^{1}

^{2}

^{1}

^{1}

This study examined adults’ frequent, efficient and adaptive use of direct subtraction (DS) and subtraction by addition (SBA) in mental multi-digit subtraction with the choice/no-choice method. Participants were offered subtractions in one choice condition (choice between DS and SBA) and two no-choice conditions (mandatory use of either DS or SBA). SBA was used as frequently as DS in the choice condition. DS was most accurate on subtractions with a large difference (e.g., 502 – 18), while SBA was fastest on subtractions with a small difference (e.g., 903 – 886). In general, participants were adaptive for task characteristics and their personal speed characteristics. We further analyzed task-based adaptivity on an individual level via a Latent Class Analysis. Results showed that two-thirds of the participants were adaptive to task characteristics, and that these adaptive participants were the most proficient in accuracy and speed in the choice condition. We further examined whether executive functions (updating, inhibition, shifting) were related to individual differences in strategy efficiency and task-based adaptivity. In line with our hypothesis, updating was related to strategy efficiency, such that participants with higher updating skills were more accurate. In contrast to our expectations, inhibition and shifting were not related to task-based strategy adaptivity. This study highlights adults’ efficient and adaptive use of arithmetic strategies, and its association with their proficiency and executive functions.

About a decade ago,

To further examine adults’ SBA use,

These two studies of

Second, the studies by

Third,

The present study aimed to address the three above-mentioned shortcomings from the studies of

First, we examined the following hypotheses regarding the frequent, efficient and adaptive use of SBA, which were all based on the results of

Our second goal was to examine participants’ task-based adaptivity on an individual level, and to verify whether adaptive strategy choices were related with higher proficiency in mental multi-digit subtraction. For the first time in adults, we investigated adaptivity for task characteristics on an individual level, using an LCA (similar to

As a third goal, we examined whether EFs were related to individual differences in strategy efficiency and adaptivity. We examined whether updating was related to strategy efficiency in the no-choice conditions (i.e., conditions not influenced by strategy selection). Because the mental multi-digit subtraction task is a complex task, which places a great demand on WM, we hypothesized that participants with higher updating skills would execute the SBA and DS strategy more efficiently (

One hundred and forty Flemish undergraduate psychology students (91% female; _{age}

All tests were computerized and group administered, except the Tempo Test Arithmetic, which is a paper-and-pencil test. We used OpenSesame (

To investigate the role of arithmetic ability, participants’ fluency in mental subtraction and addition was assessed using an extended version of the Tempo Test Arithmetic (

We used a multi-digit subtraction task similar to the one described in

After completing three practice items, all participants completed the choice condition, followed by the two no-choice conditions, for which the order was counterbalanced (as recommended by

Updating ability was measured with an N-back task (

To measure inhibition ability, we used a Simon (

Finally, shifting ability was measured using a modified version of the Wisconsin Card Sorting Task (WCST;

A more detailed description of all tasks is included in the

Measure | Min | Max | |||
---|---|---|---|---|---|

Tempo Test Arithmetic | 140 | 63.03 | 6.54 | 42 | 79 |

N-back task | 140 | 12.43 | 4.75 | 1 | 21 |

Simon task | 139 | 114.23 ms | 52.98 ms | 12.26 ms | 440.56 ms |

Flanker task | 140 | 130.48 ms | 64.30 ms | -21.19 ms | 463.74 ms |

WCST | 139 | 75% | 12% | 25% | 87% |

All tests were conducted in groups of approximately 20 participants. Testing was split up in two test sessions which took place during the same week. During the first session, we administered the multi-digit subtraction task and the Simon task. During the second session, we conducted the Tempo Test Arithmetic, the N-back task, the Flanker task and the WCST. Tasks were always administered in the order listed above. The multi-digit subtraction task and the Tempo Test Arithmetic were presented in separate sessions to eliminate subtraction practice effects. Instructions for all tests were given at the beginning of each session. The duration of a session was approximately 50 minutes.

All analyses were performed using JASP software (_{s}

Ninety percent of the participants used SBA at least once in the choice condition (RQ1a). Only 9% never used SBA, 82% used both SBA and DS, and 9% always used SBA. Furthermore, a Monte Carlo K-sample Fisher-Pitman permutation test showed no association between arithmetic achievement level and strategy repertoire (RQ1f),

Participants used SBA to solve about half of the items in the choice condition (_{s}

To evaluate differences in efficiency in the no-choice conditions (separately for strategy accuracy and speed; RQ1c), repeated measures ANOVAs with strategy (SBA, DS) and item type (SD, LD) as within-subject factors were used.

