The relationship between non-symbolic multiplication and division in childhood

Effects of intelligence and approximate number system on the non-symbolic division ability in preschoolers

Recently, it has become evident that cognitive abilities such as the approximate number system (ANS), number knowledge, and intelligence affect individuals' fundamental mathematical ability. However, it is unclear which of these cognitive abilities have the greatest impact on the non-symbolic division ability in preschoolers. Therefore, in the present study, we included 4- to 6-year-old Korean preschoolers without prior formal education of division in order to test their ability to solve non-symbolic division problems, ANS acuity, and intelligence, and to determine the interrelationships among those functions (N = 38). We used the Panamath Dot Comparison Paradigm to measure the ANS acuity, employed non-symbolic division tasks to measure the ability to solve non-symbolic division problems, and measured the intelligence using the Korean version of the WPPSI-IV (Wechsler Preschool Primary Scale of Intelligence-IV). Our results showed that, in all conditions of the non-symbolic division tasks, the 4- to 6-years old children were able to perform better than chance level. Additionally, in a relatively easy condition, the children's performance showed a significant positive correlation with full-scale intelligence quotient (FSIQ) and ANS acuity; however, in a more complex condition, only FSIQ was significantly correlated with their performance. Overall, we found significant relationships between the children's performance in the non-symbolic division tasks and verbal comprehension, fluid reasoning, and processing speed index. Taken together, our findings demonstrate that preschoolers without formal education on the arithmetic problem solving can solve non-symbolic division problems. Moreover, we suggest that both FSIQ and ANS ability play essential roles in children's ability to solve non-symbolic division problems, highlighting the significance of intelligence on children's fundamental mathematical ability.

Is Nonsymbolic Arithmetic Truly "Arithmetic"? Examining the Computational Capacity of the Approximate Number System in Young Children

Young children with limited knowledge of formal mathematics can intuitively perform basic arithmetic-like operations over nonsymbolic, approximate representations of quantity. However, the algorithmic rules that guide such nonsymbolic operations are not entirely clear. We asked whether nonsymbolic arithmetic operations have a function-like structure, like symbolic arithmetic. Children (n = 74 4- to -8-year-olds in Experiment 1; n = 52 7- to 8-year-olds in Experiment 2) first solved two nonsymbolic arithmetic problems. We then showed children two unequal sets of objects, and asked children which of the two derived solutions should be added to the smaller of the two sets to make them "about the same." We hypothesized that, if nonsymbolic arithmetic follows similar function rules to symbolic arithmetic, then children should be able to use the solutions of nonsymbolic computations as inputs into another nonsymbolic problem. Contrary to this hypothesis, we found that children were unable to reliably do so, suggesting that these solutions may not operate as independent representations that can be used inputs into other nonsymbolic computations. These results suggest that nonsymbolic and symbolic arithmetic computations are algorithmically distinct, which may limit the extent to which children can leverage nonsymbolic arithmetic intuitions to acquire formal mathematics knowledge.

Young Children Intuitively Divide Before They Recognize the Division Symbol

Children bring intuitive arithmetic knowledge to the classroom before formal instruction in mathematics begins. For example, children can use their number sense to add, subtract, compare ratios, and even perform scaling operations that increase or decrease a set of dots by a factor of 2 or 4. However, it is currently unknown whether children can engage in a true division operation before formal mathematical instruction. Here we examined the ability of 6- to 9-year-old children and college students to perform symbolic and non-symbolic approximate division. Subjects were presented with non-symbolic (dot array) or symbolic (Arabic numeral) dividends ranging from 32 to 185, and non-symbolic divisors ranging from 2 to 8. Subjects compared their imagined quotient to a visible target quantity. Both children (Experiment 1 N = 89, Experiment 2 N = 42) and adults (Experiment 3 N = 87) were successful at the approximate division tasks in both dots and numeral formats. This was true even among the subset of children that could not recognize the division symbol or solve simple division equations, suggesting intuitive division ability precedes formal division instruction. For both children and adults, the ability to divide non-symbolically mediated the relation between Approximate Number System (ANS) acuity and symbolic math performance, suggesting that the ability to calculate non-symbolically may be a mechanism of the relation between ANS acuity and symbolic math. Our findings highlight the intuitive arithmetic abilities children possess before formal math instruction.

