Edited by: Tobias Richter, Universität Würzburg, Germany
Reviewed by: Matthew Reysen, University of Mississippi, United States; Esther Ziegler, ETH Zürich, Switzerland
This article was submitted to Educational Psychology, a section of the journal Frontiers in Psychology
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Empirical findings show that students are often not capable of using numberbased strategies and the standard written algorithm flexibly and adaptively to solve multidigit subtraction problems. Previous studies have pointed out that students predominantly use the standard written algorithm after its introduction, regardless of task characteristics. Interleaved practice seems to be a promising approach to foster the flexible and adaptive use of strategies. In comparison to the usual blocked approach, in which strategies are introduced and practiced successively, they are presented intermixed in interleaved learning. Thus, the students have to choose an appropriate strategy on the basis of every task itself, and this leads to drawing comparisons between the different strategies. Previous research has shown inconsistent results regarding the effectivity of interleaving mathematical tasks. However, according to the attentional bias framework, interleaved practice seems to be a promising approach for teaching subtraction strategies to enhance the students’ flexibility and adaptivity. In this study, 236 German third graders were randomly assigned to either an interleaved or blocked condition. In the interleaved condition the comparison processes were supported by prompting the students to compare the strategies (betweencomparison), while the students of the blocked approach were encouraged to reflect the adaptivity of a specific strategy for specific subtraction tasks (withincomparison). Both groups were taught to use different numberbased strategies (i.e., shortcut strategies and decomposition strategies) and the standard written algorithm for solving threedigit subtraction problems spanning a teaching unit of 14 lessons. The results show that the students of the interleaved condition used the shortcut strategies more frequently than those of the blocked condition, while the students of the interleaved condition applied the decomposition strategies as well as the standard written algorithm less frequently. Furthermore, the students of the interleaved condition had a higher competence in the adaptive use of the shortcut strategies and the standard written algorithm. A subsequent cluster analysis revealed four groups differing in their degree of adaptivity. Being part of clusters with a comparatively high level of adaptivity was positively related to the prior arithmetical achievement and, even more so, to the interleaved teaching approach.
There is a wide consensus among mathematics researchers and educators that the abilities to use various strategies for solving a problem (flexibility) as well as to use efficient strategies (adaptivity) are important mathematical competencies students should gain (
Reviewing the literature on the strategy use of elementary school students, a wide range of usage for the terms
To assess whether a specific strategy is adaptive for solving a specific subtraction task, a more precise definition is required. To decide, whether a strategy is adaptive, i.e., appropriate/efficient, for a certain subtraction task, we take a normative perspective following several other studies (e.g.,
There are several different classifications of subtraction strategies in the literature (for an overview, see
Overview of the different subtraction strategies.
1 Numberbased strategies 


1.1 Decomposition strategies  
1.1.1 Stepwise strategy  1.1.2 Split strategy 
654  328 = 326 
756  423 = 333 
654  300 = 354  700  400 = 300 
354  20 = 334  50  20 = 30 
334  8 = 326  6  3 = 3 
547  399 = 148 
452  449 = 3 
547  400 = 147  449+ 3 = 452 
147 + 1 = 148  
725  
453  
1  
  
272 
Before the introduction of the standard written algorithm, students use the decomposition strategies most frequently to solve subtraction tasks, whereby the stepwise strategy is used most often (
Typical mistake when using the split strategy.
Unlike the stepwise and the split strategy, the shortcut strategies, compensation strategy and indirect addition, are used relatively rarely in mathematics classrooms (
Besides the mentioned numberbased strategies, children learn to solve subtraction tasks with digitbased strategies, i.e., the standard written algorithm (see
Concerning this matter, previous research has detected several reasons why students do not use calculation strategies adaptively. On the one hand, a limited strategy repertoire can have a negative impact on choosing an efficient strategy (
Although the mentioned studies detected deficiencies in the flexible and adaptive use of subtraction strategies by elementary school students, they predominantly conceptualized flexibility and adaptivity by a variablecentered view as numerical variables. The only known study following a personcentered view was carried out by
Summarizing the studies mentioned in the section above, children barely use subtraction strategies flexibly and adaptively to solve multidigit subtraction problems. This may be explained by the usual
Empirical findings regarding the effectivity of interleaved practice in mathematics are inconsistent, and this is emphasized by
Laboratory studies investigating the effectivity of interleaving mathematical tasks are predominant, whereas only few studies have been conducted in real educational settings. Two of the few studies investigated in classroom settings were carried out by
The inconsistent results regarding the effectivity of interleaved practice in mathematics lead to the assumption that the concrete implementation in the educational setting plays a major role. As the attentional bias framework (
The mentioned studies on interleaved practice indicate that it can have a positive impact on students’ learning outcomes in real educational settings, but there is still insufficient research on the subject: A first weakness of the available studies is that they were mostly conducted in laboratory and/or with university or middle school students leading to a limited transferability of the effects on elementary school mathematics. Secondly, previous studies have predominantly used the procedural knowledge as the dependent variable, whereas the effect of interleaving on the flexible and adaptive strategy choice as a major goal of mathematics education was unconsidered. Concerning this, it can be assumed that the effectivity of interleaving mathematical tasks, with studies showing inconsistent findings, is higher when the students’ discrimination processes are supported by explicit prompts to compare (
The ability to use different subtraction strategies flexibly and adaptively is a major goal of teaching arithmetic in elementary school. Even though there is a stronger consideration of numberbased strategies in classrooms nowadays, students barely use them efficiently to solve subtraction tasks, but prefer to rely on the standard written algorithm after its introduction. Interleaved practice combined with explicit prompts to compare for supporting the discrimination processes (
(1) Does interleaved practice have a positive impact on the flexible use of subtraction strategies?
