Task Propensity

I’ve really enjoyed writing about task propensity over the last few months, and I’ve learned a ton about teaching and learning in the process. The series of posts was inspired by a paper on efforts to reform mathematics curriculum in the Netherlands. Here is the an excerpt from the introduction of the paper that got me started on this path:

In each case student performance did not match expectations, and in each case textbooks analysis revealed a focus on teaching effective procedures for specific tasks, instead of developing mathematical insights at a more advanced conceptual level as the reform intended. We will argue that this is at least in part due to what we call, “task propensity,” which we define as the tendency to think of instruction in terms of individual tasks that have to be mastered by students. This task propensity entices teachers and textbook authors to capitalize on procedures that can quickly generate correct answers, instead of investing in the underlying mathematics while accepting that fluency may come later.

The authors focus on reform curriculum — curriculum that was meant to support a more conceptual mathematical focus. I’m using task propensity more broadly; in the same way that teachers and curriculum writers fall into the trap of focusing on individual tasks rather than the underlying mathematics, students working on rich, engaging tasks fall into that same trap. In short, it is easy to lose the forest for the trees, and it’s hard to step back from individual problems to see the bigger picture.

I was inspired to write this series because I see task propensity in my teaching all the time, both in the ways I structure tasks and activities and the ways students engage with them. I focused initially on Desmos activities because I think they can be particularly prone to task propensity, but I now see it in more and more places. I think task propensity is a natural human instinct, in which the practical concerns of solving the immediate problem supersede the learning that could be gained from stepping back, taking a broader perspective, and considering how the thinking in a task could be applied to new situations. I also think it’s inevitably a part of any ambitious and engaging curriculum.

I’ve explored three strategies for addressing task propensity that I think are also three useful design principles in any curriculum that focuses on students learning math by doing math — staying humble, follow-up tasks, and the pause.

Stay Humble 

When I stumble across a great lesson on the internet, it’s easy to focus on the allure of a fun activity that students will enjoy rather than the substance of the learning under the hood. That’s not to say student engagement is unimportant, just insufficient for meaningful learning. Staying humble means constantly calibrating my lens for what a great lesson is, focusing on substance over style. It means taking an ambitious lesson that is broad in scope and trimming it down to focus on a smaller number of well-defined mathematical goals. It means putting aside the big picture at times to zoom in on the building blocks that students need to support their larger understanding. Staying humble doesn’t mean teaching boring classes, but it does mean avoiding the temptation of sleek and sexy lessons when they’re just not the right tool for the job.

The Follow-Up Task 

Student engagement is great, but inevitably leads to a focus on the present rather than the future. In lots of tasks, that’s what I want, and I embrace the energy in the room. It also creates an opportunity for a deliberately designed follow-up task, where students return to a previous activity and consider its implications in a new problem. The initial task acts as an anchor to contextualize student thinking, whether they refer back to technology or manipulatives, borrow a bit of engagement from a fun experience, or reuse a useful problem type. Often during an engaging activity students are particularly engaged with the more “fun” elements of the task and not the underlying mathematics. Follow-up tasks take a step back from the incentives in the moment while returning to essential ideas that students can learn from.

Pause 

When students are caught up in an engaging activity I don’t instinctively want to stop them. I want to enjoy the moment and watch them have fun. At the same time, they’re also likely caught up in the activity in a way that prevents them from slowing down and thinking about how the math they’re doing might help them solve new problems in the future. While pausing them might elicit some groans, it also provides a great opportunity for students to think metacognitively about the connections they’re seeing and the math under the surface, rather than getting lost in the sauce of the moment-to-moment tasks. Pausing an activity is fundamentally about harnessing energy in the room to advance specific goals rather than leaving student thinking to chance.

