^{*}

^{a}

^{a}

In this study, it was investigated how domain-specific (number sense) and domain-general (working memory, creativity) factors explain the variance in mathematical abilities in primary school children. A total of 166 children aged 8 to 10 years old participated. Several tests to measure math ability, mathematical creativity, number sense, verbal and visual spatial working memory and creativity were administered. Data were analyzed with a series of correlation and regression analyses. Number sense, working memory and creativity were all found to be important predictors of academic and creative mathematical ability. Furthermore, groups with math learning disabilities (MLD) and mathematical giftedness (MG) were compared to a typically developing (TD) group. The results show that the MLD group scored lower on number line estimation and visual spatial working memory than the TD group, while the MG group differed from the TD group on visual spatial working memory and creativity. It is concluded that creativity plays a significant role in mathematics, above working memory and number sense.

Research on individual differences in the development of mathematical cognition has pointed to two main underlying cognitive factors: number sense and working memory (e.g.,

Number sense (NS) can be defined in very different ways, varying from more restricted definitions as the ability to estimate and understand approximate magnitudes (cf.

The important role of number sense in mathematics is also shown by a large body of research that studied children with MLD. Number sense deficits are found to be a main characteristic of children with serious difficulties in mathematics (e.g.,

Many studies have shown that working memory is an important domain-general predictor of mathematical performance (e.g.,

A distinction can be made between verbal working memory and visual spatial working memory skills, for the processing of verbal and visual spatial information respectively. Some authors have found that the visual spatial working memory is most strongly related to mathematics (

The role of working memory in mathematics is also shown by studies investigating children with MLD. Many studies have shown that children with MLD in general have lower working memory skills than TD children (e.g.,

How creativity comes into play in the process of mathematical development, is yet largely unknown. Creativity can be defined as the production of novel and useful products (broadly interpreted) within a social context (

The question is what is necessary to become excellent in mathematics. Gifted children have been found to differ from non-gifted children in the ability to find multiple solutions (

The goal of the current study was to compare the role of creativity to that of working memory and number sense in mathematical learning. It is the first study to combine these three predictors of mathematics into one model. This makes it possible to study these predictors not only in relation to mathematics, but also to each other. Specific attention is given to children with MLD and MG children, because the profiles of these groups can inform us about what skills are necessary to become proficient or even excellent in mathematics. Three research questions have been formulated:

How are different aspects of number sense and working memory related to mathematics in fourth graders? This first question will be answered to investigate the specific contributions of number sense and working memory to math in children from fourth grade. For number sense, both non-symbolic and symbolic tasks will be used, and for working memory verbal and visual spatial tasks. It is expected that the symbolic number sense skills are more strongly related to math than the non-symbolic number sense skills (cf.

Is creativity related to academic math tasks and to creative math tasks? The relations between a general creativity task and two types of math tasks are investigated. The first task reflects the current mathematics curriculum, the second task is designed to measure mathematical creativity, because it asks for multiple solutions. Mathematical creativity requires both creative skills and mathematical skills and can be regarded as a subcomponent of mathematics (

What are the differences in number sense, working memory and creativity between students with mathematical difficulties and typically and mathematically gifted students? To get more insight into the predictors that may be responsible for mathematical difficulties on one hand and mathematical strengths on the other, the weakest and highest performing children of the sample have been compared to the average performing children. It is expected that children with MLD show deficiencies in number sense and visual spatial working memory (cf.

Children were recruited from 11 schools in a large city in the middle of the Netherlands. All children from fourth grade were asked to participate (^{th} percentile (score V) on the CITO math test, and had a history of low performance in mathematics according to school data (>1 ^{th} percentile (score I) and a history of high mathematical performance (> 1

Statistic | TD | MLD | MG |
---|---|---|---|

130 | 16 | 20 | |

% boys | 47.70 | 25.00 | 65.00 |

9.64 | 9.89 | 9.50 | |

0.62 | 0.62 | 0.43 |

A series of tests was administered to measure students’ number sense, working memory, intelligence, creativity, and mathematical ability.

