e pur si muove

Many readers will be familiar with (or at least have some dim recollection of) the problem sums which prominently feature in the upper primary mathematics curriculum in Singapore. Many educators believe that concrete examples provide a foundation for the proper understanding of mathematical concepts, and that such concrete examples make the concepts easier for students to grasp. For example, the following New York Times illustration gives an example of the kind of thing math teachers try out on students to help them learn something concrete:

To test this prevailing wisdom, researchers and teachers at the Ohio State University conducted a statistically rigorous, randomized-trial study of whether or not concrete examples were better than going directly to the abstract mathematical representation of a particular mathematical concept.¹

To their surprise, they found that the group of students who learnt an abstract version of the mathematical concept strongly outperformed three separate groups who were taught off concrete examples. Even more surprisingly, students who were first taught a concrete example before asked to work on the abstract example did not perform any better than students who worked immediately on the abstractions, and in fact performed slightly worse on later testing.

The researchers concluded that the concrete details distracted students from the mathematical concepts that the lessons were designed to teach them, and speculated that some students simply don’t learn by abstracting from concrete examples.

It’s important to note that the study involved college students and not students of younger schooling ages, and that the different developmental stages in cognition would probably hamper the transferability of the results directly to younger students. However, this direct challenge to conventional wisdom suggests that even younger students may benefit from scrapping problem sums in favor for algebra.

There is at least some anecdotal evidence for a particular weakness for problem sums - students tend to either ‘get it’ and understand what’s going on in the problem, or they don’t and get stuck for a very long time. Time and time again, frustrated parents are left trying to help their children do problem sums without the aid of algebra, which students are sometimes expressively told not to use, yet ironically is the very thing that problem sums are supposed to prepare students for in secondary school and beyond.

References

1. Jennifer A. Kaminski, Vladimir M. Sloutsky, Andrew F. Heckler, Science, The advantage of abstract examples in learning math, 320 (5875) 454-455, 2008-04-25.
2. Jeff Grabmeier, Ohio State University News, Concrete examples don’t help students learn math, study finds,  2008-04-25.
3. Kenneth Chang, New York Times, Study suggests math teachers scrap balls and slices, 2008-04-25.

1. In this case, the study was on getting students to understand three-membered commutative groups.↩

I’m willing to bet ten bucks that the [LHC physicists] don’t see supersymmetry. Then string theory will be dead. What a wonderful thing that would be, it’s back to the philosophy department for them. Don’t get me wrong; string theory is a wonderful thing for geometry, but I think it has absolutely nothing to do with physics.

- Prof. H, a mathematician

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After a long, cold, snowy winter, spring is finally, unequivocally, radiantly in season.

Over my latest phone call home, my mum told me about the new ERP gantries that came up near our family home back in Singapore. She told me that yes, they made some mention on the news about new gantries going up on Bukit Timah Road and the ECP, but she made a very valid point that both roads were so long that she had no idea where exactly the gantries were going to be placed. They didn’t exist on Thursday, but she got hit with an extra surcharge on Friday on her way out.

The best part? The new gantries caught so many people by surprise, that many drivers were stranded without the necessary funds in their CashCards on Friday, leading to an incredible traffic jam with people trying to swerve to avoid the gantries, even pulling over on the road to run to the nearest store to buy a CashCard.

Considering how Electronic Road Pricing was (originally, at least) introduced to relieve congestion, the irony of ERP causing arguably much more congestion than before is not lost on me.