April 24, 2008

Abstract Math Produces Tangible Learning, Study Finds

When it comes to teaching mathematical concepts, abstract formulas may be more effective than the familiar examples of speeding trains and tossed coins favored by algebra instructors, according to a study published today in Science magazine.

In the study, which was described in a paper titled “The Advantage of Abstract Examples in Learning Math,” researchers at Ohio State University divided 80 undergraduates into groups and taught a mathematical rule to one group using a combination of abstract symbols. The researchers taught the same principle to the remaining students using one or more “concrete” examples that involved measuring cups of liquid, slices of pizza, or canisters of tennis balls.

When asked later to play a game using the rule, the students who had learned the concept in abstract form were better able to apply their knowledge to the new situation than were their counterparts who had studied concrete examples.

While math teachers need not abandon real-life illustrations altogether, the paper’s authors conclude, the experiment’s results suggest that relying exclusively on concrete examples might limit students’ ability to transfer what they learn to other situations. —Paula Wasley

Comments

1. I always wondered why two trains leaving their stations and heading in opposite directions was considered a concrete example of algebra. Only a railroad switchman would have any interest in such a thing in real life, and as I recall only one guy I ever went to school with—some nerdy fellow named Mooney—ever wanted to be a railroadman, and he’d set his sights on being the engineer.

   — first marci    Apr 24, 04:02 PM    #

2. In either situation, the important step is generalization. If generalization of the concept is not accomplished, then the transfer in the new context won’t happen. Relevant context does motivate the learner whether a student or faculty member.

   — Dan Apple    Apr 24, 06:00 PM    #

3. This study was done on undergraduates, who one can assume have already been exposed to algebra during high school. They are also used to thinking in abstractions by this time. These kinds of problems are usually taught to middle school and high school age persons whose cognitive abilities will also be different than college age people. So it remains to be seen whether this study is applicable to the classroom. Maybe the questions should be updated to be intuitive for students – after all, isn’t that the idea behind the real-life problems? Perhaps instead of trains meeting one could do problems based on download and upload speeds.

   — missme    Apr 24, 06:55 PM    #

4. Missme (#3). Your comments are very appropriate. I am unable to access the Science article at the moment, but your observations are to the point and I am interested to see if they have been addressed in the article.

   — henry    Apr 24, 07:29 PM    #

5. Dan Apple: good observation. I have a sixth grade son who sometimes learns through the formulae, and sometimes through the concrete problems, and you (and missme) have just clarified for me why this inconsistency occurs. Many thanks.

   — Margaret Fuller    Apr 25, 07:22 AM    #

6. Nothing new here — cognitive psychologists have known for many years that understanding a concrete example doesn’t mean that you will understand another concrete example of the same underlying rule. But practice applying the underlying rule is critical to being able to apply the underlying rule and truly learning the abstract concept. You need both for learning.

   — Ray    Apr 25, 08:21 AM    #

7. The broader point in the foregoing discussion, lest it be lost in the debate, is that Math and Science teachers are constantly exhorted to “dumb down” the teaching of abstract principles and general theories, through the use of “concrete examples”, as though the two are mutually incompatible. As others have mentioned above, both deductive and inductive reasoning are important cognitive skills and reinforce each other to complete the learning process. In a similar vein, the oft-used expression “real-world examples” seems to imply that the teaching of theory is somehow illegitimate. As the social scientist, Kurt Lewin, stated” “There is nothing quite as practical as a good theory.” Likewise, there is nothing quite as useful (meaning, generalizable) as a pure abstraction.

   — CR    Apr 25, 09:14 AM    #

8. Do you, or anyone you know, factor polynomials for a living? Anyone? Anyone?

   — formerly known as . . . .    Apr 25, 10:25 AM    #

9. “formerly know as … “ must have lingering issues with polynomial factorization (not to mention playing scales or doing pushups, which are other activities I remember from high school that I’m quite sure no one does for a living). What we as humans excel at is abstracting what we experience so that we can address related situations as they occur. Mathematics simply abstracts important characteristics of size, shape, and quantity. Humanity’s success in the development of abstract mathematics to model physical and discrete phenomena has been essential for virtually all of the science and technology we utilize today. It is often easier to understand the abstraction (say a polynomial) than a situation in which it might be applied. Our understanding of phenomena is enhanced as much by studying the model as a phenomenon, so I hardly find this study’s outcome surprising.

   — CW    Apr 25, 01:21 PM    #

10. This is a fascinating issue and discussion. I do frequently see that students have a great deal of trouble intuiting a rule from a specific example. Even though they may (initially) get upset when presented with the rule, that seems to be more a violation of their prior experience that they will be given a rule and problems that explicitly require use of that rule, rather than metarules (guidelines for when and how to apply the rule). Maybe not coincidentally, “formerly known as,” many control systems engineers and others using differential equations, as well as statisticians partitioning variance in multifactorial research studies, factor polynomials for a living on a regular basis. A student may not know that when using Matlab or SPSS, but there it is—and it helps one figure out if the analysis was correct or even appropriate.

    — Barrett Caldwell    Apr 28, 01:17 PM    #

11. In my own experience as a chemistry teacher, many students do not really learn algebra, statistics, fractions, until they see the concepts used in a venue outside of math class. I’ve had many students tell me they learned more math in chemistry class than in math class. Not that I want students to come into chemistry with no math experience, its just that I think students benefit most when they see math actually used and practiced outside of math class.

    — Robyn    Apr 28, 02:02 PM    #

12. Speaking of abstraction, “formerly known as . . . . “, you do something akin to polynomial factorization every time you do long division.

    Seriously, one of the problems with “concrete” examples is that they only infrequently have anything to do with what people do in the real world, “for a living”. But there’s no way we can introduce students to real applications that take anywhere from a quarter to a year or more of study to understand, just to motivate one aspect of mathematics. And this ignores the chicken-and-egg issue of needing the math in the first place to understand the application.

    Sometimes, we have no choice but to say, “You need to learn this. You may or may not need it in your career but, failing to learn it, you are guaranteed to rule out a large number of potential careers. Oh, and you won’t get a degree without learning it.”

    As Thomas Huxley wrote, “Perhaps the most valuable result of all education is the ability to make yourself to do the thing you have to do when it ought to be done whether or not you like it or not.” Might there be a benefit to requiring students to find internal motivation to do the work?

    — Mike Stiber    Apr 28, 07:10 PM    #

13. If you want to see a superbly satirical treatment of what can happen when concrete examples fail to lead to generaliizations, read the absurdist math instruction imparted by the Professor to the Pupil at the beginning of Eugene Ionesco’s play “The Lesson” (often performed and widely available in translation.) The Pupil cannot generalize from counting objects to numbers, and can’t get subtraction at all: asked how many noses she would have if she had three and one were taken away, she replies “one” because such abstraction is past her. She only has one nose, and that’s that.

    — Ingrid Chafee    May 13, 01:09 PM    #

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