Symbolic number ordering has been related to arithmetic fluency; however, the nature of this relation remains unclear. Here we investigate whether the implementation of strategies can explain the relation between number ordering and arithmetic fluency. In the first study, participants (N = 16) performed a symbolic number ordering task (i.e., “is a triplet of digits presented in order or not?”) and verbally reported the strategy they used after each trial. The analysis of the verbal responses led to the identification of three main strategies: memory retrieval, triplet decomposition, and arithmetic operation. All the remaining strategies were grouped in the fourth category “other”. In the second study, participants were presented with a description of the four strategies. Afterwards, they (N = 61) judged the order of triplets of digits as fast and as accurately as possible and, after each trial, they indicated the implemented strategy by selecting one of the four pre-determined strategies. Participants also completed a standardized test to assess their arithmetic fluency. Memory retrieval strategy was used more often for ordered trials than for non-ordered trials and more for consecutive than non-consecutive triplets. Reaction times on trials solved by memory retrieval were related to the participants’ arithmetic fluency score. For the first time, we provide evidence that the relation between symbolic number ordering and arithmetic fluency is related to faster execution of memory retrieval strategies.

Ordinality is a dimension of number processing that refers to the position of an item within a sequence (

The manipulation of the numerical distance between digits (e.g., distance 1: 1-2-3; distance 2: 1-3-5) has led to some initial speculations on the underlying cognitive mechanisms of number order judgment. Individuals are faster in responding when digits in ordered triplets are consecutive (i.e., distance 1; 1-2-3) compared to non-consecutive ones (e.g., 1-4-7). This phenomenon is known as the reversed distance effect (RDE) as a small distance between digits in the triplets leads to faster reaction times (

These different behavioural signatures indicate that possibly different strategies are used for different types of trial in the symbolic number ordering task. Accordingly, it has been suggested that “magnitude-based” processes are used with not-ordered triplets, whereas “memory-based” mechanisms used with ordered triplets (

Different mechanisms could explain the presence of the RDE. Order judgment may rely on a serial visuospatial item-by-item scanning on a mental number line (

The precise contribution of different strategies in a symbolic order judgment task should be examined further, though. So far, evidence for the use of different strategies in the number order judgment task is indirectly inferred from reaction time differences between different types of trials (e.g., ascending vs descending, consecutive vs non-consecutive). Retrospective self-reports after every trial constitutes an effective methodology to assess strategy implementation. This method has been successfully applied to examine the use of strategies in mental arithmetic in children and adults (e.g.,

In the first study, we aimed to identify the common solution strategies that participants implement when judging the ordinality of triplets. Previous studies based on mental reports suggested that adults are capable of describing their solution processes retrospectively for simple arithmetic problems (

Sixteen undergraduates from Psychology or Educational Sciences at KU Leuven Kulak (_{age}

Participants received a booklet, whereby each page presented a triplet (font bold Courrier view, size 40). The task included 56 triplets; 28 ordered and 28 non-ordered, that were taken from the study of Vos et al. (2017). For 28 ordered triplets, there were 14 ascending and 14 descending. Half of the ascending and descending triplets consisted of consecutive triplets (e.g., 1-2-3 or 3-2-1), the other half of non-consecutive triplets with a numerical distance of 2, 3, or 4 (e.g., 1-4-7 or 7-4-1). Triplets with distance 2, 3 and 4 were grouped together to have an equal number of trials for consecutive and non-consecutive conditions. The distance between the first two digits and the last two digits of the triplet was always identical. The remaining 28 triplets were not-ordered and matched the size and the distance of the ordered ones (e.g. 1-3-2; 4-7-1). The full list of triplets is reported in

Participants were instructed to choose between two options “correct” and “incorrect” and mark the right answer. After each triplet, they also needed to write down the strategy they implemented to decide whether the three digits were in order or not. They could choose between reporting their solutions in English or in Dutch (their first language) to avoid language-related difficulties in using math-related terminology. All the participants held nativelike proficiency in Dutch.

