http://ies.ed.gov/ncee/wwc/PracticeGuide.aspx?sid=1

But it’s also made me persistent in trying 3-Acts and other creative methods, because it gives me more levers to adjust if students seem engaged but the learning doesn’t seem to “stick”.

Here’s a depressing example from my own classroom:
2 years ago I was videotaping my lessons for my masters thesis on Accountable Talk, a discourse technique. I needed to kick off the topic of inverse functions, and I thought I had a good plan. I wrote down the formula A = s^2 for the area of a square and asked students what the “inverse” of that might mean (just intuitively, before we had actually defined what an inverse function is). Student opinions converged on the S = SqRt(A). I had a few students summarize and paraphrase, making sure they specifically hit on the concept of switching input and output, and everyone seemed to be on board. We even did an analogous problem on whiteboards, which most students got correct. Then I switched the representations and drew the point (2, 4) point on a coordinate plane. I said, “This is a function. What would its inverse be?” I expected it to be easy, but it was surprisingly difficult. Most students thought it would be (-2, -4) or (2, -4), because inverse meant ‘opposite’. Eventually a student, James (not his real name), explained that it would be (4, 2) because that represents switching inputs and outputs. Eventually everyone agreed. Multiple students paraphrased and summarized, and I thought things were good.

Class ended, but I felt good. The next class, I put up an similar problem to restart the conversation. If a function is given by the point (3, 7), what’s the inverse of that function? Dead silence for a while. Then one student (the top student in the class) piped up: “I don’t remember the answer, but I remember that this is where James ‘schooled’ us last class.” Watching the video of that as I wrote up my thesis was pretty tough.

But at least I had something to fall back on. I decided it was a case of too much cognitive load–they were processing the first discussion as we were having it, but they didn’t have the additional working memory needed to consolidate it. If I had attended to cognitive needs better, the question about (2, 4) would have been easier, and I should NOT have switched representations from equations to points until it seemed like the switch would be a piece of cake.

I also think knowing the CLT research has made me realize how much more work I need to do to spiral in my classrrom.

Factor (c) is working memory load. The main idea is found in this quote from the Sweller paper Dan linked to above, Why Minimal Instruction During Instruction Does Not Work [3]: “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.” The key here is that when your working memory is being used to figure something out, it’s not actually being used to to LEARN it. Even after figuring it out, the student may not be quite sure what they figured out and may not be able to repeat it.

Does this mean asking students to figure stuff out for themselves is a bad idea? No. But it does mean you have to pay attention to working memory limitations by giving students lots of drill practice applying a concept right after they discover it. If you don’t give the drill practice after inquiry, students do WORSE than if you just provided direct instruction. If you do provide the drill practice, they do better than with direct instruction. This is not a firmly-established result in the literature, but it’s what the data seems to show right now. I’ve linked below to a classroom study [4] and a really rigorously-controlled lab study study [5] showing this. They’re both pretty fascinating reads… though the “methods” section of [5] can be a little tedious, the first and last parts are pretty cool. The title of [5] sums it up: “Practice Enables Successful Learning Under Minimal Guidance.” The draft version of that paper was actually subtitled “Drill and kill makes discovery learning a success”!

As I mentioned in the other thread Dan linked to, worked examples have been shown in year-long classroom studies to speed up student learning dramatically. See the section called “Recent Research on Worked Examples in Tutored Problem Solving” in [6]. This result is not provisional, but is one of the best-established results in the learning sciences.

So, in summary, the answer to whether to use inquiry learning is not “yes” or “no”, and people shouldn’t divide into camps based on ideology. Still unanswered question is the question when to be “less helpful” as Dan’s motto says and when to be more helpful.

One of the best researchers in the area is Ken Koedinger, who calls this the Assistance Dilemma and discusses it in this article [7]. His synthesis of his and others’ work on the question seems to say that more complex concepts benefit from inquiry-type methods, but simple rules and skills are better learned from direct instruction [8]. See especially the chart on p. 780 of [8]. There may also be an expertise reversal effect in which support that benefits novice learners of a skill actually ends up being detrimental for students with greater proficiency in that skill.

Okay, before I go, one caveat: I’m just a math teacher in Northern Virginia, so while I follow this literature avidly, I’m not as expert as an actual scientist in this field. Perhaps we could invite some real experts to chime in?

REFERENCES:

[3]. http://projects.ict.usc.edu/itw/gel/Constructivism_Kirschner_Sweller_Clark_EP_06.pdf

[4] http://www.educationforthinking.org/sites/default/files/pdf/04-03Direct%20InstructionVsDiscovery.pdf

[5] http://act-r.psy.cmu.edu/papers/896/brunstein.pdf

[6] http://pact.cs.cmu.edu/pubs/SaldenEtAl-BeneficialEffectsWorkedExamplesinTutoredProbSolving-EdPsychRev2010.pdf

[7] http://www.cs.cmu.edu/afs/cs.cmu.edu/Web/People/bmclaren/pubs/KoedingerEtAl-IsItBetterToGiveThanToReceive-CogSci2008.pdf

[8]
http://pact.cs.cmu.edu/pubs/Koedinger,%20Corbett,%20Perfetti%202012-KLI.pdf

While people tend to debate which is better, inquiry learning or direct instruction, the research says sometimes it’s one and sometimes the other. A recent meta study found that inquiry is on average better, but only when “enhanced” to provide students with assistance [1]. Worked examples actually can be one such form if assistance (e.g., showing examples and prompting students for explanations of why each step was taken).