Closing 

Spending this time exploring task propensity has helped me to think about teaching and learning in new ways. It’s an important reminder that kids learn what they spend time thinking about, and I want to plan my lessons deliberately to promote the type of thinking that will support new mathematical knowledge. Engagement is not the same as learning, but I can use student engagement and well-structured activities to create opportunities for students to do the thinking I want them to do. And student thinking should not be left up to chance — if I have a goal for students, I should modify or restructure the student experience to make sure they meet it.

Illusions of Learning

“Have you heard of IXL? I love IXL, it’s so easy, it makes me feel so smart.” – Student

IXL is a computer-adaptive website that many teachers use for skills practice. I have nothing in particular against it. I do think that, more broadly, computer-based personalized learning platforms and the way they are used can fall into the trap of chasing what students like, rather than what’s best for their learning.

Here is an excerpt I often come back to on the science of desirable difficulties in learning:

Not long ago, the California Polytechnic State University baseball team, in San Luis Obispo, became involved in an interesting experiment in improving their batting skills.

Part of the Cal Poly team practiced in the standard way. They practiced hitting forty-five pitches, evenly divided into three sets. Each set consisted of one type of pitch thrown fifteen times. For example, the first set would be fifteen fastballs, the second set fifteen curveballs, and the third set fifteen changeups. This was a form of massed practice. For each set of 15 pitches, as the batter saw more of that type, he got gratifyingly better at anticipating the balls, timing his swings, and connecting. Learning seemed easy.

The rest of the team were given a more difficult practice regimen: the three types of pitches were randomly interspersed across the block of forty-five throws. For each pitch, the batter had no idea which type to expect. At the end of the forty-five swings, he was still struggling somewhat to connect with the ball. These players didn’t seem to be developing the proficiency their teammates were showing. The interleaving and spacing of different pitches made learning more arduous and feel slower.

The extra practice sessions continued twice weekly for six weeks. At the end, when the players’ hitting was assessed, the two groups had clearly benefited differently from the extra practice, and not in the way the players expected. Those who had practiced on the randomly interspersed pitches now displayed markedly better hitting relative to those who had practiced on one type of pitch thrown over and over. These results are all the more interesting when you consider that these players were already skilled hitters prior to the extra training. Bringing their performance to an even higher level is good evidence of a training regimen’s effectiveness.

Here again we see the two familiar lessons. First, that some difficulties that require more effort and slow down apparent gains — like spacing, interleaving, and mixing up practice — will feel less productive at the time but will more than compensate for that by making the learning stronger, precise, and enduring. Second, that our judgments of what learning strategies work best for us are often mistaken, colored by illusions of mastery.

When the baseball players at Cal Poly practiced curveball after curveball over fifteen pitches, it became easier for them to remember the perceptions and responses they needed for that type of pitch: the look of the ball’s spin, how the ball changed direction, how fast its direction changed, and how long to wait for it to curve. Performance improved, but the growing ease of recalling these perceptions and responses led to little durable learning. It is one skill to hit a curveball when you know a curveball will be thrown; it is a different skill to hit a curveball when you don’t know it’s coming. Baseball players need to build the latter skill, but they often practice the former, which, being a form of massed practice, builds performance gains on short-term memory. It was more challenging for the Cal Poly batters to retrieve the necessary skills when practice involved random pitches. Meeting that challenge made the performance gains painfully slow but also long lasting.

This paradox is at the heart of the concept of desirable difficulties in learning: the more effort required to retrieve (or, in effect, relearn) something, the better you learn it. In other words, the more you’ve forgotten about a topic, the more effective relearning will be in shaping your permanent knowledge (Make It Stick, excerpted from 79-82).

Part of my role in the classroom is to engage students in thinking about challenging ideas, monitor their learning minute by minute, day by day, and beyond, and connect concepts over time as we revisit them in more and more depth. I try to do all of that through the lens of a scientific understanding of how students learn. In 2017, too much personalized learning colors perceptions with the illusion of mastery and relies on making content feel easy as a substitute for substantive engagement, trading durable, transferable learning for hollow confidence-building and short-term skill retention.

