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The authors have declared that no competing interests exist.

This study addresses the pressing issue of how to raise the performance of disadvantaged students in mathematics. We combined established findings on effective instruction with emerging research addressing the specific needs of disadvantaged students. A sample of

Mathematics proficiency is a prerequisite not only for learning in diverse subject areas but also for many fast-growing jobs [

It is particularly the group of disadvantaged students with limited resources in their surrounding that struggle most to develop sustainable mathematics knowledge [

Understanding of fractions and rational numbers is a particularly challenging but at the same time a fundamental part of mathematical competence [

Over the past few decades, research in mathematics education, educational psychology, and cognitive science has identified principles that have the potential to promote the understanding of fractions. Effective fraction instruction should address the natural number bias [

Principle | Examples |
---|---|

exploratory tasks involving congruent gestures (e.g., cutting through pizza in terms of swiping over a touchscreen or filling tape diagrams via finger movement to represent a specific part of the whole) | |

interactive diagrams (e.g., altered iconic representation of a given fraction to demonstrate what happens, when the enumerator or the denominator is changed, or the fraction is raised) | |

adaptive adjustment of task difficulties (e.g., students have to complete sets of tasks within a certain predefined difficulty level before they are allowed to proceed to the next difficulty level; tasks answered incorrectly have to be repeated until a heuristically determined threshold of correct answers is reached) | |

graded assistance during problem-solving (e.g., students can choose to get constructive hints for solving a problem) | |

feedback for an incorrect answer is given based on students’ answers (e.g., the correct solution is shown together with the student’s answer) | |

feedback for an incorrect answer is given based on an algorithm choosing an appropriate strategy for a given problem (e.g., for comparing 7/6 vs. 8/9, benchmarking to 1 is suggested by the algorithm, while for comparing 4/9 vs. 3/5, benchmarking to 1/2 is suggested) |

In this classroom intervention study, we investigated how disadvantaged students can be supported

Of specific interest in this study is whether the benefits of those digital support principles

The studies involving human participants were reviewed and approved by the Bavarian ministry of education, Germany, reference X.7-BO5106/141/8, and the responsible local education authority “Staatliches Schulamt München für Mittelschulen”, reference SchRIII/Erh106/1—including ethics and data privacy approval. Written informed consent to participate in this study was provided by the participants’ legal guardian/next of kin.

The present classroom intervention study was conducted as a cluster randomized controlled trial with classrooms as clusters on the topic of basic fraction concepts. It followed a pre-post-follow-up constructive research strategy [

The study included 260 6th-grade students (42% female) from 15 classrooms from the German (Bavaria) public-school track

It is noteworthy that fractions are formally introduced in grade six in the mandatory German (Bavarian) curriculum, so that no formal knowledge about fractions acquired in school has to be expected by 6th-graders in this study

We created an evidence-based fractions curriculum to teach basic fraction concepts—i.e.,

The Scaffolded Curriculum group worked with an e-textbook on iPads designed within the iBooks Author framework and CindyJS as the programming environment for the interactive content [

(A) Exploratory task involving congruent gestures, i.e., cutting through pizza in terms of swiping over a touchscreen. (B) Interactive diagram showing the conceptual idea of the number line as a shrunken tape diagram—here displayed using overlay technique. (C) Task with adaptive adjustment of task difficulties that represent difficulty generating factors, i.e., number of items equals denominator in level 1, number of items is twice or thrice the denominator in level 2. (D) Task offering graded assistance during problem-solving which students can access by tapping on the traffic lights. (E) Feedback for an incorrect answer based on students’ wrong answers. (F) Feedback for an incorrect answer based on an algorithm choosing an appropriate strategy for the comparison of two fractions.

A paper-based copy of the e-textbook was used in the Curriculum group. The paper-based version differed from the e-textbook only in terms of the absence of digital learning support. Both the e-textbook [

In order to control for effects of prior knowledge of fractions, a pretest was conducted before the intervention (10 items; internal consistency of McDonald’s Omega ω = 0.82; full independent double coding with an inter-rater reliability of 0.85 ≤ κ ≤ 0.99). The original German version of the test instrument is available online [

To measure basic fraction knowledge, a second test instrument was developed (16 dichotomous items; internal consistency of McDonald’s Omega ω_{Post} = 0.68 in the posttest, and ω_{FollowUp} = 0.63 in the follow-up test; full independent double coding with an inter-rater reliability of 0.93 ≤ κ ≤ 0.99). Here, items focus on the six fraction topics given above. The developed instrument was used both for the posttest and for the follow-up test. All originally German items from this instrument can be found online [

We conducted a pilot study to ensure the quality of the instruments (pretest

The present study was conducted at the beginning of the school year 2017/2018. After review and positive evaluation by the Bavarian ministry of education and the responsible local education authority (including ethics and data privacy approval), a total number of 44 schools were asked to take part in the study on a voluntary basis and 25% of the schools answered this invitation, including eight positive and three negative responses. In voluntary agreement with the schools’ principals and the classroom teachers, students and their parents were informed about the aims and goals of the study. Taking part in the written tests was voluntary for all students, and with informed consent of them and their parents. Only after teachers agreed in taking part in the study, whole classes were randomly assigned to one of the three groups.

The intervention took place within the first 15 mathematics lessons of the school year, covering fractions only. The Curriculum group (

In order to control for effects of prior fraction knowledge, the pretest (15 minutes) was conducted before the first lesson. We assessed fraction knowledge in the posttest immediately after 15 lessons, and eight weeks after the posttest (56 ± 14 days) in the follow-up test (20 minutes). The complete research design is summarized in

The 15 participating teachers (Scaffolded Curriculum, Curriculum, and Traditional) took part in the same 90-minute teacher training before the intervention. Each teacher received an 18-page booklet with information on the research questions, the educational objectives, and the content to be taught during the 15 lessons.

