Focal Plane: Numerical applications
This is the thirteenth instalment of the GoobNet Focal Plane, an occasional series wherein we highlight an unimportant social problem, trying to make you care about it. Use the Whine Control, above and right, to view other instalments.
If Fernandela Galazzi had her way, you'd be able to add 441 and 218 without artificial aids.
"659, of course. How many of you got it in time? Don't be shy."
From her place at the fore of a conference room in Salem, OR, USA, she scans the hands in the crowd. Less than ten percent of those listening to her lecture, "A Trip to the Brain Gym: Exercising the Number One Muscle", have their hands up.
"Don't be discouraged if you didn't get it. There are plenty of great mathematicians and scientists who couldn't do arithmetic. But the point is this: The ones who can are usually better."
The phrase mathematical motivational speaker may sound like an oxymoron at first. Yet Galazzi, three years removed from a doctorate degree in applied mathematics at the University of Chicago, describes herself with exactly that phrase.
Later, she says, "Everyone likes to say that the average person uses only ten percent of their brain's capacity. Whether that's true or not I'm not qualified to say. But what that really means is that most people can think about more stuff than they are already."
Galazzi's stated goal is to encourage mental mathematics, rather than resorting to calculators. But just how much of a problem do calculators pose?
"The threat of calculators is overblown," says Alejandro Sakai. A professor of cognitive psychology at the University of Maine, Sakai's research interests currently include the mathematical portions of the human brain and how they are affected by frequent calculator use.
"Many observers are of the opinion that owning a calculator instantly precludes the human brain from doing mathematics," he continues. "Nothing could be further from the truth. Calculators are usually used to determine the solution to a real problem, rather than simply adding up esoteric numbers. As a result, the brain does the same amount of work, though different areas of the cerebrum are utilised."
Galazzi disagrees. "Things turned around in one day for me, when I was in graduate school and had a problem to work out," she recalls. "I'd left my bag, with my calculator in it, in the office when I went to lunch. Well, whilst I was consuming the Polish sausage, I thought about a new equation that might better fit the fluid mechanics data that I was working on. I tried to consider a test case to make sure it was correct, when I realised that I couldn't do it."
She says that she has never used a calculator since, and has successfully defended her thesis, balanced her checkbook, and done numerous other numerical tasks without computational aids.
Sakai, though, feels that the brain is an inefficient computational tool. "Tasks like adding or multiplying numbers are strictly memory recall tasks. For instance, when you're multiplying two multiple digit numbers, you'll start from the right. Say you have a seven and an eight. You'll have to recall that the product of seven and eight is 56, and then store that whilst you work on the next pair of digits. Each pair requires looking up a value from the multiplication table and storing the result, and then to conclude the process, one must recall the various results."
Conversely, he says, calculators do all this effectively instantaneously. "Even the time spent entering the numbers into the calculator is effective, we've found. One recent experiment that we performed here showed that at that time, the brain is already thinking ahead about what the result might be, which leads to a high success rate when catching entry errors."
Unconvinced, Galazzi believes that performing computations in the brain can enhance other types of performances. "Calculators are an educational malaise - we've seen time and again how many students have improved their scores when they ditch their calculators. And not only in mathematics, in all fields."
Galazzi discusses the brain's capacity at all of her speeches, and usually surprises those in attendance with the results.
"Let's talk about memory. Consider your favourite sports team, whichever sport it is, whether it's professional or college or high school. Hold up your hand if you can remember whether they won or lost their last game."
Most hands go up.
"What about the first game of the season? Do you still remember that game?"
Most of the hands remain up.
"All right, now how about the previous season?"
Many hands go down, but some are still up. She continues to lead her audience back one season at a time and finds that one person can remember the way that the University of Oregon's American football team began each of their past thirteen seasons.
"Did you know that you knew that?"
The person in question admits not having realised that his memory was so complete.
At this point, Galazzi continues to explain the other wonders of which the brain is capable, concluding, "So don't ask 'How can I add these numbers'. You should be asking, 'How can I not add these numbers'. I mean, you can't help but remember things. That's what your brain is there for."
A question of efficiency
There remain, however, scholarly divides on the question of calculation efficiency, and leading the pro calculator camp is Sakai, who is working on a book on the subject.
"Calculator users have consistently tested better in experiments and on examinations," Sakai declares, "whether with or without a calculator at the time." Several papers in journals such as the Journal of Mathematical Brain Functions over the past eight years largely confirm these results.
"If that's the case, how come the hundreds of people whom I've taught to free themselves from calculators can do computations more quickly now?" Galazzi replies.
Back in Oregon, her lecture is over. One person walking away, a student at a nearby high school, expresses a commonly held feeling. "I'm sure we all want to use our brains as much as we can," he says. "But if we have something that's designed to do a particular type of work, why not use it?"
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