A newly developed computer-based number sense test was used (

In the symbolic comparison task, children were confronted with two numbers below 100 on the screen, and they have to respond which one is the largest number by pressing an ‘a’ for the left number and an ‘l’ for the right number. Children were asked to respond as fast as possible. The largest number was equally presented on the left and the right. The task started with a training block of 6 items, which were not used in the scoring. After the training block 33 fixed items were randomly presented to the children. These items differed in the degree of difficulty. The difficult items included two numbers that were close to each other (ratio 0.88; e.g. 14 vs.16). Average and easy items consisted of numbers that had respectively a ratio of 0.75 (e.g. 12 vs.16) and 0.625 (e.g. 10 vs.16). Both accuracy (percentage correct) and reaction time were scored. The internal consistency of the task is high (α = .84;

The non-symbolic comparison task is comparable to the symbolic comparison task, with sets of dots instead of numbers. The amount of dots range from 1 – 100. Children were asked to respond as fast as possible but accurately. The largest size of dots was equally presented on the left and the right. The task started with a training block of 6 items, which were not used in the scoring. After the training block 43 items were presented to the children in three different blocks. After each block (of respectively 14, 14 and 15 items) there was a short break. In each block the items were randomly presented. In all items the size or the convex hull (i.e. the smallest measured surface in which the dots can fit) of the dots were manipulated. In 14 items the size of the dots and the convex hull of the dots were respectively incongruent and congruent with the amount of dots. In another 14 items this was the other way around. Furthermore in 15 items the convex hull stayed the same and only the size of the dots was manipulated (incongruent or congruent). The items within each different manipulation also differed in difficulty. Three different ratios were used (0.625, 0.75 & 0.88). An example of one item can be found in

Item of the non-symbolic comparison task. The left group of dots has 16 dots. The right group of dots has 20 dots. The convex hull of the right group of dots is incongruent with the number of dots, but congruent with regard to the size of the dots.

The number line estimation task demanded children to place a lever on a number line from 0 to 100 to position the digit that was presented. The test started with two training items in which the child had to place 0 and 100 on the number line. After the training block, 30 test items were randomly presented to the children. The proportion of explained variance (^{2}) was computed by fitting the answers of each child on a linear curve (see also

Two online computerized working memory tasks were administered (

The Lion game is a visual spatial complex span task, in which children have to search for colored lions (

The Monkey game is a verbal span-backwards task, in which children have to remember and recall different words backward. Children heard spoken one-syllable words and had to recall them backwards, by clicking on the written words presented visually in a 3 x 3 matrix. Before the task started, children were presented with four practice sets. In the first two practice sets, children were asked to recall two words forwards. After those sets, children were informed about the backward recall procedure and were presented with two more practice sets in a backward fashion. After each practice children received feedback on their performance. After the training the assessment started, which consisted of 20 items in five difficulty levels (ranging from two words in Level 1 to six words in Level 5) in which working memory load is manipulated by the number of words children have to remember and recall backward. Levels were presented in increasing order. The item sets within each level were constructed using randomization with regard to the sequence of words. The proportion of items recalled in the correct location was used as measure. The Monkey game has excellent internal consistency (Cronbach’s α between .78 and .89) and shows good concurrent and predictive validity (

The Test of Creative Thinking-Drawing Production (TCT-DP;

Two tests were used to measure mathematical achievement, a standard test that covers a broad range of mathematical tasks and that is typical for current academic math tests, and a test that measures mathematical creativity by requiring multiple solutions.

Scores from the national Cito mathematics test (

The Mathematical creativity test (MCT), developed by

IQ was estimated based on two subtests of the ‘Nederlandse Intelligentietest voor Onderwijsniveau’ (Dutch Intelligence test for Education level;

Data were collected in the fall of 2015 by research assistants with at least a bachelor’s degree in Special Education, supervised by the first author. Active informed consent was obtained from the parents of all children involved. Information about students’ age and gender was provided by the teacher. The MCT and TCT-DP were administered groupwise by two research assistants in a session of approximately one hour. The IQ tests were administered groupwise in a second 30-minute session within one week of the first session. Testing was standardized by means of elaborate written instruction for the testers. Instructions were read aloud. Additional instruction was provided for individual students when needed. Students were not allowed to copy the work of their fellow-students and to talk during test sessions. After the groupwise administration, children completed respectively the Monkey and Lion Game and the Number Sense tests individually on a laptop computer, supervised by one of the research assistants. The study was approved by the ethics committee of the faculty of Social and Behavioural Sciences of Utrecht University (FETC14-005).