The self-reports are available at

Based on the previous literature (

We then applied these categories to the dataset (N = 16; 894 trials)

Distribution of the strategies (in percentage) per condition in Study 1 (AC: ascending consecutive; AnC: ascending non-consecutive; DC: descending consecutive; DnC: descending non-consecutive; NC: not-ordered consecutive; NnC: not-ordered non-consecutive).

MEMORY (%) | DECOMPOSITION (%) | ARITHMETIC (%) | OTHER (%) | |
---|---|---|---|---|

60.71 | 24.11 | 15.18 | 0.00 | |

44.14 | 15.32 | 33.33 | 7.21 | |

43.75 | 32.14 | 17.86 | 6.25 | |

32.43 | 24.32 | 33.33 | 9.91 | |

22.77 | 55.36 | 18.30 | 3.57 | |

21.43 | 54.91 | 22.32 | 1.34 | |

The percentage of strategy use (y-axis) as a function of the direction of the triplet (x-axis; ascending, descending, not-ordered) and the numerical distance between digits [Consecutive, left panel; Non-consecutive, right panel).

We analyzed the implementation of the memory strategy compared to the three remaining strategies, which were clustered together as “not-retrieval strategies”. We ran a logistic regression on memory strategy use [0 = not-memory, 1 = memory] with distance [consecutive, non-consecutive] and direction [ascending, descending, not-ordered] as predictors. We found the main effects of distance and direction (χ^{2} = 70.24, ^{2} = 3.03,

Study 1 aimed to identify the strategy repertoire in a symbolic number ordering task and examine the frequency of strategies across task conditions. We assigned participants’ solving strategies to four categories: memory retrieval, triplet decomposition, arithmetic operations and “other” strategies. Participants used memory retrieval more often for ascending triplets compared to descending and not-ordered ones, and for consecutive compared to non-consecutive digits. This result suggests the use of memory retrieval, either through verbal or visual recognition, when triplets matched a portion of the counting list (

Decomposition, which entails the sequential magnitude comparison of digits within the sequence, was most widely used for not-ordered triplets. We speculate that participants adopted a decomposition strategy when they realized that the triplet did not belong to the counting list (forward or backwards). The strategy based on arithmetic operations was not previously contemplated (see

In summary, by assessing retrospective self-reports, Study 1 demonstrated that participants use a variety of solution strategies in a symbolic number ordering task. We propose the presence of three main strategies: memory retrieval, decomposition, and arithmetic operations. Descriptive statistics showed that memory retrieval is used more often for ordered triplets with consecutive digits, whereby a direct recognition that the triplet belongs to the counting list can be made. Conversely, not-ordered triplets with non-consecutive digits prompted the use of more sequential strategies, such as comparing the magnitude of digits or performing some arithmetical operations.

In Study 2, we aimed at replicating these results in a larger sample, using a choice menu for assessing strategies instead of retrospective self-reports. We also registered reaction times to analyse the speed of execution for each strategy. Finally, we measured participants’ arithmetic fluency to evaluate whether the frequency and the speed of execution of one or more strategies is related to arithmetic skills.

Sixty-one undergraduates from Applied Economics at KU Leuven Kulak (_{age}

At the beginning of the experimental session, participants completed the Tempo Test Rekenen (TTR; Vos, 1992). The TTR is a time-limited test consisting of five columns on a sheet of paper: addition, subtraction, multiplication, division and one with mixed operations. Each column consists of forty arithmetic problems (e.g., 12 × 3 = ___) presented in increasing difficulty. Participants completed one column at time and had one minute to solve as many problems as possible in a column. We calculated the total number of correct responses.

After completing the arithmetic fluency test, participants were introduced to the number order judgement task and the strategies derived from Study 1. They received sheets with explanations of the four strategies and some examples (The description of the strategies can be found at

We removed responses below 200 msec (i.e., anticipations; 6 trials) and extremely slow responses (44 trials) that were three standard deviations above the grand mean. We then calculated the individual median reaction times of correct responses. We removed one participant from further analysis because had a median response time that was more than three standard deviations above the sample mean. Accuracy was close to ceiling (

First, we examined the presence of reversed distance effect (RDE) and distance effect (DE) in the response times (RTs) across conditions. Second, we evaluated the frequencies of strategies and execution times of memory compared to the remaining strategies. Finally, we examined the relation between the performance in the order judgement task and arithmetic fluency.

