# How not to report Pearson's r: a cautionary tale in mathematical cognition

Posted by: Gary Ernest Davis on: January 27, 2010

investigates issues in mathematical cognition, asking questions such as “What do people know about numbers, arithmetic, and math?” and “How do we learn math?”

In 2007 Mark Ashcraft published a paper Â in the Psychonomic Bulletin & Review with Jeremy Krause. You can download it here: ASHCRAFT_KRAUSE_2007

The issues discussed in this paper are the connections between working memory, mathematics anxiety, and mathematics performance.

Two main aspects to the argument in the paper, as I understand it, are as follows:

1. Many other studies have shown a decrease in mathematics performance in arithmetic problems as working memory is stressed. This seems not only to be true but to be fairly common-sensical: arithmetic problems that have more steps and do not rely on automatic recall make greater demands on working memory. This is associated with longer times to complete problems and an increase in errors.
2. Mathematical performance and mathematics anxiety are correlated. They write:

The higher oneâ€™s math anxiety, the lower oneâ€™s math learning, mastery, and motivation;”

highly math-anxious individuals get poorer grades in the math classes they take …”

and:

These correlations mean, simply but importantly, that as math anxiety increases, math achievement declines.”

On what do they base these latter assertions? They state there is:

• a math anxiety correlation of -0.30 with high school grades,
• a math anxiety correlation of -0.75 with enjoyment of math,
• a math anxiety correlation of -0.64 with motivation to take more math or do well in math;
• a math anxiety correlation of -0.31 with the extent of high school math taken.

They also state that:

The overall correlation between math anxiety and individualsâ€™ math achievement, as measured by standardized tests, is -0.31.”

Correlations are typically expressed as “r” values. This correlation coefficient r can vary from -1 to 1, with the extremes of -1 and 1 indicating 100% correlation.

So what does a correlation of Â -0.30 mean? What interpretation can we give to this? One commonly accepted interpretation is that $r^2$ tells us the degree of variation in the dependent variable (math performance) accounted for by variation in the independent variable (math anxiety).

If $r=-0.30$ what is $r^2$? A quick calculation tells us that in this case $r^2=0.09$ which is 9%.

Well, this pretty good, no? 9% of the variation in mathematics performance can be accounted for by variation in math anxiety.

But think of this the other way around: over 90% of the variation in mathematics performance is NOT accounted for by variation in math anxiety. In other words, math anxiety, as an independent variable, does not have much to do with variation in mathematics performance.

How about the highest correlation $r=-0.75$? Â In this case $r^2\approx 0.563$ so over 56% of the negatively correlated variation in enjoyment of mathematics is accounted for by variation in math anxiety. This does not seem surprising to me: I expect people who have math anxiety to not enjoy mathematics.

The other low r values all indicate one thing, and that is contrary to Ashcraft and Krause’s conclusion: math anxiety, as an independent variable, does not have much to do with variation in mathematics performance.

The authors go on to draw some speculative educational implications based on these unsound interpretations of low r values.

A danger, dear reader, is that this work may well make its way into educational psychology books and be taught to prospective mathematics teachers as proven fact.

Be wary dear reader. Be numerate and literate, read carefully, and critically. Â It’s a jungle out there!

### 3 Responses to "How not to report Pearson's r: a cautionary tale in mathematical cognition"

I teach high school Stats. I spend an entire class period (90 minutes) discussing the misuse and misrepresentation of statistics in the world. Why it happens, how it happens, who does it, etc.

This is just scary *shivers*

Pearson correlation is THE most abused statistical parameters. It is one of the “statistical rituals” many are observing without knowing their meaning. BTW, I am a part of this “many”, mainly due to insufficient statistical education.

Interestingly, last week I started a blog series on the same topic: http://www.inthehaystack.com/blog/correlation-matrices-part-1-euphoric-prelude/

I am always dubious when effect follows cause in a circle (or cycle). In another dimension you could have a ball. Fun learning can be more of both fun and learning, in a happy cycle. The music of math is more than just an allegory.