h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

The purpose of the OU residential school week I did last week was essentially to design and execute an experiment. The particular thing I was looking at was numerical comparison and how we humans actually compare two numbers.

It is an area still not fully tied down as there are differences depending on how many digits the number has and whether you are doing the comparison externally (two numbers written down or displayed onscreen) or internally (holding one number in memory and then comparing others against it).

Theres a couple of theories and the papers I was working from had a bit of a gap in it so I checked up on that. The main area of investigation is to do with whether or not we compare external numbers (both displayed on screen) as a whole or by their individual digits. I.e. do we tokenise '49' and '55' and then compare those tokens or do we compare 4 with 5 and 9 with 5?

One way to look for this is what has been called the Stroop like effect. Stroop being the bit where the word 'Green' is displayed in red ink and you have to name the ink colour not the word. The stroop effect is therefore really an interference effect.

With numbers this interference effect was hypothesised as coming into play when the decade digits had the opposite relation to the unit digits.

I.e. in the case above of 49 and 55;
4 is less than 5
9 is greater then 9

If we compare numbers by individual digits then it should take a bit longer to work this out because the 9 > 5 gives a misleading answer than has to be overridden by the 4 < 5.

Compare this to 41 and 55 where both digits of 41 are less than their equals in 55 and thus no effect should be seen.

Well, we did our experiment and lo and behold there 8is* a stroop like effect, 49&55 does take longer to compare than 41 & 55. Not by much, but it is strongly statistically significant (p<0.001). Which is great, we spent a week proving that the stroop effect, one of the most robust findings in psychology, works. I bet they're all glad we helped them out

Anyway, I was reading a 'stupid things customers say' website and I found this:
http://notalwaysright.com/math-saves-the-day-yet-again/1751

where a customer was complaining that they were being charged $299 but that the advert said $315 - until the store assistant pointed out that 299 < 315

And I thought - that's exactly the experiment we did! Except presumably with 3 digit numbers where 2 are greater the effect will be reinforced!

Rather a long winded post but I found it interesting to see a practical display of the effect we had found in our experiment

(p.s. original paper we used was:
Zhang, J. and Wang, H. (2005) 'The effect of external representations on numeric tasks', The Quarterly Journal of Experimental Psychology, vol.58A, no.5, pp.817-38

if you are interested.)

<<where a customer was complaining that they were being charged $299 but that the advert said $315 - until the store assistant pointed out that 299 < 315>.>>

At first i understood that as the customer being "right" if you see what I mean. Interesting post though, even if I don't fully understand all of it.

s'ok, ask away if you want anything clarified

< 5 gives a misleading answer than has to be overridden by the 4 < 5.>>

First question shouldn't "9 is greater than 9 be 9 is greater than 5? and how does 9>5 give a misleading answer?

"shouldn't "9 is greater than 9 be 9 is greater than 5?"
Erm, yes, well spotted, have a choccy sweet

"and how does 9>5 give a misleading answer?"
Because if we do process the number digit by digit then we would be keeping a tally of greater thans and less thans.

So for example say we compare units first, we'd go
for 49 & 55

9 > 5 - ooh, looks like the first number is larger!
4 < 5 - ahh, no it isn't, it's less.

In truth the actual theory is not sequential but parallel, so the brain takes the incoming digits and streams each digit to a comparison process and then feeds all those comparisons together weighting each one by position.

It is in a way a negative example as well as a positive one. If there is any effect here (and there is) then we can't be processing them holistically (ie tokens).

I think it is a good deal more complex than that as most people will say they just look at the decades digit and compare those. And one other effect we found was that when the two numbers to compare are in the same decade then the response time is even longer. So clearly there is more to it than just this.

But the stroop like effect shows that there is some sort of interaction going on with the individual digits as if we used tokens OR if we just used the decades then there would be no response difference for the stroop effect.

BTW, the non stroop condition was both digits having the same relationship, i.e. 41 with 55 and 79 with 55 since in those cases both digits are larger or smaller than 55.

So effectively we compared the response time for 41 vs 55 with the response time for 49 vs 55 (and other combinations, 192 comparisons per participant overall) and there is indeed a difference so the units must be playing some part in the processing.

Thanks for explaining that. Sounds quite interesting. So where does stuff like dyscalculia come into this then?

this is interesting basic on discalculia:
http://www.bbc.co.uk/skillswise/tutors/expertcolumn/dyscalculia/
"Dyscalculia is an individual's difficulty in conceptualizing numbers, number relationships, outcomes of numerical operations and estimation - what to expect as an outcome of an operation."

So if there is a problem with conceptualising the number, recognising it and indeed in relating it to other numbers one would expect either a higher response time or more errors in the responses. These results would skew the overall averages and thus affect any apparent effects. Removing this allows us to concentrate more on the specific thing we are looking at.

I.e when designing and performing an experiment the aim is to remove as many variables from the equation as possible so we can get a clear answer that is not skewed by some other effect we are not looking at.