To check to which extent our data replicated behavioral effects from previous studies (^{2}_{g} = .003, whereas the main effect of direction was, _{[gg]} = .002, η^{2}_{g} = .02. Crucially, the interaction between distance and direction was also significant, ^{2}_{g} = .01. Participants were faster with consecutive (_{[bonf]} = .015. A similar tendency for a RDE was observed for descending triplets (consecutive: _{[bonf]} = .075). No such difference was observed between consecutive and non-consecutive triplets in the not-ordered trials (consecutive: _{[bonf]} = .141).

In

Boxplots represent the distribution of percentages of strategy use (y-axis); dots represent individual values.

In

The percentage of strategy use (y-axis) as a function of the direction of the triplet (x-axis; ascending, descending, not-ordered) and the numerical distance between digits [Consecutive, left panel; Non-consecutive, right panel).

Distribution of the strategies (in percentage) per condition in Study 2 (AC: ascending consecutive; AnC: ascending non-consecutive; DC: descending consecutive; DnC: descending non-consecutive; NC: not-ordered consecutive; NnC: not-ordered non-consecutive).

MEMORY (%) | DECOMPOSITION (%) | ARITHMETIC (%) | OTHER (%) | |
---|---|---|---|---|

70.68 | 18.55 | 8.77 | 2.01 | |

49.74 | 29.64 | 15.98 | 4.64 | |

63.73 | 42.35 | 9.59 | 2.38 | |

44.09 | 38.22 | 14.70 | 2.89 | |

31.66 | 54.44 | 7.72 | 6.18 | |

33.84 | 50.88 | 8.08 | 7.20 | |

We analyzed the implementation of the memory strategy compared to the three remaining strategies, which were clustered together as “not-retrieval strategies”. We ran a logistic regression on memory strategy [0 = not-memory, 1 = memory] use with distance [consecutive, non-consecutive] and direction [ascending, descending, not-ordered] as predictors. We found evidence for the interaction model compared to the model with the two main effects (χ^{2} = 39.81,

We also analyzed the execution times of the different strategies. The implementation of the memory strategy, which is based on the immediate retrieval, should yield faster reaction times compared to the remaining non-memory strategies, which, by definition, require a series of additional procedures or manipulations in order to verify whether a triplet was in order or not. In

Boxplots of reaction times (y-axis) for trials in which the memory strategy was used (Yes; white) or not (No; dark grey) across direction (left: Ascending; middle: Descending; right: Not-ordered) and distance (x-axis). The diamonds represent the mean of the distribution whereas transparent dots represent single trials.

We examined the zero-order correlations between arithmetic fluency and the overall median RT and the median RTs for each condition separately (

Zero-order correlations between overall median RTs (All), median RTs across conditions (AC: ascending consecutive; AnC: ascending non-consecutive; DC: descending consecutive; DnC: descending non-consecutive; NC: not-ordered consecutive; NnC: not-ordered non-consecutive), and arithmetic fluency score (TTR). **

MEASURE | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

1. All | |||||||

2. AC | .79** | ||||||

3. AnC | .88** | .76** | |||||

4. DC | .86** | .75** | .72** | ||||

5. DnC | .85** | .71** | .85** | .76** | |||

6. NC | .88** | .59** | .70** | .69** | .63** | ||

7. NnC | .85** | .54** | .72** | .59** | .65** | .78** | |

8. TTR | –.35** | –.37** | –.36** | –.44** | –.44** | –.12 | –.19 |

To get a better insight into the relation between the frequency and efficiency of strategy use, we correlated the proportion of the memory strategy use and median RTs for memory retrieval and decomposition strategies with the TTR scores (

Zero-order correlations between proportion of memory strategy use, median RTs for memory, median RTs for decomposition, and arithmetic fluency (TTR). *

