Where does human knowledge begin? Research on human infants, children, adults, and nonhuman animals, using diverse methods from the cognitive, brain, and computational sciences, provides evidence for six early emerging, domain-specific systems of core knowledge. These automatic, unconscious systems are situated between perceptual systems and systems of explicit concepts and beliefs. They emerge early in infancy, guide children's learning, and function throughout life.

Previous research reported that college students' symbolic addition and subtraction fluency improved after training with non-symbolic, approximate addition and subtraction. These findings were widely interpreted as strong support for the hypothesis that the Approximate Number System (ANS) plays a causal role in symbolic mathematics, and that this relation holds into adulthood. Here we report four experiments that fail to find evidence for this causal relation. Experiment 1 examined whether the approximate arithmetic training effect exists within a shorter training period than (...) originally reported (2 vs 6 days of training). Experiment 2 attempted to replicate and compare the approximate arithmetic training effect to a control training condition matched in working memory load. Experiments 3 and 4 replicated the original approximate arithmetic training experiments with a larger sample size. Across all four experiments (N = 318) approximate arithmetic training was no more effective at improving the arithmetic fluency of adults than training with control tasks. Results call into question any causal relationship between approximate, non-symbolic arithmetic and precise symbolic arithmetic. (shrink)

The Approximate Number System (ANS) is a system that allows us to distinguish between collections based on the number of items, though only if the ratio between numbers is high enough. One of the questions that has been raised is what the representations involved in this system represent. I point to two important constraints for any account: (a) it doesn’t involve numbers, and (b) it can account for the approximate nature of the ANS. Furthermore, I argue that representations of pure (...) magnitude with vehicles that have an imprecision in the value of the unit of measurement (further clarified through a formal model from measurement theory) fit both these requirements. (shrink)

The goals of the present study were to test whether (and which) basic numerical abilities can be improved with training and whether training effects transfer to improvement in children’s math achievement. The literature is mixed with evidence that does or does not substantiate the efficacy of training basic numerical ability. In the present study, we developed a child-friendly software named ‘123 Bakery’ which includes four training modules; non-symbolic numerosity comparison, non-symbolic numerosity estimation, approximate arithmetic and symbol-to-numerosity mapping. Fifty-six first graders (...) were randomly assigned to either the training or control group. The training group participated in 6 weeks of training (5 times a week, 30 minutes per day). All participants underwent pre- and post- training assessment of their basic numerical processing ability (including numerosity discrimination acuity, symbolic/non-symbolic magnitude estimation, approximate arithmetic, symbol-to-numerosity mapping), overall math achievement and intelligence, 6 weeks apart. The acuity for numerosity discrimination (approximate number sense acuity; ANS acuity) significantly improved after training, but this training effect did not transfer to improvement in symbolic, exact calculation, or any other math ability. We conclude that basic numerical cognition training leads to improvement in ANS acuity, but whether this effect transfers to symbolic math ability remains to be further tested. (shrink)