(2) Does interleaved practice have a positive impact on the adaptive use of each subtraction strategy?
We supported the discrimination processes evoked by interleaved practice through explicit prompts to compare in order to direct the attention of the students to the differences between the strategies. The flexible and adaptive application of subtraction strategies is expected to benefit from the intervention. A substantial amendment of this research consists in examining the adaptive use for each strategy separately facilitating a differentiated insight into the effectivity of interleaved practice.
(3) Are there clusters of students differing in the adaptive use of the newly acquired subtraction strategies?
Another goal of this study is to identify students with different adaptivity profiles. In addition to the first two research questions following a variablecentered approach, the third research question is taking a personcentered view. By this personcentered view which takes variability between and within the students into account, adaptivity profiles can be generated. Thus, it can be shown whether student subgroups can be identified that differ in the adaptive application of the different subtraction strategies. An exploratory approach will be used to pursue this question since no hypotheses about possible adaptivity profiles can be formulated in advance.
(4) Do the teaching approach and the prior arithmetical achievement predict the adaptivity profile of students?
On the basis of the cluster analysis, the fourth research question explores if being taught subtraction strategies interleaved or blocked is related to the cluster membership. It is expected that the probability of being grouped in a cluster with a high level of strategyspecific adaptivity is higher when having been taught subtraction strategies interleaved. Moreover, previous research has shown that the knowledge about numbers, number relations, and the arithmetic operations are central prerequisites for using subtraction strategies efficiently (
In a 2 (group: interleaved vs. blocked) × 4 (time: before intervention, 1 day later, 1 week later, 5 weeks later) experimental study, German elementary school students were taught in either an interleaved or blocked condition in solving threedigit subtraction problems with different strategies. A total sample of 236^{1} German third graders from 12 different classes attending four Hessian elementary schools participated in this study. The classes were split, and the students were randomly assigned to one of the conditions. In this way, one half of the class learned the subtraction strategies blocked and the other half interleaved. The students themselves did not know they were taught differently. A precondition to be part of the study was that the subtraction up to 1,000 had not previously been introduced in class. The addition up to 1,000 had to be introduced. During the intervention (until T2), no regular mathematics lessons were held.
The prior arithmetical achievement was measured at T0 in November 2016, i.e., before the intervention took place. The variables flexibility and strategyspecific adaptivity were measured immediately before the intervention (T1), immediately after the intervention (T2), and in two followup tests – 1 week (T3) and 5 weeks (T4) after the treatment (
Design of the study.
The students involved in the study were aged from 8 to 10 years old (
Prerequisites of the students separately for the interleaved and the blocked condition.
Age 
9.04 (0.40)  9.10 (0.43)  
Female (%)  45.38%  45.30%  
Arithmetical achievement 
12.04 (5.65)  12.17 (6.02)  
Quantity of strategy use 
Written algorithm  0.68 (2.29)  Written algorithm  0.70 (2.47) 
Split strategy  0.66 (2.13)  Split strategy  1.05 (2.60)  
Stepwise strategy  6.40 (5.81)  Stepwise strategy  5.90 (4.86)  
Compensation strategy  0.68 (1.98)  Compensation strategy  0.61 (1.81)  
Indirect addition  0.12 (0.96)  Indirect addition  0.16 (1.00)  
Strategyspecific adaptivity 
Written algorithm  3.08% (12.22%)  Written algorithm  4.08% (14.71%) 
Stepwise strategy  30.77% (24.73%)  Stepwise strategy  30.66% (25.69%)  
Compensation strategy  6.41% (20.22%)  Compensation strategy  7.15% (21.39%)  
Indirect addition  2.65% (16.15%)  Indirect addition  3.59% (17.42%) 
The treatment included 14 lessons (à 45 min) and was conducted by four trained staff members who studied mathematics for elementary school. Each staff member taught the blocked as well as the interleaved condition in the same quantity. For an increased comparability of the lessons, a precise script was developed for each condition. This script contained detailed information on the time course of the lessons, the tasks, the expected behavior of the students, and possible teacher reactions, teacher questions, and possible action alternatives.