I am interested in computer-based platforms for supplemental practice if they make my life easier, but personalized learning is far from where it needs to be to take on a primary role in the classroom.

Introducing Concepts in Calculus

Thanks to Jonathan Claydon for starting some great conversation about tough concepts in calculus, first with his post on avoiding magic tricks, and then starting work on a calculus chapter in Nix the Tricks. I just learned some cool new ways to introduce tricky ideas in calculus, and my teaching will be better for it. Check out this cool visual explanation of the Product Rule!

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Jonathan’s post also brought to mind a neat exchange I saw on Twitter recently on explaining where the Chain Rule comes from, also very cool!

Alright, now it’s time for a confession.

I don’t like introducing complex ideas like the chain rule by proving why they work.

I think this type of introduction-by-proof appeals to a subset of my students, but it tends to turn off others, and the kids who turn away are the ones I most want to engage.

Here is a different approach I’ve used for the Chain Rule:

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(doc) (Yes, I introduce transcendental functions before the Chain Rule. I realize many calculus teachers follow a different sequence, but this approach still works with different examples.)

I give students the handout and tell them that it shows functions on the left, and each function’s derivative on the right. I ask them what patterns they notice, and how they could use those patterns to find other derivatives in the future.

Students don’t usually figure everything out on their own. That’s not my goal. Instead, students have a chance to think about “inside functions” and “outside functions” and describe this funny derivative rule informally before we describe it formally. I build off of students’ language describing the types of functions they see and their connection with the derivatives. Then, after offering some explicit instruction in informal language, talking about inside and outside functions and multiplying by the inside derivative on the outside, I might offer them a more formal definition of the Chain Rule, like this one:

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I ask students to discuss in pairs or groups how this definition is connected to the examples they just looked at. After a bit of informal discussion on their own terms, I ask them to identify both f(x) and g(x) in several of the initial examples and I annotate a few functions on the board, using color to emphasize the different pieces and how they fit together.

The purpose of this sequence is to move from informal to formal, to give students a chance to make sense of an abstract rule on their terms before Leibniz’s, and to use worked examples to illustrate an idea while still putting the thinking on students. I have no illusions that this is sufficient to teach the Chain Rule, but hopefully at this point students are set up to be successful in engaging with some practice.

After this sequence, students are hopefully thinking, “ok, that kinda makes sense, but why is that the case?”. That’s where I think the conceptual explanations I referenced above come in. Once students have a basic grasp of a rule, I think they are in a much better position to grapple with the complexities of where it comes from. Even better, the initial exploration could happen on one day, stew overnight, and the next day I share a way of understanding where the rule comes from. Students’ informal understandings and experiences with a few concrete examples of a concept will hopefully help them better understand and make sense of an explanation of where that concept comes from.

I like this approach because I think it honors Jonathan’s desire, and my desire, to help math make sense to students, while also prioritizing informal thinking before formal thinking. I also like that, at every step, I can give students who struggle with calculus opportunities to engage on their own terms and feel like they can make sense of new ideas. I don’t think this is the right approach for every new mathematical idea, but in calculus lots of concepts have algebraically or computationally complex proofs, and this approach hopefully minimizes the challenges of that complexity.

Differentiation Strategies

I don’t like the word differentiation. Teaches assume it means “give students different work”, when I see differentiation as a set of tools used to make sure all students learn, regardless of academic background, prior experiences, or identity. I think setting ambitious goals for all students to work toward together is one of the best things about a classroom community. I think there are a range of strategies I can use to do that, and I don’t have a better word to lump these strategies together than “differentiation”.

