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Edited by: Michael S. Dempsey, Boston University, United States

Reviewed by: Peter Verkoeijen, Erasmus University Rotterdam, Netherlands; Veit Kubik, Stockholm University, Sweden

This article was submitted to Educational Psychology, a section of the journal Frontiers in Psychology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The purpose of the present study was to investigate the effect and use of distributed practice in the context of self-regulated mathematical learning in high school. With distributed practice, a fixed learning duration is spread over several sessions, whereas with massed practice, the same time is spent learning in one session. Distributed practice has been proven to be an effective tool for improving long-term retention of verbal material and simple procedural knowledge in mathematics, at least when the practice schedule is externally guided. In the present study, distributed practice was investigated in a context that required a higher degree of self-regulation. In total, 158 secondary school students were invited to participate. After motivational and cognitive characteristics of the students were assessed, the students were introduced to basic statistics, a topic of their regular curriculum. At the end of the introduction, the students could sign up for the study to further practice this content. Eighty-seven students did so and were randomly assigned either to the distributed or to the massed practice condition. In the distributed practice condition, they received three practice sets on three different days. In the massed practice condition, they received the same three sets, but all on one day. All exercises were worked in the context of self-regulated learning at home. Performance was tested 2 weeks after the last practice set. Only 44 students finished the study, which hampered the analysis of the effect of distributed practice. The characteristics of the students who completed the exercises were analyzed exploratory: The proportion of students who finished all exercises was significantly higher in the massed than in the distributed practice condition. Within the distributed practice condition, a significantly larger proportion of female students completed the exercises compared to male students. Additionally, among these female students, a larger proportion showed lower concentration difficulty. No such differential effects were revealed in the massed practice condition. Our results suggest that the use of distributed practice in the context of self-regulated learning might depend on learner characteristics. Accordingly, distributed practice might obtain more reliable effects in more externally guided learning contexts.

Teachers at school and university should generally be interested in learning techniques that promote long-term retention of the learned contents. The reason for this is that most topics taught in school or at university are rather complex and many advanced topics rely – sometimes more, sometimes less so – on prior knowledge. In mathematics, for example, in order to grasp stochastics, solid knowledge of fractional arithmetic is helpful. That is why in most cases it is important to use learning strategies that help to store knowledge in a way that facilitates long-term retention, rather than learning strategies that result in knowledge that can be retrieved only for a short period after it was taught and acquired.

One branch of learning strategies that promote long-term retention is based on so-called desirable difficulties (

Most of these learning strategies have been studied extensively in the laboratory and are known to result in robust and strong performance improvements as compared to the respective control conditions (

Distributed practice means that a given learning or practice duration is distributed across more than one learning session, whereas in massed learning, the same time is spent in one learning session only (

However, there are still contexts and conditions that have been considered less so far. These include the effect of distributed practice on mathematical learning in general and, in particular, on mathematical learning in school. An exception are the studies by

Another branch of studies investigated a combination of interleaved and distributed practice, termed “mixed review,” in school (

Only few studies have investigated the isolated effect of distributed practice on mathematics learning in school:

In sum, although only few studies have investigated the isolated effect of distributed practice on mathematics learning in school, most findings suggest a positive impact. However, in previous studies, sessions including distributed practice were highly structured by the experimenters or teachers and took place solely in the classroom. In real-world learning settings, especially older students at school and students at university have to do a lot of learning and practice outside the classroom in a more self-regulated manner. The aim of the present study was to investigate whether distributed practice can also be implemented using online exercises in order to improve mathematical performance in a real-world learning context, in which practice relies more heavily on self-regulation. In addition, we examined which learners actually followed the distributed practice schedule.

One question that has hardly been investigated in studies examining the effects of desirable difficulties in general, and of distributed practice in particular, is whether all learners profit from such learning strategies in a similar way or whether individual differences moderate the effectivity of distributed practice (

As noted previously, most of the aforementioned studies employed a highly teacher-guided practice procedure. In a more self-regulated learning scenario, the question of whether and how individual learner characteristics affect the use and effectivity of distributed practice actually becomes even more important: Individual motivational and cognitive traits could not only affect the effect of distributed practice on final test performance, but also determine if and how the students follow the respective practice schedule. Distributed practice requires learners to repeatedly engage with a topic or procedure, which may be difficult to retrieve given the temporal delay between learning sessions. In fact, this is why distributed practice is related to desirable difficulties: It is assumed that the lags between practice sessions make the learning process more difficult, which in turn should improve long-term retention. Learners with low mathematical self-efficacy, however, could suffer from this additional difficulty and decide to stop to engage mentally or in practice with the topic (^{1}.

