Biology in the News Explained

“Number sense” and math ability: what assumptions drive the science?

Here’s the headline from Science Daily:  You can count on this: Math ability is inborn, new research suggests.

It’s a headline (and story) loaded with assumptions.  While the paper itself has flaws, even worse is a poorly written article that seems to be not much beyond an inflated press release promoting a study that really doesn’t show much. The article is certainly no journalistic attempt to objectively evaluate the study’s significance (only the authors were interviewed).  Sadly, with both public scientific knowledge and journalism itself in such steep decline, this unvarnished promotion of work based on naive assumptions and a misunderstanding of what statistics tell us is what passes for a mainstream science article these days.

To understand the problem with this paper, the concepts of “number sense” and “formal mathematical ability” need to be explained.  ”Number sense” is a rather general term whose meaning often varies with whoever is using it (Berch, 2005) but in this case it refers to a more precise phenomenon better termed as ability in “approximate arithmetic,” (“approximate number system acuity” in the paper) which refers to the widespread and innate ability in both humans and many animals (De Cruz et al., 2010) to determine relative numbers of objects.  For example, it is the sense we use to know which of two piles of rocks contains more.

This is to be distinguished from “exact arithmetic,” or calculating exactly that there are six more rocks in one pile than in the other.  This ability is considered to be learned, and unique to humans.



In the study, 3- to 5-year-olds (pre-K) were tested both for approximate arithmetic ability and “mathematical ability” (the Test for Early Mathematics Ability (TEMA), which includes the following measures of learned ability:

numbering skills (e.g. verbally counting the number of objects on a page), number comparison facility (e.g. determining which of two spoken number words is larger), numeral literacy (e.g. reading Arabic numerals), mastery of number facts (e.g. retrieving multiplication facts), calculation skills (e.g. solving written addition and subtraction problems), and number concepts (e.g. answering how many 10s are in 100).

Approximate mathematical ability scores were estimated for accuracy, response time (RT) and “w” (essentially a model of variance in response accuracy)  and the results were correlated with the TEMA results (controlling for age and verbal ability) using regression analysis.  Here is the key result:

each of these estimates significantly correlated with math ability (accuracy: R2 = 0.18, p < .001; w: R2 = 0.07, p < .01; RT: R2 = 0.08, p <.001). This means that faster RT and greater accuracy on the ANS acuity task are associated with higher math ability.

And with these results we see the researchers fall into the same trap that has captured so many (especially in medical and social sciences): equating statistical significance with meaningful results.  Look again at the numbers.  For their three different regressions, the range in the percentage of results explained by their model is 7-18%, and yet their final conclusion in the paper is “Our study thereby supports the notion of a tight link between a primitive sense of number and more formal math abilities.”  Really? Look past the statistically significant trend.  Of what possible use is knowing that a measurement of “number sense” explains (on average) such a small amount of variation in exact arithmetic understanding? This result brings us no closer to understanding the drivers of mathematical competence in an individual student.

The Science Daily article goes much further, suggesting that because approximate arithmetic sense is “innate” and “mathematical ability” is learned, therefore “mathematical ability” is inborn.  This is absurd, because (thankfully) the authors themselves say nothing of the kind in their paper.  Even if you take the regression results as meaningful, there is no demonstration of causation.

Indeed, the authors do point out in the discussion that there could easily be a co-variate which affects both approximate and exact arithmetic abilities, but as usual this obvious caveat gets lost in the media promotion of the paper.  A glaring hole that suggests this is not a trivial concern is the apparent assumption that testing kids before “formal mathematics instruction” draws some sort of a valid line between being previously exposed to math vs. not.  What if informal mathematics instruction (e.g. talking about numbers and their relationships a lot at home or in preschool, practice counting, etc.) is just as important or more so?  Unfortunately there is no assessment in the paper of what sort of previous exposure to numbers and arithmetic the children have had up to that point.

But the authors downplay the concept that just because an approximate arithmetical sense is innate doesn’t mean that it is unchanging.  As summarized in a review by Berch (2005):

…most theorists who adhere to the view that number sense has a long evolutionary history and a specialized cerebral substrate do not judge that it thereby constitutes a fixed or immutable entity.  Rather, the emergence of rudimentary components of number sense in young children is thought to occur “spontaneously without much explicit instruction”… at least under nondisadvantaged rearing conditions.

Furthermore… the neurocognitive systems supporting these elementary numerical abilities include what has been referred to as skeletal principles…, because they provide just the foundational structure for the acquisition of these abilities. Concomitantly, engaging in numerical kinds of games and activities is thought to “flesh out” these principles.

If both types of arithmetic sense are indeed affected by learning, then that also reduces the significance of finding them to be correlated.  The Science Daily article, though, wants to go even further and suggest that the paper shows that “mathematical ability” is determined before birth, when of course it shows nothing of the kind.  But the media these days seem to love to tell stories that absolve us of any personal responsibility for our success, reinforcing underachievement by telling people that if they’re not good at something, it’s just not their fault.  Math of course is particularly prone to this type of reasoning already; the United States seems to have a uniquely math-averse culture, in which people routinely state that they “can’t do math” as if it were a point of pride.


(Incidentally, the researchers checked for but found no difference based on gender in their number sense test, an unsurprising result — but one that will probably be ignored by those pushing their pro-gender-differences agenda.)



Berch, D. B. 2005. Making sense of number sense: implications for children with mathematical disabilities. Journal of Learning Disabilities 38(4):333-339

De Cruz H., Neth H., and Schlimm D. 2010. The cognitive basis of arithmetic. In: B. Löwe, T. Müller (eds.). PhiMSAMP. Philosophy of Mathematics: Sociological Aspects and Mathematical Practice. College Publications, London. Texts in Philosophy 11; pp. 59–106.

Libertus M.E., Feigenson L. and Halberda J., 2011. Preschool acuity of the approximate number system correlates with school math ability. Developmental Psychology. Article first published online: 2 AUG 2011, DOI: 10.1111/j.1467-7687.2011.01080.x



Image By Chanchicto (Own work) [CC-BY-SA-3.0 ( or GFDL (], via Wikimedia Commons


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