The dataset was not complete. Especially in the computerized number sense tests, a relative large amount of the data was missing (17%), mostly due to computer problems. There were also data missing in the creativity test (9.6%), because it was impossible to score some tests. This was for example because children erased many lines and it was difficult to see which ones were erased or not, or they used a very thick pencil, or had very bad handwriting. Missing data were handled by using pairwise deletion. Pairwise deletion of the missing data was not considered problematic since the data in this study were missing in a random pattern (Little’s MCAR test (^{2} (60) = 53.59,

To examine the relations between mathematics, working memory, number sense, and creativity, Bayesian correlation analyses were conducted, with the statistical package JASP (

For each analysis, specific hypotheses were formulated and tested. Bayesian analyses involve the calculation of the Bayes Factor (or BF) of an informative hypothesis versus the alternative hypothesis. The BF represents the amount of support from the data in favor of one hypothesis compared to another hypothesis. In addition to the BF, posterior model probabilities (PMPs) can be computed, representing the relative support for a specific hypothesis within a set of hypotheses. If the PMP of one hypothesis is larger than the PMP of the unconstrained hypothesis, the constraints used to specify the hypothesis are supported by the data; and if the PMP of a first hypothesis is larger than the PMP of a second hypothesis, the support in the data is larger for the first than for the second hypothesis. Note that the sum of the PMPs for a set of competitive hypotheses is always one. Moreover, in this type of analysis it is not necessary to provide additional estimates of effect size. The effect size is incorporated into the BF in the sense that a larger effect size results in a larger BF (

In

_{NS <} μ_{S}, μ_{NL}). There was little support for this hypothesis in the data regarding the MCT (BF = 2.15, PMP = .68) and some support regarding the CITO (BF = 2.98, PMP = .75).

Variable | Range | |||
---|---|---|---|---|

IQ (averaged z-score) | 166 | 0.00 | 0.80 | -1.86 – 1.98 |

CITO Ability Score (0 - 150) | 166 | 86.49 | 14.57 | 35 – 133 |

MCT (z-score) | 162 | 0.00 | 0.91 | -2.03 – 3.03 |

Numberline (linear fit 0.00 – 1.00) | 138 | .91 | .10 | .47 – .99 |

Comparison NS (% correct 0-1) | 140 | .63 | .12 | .14 – .81 |

Comparison NS (rt) | 140 | 1.16 | 0.36 | 0.26 – 2.07 |

Comparison S (% correct 0-1) | 139 | .91 | .08 | .48 – 1.00 |

Comparison S (rt) | 139 | 1.01 | 0.22 | 0.52 – 1.67 |

Visual spatial WM (proportion 0-1) | 165 | .72 | .10 | .36 – .94 |

Verbal WM (proportion 0-1) | 165 | .56 | .08 | .26 – .79 |

Creativity (raw score 0 - 72) | 150 | 23.19 | 8.38 | 5 – 51 |

Variable | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|

1. IQ | — | .41*** | .20* | .17 | .02 | .02 | -.28** | .17 | .19 | .59*** | .53** |

2. Verbal WM | — | .49*** | .01 | .07 | .02 | -.10 | .14 | .15 | .46*** | .32*** | |

3. Visual spatial WM | — | .01 | .09 | .13 | -.09 | .10 | .20* | .33*** | .23** | ||

4. Comparison NS ACC | — | .30*** | .13 | -.11 | .21* | .05 | .19 | .17 | |||

5. Comparison NS RT | — | .28*** | .29*** | .13 | .08 | .01 | -.08 | ||||

6. Comparison S ACC | — | .45*** | .13 | .04 | .07 | .02 | |||||

7. Comparison S RT | — | .04 | -.14 | -.27** | -.21* | ||||||

8. Numberline | — | .02 | .39*** | .24** | |||||||

9. Creativity | — | .29*** | .28*** | ||||||||

10. CITO | — | .69*** | |||||||||

11. MCT | — |

*BF > 3. **BF > 10. ***BF > 30.