The objectives of the present study were to investigate distributed practice in a real learning context including a relatively high degree of self-regulated learning. The main questions were whether distributed practice is used reliably by learners, and which learner characteristics promote (or hinder) its use. The sample consisted of high school students, and the material was relevant for their mathematics curriculum. We expected that distributed practice might not consistently be applied by the students in the context of their self-regulated learning (see also

This study was carried out in accordance with the recommendations of the ethics committee of the Faculty of Human Sciences of the University of Kassel and with written informed consent from all legal guardians of the subjects in accordance with the Declaration of Helsinki.

In total, 158 students of eight courses from Grades 10 and 11 (first year of senior classes), attending three schools, were requested to participate in the current study. These students were enrolled either in regular math courses or in intensive math courses, depending on their own choice. All schools were located around a medium-sized German city in neighborhoods with inhabitants of a medium socio-economic status. Participation was voluntary and could be terminated at any time. Only students who had written consent from their parents could participate. They were told that they would receive 10 Euro if they completed the study. Signing up for the study required providing an e-mail address, because the experimental part of the study took place online. Of the 87 students who signed up (40 female, 47 male; 58 of regular math courses, 29 of intensive math courses; _{age} = 16 years 5 months, age range: 15–17 years), 43 terminated their participation ahead of time, and only 44 students finished it completely (25 female, 19 male; _{age} = 16 years 6 months, age range: 15–17 years).

The independent variable was practice condition with two between-subjects levels: One group of students worked the exercises massed in one session and the other group worked the same exercises distributed across three sessions. Both conditions worked a total number of twelve practice exercises (three sets with four exercises each). An expanding interval schedule was used for the practice sessions of the students in the distributed practice condition (see Procedure;

Of the 87 students who initially signed up, 49 were assigned to the distributed practice condition (25 female, 24 male; _{age} = 16 years 5 months, age range: 15–17 years) and 38 were assigned to the massed practice condition (15 female, 23 male; _{age} = 16 years 6 months, age range: 15–17 years). A slightly larger proportion of students was assigned to distributed practice as we expected a larger dropout in this condition. To ensure that the overall math performance level was roughly equal in both practice condition groups before the manipulation, students who signed up for the study were ranked by their most recent mathematics grade and then, within each class and grade level, randomly assigned to one of the two practice conditions. In order to minimize potential effects of class, the ratio of massed and distributed practicing students was similar in each class. As mentioned above, in total only 44 students completed the study (i.e., 17 distributed practicing students with a median math grade of 2.0 and 27 massed practicing students with a median math grade of 2.3^{2}).

Additionally, a questionnaire assessing some motivational and cognitive characteristics of the students (see Table

Instruments used to assess potential moderators.

Motivational characteristics | Employed instrument | Reliability |
---|---|---|

Mathematical self-efficacy | Seven items of a German |
α = 0.88 |

Performance avoidance goals | Eight-item German |
α = 0.86 |

Work avoidance | Eight-item German |
α = 0.85 |

Concentration difficulty | Six items of the German |
α = 0.92 |

In the course of the experiment, the students were introduced to basic statistics by student assistants with teaching experience, who were supervised by the authors. More specifically, the students were taught the definition of variables and their manifestations, the law of large numbers, the sum rule and the interpretation and creation of diagrams. The topic of basic statistics is generally part of the following school year, thus, no class had covered the topic in their current school year prior to the study. The lessons and practice material were prepared with the support of didactics experts with teaching experience in order to make the learning environment as realistic as possible. The complete material (lesson scripts, practice and test sets, and the scoring scheme) is provided in German online ^{3}.

Each practice set for the students consisted of four exercises and involved calculating absolute from relative frequencies, using the sum rule for probability calculation, naming variables and values, and preparing calculations for a diagram. The practice sets contained conceptually similar but not identical exercises, that is, solutions could not be learned by heart. Each practice set could easily be finished in less than 30 min. An example of a practice set can be found in the

The students were asked to work through all of the practice and test exercises that followed the lecture at home; only the questionnaire and the lecture sessions at the beginning of the study were completed at school. Practicing at home resembles real-world learning settings in that students usually have to do their homework outside the classroom in a self-regulated manner. In order to avoid students being particularly prepared or relying on help for the test at home, the test sheet was announced as “further exercises.”

