Sunday, 8 June 2008

A Little Arithmetic Test


OK, quick, evaluate the two arithmetic expressions below






3 + 5*7............. 2+5 * 8



Ok, if you remember high school algebra, the answers are 38 and 42. If you don't remember high school algebra, you may have gotten 56 on both. The formal order of operations that we try to get across to high school students is that these questions should be done by first completing the multiplication or division operations, and then do the addition or subtractions. (for the one on the left, you should multiply 5 * 7 first, then add the 3)


The amazing thing, according to research by cognitive psychologists David Landy and Robert Goldstone from Indiana University, is that you were more likely to miss the second than the first, and even if you are an old hand at simple arithmetic, the second may have taken a micro-scosh longer... because of the way they are spaced. Notice that in the first I have grouped the multiplication operation more tightly and in the second the addition operation is grouped tightly. The researchers write,"When physical spacing was inconsistent with
order of precedence rules, six times as many errors were made relative to when the spacing was
consistent."

The grouping by proximity seems to be a natural process affecting how we read and write expressions. In the authors own words:
"Specifically, we propose that formal notations are diagrammatic as well as sentential, and that the property conventionally described as syntactic structure is cognitively mediated, in part, by spatial information. Elements of expressions are “bound” together through perceptual grouping— often induced by simple spatial proximity."

According to the research, it even effects the way you write the expressions. They tested people by giving them random mixed expressions using words with addition and multiplication in both orders, such as "Four plus seven times three". and asking people to write them down. It seems that people wrote them down with the multiplication operation more closely grouped than the addition.

The suggestion of the researchers is that this grouping tendency may be observed from parents and teachers and become part of our reading and writing style even if we do NOT remember the order of operations. However, there was a greater accuracy of the results of people who tended to group, "Consistently spaced submissions were also more likely than either inconsistent or
unspaced expressions to be correct; 53% of consistent equations were correct, compared with
only 37% of inconsistently spaced and 50% of unspaced equations." To me this would seem to support a written response that indicates a cognitive bias based on knowing the order of operations more profoundly.

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