Creativity is also related to both math tasks, but there is no difference in strength between the two correlations. The hypothesis that creativity is more strongly linked to the creative math task than to the academic math task, was not confirmed when controlled for IQ (BF = 1.15, PMP = .13), because the hypothesis that the effects were the same received much more support (BF = 6.80, PMP = .76). Neither working memory nor creativity was related to any of the number sense tasks.

In a comparison of regression analyses (see

Model | CITO |
MCT |
||
---|---|---|---|---|

BF | BF | |||

Unconstrained model | 1.00 | .01 | 1.00 | .04 |

Model 1 (WM) | 3.52 | .04 | 2.44 | .10 |

Model 2 (WM+NS) | 31.83 | .33 | 6.77 | .29 |

Model 3 (WM+NS+creativity) |

Predictor | CITO | MCT |
---|---|---|

Intelligence^{a} |
.39 | .42 |

Visual spatial WM | .08 | .05 |

Verbal WM | .24 | .11 |

Numberline | .29 | .15 |

Comparison | .10 | .04 |

Creativity | .11 | .15 |

^{a}Variable included as covariate in the model.

Finally, the differences between the three performance groups were tested (see

Variable | TD ( |
MLD ( |
MG ( |
|||
---|---|---|---|---|---|---|

IQ | 0.02 | 0.77 | -0.65 | 0.42 | 0.69 | 0.75 |

CITO | 85.64 | 11.01 | 67.19 | 12.38 | 107.40 | 10.34 |

MCT | -0.11 | 0.78 | -0.74 | 0.41 | 1.25 | 0.82 |

Numberline | 0.91 | 0.09 | 0.80 | 0.15 | 0.94 | 0.07 |

Comp NS ACC | 26.67 | 5.01 | 28.00 | 4.10 | 28.41 | 5.52 |

Comp NS RT | 1.18 | 0.38 | 1.07 | 0.28 | 1.14 | 0.29 |

Comp S ACC | 29.99 | 2.61 | 29.30 | 3.47 | 30.29 | 3.04 |

Comp S RT | 1.03 | 0.22 | 0.97 | 0.23 | 0.92 | 0.20 |

Visual spatial WM | 0.72 | 0.10 | 0.67* | 0.09 | 0.75 | 0.08 |

Verbal WM | 0.56 | 0.08 | 0.53 | 0.09 | 0.58 | 0.08 |

Creativity | 22.94 | 8.49 | 20.14 | 6.51 | 27.75 | 7.54 |

Variable | Model 0 | Model 1 | Model 2 | Model 3 | ||||
---|---|---|---|---|---|---|---|---|

μ_{TD}, μ_{MLD}, μ_{MG} |
μ_{TD} = μ_{MLD} = μ_{MG} |
μ_{MLD} < μ_{TD} < μ_{MG} |
μ_{MLD} < μ_{TD} = μ_{MG} |
|||||

BF | PMP | BF | PMP | BF | PMP | BF | PMP | |

Comp NS ACC | 1 | .22 | 0.49 | .11 | 0.58 | .13 | ||

Comp NS RT | 1 | .12 | 1.14 | .14 | 2.11 | .26 | ||

Comp S ACC | 1 | .08 | 3.07 | .26 | ||||

Comp S RT | 0.23 | .10 | 0.16 | .07 | ||||

Numberline | 1 | .10 | 0.04 | .00 |

For the domain general variables, the hypothesis that the MLD group scored below the TD group and the MG group above the TD group was compared to a model in which no differences between the group exist and an unconstrained model (see

Variable | Model 0 | Model 1 | Model 2 | Model 3 | ||||
---|---|---|---|---|---|---|---|---|

μ_{TD}, μ_{MLD}, μ_{MG} |
μ_{TD} = μ_{MLD} = μ_{MG} |
μ_{MLD} < μ_{TD} < μ_{MG} |
μ_{MLD} = μ_{td} < μ_{MG} |
|||||

BF | PMP | BF | PMP | BF | PMP | BF | PMP | |

IQ | 1 | .14 | 0.00 | .00 | ||||

MCT^{a} |
1 | .14 | 0.00 | .00 | ||||

CITO^{a} |
1 | .15 | 0.00 | .00 | ||||

Visual spatial WM^{a} |
1 | .14 | 2.02 | .29 | ||||

Verbal WM^{a} |
1 | .10 | 1.49 | .15 | ||||

Creativity^{a} |
1 | .09 | 1.80 | .16 | 3.77 | .33 |

^{a}Controlled for IQ.

