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Mental abacus math: No words required

STANFORD (US) — A 400-year-old style of abacus known as a soroban is the genesis for a method of learning math mentally without the use of language.

Although animals and pre-language infants are able to keep track of small numbers and can approximately judge large quantities, they can’t conceptualize exact larger numbers.

This inability begins to disappear as infants develop the language skills necessary for counting, suggesting that mental representations of large, exact quantities are often tied to language.


“All of that leaves open the question of whether language is really the only way to represent numbers,” says Michael Frank, assistant professor of psychology at Stanford University.

Mental Abacus, or MA, suggests the answer is no. Students are taught to visualize an abacus and often flick their fingers when they calculate, miming the movement of the beads, making use of the visual—rather than verbal—working memory.

“There are a limited number of things that we explicitly remember,” says Frank. People are only able to hold three to four separate items in the visual working memory at any one time.

In contrast, an MA calculation might involve the manipulation of fifteen beads. “Given these limitations, we were confused about how a whole abacus could be represented in working memory.”

Frank and co-author David Barner, assistant professor of psychology at the University of California, San Diego, explore the mystery in a new study published online in the Journal of Experimental Psychology: General, and demonstrate that MA does, in fact, involve visual manipulations of an imagined abacus—but storing visual information about each abacus column, not each bead.

The researchers examined elementary school students in India’s Gujarat Province, where MA is taught in a three-year afterschool program. Children went through a series of timed addition games that adjusted their difficulty to the user’s skill level.
The children’s calculating abilities dropped off sharply when they were asked to add four-digit numbers.

Each new place value requires a new abacus column—the rightmost column is the ones place, the next is the tens place, and so on. The result suggests that MA users are unable to imagine more than three abacus columns at once.

On the other hand, increasing the number of imaginary beads necessary for a problem without increasing the number of columns had no effect. And when it came to counting how many beads were present on a flashcard, MA users were no faster than untrained adults.

Therefore, rather than increasing ability to hold a mental image of an abacus, the method makes use of standard human visual memory.

“Clearly, the mental image doesn’t carry all the details of the abacus itself,” Frank says. “But we’re zeroing in on what the image consists of.”

The researchers also directly tested whether verbal or motor memory was in play during mental calculation. Participants were asked to calculate while drumming their fingers on the desk or repeating a book on tape.

Verbal distractions significantly affected the accuracy of the research subjects who had no experience with MA, but motor distractions had little effect.

MA users, on the other hand, showed only slight effects during both tasks, suggesting that verbal working memory plays at most a minor role.

“The process is similar to what electronic calculators do,” Frank explains. In an electronic calculator, the representation is binary, but in MA it’s an imaginary abacus.

“You start by reading out the problem in Arabic numerals or words, but then you convert it to a representation that’s really good for calculations.”

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