Discovering an illusion

“Discovery learning” advocates claim Japan’s high-scoring math students come up with original answers without being guided by the teacher. That’s not so, says Alan Siegel, a computer science professor at NYU who’s studied the videotapes of Japanese math lessons. Columnist Linda Seebach explains:

The eighth-grade geometry lesson Siegel discusses is based on the theorem that two triangles with the same base and the same altitude have the same area, and it is framed in nominally “real world” terms as a problem in figuring out how to straighten the boundary fence between two farmers’ fields so that neither farmer loses any land.

. . . The teacher first primes the class by reminding them of the theorem, which they had studied the previous day. Then he playfully suggests with a pointer some ways to draw a new boundary, most of them amusingly wrong but a couple that are in fact the lines students will have to draw to solve the problem (though they aren’t identified as such).

Then he gives the students a brief time, three minutes, to wrestle with the problem by themselves, and another few minutes for those who have figured out a solution based on his broad hints to present it. Then he explains the solution, and then he extends the explanation to a slightly more complex problem, and finally assigns yet another extension for homework.

As Siegel describes it, “The teacher-led study of all possible solutions masked direct instruction and repetitive practice in an interesting and enlightening problem space.

“Evidently, no student ever developed a new mathematical method or principle that differed from the technique introduced at the beginning of the lesson. In all, the teacher showed 10 times how to apply the method.”

A U.S. Department of Education report claims Japanese students devise their own solutions to “mathematics problem employing principles they have not yet learned.” Siegel says analysts who watched the videos were poorly trained. They came up with “10 student-generated alternative solution methods, even though it contains no student-discovered methods whatsoever.”

Discovery learning is fashionable in math reform circles, writes Seebach. The Japanese are supposed to be the models. But the Japanese teach traditionally — with “beautifully designed and superbly executed” lessons.

The videotape shows, Siegel says, that “a master teacher can present every step of a solution without divulging the answer, and can, by so doing, help students learn to think deeply. In such circumstances, the notion that students might have discovered the ideas on their own becomes an enticing mix of illusion intertwined with threads of truth.”

We’re short of master teachers, especially in math.

Update: According to Chris Correa, Japanese teachers believe that students are less serious about learning math.

Overall, students had become weaker in nine of the 11 areas that the survey asked about. The most striking declines in students’ scholastic aptitude were in the ability to calculate, which did not feature in the society’s survey in the mid-1990s, and in logical expression. The most alarming deterioration was among students at teacher-training universities.

That bodes ill.

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  1. This was pretty standard at my university. The prof would show how a certain class of theorem could be proved, and then he would ask us to prove similar theorems that required the same approach, and varying degrees of creative thinking on our part.

    There is a clear difference between this and asking students to come up with entirely original stuff on their own.

    It shouldn’t be too difficult for anyone trained this way to come up with lesson plans. But I suspect our textbooks aren’t written this way.

  2. Bluemount says:

    An interesting teacher can have a profound impact on the lives of their students. Not only do they set a standard of excellence, but an appreciation for the rigors of discipline and humility. True excellence should leave us in awe of the history and growth of field of study.

    The idea there is a creative approach that skips the gruel of hard work is a shabby confidence game. While the creative evolution of society may be improved by change, closer inspection will always find effective change has a stable foundation. What is disturbing is selling the idea that we don’t value the stability gained by hard work.

  3. It’s my understanding that teachers in Japan have much more time to collaborate on lesson plans.

  4. Steve LaBonne says:

    An excellent point. But it’s really too bad that the ill effects of insufficient preparation time and lack of encouragment for collaboration are exacerbated by the constant stream of incompetent, counterproductive advice that teachers and administrators recieve from education “researchers” like the ones pilloried in this item.

  5. Lesson planning can help, but not if the basic assumptions behind the curriculum are a mess. The aforementioned professor that I’m thinking of was indifferent to teaching, and put almost no effort into it. But he knew his subject matter inside out, and chose an excellent textbook that did most of the work for him.

    And I learned a great deal in his class not just about the narrow subject being taught but about the nature of mathematics and am profoundly grateful to him to this day.

  6. Back in grad school (ed) after five years of teaching second grade. Started off wanting to do well by my students, thought this “discovery” method was the way to go. Began teaching and learned v quickly how students (um, duh) need more guidance. The textbooks quickly co-opted the ideas espoused then (and now, I presume)… and teachers were under a deluge of lessons designed around exploration. There is a time, there is a place, but direct teaching isn’t a four-letter word. Or action, as it were. Sad part is how anyone who took this to heart really needed to close doors to do it, and would make up some very tricky dog-and-pony shows when it came time for the observation cycle.