Low-Floor, High-Ceiling Tasks 

A task with a low floor and a high ceiling doesn’t need to be anything fancy. It can be a pretty humble task that has features to provide access and extension to more students. One example I like is a card sort. I often use a lesson from the Shell Center in a unit on exponential functions, and have students complete this card sort:

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Students match graphs, tables, equations, and descriptions together, each describing a situation of either simple or compound interest. The sorting makes this task accessible for more students, as they can start to make associations even without much fluency with exponential functions. There are two blank cards for students to fill in missing equations, and blank spaces in the tables for students to fill in missing values. Some of the descriptions are also trickier than others, and lead to analysis comparing how often different situations take to double and other potential extensions to challenge students. The entire activity helps students to connect different ideas and make explicit the different ways that exponential structure plays out in compound interest while distinguishing exponential growth from linear growth. The lesson also includes thoughtful follow-up tasks for students to do more work with these ideas and apply them in different situations.

Teaching to Big Ideas 

Teaching to big ideas means exposing students to key understandings early in a unit, and returning to them over and over again. I just taught a unit on conic sections, and one of the big ideas is that replacing every x with an x – 2 moves a graph two units in the positive x direction, and replacing every y with a y + 3 moves a graph three units in the negative y direction. More broadly, these transformations can be used to translate a graph in any direction.

Students first see these ideas with circles, ellipses, hyperbolas, and parabolas. I like to give students an unusual example to further emphasize the big idea. This equation creates an interesting graph:

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While it’s not a conic section, it does behave in similar ways. I ask students to describe how to graph a general version of this equation using similar strategies to the ones we use with conic sections. Some students explore with vertical and horizontal translations; others get into different types of dilations, and some can be challenged to analyze changing the signs and exponents to see what happens. There’s lots of great structure here and connections to other ideas, and students can engage with the big ideas on multiple levels through the same problem.

One advantage of this approach is that a focus on this big idea extracts the essential mathematics from conic sections while putting less emphasis on things I care less about. It’s easy in that unit to get caught up in manipulating conic sections algebraically. While that can be a useful goal, it’s also one that requires much more background knowledge, and serves less of a purpose in the broader mathematics progression. Focusing on this big idea, important in conic sections and recurring in other topics, helps to make sure that what students are thinking about is the math that will be most helpful to them in the future.

Making Big Ideas Explicit

While the task I used above provides students with a useful opportunity to engage with a big idea, working with that big idea without making it explicit leads to a less equitable classroom. Many students will make connections on their own, and figure out that the structure of this function has a lot in common with other conic sections. But some won’t, and those who won’t are likely to be students who have struggled with math in the past, compounding gaps that already exist. Making the big idea explicit means pausing during the lesson to discuss what students notice and what strategies they are using, and reflecting at the end of the lesson on how these ideas are connected with ellipses, hyperbolas, and other objects. These discussions and reflections are great learning opportunities for everyone, whether students are practicing articulating their ideas using mathematical language, pausing to notice features they might have rushed past while focused on completing the task, or seeing a connection for the first time.

In the same way that I need to structure that task to make big ideas explicit, I need to make them explicit at every other step of the way as well, both leading up to this task and as this idea comes up in the future. It’s easy to forget that students see math differently than I do — that they don’t have the background knowledge I have and don’t see the broader structure of math as readily. I need to constantly remember to make essential ideas explicit so that every student has an opportunity to engage with them, rather than leaving it to chance. For more thoughts on big ideas, read David Wees’s piece here.

Aim for Relevance 

I don’t want to assume that every student finds math worth learning. I want to do what I can to help all students feel a sense of ownership and enjoyment in my class. One way of doing that is offering multiple perspectives on why students should learn math. I try to capture moments of wonder and curiosity, and help students have those a-ha moments that will make math class enjoyable. I frame math as a series of puzzles to be solved, practicing skills that will help them solve more puzzles in the future. I dive into applications of topics like exponential growth with examples of financial advice for students to evaluate based on what they’ve learned. I give students a window into how different ideas are relevant to higher math that they might study in the future. And I talk about how some math is worth learning just because it’s fascinating — mathematicians studied prime numbers largely out of curiosity for thousands of years before they became the building blocks of today’s web encryption standards. None of these arguments for learning math is sufficient on its own — each appeals to different students at different times. And not every student will be engaged every day; I need to be realistic in my goals. But offering a range of ways for students to make meaning of math class helps more students to come to class motivated and ready to engage.