For creativity, an extra hypothesis was tested, that the MLD group does not differ from the TD group, but that the MG group outperforms both other groups. Model 3, no differences between the MLD and TD group, but higher scores in the MG group, received most support from the data.

In this study, the role of domain-specific (number sense) and domain-general (working memory and creativity) factors in academic and creative mathematics ability of primary school children was investigated. First it was investigated how different aspects of number sense and working memory are related to mathematical ability in fourth graders. The results confirmed our hypothesis that all three number sense tasks were related to mathematics (cf. ^{2}) and the symbolic comparison task (reaction time) were found to be good predictors of mathematical skills, even in fourth grade students. Furthermore, we found both verbal and visual spatial working memory to be predictors of math ability, with verbal working memory the strongest predictor, as expected at this age at which children start memorizing basic mathematical knowledge and procedures and because the problems in the tests were presented in a verbal format (cf.

The focus of our study, however, was on the specific role that creativity has in mathematics, because this has not been studied before in relation to other cognitive predictors of mathematics. It was investigated how creativity was related to academic math tasks and to creative math tasks. Creativity showed a medium correlation to both math tasks. Even when corrected for intelligence, creativity was not more strongly related to the creative math tasks than to the academic math task. In this study, intelligence was estimated based on two subtests of a groupwise administered intelligence test, to control for a domain general effect underlying both creativity and mathematics. Intelligence was indeed found to be the strongest predictor of mathematics. The relation between intelligence and creativity, however, was relatively low in this study. Although it was comparable to correlations found in other studies (see

Furthermore, it was found that a regression analysis including creativity better fits the data than a regression with only IQ, working memory and number sense included. These results confirm our hypothesis that creativity is an important domain-general factor in mathematics that deserves more attention in education (

Another aspect that should be taken into account is that we have used one single measure of creativity. Since creativity is a complex, multidimensional construct (

Thirdly, we have investigated differences between MLD, MG, and TD children on number sense, working memory and creativity. Our hypothesis that children with MLD have deficits in visual spatial working memory and number sense was confirmed (cf.

Furthermore, children with MLD were not found to differ from TD children in creativity. An interesting finding from this study is that MG children did not significantly differ from TD children in number sense, but that they did differ in visual spatial working memory and creativity. It seems that a certain level of number sense is needed to perform well in mathematics, but that above this threshold other factors, such as creativity, come into play. It should be noted of course that the sample size in this study was relatively small, which make generalizations difficult. It is advised to further investigate the differences between MG and TD children with larger samples. Furthermore, both creativity and mathematical ability may be influenced by group factors. When this study is replicated, larger samples would also make multilevel analyses possible. Another factor that should be taken into account is that this study does not tell us how we should interpret the directions of the relation between creativity and mathematics. Based on literature, we assume that creativity supports mathematical thinking and problem solving during mathematical tasks, but the experience with mathematical problem solving also might influence children’s creative skills, and other factors may also have a mediating or moderating role in these relations. This is one of the first studies that examined relations between creativity and mathematics in relation to working memory and number sense, but further research is necessary to give insight in the direction of these relations.

To conclude, symbolic number sense, working memory and creativity are all related to academic and creative mathematical ability. Furthermore, groups with math learning disabilities (MLD) and mathematical giftedness (MG) were compared to a typically developing (TD) group. The results show that the MLD group scores significantly lower on number line estimation and visual spatial working memory than the TD group, while the differences between the MG group and the TD group were found in visual spatial working memory and creativity. It is concluded that visual spatial working memory is a domain general factor that discriminates between the three ability groups, that number sense discriminates between MLD and TD children and that creativity discriminates between TD and MG children. This study confirms the hypothesis that creativity plays a significant role in mathematics.

The authors have no funding to report.

The authors have declared that no competing interests exist.

The authors have no support to report.