Approximate multiplication in young children prior to multiplication instruction

Prior work indicates that children have an untrained ability to approximately calculate using their approximate number system (ANS). For example, children can mentally double or halve a large array of discrete objects. Here, we asked whether children can perform a true multiplication operation, flexibly attending to both the multiplier and multiplicand, prior to formal multiplication instruction. We presented 5- to 8-year-olds with nonsymbolic multiplicands (dot arrays) or symbolic multiplicands (Arabic numerals) ranging from 2 to 12 and with nonsymbolic multipliers ranging from 2 to 8. Children compared each imagined product with a visible comparison quantity. Children performed with above-chance accuracy on both nonsymbolic and symbolic approximate multiplication, and their performance was dependent on the ratio between the imagined product and the comparison target. Children who could not solve any single-digit symbolic multiplication equations (e.g., 2 × 3) on a basic math test were nevertheless successful on both our approximate multiplication tasks, indicating that children have an intuitive sense of multiplication that emerges independent of formal instruction about symbolic multiplication. Nonsymbolic multiplication performance mediated the relation between children's Weber fraction and symbolic math abilities, suggesting a pathway by which the ANS contributes to children's emerging symbolic math competence. These findings may inform future educational interventions that allow children to use their basic arithmetic intuition as a scaffold to facilitate symbolic math learning.

First and Second Graders Successfully Reason About Ratios With Both Dot Arrays and Arabic Numerals

Children struggle with exact, symbolic ratio reasoning, but prior research demonstrates children show surprising intuition when making approximate, nonsymbolic ratio judgments. In the current experiment, eighty-five 6- to 8-year-old children made approximate ratio judgments with dot arrays and numerals. Children were adept at approximate ratio reasoning in both formats and improved with age. Children who engaged in the nonsymbolic task first performed better on the symbolic task compared to children tested in the reverse order, suggesting that nonsymbolic ratio reasoning may function as a scaffold for symbolic ratio reasoning. Nonsymbolic ratio reasoning mediated the relation between children's numerosity comparison performance and symbolic mathematics performance in the domain of probabilities, but numerosity comparison performance explained significant unique variance in general numeration skills.

Approximate arithmetic training does not improve symbolic math in third and fourth grade children

BACKGROUND

Prior studies reported that practice playing an approximate arithmetic game improved symbolic math performance relative to active control groups in adults and preschool children (e.g. Park & Brannon, 2013, 2014; Park et al., 2016; Szkudlarek & Brannon, 2018). However, Szkudlarek, Park and Brannon (2021) recently failed to replicate those findings in adults. Here we test whether approximate arithmetic training yields benefits in elementary school children who have intermediate knowledge of arithmetic.

METHOD

We conducted a randomized controlled trial with a pre and post-test design to compare the effects of approximate arithmetic training and visuo-spatial working memory training on standardized math performance in third and fourth grade children.

RESULTS

We found that approximate arithmetic training did not yield any significant gains on standardized measures of symbolic math performance.

CONCLUSION

A Bayesian analysis supports the conclusion that approximate arithmetic provides no benefits for symbolic math performance.

Shared and distinct neural circuitry for nonsymbolic and symbolic double-digit addition

Symbolic arithmetic is a complex, uniquely human ability that is acquired through direct instruction. In contrast, the capacity to mentally add and subtract nonsymbolic quantities such as dot arrays emerges without instruction and can be seen in human infants and nonhuman animals. One possibility is that the mental manipulation of nonsymbolic arrays provides a critical scaffold for developing symbolic arithmetic abilities. To explore this hypothesis, we examined whether there is a shared neural basis for nonsymbolic and symbolic double-digit addition. In parallel, we asked whether there are brain regions that are associated with nonsymbolic and symbolic addition independently. First, relative to visually matched control tasks, we found that both nonsymbolic and symbolic addition elicited greater neural signal in the bilateral intraparietal sulcus (IPS), bilateral inferior temporal gyrus, and the right superior parietal lobule. Subsequent representational similarity analyses revealed that the neural similarity between nonsymbolic and symbolic addition was stronger relative to the similarity between each addition condition and its visually matched control task, but only in the bilateral IPS. These findings suggest that the IPS is involved in arithmetic calculation independent of stimulus format.