Prior to the experimental manipulation, the study started with a survey in school assessing students’ mathematical self-efficacy, self-rated difficulty to concentrate, performance avoidance goals and work avoidance. The questionnaire was programmed with LimeSurvey (

Between 5 and 7 days after the last lesson, the students received their first practice exercises, provided via a personalized link that was sent by e-mail. The students were not allowed to keep the lesson material, that is, the material was not available for the practice exercises. However, after a practice exercise was completed, the correct solution was displayed on the screen. The test set was similar to the practice sets, but no correct solutions were provided after test exercises. It was not possible to go back to previous pages at any time. The practice and test sheets were created and distributed with the research tool formr (

Each given answer (mostly numbers or single words) was either correct or wrong, that is, no partial points were granted. For each practice and test set, the maximum score was 15 points. Two raters scored the answers independently from each other according to a predefined scheme. Afterward, the scores of both raters were compared (Cohen’s Kappa = 0.92) and differences were discussed and resolved by these raters. To ensure the reliability of the final rating, a third rater rated the answers independently as well. The final ratings of the first two raters were nearly identical to the third (control) rater (Cohen’s Kappa = 0.96). Therefore, the final scores of the two first raters were analyzed.

Because of the severe dropout in the course of the study (of 158 eligible students, only 44 finished the study), analyses concerning the effect of practice condition on retention performance turned out to be rather inappropriate: First, the remaining groups were rather small and not of equal size, and second, there seemed to be a selection bias concerning the dropouts, because the rate of completion was much higher in the massed practice condition (71%) than in the distributed practice condition (35%). We nevertheless report the analysis concerning the effect of practice condition for the sake of completeness, keeping in mind these limitations and that the results should be interpreted with caution. We used a Bayesian linear regression model to analyze the test performance, among other reasons because of the particularly small resulting sample size. One advantage of Bayesian modeling is that it provides a range of possible values for each estimated parameter and assigns probabilities to them, which facilitates interpretation especially when the results are not conclusive in classical statistical modeling (

As the main aim of the study was to investigate whether distributed practice works in self-regulated learning, subsequently exploratory analyses were conducted to examine which students completed the exercises in the context of their self-regulated learning. These analyses address two other questions that are important when implementing distributed practice in school, besides its general effect: (a) Which students are in general willing to invest additional effort into their mathematics learning by signing up for such a study, and, more specifically, (b) which students actually complete the distributed practice condition? On that account, conditional inference tree models were calculated (

Conditional inference tree models seek to identify independent variables that can be used to split up the respective sample into groups that are maximally different with regard to the dependent variable (e.g., test performance score or participation in the study). This is accomplished by recursive binary partitioning: First, the model checks if the distribution of the dependent variable is unrelated to all independent variables. If this null hypothesis can be rejected, the model selects the independent variable that has the strongest relationship with the dependent variable (e.g., work avoidance). The sample then is split into two groups, based on the selected independent variable, in a way that minimizes the

First, in the Bayesian linear regression model mentioned above, test performance served as dependent variable and practice condition (distributed vs. massed practice) and performance in the first practice set (sum score) as independent variables. No priors were specified, that is, an improper flat distribution over the reals was used as prior distribution, which means that the results were highly data-driven and hardly influenced by the priors (

Because participation in the current study was voluntary, a first conditional inference tree model was performed with the

Proportion of students who enrolled in the study, separated by the level of their last math grade.

Enrollment |
||
---|---|---|

Last math grade | Yes | No |

Median or above | 69% ( |
31% ( |

Below median | 38% ( |
62% ( |

A second conditional inference tree model was performed with the

Proportion of students who finished the study, of those who initially enrolled, separated by the statistically relevant variables.

