spectrum of struggle – emergent math

It’s been a while since I’ve revisited the Taxonomy of Problems I threw together a while back, but I think it’ll be helpful to spend some time there when considering the following Most-Wanted question around Problem-Based Learning:

At what point after allowing the students to work on a problem do I scaffold the content knowledge?

It’s probably important to identify exactly what type of problem you’re implementing before deciding this.

One of the reasons I wanted to think about this as a potential framework is to address scaffolding (I’ve already addressed assessment). It might not be perfect or precise, but here’s what I basically envisioned.

Unintentionally, this kind of mirrors the ideal progression of both a PrBL Unit as well a classroom and high school experience.

So once you’ve figured out where you are on the taxonomy, where you are in the unit, you can think about your scaffolding.

What & When

I’ll toss out a couple broad-brush rules that oughtn’t be universally applied.

When you’re at the left end of the spectrum – the Content Learning Problems, I’d suggest the following.

If the need for the content is germane to the problem, intervene relatively quickly and with the entire class.

If the need us for an ancillary concept or “side-topic”, consider holding back and/or offering small, differentiated workshops.

For example, I threw Dan’s Taco Cart task into my unit on Linear Equations.

However, use of the Pythagorean Theroem is required to develop your linear equations to model. There will no doubt be a need for some – probably not all – students to revisit or relearn the Pythagorean Theorem. That is ancillary content knowledge: essential, but not the targeted content knowledge skill. Consider holding off on scaffolding that – another groupmate might be the better vessel to explain the concept. Or, if you deem yourself the ideal vessel, consider jigsawing that concept or holding a small pullout workshop with one groupmember per group (the groups’ “student-teacher liaison” as it were).

If the knowledge is germane and is the targeted content knowledge of the task, the scaffolding might need to be more prescriptive, more whole-group. You certainly could lecture (Grant Wiggins has an exceptional post on that), but you could also offer one of these scaffolding tasks. I’m a huge fan of manipulatives and students evaluating student work samples.

Ah, but when do you offer that scaffolding? How much productive struggle should we allow students before intervening? This is where teaching is more of an art than a science. Although if it is truly germane to the problem and it’s a Content Learning problem, I’d err on the side of quick-intervention. Twenty minutes after a problem is launched, perhaps? Thirty?

More important than a time demarcation for instruction is probably some classroom behavioral evidence. Here’s a short list of things to look for to initiate INSTRUCTION MODE:

Over half the groups or students asking the same or similar thing
Loss of cognitive demand in the attempted solutions
Attempted solutions going totally off the rails

What have I missed? What are some indicators that it’s time for you to intervene with scaffolding? Or do you have a particular system or time-frame when considering when to cease the productive struggle time?

If your problem is more to the right on that arrow above – Exploratory or Conceptual Understanding problems – the question might not be “what and when” to scaffold but “if”. There is inherent value in an unscaffolded, nonroutine, “ill-structured” problem with a lugubrious associated standard. For these problems consider restricting yourself solely to small workshops devoted to ancillary content knowledge. Or perhaps follow up the problem with a standalone scaffolding task – perhaps, again, a manipulative or evaluation of work samples. Scaffolding for Assessment problems should focus on revision and peer-editing.

The tension between inquiry and instruction shifts from day-to-day, problem-to-problem, so I wouldn’t hold anyone to a hard-and-fast rule. I hope you’ve appreciated my self-indulgence as I continue to try to figure this out and establish a few basic tenets of solid PrBL practice. As always, feedback and commentary is appreciated.

This NPR radio spot confirms much of what we already know about struggle. There’s so much good stuff in this report, I’d encourage you to go listen to it or read it. Here are a couple nuggets I found particularly illuminating (emphasis mine).

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For example, Stigler says, in the Japanese classrooms that he’s studied, teachers consciously design tasks that are slightly beyond the capabilities of the students they teach, so the students can actually experience struggling with something just outside their reach. Then, once the task is mastered, the teachers actively point out that the student was able to accomplish it through hard work and struggle.

“And I just think that especially in schools, we don’t create enough of those experiences, and then we don’t point them out clearly enough.”

…

“We did a study many years ago with first-grade students,” he tells me. “We decided to go out and give the students an impossible math problem to work on, and then we would measure how long they worked on it before they gave up.”

The American students “worked on it less than 30 seconds on average and then they basically looked at us and said, ‘We haven’t had this,’ ” he says.

But the Japanese students worked for the entire hour on the impossible problem. “And finally we had to stop the session because the hour was up. And then we had to debrief them and say, ‘Oh, that was not a possible problem; that was an impossible problem!’ and they looked at us like, ‘What kind of animals are we?’ ” Stigler recalls.

“Think about that [kind of behavior] spread over a lifetime,” he says. “That’s a big difference.”

…

Great, teachers say, I know students learn from struggling through a complex problem, but my kids won’t do that. They won’t participate in productive struggle. They don’t come pre-loaded with productive struggle software. Therefore, the thinking goes, I need to address the kids where they’re at in terms of willing to struggle through a task.

Talk to any math teacher, and one of their frustrations will no doubt be that students give up too quickly on a complex problem. Mathematics is unique among all subject areas in its potential for cultivating a positive attitude toward struggle. It’s also the subject that tends to shut kids down for that very reason.

In other words, it’s a struggle to get kids willing to struggle.

And it’s one we need to struggle through.

Teachers, particularly math teachers, know that most learning occurs in the midst of struggle and overcoming struggle. But all students’ life they’ve been assessed and encouraged to solve rote problems quickly. How can we undo years of low-complexity problems? How might we develop students’ desire, ability, and disposition to struggle? What should students be doing? What should teachers be doing? What kind of tasks should kids be engaged with to encourage positive struggle behavior?

Even though dubbing this the “East” vs. “West” mindset is lazy terminology, it does conjure up a sort of spectrum of student and teacher attitude toward struggle. How can we nudge students (and teachers) a bit eastward on the spectrum? What kinds of tasks can we provide for kids so they may learn persistence.

* – “My Favorite No” refers to this video from TeachingChannel.org.

** – “The Mistake Game” refers to this post from Kelly O’Shea.

While, as the article cited suggests, I hate to place a value judgment on the East-West mindsets presented. That said, I don’t think we’re particularly lacking of Western-style classrooms. Do you have particular tasks that help kids get away from a Western attitude toward struggle and smarts? Even if it nudges kids just a bit further “eastward”, please share. Because we all know that struggling and persisting are good things, but we’re not always sure on how to engage students in this manner.

Category: spectrum of struggle

When to scaffold, if at all

The Struggle for Productive Struggle