^{*}

^{a}

^{a}

^{b}

^{b}

Large individual differences in children’s mathematics achievement are observed from the start of schooling. Previous research has identified three cognitive skills that are independent predictors of mathematics achievement: procedural skill, conceptual understanding and working memory. However, most studies have only tested independent effects of these factors and failed to consider moderating effects. We explored the procedural skill, conceptual understanding and working memory capacity of 75 children aged 5 to 6 years as well as their overall mathematical achievement. We found that, not only were all three skills independently associated with mathematics achievement, but there was also a significant interaction between them. We found that levels of conceptual understanding and working memory moderated the relationship between procedural skill and mathematics achievement such that there was a greater benefit of good procedural skill when associated with good conceptual understanding and working memory. Cluster analysis also revealed that children with equivalent levels of overall mathematical achievement had differing strengths and weaknesses across these skills. This highlights the importance of considering children’s skill profile, rather than simply their overall achievement.

Individuals’ success in dealing with numbers and quantities is related to their job prospects, income and quality of life (

As a result, researchers have increased their efforts to understand the cognitive bases for mathematical achievement with a view to developing more effective teaching strategies. There has been a particular focus on the early years of education, based on an assumption that early interventions may be more effective and efficient (

Research to date has made progress in identifying a set of skills that are related to success in mathematics achievement. Many researchers have adopted a multi-component framework which recognizes that both domain-specific and domain-general processes contribute to mathematical skills (

Procedural skill is the ability to carry out a sequence of operations accurately and efficiently (

Conceptual knowledge encompasses understanding of the principles and relationships that underlie a domain (

Much research has explored how conceptual understanding and procedural skill develop, with evidence supporting a model of iterative development, such that advances in conceptual understanding lead to developments in procedural skill and vice-versa (

Importantly, while previous research has demonstrated that procedural skill and conceptual understanding are both predictors of mathematics achievement, we do not know if they have independent effects. In fact, the iterative model of development would predict that these effects are not independent, but rather levels of conceptual understanding moderate the relationship between procedural skill and mathematics achievement. We return to this point below.

Alongside recognition of the importance of domain-specific skills has come an increased understanding of the importance of domain-general skills for mathematics learning. In particular, an extensive body of research has identified executive function skills, including working memory, inhibition and shifting, as being critical for success with mathematics. The majority of research has focused on working memory, the ability to monitor and manipulate information held in mind, and revealed that this is related to both concurrent and future mathematics achievement across development (see reviews by

The studies described above have demonstrated the importance of procedural skills, conceptual understanding and working memory for mathematics achievement. One limitation of this body of work is that little attention has been paid to the moderating effects of one skill on the relationship between other skills and mathematics achievement. Although studies have established that procedural skill, conceptual understanding and working memory each make independent contributions to mathematics achievement, they have tended to focus on main effects and not interactions (

In fact, we predict that differences in mathematics achievement arise not only from children’s differing levels of skill in each of these areas, but also from the interactions amongst them. These interactions might be crucial to understanding why some children succeed with mathematics and others struggle. The importance of one skill may depend on the levels of proficiency with other skills. This could come about because children may be able to compensate for weakness in one skill with strengths in another. For example, good conceptual understanding may help children overcome working memory difficulties by allowing them to identify and use shortcut strategies which have lower working memory demands than simple computation. In contrast, it may be that weaknesses in one skill exacerbate difficulties with another, e.g. poor conceptual understanding combined with poor procedural skill may be particularly problematic. It is therefore important to consider different profiles of skills and how these relate to differences in mathematics achievement.

There is some previous evidence to suggest that it is important to consider combinations of skills, and not just individual skills in isolation.

A small number of studies have begun to explore the interactions among procedural skill, conceptual understanding and domain-general skills.

Here we explore the association between mathematics achievement and procedural skill, conceptual understanding and working memory in children who are at the early stages of learning mathematics. We predict that mathematics achievement will not only be associated with the main effects of procedural skills, conceptual understanding and working memory, but also with the interaction between them. Specifically, we predict that conceptual understanding and working memory will moderate the relationship between procedural skill and mathematics achievement. We focus on children who are at the earliest stages of formal mathematics learning in an effort to understand how early differences in mathematics achievement arise.

Participants in the study were 75 children (41 male) in Year 1 of primary school (mean age = 6.2 years

The children completed tasks to assess their mathematics achievement, procedural skills (counting and arithmetic), conceptual understanding and working memory (verbal and visuo-spatial). The working memory tasks were presented on a laptop computer, the other tasks were presented verbally, or with cards and props as described below.

Mathematics achievement was measured using the Wechsler Individual Achievement Test–II UK (

Children’s procedural arithmetic skills were assessed using two tasks. A total procedural skill score was calculated by adding accuracy scores across the two tasks. Cronbach’s alpha for this combined score was 0.73.

Children were asked to complete five counting sequences. They were asked to count up to 10, on from 28, on from 45, backwards from 12, and backwards from 33. With the exception of the first trial children were stopped after they had given seven numbers, which ensured that they had crossed a decade boundary in each case. Each trial was scored as correct if children completed the whole series correctly and proportion correct scores were recorded.

