Let me highlight another conversation from the comments, this time between Kevin Hall, Don Byrd, and myself, on the merits of direct instruction, worked examples, inquiry learning, and some blend of the three.
Some biography: Kevin Hall is a teacher as well as a student of cognitive psychology research. His questions and criticisms around here tend to tug me in a useful direction, away from the motivational factors that usually obsess me and closer towards cognitive concerns. The fact that both he and Don Byrd have some experience in the classroom keep them from the worst excesses of cognitive science, which is to see cognition as completely divorced from motivation and the classroom as different only by degrees from a research laboratory.
While people tend to debate which is better, inquiry learning or direct instruction, the research says sometimes itâ€™s one and sometimes the other. A recent meta study found that inquiry is on average better, but only when â€œenhancedâ€ to provide students with assistance . 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 â€œdirect instructionâ€ 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.
Hereâ€™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 youâ€™re testing the quality of the explanations and not the generation effect itself. However, a pretty brilliant (in my opinion) study controlled for this and verified the effect . We need more studies to confirm. Here is a really portent paragraph from the second page of the paper: â€œBecause examples are often addressed in Cognitive Load Theory (Paas, Renkl, & Sweller, 2003), it is worth a moment to discuss the theoryâ€™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.â€
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 : â€œInquiry-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.â€ The key here is that when your working memory is being used to figure something out, itâ€™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 donâ€™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 itâ€™s what the data seems to show right now. Iâ€™ve linked below to a classroom study  and a really rigorously-controlled lab study study  showing this. Theyâ€™re both pretty fascinating readsâ€¦ though the â€œmethodsâ€ section of  can be a little tedious, the first and last parts are pretty cool. The title of  sums it up: â€œPractice Enables Successful Learning Under Minimal Guidance.â€ The draft version of that paper was actually subtitled â€œDrill and kill makes discovery learning a successâ€!
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 â€œRecent Research on Worked Examples in Tutored Problem Solvingâ€ in . 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 â€œyesâ€ or â€œnoâ€, and people shouldnâ€™t divide into camps based on ideology. Still unanswered question is the question when to be â€œless helpfulâ€ as Danâ€™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 . His synthesis of his and othersâ€™ 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 . See especially the chart on p. 780 of . 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: Iâ€™m just a math teacher in Northern Virginia, so while I follow this literature avidly, Iâ€™m not as expert as an actual scientist in this field. Perhaps we could invite some real experts to chime in?
Thanks a mil, Kevin. While weâ€™re digesting this, if you get a free second, Iâ€™d appreciate hearing how your understanding of this CLT research informs your teaching.
The short version is that CLT research has made me faster in teaching skills, because cognitive principles like worked examples, spacing, and the testing effect do work. For a summary of the principles, see this link.
But itâ€™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 doesnâ€™t seem to â€œstickâ€.
Hereâ€™s a depressing example from my own classroom:
Two 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 â€œinverseâ€ 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, â€œThis is a function. What would its inverse be?â€ 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 â€˜oppositeâ€™. 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), whatâ€™s the inverse of that function? Dead silence for a while. Then one student (the top student in the class) piped up: â€œI donâ€™t remember the answer, but I remember that this is where James â€˜schooledâ€™ us last class.â€ 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 loadâ€“they were processing the first discussion as we were having it, but they didnâ€™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 classroom.
Then in another thread on adaptive math programs:
My intention was to respond to your critique that a computer canâ€™t figure out what mistake youâ€™re making, because it only checks your final answer. Programs with inner-loop adaptivity do, in fact, check each step of your work. Before too long, I they might even be better than a teacher at helping individual students identify their mistakes and correct them, because as as teacher I canâ€™t even sit with each student for 5 min per day.
I have only a modest amount of experience as a math teacher; I lasted less than two years â€” less than one year, if you exclude student teaching â€” before scurrying back to academic informatics/software research. But I scurried back with a deep interest in math education, and my academic work has always been close to the boundary between engineering and cognitive science. Anyway, I think Kevin H. is way too optimistic about the promise of computer-based individualized instruction. He says â€œIt seems to me that if IBM can make Watson win Jeopardy, then effective personalization is also possible.â€ Possible, yes, but as Dan says, the computer â€œstruggles to capture conceptual nuance.â€ Success at Jeopardy simply requires coming up with a series of facts; thatâ€™s highly data based and procedural. The distance from winning Jeopardy to â€œcapturing conceptual nuanceâ€ is much, much greater than the distance from adding 2 and 2 to winning Jeopardy.
Kevin also says that â€œbefore too long, [programs with inner-loop adaptivity] might even be better than a teacher at helping individual students identify their mistakes and correct them, because as as teacher I canâ€™t even sit with each student for 5 min per day.â€ Iâ€™d say itâ€™s likely programs might be better than teachers at that â€œbefore too longâ€ only if you think of â€œidentifying a mistakeâ€ as telling Joanie that in _this_ step, she didnâ€™t convert a decimal to a fraction correctly. Itâ€™ll be a very long time before a computer will be able to say why she made that mistake, and thereby help her correct her thinking.