Wednesday, December 5, 2018

When fractions are better than decimals

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.