Regarding strategy accuracy (_{holm}_{holm}_{holm}_{holm}_{holm}_{holm}_{holm}

Arithmetic Achievement group | No-choice SBA |
No-choice DS |
|||
---|---|---|---|---|---|

Small difference | Large difference | Small difference | Large difference | ||

Low | 29 | 0.82 (0.25) | 0.67 (0.28) | 0.85 (0.23) | 0.77 (0.27) |

Below-average | 31 | 0.96 (0.10) | 0.80 (0.21) | 0.90 (0.13) | 0.90 (0.12) |

Above-average | 29 | 0.88 (0.20) | 0.81 (0.20) | 0.88 (0.18) | 0.88 (0.19) |

High | 33 | 0.96 (0.08) | 0.86 (0.24) | 0.91 (0.17) | 0.87 (0.21) |

Total | 122 | 0.91 (0.18) | 0.79 (0.24) | 0.89 (0.18) | 0.86 (0.21) |

For strategy speed (_{holm}_{holm}_{holm}_{holm}_{holm}_{holm}_{holm}_{holm}

Arithmetic achievement group | No-choice SBA |
No-choice DS |
|||
---|---|---|---|---|---|

Small difference | Large difference | Small difference | Large difference | ||

Low | 29 | 10.58 (6.64) | 17.53 (7.17) | 18.50 (8.33) | 17.45 (7.12) |

Below-average | 31 | 8.46 (5.13) | 15.69 (7.51) | 13.71 (4.87) | 13.57 (6.04) |

Above-average | 29 | 8.14 (4.12) | 14.28 (5.71) | 14.51 (5.91) | 12.78 (5.10) |

High | 33 | 5.41 (1.95) | 10.14 (3.90) | 13.48 (6.26) | 10.22 (4.39) |

Total | 122 | 8.06 (5.01) | 14.29 (6.72) | 14.98 (6.67) | 13.40 (6.22) |

To examine participants’ adaptivity, we investigated task-based adaptivity (RQ1d) on a general level, using a repeated measures ANOVA, and on an individual level, using an LCA (RQ2a). Furthermore, subject-based adaptivity (RQ1e) was examined using ANCOVAs.

A repeated measures ANOVA, with frequency of SBA in the choice condition as dependent variable and item type as within-subject factor showed a main effect of item type,

Two ANCOVAs were used to examine whether participants took into account their individual strategy efficiency (separately for accuracy and speed) when selecting a strategy in the choice condition. The analysis, with the frequency of SBA as dependent variable and either the accuracy or speed differences between the no-choice conditions as covariate, revealed that participants were adaptive for their personal speed characteristics,

We performed an LCA on the strategy choices in the choice condition to assess participants’ adaptivity to task characteristics on an individual level (RQ2a). A model with three classes was retained based on theoretical considerations and statistical evidence. Theoretically, a three-class model, including a class of participants who consistently use DS, a class of consistent SBA users, and a class of participants who adapt their strategy choices to the task characteristics, was considered most likely. These were actually also the three classes that

The estimated class population shares indicated that 18.5% of the participants belonged to the DS class, 15.7% to the SBA class, and 65.8% to the adaptive class (Entropy ^{2} = .97, posterior error = .03). Subsequently, the software predicted the class membership for each individual participant, based on modal posterior probability, leading to the final distribution into classes that was used in the remainder of this article. _{prob}_{prob}_{prob}_{prob}

A Monte Carlo K-sample Fisher-Pitman permutation test showed that participants from different arithmetic achievement levels were not equally distributed (

Arithmetic achievement groups | Latent class |
|||
---|---|---|---|---|

DS class | Adaptive class | SBA class | ||

Low | 35 | 11 (31%) | 15 (43%) | 9 (26%) |

Below-average | 35 | 6 (17%) | 26 (74%) | 3 (9%) |

Above-average | 34 | 4 (12%) | 25 (74%) | 5 (15%) |

High | 36 | 4 (11%) | 27 (75%) | 5 (14%) |

Total | 140 | 25 | 93 | 22 |

To examine the relation between adaptivity and the proficiency in the choice condition (RQ2b), we used repeated measures ANOVAs to compare the accuracy and speed in the choice condition (in separate analyses; dependent variable) per item type (within-subject variable) between the three latent classes found in the LCA (between-subjects variable).

For accuracy, there was a main effect for latent class _{holm}_{holm}_{holm}_{holm}_{holm}_{holm}

Latent class | Accuracy |
Speed |
||
---|---|---|---|---|

Small difference | Large difference | Small difference | Large difference | |

DS class | 0.72 (0.29) | 0.79 (0.20) | 16.61 (6.12) | 17.91 (8.02) |

Adaptive class | 0.93 (0.11) | 0.81 (0.23) | 8.87 (4.83) | 15.66 (6.74) |

SBA class | 0.85 (0.22) | 0.76 (0.27) | 9.63 (5.08) | 16.80 (7.26) |

Total | 0.88 (0.19) | 0.80 (0.23) | 10.37 (5.87) | 16.24 (7.06) |

Turning to speed, again, a main effect for latent class was found (_{holm}_{holm}_{holm}_{holm}_{holm}_{holm}_{holm}_{holm}

We first examined whether updating was related to strategy efficiency in the no-choice conditions (RQ3a). We used the same analyses as described in section ‘Efficiency’, i.e., two repeated measures ANOVAs with strategy and item type as within-subject factors and arithmetic achievement level as covariate, but included participants’ score on the N-back task as an additional covariate. No significant effect for N-back score was found in the model for strategy speed,

We then examined whether inhibition and shifting were related to task-based strategy adaptivity in the choice condition (RQ3b). Multiple ANOVAs were used to compare the scores on the tasks for inhibition and shifting (dependent variable) between the three latent classes (between-subjects variable; see

