^{1}

^{2}

^{3}

Individuals solve arithmetic problems in different ways and the strategies they choose are indicators of advanced competencies such as adaptivity and flexibility, and predict mathematical achievement. Understanding the factors that encourage or hinder the selection of different strategies is therefore important for helping individuals to succeed in mathematics. Our research contributed to this goal by investigating the skills required for selecting the associativity shortcut-strategy, where problems such as ‘16 + 38 – 35’ are solved by performing the subtraction (38 – 35 = 3) before the addition (3 + 16 = 19). In a well-powered, pre-registered study, adults completed two tasks that involved ‘a + b – c’ problems, and we recorded a) whether and b) when, they identified the shortcut. They also completed tasks that measured domain-specific skills (calculation skill and understanding of the order of operations) and domain-general skills (working memory, inhibition and switching). Of all the measures, inhibition was the most reliable predictor of whether individuals identified the shortcut, and we discuss the roles it may play in selecting efficient arithmetic strategies.

The way that individuals solve arithmetic problems reflects their knowledge of mathematical principles and procedures, potentially more so than the accuracy of their answer or time taken to reach it. Strategy use also predicts future expertise with mathematics, and education and employment success (

Our research aimed to fill this gap by focusing on three-term addition and subtraction problems of the form ‘a + b – c’ and the strategies individuals use to solve them. More specifically, we were concerned with strategy selection, which is one component of strategy use (

Arithmetic knowledge can be crudely divided into procedural and conceptual: conceptual knowledge refers to understanding the principles that underlie a domain, while procedural knowledge refers to knowledge of the mathematical symbol system and the rules for solving problems (

We focused on ‘a + b – c’ problems, and two strategies that can be used to solve them, a) a left-to-right strategy and b) a right-to-left strategy. For example, ‘16 + 38 – 35’ can be solved left-to-right if the addition (16 + 38) is performed before the subtraction (54 – 35), or right-to-left if the subtraction (38 – 35) is performed before the addition (3 + 16). For this problem, the right-to-left strategy is quicker and easier, and we refer to it as a shortcut. Compared to strategies derived from other arithmetic principles, the associativity shortcut is used infrequently (

To encourage individuals to use efficient strategies we must first understand the skills that are required to identify those strategies. Difficulties with particular cognitive skills (e.g., working memory, inhibition, calculation skill) could make different aspects of mathematics challenging; for example, an individual might be able to perform the calculations in the associativity shortcut (‘b – c’, ‘a + b’) but struggle to identify the strategy because they have a difficulty with other cognitive skills. By understanding the factors involved in the process of identification can we help individuals to make efficient strategy selections.

Here, we focus on two domain-specific skills (calculation skill and knowledge of the order of operations) and three domain-general skills (visuospatial working memory, inhibition and switching) that might help in identifying the associativity shortcut.

Calculation skill refers to the accuracy and speed with which an individual solves arithmetic problems and is often used to measure procedural knowledge. While the precise relationship between procedural and conceptual knowledge remains debated (

A second domain-specific skill that might be important for identifying the associativity shortcut is knowledge of the order of operations, the convention that for problems with mixed operations (e.g., ‘2 + 4 × 5’), multiplication and division should be performed before addition and subtraction, but within multiplication and division and within addition and subtraction order does not alter the result. Individuals who understand the order of operations are more likely to be aware of different permissible ways that problems can be solved, which in turn could help them to identify the shortcut. For example,

However, many individuals have a limited or misconceived understanding of operation order (

Recent years have seen an increased recognition that executive functions (working memory, inhibition and switching) are important for mathematics achievement, and this might be particularly so for selecting efficient arithmetic strategies. While both verbal and visuospatial working memory are associated with mathematics performance (

To date, three studies have investigated whether VSWM relates to the use of conceptually-derived arithmetic strategies.

A second executive function is inhibition, which refers to “the stopping or overriding of a mental process, in whole or in part, with or without instruction” (

Indeed, two studies provide supporting evidence for the role of inhibition in strategy selection (

Finally, switching, the ability to shift attention between different tasks (

Understanding why individuals fail to select efficient strategies is important if we are to improve current standards and outcomes in mathematics education (

The study was approved by Loughborough University’s Ethics Approvals (Human Participants) Sub-Committee. Before data collection, the study hypotheses, design, sample size, exclusion criteria and analysis plan were pre-registered. The pre-registration protocol is provided in the

120 adults aged 18 – 59 years (

The sample size was determined by the minimum number of participants required to detect an odds ratio of 2.15 for a binary logistic regression (where 1.49 = small, 3.45 = medium and 9.00 = large) with 5 predictors, 90% power, and an alpha level of 0.05. The effect size was guided by previous studies (

Participants completed seven tasks. Two tasks measured the identification of the associativity shortcut, a shortcut self-report task and a trial-by-trial shortcut task. Two tasks measured domain-specific skills, a calculation skill task and an instrument that measures knowledge of the order of operations. Three tasks measured domain-general skills, an odd-one-out task (working memory), a Go/No-Go task (inhibition) and a letter-number categorisation task (switching).