Scaffolds 

Scaffolds are probably the strategy I think about the most, but they are also the hardest to master. I need to provide scaffolds that help students to access content, while also taking them away when possible. One strategy I’ve found useful is scaffolding different parts of a task at different times. For instance, after introducing the different parts of rational functions, I might ask students to analyze the end behavior of a group of functions. Then, I’ll give them this task, graphing rational functions when already given end behavior, vertical asymptotes, and intercepts:

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Then, I’ll ask students to find all relevant features of a function without graphing. Then they will put it all together. Varying the scaffolds helps students to focus their attention on different parts of the whole, and builds fluency in chunks that are manageable before attempting the entire task.

Building Relationships 

Building relationships is both important and subtle. A lot of relationship building happens in small ways, inside and outside of class. One way I approach relationships systematically is to pay attention to particular students that I know often feel bored in class, or often struggle and feel confused. Say I’m having students engage with an Illustrative Mathematics task in a unit on exponential functions and logarithms:

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Every student will experience this task differently. While students are working in groups, I am monitoring for different strategies to share with the class and making sure students stay on track. I’m also paying particular attention to students who often feel bored or students who make be bored in that task, trying to offer them an extension and doing what I can to keep them engaged. I’m also paying particular attention to students who often struggle and feel confused. My goal is that every student engages with the big ideas of the task to an extent where, when we go over it, they will be able to engage and learn from others’ strategies and understanding of the math. By timing student work around these students, I can make decisions deliberately that help every student engage and help every student feel like they can be successful. There’s no perfect solution, but by keeping tabs on those students’ experiences, I can make those decisions a little better.

Conclusion 

These strategies are the result of a paradigm shift for me. One paradigm is that students come into math class with different levels of knowledge and skill, and I need to offer them different experiences to meet them where they are. A different paradigm is that students come in with different levels of knowledge and skill, and if I look at all of my instructional decisions through that lens I can provide meaningful, common experiences that help every student engage with mathematical thinking.

None of these strategies alone solves the problem of students with different backgrounds and different experiences trying to learn the same concepts. I still think that is a goal worth working toward, and I think that these strategies together can make a difference in moving in that direction. None are easy to implement — I’m still getting better at all of them. But I’m optimistic that that improvement will make me a better teacher for the students who most need it.

 

Differentiation

I dislike the word differentiation. I think the goal is important — to teach so that all students learn, regardless of their academic background, prior experiences, identity, or any other factor. It’s really hard to teach lessons that effectively support learners who are struggling as well as learners who find the content easy.

But differentiation is often conflated with “give different work to different students”. When teachers describe a lesson as “fully differentiated”, they usually mean “every student did different work”. I worry that this implicitly lowers expectations, prevents students who are behind from catching up with their peers, and creates far more work for teachers than it’s worth.

I do give students different work at times, but as one of the last strategies I try. I instead try to find tasks with a low floor and a high ceiling, to teach toward big ideas that students can engage with on multiple levels, to make those big ideas explicit and ensure students engage with them multiple times in multiple contexts, to make learning feel relevant to students with a range of backgrounds, to incorporate scaffolds for tasks that allow all students to access them, to provide extra support and extension either inside or outside of class, and to build relationships so that students are more likely to engage with challenging ideas and buy into classroom routines.

I think that those strategies are incredibly important to my teaching, and I think that they should be grouped together one idea. But I don’t like calling them differentiation, because when I talk about these strategies as differentiation, other teachers just assume I’m talking about giving students different work.