The main teaching goal of both conditions was to teach the students how to solve subtraction tasks adaptively. Therefore, the numberbased subtraction strategies, including the decomposition strategies (split strategy and stepwise strategy) and the shortcut strategies (compensation strategy and indirect addition), and the standard written algorithm as a digitbased strategy were introduced and practiced in class. In addition to the introduction and use of the technical terms of the subtraction strategies, pictorial representations of animals^{2} were assigned to the different strategies as previous research has shown that labeling categories can support comparison mechanisms (
In both conditions, the time spent on the strategies in classroom discussion and individual work was nearly equal. However, the time percentages differed between the strategies in both conditions: The time spent on the split strategy (about 55 min) was comparatively low in both conditions, since this strategy is errorprone (see the section “Subtraction Strategies”) and therefore, was only part of the teaching unit used to sensitize the students for potential difficulties. The time spent on the stepwise strategy, the compensation strategy, and the indirect addition was about 100 min each, and on the standard written algorithm with about 190 min even higher. While the time percentages for the strategies were equal in the two conditions, they differed in the order of the introduction and practice of the strategies. The first two lessons were equal for both conditions to activate relevant previous knowledge (knowledge of numbers: e.g., number relations on a number line, greater/lesscomparisons) and to initiate a first approximation of using subtraction strategies in a clever way in a math conference, i.e., groups of students discussed which strategy is the most appropriate for solving a specific subtraction task. In the following lessons, the two conditions differed in the order of the introduction and practice of the strategies and the teaching activities (
Overview of the activities of each lesson.
Activities  

Lesson  Blocked  Interleaved 
1 & 2  • Activation of relevant previous knowledge about numbers (e.g., number relations on a number line, greater/lesscomparisons) 

3  • Introduction and practice of the split strategy 
• Introduction and practice of the split strategy 
4  • Introduction and practice of the stepwise strategy 
• Successive repetition and practice of the split strategy and the stepwise strategy 
5  • Repetition and practice of the stepwise strategy  • Successive repetition and practice of the split strategy and the stepwise strategy 
6  • Introduction and practice of the compensation strategy 
• Successive repetition of the compensation strategy, stepwise strategy, and the split strategy 
7  • Repetition and practice of the compensation strategy 
• Introduction and practice of the indirect addition 
8  • Introduction and practice of the indirect addition 
• Repetition of the indirect addition 
9  • Repetition and practice of the indirect addition 
• Introduction of the standard written algorithm 
10  • Introduction of the standard written algorithm  • Repetition and practice of the standard written algorithm 
11  • Repetition and practice of the standard written algorithm  • Successive repetition of the standard written algorithm and the compensation strategy 
12  • Repetition and practice of the standard written algorithm 
• Successive repetition of the standard written algorithm and the indirect addition 
13  • Repetition and practice of the standard written algorithm 
• Successive repetition and practice of the standard written algorithm and the split strategy 
14  • Successive repetition of the split strategy, the compensation strategy, the indirect addition, and the standard written algorithm 
• Successive repetition of the split strategy, the compensation strategy, the indirect addition, and the standard written algorithm 
Due to the fundamental importance of the discrimination of contents for the interleaved practice (
Examples for withincomparisons in the blocked approach and betweencomparisons in the interleaved approach in classroom discussion.
Blocked  Interleaved  

Material  Subtraction tasks (413 – 409, 287 – 152, 579 – 348) solved solely with the indirect addition  Subtraction tasks (413 – 409, 287 – 152, 579 – 348) solved with the indirect addition and the stepwise strategy 
Instruction  “You have solved many tasks using the frogstrategy. Now we want to find out, for which tasks it is clever to use the frogstrategy. Let’s have a look at the following tasks. When is it clever to use the frogstrategy?”  “You have solved many tasks using the frogstrategy. Now we want to compare the frogstrategy and the mousestrategy. Let’s have a look at the first task. How did the frog solve the task? How did the mouse solve the task? Which strategy is more clever?” 
Expected student behavior  The students recognize that the indirect addition is adaptive for tasks with a small difference between the minuend and the subtrahend. The students argue for or against the application of the indirect addition based on the discussed criteria (number of solution steps, error rate, cognitive effort).  The students recognize that the indirect addition is more adaptive than the stepwise strategy for tasks with a small difference between the minuend and the subtrahend. The students argue for or against the application of a specific strategy based on the discussed criteria (number of solution steps, error rate, cognitive effort). 