One difficulty with just discussing this topics that people tend to disagree about what constitutes inquiry-based learning. I heard David Klahr, a main researcher in this field, speak at a conference once, and he said lots of people considered his “direct instruction” conditions to be inquiry. He wished he had just labelled his conditions as Condition 1, 2, and 3 because it would have avoided lots of controversy.

Here’s where Cognitive Load Theory comes in: effectiveness with inquiry (minimal guidance) depends in the net impact of at least 3 competing factors: (a) motivation, (b) the generation effect, and (c) working memory limitations. Regarding (a), Dan often makes the good point that if teachers use worked examples in a boring way, learning will be poor even if students cognitive needs are being met very well.

The generation effect says that you remember better the facts, names, rules, etc that you are asked to come up with on your own. It can be very difficult to control for this effect in a study, mainly because its always possible that if you let students come up with their own explanations in one group while providing explanations to a control group, the groups will be exposed to different explanations, and then you’re testing the quality of the explanations and not the generation effect itself. However, a pretty brilliant (in my opinion) study controlled forvthis and verified the effect [2]. We need more studies to confirm. Here is a really portent paragraph from the second page of the paper: “Because examples are often addressed in Cognitive Load Theory (Paas, Renkl, & Sweller, 2003), it is worth a moment to discuss the theory’s predictions. The theory defines three types of cognitive load: intrinsic cognitive load is due to the content itself; extraneous cognitive load is due to the instruction and harms learning; germane cognitive load is due to the instruction and helps learning. Renkl and Atkinson (2003) note that self-explaining increases measurable cognitive load and also increases learning, so it must be a source of germane cognitive load. This is consistent with both of our hypotheses. The Coverage hypothesis suggests that the students are attending to more content, and this extra content increases both load and learning. The Generation hypothesis suggests that load and learning are higher when generating content than when comprehending it. In short, Cognitive Load Theory is consistent with both hypotheses and does not help us discriminate between them.”

I will continue in a separate post below..don’t want to lose what I wrote so far.

[1]. http://204.14.132.173/pubs/journals/features/edu-103-1-1.pdf

[2]. https://ad84e4ec-a-62cb3a1a-s-sites.googlegroups.com/site/bobhaus/home/vita/Hausmann2010.pdf?attachauth=ANoY7cpR1nTd3fgRsSfU-UM-j5NS9caTjPi7vXJMc4gTLUnup6KU4xtC6ChZi14wq45EmERQOVj9PeqwTHdViU82XTLzNpEh7CL34pGfoAIJVeDnjVzUKEBqso2ZNKNbiLql0bD3W5haudNbDzGZnXJxCgxz_GGhvy903AsOsP8IQZA_fSK-YVRVX7L5B2EPwCUQbhOa9lBPCQKX79LLiAzZJQcdZh6-HgUHigBFaXFCA55UzVYFde4%3D&attredirects=1

I suppose it’s no surprise I agree with you here. But catch this exchange with commenter Kevin H. He has a good command of the research and seems every ounce a reasonable broker of these ideas. I’ll ping him on this thread and see if he wants to weigh in.

(Even setting aside problem solving, how many worked examples comprise “Algebra 1”? I read a paper that quoted 97, though the citation eludes me. 97 worked examples times 3 instances of each example times 20 practice problems. My word.)

For all the noise about whether inquiry is the best way to learn how to answer mathematical questions, does anyone deny that problem-solving is the best way to learn mathematical problem-solving?

Wouldn’t the worked-examples/practice model, when applied to learning how to solve new problems, involve receiving an explicit model of how to solve a new problem and then being given the chance to engage in lots of genuine problem-solving?

And don’t the CCSS require that kids learn how to solve new problems?

Jeremy: However, much of what I read from Sweller is certainly compatible with your approach and my own general lean towards inquiry.

Thanks. I doubt Sweller would agree, though. Two of his titles from the last five years:

Why Minimal Guidance during Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching

And in case you’re wondering if that paper is too general to apply to math education in particular:

Teaching General Problem-Solving Skills Is Not a Substitute for, or a Viable Addition to, Teaching Mathematics

They’re very clear that if it’s anything less explicit than worked examples, it isn’t good learning.

As I read your post I was caught off guard by the authors of the article, Sweller and Cooper. I was caught off guard as I much of my graduate research (in physics education) was based on some of Sweller’s work. After some deep belly-breathing, I remembered that I often had my own conflicts with Sweller, both in what he said and the general lab rat style of educational research. In particular I remember an article talking about the failure of inquiry based learning…

(pdf) –> http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.169.8810&rep=rep1&type=pdf

My advisor had to talk me down after reading that article. I felt as if the rug had been pulled out from under me.

However, much of what I read from Sweller is certainly compatible with your approach and my own general lean towards inquiry. I was often in awe or at least disbelief in the general willingness of researchers in general to leap to conclusions based on 10 or 20 paid volunteers.

My research was designed around taking some of the ideas from Cognitive Load Theory (John Sweller, being a key player there) and applying it to a large classroom (200+ college students) over a longer period of time (half a semester). Unfortunately, I was only in for the junior graduate degree (masters) and didn’t get a chance to refine and repeat. We saw potentially great results (nearly 7-8% improvement, almost a letter grade!) but still nowhere near lab quality results…

This year I successfully brought in “completion problems” into my physics classroom. (I have not tried them with math. Which might be a different beast.) The ability for my students to solve problems, traditional and no so much so, seems to be greatly improved, but it is a different group of kids…