The odd-one-out, Go/No-Go, letter-number categorisation tasks and the trial-by-trial shortcut task were presented on a 15” laptop using PsychoPy (

Materials for the shortcut identification tasks can be found in the

The shortcut self-report task measured early identification of the shortcut, i.e., identification on the first three-term problem that was presented in the study. Participants were presented with five arithmetic problems one at a time, and were asked to solve each problem mentally, write down their answer in an A4 booklet, and respond to the open-ended question “How did you solve the problem?” They were instructed to report how they were trying to solve the problem, even if they did not compute an answer. The third problem was ‘6 + 38 – 35’, which contains an associativity shortcut, while the other four problems were two-term problems with no associativity shortcuts. We were interested in responses to the associativity problem, which measures spontaneous use of the shortcut on a single item. Each problem was presented for 15 seconds, and the problems were presented in the same order for all participants.

Participants were classed as an identifier of the shortcut if their self-report contained any indication of having performed the subtraction before the addition (e.g., “I did the right-hand side first”). If participants reported using both strategies (left-to-right and right-to-left), they were classed as an identifier because they had technically identified the shortcut.

The trial-by-trial shortcut task (

Thirty ‘a + b – c’ arithmetic problems were presented one at a time on the screen. Half of the problems were ‘conducive’ to a shortcut strategy (e.g., ‘15 + 48 – 44’) and half were non-conducive (e.g., ‘36 + 27 – 44’), see

At the end of the task, participants were interviewed about the strategies that they had used to solve the problems (recorded using a voice recorder). Their responses were used to categorise them as an identifier of the shortcut or a non-identifier of the shortcut on the task in the same way as the shortcut self-report task, and most people could be easily categorised. Performance on the trial-by-trial shortcut task was therefore measured by two variables, 1) self-reported use of the shortcut after all of the trials had been presented (categorical), and 2) the percent of trials that the shortcut was used on (continuous), calculated from the IP.

Materials for the domain-specific tasks can be found in the

This was a paper and pencil task where participants solved as many two-term addition and subtraction problems as possible in 90 seconds without a calculator. There were 64 problems in total, which varied by problem size (small: contained one double-digit and one single-digit; large: contained two double-digits), operation (addition or subtraction) and the presence of carries or borrows in the units (with or without a carry/borrow). The problems were presented in the same order to all participants, which was pseudorandomised prior to the study with the constraint that the correct answer was not the same on consecutive problems. Performance was measured by the total number of problems correctly solved in the time-limit.

This instrument consists of 14 multiple-choice items that measures individuals’ understanding of the order of operations (

The instrument contained three types of items (brackets, precedence, misconception); bracket and precedence items had one correct response, which was the option that contained the bracket and the operation with precedence, respectively. For the misconception items, the response options distinguished between different types of understanding and misconceptions of BODMAS. For example, in the item above (‘46 – 39 + 14 + 22’), the response options ‘39 + 14’ and ‘46 – 39’ are consistent with literal and left-to-right misconceptions of BODMAS respectively. ‘46 – 39 or 14 + 22’ is the correct answer, and ‘46 + 39 or 46 – 14’ intends to capture guessing behaviour. The instrument contained 3 bracket problems, 3 order problems and 8 diagnostic problems. The items and response options were presented in the same pseudorandomised order to all participants and performance was measured by the percent of literal responses on the misconception items. See

Our domain-general tasks measured working memory (

The odd-one-out task measured visuospatial working memory (VSWM;

Each trial began with a blank screen (500 ms) followed by the presentation of a 3 x 3 grid of nine black squares. Four of the squares contained pictures of objects (e.g., circles, arrows, stars), three of which were identical, and one was different (e.g., an empty circle). Images remained on the screen until the participant identified the odd-one-out by clicking on one of the squares. Once all items in the set had been presented, “?” was displayed at the centre of the screen and participants recalled the locations of the odd stimuli in order by clicking on an empty 3 x 3 grid. Each set length was presented three times, and if the participant recalled two of the sets correctly, the length increased by one. If they did not, the task ended. The stimuli and their location were pseudorandomised before the study and were the same for each participant.

A Go/No-Go task (

The letter-number task (adapted from

There were 65 trials; 31 were ‘no-switch’ trials and 33 were ‘switch’ trials (the first trial could not be classed as switch or no-switch because there was no prior stimulus to compare it to). On no-switch trials, the vertical location of the current stimulus (upper or lower) was the same as the previous stimulus. On switch trials, the vertical location of the current stimulus (e.g., upper), and therefore categorisation type, was different to the previous stimulus (e.g., lower). Stimuli appeared in the upper quadrants on half of the trials, in the lower quadrants on the other half and were presented in the same pseudorandom order for all participants. Performance was measured by subtracting the median reaction time of correct responses on no-switch trials from the median reaction time of correct responses on switch trials. Cronbach’s alpha for reaction times on the switch trials was 0.87 and for reaction times on the no-switch trials was 0.76.

