{"id":2528,"date":"2008-04-25T09:04:42","date_gmt":"2008-04-25T09:04:42","guid":{"rendered":"http:\/\/scienceblogs.com\/principles\/2008\/04\/25\/abstraction-compartmentalizati\/"},"modified":"2008-04-25T09:04:42","modified_gmt":"2008-04-25T09:04:42","slug":"abstraction-compartmentalizati","status":"publish","type":"post","link":"http:\/\/chadorzel.com\/principles\/2008\/04\/25\/abstraction-compartmentalizati\/","title":{"rendered":"Abstraction, Compartmentalization, and Education"},"content":{"rendered":"

Given the amount of time I’ve spent writing about academic issues this week, it’s only fitting that the science story getting the most play is about math education. Ed Yong provides a detailed explanation<\/a>, and Kenneth Chang summarizes the work in the New York Times<\/cite><\/a>. Here’s Ed’s introduction:<\/p>\n

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Except they don’t really work. A new study shows that far from easily grasping mathematical concepts, students who are fed a diet of real-world problems fail to apply their knowledge to new situations. Instead, and against all expectations, they were much more likely to transfer their skills if they were taught with abstract rules and symbols. <\/p>\n

The use of concrete, real-world examples is a deeply ingrained part of the maths classroom. Its advantages have never really been tested properly, for they appear to be straightforward. Maths is difficult because it is a largely abstract field and is both difficult to learn and to apply in new situations. The solution seems obvious: present students with many familiar examples that illustrate the concepts in question and they can make connections between their existing knowledge and the more difficult concepts they are trying to pick up. <\/p>\n

The train problem is a classic example. Another is the teaching of probability with rolls of a die, or by asking people to pick red marbles from a bag containing both blue and red ones. The idea is that, armed with these examples, students will recognise similar problems and apply what they have learned. It’s a technique deeply rooted in common sense, which is probably as good an indicator as any that it might be totally wrong.<\/p>\n<\/blockquote>\n

As with any research involving subjects more complicated than a diatomic molecule, of course, there are a billion ways that the study could be going wrong. After the initial shock of seeing the findings, though, it’s not actually that surprising. I wonder if this isn’t related to the well-known problem of compartmentalization in science education.<\/p>\n

<\/p>\n

Well, ok, I’m not sure that’s the technical name for it, but anybody teaching in a mathematical science is probably familiar with what I’m talking about. When we present new mathematical apparatus for solving physics problems, students will often get tripped up on basic elements of calculus. When asked “Haven’t you done this in your math classes?” they often respond with an answer that amounts to “Yeah, but that’s math class<\/strong>. It doesn’t have anything to do with physics.” We also hit some student resistance when we ask them to do computer simulations of physical situations– they grumble about having to write fairly trivial bits of code, despite the fact that they’re taking computer programming classes at the same time.<\/p>\n

This is a persistent source of annoyance for college science faculty– many students seem unable or unwilling to take things learned in a class in one department and apply them to subjects studied in a class in another department. The problem with concrete examples noted in the current work seems like just a more extreme version of the same thing. If you teach algebra using the classic example of trains leaving stations and so on, that goes into the mental compartment for “math problems with trains” and when they get to a later problem about chemical reaction rates, they won’t think to apply the same technique, even though the two situations may be mathematically identical. After all, that was about trains, not chemicals.<\/p>\n

Of course, then I wonder if this isn’t a confounding factor in their study– the stuff I’ve read says that the students who learned a mathematical technique using abstract symbols did a better job of applying it to a new situation than students who learned using real-world examples. The “new situation” described by Ed sounds like it maps pretty easily onto the abstract symbols of the training– they’re games that fall into the same compartments. What would be interesting to see (and I don’t know if it’s in the actual paper) is how well the abstract symbol people did when asked to apply their knowledge to one of the concrete examples.<\/p>\n

The issue might be that “problems with abstract symbols” is mentally closer to “games with odd objects” than “problems with measuring cups” is. In which case, it’s be interesting to see how students who learned using “problems with abstract symbols” do when given “problems with measuring cups” on a test.<\/p>\n

That may be in there– I can’t access the paper itself from home, and I’m kind of busy<\/a> at work. I’ll try to find time to look at it this weekend.<\/p>\n

Regardless, there’s a tricky balancing act here. Even if it’s true that the best method of teaching problems is to use abstraction rather than concrete examples, you leave out the concrete examples at your own peril. The point of giving concrete examples is not just to aid learning, but also to stave off questions of the form “Why do we have to learn this, anyway?” and more importantly, student evaluation comments of the form “This class didn’t have anything to do with the real world.”<\/p>\n","protected":false},"excerpt":{"rendered":"

Given the amount of time I’ve spent writing about academic issues this week, it’s only fitting that the science story getting the most play is about math education. Ed Yong provides a detailed explanation, and Kenneth Chang summarizes the work in the New York Times. Here’s Ed’s introduction: Except they don’t really work. A new… Continue reading Abstraction, Compartmentalization, and Education<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"1","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8,13,7,11],"tags":[],"class_list":["post-2528","post","type-post","status-publish","format-standard","hentry","category-academia","category-education","category-physics","category-science","entry"],"_links":{"self":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/posts\/2528","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/comments?post=2528"}],"version-history":[{"count":0,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/posts\/2528\/revisions"}],"wp:attachment":[{"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/media?parent=2528"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/categories?post=2528"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/chadorzel.com\/principles\/wp-json\/wp\/v2\/tags?post=2528"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}