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We investigated the use of the subtraction by addition strategy, an important mental calculation strategy in children with different levels of mathematics achievement. In doing so we relied on Siegler’s cognitive psychological model of strategy change (Lemaire & Siegler, 1995), which defines strategy competencies in terms of four parameters (strategy repertoire, distribution, efficiency and selection), and the choice/no-choice method (Siegler & Lemaire, 1997), which is essentially characterized by offering items in two types of conditions (choice and no-choice). Participants were 63 11-12-year-olds with varied mathematics achievement levels. They solved multi-digit subtraction problems in the number domain up to 1,000 in one choice condition (choice between direct subtraction or subtraction by addition on each item) and two no-choice conditions (obligatory use of either direct subtraction or subtraction by addition on all items). We distinguished between two types of subtraction problems: problems with a small versus large difference between minuend and subtrahend. Although mathematics instruction only focused on applying direct subtraction, most children reported using subtraction by addition in the choice condition. Subtraction by addition was applied frequently and efficiently, particularly on small-difference problems. Children flexibly fitted their strategy choices to both numerical item characteristics and individual strategy speed characteristics. There were no differences in strategy use between the different mathematical achievement groups. These findings add to our theoretical understanding of children’s strategy acquisition and challenge current mathematics instruction practices that focus on direct subtraction for children of all levels of mathematics achievement.

Cumulative evidence indicates that children and adults rely on a rich diversity of mental calculation strategies to solve elementary arithmetic tasks (e.g.,

On the other hand, previous studies have revealed that children hardly report the subtraction by addition strategy on multi-digit subtraction problems (

The theoretical and methodological framework of R. S. Siegler (

Since the end of the previous century, the acquisition of various strategies, which can be applied efficiently and flexibly on different types of mathematical tasks, has become a major goal of primary mathematics education worldwide (

The first classification, the

The second classification of multi-digit subtraction strategies, the ^{i} As indicated by the names of the strategies and as outlined above, direct subtraction strategies are characterized by the straightforward use of the subtraction operation to solve subtraction problems: they involve the direct subtraction of the subtrahend from the minuend (e.g., solving 628 - 313 = __ via 628 - 300 = 328, 328 - 10 = 318, 318 - 3 = 315). By contrast, subtraction by addition is characterized by the use of the complementary addition operation to solve a given subtraction problem: subtraction by addition strategies require to determine how much should be added to the subtrahend to arrive at the minuend (e.g., solving 628 - 313 = __ via 313 + 87 = 400, 400 + 200 = 600, 600 + 28 = 628, so the answer is 87 + 200 + 28 or 315). Whereas the direct subtraction strategy fits nicely with the operational model of subtraction as “taking away”, subtraction by addition corresponds with the “bridging the difference” model of subtraction (

Following a choice design, with subtraction problems offered in only one choice condition, previous investigations on children’s direct subtraction versus subtraction by addition strategy use revealed that children hardly report using the latter strategy to solve multi-digit subtraction problems (

Following a choice/no-choice design,

Against the background of the above-mentioned findings in adults, we aimed to analyse children’s use of the direct subtraction versus subtraction by addition strategy relying on the same theoretical framework and research method. As outlined above, previous investigations on children’s direct subtraction versus subtraction by addition strategy use (

Using the four parameters of the model of strategy change, we formulated the following research questions (RQ).

RQ1: Do primary school children verbally report subtraction by addition on subtractions up to 1,000 in the choice condition (= strategy repertoire)?

RQ2: How frequently do they apply subtraction by addition on subtractions up to 1,000 in the choice condition (= strategy distribution)?

RQ3: How accurately and quickly do they apply subtraction by addition compared to direct subtraction on subtractions up to 1,000 in the no-choice conditions, in general and on the different types of subtractions (= strategy efficiency)?

RQ4: Do they fit their strategy use to the numerical characteristics of the subtraction problems and/or to their individual strategy efficiency competencies (= strategy selection)?

RQ5: To what extent do the answers to these first four research questions differ between children of varying mathematical achievement levels?