Perspectives on Formative Assessment

Two quotes that are on my mind, on the topic of formative assessment:

Imagine what would happen if a pilot flew like many teachers assess. So I flew back from Seattle a few weeks ago. Just imagine what the pilot would have done. He would have flown east for nine hours. And then after nine hours he’d say it’s time to land so he’ll put the plane down and he’ll ask, “Is this London?” And of course even if it’s not London, he says “well everybody’s got to get off because I’ve got to get on the the next journey”. That’s exactly the way we’ve assessed in the past. We teach students material, and at the end of that teaching we find out if they’ve learned it or not, and if they haven’t we say too bad because we’re on to the next unit. So what formative assessment does is encourages teachers to take constant readings about where students are, just as a pilot takes constant readings about their position, and if the learning isn’t proceeding as you’d planned then you make adjustments.

-Dylan Wiliam (source)

Picture this scene: Dr. Gillupsie has grouped around him several of the young resident surgeons at Blear General Hospital.They are about to begin their weekly analysis of the various operations they have performed in the preceding four days.

GILLUPSIE: Well, Carstairs, how have things been going?
CARSTAIRS: I’m afraid I’ve had some bad luck, Dr. Gillupsie. No operations this week, but three of my patients died.
GILLUPSIE: Well, we’ll have to do something about this, won’t we? What did they die of?
CARSTAIRS: I’m not sure, Dr. Gillupsie, but I did give each one of them plenty of penicillin.
GILLUPSIE: Ah! The traditional ‘good for its own sake’ approach, eh, Carstairs?
CARSTAIRS: Well, not exactly, Chief. I just thought that penicillin would help them get better.
GILLUPSIE: What were you treating them for?
CARSTAIRS: Well, each one was awful sick Chief, and I know that penicillin helps sick people get better.
GILLUPSIE: It certainly does, Carstairs. I think you acted wisely.
CARSTAIRS: And the deaths, Chief?
GILLUPSIE: Bad patients, son, bad patients. There’s nothing a good doctor can do about bad patients. And there’s nothing a good medicine can do for bad patients, either.
CARSTAIRS: But still, I have a nagging feeling that perhaps they didn’t need penicillin, that they might have needed something else.
GILLUPSIE: Nonsense! Penicillin never fails to work on good patients. We all know that. I wouldn’t worry too much about it, Carstairs.

-Neil Postman & Charles Weingartner, Teaching as a Subversive Activity

I’ve come to bristle a bit at the phrase “teaching style”. I do have a style of teaching — different habits and structures I tend to use. But that phrase is also used as a justification for particular practices, regardless of whether they help students learn. “Well, that’s my teaching style, [Student X] will just have to adjust”. I’m happy I have my quirks and preferences in the classroom, but I need to balance my preferences with what works best for my students, and be willing to be wrong and try something new.

Task Propensity: Beyond Desmos

Task propensity refers to situations where students are so focused on the features of a specific task that they don’t generalize their thinking in a way that is useful to solve different problems in the future. In short, they lose the forest for the trees. I’m exploring how task propensity relates to Desmos activities and how this thinking could help me teach more thoughtfully with Desmos tools. I first learned about task propensity through this paper, and you can read the rest of my series on the topic here.


 

When I first read about task propensity, I was interested because it described one of my hesitations with Desmos activities. The activities are engaging and fun for students, but that engagement didn’t always lead to the thinking that I wanted it to. I have spent some time this fall thinking about strategies to counteract that phenomenon — keeping activities humble, designing focused follow-up tasks, and pausing the activity.

At the same time as I’ve been practicing those strategies when I use Desmos activities, I’ve seen task propensity in other areas of my teaching. Any rich task can fall victim to a focus on the task itself rather than the broader mathematical thinking that goes into solving it. I want to explore two examples of task propensity in activities I’ve used, and how I might modify those activities the next time I teach them.

Trig War 

I want students to practice evaluating sine and cosine functions, but I don’t want that practice to be any more soul-deadening than it needs to be. Inspired by Kate Nowak’s Log War, I put together Trig War. Students pair up and divide the stack of cards in half. They then each flip over a card, figure out whose value is larger, and that person keeps both cards. Wash, rinse, repeat. It’s pretty fun and gets a lot of practice in. At the same time, some pairs end up relying on one person to evaluate tricky values, especially those outside the unit circle, or they end up rushing and missing opportunities to think about the structure of the unit circle and sine and cosine functions, focused more on the War than on the Trig.

