Launchings

Abstract versus Concrete Examples in Teaching Math

David M. Bressoud December, 2008

"Students acquire higher levels of mathematical proficiency when they have opportunities to use mathematics to solve significant problems as well as to learn the key concepts and procedures of that mathematics. Although mathematics gains power and generality through abstraction, it finds both its sources and applications in concrete settings, where it is made meaningful to the learner. There is an inevitable dialectic between concrete and abstract in which each helps shape the other. […] Learning begins with the concrete when meaningful items in the child's immediate experience are used as scaffolding with which to erect abstract ideas. To ensure that progress is made toward mathematical abstraction, we recommend the following:

"Links among written and oral mathematical expressions, concrete problem setting, and students' solution methods should be continually and explicitly made during school mathematics instruction." [6, p. 426]


[1] Jennifer A. Kaminski, Vladimir M. Sloutsky, and Andrew F. Heckler. 2008. The Advantage of Abstract Examples in Learning Math. Science. 25 April. 320: 454–455.

[2] Kenneth Chang. 2008. Study Suggests Math Teachers Scrap Balls and Slices. New York Times. 25 April.

[3] Jennifer A. Kaminski, Vladimir M. Sloutsky, and Andrew F. Heckler. 2008. Supporting Online Material for "The Advantage of Abstract Examples in Learning Math." URL: www.sciencemag.org/cgi/data/320/5875/454/DC1/1

[4] They were given six rules:

    1. The order of the two symbols does not matter: (diamond, flag) -> diamond; (flag, diamond) -> diamond
    2. Any symbol combined with flag results in that other symbol: (flag, diamond) -> diamond, (circle, flag) -> circle
    3. (circle, diamond) -> flag
    4. (circle, circle) -> diamond
    5. (diamond, diamond) -> circle
    6. The result does not depend on which two symbols are combined first: (diamond, flag, circle) -> flag. It does not matter if we do (diamond, flag) first, then circle, or (flag, circle) first and then diamond.

Comparable sets of six rules were given for each of the three concrete examples.

[5] For Generic 2, they were told that the rules are like the rules of their example, and that (vase, ring) -> vase; (lady bug, vase) -> ring; (lady bug, lady bug) -> vase; (vase, vase) -> lady bug.

[6] National Research Council. 2001. Adding It Up: Helping children learn mathematics. Jeremy Kilpatrick, Jane Swafford, and Bradford Findell (eds). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.


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David Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, and president-elect of the MAA. You can reach him at bressoud@macalester.edu. This column does not reflect an official position of the MAA.

 

 

 


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