Children were shown a series of addition and subtraction problems (e.g. 3 + 2) printed onto cards and were asked to solve them using any strategy of their choice. There were four practice trials and eight experimental trials. All problems involved single digit addends. The items were presented in one of two orders, counterbalanced across participants. Number lines marked from 1 – 10 and 1 – 20 and a set of counters were provided for children to use if they wished. Proportion correct scores were recorded.

To assess conceptual understanding children played a game involving a puppet, adapted from

Children completed two practice example problems, each with one identical, one unrelated and one related (addend + 1 rule) probe problem, followed by 24 experimental trials (six example problems each with 4 probe problems). Feedback was provided during the practice trials to help children understand the task. All trials were audio recorded. For each item children were given a score (0 or 1) for answering whether the items were related and a score (0 or 1) for identifying the relationship. If children correctly answered that the items were unrelated then they were also given credit for a correct explanation. Therefore, the total score for accuracy was out of 24 and for explanations was out of 24. Proportion correct scores were calculated for each measure and a total conceptual understanding score was calculated by adding the proportion correct scores for answers and explanations. Cronbach’s alpha for this combined score was 0.77.

Children completed separate verbal and visuo-spatial working memory tasks. A total working memory score was calculated by adding the scores across the two tasks.

Verbal working memory was assessed via a sentence span task. Children heard a sentence with the final word missing and had to provide the appropriate word. After a set of sentences children were asked to recall the final word of each sentence in the set, in the correct order. There were three sets at each span length, beginning with sets of two sentences, and children continued to the next list length if they responded correctly to at least one of the sets at each list length. The total number of correctly recalled words was recorded.

Visuo-spatial working memory was assessed via a complex span task. Children saw a series of 3 x 3 grids each containing three symbols and they had to point to the symbol that differed from the other two. After a set of grids children were asked to recall the position of the odd-one-out on each grid, in the correct order. There were three sets at each span length, beginning with sets of two grids, and children continued to the next span length if they responded correctly to at least one of the sets at each span length. The total number of correctly recalled locations was recorded.

In the following sections we first report children’s performance on the set of arithmetic tasks and working memory measures and the associations between them. The relationship between procedural skill, conceptual understanding and working memory and mathematics achievement is then investigated using linear regression including moderation (interaction terms). Finally, cluster analysis is used to examine the performance of subgroups of children.

Descriptive statistics for performance on the arithmetic and working memory tasks are presented in

Task | Min | Max | ||
---|---|---|---|---|

WIAT Numerical Operations (raw score) | 8.76 | 3.02 | 3 | 24 |

WIAT Mathematics Reasoning (raw score) | 18.00 | 5.20 | 8 | 36 |

Counting (proportion correct) | 0.69 | 0.27 | 0.20 | 1 |

Arithmetic (proportion correct) | 0.74 | 0.27 | 0 | 1 |

Conceptual understanding accuracy (proportion correct) | 0.60 | 0.12 | 0.25 | 0.83 |

Conceptual understanding explanations (proportion correct) | 0.52 | 0.14 | 0.08 | 0.75 |

Verbal working memory (total item score) | 5.39 | 3.34 | 0 | 15 |

Visuo-spatial working memory (total item score) | 14.59 | 9.07 | 0 | 42 |

Zero-order correlations among the measures of conceptual understanding, procedural skill, working memory and mathematics achievement are reported in

Measure | Mathematics Achievement | Procedural skills | Conceptual understanding |
---|---|---|---|

Mathematics Achievement | |||

Procedural skills | .624** | ||

Conceptual understanding | .620** | .426** | |

Working Memory | .639** | .410** | .354** |

**

We then sought to understand the relationship between children’s working memory, conceptual and procedural scores with overall mathematics achievement. We conducted a multiple linear regression including the main effects of procedural skill, conceptual understanding and working memory as well as all two-way interactions and the three-way interaction. Specifically, we tested whether levels of conceptual understanding and working memory moderated the relationship between procedural skill and mathematics achievement. This was conducted using the PROCESS macro for SPSS (

The model was significant,

Predictor | b | ||
---|---|---|---|

Procedural skill | 0.39 | 4.35 | < .001 |

Conceptual understanding | 0.23 | 2.63 | .010 |

Working memory | 0.27 | 3.12 | .003 |

Procedural * Working memory | 0.09 | 0.84 | .407 |

Conceptual * Working memory | -0.00 | -0.05 | .964 |

Procedural * Conceptual | 0.25 | 2.39 | .020 |

Procedural * Conceptual * Working memory | 0.22 | 2.65 | .010 |

^{2} = .71.

To explore these interaction effects we plotted predicted mathematics achievement scores (

Predicted mathematics achievement (WIAT mathematics composite

The three-way interaction between procedural skill, conceptual understanding and working memory on mathematics achievement is depicted in

Predicted mathematics achievement (WIAT mathematics composite

Finally, we further explored the influence of these three skills on children’s mathematics achievement by examining subgroups of children within the whole sample. We conducted a hierarchical cluster analysis (using Ward’s method) on children’s procedural skill, conceptual understanding and working memory. Examination of the dendogram indicated that 5 distinct clusters could be identified, and this solution accounted for 67.0% of the variance in scores.

