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On April 25th, Nature published an article titled LEARNING THEORY: The Advantage of Abstract Examples in Learning Math that swarmed across the Internet with the essential claim that abstract examples were more useful to learning than concrete examples.
The researchers in the experiment were teaching the structure of “clock arithmetic” (also known as modular arithmetic).
Notice that the numbers when added “wrap around”, so while 2 + 2 normally equals 4 here it equals 0. The same effect can be achieved by adding the numbers, dividing by 3, and keeping only the remainder.
The above example with the clocks was my own. Of the Nature article’s numerous issues I want to focus on just one: The concrete example chosen was not a very good one.*
What’s being conveyed here is when the first and second glasses are combined sometimes there’s an overflow; that’s what the “leaves” is about. The fact it took me several minutes to parse this meaning threw up immediate warning bells. Even when the problem is understood, the mental sequence in visualizing the clock example and the water examples are quite different.
Clock: Rotate, then rotate again. Look at what you have.
Water: Pour one glass into the other glass, if you can. If you can pour the entire glass, look at the glass with the water in it. If you can’t pour the entire glass, look at the glass you’re holding.
I believe what the researchers were ignoring was that concrete examples are a rhetorical device. A rhetorical device should be a hook to the audience, should be easy to visualize, should form an easy analogy with the intended goal, and should be a reliable anchor for students to fall back on should they get puzzled later. The concrete example of the water glasses was none of these.
Rhetoric is hard to analyze scientifically, because so much is dependent on context. The context is not just of the material being presented, but the presenter (charisma, appearance), the audience (older or younger?), the setting (large auditorium or small circle?), and the placement in the overall lesson (does it start the class or is it somewhere in the middle?)
I know researchers would love to find the “best” way to teach any material. I do believe it possible to weigh teaching methods to an extent, but any effort needs to be pragmatic and aware of the difficulties I outlined above.
* If you want to hear about the others look at this graphic first. Notice how the “abstract” version matches closely with what is being tested. Since the “mental map” is closer it is much easier to extrapolate to the questions about the game. If the questions were instead numerical (either using a clock, or using numbers and asking 2 + 2 = ?) I have no doubt those exposed first to the “concrete” example would score higher. On top of that, the “abstract” example is hardly the sort of abstract mathematicians talk about; it might as well be a picture of an apple, a pear, and a banana.
 Jason Dyer holds degrees in Fine Arts Studies and Math and teaches at Pueblo High School in Arizona. His school mascot is the Warriors and his other blog of residence is The Number Warrior.
POSTED ON HOTCHALK.COM
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