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Edited by: Ann Dowker, University of Oxford, United Kingdom

Reviewed by: EeLynn Ng, Nanyang Technological University, Singapore; Annemie Desoete, Ghent University, Belgium; Lu Wang, Ball State University, United States

This article was submitted to Developmental Psychology, a section of the journal Frontiers in Psychology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Mathematical word problem solving (MWPS) involves multiple steps, including comprehending the problem statements, determining the arithmetic operations that have to be performed, and finding the answers. This study investigated the relative contributions of different cognitive and affective variables to children’s MWPS. To achieve this goal, 116 third-grade Chinese children were tested. Results showed that after controlling for age and non-verbal intelligence, the abilities to solve direct and indirect mathematical word problems were positively correlated with the working memory component of executive function, reading comprehension ability, math fact fluency and math anxiety. Moreover, math anxiety was found to fully mediate the relationships between reading anxiety and MWPS. Implications of the findings on how to promote children’s MWPS skills were discussed.

One of the major goals of mathematics learning is to know how to apply mathematical concepts to solve problems in everyday life (

Mathematical word problems refer to mathematical problems that are embedded in story contexts. Children are thus required to integrate their linguistic and basic calculation skills to find out their solutions (

Several past studies have demonstrated the significant role of language and literacy skills in MWPS.

On the other hand, MWPS is, no surprise, a good indicator of mathematical proficiency. In the study of

Anxiety can impair cognitive functioning. As sub-types of anxiety, reading anxiety and math anxiety are not exceptions and are found to be linked with individuals’ performance in respective domains.

To account for the mechanisms of how anxiety hampers cognitive performance,

Despite the fact that correct comprehension of the problem statements was the very first step for successful MWPS, no existing studies have examined the relative roles of reading anxiety and math anxiety in MWPS. The present study thus seeks to fill in this research gap. As demonstrated in past studies, the nature and effects of reading anxiety and math anxiety seemed to be intertwined with each other.

Executive function can be defined as the abilities to control and shift attention in a flexible manner, inhibit impulsive responses and retain information in working memory (

Numerous past studies have shown that children’s executive function was correlated with mathematical proficiency.

As suggested by

Even more, with emotional control as one of its components, executive function may help children regulate the negative emotions, such as anxiety, induced during the learning process.

As discussed, MWPS is a crucial part of mathematics learning. Despite the attention received by various researchers, the contributions of different cognitive skills (including general and domain-specific ones) to children’s MWPS have seldom been compared, and the potential role of affective variables in children’s MWPS has often been overlooked. The present study thus sought to examine Chinese children’s MWPS in relation to an array of cognitive and affective variables.

The cognitive variables under investigation included: (1) non-verbal intelligence, (2) executive function (including five components, namely inhibit, shift, emotional control, working memory, and plan/organize), (3) math fact fluency, and (4) reading comprehension. The first two were selected because these general cognitive skills play an important role in many types of cognitive processing (

To obtain a more comprehensive capture of children’s MWPS skills, their performance was assessed with two tasks, namely direct and indirect problems. Compared to direct problems, indirect problems may be more challenging. This is because children have to be careful not to misinterpret the relational statements and perform arithmetic operations that are inconsistent with the relational term used in the problem statements (

Behavioral rating scales would be used to measure executive function, math anxiety and reading anxiety, whereas performance-based measures would be adopted to assess the remaining variables. We rely on behavioral rating scale rather than performance-based measure for executive function because two of the variables under focus were related to anxiety. We thus wanted to assess executive function skills as displayed in natural everyday life rather than stressful test situation. The behavioral rating scale could allow us to assess the “emotional control” component of executive function.

Based on results of past studies (e.g.,

H1a: Children’s MWPS is positively related to their non-verbal intelligence, executive function, math fact fluency, and reading comprehension ability.

H1b: Children’s MWPS is negatively related to their math anxiety and reading anxiety.

H2: Compared to direct MWPS, indirect MWPS has stronger correlations with executive function and reading comprehension ability.

H3: Children’s math anxiety mediated the relationship between their reading anxiety and MWPS.