Mean scores for each cluster on the measures of procedural skill, conceptual understanding and working memory are depicted in

Procedural skill, conceptual understanding and working memory (mean

The differences among children in each of these subgroups can therefore be summarised as follows. Cluster 5 is a small group of children with generally low performance and Cluster 4 is a large group of children with generally high performance. Children in Cluster 1 had average levels of procedural skill and conceptual understanding, but poorer working memory. Children in Cluster 2 had average levels of procedural skill and working memory, but poorer conceptual understanding. Children in Cluster 3 had average levels of conceptual understanding, but poorer procedural skill and working memory. Despite these differences, the performance of children in these three clusters did not differ on overall mathematics achievement.

In line with multi-component frameworks of mathematics achievement (

The significant 3-way interaction indicated that the relationship between procedural skill and mathematics achievement was moderated by levels of conceptual understanding and working memory. Good conceptual understanding is important to be able to identify and select computationally less demanding strategies, which can in turn reduce the reliance on procedural skills. However, this may only be possible if an individual has sufficient working memory resources. In line with this we found that for children with lower levels of working memory, procedural skill and conceptual understanding had independent effects on mathematics achievement and there was no additional benefit of having both good conceptual understanding and procedural skills. However, for children with higher levels of working memory, additional procedural skill conferred an extra benefit for mathematical achievement when associated with good conceptual understanding, compared to the effect on children with lower levels of conceptual understanding. This suggests that children are only able to apply good conceptual understanding in their problem solving, and thereby reduce computational demands, if they have the domain-general resources to allow them to identify how conceptual understanding is relevant. These results are consistent with those of

Models of mathematical cognition (e.g.

Our cluster analysis highlighted subgroups of children with different profiles of performance. Interestingly, we found that weaknesses in different skills appeared to have equivalent impact on overall mathematics achievement. Specifically, there was no difference in mathematics achievement, measured by mathematics composite WIAT scores, for children in Clusters 1, 2 and 3, despite the fact that they had different profiles of procedural skill, conceptual understanding and working memory. This adds to previous evidence suggesting that there are multiple pathways to mathematical competence (

The results of our cluster analysis also have important implications for education. In a classroom setting we would expect groups of children such as those identified by our cluster analysis, to require different types of support for their mathematics learning. However, simply observing their scores on a mathematics achievement measure would not indicate this. This highlights the importance of examining more detailed profiles of children’s performance when determining the types of support that they require. Our findings also have implications for the development of interventions. The moderating effect of working memory indicates that children’s ability to benefit from interventions targeting either procedural skill or conceptual understanding is likely to depend on levels of working memory.

Here we focused on working memory as a measure of domain-general skills. Research which measures a broader range of executive functions, including inhibition and cognitive flexibility may reveal a more complex pattern of interactions between different executive function skills and arithmetic skills such that working memory, inhibition and cognitive flexibility interact in different ways with procedural skill and conceptual understanding. Research exploring these patterns may help to identify the roles played by inhibition and cognitive flexibility in mathematics learning, which have received less attention in comparison to the role of working memory (

One limitation of the current study is the cross-sectional and correlational nature of the data. It is therefore not possible to identify the extent to which these patterns represent causal relationships, or how they might change over time. For example, while children in Clusters 1, 2 and 3 had equivalent mathematics achievement at a single time point despite different patterns of skills, it is possible that the future learning trajectories for these groups may be very different. Recent use of latent growth modelling has started to identify the patterns of skills associated with faster growth in mathematics learning (e.g.

Here we found that procedural skill was an important component of mathematics, with both independent and interactive effects. Inevitably because procedural skill is important for mathematics achievement there is some overlap in the items used to measure both procedural skill and overall achievement. This problem may be particularly the case for young children who have a limited range of mathematics knowledge. Therefore, it would be informative to replicate these findings in older children, for whom it may be possible to more clearly separate measures of procedural skill and overall mathematics achievement.

In conclusion, we have extended multi-component frameworks of mathematics by demonstrating that not only do procedural skill, conceptual understanding and working memory independently account for differences in mathematics achievement, but the interaction amongst these skills is also important. We have identified how the profile of children’s skills has impact on their mathematics performance, thereby elucidating multiple pathways to mathematics achievement. In particular, children who had strengths in each of these three areas were particularly advantaged in terms of their overall achievement. This emphasizes the complex nature of mathematics and the wide range of skills that are needed to succeed. This may provide one explanation for why so many children struggle with learning mathematics, because lower levels of proficiency in just one area can impact on overall mathematics performance. Consequently, there is a multitude of different ways in which children can have difficulties with mathematics. Researchers and educators should take account of the wide range of skills involved when attempting to understand differences in mathematics achievement and provide appropriate support to learners.

This work was funded by grant RES-062-23-3280 from the Economic and Social Research Council, UK. CG is funded by a Royal Society Dorothy Hodgkin Research Fellowship. The funders had no role in the research design, execution, analysis, interpretation or reporting.

The authors have declared that no competing interests exist.

The authors have no support to report.