Participants were 68 sixth-graders from two primary schools in Flanders (Belgium) (_{age} = 11y8m [

We analysed children’s mathematics instruction history via scrutinized textbook analyses and structured teacher interviews. These revealed that all children had received instruction on multi-digit subtraction problems from 2^{nd} grade on. In line with typical mathematics practices in Flanders, instruction focused on mental calculation through direct subtraction, and more specifically on the sequential direct subtraction strategy. After sufficient practice of this sequential direct subtraction strategy, children received instruction on other direct subtraction strategies, including varying strategies such as compensation. However, as is typically the case in Flanders (

We assessed children’s general mathematical achievement level with a standardized mathematical achievement test, addressing all domains of the mathematics curriculum in sixth grade (^{th} [low] and 84^{th} [high] percentiles). As elaborated in the Procedures section, the data of five participants (two low achievers, two above-average achievers, one high achiever) had to be discarded. The final sample consisted of 63 participants (_{10} = 9.8 x 10^{31}.

Mathematics Achievement Percentile Score |
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Achievement Group | Boys | Girls | Mean | Standard Deviation |

Low | 2 | 6 | 8.25 | 5.20 |

Below-average | 9 | 6 | 38.33 | 9.94 |

Above-average | 9 | 9 | 66.94 | 9.10 |

High | 8 | 14 | 93.59 | 5.21 |

Children were offered three parallel series of 13 multi-digit subtraction problems up to 1,000. In line with

All subtraction problems required crossing-over both tens and units. To dissuade children from using computation strategies that differ from the requested direct subtraction and subtraction by addition strategy (see below, conditions), we did not include subtraction problems with unit values of 0, 5 or 9. To match the difficulty of the three series of subtraction problems, we equated the mean sizes of the minuends and subtrahends as well as the mean sizes of the differences across the series.

In each of the three series subtraction problems were ordered based on three criteria: (1) unpredictable order of the different types of problems; (2) no repetition of problem type on two consecutive trials; (3) the answer to each subtraction problem could not easily be derived from the previous one. We created two orders per series, with order 2 being the reverse of order 1.

Children were interviewed individually in one choice and two no-choice conditions. In the choice condition, they could choose between direct subtraction and subtraction by addition on each problem. In the experimental instructions, animations (a boy and a girl) were used to explain the strategies and steps that were allowed. To allow valid comparisons of children’s direct subtraction versus subtraction by addition strategy competencies, children had to apply one specific sequential variant of direct subtraction and subtraction by addition (and were thus not allowed to use a rich variety of decomposition, sequential and varying direct subtraction versus subtraction by addition strategies). The solution steps were visible on the screen, and the experimenter explained what the boy and girl did to solve the example problem. In line with the adults studies of

In the

All children were tested individually in a quiet room at their school. They all solved the three parallel series of problems in a within-subject design with different sessions on different days, and at least one day in between two sessions. All conditions were offered on a laptop computer. In each condition, children were told the strategy/strategies they were allowed to use by using the above-mentioned boy/girl figure(s). Afterwards, they were asked to solve one practice item for all allowed strategies. They were not allowed to use pen and paper. They had to verbally report their solution steps immediately after solving each subtraction problem to make sure that they understood the strategy/strategies. Next, a series of 13 trials (10 experimental items and 3 buffer items) was presented. Each trial started with an asterisk that appeared for 1000 ms in the centre of the screen, followed by the item presented in the same position. The characters were 2 × 2 cm large, black, separated by adjacent spaces and presented against a white background. The item remained on the screen until the child uttered the answer. Time started to run when the item appeared and ended when the experimenter pushed the spacebar of an external keyboard. Next, the experimenter entered the child’s answer and the child was asked to verbally report his/her calculation steps. After this the next trial was initiated. No feedback was given. All conditions were audio-taped to allow a reliable analysis of the verbal strategy reports.

As recommended by

Only the accuracy, speed and verbal report data on the 2040 SD and LD subtraction problems (10 items in 3 conditions for 68 participants) were included in the dataset. Before starting analyses we excluded 64 trials (3.14%) due to technical problems during data collection. Next, we coded all verbal strategy reports to check whether children only applied the strategy/strategies they were allowed to use. Children’s verbal reports were coded as (a) direct subtraction, including all solutions in which the child sequentially took away the hundreds, tens and units of the subtrahend from the minuend; (b) subtraction by addition, including all reports in which the child used the complementary addition operation to solve the subtraction problem by first adding up from the subtrahend to the next hundred, then up to the hundreds of the minuend, then up to the minuend and finally computed the sum of all these intermediate results; or (c) other, consisting of all reports that could not be classified as direct subtraction or subtraction by addition as defined above and of all unclear strategy reports. Two researchers independently scored 10% of the strategy reports in the different conditions. They agreed on 85.38% of the trials and the interrater reliability was sufficient (Cohen’s κ = .75). Trials with no agreement were reconsidered and in this second round consensus was found on all trials. No items had to be removed due to not following instructions (using direct subtraction in the no-choice subtraction by addition condition or vice versa). All 66 strategy reports classified as other strategies were removed from the dataset (3.30%). Finally, we controlled for whether participants (in all three conditions) had at least 2 out of 5 SD and LD subtraction problems remaining in the dataset. Five participants (two low achievers, two above-average achievers, one high achiever) for which this was not the case were removed from further analyses. The final dataset thus consisted of data of 63 participants (1829 items, 89.66% of the initial dataset).

