During Dan’s series of lesson makeovers last summer, I intended to write up a critique of the Least Common Multiple makeover he and Dave Major created. Dan’s write-up seemed to illustrate his view of the respective roles of inquiry and direct instruction, and I’ve always thought it showed a misinterpretation of the cognitive research that cautions against unguided inquiry. I didn’t bother chiming in until Grant Wiggins wrote this blog post, bringing up the issue of cognition, working memory, and inquiry vs. direct instruction again. So here are my thoughts.
For context, Shipping Routes shows students the clip below and asks whether the two boats will ever get back to port at the same time:
Students are then sent to a simulator programmed by Dave Major, allowing them to choose different round-trip times for the two boats and showing their cycles back and forth. Students can discover for themselves when the boats get back in sync, and what that has to do with least common multiples. For example, boats with round-trip times of 2 min and 5 min will be back in sync after 10 min. Sadly, Dave’s simulator seems to be offline right now, so you can’t try it for yourself.
In his write-up, Dan shies away from direct instruction right from the get-go:
I could tell students what to look for here and how to approach the problem. I could show a few worked examples…
Two problems there:
- Some students will need more than just three examples to determine a pattern.
- My selection of those particular examples – that is, my decomposition of the entire solution space into just three categories – did a lot of the intellectual heavy lifting for my students. They need to decide on those three categories and come up with a rule that takes them all into account.
Regular readers of Dan’s blog (cuz, yeah, there really aren’t regular readers of my blog) may know that worked examples are an interest of mine. Here’s how the stuff I’ve learned applies to Shipping Routes.
Science of Memory and Learning
Humans have 2 kinds of memory, working memory and long-term memory, and they function completely differently. Retrieving information from long-term memory is almost effortless, and we can process huge amounts of information from long-term memory simultaneously. In contrast, working memory can only hold on to 4-7 chunks of information, and only for about 30 seconds or so. The word “chunks” is important. For a demonstration, watch the first 1:30 minutes of this video:
There are various theories about how information gets encoded in long-term memory, but the main idea is that the information has to stay in working memory for long enough to get practiced/recalled several times before the working memory dumps it–each practice opportunity strengthens its foothold in long-term memory. Working memory will dump it when new information comes in, so if you keep throwing new information at someone who’s still encoding the previous information, you make it very hard for them to form long-term memories, even if they seem to be processing what you’re saying as you’re saying it. It could have made sense to them when you said it and then disappeared from their mind when you said the next thing. (Sound familiar to any teachers?)
For that reason, inquiry instruction can make it very hard to encode information into long-term memory. As Sweller et al (2006) say in paper that spawned much controversy, “Inquiry-based instruction requires the learner to search a problem space for problem-relevant information. All problem-based searching makes heavy demands on working memory. Furthermore, that working memory load does not contribute to the accumulation of knowledge in long-term memory because while working memory is being used to search for problem solutions, it is not available and cannot be used to learn.”
Grant Wiggins really does not like this paper–much of his most recent post (which inspired me to dust off my keyboard here) is spent picking it apart, particularly the “ludicrous” emphasis Sweller et al place on novice vs. expert learners. I think Wiggins misunderstands the authors on that point. Granted, I’m a layman, but I think the difference between novice and expert learners is pretty simple: isn’t it the difference between someone who’s encoded the information being studied into long-term memory and someone who’s still processing it from working memory?
The strong opinions (not limited to Wiggins by any means) reflect the fact that this theory of memory and cognition points out shortcomings of inquiry learning. As I’ll describe at the bottom of this post, it does not actually mean that you should never use inquiry, but some people on both sides of the debate say it does, particularly when the theory itself is misunderstood.
Critiquing Shipping Routes
How would this play out in Shipping Routes? In order to be successful with Dan’s lesson, students need to try different pairs of round-trip times on the simulator to discover what governs when the boats are back in sync. Students can set times to the tenth of a second, e.g, 3.2 min for the first boat and 4 min for the second boat. These boats would by in sync for the first time after 16 minutes. Would they notice how 16 arises from 3.2 and 4 as the least common multiple?
Let’s give the lesson the benefit of the doubt and say that a teacher would suggest (or have a student suggest) that everyone try whole-number times to start. So one group of students might, for example, try 2 min and 6 min. and discover that the answer, 6 min, is the larger number. At this point, Dan would have the teacher challenge this group to see if that rule always works, with an eye toward finding counterexamples. Dan says,
If a student just tries the first example and says, “It’s easy. It’s always the longer of the two times.” I can then say, “Great. But try that on several more examples and make sure it works.” (It won’t.) Or I can suggest one of the other two categories. But I’d rather not offer those categories before the student has even considered why she might need them, or even the fact that there are different categories.
The other categories are coprime numbers (like 2 and 5) or numbers with a common factor (like 6 and 10). Students have to identify the three situations and find the rule for each, with minimal guidance.
Do you see how this might overtax working memory to the point of inhibiting long-term memory formation…how students might successfully find one or two of the three rules on Monday and then walk in on Tuesday having forgotten what they’d discovered? Or find all three rules, but be unable to remember the first one by the time they find the last one? If you don’t think this would tax their memories, keep in mind that many students have to stop and think or count on their hands just to remember that (3)(6) = 18.
Modifying Shipping Routes
How could Shipping Routes be adapted so it accounted for working memory limitations? Here’s what I would do. Whatever rule students discover first, before encouraging them to find cases where their new rule DOESN’T work and where they need to create a new rule, I’d have them fully digest the math behind the rule that they have found. They’ve just found this rule. It’s fragile, sitting in their memory buffer temporarily–this is our chance to have them encode it into their long-term memory and harden it by connecting it to other conceptual schema already in their brains. (Yes, I do believe strongly in developing conceptual understanding in addition to mere skill development). If we have them switch to a new search for a new rule, we throw away most of the benefit of the discovery they just made.
How would I help them practice it and connect it to other concepts? After students played around with the simulator for a while, I’d present them with text input boxes like this and ask students to fill in the blanks:
Boat 1: 2 min
Boat 2: ______
Time until they’re in sync: _____
A computer could easily analyze student responses and figure out which rule they’ve discovered. Let’s take the example of a student who discovered that for coprime numbers like 2 and 5, you multiply the numbers. Now I’d want to have the students really dig into that rule.
Then I’d ask them to self-explain the connection via drop-down menus, as in Martina Rau’s paper finding that multiple representations didn’t help students learn fractions unless students were prompted to self-explain the connections between the representations. (The picture shows drop-down menus for self-explaining adding fractions, but you could do something similar for explaining why 2 and 5 have a common multiple at 10):
Then I’d have students practice this rule a minimum of 5-10 times, perhaps following the approach of this other paper by Martina Rau.
And then, finally, I’d have the online lesson challenge these students to find a case that breaks their rule, and the learning cycle would begin again.
So When & How Should We Use Inquiry?
As I said above, understanding working memory limitations doesn’t mean you should never use inquiry. Rather than spelling out my views on this here, I’ll just point you all again to Dan’s posting of my thoughts on this topic over at his blog. Incidentally, I did try to convince some researchers in the learning sciences to use Dan’s makeover lessons as a test-bed for studying instructional principles, and I still think that would be a great idea (see my first blog post ever). I mean, researchers, if you want teachers to actually pay attention to your work, why not conduct your studies in the context of the lessons we’re all talking about online?