And I am not talking about the specific content standards. Even if we accept the Common Core as a perfect document (which the authors certainly never intended), we still need to make decisions about emphasis and about how we understand mastery. I think this conversation will be much more grounded if we look at specific learning outcomes and discuss the teaching strategies in that context. I suspect that there will be a surprising amount of common ground in approaches once we decide on a particular learning outcome.

It really seems to be more about the problems than the methodology.

An an example, I give a question like this as part of my summative assessment near the end of the year. There is no additional data provided, students are assessed with a rubric according to the competency of their explanation of their reasoning.

—
The population of bacteria in a sample is estimated over a period of several hours. You have access to the population data. We would like to build a mathematical model of the bacteria population so that we can make predictions.

We’ve worked with a variety of types of functions this year, including these eight: linear, quadratic, cubic, indirect, square root, cube root, exponential and logarithmic. How would you choose which of these functions or combination of functions to use to model the scenario? What justifications would you use?
—

Some students immediately respond with exponential and give competent arguments for why. But many of the functions could be appropriate choices, depending on the data. There are many different aspects of the real world that may be affecting the real world data that a competent student will consider and discuss.

I am not looking for one particular answer. I do not think that a problem is an authentic modeling problem if there is a single correct answer. I think we get good at modeling by working on modeling-ish problems. And oftentimes, DI is an appropriate technique for teaching modeling-ish problems. But sometimes students need an opportunity to solve real modeling problems. And the process of solving any modeling problem gets outside the bounds of strict DI. It has incongruity and casual bridging inferences written directly in to the problem.

In other words, to say that a class exclusively uses explicit instruction is to say that the class doesn’t cover modeling.

(An exclusive non-explicit instruction class is problematic for its own reasons but no one is arguing for that here.)

Just sayin’.

The fact that Schwartz and Martin were able to replicate their own study with different teachers tells us something. However, it probably does not tell us as much as a replication by other researchers who may have used a different method / control.

“The science of learning” is an interesting phrase to use. Some of those who I tend to disagree with would dispute that such a thing is even possible. They would call it “positivism”. However, I do think that we have learnt a lot. This does not necessarily mean that teachers from the 1960s are all very different from teachers today. And it does not mean that we should dismiss a whole body of experimental research just because it is a bit old. We don’t do that with any other science.

The PF studies tend to utilise a context that is particularly suited to PF e.g. standard deviation. By asking ‘which is the most consistent baseballer?’ we have a question that students can comprehend and attempt, even if they don’t know the canonical method. It would be hard to think of an equivalent situation in many abstract areas of maths (although I suspect that Dan might dispute this).

Your comment here surprises me. I thought I understood your position as “all explicit, all the time when learning something new.” (Not the same as “all explicit, all the time.”) Here you seem to have carved out a genre of content where other instructional strategies are more effective than EI. That’s interesting.

As you guessed, I think the category of “questions that students can comprehend and attempt, even if they don’t know the canonical method” is quite large, from subtraction (Saxe) through linear functions (Moschkovich) and on. My dissertation research was a variant on IPL and was purely abstract. Just coordinates, no context. Students can understand each of these concepts informally and the key in IPL is to “transfer in” that knowledge to help the students learn from new instruction that’s explicit and formal. (“When preparing students to learn,” write Schwartz & Martin, 2004, p. 132, “the instructional challenge is to help students transfer in the right knowledge.”)

However, the process-product research shows that a form of explicit instruction was used by the more successful teachers. Nothing like PF emerges. This could be either because no teachers thought of using such a model at this time, some used the model but it was less effective or some teachers used it effectively but not at a scale captured by the research. It may be that well-designed PF works in the lab but is hard to implement at scale.

Could be:

- The science of learning has advanced a great deal since the 1970s.
– Better pedagogy is harder pedagogy.

If it’s that second case, I’m happy to start making compromises. I wouldn’t want to make the best pedagogy the enemy of good pedagogy.

But Schwartz & Martin recreated the results they saw from their first study (in which they taught the classes) with other teachers of different pedagogical abilities and preference. The results replicated.

You are correct about my tentative theory. It’s a hunch.

1. Regarding the use of low guidance, I do not necessarily recommend PF alone. I do, however, think that PF tasks could be coupled with metacognitive instruction (read the JOEP paper above). Giving metacognitive prompts and problem-solving activites has excellent learning outcomes. The intervention I linked above was brief (several hours) and resulted in far transfer and increased motivation. Of course, I would always like to see more empirical evidence. It is also unclear how much metacognitive training is optimal. My guess is at most 15-20% of a school year.

3. It may be that well-designed PF works in the lab but is hard to implement at scale.

Some work on MetaCognitive Tutors may be helpful http://bit.ly/1UB3i78

This has been a nice chat. I’m currently working on a master’s thesis in this area. Let me know if you would like to discuss this further through email (vpletap@outlook.com).