Posted by: Alexandre Borovik | April 27, 2008

Quite a splash

muriel and Scott Carter brought to my attention to a recent paper in Science, The Advantage of Abstract Examples in Learning Math, by Jennifer A. Kaminski, Vladimir M. Sloutsky and Andrew F. Heckler. It appears to make quite a splash. From abstract:

Undergraduate students may benefit more from learning mathematics through a single abstract, symbolic representation than from learning multiple concrete examples.

The conclusion could be hardly characterised as surprising, but the redeeming quality of the paper is its experimental confirmation. Here I have some difficulty. The experiment was concerned with symbolic and concrete representation for cyclic group of order 3:

Two representations

Unfortunately, the “concrete” representation, by measuring cups of liquids, looks unnecessary complicated and therefore methodologically flawed: it is much more natural to represent the identity element by the empty cup. BTW, why the empty cup is not present in the scheme? In the bottom row, the most natural “remaining” is the empty cup. Maybe this is the reason why Concrete A representation on the right is harder than the Generic one on the left? Concrete B and Concrete C examples were formulated in terms of slices of pizza or tennis balls in a container, rather than portions of a measuring cup of liquid. Why not in terms of a switch which could be rotated through angles 0,\; \pm 2\pi/3?

Basically, the paper proves that a symbolic representation void of real-world connotations is better than bad and overloaded with unnecessary details “real world” representation. Not much to prove.



  1. But surely the other concrete conditions aren’t susceptible to your critique? The identity element in the tennis ball condition was three empty circles. I’m not sure I see the distinction between your suggestion and either the B or C conditions.

  2. i saw this reportd in the Times. i would agree–they are conflating syntax with semantics. ( i think this is the ‘stroop effect’ (mixing color categories with their names) . i think this mostly shows really shoddy research.

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