  7. mike from oregon says:

    Do we lack ‘master teachers’? I don’t know, but I do know that I’ve tried to help teach my daughter (who is about to enter high school as a freshman) algebra I over the summer, so that taking it in school won’t be that hard for her. Bottom line is (at least at this level) drill, drill, drill, drill, if you do it over and over and over again, eventually it sinks in.

    I’m not a math whiz, to this day I would have difficulty telling you what the associative or cummunitive laws of math are. I’ve seen people who understand them, they seem to get math much easier, I didn’t and the majority of people that I know and have worked with (I use to tutor in college) don’t understand those laws. However, allow me to put up sample after sample after sample and eventually they understand whatever concept we’re trying to learn.

    Would it be better if there were a simpler, more interesting, more intuitive method of learning? Probably, but while they try to impliment these ‘new and improved’ learning methods (that often don’t work). They end up cheating those experimental students out of a good education (if the ‘new’ methods don’t work). It’s a shame.

  8. When I was in grad school in math I had a number of friends in grad school in education, and I was appalled (as were my friend in the School of Education) by the venality and/or political goofiness of many of the ed professors and graduate students. They were more entertaining that students and faculty in the math department, but it did scare me about the future. However, they’ve exceeded even my pessimistic forecast.

    I suppose that the “discovery of math” push can be motivated by a genuine desire to do a better job of teaching, but it’s just so stupid that I refuse to believe that this is the primary motivation. I think that venal education professors propose bold new initiatives to gain stature and grant money so they can get promoted and buy a nicer car. The trouble is that somehow these stupid initiatives wind up in the classroom, and kids are taught that all approaches to solving a math problem are equally valid. (Too bad a bridge won’t stay up because of the good intentions of its designer.)

    My kids go to a good private school and even so my wife and I have supplemented their math instruction. Part of the problem, I think, is that it is a unusual child that doesn’t love to read once they learn how, but loving to play with numbers is a less common trait. And since math builds on itself in way that no other discipline does, there is a lot of memorization (i.e., drill) for every conceptual payday. My kids take to it — my eldest daughter is going into 8th grade shortly and has already started learning calculus — but even with them we spent a lot of time practicing multiplication and division.

  9. I’m good at math, and I think it’s =because= I spent a lot of time drilling in arithmetic. I was faster than other kids, but I still drilled with flashcards every school night for at least 2 years.

    Once I got to algebra, it was nothing to do 30 problems for homework, just to solve for x. When I took Calculus, it was also “take the derivatives of these 30 functions”. I don’t have to think about these things – they’re instinctual now. The difference is, I enjoyed math. I didn’t think the drilling was particularly fun, but I knew it would take me to the really interesting stuff (as I was reading stuff like Martin Gardner on the side, starting in middle school). It’s similar to studying music performance – you’ve got to do finger exercises and scales, which nobody particularly thinks of as fun … or athletic training, which involves lots of boring drills … but those doing it know it’s necessary for true proficiency at upper levels.

    As well, I went to math grad school. We didn’t have any of this “discovery method” nonsense in classes. There’s always some direction. Definition, theorem, proof (or proof is left to the reader). There is something called the Moore Method, but it’s not used often, and, just as in the Japanese case, does include guidance.

  10. It’s silly to suppose that rediscovery of mathematics is an effective way to learn it. It took a very long time to get the basic idea of numbers, and people don’t appear to have gotten all that much smarter. I would be impressed by any kid who could figure out the idea of counting and arithmetic without guidance, just as I would be impressed if they could teach themselves to read spontaneously. But evem in the cases where it worked, you’d have a brilliant 12 year old that had learned how to count to 10…

  11. I had posted earlier re: my experiences in teaching second grade. I want to echo what Mike from Oregon said, that lots of teachers feel, that so much time is spent on experimental methods and kids end up getting cheated. I am trying hard to strike a balance between respect for my own Dept of Ed (I am, after all, still in my twenties, largely inexperienced, and most certainly eager to learn). However, when I see the assigned reading and methods these young women (and one man) are being taught, I can’t help but cringe. I know how little good these things will do for them when they’re in the real teaching world. Good intentions do not a good curriculum make. But there IS a group of teachers out there who want to help remedy some of the appalling backward steps education has taken. I need this blog because I want to keep the faith.

  12. Andy Freeman says:

    I’m confused.

    Significant science and math artifacts, no matter how obvious in hindsight, are the result of significant insight/work by very smart folk who typically knew the state of the art to an excruiating detail, even if said folk didn’t have digital watches.

    Is it really sane, let alone efficient, to expect typical kids to duplicate that feat as part of teaching them about those artifacts?

  13. I don’t think that you are confused at all. Of course, a process that involves students so deeply that they figure out some of the steps as the material is being presented is far preferable to having them be talked at, but it requires great skill and preparation on the part of the teacher — it’s definitely not asking students to discover things for themselves. It sounds as though this is what the Japanese math teachers are doing, and I know that math courses where this happened were my favorites.


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