This time the recommendations come from Natalie Angier, a science reporter with the New York Times.
In this week's Science Section, she reports on two studies showing connections between two cognitive number mechanisms:
1. The approximate number system: in Angier's words, "an ancient and intuitive sense that we are born with and that we share with many other animals."
2. The abstract, symbolic system that allows us to "manipulate representations of numbers" and make precise calculations.
One study shows that a person's facility with the approximate system is connected to his/her facility with the symbolic system. The other shows that, in Angier's words:
[P]reschool children are remarkably good at approximating the impact of adding to or subtracting from large groups of items but are poor at translating the approximate into the specific.All this, Angier reports, has "potentially broad implications for math education:"
Taken together, the new research suggests that math teachers might do well to emphasize the power of the ballpark figure, to focus less on arithmetic precision and more on general reckoning.Huh? Less on symbolic and more on approximate? This is a non-sequitor, unless we know that the causality flows from approximate to symbolic.
But as Angier quotes one of the researchers (Lisa Feigenson of Johns Hopkins) as saying, “We can’t draw causal arrows one way or another" between "your evolutionarily endowed sense of approximation" and "how good you are at formal math.”
And, as Angier herself writes: "The researchers caution that they have no idea yet how the two number systems interact," and that it's currently an "open question ...how malleable our inborn number sense may be, whether it can be improved with training, and whether those improvements would pay off in a greater appetite and aptitude for math."
So how does Angier leap to the conclusion that schools should be stressing approximate number sense over symbolic numerical reasoning?
It seems that our science reporter has managed to:
1. avoid visiting actual classrooms, where she would see how much symbolic math has been jettisoned the sake of "number sense," and by how much overall levels of math achievement have therefore declined.
2. fall under the influence of the reigning ed school orthodoxy, which is as enamored of intuition as it is contemptuous of abstract reasoning.
3. take, on faith, the bizarre claims by one of the researchers about the parlor games played by mathematicians:
“When mathematicians and physicists are left alone in a room, one of the games they’ll play is called a Fermi problem, in which they try to figure out the approximate answer to an arbitrary problem,” said Rebecca Saxe, a cognitive neuroscientist at the Massachusetts Institute of Technology who is married to a physicist. “They’ll ask, how many piano tuners are there in Chicago, or what contribution to the ocean’s temperature do fish make, and they’ll try to come up with a plausible answer.”Not the mathematicians I know!
Why doesn't anyone ask them about what they think of what's going on in today's actual classrooms?