The current dataset was analysed by using both frequentist and Bayesian statistics with JASP software (version 0.8.1.2), using the default JASP settings. We applied repeated measures ANOVA with Tukey-Kramer adjustments for post-hoc comparisons. In addition to these classic statistical analyses, we calculated Bayes Factors (BF), which quantified the evidence in the data for the alternative hypothesis (H_{1}) of a given effect compared to the evidence for the null hypothesis (H_{0}): BF_{10}. If BF_{10} is larger than 1, the data contains more evidence for H_{1} than for H_{0}. If BF_{10} is between 1 and 0, the data contains more evidence for H_{0}. The size of the BF indicates the evidential strength for a given hypothesis, or stated differently, how many times more likely one hypothesis is than the other. We used Jeffrey’s interpretation of BF, as clarified by _{1} and < 1/100 for H_{0}).

Preliminary analyses involved the evaluation of order effects of the two no-choice conditions. Repeated measures ANOVAs revealed that there was no effect of order on the strategy distribution (_{10} values were below 1, indicating that the H_{0} of no order effect was more likely (strategy distribution: BF_{10} = 0.204; strategy efficiency: accuracy: BF_{10} = 0.273, speed: BF_{10} = 0.606; strategy selection: BF_{10} = 0.543). This all indicated that the order of the no-choice conditions did not affect task performance. We grouped the data of both orders of no-choice conditions for further analyses. We present the results along the four parameters of the model of strategy change and investigate for each parameter how it was moderated by mathematics achievement level.

In the choice condition, subtraction by addition was used at least once by most children. Only 9 children (14%) never used subtraction by addition; 17 children (27%) used it all the time; and 37 children (59%) used both direct subtraction and subtraction by addition. We observed no differences in strategy repertoire between the four achievement groups (Fisher’s Exact ^{2} = .065. The analysis of the BF, however, indicated that this should be interpreted with caution as the evidence for this difference was only anecdotal (BF_{10} = 1.030).

We analysed differences in the accuracy and speed (on both correctly and incorrectly answered items) of strategy execution between the no-choice conditions by means of repeated measurements ANOVA, with condition (no-choice direct subtraction vs. no-choice subtraction by addition) and problem type (SD vs. LD) as within-subject factors and mathematical achievement group (low, below-average, above-average, high) as a between subjects factor.

Achievement Group | No-Choice Direct Subtraction |
No-Choice Subtraction by Addition |
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SD |
LD |
SD |
LD |
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Low | 55.62 | .10 | 75.83 | .10 | 95.00 | .06 | 71.87 | .10 |

Below-average | 82.33 | .07 | 75.67 | .07 | 96.00 | .05 | 80.78 | .07 |

Above-average | 87.22 | .07 | 70.83 | .07 | 93.33 | .04 | 73.33 | .07 |

High | 84.55 | .06 | 71.59 | .06 | 89.09 | .04 | 75.45 | .06 |

All | 77.43 | .04 | 73.48 | .04 | 93.36 | .02 | 75.36 | .04 |

Focusing on ^{2} = .136, BF_{10} = 4.040. Subtraction problems were more accurately solved when subtraction by addition had to be used (^{2} = .212, BF_{10} = 8414.022, with SD subtraction problems (^{2} = .152, but the analysis of the BF indicated that this should be interpreted with caution as BF_{10} = 0.273.

There was no main effect of mathematical achievement group, ^{2} = .019. The BF_{10} was equal to 0.110, which indicates substantial evidence for H_{0} of no difference between the achievement groups. There was a condition × problem type × mathematical achievement group interaction, ^{2} = .160, but the analysis of the BF indicated that this interaction should be interpreted with great caution as BF_{10} = 0.067.