Latent class | Flanker task | Simon task | WCST |
---|---|---|---|

DS class | 126.69 ms (71.92 ms) | 119.55 ms (77.83 ms) | 75% (8%) |

Adaptive class | 133.65 ms (65.44 ms) | 115.75 ms (46.28 ms) | 75% (13%) |

SBA class | 121.37 ms (50.44 ms) | 101.86 ms (45.16 ms) | 74% (12%) |

Total | 130.48 ms (64.30 ms) | 114.23 ms (52.98 ms) | 75% (12%) |

The current study examined adults’ use of SBA in mental multi-digit subtraction, expanding previous research by

Concerning strategy repertoire (RQ1a) and distribution (RQ1b), we hypothesized that most adults would use SBA at least once in the choice condition, and that SBA would be used in this condition about as frequently as DS. Our expectations were met, as 90% of participants used SBA at least once, and 51% of the items in the choice condition were solved using SBA. As expected, based on the findings of

For strategy efficiency (RQ1c), we expected adults to perform SBA faster than DS and at least as accurately, and SBA to be more efficient than DS on SD items, in the no-choice conditions. Concerning accuracy, our hypothesis was only partially confirmed, as on SD items both strategies were equally accurate. However, contradicting the results of

Concerning strategy speed, our hypothesis was also only partially confirmed, since, while SBA was, as expected, executed faster than DS on SD items, there was unexpectedly no speed difference between the two strategies on the LD items. The efficiency on SD items might reflect a speed-accuracy trade-off. Although participants were instructed to solve the items as accurately and fast as possible, they might have prioritized accuracy at the expense of speed for the SD items. This could explain why there was an advantage of SBA over DS on the SD items for speed, but not for accuracy. As expected, the lowest arithmetic achievement groups were the slowest (RQ1f). Notably, while all other groups solved SD items faster than LD items, the high arithmetic achievement group was the only group for which the solution times for SD and LD items did not differ. This result could be explained by their overall fast responses, which reveals their efficient use of both strategies on both item types.

We examined adults’ adaptivity for task and subject characteristics at group level. Confirming our hypothesis (RQ1d), participants were adaptive for task characteristics, as they used SBA more frequently on SD than on LD items in the choice condition. This effect was significant for participants from all arithmetic achievement levels. Turning to subject-based adaptivity (RQ1e), as expected, participants were adaptive for their strategy speed, but not for their strategy accuracy. This contrastive finding could be explained by the lack of direct feedback about accuracy, compared to speed, when solving subtractions. When solving some items with DS and other items with SBA, there is a possibility that participants notice that for certain types of items, one strategy leads to a faster answer than the other, whereas they do not get such feedback for accuracy. Finally, there was no effect of arithmetic achievement on speed-based adaptivity (RQ1f), contrasting the surprising result from

For the first time in adults, we examined task-based adaptivity in the choice condition also on an individual level, with an LCA (RQ2a). In line with previous results in ten-year-olds (

As stated above, both previous studies of

We examined whether the EFs updating, inhibition and shifting were related to individual differences in strategy efficiency and task-based adaptivity. On the one hand, our results showed that updating was related to strategy accuracy (but not speed) in the no-choice condition (RQ3a). Our study thus provides additional evidence that the ability to hold information in WM and to flexibly manipulate it was related to overall mental multi-digit arithmetic efficiency (

Why did our study provide only limited evidence for an association between EFs and strategy efficiency and adaptivity in mental multi-digit subtractions? At a general level, this might be due to the fact that our sample was a rather homogeneous group of university students, all belonging to a similar age range. A more heterogeneous sample would contain a better representation of the distribution of arithmetic achievement and EFs in the general population, and this larger variation has the potential to find stronger associations between EFs and strategy efficiency and adaptivity. Moreover, most studies that did find a relation between EFs and mathematics achievement involved children (

While our expectations concerning the association between inhibition and shifting, on the one hand, and strategy selection, on the other hand, were theoretically grounded – as it is reasonable to expect that participants have to inhibit the use of a dominant subtraction-based process when confronted with a minus sign, and need to switch between different strategies within a given set of items – no evidence was found for these associations. In this respect, we point to a recent review by

In addition to confirming findings from previous research regarding the frequent, efficient, and adaptive use of SBA (

As argued by several previous researchers (e.g.,

Lieven Verschaffel is one of the Guest Editors of this JNC Special Issue but played no editorial role in this particular article or intervened in any form in the peer review process.

This research was supported by grant G0C7217N “Subtraction by addition. A most efficient strategy for solving symbolic subtraction problems?” from the Research Foundation - Flanders.

From every participant, a written informed consent was obtained. The study was approved by the local ethical board of the university (SMEC; registration number: G- 2017 12 1033).

The Supplementary Materials include the following items (for access see

a table with all multi-digit subtraction items that were used in this study,

a detailed description of the three tasks that were used to measure executive functions,

a table with the exact probability values to use SBA (and SE) for each class from the LCA, and

figures displaying the results for strategy efficiency and proficiency.

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