To test our hypotheses, six-step hierarchical linear regression analyses would be performed on direct and indirect MWPS, respectively. In the first step, age and non-verbal IQ would be entered, so as to control their effects on direct and indirect MWPS. In the second step, executive function would be entered because this could allow us to understand the extent to which the contribution of the domain-general variable (i.e., executive function) to MWPS was shared by the domain-specific ones. Instead of combining the five components of executive function into one latent variable and entering it into the regression equation, the five components would be entered in stepwise fashion. This was important because indirect MWPS might require higher levels of inhibition of prepotent response than direct MWPS, and such analytic approach could help us know whether different component(s) of executive function had differential associations with direct and indirect MWPS. In the third step, math fact fluency would be entered because it assessed one’s basic mathematical abilities and was apparently the most relevant to MWPS. Adding it into the regression equations at this stage could let us examine at later stages whether other domain-specific cognitive and affective variables could make unique contributions to MWPS after controlling one’s basic mathematical abilities. The remaining three variables (i.e., math anxiety, reading anxiety, and reading comprehension ability) would be entered alternatively in the fourth to sixth steps. This allowed us to investigate whether each of them could make unique contributions to children’s MWPS. To test Hypothesis 3,

The participants were 116 third-grade primary students (61 boys and 55 girls, mean age = 9.60 years,

Non-verbal intelligence was assessed using the Raven’s Progressive Matrix (Set A and Set B) (

The Behavior Rating Inventory of Executive Function (BRIEF) (

During the task (

This test was adopted and modified from the tasks used by

With reference to related past studies (

The Test Anxiety Scale of

This scale consisted of five items. The items and instructions given to participants were exactly the same as those of the math anxiety scale, except that the word “math” was replaced by the word “reading.” The maximum score was 35 (Cronbach’s α = 0.89).

The descriptive summary of all the variables is displayed in Table

Descriptive statistics for all measurements.

Minimum | Maximum | Mean | ||
---|---|---|---|---|

Non-verbal IQ | 12.00 | 24.00 | 20.99 | 2.56 |

EF – Inhibit | 37.00 | 69.00 | 49.96 | 7.94 |

EF – Shift | 36.00 | 74.00 | 50.32 | 9.24 |

EF – Emotional control | 36.00 | 67.00 | 48.92 | 8.20 |

EF – Working memory | 38.00 | 74.00 | 52.68 | 8.24 |

EF – Plan/Organize | 33.00 | 58.00 | 51.66 | 5.33 |

Math fact fluency | 42.00 | 150.00 | 97.37 | 24.19 |

Direct MWPS | 0.00 | 9.00 | 3.78 | 2.42 |

Indirect MWPS | 0.00 | 10.00 | 3.11 | 2.20 |

Math anxiety | 1.00 | 7.00 | 2.82 | 1.68 |

Reading comprehension | 4.00 | 28.00 | 20.99 | 4.53 |

Reading anxiety | 1.00 | 7.00 | 3.07 | 1.74 |

Table

Correlations and partial correlations among variables.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1. Age | 1.00 | ||||||||||||

2. Non-verbal IQ | −0.10 | 1.00 | |||||||||||

3. EF – Inhibit | −0.01 | −0.09 | 1.00 | 0.60^{∗∗∗} |
0.64^{∗∗∗} |
0.74^{∗∗∗} |
0.01 | −0.16 | −0.26^{∗∗} |
−0.24^{∗∗} |
0.12 | −0.13 | 0.21^{∗} |

4. EF – Shift | −0.18 | −0.10 | 0.60^{∗∗∗} |
1.00 | 0.70^{∗∗∗} |
0.59^{∗∗∗} |
0.08 | −0.17 | −0.13 | −0.18 | 0.14 | −0.09 | 0.17 |

5. EF – Emotional control | −0.13 | 0.08 | 0.62^{∗∗∗} |
0.69^{∗∗∗} |
1.00 | 0.59^{∗∗∗} |
0.02 | −0.07 | −0.09 | −0.09 | 0.05 | −0.12 | 0.09 |

6. EF – Working memory | 0.10 | −0.07 | 0.74^{∗∗∗} |
0.60^{∗∗∗} |
0.59^{∗∗∗} |
1.00 | 0.07 | −0.13 | −0.31^{∗∗} |
−0.29^{∗∗} |
0.07 | −0.17 | 0.15 |

7. EF – Plan/Organize | −0.51^{∗∗} |
−0.07 | 0.02 | 0.17 | 0.07 | 0.12 | 1.00 | 0.16 | −0.04 | −0.08 | −0.08 | 0.04 | 0.07 |

8. Math fact fluency | −0.31^{∗∗} |
0.05 | −0.15 | −0.10 | −0.03 | −0.10 | −0.28^{∗∗} |
1.00 | 0.46^{∗∗∗} |
0.44^{∗∗∗} |
−0.15 | 0.18 | −0.08 |

9. Direct MWPS | −0.12 | 0.12 | −0.27^{∗∗} |
−0.12 | −0.07 | −0.30^{∗∗} |
−0.01 | 0.47^{∗∗} |
1.00 | 0.79^{∗∗∗} |
−0.40^{∗∗∗} |
0.35^{∗∗∗} |
−0.30^{∗∗} |

