since one of the pillars of a Thinking Classroom is that you're never "done."
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"Who knew there could be so much math to think about in a box of Lucky Charms?"

3. Figure-It-Out Thin-Sliced Curricular Tasks

What are they?  These are the bread-and-butter curricular tasks that Liljedahl gives a fabulous example of on page 161.  In this type of task, you remind students of a type of problem they can already solve, then challenge them to extend that knowledge to solve a new, related type of problem.  I've already written a whole post on this type of task, so I'll send you there for more details (or to page 161 in the book.  Or page 27).

When do I use them?  ALL THE TIME.  These are far and away the most common type of task I schedule.  In fact, my ability to schedule these is a sign that I'm planning well - if I am, then most days within a topic of study should build off the prior day in a way students can discover on their own, meaning a well sequenced unit of study should have long stretches of days with this type of task.
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How do I plan them?  Painstakingly, unfortunately.  They take a long time and a lot of work.  I have to find the entry point (the magic is in the mild, after all), figure out all the different manifestations of the type of problem and increases in complexity, and then make up all the questions.  It's hard, but it makes all the difference.
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"Aren't we lucky that he's willing to spend all that time slicing these tasks just right?"
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4. Introduce the Minimum, Then Thin-Slice To THe Maximum Curricular Tasks

What are they?  There are certain topics where I can't reasonably expect students to figure it out on their own, but I can reasonably expect students to take the briefest possible example and run with it.

Take, for example, changing between fractions, decimals, and percentages.  I had only a day to address this, and this is a topic that I really need them to understand in a very certain, conceptual way.  My chances of having them figure this out on their own in just a day are slim, and my chances of having them figure it out the way I need them to understand it are basically zero.

For tasks like these, I do a very, very short introduction to the concept or procedure with the mildest possible example, then let them run with it through thin-slicing.  They get the basic idea from me, but then they have to make sense of it on their own and thin-slice their way through more advanced versions of the skill.

I introduce the minimum.  

They thin-slice their way to the maximum.

For the fractions-decimals-percentages topic I referenced above, here's a recording of our class that day.  What you'll see is, with just four or five minutes of "kickstarter" questions, I introduce them to the idea that fractions, decimals, and percentages are all ways of representing the same concept, and that with some basic thinking, we can interchange between which representation we use using nothing more than our knowledge of conventions.  It isn't much, but it is just enough to get them thinking about the concept the way I need them to with basic examples.

If it helps, here are the slides the kids see on the screen and here is the task they move into afterward, where they take the very basic introduction I offered and extend it into more and more challenging examples.
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When do I use them?  I use this type of task in situations like the one I explained above, where either a) I don't think the students are likely to be able to figure out what to do entirely on their own, and/or b) when I need them to understand a topic a certain way.  I don't need to do them very often, but when I do, it tends to be the first day in a topic.  My preferred way to launch a thinking task is with a "you can already... but what about...," but often on the first day, there simply isn't a "you can already" to connect back to if the idea is brand new.
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"He really didn't give us much to work with here..."

How do I plan them?
The planning for these is the same as the other thin-sliced thinking tasks (see #3).  The only difference is that I plan the additional, brief introduction to the topic to help get the thinking started in the direction I need it to go.
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5. Non-Curricular Task + Direct Instruction

What are they?  There are days when I simply need to teach something directly, but I don't want to have a day go by without a thinking task, so I just schedule a non-curricular task beforehand.  This way, at least, I get the energy level up and get the kids' brains turned on before the direct instruction begins.

When do I use them?  The only days where I've used this format this year are those when I need to teach vocabulary or conventions - days when there isn't anything to "figure out."  The only two days I could find where I did so were:

• The conventions behind representing a ratio and the accompanying vocabulary
• The conventions behind absolute value and the accompanying vocabulary

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These aren't days where there are problems to solve or ideas that are "figure-out-able;" they're days where I need them to learn certain words, symbols, and conventions.  Not much thinking to do with those, so we think about something else instead.  The kids love non-curricular tasks, anyways, so these are a nice chance to sneak in a few extra ones.

How do I plan them?  I choose a non-curriuclar task from Liljedah's website, or I use a Ted-Ed riddle (my kids love these!), then I plan a traditional lesson to teach the content afterwards.
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"Does anybody even remember how to learn from direct instruction like this?"

6. Test-Day Non-Curricular Tasks

What are they?  Just like I plan non-curricular thinking tasks to energize the kids ahead of direct instruction, I do the same on test days.  Non-curricular tasks get them up, awake, and engaged before sitting down to dig into a test.  They also give the kids something to keep thinking about if they finish their test early.

When do I use them?  Test days!  I even did this on final exam days last week.  My school blocked out 2-hour classes for our 90 minute final exams, so the first 30 minutes of class were devoted to the Ted-Ed Virus Riddle so that we could have some fun and get into a good mental place before exam time.  Coupling a non-curricular thinking task with our typical "what should I be forgetting today?" class starter makes test and exam days feel more like 'regular' class days, too, and gets kids into tests with their 'regular' thinking prowess activated.

"Says here that there's supposed to be a test today... what's with all this non-curricular thinking?"

How do I plan them?  I choose a non-curriuclar task from Liljedah's website, or I use a Ted-Ed riddle (my kids love these!), then we test.

There you have them - the six different ways I use the two different types of thinking tasks.  In my first year teaching through thinking, developing this bank of task formats ensures that every day - every day - we think. 

I'd love to hear from you in the comments, too.  Are there types of tasks or class formats you use that are different from mine?  How do you keep the thinking alive in circumstances I haven't thought of? 

"Is it just me, or is it starting to feel like we might never get a break from all this thinking?"

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