In each lesson, the students had to work on one to two worksheets that were developed for this teaching unit. The subtraction tasks of the work sheets were the same for both groups. Based on the worksheets, the students practiced either the application of the strategies procedurally or they were prompted to draw comparisons between (interleaved condition) or within (blocked condition) the strategies.
Examples for withincomparisons in the blocked approach (on the left) and betweencomparisons in the interleaved approach (on the right) in individual work.
Furthermore, posters of the subtraction strategies including the animal illustrations and worked examples with complete solution procedures were hung up during the relevant lessons since they can support the students in discovering the characteristics and underlying rules of each subtraction strategy (
The arithmetical achievement of the students regarding their knowledge about numbers, number relations, about the relation of addition and subtraction, and competencies in calculating were measured at T0 (
Sample tasks of the arithmetical achievement test. H, hundreds; T, tens; O, ones.
The test consisted of 25 tasks and the students could have achieved a maximum of 25 points. To ensure that all students understood every task, the survey headers explained each task with a standardized test instruction. Students were required to solve the test in 36 min. On average, the students reached 12.10 points (
The dependent variables flexibility and adaptivity were measured at T1, T2, T3, and T4 using a subtraction strategy test. The test contained 11 items on each point of measurement assessing how (i.e., with which subtraction strategy) the students solve subtraction problems^{4}. Six out of the 11 items were included in the test of each point of measurement, while the other five items varied to reduce potential memory effects. The varying items were developed parallel in respect of task characteristics and therefore, should represent the same competence (e.g., T1: 469 – 283, T2: 745 – 271, T3: 629 – 372, T4: 836 – 352; in all tasks, the tensdigit of the minuend is smaller than the tensdigit of the subtrahend). The prompt of the test was “Solve the tasks in a clever way. Write down how you solved the tasks.” The test took 28 min. The selected tasks evoked the mentioned numberbased strategies (except of the split strategy) as well as the standard written algorithm. For most of the items using the indirect addition (four items at each point of measurement, e.g., 663 – 656) or the compensation strategy (four items at each point of measurement, e.g., 534 – 399) was most adaptive. The stepwise strategy and the standard written algorithm were considered to be almost equally adaptive for the three remaining items (e.g., 532 – 476). One exception here was the item which had a zero in the minuend (720–269) because zeros in the minuend often lead to calculation errors when using the standard written algorithm (
To assess the students’ flexibility, their strategy use was coded by four trained coders independently guided by a standardized coding manual. This coding manual had been developed based on the coding manual of the TigeRstudy (
Besides coding the applied strategies, the adaptivity of all subtraction strategies was rated for each task in the tests. Two independent raters estimated the adaptivity dichotomously (0 = nonadaptive, 1 = adaptive). For the normative adaptivity rating, the following criteria were taken into consideration: number of solution steps, mental effort, and error rate. The interrater reliability was overall satisfactory (0.69 ≤ κ ≤ 1.00). If the raters did not agree, a consensus was negotiated.
In order to be able to assess the effectivity of interleaved practice on each subtraction strategy, the raw data of the adaptivity rating were restructured and the strategyspecific adaptivity was calculated. Since every strategy could not have been used adaptively in the same quantity, an index of the adaptive use of the different subtraction strategies at each point of measurement was generated by relativizing the sums of the actual adaptive use in consideration of (1) the potential adaptive and nonadaptive application at one point of measurement as well as (2) the actual, individual sums of the adaptive and nonadaptive use at one point of measurement.
This led to the following equation:
with:
The procedure for calculating the strategyspecific adaptivity index is shown in the following example: The standard written algorithm could have been applied nine times nonadaptively and twice adaptively in the test 1 day after the intervention. If one student solved five subtraction tasks nonadaptively using the standard written algorithm and once adaptively, the relative proportion of the strategyspecific adaptivity would have been
If students did not use a specific strategy at one point of measurement, even though it would have been adaptive, their strategyspecific adaptivity was set 0.00% for this specific strategy.
To address the first research question, whether interleaved practice has a positive impact on the flexible use of subtraction strategies, the frequency of use was summed up for every subtraction strategy at every point of measurement. The differences of the strategy distributions between the two conditions were determined by χ^{2}homogeneity tests for each point of measurement (T1, T2, T3, T4).
To address the second research question, whether interleaved practice has a positive effect on the adaptive use of the standard written algorithm, the stepwise strategy, the compensation strategy, and the indirect addition, 2 (group) × 4 (time) ANOVAs with repeated measures (T1, T2, T3, T4) were conducted for each strategy. When the assumption of sphericity was violated, the Greenhouse–Geisser correction was used. Pairwise comparisons between the points of measurement were calculated in cases of a significant time effect with Bonferroni adjustments for multiple comparisons to identify between which points of measurement the significant differences occurred. In cases of a significant group effect,
To address the third research question, a hierarchical cluster analysis (Ward’s method with squared Euclidean distances) was conducted to find out whether there are specific subgroups of students that differ in using the standard written algorithm, the stepwise strategy, the compensation strategy, and the indirect addition adaptively at the points of measurement. The split strategy was again not part of the analysis since it could not have been used adaptively in the strategy test.