10. Indirect MWPS | −0.22^{∗} |
0.19^{∗} |
−0.25^{∗∗} |
−0.15 | −0.04 | −0.27^{∗∗} |
−0.03 | 0.48^{∗∗} |
0.80^{∗∗} |
1.00 | −0.39^{∗∗∗} |
0.33^{∗∗∗} |
−0.32^{∗∗} |

11. Math anxiety | 0.08 | −0.13 | 0.130 | 0.14 | 0.04 | 0.07 | −0.09 | −0.17 | −0.41^{∗∗} |
−0.41^{∗∗} |
1.00 | −0.18 | 0.77^{∗∗∗} |

12. Reading comprehension | −0.08 | 0.21^{∗} |
−0.15 | −0.09 | −0.09 | −0.17 | 0.06 | 0.20^{∗} |
0.36^{∗∗} |
0.36^{∗∗} |
−0.20^{∗} |
1.00 | −0.16 |

13. Reading anxiety | −0.02 | −0.10 | 0.21^{∗} |
0.18 | 0.08 | 0.16 | 0.09 | −0.08 | −0.30^{∗∗} |
−0.32^{∗∗} |
0.77^{∗∗} |
−0.17^{∗} |
1.00 |

^{∗}p < 0.05;

^{∗∗}p < 0.01;

^{∗∗∗}p < 0.001.

To further examine the contributions of executive function, math fact fluency, reading anxiety, math anxiety, and reading comprehension to direct MWPS and indirect MWPS, six-step hierarchical regressions were conducted (see Tables

Hierarchical regression predicting direct math word problem solving.

Step | ^{2} |
^{2} |
||
---|---|---|---|---|

1 Age | −0.11 | −1.16 | 0.03 | 0.03 |

Non-verbal IQ | 0.11 | 1.19 | ||

2 EF – Working memory | −0.31 | −3.419^{∗∗} |
0.12 | 0.09^{∗∗} |

3 Math fact fluency | 0.44 | 5.22^{∗∗∗} |
0.29 | 0.17^{∗∗∗} |

4 Reading anxiety | −0.23 | −2.92^{∗∗} |
0.34 | 0.05^{∗∗} |

5 Math anxiety | −0.37 | −3.07^{∗∗} |
0.39 | 0.05^{∗∗} |

6 Reading comprehension | 0.19 | 2.49^{∗} |
0.42 | 0.03^{∗} |

4 Reading comprehension | 0.24 | 2.95^{∗∗} |
0.34 | 0.05^{∗∗} |

5 Reading anxiety | −0.20 | −2.63^{∗} |
0.38 | 0.04^{∗} |

6 Math anxiety | −0.34 | −2.91^{∗} |
0.43 | 0.05^{∗∗} |

4 Reading comprehension | 0.24 | 2.95^{∗∗} |
0.34 | 0.05^{∗∗} |

5 Math anxiety | −0.30 | −3.97^{∗∗∗} |
0.42 | 0.08^{∗∗} |

6 Reading anxiety | 0.06 | 0.50 | 0.42 | 0 |

^{∗}p < 0.05;

^{∗∗}p < 0.01;

^{∗∗∗}p < 0.001.

Hierarchical regression predicting indirect math word problem solving.

Step | ^{2} |
^{2} |
||
---|---|---|---|---|

1 Age | −0.21 | −2.27^{∗} |
0.08 | 0.08^{∗} |

Non-verbal IQ | 0.17 | 1.86 | ||

2 EF – Working memory | −0.28 | −3.24^{∗∗} |
0.16 | 0.08^{∗∗} |

3 Math fact fluency | 0.42 | 4.99^{∗∗∗} |
0.31 | 0.15^{∗∗∗} |

4 Reading anxiety | −0.25 | −3.23^{∗∗} |
0.37 | 0.06^{∗∗} |

5 Math anxiety | −0.29 | −2.44^{∗∗} |
0.40 | 0.03^{∗} |

6 Reading comprehension | 0.18 | 2.26^{∗} |
0.43 | 0.03^{∗} |

4 Reading comprehension | 0.22 | 2.73^{∗∗} |
0.36 | 0.05^{∗∗} |

5 Reading anxiety | −0.23 | −2.96^{∗∗} |
0.40 | 0.04^{∗∗} |

6 Math anxiety | −0.27 | −2.29^{∗} |
0.43 | 0.03^{∗} |

4 Reading comprehension | 0.22 | 2.73^{∗∗} |
0.36 | 0.05^{∗∗} |

5 Math anxiety | −0.28 | −3.80^{∗∗∗} |
0.43 | 0.07^{∗∗∗} |

6 Reading anxiety | −0.02 | −0.17 | 0.43 | 0 |

^{∗}p < 0.05;

^{∗∗}p < 0.01;

^{∗∗∗}p < 0.001.