Turning to ^{2} = .264, BF_{10} = 365.314. This indicates decisive evidence that subtraction problems were solved faster in the no-choice subtraction by addition condition (^{2} = .670, BF_{10} = 8.750 x 10^{13}. This provides decisive evidence that SD subtraction problems (^{2} = .506, BF_{10} = 73239.045. As shown in ^{2} = .723, BF_{10} = 7996.836) and the no-choice direct subtraction condition (^{2} = .213, BF_{10} = 168.493). On the other hand, there was no difference in the LD subtraction problems in the two no-choice conditions (^{2} = .000, BF_{10} = 0.245), whereas SD subtraction problems were solved faster in the no-choice subtraction by addition condition compared to the no-choice direct subtraction condition (^{2} = .487, BF_{10} = 2.091 x 10^{8}).

Achievement Group | No-Choice Direct Subtraction |
No-Choice Subtraction by Addition |
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SD |
LD |
SD |
LD |
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Low | 21.11 | 3.21 | 23.43 | 3.21 | 8.03 | 2.09 | 25.39 | 3.29 |

Below-average | 17.12 | 2.35 | 24.41 | 2.35 | 7.58 | 1.53 | 21.13 | 2.40 |

Above-average | 15.25 | 2.14 | 18.07 | 2.14 | 6.95 | 1.40 | 20.43 | 2.19 |

High | 13.47 | 1.94 | 16.41 | 1.94 | 7.73 | 1.26 | 15.99 | 1.98 |

All | 16.74 | 1.23 | 20.58 | 1.23 | 7.57 | .80 | 20.73 | 1.26 |

We found no differences in speed in the no-choice conditions between the four mathematical achievement groups, ^{2} = .097. An analysis of the BF indicated that this evidence was only anecdotal (BF_{10} = 0.554). There was an effect of the condition × problem type × mathematical achievement group interaction, ^{2} = .135. The analysis of the BF, however, indicated that this finding should be interpreted with great caution (BF_{10} = 0.221).

In line with

There was an effect of problem type, ^{2} = .412, BF_{10} = 2.907 x 10^{7}. Children more frequently selected subtraction by addition on SD subtraction problems (^{2} = .019, BF_{10} = 0.104 nor an interaction between achievement group and problem type, ^{2} = .017, BF_{10} = 0.066. The BFs indicated that there was substantial evidence for the hypothesis of no difference between the achievement groups (H_{0}). This suggests that the achievement groups did not differ in their flexible application of subtraction by addition on the different problem types.

Second, we analysed whether children took into account their individual strategy efficiency competencies during the strategy selection process by correlating subjects’ frequency of subtraction by addition in the choice condition with the accuracy and speed differences between the no-choice direct subtraction and no-choice subtraction by addition conditions. The correlation between children’s subtraction by addition strategy frequency in the choice condition and the accuracy differences between the no-choice conditions was not significant, _{10} = 0.157. These results indicate that children did not take into account their individual strategy accuracy characteristics during strategy selection. By contrast, they flexibly fitted their strategy choices to their individual strategy speed competencies, _{10} = 6622.335: The larger the speed difference between the no-choice subtraction by addition and the no-choice direct subtraction condition in favor of subtraction by addition, the more frequently children reported subtraction by addition in the choice condition. An ANCOVA on the accuracy and speed differences between the no-choice conditions, with the frequency of subtraction by addition in the choice condition and mathematical achievement group as predictor variables, indicated no differences between the four achievement groups. In line with the correlation analysis, we observed no effect of the frequency of subtraction by addition on the accuracy differences, _{10} = 0.257. We neither found an effect of mathematical achievement group, _{10} = 0.210. Although we observed an effect of the frequency of subtraction by addition on the speed differences, _{10} = 4943.223, there was no effect of mathematical achievement group, _{10} = 0.227. In all, both frequentist and Bayesian analyses indicated that there were no effects of mathematical achievement level on the children’s strategy selection.