The cluster analysis detected four clusters since there was a comparatively big change regarding the distance coefficients between the four (224.02) and the three cluster solution (242.42). The results of the quality check of the cluster analysis were satisfying. Conformance checks with a hierarchical cluster analysis with Ward’s method and cityblock distance (82.05%, κ = 0.74) as well as with Kmeans clustering as a confirmatory method (87.18%, κ = 0.82) showed a high validity of the allocation of the students to the clusters. Moreover, the clustering was examined with a discriminant analysis. The first discriminant function had a canonical correlation of 0.98 (eigenvalue = 20.15, explained variance = 84.24%, Wilk’s λ = 0.06,
Standardized canonical discriminant functions and average discriminant coefficients of the cluster solution.
Discriminant coefficient  

Variable  Function 1  Function 2  Function 3  Average 
Standard written algorithm T1  0.00  –0.06  0.04  –0.07 
Standard written algorithm T2  0.07  0.04  –0.01  0.03 
Standard written algorithm T3  0.11  0.14  –0.04  0.02 
Standard written algorithm T4  0.10  0.05  –0.17  –0.04 
Stepwise strategy T1  0.00  –0.06  –0.01  –0.02 
Stepwise strategy T2  0.05  –0.03  0.41  0.14 
Stepwise strategy T3  0.02  –0.04  0.34  0.11 
Stepwise strategy T4  0.04  –0.07  0.37  0.13 
Compensation strategy T1  0.03  0.07  –0.06  0.01 
Compensation strategy T2  0.33  0.08  0.12  0.18 
Compensation strategy T3  0.89  –0.15  –0.03  0.24 
Compensation strategy T4  0.28  –0.02  0.04  0.10 
Indirect addition T1  0.02  0.09  –0.05  0.02 
Indirect addition T2  0.25  0.62  0.50  0.46 
Indirect addition T3  0.21  0.30  0.13  0.21 
Indirect addition T4  0.23  0.55  –0.30  0.16 
To determine differences in the development of the strategyspecific adaptivity between the identified clusters,4 (group) × 4 (time) ANOVAs with repeated measures were conducted in consideration of all four points of measurement including
To address the fourth research question, to analyze in how far being part of a specific cluster depends on the prior arithmetical achievement and the teaching approach, a multinomial logistic regression was used, whereby the identified clusters were the dependent variable and the teaching condition as well as the prior arithmetical achievement the independent variables.
To address the first research question, the strategy distributions of the two conditions were compared to establish whether the students of the interleaved practice use the subtraction strategies more flexibly after the treatment than the students of the blocked approach.
Distribution of the strategies used for solving the subtraction tasks.
A χ^{2}homogeneity test revealed just a marginally significant difference between the interleaved and the blocked group at T1 with a small effect size, χ^{2}(5,
The two groups differed significantly at all points of measurement after the intervention, at T2, χ^{2}(5,
In summary, the students of the interleaved practice showed a higher percentage in the use of the compensation strategy and the indirect addition, whereas the students of the blocked condition used the standard written algorithm and the stepwise strategy more frequently.
The results of the strategy distributions show that the students of the interleaved approach used the two shortcut strategies more often and the standard written algorithm as well as the stepwise strategy less often than the students of the blocked condition. However, these results do not implicate how much more adaptively the strategies were used. The second research question investigates whether the two conditions differ in their strategyspecific adaptivity.
Means and standard deviations of the strategyspecific adaptivity at T1, T2, T3, and T4 and results of the
Interleaved 
Blocked 


Standard written algorithm T1  113  3.08%  12.22%  109  4.08%  14.71%  
Standard written algorithm T2  112  38.13%  34.19%  110  21.72%  25.31%  interleaved > blocked 
Standard written algorithm T3  116  60.69%  36.04%  113  41.94%  33.10%  interleaved > blocked 
Standard written algorithm T4  115  53.37%  36.62%  111  40.64%  29.03%  interleaved > blocked 
Stepwise strategy T1  113  30.77%  24.73%  109  30.66%  25.69%  
Stepwise strategy T2  112  27.35%  42.65%  110  27.33%  36.35%  
Stepwise strategy T3  116  24.06%  41.04%  113  19.86%  34.26%  
Stepwise strategy T4  115  16.92%  33.50%  111  15.11%  29.46%  
Compensation strategy T1  113  6.41%  20.22%  109  7.15%  21.39%  
Compensation strategy T2  112  64.96%  36.88%  110  20.48%  37.43%  interleaved > blocked 
Compensation strategy T3  116  64.20%  37.66%  113  30.11%  41.52%  interleaved > blocked 
Compensation strategy T4  115  55.93%  39.19%  111  20.97%  35.39%  interleaved > blocked 
Indirect addition T1  113  2.65%  16.15%  109  3.59%  17.42%  
Indirect addition T2  112  63.03%  47.25%  110  22.54%  40.46%  interleaved > blocked 
Indirect addition T3  116  63.15%  47.51%  113  25.20%  42.39%  interleaved > blocked 
Indirect addition T4  115  40.00%  49.20%  111  15.24%  36.02%  interleaved > blocked 
ANOVAs with repeated measures revealed that the students of the interleaved approach had an advantage regarding the adaptive use of the
Regarding the
The students of the interleaved condition were superior in the adaptive use of the
Furthermore, there were significant differences between the conditions regarding the strategyspecific adaptivity of the
Summarizing the results, the students of the interleaved practice showed a higher strategyspecific adaptivity at T2, T3, and T4 regarding the standard written algorithm, the compensation strategy, and the indirect addition, while both conditions had the same low level in the strategyspecific adaptivity of the stepwise strategy.
