Dehaene (2011). Small heads for big calculations.” – Article summary

People do not have innate mechanisms for complex calculations (e.g. math). However, children typically do spontaneously come up with some form of counting without being explicitly taught. It may be that counting knowledge is innate but it is also possible that this develops through imitation and that children, while being able to count, do not know the meaning of these numbers.

At three and a half years of age, children know that the order in which one recites numerals is crucial. Young children are able to point out subtle counting errors (e.g. count something twice) and by four years of age, children have mastered the basics of how to count. Children do not know the meaning of counting until the end of their fourth year.

Children tend to come up with calculation algorithms without explicit instruction (e.g. adding numbers up using their fingers). Children initially have big difficulties in counting without their fingers and may explicitly state their process (i.e. name what they are doing), though this requires a lot of effort and concentration. The minimum strategy refers to starting an addition equation with the larger quantity and count a number of times equal to the smaller of the two numbers (e.g. 4 + 2 = four, five, six, six!). This strategy underlies most of children’s calculations before the onset of formal schooling. Children develop an intuitive understanding of commutativity (i.e. a+b=b+a) without any formal schooling. Between the ages of 4 and 7, children exhibit an intuitive understanding of what calculations mean and how they should best be selected.

Young adults rarely solve addition and multiplication problems by counting but retrieve results from a memorized table. Accessing this table takes longer as the operands get larger. This may be due to the accuracy of mental representations dropping with number size (1), order of acquisition influencing memory (2) and the amount of drilling (i.e. less training with larger number sizes because they are less common) (3). This means that memory plays a central role in adult mental arithmetic.

The multiplication table is more difficult to retain in memory because arithmetic factors are not arbitrary and independent of each other. Human memory is associative (i.e. it links events with each other). This permits the reconstruction of memories on the basis of fragmented information. It allows one to remember a lot with only a small piece of information (1), it allows one to take advantage of analogies (2) and it allows one to apply knowledge acquired under other circumstances to a novel situation (3). However, associative memory is a weakness when knowledge must be kept from interfering with each other. This means that when one wants to remember the answer to 7x5, all the associated arithmetic functions (e.g. 7x6) are also activated, making it more difficult to remember.

The automatization of arithmetic memory (e.g. seeing two numbers and automatically adding them up) starts at age seven. It is possible that children eventually do learn (part of) the arithmetic table through recording it in verbal memory. Calculation becomes tied to the language in which it is learned at school because arithmetic tables are learned verbatim. Reading errors do not occur after multiplication but during it. In parallel to calculating the exact result, the brain also calculates an estimate of its size. To calculate fast, the brain is forced to ignore the meaning of the computations it performs.

Children’s subtraction algorithms have some systematic errors (i.e. bugs). This is because textbooks do not cover all forms of subtraction and the child needs to come to one’s own conclusions following the teacher’s behaviour and the examples.

Allowing children to use calculators rather than having to memorize the arithmetic table may allow them to concentrate on meaning and sharpen their natural sense of approximation. It may help children adore numbers rather than despise them. However, there are alternatives for calculators and children do need to memorize basic arithmetic (e.g. 2x3 = 6). The goal of math curricula should be to improve children’s fluency in arithmetic and not perpetuate a ritual (i.e. memorize the arithmetic table).

It is possible that many adults do not really understand when to apply mathematics knowledge appropriately. They do not have a deep understanding of arithmetic principles yet they are able to compute. Innumeracy refers to people without such a deep understanding and this puts one at risk to draw conclusions based on a reasoning that seems mathematical but is only so in appearance (e.g. two tubs of 35 degree water makes one tub of 70 degree water; base rate fallacy). Errors as a result of innumeracy occur because people jump to conclusions without considering the relevance of the computations they perform.

Arithmetic makes use of the prefrontal cortex and this develops until adulthood, putting adolescents (and children) at risk of arithmetical impulsiveness. Innumeracy may result from the difficulty of controlling the activation of arithmetic schemas distributed in multiple cerebral areas. Arithmetic makes use of a lot of brain networks and is controlled by the prefrontal cortex and as this is not fully developed, people show arithmetic impulsiveness. Schooling plays a crucial role in battling innumeracy because it helps children draw links between the mechanics of calculation and its meaning.

Schools often ignore and do away with one’s precocious math abilities (e.g. finger counting) and this can have a negative effect on their subsequent opinion of mathematics. It gives the idea that it is detached from intuition and ruled by arbitrariness. The insistence on mechanical computation at the expense of meaning is detrimental for interesting children in mathematics. Intuition needs to be encouraged. Mathematical knowledge should be grounded on concrete situations rather than on abstract concepts.