I opened this semester using a task for my elementary math methods course that I’ve used many times before. Each time I present it I’m struck by how engaged the PSTs are. They persevere for an extended period of time and I frequently hear that they continued to think about the task and work on it after the class has been dismissed. From an engagement standpoint, the task isn’t all that impressive. In fact, the side of me that really loves socio-cultural perspectives on learning gets mad every time I use this task. It’s not “real-life”, I mean, not really. It’s not a situation they’re going to run into again, thus giving students prior knowledge that would transfer. So why was it engaging? Why did they persevere? As I share the task I encourage the reader to avoid making a “Here’s why it’s engaging” list or “Here’s why they persevered” list. I feel like we all know what those lists would look like and that they may ultimately be hollow. Instead, I’m interested in how you might alter the task in a way that would kill engagement or cause persistence to falter. It’s a weird goal, I know. Attending to what hurts persistence can tell us just as much as attending to what helps it. Shoot for tiny alterations. The most minute alterations would tell us a lot about lesson design. In my experience, it’s the small differences that make it so one lesson thrives while another bombs. I mean, from one day to the next I don’t say, “You know, I think this technology thing is a fad and I don’t really want to build it into my lessons.” Telling me that technology use increases engagement and can support perseverance is the sort of broad-strokes advice that isn’t super helpful. I get it. That’s not a tool I’m likely to mistakenly toss aside. I’m interested in those small tweaks that I may not even be aware of. Bringing those to the surface is what improving teaching is all about.

So, here’s the task.

**Launch:** I start by telling a (mostly) fabricated story about watching my little niece play with blocks. I notice that she was building a pattern, as kids are wont to do, so I watched more closely. I present the pattern as my adult brain made sense of it. As a pattern that started with a single blue block and then grew outward in concentric rings. I label each new ring a “step”. I use a simple presentation to show the first 3 steps as they develop.

I ask students to list any questions they might have if they were watching this pattern unfold, giving them a voice in the problem formulation. If they didn’t ask it, I’ll usually ask a question about whether or not my niece would run out of blocks. I then ask PSTs to find how many total red blocks would there be in the entire figure once step 5 has been completed. I generally have multi-link cubes, tiles, and graph paper available. I let students work alone or in groups.

*Side note:* My elementary math methods students generally don’t have go-to strategies regarding linear or quadratic functions, so this task tends to put them at the edge of their comfort level without pushing them overboard.

**Explore:** As students problem solve, I walk around asking questions about how they found their answer, making note of the different strategies they tried. I only redirect their thinking if I think it will lead to unproductive failure. I’m more thank okay with incorrect answers, but some incorrect approaches do not support productive struggle.

**Discuss:** I choose students to share with the attempt to build toward greater efficiency. So, if I had a student who built the whole figure and then counted each red block, I have them go first. I ask if that strategy will always work. We talk about strengths of the approach as well as any potential obstacles. A student who built it (or drew it) and only counted up a quarter of the figure and then mulitplied by 4 might go next. We identify similarities and differences, strengths and challenges. Some students start noticing patterns, we get those out there. Some start tracking information through tables, we talk about those. Pretty foundational stuff.

We then repeat the launch, explore, discuss cycle 2 more times. During the second cycle they answer the same question for step 10 (does it work just to double step 5?) and on the third, they tackle step 100. Each Discuss section involves making connections, justifying work, asking questions to presenters (student to student interactions), and challenging one another’s thinking.

There are quite a few fits and starts. Strategies are tried and then abandoned. No one builds step 10 in its entirety. Many find that each new row of red grows by 8 blocks. That makes it easy to track the new red and then add each layer together. It’s fairly efficient for 10 steps, but not 100. By step 100 they either need an algebraic expression (formal or informal), or enough patience to number crunch.

So… Nothing special. Students really persist, though. Why? Is it just because they’re wonderful students? I mean, probably, but that can’t be the only reason.

So, let’s hear those alterations. Shoot for boring. What would shift this from a dynamic experience to a mundane or tedious one? I’ll start the ball rolling with some obvious, broad-strokes alterations.

- Jump right to step 100. Skip the steps 5 and 10 cycles. By requiring abstraction right off the bat I would lose all of my less confident students. Frustration would replace persistence.
- Remove the tools and manipulatives. This would also implicitly communicate that the only type of thinking that is valued is abstract representation.
- Expect all students to work independently, because, you know, who cares about Vygotsky? No need for all this ZPD stuff.

These would surely be terrible alterations, but they’re also not ones I’m likely to make. Can you think of subtle changes that could really impact engagement and persistence?

*Other thoughts about the task generally:*

Things I don’t like about the task:

- The problem is not compelling. If she runs out of blocks, who cares? Is this something that really needs to be solved?
- As far as PSTs’ mathematics is concerned, it doesn’t function well as a stand-alone experience. Students do not gain deep insights into different functions. They would need follow up experiences to deepen and connect their knowledge.

Reasons I’ll likely use it again next semester:

- Did I mention students persisted?
- It gives me great insights into their problem-solving approaches. Some students needed very concrete approaches. Some loved to delve into the abstract but got lost in their own thinking. Some were brilliant, but lacked confidence. As a formative assessment piece, it was very rich.
- It engages students in a number of important mathematical practices.