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

Calculation skill (no. problems correct) | 19.69 | 7.18 | 7.00 | 40.00 |

Order of operations (% of literal responses) | 24.82 | 36.53 | 0.00 | 100.00 |

VSWM (total item score) | 44.54 | 13.32 | 16.00 | 80.00 |

Inhibition (% of false alarms) | 36.56 | 18.12 | 0.00 | 81.25 |

Switching (RT difference between switch and no-switch trials, ms) | 504.20 | 269.18 | -274.81 | 1257.80 |

Relationships between performance on the shortcut identification tasks, domain-specific tasks and domain-general tasks were explored with zero-order point-biserial correlations for the shortcut self-report variables and linear correlations for the percent use variable on the trial-by-trial shortcut task (

Variable | Shortcut self-report task (identifier, non-identifier)^{a, b} |
Trial-by-trial shortcut task (identifier, non-identifier)^{a, b} |
Trial-by-trial shortcut task (% use)^{c} |
Calculation skill | Order of operations | VSWM | Inhibition |
---|---|---|---|---|---|---|---|

Calculation skill | 0.155 | 0.225* | 0.219* | — | — | — | — |

Order of operations | -0.037 | -0.111 | -0.181 | -0.113 | — | — | — |

VSWM | 0.054 | 0.096 | 0.127 | 0.319** | 0.034 | — | — |

Inhibition | -0.125 | -0.263** | -0.240** | -0.297** | 0.059 | -0.119 | — |

Switching (RT difference between switch and no-switch trials) | -0.213* | -0.037 | -0.021 | -0.032 | 0.067 | -0.055 | 0.162 |

^{a}Identifiers were coded ‘1’ and non-identifiers ‘0’. ^{b}point-biserial correlations. ^{c}linear correlations.

*

There were 33 identifiers and 86 non-identifiers (one participant could not be categorised). We checked whether the data met the assumptions of a logistic regression including a) linearity between each predictor and the logit of the self-report variable (

A hierarchical logistic regression was performed to assess whether identification of the shortcut on the shortcut self-report task was associated with the domain-specific and domain-general measures. Two regression models were computed, one with the domain-specific skills (calculation skill and knowledge of the order of operations) as predictors, and a second with the domain-specific skills and domain-general skills (VSWM, inhibition and switching) as predictors.

Predictor | Model 1 |
Model 2 |
||||
---|---|---|---|---|---|---|

Exp(β) | Wald | Significance | Exp(β) | Wald | Significance | |

Calculation skill | 1.05 | 2.78 | 0.096 | 1.04 | 1.67 | 0.196 |

Order of operations | 1.00 | 0.05 | 0.822 | 1.00 | 0.00 | 0.998 |

VSWM | 0.99 | 0.13 | 0.714 | |||

Inhibition | 0.99 | 0.21 | 0.649 | |||

Switching | 0.12 | 4.45 | 0.035 | |||

Model statistics | χ^{2} = 3.04, |
χ^{2} = 8.53, |

The overall rate of identification on the trial-by-trial shortcut task was 53%, with 64 identifiers and 56 non-identifiers. Comparing this rate with the rate of identification on the shortcut self-report task indicates that there were 33 participants who reported using the shortcut on the self-report task, 27 of whom were also identifiers on trial-by-trial shortcut task. Furthermore, 86 participants did not report using the shortcut on the shortcut self-report task, 49 of whom also did not report using the shortcut on the trial-by-trial shortcut task. The trial-by-trial shortcut task includes many more opportunities to identify the shortcut and therefore we would expect the rate of identification to be higher.

The data met the assumptions of a hierarchical logistic regression (above), and

Predictor | Model 1 |
Model 2 |
||||
---|---|---|---|---|---|---|

Exp(β) | Wald | Significance | Exp(β) | Wald | Significance | |

Calculation skill | 1.07 | 5.33 | 0.021 | 1.05 | 2.55 | 0.110 |

Order of operations | 1.00 | 0.92 | 0.336 | 1.00 | 0.95 | 0.330 |

VSWM | 1.00 | 0.02 | 0.880 | |||

Inhibition | 0.98 | 4.76 | 0.029 | |||

Switching | 1.41 | 0.21 | 0.644 | |||

Model statistics | χ^{2} = 7.19, |
χ^{2} = 12.28, |

For the self-reported identifiers who had an IP (

There was a significant correlation between calculation skill and inhibition, suggesting that the tasks may have partly measured the same construct. To investigate which variable drove the relationship with shortcut identification, Bayesian analyses were conducted to see if the Bayes factors were substantially larger for one variable (calculation skill or inhibition). Independent Bayesian _{10} of 3.090, and in inhibition was supported by a BF_{10} of 9.297 (Bayes factors of 1-3 are ‘anecdotal’ and 3-10 are ‘moderate’ evidence for the alternative hypothesis; _{10} for inhibition is approximately three times that of calculation skill, suggesting that inhibition is the more reliable predictor of identification.