The goal of the third research question was to detect different adaptivity profiles capturing variability between and within the students to ascertain whether clusters of students can be determined that differed in their adaptive use of the standard written algorithm, the stepwise strategy, the compensation strategy, and the indirect addition. The split strategy was again not part of the analysis since it could not have been used adaptively (see the section “Flexibility and StrategySpecific Adaptivity”). A hierarchical cluster analysis revealed four subgroups of students varying in their degree of strategyspecific adaptivity. As
Result of the cluster analysis.
In
Means and standard deviations of the strategyspecific adaptivity at T1, T2, T3, and T4 for the four clusters and results of the
Cluster 1 ( 
Cluster 2 ( 
Cluster 3 ( 
Cluster 4 ( 


Variable  
Standard written algorithm T1  1.39%  8.33%  4.96%  15.54%  5.98%  16.91%  3.34%  13.59%  
Standard written algorithm T2  43.72%  32.62%  39.22%  38.33%  33.58%  29.89%  19.69%  22.28%  1, 2 > 4 
Standard written algorithm T3  79.86%  24.88%  65.27%  36.77%  50.26%  37.91%  37.24%  27.85%  1 > 3, 4; 2 > 4 
Standard written algorithm T4  70.13%  27.40%  48.91%  37.86%  54.19%  32.49%  32.01%  26.53%  1, 2, 3 > 4 
Stepwise strategy T1  28.32%  23.27%  30.31%  23.76%  36.18%  28.69%  29.44%  25.82%  
Stepwise strategy T2  13.13%  33.42%  61.03%  47.49%  23.13%  38.06%  18.68%  29.63%  2 > 1, 3, 4 
Stepwise strategy T3  9.91%  28.71%  48.78%  50.61%  20.05%  37.70%  12.99%  26.09%  2 > 1, 3, 4 
Stepwise strategy T4  0.00%  0.00%  37.71%  46.49%  14.74%  30.40%  12.07%  23.44%  2 > 1, 3, 4 
Compensation strategy T1  15.43%  31.08%  8.05%  21.50%  5.83%  14.92%  4.13%  18.74%  
Compensation strategy T2  81.78%  26.68%  82.28%  21.67%  58.69%  35.44%  4.15%  18.82%  1, 2 > 3; 1, 2, 3 > 4 
Compensation strategy T3  89.13%  11.70%  85.38%  17.85%  80.08%  8.30%  0.47%  4.27%  1, 2 > 3; 1, 2, 3 > 4 
Compensation strategy T4  70.36%  35.40%  70.18%  29.27%  60.15%  34.04%  2.29%  11.97%  2 > 3; 1, 2, 3 > 4 
Indirect addition T1  9.60%  28.67%  3.55%  17.00%  0.00%  0.00%  2.41%  15.43%  
Indirect addition T2  92.32%  24.54%  95.72%  17.74%  1.43%  8.45%  14.32%  33.94%  1, 2 > 3, 4 
Indirect addition T3  95.52%  17.68%  84.15%  36.12%  31.43%  47.10%  11.56%  30.93%  1, 2 > 3, 4 
Indirect addition T4  100.00%  0.00%  41.46%  49.88%  8.35%  27.68%  1.20%  10.98%  1 > 2; 1, 2 > 3, 4 
ANOVAs with repeated measures including
Results of the
Cluster 1 
Cluster 2 
Cluster 3 
Cluster 4 


comparisons  
Standard written algorithm  
T1 – T2  1.13***  0.76***  0.82***  0.71***  
T1 – T3  2.75***  1.45***  1.22***  1.23***  
T1 – T4  2.39***  1.13***  1.47***  1.08***  
T2 – T3  0.98***  0.52***  0.47*  0.71**  
T2 – T4  0.92***  0.62**  0.42*  
T3 – T4  –0.35*  
Stepwise strategy  
T1 – T2  0.65***  
T1 – T3  0.34*  –0.48**  
T1 – T4  –1.22***  –0.54**  –0.56***  
T2 – T3  
T2 – T4  –0.42***  
T3 – T4  
Compensation strategy  
T1 – T2  1.64***  2.66***  1.57***  
T1 – T3  2.25***  2.85***  4.25***  
T1 – T4  1.20***  1.83***  1.61***  
T2 – T3  0.62***  
T2 – T4  
T3 – T4  –0.53***  –0.66***  –0.55***  