Study completed |
||
---|---|---|

Practice condition, gender, and self-rated concentration difficulty | Yes | No |

Massed practice | 71% ( |
29% ( |

Female students | 60% ( |
40% ( |

Concentration difficulty | 1.69 (0.7) | 3.01 (1.1) |

Male students | 8% ( |
92% ( |

Finally, the mean perceived practice difficulty regarding the practice sets was analyzed using Bayesian

One of the main purposes of the present study was to investigate the effect of distributed practice on the mathematical performance of high school students using curriculum-relevant material. However, due to a severe dropout rate over the course of the study as a consequence of the study relying on self-regulated-learning, the effect of practice condition on final test performance has a low validity. In the sample that could be analyzed, however, the effect of practice condition unexpectedly indicated a negative effect of distributed practice as compared to massed practice. The main focus, then, was on exploratory analyses that were performed to identify factors that contributed to the students’ participation in the study and completion of the study exercises. This is especially important with regard to self-regulated learning, as these factors can give insight into the question of whether students are willing to implement distributed practice in their own learning schedule and, more specifically, which students are willing to do so.

These exploratory analyses revealed some interesting results. First of all, the proportion of students who finished the study was significantly higher in the massed practice condition than in the distributed practice condition. Furthermore, within the distributed practice condition, additional differential effects were found: The proportion of students who finished their exercises was significantly higher among female students than among male students. In addition, within female students who practiced in a distributed manner, the proportion of students who finished the study was significantly higher for girls with low concentration difficulty than for girls with high concentration difficulty. None of these differential effects were found for the massed practice condition. That is, not only did the students complete their exercises more often in the massed practice condition, but for the distributed practicing students, personal characteristics had an additional influence on the completion of the exercises. Taken together, these results imply that distributed practice in self-regulated learning, contrary to massed practice, favors specific students in terms of their willingness to realize this strategy, while others are at a disadvantage. Finally, the perceived difficulty of the exercises was compared between the groups, but there was no difference regarding the difficulty judgments between the distributed and massed practicing students.

Despite the exploratory character of the present results, the differential effects on exercise completion – which were found only within the distributed practice condition – are relevant when implementing distributed practice in school learning. The observed differences between the massed and distributed practice conditions concerning the effects of individual characteristics on the completion of the exercises could be explained by different challenges posed by massed and distributed practice. In contrast to massed practice, with distributed practice the students have to actively decide to resume working on the exercises on multiple occasions, instead of being able to just continue working. That is, action has to be initiated more often in distributed practice than in massed practice, potentially resulting in a higher influence of personal characteristics related to study management on the completion of the exercises (

The finding that among females, lower concentration difficulty was associated with a greater success at completing distributed practice tasks contradicts the assumption that learners with poor concentration ability might profit from distribution. However, concentration ability is associated with a higher engagement in learning in general (

The analyses of the perceived difficulty of the practice and test sets provided no evidence for the fact that the perceived difficulty differed between massed and distributed practicing students. However, it should be noted that the students did not explicitly rate the difficulty of the practice strategy but only the difficulty of the exercises. The students were not able to directly compare the alternative practice strategies and, thus, could not rate the relative but only the absolute difficulty. Especially because the exercises generally were not perceived as particularly difficult, potential differences could have been minimized. Additionally, the lack of a meaningful difference could also be due to prior self-selection, because students who perceived the exercises as particularly difficult may have stopped working on them in the course of the study. That is, the question of whether distributed practice is in fact perceived as more difficult than massed practice should ideally be investigated in studies with a within-subjects design.

First of all, the main results of the present study are rather exploratory and hence should be verified by further studies. The question of whether distributed practice generally improves performance in mathematics compared to massed practice, however, should be investigated in studies with less emphasis on self-regulated learning in order to maintain sufficient sample sizes and reduce potential selection bias. Additionally, though the topics were picked from the regular curriculum and hence were generally relevant for the students, their performance in our study did not influence their math grades and was not even shared with the teacher. This could have negatively impacted the motivation to participate in and complete the study. Ideally, in future studies on distributed practice in a self-regulated learning context, the personal relevance of the learned content should be increased compared to the current study – for example, by grading the performance. Finally, the students worked on the exercises at home, that is, the context and state in which the students participated in the study was barely controlled. However, this limitation should apply to both conditions equally and is no explanation for the differences between practice conditions.