Strategy identification may be a key component of advanced strategy competencies such as adaptivity and flexibility, which in turn predict mathematical expertise (

Our research contributes to this goal by investigating whether domain-specific and domain-general skills are important for identifying shortcuts, and our findings suggest that of those skills, inhibition is the most important. In the next section we discuss the theoretical contribution of our findings to the literature on a) arithmetic strategy choice and b) arithmetic principles. We then discuss how the three functions of inhibition, a) resisting distractor interference, b) resisting proactive interference, and c) resisting prepotent responses, might help in the identification of the shortcut.

A variety of computational models predict how and when individuals select among a repertoire of strategies and shift from using one strategy to another (e.g.,

Some studies have inferred a role of executive functions by comparing strategy selection of individuals of different ages, and individuals with and without cognitive difficulties (e.g.,

In the arithmetic principle literature, studies have begun to explore the role of executive functions in children’s use of arithmetic principles (e.g. shortcut use on ‘a + b – b’ inversion problems). For example, studies have investigated whether children’s working memory and switching skills relate to solution accuracy and self-reported use of strategies derived from a mixture of arithmetic principles (e.g.,

It is possible that other factors, beyond those measured here, may also be involved in selection of an arithmetic shortcut. Regarding domain-general factors, we considered visuospatial working memory rather than verbal working memory, because our arithmetic expressions were presented visually and we were specifically interested in skills required to select an arithmetic shortcut rather than to calculate the shortcut. However, it is also possible that verbal working memory may play a role. Verbal working memory is involved in arithmetic calculation (

Regarding domain-specific factors, we found that knowledge of operation order was not involved in selection of an arithmetic shortcut. This study was conducted in the UK, and the majority of previous research on associativity shortcuts has been conducted in Canada (e.g.,

We note that 6 participants who identified the shortcut on the self-report task then did not identify the shortcut on the trial-by-trial task. This could be because the practice trials for the trial-by-trial shortcut task were non-conducive, and could have caused self-doubt as to whether the shortcut was appropriate (i.e., because it conflicts with taught acronyms). Further domain-specific factors that were not measured here may also be involved in associativity shortcut selection. Identifying problems on which the associativity-based shortcut is advantageous requires estimation of the size of the quantities involved. Consequently, an additional domain-specific skill that may be relevant is estimating the magnitude of symbolic quantities. Tasks such as number line estimation or symbolic magnitude comparison would be worth considering as predictors in future research.

Go/No-Go tasks are established measures of the ability to a) pause before response selection and b) pause during response execution, both of which are relevant to identification. For example, on three-term inversion problems some individuals discover the shortcut before initiating any calculation, while others discover it part-way through executing a left-to-right procedure (

Other functions of inhibition include a) resisting distractor interference and b) resisting proactive interference (

Our findings highlight the importance of inhibition in selecting efficient arithmetic strategies. Although our research was conducted with adults, it is likely that similar factors influence strategy selection in younger learners. These factors may be more important for younger children in whom executive function skills, such as inhibition, are still developing. Our findings can be used to raise teachers’ awareness that the reason children sometimes choose suboptimal strategies is not because they lack an understanding of mathematics, but because they lack sufficient cognitive skills. As a result, teachers could devise techniques to lessen those cognitive demands, such as encouraging children to pause and think about how they are going to solve a problem before they begin. There is some evidence that “stop and think” interventions of this form can be beneficial (

Identifying arithmetic strategies is necessary if individuals are to become adaptive and flexible in arithmetic strategy use. However, relatively few individuals select efficient ‘shortcut’ strategies on problems of the format ‘a + b – c’ and in a well-powered correlational study we investigated why. Of five domain-specific and domain-general skills, inhibition was the most reliable predictor of identifying the shortcut. We propose different functions that inhibition may serve, which could be communicated with educators to make them aware of the importance of cognitive resources in the selection of efficient strategies.

This work was supported by a PhD Studentship from Loughborough University Doctoral college. C.G. is supported by a Royal Society Dorothy Hodgkin Fellowship.

For this article, a data set is freely available (

The Supplementary Materials contain the following items (for access see

Pre-registration protocol

Research data

Materials for the tasks

The authors have declared that no competing interests exist.

The authors have no additional (i.e., non-financial) support to report.