Indirect addition  
T1 – T2  1.79***  3.85***  0.31**  
T1 – T3  2.67***  2.12***  0.67***  
T1 – T4  3.15***  0.70***  
T2 – T3  0.59***  
T2 – T4  –0.96***  –0.43***  
T3 – T4  –0.76***  –0.46*** 
Concerning the
For the
Concerning the
Summarizing the results, four clusters were detected differing in their strategyspecific adaptivity of the subtraction strategies. Cluster 2 grouped those students together with a comparatively high adaptivity in the use of all subtraction strategies. In comparison, students in cluster 1 showed a high level of adaptive strategy use in all strategies except for the stepwise strategy and cluster 3 is characterized by a strategyspecific adaptivity which is limited to the written algorithm and the compensation strategy. The advantage of the strategyspecific adaptivity of cluster 1 (except the stepwise strategy) and cluster 2 could be shown for all points of measurement after the treatment. Finally, the students of cluster 4 had a comparatively low strategyspecific adaptivity of all strategies at all points of measurement.
Based on the four clusters, the fourth research question explored whether belonging to a specific cluster depends on the teaching approach and the prior arithmetical achievement. A descriptive view on the distribution of the students of the two conditions to the clusters showed that the students of the interleaved approach were the predominant part of cluster 1 (interleaved:
A subsequent multinomial logistic regression with cluster 4 as reference category supported the descriptive findings. The model fit, χ^{2}(6) = 90.79,
Multinomial logistic regression predicting the affiliation to a specific cluster (reference category: cluster 4).
Dependent variable  Independent variable  Wald  

Cluster 1  Treatment (reference category: blocked)  2.89  0.57  25.69  17.75  <0.001 
Arithmetical achievement (T0) (zscore)  0.25  0.05  24.73  4.21  <0.001  
Cluster 2  Treatment (reference category: blocked)  3.13  0.57  30.33  22.89  <0.001 
Arithmetical achievement (T0) (zscore)  0.26  0.05  28.47  4.61  <0.001  
Cluster 3  Treatment (reference category: blocked)  1.70  0.48  12.33  5.46  <0.001 
Arithmetical achievement (T0) (zscore)  0.17  0.04  14.58  2.67  <0.001 
The results reveal that the students of the interleaved practice had a 17.75 times higher chance of belonging to cluster 1 with reference to cluster 4. The likelihood of being in cluster 1 increased by 4.21 times when having an arithmetical achievement of one standard deviation above the total mean. As a result, the independent variable treatment makes a much greater contribution for predicting the affiliation to cluster 1 than the prior arithmetical achievement at T0. Regarding cluster 2 with reference to cluster 4, the
Summarizing the results, the cluster membership was strongly related to the teaching approach: Being taught interleaved was a strong predictor for the affiliation to clusters with a higher strategyspecific adaptivity in all/some strategies with reference to a cluster with a comparatively nonadaptive use of all strategies. The prior arithmetical achievement had a much smaller influence than the teaching approach.
