… that teaching mathematical concepts with confusing real-world examples is not a good idea.

Last month, the journal Science published an article about learning mathematics. Many newspapers and magazines picked up the story and pitched it at readers, summarizing the research result more or less like ars technica did [here]: “[S]tudents who learn through real-world examples have a difficult time applying that knowledge to other situations.”

This doesn’t agree with my experience in the classroom, and I sought the source. The full text of the Science article isn’t free, and the following remarks are based on what seems to be an earlier report of the same research by the same authors: “Do Children Need Concrete Instantiations to Learn an Abstract Concept?,” in the Proceedings of the XXVIII Annual Conference of the Cognitive Science Society [accessed at http://cogdev.cog.ohio-state.edu/fpo644-Kaminski.pdf on May 6, 2008.] Here the Ohio State researchers described their experiment.

In the first phase, subjects learned a mathematical concept.

[The concept] was a commutative group of order three. In other words, the rules were isomorphic to addition modulo three. The idea of modular arithmetic is that only a finite number of elements (or equivalent [sic] classes) are used. Addition modulo 3 considers only the numbers 0,1, and 2. Zero is the identity element of the group and is added as in regular addition: 0 + 0 = 0, 0 + 1 = 1, and 0 + 2 = 2. Furthermore, 1 + 1 = 2. However, a sum greater than or equal to 3 is never obtained. Instead, one would cycle back to 0. So, 1 + 2 = 0, 2 + 2 = 1, etc.

Subjects learned this concept through one of two scenarios: a scenario using geometric symbols with “no relevant concreteness” or a scenario “with relevant concreteness” using measuring cups containing various levels of liquid.

Ok, I’ll bite. I was a New Math child, and Mrs. Szeremeta taught us modular arithmetic with (analog) clocks. A “liquid left over” scenario with measuring cups ought to work, too. Most students “know” the idea of measuring cups.

Mrs. Szeremeta’s clocks worked, because the clock scenario contained “relevant concreteness,” and because its concreteness was familiar. For a concrete example to be a good teaching tool for an abstract concept, the concreteness has to be both relevant and familiar. Relevant means the concrete instantiation has to behave in real life more or less according to the rules of the abstract concept. Familiar means students know or can quickly learn how it works in real life. As the authors correctly observe, “the perceptual information communicated by the symbols themselves can act as reminders of the structural rules.”

Oops.

In addition to scenarios “with no concreteness” and “with relevant concreteness,” other kinds of scenario exists: ones “with confounding concreteness,” or ones with “irrelevant concreteness,” or ones with “distracting concreteness.” A scenario with confounding concreteness is one that draws on the familiar, but where the familiar behavior works contrarily to the rules of the abstract concept.

Here is the authors’ concrete scenario:

To construct a condition that communicates relevant concreteness, a scenario was given for which students could draw upon their everyday knowledge to determine answers to test problems. The symbols were three images of measuring cups containing varying levels of liquid. Participants were told they need to determine a remaining amount when different measuring cups of liquid are combined.

So far so good, but unfortunately, the three images used to represent the equivalence classes were those of a 1/3-full, a 2/3 -full, and a full measuring cup, representing the equivalence classes [1], [2], and [0] respectively.

Yes, the authors used a full measuring cup, not an empty one, to represent the additive identity, zero. The upshot of the experiment, then, in my opinion, was to compare an abstract implementation (geometric symbols with no concreteness), with a second implementation having both relevant and confounding concreteness. Relevant, because combining liquid and considering remainders works like addition in this group, but confounding, because one notion of the abstract concept is the idea of an additive identity (zero, nothing), which students learned to equate with a full measuring cup, which contains something, not nothing.

Students were expected to report the amount of liquid remaining after combining two amounts (but the result could not be “none”). In both the “cups” domain and the “no relevant concreteness” domain where squiggle = 0, disk = 1, and rhombus = 2, students learned to report correct answers (called “remainders” or “results”, depending on the domain) from the action of combining items.

The now combining-savvy students were challenged to learn similar rules in a new domain. The new symbols were images of a round ladybug, a tallish vase, and (I think) a cabochon ring viewed so its projection on the page is an eccentric ellipse with larger axis horizontal. They were told that the rules of this new system were similar to what they had previously learned, and that they should figure them out using their new knowledge.

Coincidentally, or perhaps not, the three images somewhat resembled baroque renditions of the earlier disk, rhombus, and squiggle, but bore little similarity to variously-filled measuring cups. They were taught a “game” where two students each pointed to an item, and a “winner” then pointed to one. Subjects were expected to learn which object the winner would point to. Students might have discovered that the visual similarities disk~ladybug, rhombus~vase, and squiggle~cabochon explained the new behavior.

Certainly the new domain was an abstract one. A pointing game like this, or real-world “ways” vases, ladybugs, and cabochons combine, are not part of the real world. I’d consider this new domain a purely symbolic one. When we teach mathematics, we want students to be able to transfer their knowledge to new domains both applied (with relevant concreteness) and abstract (symbolic, or with no relevant concreteness).

It doesn’t surprise me that students masters a new abstract domain more easily if they’d previously mastered one; and I’d expect it easier to master a new domain with relevant concreteness if you’d previously mastered one of those.

The short of it? Interesting research, but the experimental design is flawed as far as answering the question posed. Certainly this is no reason to give up finding creative, relevant, and familiar examples for abstract mathematical ideas.