Previous research with

In contrast to previous studies on children’s use of the subtraction by addition strategy, which followed a simple choice design (

In contrast with the strong instructional focus on direct subtraction and with the findings from previous studies with children (^{nd}- to 4^{th}-graders, we included 6^{th}-graders. They all had practiced multi-digit subtraction problem solving for more than three years, as evidenced by our textbook analyses and teacher interviews, and (thus) presumably mastered the explicitly taught direct subtraction strategy sufficiently well to be able to invent other strategies, including the subtraction by addition strategy (

Our study demonstrated not only that the subtraction by addition strategy was part of children’s strategy repertoire, but also that they applied the untaught subtraction by addition strategy frequently and efficiently (= RQ2 and RQ3). Children solved multi-digit subtraction problems surprisingly efficiently via subtraction by addition. Despite the highly frequent practice of direct subtraction in children’s classrooms, they were slower in executing this strategy compared to the untaught subtraction by addition strategy in the no-choice conditions. Moreover, they did not only answer small-difference subtraction problems but also large-difference subtraction problems faster via subtraction by addition than via direct subtraction. This finding challenges current mathematics instruction, which capitalizes on the routine mastery of direct subtraction and pleas for more instructional attention to subtraction by addition as alternative - and maybe even more obvious - strategy in this domain (cf.

In addition to these further descriptive studies on the superior efficiency of subtraction by addition compared to direct subtraction, it is also of utmost interest to systematically investigate the reasons for these observed differences in efficiency. As a first explanation, we refer to the intrinsic difficulty of the underlying addition and subtraction processes. It can be argued that the process of adding a number to a given number is inherently easier than the process of taking away a similar number from a given number. Although it is difficult to provide direct evidence for this claim, there are several elements from various sources in support of it (see also

A second explanation for the efficiency of subtraction by addition that requires further research attention relates to the number of carry and borrow operations that had to be performed when using subtraction by addition or direct subtraction in the present study. Cognitive psychological studies revealed that the number of trades influences the accuracy and speed of responding in complex addition and subtraction (

All children also flexibly fitted their strategy choices to both numerical characteristics of the subtraction problems and to their individual strategy speed competencies (= RQ4). This suggests that they were also able to demonstrate some flexibility in their use of the subtraction by addition strategy, despite the absence of any instructional attention to it. This finding complements previous results on strategy flexibility in single-digit addition and subtraction, evidencing more flexible strategy choices with increasing experience in the domain (

The present study revealed that both mathematically higher achieving children and their lower achieving peers frequently, efficiently and flexibly applied the untaught subtraction by addition strategy to solve multi-digit subtraction problems (RQ5). Just like their above-average and high achieving peers, low and below-average achievers solved multi-digit subtraction problems frequently and efficiently via subtraction by addition. Moreover, even the lower achieving children flexibly fitted their strategy choices to both numerical characteristics and individual strategy competencies. This is in sharp contrast with the assumption that the acquisition of strategy flexibility is most difficult for the lower achieving children (

The results of the present study are in line with previous findings on low achievers’ subtraction by addition strategy use (

Combining theoretical and methodological tools from cognitive psychology and insights from mathematics education research, the findings of the present study raise questions about the adequacy of current mental multi-digit subtraction instruction with its strong focus on direct subtraction. Despite the strong instructional effort for mastering direct subtraction and the intensive practice of this strategy in classrooms, children answered multi-digit subtraction problems more efficiently with the untaught subtraction by addition strategy than with the explicitly taught and intensively practiced direct subtraction strategy. Moreover, and in line with the adult studies of

In addition to ascertaining studies, we need intervention studies wherein new instructional approaches to the introduction and development of the subtraction by addition strategy are designed, implemented and tested. In this respect, one should be aware that the categorisation of multi-digit subtraction strategies on the basis of their underlying operation (differentiating between direct subtraction and subtraction by addition) might be less intuitive for children - especially for the younger and lower achieving ones - than their categorisation in terms of the concrete solution steps involved in the solution process (differentiating between decomposition, sequential and varying strategies). However, the mathematics education literature already contains several appealing proposals for introducing the former classification to children, such as the inclusion of clear external models to represent the different strategies including the empty number line. These proposals and models might serve as promising didactic tools to foster children’s understanding of the differences between and value of the different strategies (e.g.,

This research was partially supported by Grant 3H160242 “Early development and stimulation of core mathematical competencies” from the Research Fund KU Leuven, Belgium.

It should be noted here that the operation perspective distinguishes also a third type of strategy, namely indirect subtraction. In case of indirect subtraction, one finds the solution by determining how much has to be decreased or subtracted from the minuend to get the subtrahend (e.g., solving 628 – 313 = __ via 628 - ? = 313) (

The authors have declared that no competing interests exist.

The authors have no support to report.