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The First Rule of Thoughtful Learning

The first rule of thoughtful learning as I see it is that, short of abuse, pretty much any pedagical technique is sometimes appropriate. The...

Sunday, March 11, 2018

Why I Have a Soft Spot for Inquiry-Based Learning (in Math)

Someday, I am going to get punched for saying, "That's a terribly designed experiment" or "How could they analyze their data this way?" one too many times. My long-suffering colleagues routinely listen to me rant about papers with avoidably confounded variables, uninterpretable multiple regressions (there's a paper in the works) and pseudoreplication in the education literature. If you claim to have data supporting something, I want to look at it, pick it apart and think of five alternative explanations for it. Not surprisingly, I'm a big fan of the Kirscher, Sweller and Clark paper "Why Minimal Guidance During Instruction Does Not Work" (I found this link by typing the title verbatim into Google) and definitely not a fan of discovery learning. However, there is one exception. Sort of.

This exception, as the reader already knows, is inquiry-based learning in math. Specifically, and this is important, it is inquiry-based learning (IBL) in upper division or majors-oriented lower division college math classes. What I am about to say is not meant to apply in any other context.

Inquiry-based learning in college math consists of having students learn math largely or entirely by working through sequences of problems and proofs. The version in which all proofs are done by the students, who are not allowed to use any outside references, is often called the Moore method. Less pure versions also exist and seem to be more commonly used.


The basic framework of an IBL math course has students work on proofs outside of class. Class sessions consist mainly of having students present their work and other students critiquing it as necessary. The instructor provides the problems or theorem statements and a bit of guidance during discussions but otherwise stands back.

An IBL math class embodies thoughtful learning in a way few other teaching methods in any subject do. Thinking is the entire point. Furthermore, since most of the actual work is done outside of class, students have time to think deeply and to grapple with serious problems. There is an alternation between solitary and communal thinking that takes advantage of the strengths of both -- the concentration possible alone and the error-checking and fresh viewpoints provided by others. Indeed, this is how real scientific collaborations often work.

Also, IBL fits its subject in a way that is rarely possible in the sciences. (It does bear some resemblance to seminars and writing workshops in the humanities.) While it may not be (and probably isn't) the most effective way to teach specific mathematical content because of the high cognitive load imposed by figuring out a proof and the very real possibility of proving something without understanding it, if teaching a particular way of thinking is an important goal, IBL succeeds admirably.

There is another, idiosyncratic reason why I have a soft spot for IBL. A few years ago, I was sent to a week-long workshop on the subject. Of that week, no more than 20-30 minutes were devoted to reviewing research, most of which was on active learning in general. The rest was looking at implementation and the details of what actually happens in the classroom. Rather than using bad data to try to show that a particular method of teaching was best, the workshop leaders in effect said, "Here's a way to teach we think is good and here are ways to do it". In keeping with the (apocryphal but frequently misattributed to Mark Twain) principle that "It ain't what we know that gives us trouble, it's what we think we know that just ain't so," no data can be better than bad data because it doesn't cause false confidence in the way bad data does. Perhaps grist for a future post?
Read more at: https://www.brainyquote.com/quotes/mark_twain_109

3 comments:

  1. I also attended that IBL workshop. Remember our stroll through the architecture graveyard?

    Here you're not commenting one way or another on lower-level, non-major math courses. But FWIW, IBL-ish methods can work there. There are no proofs, since the students don't know what those are. But given the right amount of direction, the students can work out the details of key examples and counterexamples, and many of them will get to the point where they can solve the kind of problems that cannot be done simply via pattern matching. As one of the luminaries put it, "It doesn't go as well as you hope, but it goes better than you expect."

    I learned many things at that workshop, only one of which I've since needed to unlearn.

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    1. Great to hear from you and thanks for providing my first non-Facebook comment! I remember that exploration very well.

      I do have doubts about IBL as a way of teaching actual content to students below the college math major level. Some can learn from such a setting, but many won't make contact with the idea to be learned or will be so overwhelmed (cognitive load) that they solve the problem but don't generalize from it. And if students do learn, do the advantages justify the extra time? Greg Ashman writes a lot about this on his blog. https://gregashman.wordpress.com/2018/03/16/inquiry-learning/ Check it out and let me know what you think.

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  2. Took me a while to get back here, as I lost the link, and I couldn't find it in your FB profile. The criticism you link to sounds plausible. However, I haven't had to think about IBL-in-lower-level-math all that deeply because I've never applied it. The "ish" in "IBL-ish" above is significant. In my case, it implies having some lectures, just shorter ones than would be typical. (For example, if it turns out that a significant number of students display a particular misconception during their work, then I can give a 5-minute talk on it on the spot.) Ashman calls this "active learning", and he considers it distinct from IBL, as do I.

    Apropos of nothing, a few months after we met, I found myself in a wheelchair for a while, and in a country with a different notion of accessibility, and so became even more impressed with your ability to get around on grassy hills.

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