In order to graduate from high
school, people usually spend about13
years (K-12) learning about math concepts.
One of the things that math does is to give us a universal way to think
about quantities. The beauty of the
number 2, for example, is that it refers to a pair of items regardless of
whether those items are bowling balls, butterflies, or beer barrels.
Even though math is about abstract
concepts, the human mind is often focused on specific situations in the
world. As a result, the mathematical
notations we learn may not always make it easy for us to reason about the
world. Psychologists have begun to
explore the relationship between the way people naturally want to reason and
ways that we can represent situations using numbers.
An interesting paper in the
February, 2015 issue of the Journal of
Experimental Psychology: General by Melissa DeWolf, Miriam
Bassok, and Keith Holyoak explored differences in the way that fractions and
decimals affect thinking.
A fraction takes two integers and
places them in a ratio (like 3/4 or 10/15).
A decimal can express the same numerical quantity, but it does so with a
single number (like 0.75 or 0.66). So,
the fraction makes the relationship among numbers clear in the way it is
written, while the decimal does not.
In one study, the researchers
showed college students displays in which a fraction or decimal could be used
as a description. In the continuous
displays, there was a rectangle and some of it was shaded red, while the rest
was shaded green. In the discrete displays,
there were several objects, some of which were red, and some of which were
green. Finally, in the discretized
displays, there was a rectangle (like in the continuous display), but it was
divided into regions of equal size. Some
of those regions were red, and some were green.
With displays like this, there are
two kinds of comparisons people can make.
There are part-to-whole comparisons. For example, if there are three red
squares and five green squares, then the part-to-whole relationship of red
squares is 3/8. There are also
part-to-part comparisons. In this same
display, the relationship between red squares and green squares is 3/5.
In one study, participants were
shown a continuous, discrete, or discretized display and were asked whether they
would prefer to describe either a part-to-whole or part-to-part relationship
for that display with a fraction or a decimal.
They weren’t asked which fraction or decimal they would use for the
displays, just whether a fraction or a decimal would feel more
appropriate. For both kinds of
relationships, participants preferred to use decimals for continuous displays,
and to use fractions for discrete and discretized displays. A second study asked people to identify the
specific relationship shown in a display and found that people were equally
good at using fractions and decimals for continuous displays, but much better
at using fractions than decimals for discrete and discretized displays.
So far, these results are pretty
straightforward. The rest of the studies
explored the ability to perform mathematical analogies. In these studies, participants saw a display
and either a fraction or a decimal that described the part-to-part or
part-to-whole relationship in that display.
For example, if they saw 3 red squares and 5 green ones, and the
fraction 3/5, that was describing the part-to-part relationship.
Next, they saw a second display of
the same type, and two descriptions of the relationship. For example, this time they might see 5 red
stars and 7 green stars. They would see
one mathematical description of the part-to-part (5/7) relationship and one
description of the part-to-whole relationship (5/12), and they had to pick the
response that referred to the same relationship that was shown in the first display. In this case, they would have to pick the
part-to-part display.
They found that people had a hard
time with this task for both fractions and decimals when the displays were
continuous. They were much better at the
discrete and discretized analogies with fractions than with decimals. This was true, even when the fractions
referred to the same ratio, but did not map directly onto the number of items
in the display. For example, if the
part-to-part relationship in the display with 3 red squares and five green
squares was described with the fraction 6/10, that would be equivalent to 3/5,
but the numbers would not map directly to the display.
This work fits with a lot of
previous research suggesting that people like to reason about frequencies of
things in the world rather than proportions.
We experience the world in terms of the numbers of objects we see and
the numbers of events we experience.
Math allows us to create other representations like decimals that are
great for calculations, but they can make it harder to reason about what has
happened in the world.
This work also suggests that if
you are staring at a numerical description of a situation and that description
does not make sense to you, consider trying another way to think about it. You are often asked to make decisions based
on information that involves numbers.
Often, those numbers are decimals or proportions. Consider turning those numbers into
frequencies or fractions when reasoning about them.
