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Fractional Intelligence

June 29, 2012

One measure that scientists use to gauge the utility of technologically significant devices is the figure of merit, commonly referred to as FOM. FOMs are useful, since they fold a lot of information into a single number.

The nature of an FOM depends on what system is being studied. For example, a figure of merit for the display on which you are reading this article is the contrast ratio; that is, the range of light to dark that's rendered. Obviously, higher contrast ratios are desirable.

When I was doing research on magneto-optical materials, my figure of merit was degrees/dB, which was a ratio of the optical rotation of the Faraday rotator material (measured in degrees) and its optical attenuation (measured in decibels). Again, large ratios are good.

When we were in our Total Quality Management Morass (TQMM), the Voice of the Customer was supposedly of paramount importance. As a result, we were encouraged to survey our customers, and our analytical department surveyed its customers, including me.

I responded by suggesting an FOM for their services that would combine such features as their turn-around-time on analyses. My FOM was apparently too precise a performance metric for them to bear, since they ignored it completely. As they say, you can't argue with science (except when global warming is involved?)

Marble block used to stabilize an analytical balanceNon-scientists sometimes think we have rocks in our heads, but our rocks are usually on table tops.

A marble block used to stabilize an analytical balance.

(Via Wikimedia Commons).

the Intelligence Quotient (IQ) is used as the FOM for intelligence. The IQ is one number that's presumed to quantify a person's intelligence; or, at least, what's tested on intelligence tests. As I wrote in a previous article (Extreme Intelligence, October 14, 2011), Albert Einstein and Stephen Hawking have estimated IQs of 160, while Richard Feynman had a measured IQ of "only" 126.

Unfortunately, IQ is not a good predictor of success in any given profession. There's also evidence that a high IQ comes with emotional baggage that actually inhibits success. However, where the sciences are concerned, there is one good predictor of success, and that's facility with mathematics.[1] According to a 2006 Harvard study, "Mathematical fluency is the single best predictor of college performance in biology, chemistry, and physics..."[2]

As a practicing scientist, I know how important mathematics is in my daily activities. I'm always making "back-of-the-envelope" calculations to steer my experiments into more productive or more stable regimes. Is there an easier way to infer mathematical ability, short of a full battery of SAT or GRE mathematics tests?

Figure captionThe headquarters of the Educational Testing Service, creators of various standarized tests, are about an hour's drive from my house.

(Via Wikimedia Commons).

Apparently, knowledge of long division by children, and how they deal with fractions, is a predictor of mathematical acumen in later life.[3-6] This is the conclusion of a study by a research team led by Robert Siegler, a professor of cognitive psychology at Carnegie Mellon University. This study, "Early Predictors of High School Mathematics Achievement," is published in Psychological Science.[5] Other team members are from the University of California-Irvine, the University of Michigan, the University of London, the University of Chicago, and Vanderbilt University.[3,5]

In a study such as this, controls are important. The research team statistically controlled for parents' education and income, as well as the children's age, gender, IQ, reading comprehension, working memory, and knowledge of whole number addition, subtraction and multiplication.[3] It was apparent that an understanding of fractions and division at the fifth grade level was a predictor of high school achievement in algebra and other mathematics.[3]

This hypothesis, that success with long division and fractions can predict subsequent math performance, had its origin in Siegler's work on the National Mathematics Advisory Panel in 2006-2008.[4] This study found that students who "start ahead in math generally stay ahead" and that those who "start behind generally stay behind."[4] Reinforcing this idea was the fact that many college students have difficulty with division and fractions, although these were supposedly mastered in elementary school.[3]

The study involved two data sets, a decades old data set from the United Kingdom, and a more recent data set from the United States.[3-4] The UK data were for 3,677 children, first tested in 1980 when they were ten; and in 1986, at age sixteen.[3-4] The US data set was much smaller, with only 599 children involved. These were tested in 1997, at age 10-12; and in 2002, at age 15-17.[3-4] The UK data noted the strong predictive value of knowledge of fractions at the earlier age upon success in algebra. The US data indicated the same.[4]

The study authors blame the teachers, not the students, citing a "lack of a firm conceptual understanding" of the concepts by teachers. The report cited examples in which American teachers were unable to explain the reasoning behind the mathematical operation, whereas most teachers in Japan and China could offer two, or three, explanations.[4] Poor teaching is likely the reason why the math scores for U.S. high school students have been stagnant for more than 30 years.[4]

This research was supported by the U.S. Department of Education's Institute of Education Sciences; and by the National Science Foundation's Social, Behavioral, and Economic Directorate.[3]


  1. Philip M. Sadler and Robert H. Tai, "Education Forum - The Two High-School Pillars Supporting College Science," Science, vol. 317, no. 5837 (July 27, 2007), pp. 457-458.
  2. Steve Bradt, "High school AP courses do not predict college success in science," Harvard Gazette, February 17, 2006.
  3. Shilo Rea, "Carnegie Mellon-Led Research Team Finds Knowledge Of Fractions and Long Division Predicts Long-Term Math Success," Carnegie Mellon University Press Release, June 15, 2012.
  4. Mary Niederberger, "Formula written for math success - Learning fractions and division early will help America's students as they tackle tougher problems ahead, a research study finds," Pittsburgh Post-Gazette, June 19, 2012.
  5. Robert S. Siegler, Greg J. Duncan, Pamela E. Davis-Kean, Kathryn Duckworth, Amy Claessens, Mimi Engel, Maria Ines Susperreguy and Meichu Chen, "Early Predictors of High School Mathematics Achievement," Psychological Science, Published online before print (June 14, 2012), Document No. 0956797612440101.
  6. Early Knowledge of Fractions and Long Division Predicts Long-Term Math Success, YouTube video, June 7, 2012.

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Linked Keywords: Scientist; Technology; technological; figure of merit; display device; contrast ratio; research; magneto-optical materials; degree; decibel; dB; optical rotation; Faraday rotator; optical; attenuation; Total Quality Management; Voice of the Customer; analytical chemistry; performance metric; global warming; analytical balance; Wikimedia Commons; Intelligence Quotient; intelligence; Stanford-Binet; intelligence test; Albert Einstein; Stephen Hawking; Richard Feynman; intellectual giftedness; high IQ; emotional baggage; science; mathematics; Harvard; biology; chemistry; physics; Fermi problem; back-of-the-envelope calculation; experiment; SAT; GRE; Educational Testing Service; standarized test; long division; child; children; fraction; Robert Siegler; cognitive psychology; Carnegie Mellon University; Psychological Science; University of California-Irvine; University of Michigan; University of London; University of Chicago; Vanderbilt University; scientific control; statistics; education; income; gender; reading comprehension<; working memory; natural number; whole number; addition; subtraction; multiplication; fifth grade; high school; algebra; National Mathematics Advisory Panel; college; elementary school; United Kingdom; United States; teacher; Japan; China; United States Department of Education; Institute of Education Sciences; National Science Foundation; Social, Behavioral, and Economic Directorate.

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