Number sense, age and school maths

If someone offers you a bag of candies and there are two to pick from, the two not containing the same number of sweets, you will most likely try to estimate which bag contains the most candies and pick that one. To do that, you use something called the approximate number system (ANS), which is the cognitive system that gives rise to basic numerical intuitions.

The ANS generates nonverbal representations of numerosity not only in humans (adults, children and infants) but also in nonhuman animals. At the cellular level, imaging studies suggest that these basic numerical intuitions are supported by neurons located in a region of the brain called the intraparietal sulcus.

  • The question

In this study published in the scientific journal PNAS in June 2012, the researchers were interested in two questions regarding the ANS:

– how does the precision of the ANS vary in humans depending on their age?
– is there a relationship between the ANS and more formal mathematical abilities, not only during the school years (when we acquire these abilities), but also beyond that (long after these abilities have been acquired)?

  • The method

To try and answer such questions, the research team needed to gather data from a huge number of people across a wide age range. They therefore set up a Web-based experiment instead of recruiting people to come in a lab. They posted an ANS test online, which was designed to assess the precision of an individual’s number sense, and within 3 months a bit more than 10,000 people had freely chosen to take the test. Before taking the ANS test, the participants had also agreed to fill out a questionnaire, reporting on their age and their own sense of how well they performed relative to their peers in various school subjects (mathematics, science, writing, etc.).

The researchers used a simple method to assess the participants’ number sense: they asked the subjects to judge which of two arrays of dots was more numerous. More precisely, the ANS dots test was made of 300 trials and took about 8 minutes to complete. For each trial, a mix of blue and yellow dots briefly flashed on the screen, and the participants were asked whether there were more blue or yellow dots. From one trial to the other, the ratio of blue to yellow dots varied. Of course if you look at a mix of 10 yellow and 20 blue dots, it will be easy to answer. However, if you see only briefly a mix of 15 yellow and 14 blue dots, then it’s a lot more difficult to estimate which of the yellow or blue dots are more numerous.

If you’re interested in testing your own number sense, a version of this test is available online at Once you have completed the test, you can also see how precise your number sense is compared to the general population.

  • The results

1) Number sense precision and school maths abilities

The researchers used the results of the ANS dots test to assess the precision of each participant’s ANS, which they indexed using two variables: a Weber fraction (w – and if you really want to know what it represents, look here) and an average response time (RT). They observed that these two variables were uncorrelated, suggesting that they may represent different abilities. According to the authors of the study, w is an “estimate of the internal noise, or confusability” of someone’s ANS, whereas RT represents the time the participants take to make their decision.

Analyzing the data collected from the 10,548 people who took the online ANS dots test and reported their school mathematics abilities, the researchers observed that individuals with better ANS precision tended to report being better in school maths relative to their peers, whereas participants with poorer ANS precision tended to report being worse in school maths than their peers. This relationship appeared to be independent of the age of the subject.

To test the reproducibility of their results, the researchers conducted another web-based experiment. This time, the ANS dots test was shorter (200 trials done in about 5 minutes), and 3,006 people completed the test and accompanying questionnaire. In this new sample of individuals, the same “modest, yet stable” (as the authors of the study put it) relationship between ANS precision and self-reported school maths abilities was observed.

For this new round of experiment, the researchers had also asked the participants to report their scores on the maths and verbal sections of the SAT (the SAT is the most widely used standardized examination for gaining entry to colleges in the US). 458 people reported their scores and the researchers observed that individuals with a more precise ANS tended to report higher scores on the SAT maths section (but not on the verbal one). Of note, the participants’ maths scores on the SAT correlated with their self-reported maths abilities relative to their peers, providing some degree of validation for the use of self-reported assessments of maths abilities in the first test group of over 10,000 people.

2) Number sense precision and age

Taking advantage of the size and heterogeneity of the group of people whose ANS precision had been tested (over 10,000 people ranging from 11 to 85 years of age), the researchers analyzed how the precision of the ANS changed according to age. They found that ANS representations were gradually improving during the school-age years, but that, surprisingly, the optimal precision of ANS representations was reached somewhat later in life than what they had expected, somewhere around 30 years of age.

Of course, despite the observation of developmental improvements as people aged, there remained large individual differences. The researchers thus found that about 1 in 8 adults had a number sense less precise than a typical 11-year old. So if you think that you have a better chance than your kid to pick the bag with the most candies, you are probably right. But then again, it might also turn out that you’re not smarter than a 5th grader.

Number sense across the lifespan as revealed by a massive Internet-based sample. Halberda J, Ly R, Wilmer JB, Naiman DQ, Germine L. Proc Natl Acad Sci U S A. 2012 Jul 10;109(28):11116-20. doi: 10.1073/pnas.1200196109
PMID: 22733748

ResearchBlogging.orgHalberda J, Ly R, Wilmer JB, Naiman DQ, & Germine L (2012). Number sense across the lifespan as revealed by a massive Internet-based sample. Proceedings of the National Academy of Sciences of the United States of America, 109 (28), 11116-20 PMID: 22733748


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