Thinking about critical thinking

Cognitive scientist Daniel Willingham talks about critical thinking at the Harvard Initiative for Learning and Teaching.

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  1. Exhibit A for the defense of leaving classrooms to the teachers (and arguing to keep college theorists and their theories from becoming mandates)

    Aside: Help me out here with his supporting example (~6:35 min); “If someone comes to you and they [sic] have a positive mammography, what are the odds that this person has breast cancer?” Hmmm She HAS a positive mammography and either has or doesn’t have breast cancer – if she doesn’t have it, then she’s part of the 10% with a false positive. We weren’t asked what are the odds of ANY woman at age 40 going in and getting a positive mammography (which would be the base rate of 1%). I’m no mathematician and his explanation was not satisfactory to me and seemed presumptuous.

    • The properly phrased question is as follows:
      A randomly chosen women tests positive; what is the probability she has cancer?

      • “A randomly chosen women tests positive; what is the probability she has cancer?”

        This is definitely much more likely to get me to name the base rate as the answer, but as a non-statistician trying to answer this question the best I can based, I’m still stuck here, because according to the prof the statistics only apply to women who get the test – not ALL women – and I would think that there are a great number of women out there not getting the test, because they have no history and/or no desire to take the test, which, to me, would make the base rate for ALL women to be something less than 1 percent.

        Honestly, how does his analysis of “critical thinking” help teachers? The audience looks bored,confused, and unimpressed. They brighten up when they can talk amongst themselves. I’d love to listen to them figuring out what he wants to hear.

    • I don’t know if I would call his explanation of this problem presumptuous, but it was confusing. This was compounded by the fact that he never does share the correct answer. The odds that the 40 year old woman with a positive mammography actually has breast cancer will be about 7.5%.

      The quail analogy doesn’t really apply here. What we are looking at is the application of Baye’s Rule in statistics. But you don’t have to be a statistician to get this. Just think about the numbers involved. Imagine you have a group of 1000 forty year old women.

      1) About 1% of women at age 40, will have breast cancer, so 10 of those 1000 women will have breast cancer and 990 will not.

      2) 8 of the 10 with breast cancer will test positive.

      3) Because 10% of women who test positive have a false positive, 99 of the remaining 990 women who do not have breast cancer will have a false positive.

      • Richard Aubrey says:

        Part of the real-world question depends on the seriousness of the issue. WRT cancer, the next step in diagnosis is imperative. Same numbers regarding something unimportant, no big deal.
        Now, if you were doing a “story problem” just to see if a person can set it up, that’s another issue.

  2. What you’re looking for in the cancer example is Bayes Theorem. The math is just simple algebra, but the theorem itself is so counter-intuitive as to be difficult to understand. See:'_theorem

    The birding example is a poor one, because there are all sorts of birds whose ranges don’t match up to their names.

    • “the theorem itself is so counter-intuitive as to be difficult to understand”

      I couldn’t agree more! This type of theory-thinking is fine between people who find this stimulating, but as for real-world application in the classroom? What a waste of time and effort (and $$$$) trying to apply it broadly to curriculum development and specifically to classroom teaching!

      • Ann in L.A. says:

        I don’t think that was his point. The point was that we want to get students to know enough about the world that they have the tools to think critically. The case where a professor gave the same test before and after the semester was interesting, because it showed how hard it is to get students up to a competent level. The obscure statistics question was there just to show how hard it could be, not to suggest that everyone should be able to answer such questions. He mentioned that such questions were a big problem in the medical field, and that medical schools were working on teaching how to deal with numbers like this, but getting med students and doctors competent is actually difficult.

        That said, people are woefully educated when it comes to statistics. It’s such an important subject–everyone hears statistics almost every day–and yet it isn’t part of most school’s math curriculum–even in college prep, or is it especially in college prep–or a required stand-alone class. One semester of statistics in high school would at least help people to ask basic questions when they hear something like: eating xxxxx will double your risk for yyyyyy cancer! The obvious question there would how prevalent is the cancer to begin with. I would love if people were taught how to ask critical questions about the statistics they read in the news every day.

        • Roger Sweeny says:

          That’s a great idea but (at least) two things hold it back.

          1. There is no college course like it, and high school courses are basically college courses modified for younger students. College math departments give statistics courses that are basically math, the theory behind statistics. Other departments may give “Statistics for ______” courses, showing how people in that field do research. Neither is what you are thinking of.

          2. Related is the question of what department should offer such a course and who should teach it. The math department is the obvious candidate but you very much don’t want the course to involve any advanced math or to emphasize mathematical techniques. The first will lose most of the students. The second will not get them practice in asking critical questions. The teacher should be someone with a broad knowledge of lots of good and bad research. That person may be in the science or social studies department.

    • JaneC: I disagree. Bayes Theorem is quite useful in many disciplines. In fact, that “Data Analytics” you hear so much about when reading about “Cloud Computing” often uses Bayesian statistics and methods. The spam filter your email provider uses may be based on Bayes Theorem. It’s vital when analyzing drug testing results. It shows up in a lot of places these days.

      That said, it seems to me more appropriate for a college course, not so much a high school course. We are, after all, talking about theory started by Reverend Bayes in the 1700s and extended by Laplace in the 1800s. Not exactly cutting edge stuff…