MEASURE | 1 | 2 | 3 |
---|---|---|---|

1. Proportion memory strategy | |||

2. Median RTs for memory strategy | –.04 | ||

3. Median RTs for decomposition strategy | .12 | .67** | |

4. TTR | .13 | –.31* | –.12 |

The aim of Study 2 was twofold. First, we examined whether the findings from Study 1 could be replicated in a larger sample by assessing strategies with a retrospective choice menu instead of free self-reports. All categories of strategies derived on the basis of free retrospective reports in Study 1 were selected on several occasions. In general, the memory retrieval strategy (45%) and the decomposition strategy (40%) were reported more often. The arithmetic operations strategy was reported in only 10% of the trials. This pattern is similar to the one observed in Study 1. Moreover, as in Study 1, the frequency of each strategy varied across conditions: memory retrieval was used more in ordered conditions, especially with consecutive triplets (i.e., consecutive triplets;

A second aim of Study 2 was to examine whether frequency and efficiency of strategies related to arithmetic fluency. The frequency of the memory retrieval strategy was not related to arithmetic fluency. In contrast, participants doing well on the arithmetic fluency task were faster on number ordering trials in which they reported using memory retrieval. We return to the relevance of this finding in our general discussion.

Recently, a lot of research has been devoted to the symbolic number ordering task judgement because of its relation with arithmetic fluency (for reviews see

The first aim of this study was to identify the strategies applied in the symbolic number order judgment task. In contrast to the field of arithmetic, in which strategies are introduced as a part of the educational curriculum at schools, we needed to define a repertoire of strategies in number ordering. Therefore, in Study 1, we analyzed participants’ self-reports on solving strategies. We identified three main strategies: memory retrieval, triplet decomposition, and arithmetic operations. A few self-reports did not fit in one of these three categories and were labelled as “

Memory retrieval was more frequently used for ordered triplets, especially with consecutive digits. In contrast, decomposition was applied more often for non-ordered triplets. Presumably, memory strategies are more often used when the triplet matches the counting list either forward or backwards. When the triplet does not match the counting list, participants used more sequential strategies like decomposition. It remains an open question whether participants compare the triplet with the counting list and, in case of a non-match, apply decomposition or arithmetic strategies or whether the strategies are run in parallel.

Our findings are in line with previous studies (Vos et al., 2017;

As expected, response times were faster when memory retrieval strategy was applied compared to non-memory retrieval strategies. This has important consequences for the origin of the reversed distance effect. Previous studies argued that the reversed distance may be the consequence of different association strengths between consecutive and non-consecutive digits (

The second aim was to get a better insight into the association between symbolic number ordering and arithmetic proficiency. In line with previous studies, we found that fast response times in the order judgment task are associated with better arithmetic fluency (

The findings of the present study should be considered in light of some limitations. First, the distance between the first two and last two digits of each ordered triplet was always the same. The regular pattern of the ordered triplets may have triggered the memory retrieval strategy, resulting in an overestimation of this strategy in the current study. Further studies are needed to examine whether including triplets with varying inter-item distances (e.g., 2-4-7) influences the frequency of strategies.

Second, the homogeneity of our sample may have resulted in participants using the different strategies with similar frequency. Therefore, whether arithmetically skilled and less-skilled participants differ present a different frequency distribution of strategies remains to be further investigated in a more heterogeneous sample. Related, our findings cannot be generalized to children, who are still learning associations between single digits (i.e., ordering) and associations between arithmetic problems and solutions (i.e., arithmetic fluency).

The final limitation concerns the validity of the retrospective self-reports as an empirical method. For instance, participants can differ with respect to the ability to give these retrospective reports; there is no consistent evidence yet for a relation between the awareness level of the strategy usage and its further verbalization (for the discussion, see

Using retrospective self-reports, this study demonstrated the distribution of strategies across different conditions of the symbolic number ordering task. Memory retrieval was used more often for ordered sequences and decomposition for not-ordered sequences. The well-established relation between symbolic number ordering and arithmetic fluency is due to faster execution of memory retrieval strategies, suggesting that the common core between symbolic number ordering and arithmetic is memory retrieval.

OSF project link:

The additional file for this article can be found as follows:

Stimulus list. DOI:

Helene Vos. PhD in Psychology.

One observation was missed due to a participant skipping a trial.

Two observations were missing because two participants skipped a trial.

The experimental protocol was approved by the university’s ethical committee (SMEC: G-2016 12 703). All participants gave written informed consent before participating in the study.

The authors have no competing interests to declare.