The results of this study suggest that an interleaved approach extended by prompts to compare (1) is practicable and can be well integrated into regular elementary school classrooms. Moreover, (2) it enhances the flexible and adaptive use of subtraction strategies among third graders compared to a blocked approach with prompts for withincomparisons. The analysis of the strategy distributions showed a lower level of flexibility in the blocked condition: The students of the blocked approach predominantly used the standard written algorithm after its introduction to solve subtraction tasks, whereas the compensation strategy and the indirect addition were used comparatively rarely. The dominance of the standard written algorithm even increased over time. As a result, our study replicates the findings of previous research regarding the dominance of the standard written algorithm after its introduction (
The students of the interleaved condition showed not only a higher level of flexibility but also a higher level of strategyspecific adaptivity of almost all subtraction strategies. The only strategy in which the students of the interleaved condition were not superior was the stepwise strategy. This could be explained by the characteristics of the stepwise strategy itself: While the use of the compensation strategy and the indirect addition is predestined for specific types of subtraction tasks that are comparatively easy to identify, there are no explicit task characteristics showing that the stepwise strategy is adaptive – instead it is more a procedure of exclusion in consideration of the other strategies (e.g., 354 – 227: There is not a small difference between the minuend and the subtrahend, the subtrahend is not close to a full hundred, and two digits of the subtrahend are bigger than those of the minuend; ergo the indirect addition, the compensation strategy, and the split strategy are not adaptive, while the stepwise strategy and the standard written algorithm are adaptive). The students of both conditions might have used the stepwise strategy only if they have ruled out the other strategies erroneously leading to a comparatively nonadaptive use. Since the students of the interleaved practice did not use the stepwise strategy very often, it may be the case that this strategy was only then applied if the students did not know which of the other strategies would have been adaptive and therefore, they did not use it efficiently. Regarding the adaptive use of the standard written algorithm, the students of the interleaved condition benefitted significantly at all points of measurement after the intervention. This result supports the assumption of the standard written algorithmresilience that can be caused by interleaving subtraction strategies. Moreover, the students of the interleaved condition showed a higher level of adaptive use of the compensation strategy and the indirect addition. For both subtraction strategies the effects were even more substantial than for the standard written algorithm. However, there was a huge decrease of the effect over time, especially for the indirect addition. Since there was a decrease of the adaptive use over time of not only the indirect addition but all subtraction strategies, it seems advisable to integrate additional booster sessions refreshing the students’ knowledge of the adaptive application of the strategies.
Starting from a personcentered view, a subsequent hierarchical cluster analysis revealed four different subgroups of students differing in their adaptive use of the stepwise strategy, the compensation strategy, the indirect addition, and the standard written algorithm. A multinomial logistic regression with cluster 4, i.e., the cluster with a low strategyspecific adaptivity regarding all strategies, as reference category revealed that being part of the others was positively related to (1) the treatment, with interleaving having a positive impact, and (2) the prior arithmetical achievement. For all clusters the teaching approach was the major predictor. Especially for cluster 1 grouping students together with a high level of adaptivity regarding all strategies except for the stepwise strategy and cluster 2, i.e., the cluster characterized by a high strategyspecific adaptivity in all subtraction strategies, the probability of the affiliation to these clusters was highly related to the teaching approach.
Summarizing the results, interleaving subtraction strategies with supporting discrimination processes by prompts to compare seems to foster the flexible strategy use and the ability to choose an appropriate strategy based on specific tasks and their characteristics sustainably. Therefore, this study supplements previous research on interleaved practice in mathematics, which did not thoroughly show positive effects (
As stated, interleaved practice may require a higher cognitive effort from the students. Hence, further research should investigate whether all students benefit equally from interleaving subtraction strategies. On the one hand, it is conceivable that the positive impact of interleaving subtraction strategies is affected by the arithmetical achievement since multiple comparisons can cause a cognitive overload for students with a low prior knowledge (
Furthermore, it has to be taken into consideration that we took a normative perspective when rating the adaptivity of strategy use which is partially criticized in the literature (
This study demonstrated that interleaved practice including explicit prompts to compare can foster the flexible and adaptive application of subtraction strategies as highsimilarity categories by third graders. However, further research should explore whether these positive findings are transferable to (1) other mathematical contents, (2) other school subjects, and (3) whether elementary school students also benefit from interleaving lowsimilarity categories as the study by
This study was carried out in accordance with the recommendations of the Declaration of Helsinki as well as the ethical guidelines of the German Psychologists Association (BDP) and the German Psychological Society (DGP). The protocol was approved by the Ethics Committee of the Faculty of Human Sciences (University of Kassel). All parents gave written informed consent in accordance with the Declaration of Helsinki.
FL supervised the project. KW, JA, and FL conceived and planned the experimental study. KW and LN were part of the teacherteam. KW, JA, and LN performed parts of the measurements. SV had the idea to investigate the effectivity of interleaved practice for each subtraction strategy. LN performed the calculations and drafted the following parts of the manuscript: Introduction, Materials and Methods (Design and Participants, Instruments: Calculation of the StrategySpecific Adaptivity, Analysis), Results, and Discussion. KW drafted the other parts of the section ‘Materials and Methods.’ KW, JA, SV, and FL peer reviewed the manuscript critically. All authors approved the article for publication.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Due to missing values (listwise deletion), the size of the sample is lower in some analyzes than stated in this section.
The pictorial representations of the animals highlighted the features of each strategy (split strategy as monkey, stepwise strategy as mouse, compensation strategy as squirrel, indirect addition as frog and standard written algorithm as owl). For instance, the indirect addition was labeled as the frog strategy since it just needs a small “jump” from the subtrahend to the minuend to solve suitable subtraction tasks.
The withincomparisons in the blocked condition and the betweencomparisons in the interleaved condition were carried out using several subtraction tasks in each case.
All subtraction tasks were threedigit except of two twodigit tasks in the pretest.