One idea I want students to take away from Trig War is a stronger intuitive understanding of where things are on the unit circle. Students might look at two values and, without evaluating them, know by visualizing that one value is positive and the other is negative. They might recognize that cos(x) = cos(-x) based on the structure of the unit circle and use that property to quickly evaluate negative values for the cosine function. They might compare two values that are very close together on the unit circle without evaluating by thinking about whether that function is growing or shrinking on the relevant interval.

But in most cases, students don’t do this thinking. They’re too wrapped up in the game, and don’t want to slow down and look for mathematical structure. I can instigate the thinking I want using the same strategies I identified for task propensity. I can pause the game, have students put their cards down, and pose a sample comparison that elicits strategies I’m interested in. By slowing down and focusing on one question as a class, I avoid leaving student strategies to chance, and share with them some of the thinking I’d like to see from them. I can do something similar with a follow-up task — reinforcing a strategy we discussed as a group, and provoking extended reasoning and generalization in a way that is hard in a game context in the moment. And finally, I can try to keep the game as focused as possible. The version I have used in the past is ambitious — it addresses both sine and cosine functions, including negative values and positive values beyond the domain of the unit circle. I could consider splitting this up into two games — the first focused strictly on the unit circle, and the second expanding the domain to other values and stretching student thinking, so that each game can be laser focused on the specific strategies I want to elicit and the goals I set for students.

Sequences and Skittles 

I often begin a unit on sequences and series with a Skittles activity adapted from Julie Reulbach. Each group gets a package of Skittles, a plate, and a cup. They put a certain number of Skittles in the cup, shake them, place them on the plate, and then remove Skittles according to a rule — remove all Skittles with the “S” up, remove all Skittles with the “S” up then add five back, and more. Julie uses this activity to focus on decay and recursive functions, but I’ve adapted it to address other ideas as well.

It’s a ton of fun. There’s plenty of management involved to avoid making a complete mess, but it gets at neat examples of recursive functions that provoke some useful mathematical thinking. At the same time, that engagement can mean students are thinking more about Skittles and the excitement in the room than they are about the math.

The follow-up task becomes particularly important with this activity. Students can record data while they play with the Skittles, but asking them to do much more thinking while playing with the cup is likely to lead to rushed work and shallow reasoning. Instead, I see this activity as having two stages — the high-engagement initial task of playing with the Skittles and recording data, and then a follow-up task, once the Skittles are away, where students analyze and make connections with the specific math concepts I want to get at. This is also a great example of a place where I need to stay humble. I’ve tried to run this with multiple different versions of growth and decay with different recursive rules, but I think Julie was onto something by focusing on just two rules, both decaying. The more focused the investigation is, the more likely I am to get all students to reach my mathematical goals, and the less is left to chance. Finally, there’s the potential for some really great thinking while students are playing with the second rule Julie used — removing Skittles with the “S” up and adding five back. But just letting the activity run and stepping back leaves those a-ha moments to chance. Instead, I can time the activity more deliberately, get the majority of groups starting that experiment at the same time, and pause as they start to realize that the number of Skittles will likely never reach 0 to discuss with the whole class. These are subtle changes, but they’re changes that do a ton to focus student thinking on mathematical goals and minimize the task propensity of an engaging activity.

In Closing 

If there’s one lesson I want to take away from this, it’s that I need to constantly ask myself, “what are students thinking about right now?” Memory is the residue of thought. Task propensity happens when student thinking is focused too much on a specific task, and less on the mathematics behind that task. This was something I missed for a long time. I thought, “students are engaged, and there’s a lot of god thinking that could come out of this task”, and left it there. Now, I’m trying to think more about how to harness that engagement to make sure all students do exactly the thinking I’m interested in.