One of the original questions of this study of whether distributed practice improves performance in mathematical learning in high school can hardly be answered based on the present study. The moderate evidence for a negative effect of distributed practice should not be overemphasized due to a high likelihood of self-selection in the course of the study. As long as there is no further empirical confirmation of this unusual result, the general assumption that distributed practice improves performance in later tests compared to massed practice (

In future research it should be investigated if the implications of the exploratory results of this study can be replicated and whether and how students can be supported by implementing distributed practice effectively in their self-regulated learning. A potential measure to motivate students to keep on working even in the distributed practice schedule could be to inform them prior to the practice phase about the positive effects of distributed practice. For example, at least in completely self-regulated learning,

The datasets and analysis scripts for this study are part of the Supplementary Material repository (

Both authors developed the basic idea of the present study. KBN was responsible for the study design and data collection procedure, conducted the analyses, and wrote a first version of the manuscript, which was then repeatedly revised by both authors.

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.

We would like to thank Wiebke Eden, Carina Haines, Ulrike Haßfurther, Fabian Kinzl, Maike Siegle, Amrei Schmieg, and Christina Vogelsang for their support in conducting the experiments and Jan Bender and Klara Schelbert for also teaching the participating classes. Our thanks for proofreading and valuable comments on this manuscript goes to Maj-Britt Isberner.

Each exercise set consisted of four exercises similar to the following. The complete material was provided in German language online (

The so called Sunday question (‘Sonntagsfrage’) is used to determine the political mood in Germany. Last Sunday, 1,300 citizens answered the following question: ‘Which political party would you vote for if there was an election next Sunday?’

How many citizens would vote for the ‘CDU’ and ‘Die Grünen’? How many citizens would vote for the ‘AfD’?

Pie chart for exercise 1.

CDU: 1300/100 = 13 → 13 ^{∗} 33 = 429 voters

Die Grünen: 1300/100 = 13 → 13 ^{∗} 9 = 117 voters

AfD: 1300/100 = 13 → 13 ^{∗} 14 = 182 voters

The six sides of a quadratic cube are labeled with numbers from 1 to 6. The following probabilities apply to the different sides:

P(1) = P(6) = 0.07

P(2) = P(5) = 0.13

P(3) = P(4) = 0.3

Calculate the probability for the following events if the dice is rolled once:

The dice rolls a 1 or a 6.

The dice rolls an even number.

The dice rolls a 3 or a 5.

The dice rolls a 1 or a 6: P(1) + P(6) = 0.07 + 0.07 = 0.14

The dice rolls an even number: P(2) + P(4) + P(6) = 0.13 + 0.3 + 0.07 = 0.5

The dice rolls a 3 or a 5: P(3) + P(5) = 0.3 + 0.13 = 0.43

In the winter term 2015/2016 there were 244.322 students enrolled in different types of universities in Hessen. The distribution was as follows:

Table for exercises 3 and 4.

University | 154274 |
---|---|

Theological University | 764 |

University of Arts | 1737 |

University of Applied Science | 83411 |

University of Management | 4136 |

Please indicate the sample, name the characteristic (variable) and the possible values.

Sample: all students = 244322

Characteristic (variable): type of university

Values: University, Theological University, University of Arts, University of Applied Science, University of Management

Table for exercises 3 and 4.

University | 154274 |
---|---|

Theological University | 764 |

University of Arts | 1737 |

University of Applied Science | 83411 |

University of Management | 4136 |

Calculate the missing relative frequencies and the angles for a pie chart:

University: ___

Theological University: 0.003

University of Arts: 0.007

University of Applied Science: ___

University of Management: ___

University: ___

Theological University: 1.08°

University of Arts: 2.52°

University of Applied Science: ___

University of Management: ___

Relative frequencies:

University: 154274/244322 = 0.63

Theological University: 0.003

University of Arts: 0.007

University of Applied Science: 83411/244322 = 0.34

University of Management: 4136/244322 = 0.02

Angles:

University: 0.63 ^{∗} 360 = 226.8°

Theological University: 1.08°

University of Arts: 2.52°

University of Applied Science: 0.34 ^{∗} 360 = 122.4°

University of Management: 0.02 ^{∗} 360 = 7.2°

Originally, another purpose of the present study was to investigate the effect of the practice condition on the test performance and possible interactions with the mentioned characteristics. However, as the study suffered a severe dropout, the final sample was too small to examine interaction effects on the test performance. Therefore, the exploratory analyses are limited to the dependent variables of participation and adherence to the practice schedule (see the section “Results”).

In Germany, grades range from 1 (very good) to 6 (inadequate).

The data as well as analysis scripts are provided online (