# Launchings

## Abstract versus Concrete Examples in Teaching Math

David M. Bressoud December, 2008

I took some flak from a comment I made to a New York Times reporter last spring in response to the Science article, The Advantage of Abstract Examples in Learning Math [1]. My actual words, quoted by the New York Times [2], were "It's a fascinating article. In some respects, it's not too surprising." That is a pretty innocuous statement, but it suffered from context. The Times's article recast the findings of the study as demonstrating that concrete examples in math class are a waste of time, and my words appeared to affirm that conclusion. Now, with access to the study behind the Science article and a bit more time to choose my words carefully, I'd like to revisit my reaction to the original study and comment on the Times article.

The study on which all of this is based was conducted at Ohio State by Kaminski, Sloutsky, and Heckler of the Center for Cognitive Science, The online supporting materials [3] give full details of the study and the actual training and testing materials that were used. In the introduction to their study, Kaminski et al. describe a mathematical concept as an abstract entity, any attempt to communicate this concept with as few extraneous details as possible as a generic instantiation, and a situation that introduces many more extraneous details as a concrete instantiation. The example of a generic or abstract instantiation given in the introduction is the derivative defined as the limit as h approaches 0 of (f(x+h) - f(x))/h. A concrete instantiation would be to describe the derivative as the speed of a particular red car. The outcome they expected to see, and one that would get no argument from researchers in mathematics education, is that extraneous details often create confusion for students who have trouble recognizing what is relevant when a new concept is introduced.

The actual example that they used introduced additional complications. They decided to test understanding of the notion of an abstract group of order three. The concrete instantiation involved addition, modulo 3, in one of three specific contexts: Concrete 1 involved adding thirds of a measuring cup and only keeping track of how much was left over, Concrete 2 used burnt thirds of a pizza, keeping track of what fraction over a whole pizza gets burned, and Concrete 3 used tennis balls in cans (three to a can), keeping track of how many are left over. In all three concrete cases, the elements were represented pictorially as either partially filled measuring cups, a circle divided in thirds, or an arrangement of three circles with some circles filled in. Against these concrete examples were rules for combining elements in a group of order three where the elements were either diamond, flag, circle (Generic 1, these were shown as symbols rather than words) or lady bug, ring, vase (Generic 2, again shown as symbols). A total of eighty students unfamiliar with groups of order 3 were randomly assigned to one of four training rooms, either Concrete 1, 2, or 3 or Generic 1. They were given practice problems on their setting and tested to ensure that they understood its rules [4]. All students were then introduced to a minimal set of rules for combining elements in Generic 2 [5] and tested on it. Students who had practiced on Generic 1 did significantly better (mean of 76% with standard deviation of 21.6%) on Generic 2 than did the students who had practiced on Concrete 1, 2, or 3 (respective means of 44%, 44%, 51% and standard deviations of 16.0%, 17.2%, and 20.3%). In fact, the students who had practiced on the concrete examples had responses on Generic 2 that were comparable to those of students who had had no preparation before being tested on Generic 2.

The experiment was repeated with twenty new students who were trained on both Concrete 1 and 2 with the parallels pointed out (1/3 of a cup corresponds to 1/3 of a pizza). Their performance on Generic 2 was comparable to those who had studied a single concrete example (mean = 41%, SD =16.7%).

A final experiment divided forty students into two groups. One group was trained on just Generic 1. The other group was trained on Concrete 1 and then on Generic 1. Both groups did significantly better on Generic 2 than those trained only on concrete examples. What was interesting about this experiment was that the group that had not received training on Concrete 1 did significantly better (mean = 83.3%, SD = 10.6%) than the group that had (mean = 65.5%, SD = 26.2%). It appears that first talking about cups of water confused more students than it helped, especially since no attempt was made to explain how it related to the situation presented by Generic 1.

I do still think this study is interesting (maybe "fascinating" is a bit of a stretch). It shows that there is a real conceptual leap from addition modulo 3, where it is possible to make some sense of the rules for combining elements, to an abstract group of order 3 where the rules appear much more arbitrary. And the last experiment should be cautionary. Students learning new material are struggling to identify what the relevant features really are. Giving two examples that superficially look quite different and providing no assistance in making the connection between them can create more confusion than insight.

The Science article described this study quite accurately, though the tag line was a bit provocative: "Undergraduate students may benefit more from learning mathematics through a single abstract, symbolic representation than from learning multiple concrete examples." It sets it up as a dichotomy when most research in mathematics education makes it quite clear that both are needed, as well as extensive work on identifying the common conceptual framework and providing opportunities to apply the abstract concept in unfamiliar and superficially dissimilar concrete situations. The New York Times article made a travesty of the insights of this study, suggesting that "it might be better to let the apples, oranges and locomotives stay in the real world and, in the classroom, to focus on abstract equations."

The balance that is needed is nicely summarized in one of the recommendations of the National Research Council's Adding It Up. Though this book addresses only preK-8 mathematics, it contains a lot of wisdom that is valid right through college. Addressing the issue of abstract versus concrete examples, it counsels,

"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]

[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.