To provide more direct evidence confirming that reading anxiety was associated with direct MWPS and indirect MWPS with the mediation of math anxiety, we combined direct MWPS and indirect MWPS as math word problem solving (WPS) and conducted additional hierarchical regression analysis. Four conditions should be met to confirm the full mediation effect (

Hierarchical regression examining the mediating effect of math anxiety.

Condition 1 |
Condition 2 |
Condition 3 |
Condition 4 |
|||||
---|---|---|---|---|---|---|---|---|

Reading anxiety | −0.33 | −3.69^{∗∗∗} |
– | – | 0.77 | 12.94^{∗∗∗} |
0.02 | 0.12 |

Math anxiety | – | – | −0.43 | −5.11^{∗∗∗} |
−0.44 | −3.33^{∗∗∗} |

^{∗}p < 0.05;

^{∗∗}p < 0.01;

^{∗∗∗}p < 0.001.

Math anxiety mediating reading anxiety and math WPS.

The present study examined the relative contributions of different cognitive and affective variables to children’s MWPS. Our findings suggested that after controlling for age and non-verbal IQ, children’s MWPS (no matter for direct or indirect problems) was only significantly correlated with the working memory component of executive function, math fact fluency, reading comprehension and math anxiety. Regarding the relationships of executive function and reading comprehension to MWPS, their strengths were similar across direct and indirect problems. Moreover, the association between reading anxiety and MWPS was fully mediated by math anxiety.

Partially different from our initial speculations, only two of the five components of executive function (i.e., inhibit and working memory) had significant but weak zero-order correlations with MWPS (including direct and indirect problems). No significant associations were found for the remaining three components (i.e., shift, emotional control, and plan/organize). However, when age, non-verbal intelligence and the five components were considered together in the regression equations, only working memory remained as a significant correlate. This is somehow similar to the findings by

Consistent with our initial speculation and results of past studies (

As expected, reading comprehension was a significant correlate of both types of MWPS. This again shows that it is important for children to have a deep comprehension of the problem statements, instead of just relying on the keywords in the problem statements (e.g., the number words, the relational terms) to solve the mathematical word problems (

Of the two affective variables under investigation, math anxiety seemed to be more relevant to MWPS than reading anxiety was. On the one hand, though reading anxiety had significant zero-order correlations with both types of MWPS, it was not a significant correlate when math anxiety was included in the regression equations. On the other hand, even after controlling for all the cognitive variables, math anxiety could still account for additional variance in both types of MWPS. These results, somehow, are not surprising for at least two reasons. First, as proposed by

Findings of the present study can provide educators and parents with insights on how to promote children’s MWPS skills. First, when teachers observe that a child makes mistakes frequently when solving mathematical word problems, teachers have to examine more closely the reason(s) for such a situation. As shown in the present study, it might happen because the child shows difficulties in processing multiple pieces of information in the mind, comprehending the problem statements, and/or retrieving basic math facts to find out the answers. These different reasons indeed call for different approaches to help the child.

Second, the present study shows that teachers have to find out effective strategies to help children reduce math anxiety. This is because the fear induced by the necessity of tackling math problems might be so overwhelming that it can hinder children’s math performance, even though the children might have already possessed the required math knowledge and skills. Teachers and parents should thus talk to children who show high levels of anxiety, so as to understand the reasons underlying their anxiety and adopt corresponding strategies to relieve their stress.

The present study had several limitations that required attention. First, given that all variables were measured at one time point only, no causal relationships between the variables can be drawn. Future researchers can thus conduct longitudinal studies to examine the extent to which various cognitive and affective variables and their growth can predict children’s performance in MWPS in future.

Second, the present study only relied on parental report questionnaire to measure children’s executive function. Some recent studies (e.g.,

Third, the present study only focused on two types of word problems (i.e., direct and indirect two-step arithmetic word problems) and the outcome of children’s MWPS (i.e., the accuracy of their answers). In fact, the difficulty level of arithmetic word problems depends on a range of linguistic, numerical and contextual factors (

In summary, the present study is one of the few studies to investigate the relative contributions of different cognitive and affective variables to children’s MWPS. Our findings showed that children’s MWPS was significantly correlated with their working memory, reading comprehension, math fact fluency and math anxiety. In order to provide more effective support to children struggling with MWPS, it is essential for teachers and parents to figure out in which aspect these children show difficulties and adopt corresponding strategies to help them overcome their barriers.

This study was carried out in accordance with the recommendations of ethic guidelines, Institutional Review Board in the University of Macau with written informed consent from all subjects. The protocol was approved by the Institutional Review Board in the University of Macau.

JZ and SKC developed the research idea and research design. CW analyzed the data. JZ, SKC, CW, and YM wrote and reviewed the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We would like to thank all the children and parents who had participated in our study.