We conducted structured and formal interviews with all 15 teachers after the intervention to ensure that they covered all topics, did not differ in total instruction time, and used the material provided for the Scaffolded Curriculum group and the Curriculum group as intended. Relying on the interviews, we considered the implementation adequate if and only if …

Traditional group: … the content listed above was completely addressed during the 15 lessons.

Curriculum group: … the content listed above was completely addressed during the 15 lessons; our exploratory introductions were used to start with new topics; all 90 tasks (i.e., the printed version of the widgets in the e-textbook) were worked on at least once.

Scaffolded Curriculum group: … the content listed above was completely addressed during the 15 lessons; our exploratory introductions were used to start with new topics while using the e-textbooks interactive content; all 90 widgets were worked on at least once with recourse to the e-textbooks scaffolds.

The teachers’ statements indicated that all study conditions were implemented as intended. In addition, informal interviews with all 15 teachers were conducted at the follow-up test to ensure that the teachers did not teach the content of the intervention between posttest and follow-up test—which was not the case.

A generalized linear mixed model (GLMM) was used to estimate differences between the three intervention groups in answering the test items. For the aim of the current study, they have several advantages over other statistical approaches, i.e., handling of unbalanced designs, as well as nested data structures, and dichotomous data [

The Null model contains only a fixed effect for Time (0 = posttest, 1 = follow-up test), estimating an overall effect of the two different time points on students’ ability to answer items regarding fraction concepts. In the Full model, fixed effects for specific control variables—i.e., Prior Knowledge (pretest result, standardized at the sample), and Gender (-0.5 = female; 0.5 = male)—are added. To estimate effects of the intervention, a fixed effect for Intervention Group (baseline: Scaffolded Curriculum) is added, predicting differences in the posttest. Differences between the three intervention groups in the follow-up are represented in the Time × Intervention Group interaction effect. To control for the variance in the time between posttest and follow-up in the 15 different classrooms, a Time × Delay (number of days between posttest and follow-up, standardized at the sample) interaction effect is added. As the effect of Delay is not meaningful for estimating posttest results, the main effect of Delay is omitted in the Full model. Both the Null model and the Full model allow for a Student, a Classroom, and an Item random intercept—to account for the nested data structure (students clustered in classrooms) as well as difference in item difficulty and student ability. Data used for the analysis is given in the

Estimates are given as Odds Ratios, reflecting unique changes in the estimated probability to answer an average item in the test correct, when a categorial predictor changes from the baseline to another level, or when a metric predictor is 1 SD above the sample mean. Odds ratios and their confidence intervals were calculated using the

As students were nested in classrooms and classrooms were randomly assigned to one of the three intervention groups, the Proportion Change in Variance [PCV, see

For less than 4% of the sample (

The groups showed no significant differences in terms of prior knowledge,

260 | 2.28 | 2.02 | |

female | 110 | 2.21 | 2.04 |

male | 150 | 2.33 | 2.01 |

Scaffolded Curriculum | 107 | 2.55 | 2.18 |

Curriculum | 71 | 2.17 | 1.96 |

Traditional | 82 | 2.02 | 1.81 |

There was a significant effect of prior knowledge on posttest fraction knowledge (

Null Model | Full Model | |||||
---|---|---|---|---|---|---|

Fixed effects | OR | CI | p | OR | CI | p |

Prior Knowledge | 1.23 | [1.13, 1.35] | < 0.001 | |||

Gender | 1.00 | [0.84, 1.19] | 0.990 | |||

Scaffolded Curriculum → Curriculum | 0.66 | [0.47, 0.91] | 0.012 | |||

Scaffolded Curriculum → Traditional | 0.50 | [0.37, 0.69] | < 0.001 | |||

Time | 0.98 | [0.88, 1.09] | 0.721 | 0.98 | [0.83, 1.16] | 0.818 |

Time × Curriculum | 1.03 | [0.79, 1.35] | 0.839 | |||

Time × Traditional | 0.97 | [0.74, 1.26] | 0.800 | |||

Time × Delay | 0.94 | [0.83, 1.05] | 0.274 | |||

Student ( |
0.32 | 0.28 | 11.81% | |||

Classroom ( |
0.16 | 0.03 | 81.41% | |||

Item ( |
1.49 | 1.49 | 0.57% | |||

Observations | 7888 | 7888 | ||||

AIC | 8441 | 8421 |

We expected digital learning support to be beneficial for disadvantaged students—in addition to evidence-based fraction instruction. In line with this hypothesis, students from the Scaffolded Curriculum group yielded the highest outcomes, both in the posttest and the follow-up (

Development is shown from the posttest (immediately after the four weeks of instruction during the intervention, i.e., 15 lessons) to the follow-up test (additional eight weeks after the posttest) as estimated average item-solution probabilities, predicted by the generalized linear mixed model. Error bars represent 95% confidence intervals.

Most important for the present study, in none of the three groups, students’ outcomes did not differ significantly between the posttest and the follow-up test in any of the three groups (Scaffolded Curriculum group,

The

Our results yield unique ecologically-valid empirical evidence that digital learning support—implemented in evidence-based fraction instruction—helps disadvantaged students to acquire—and most important to maintain—fraction knowledge in real mathematics classrooms. With regards to the constructive research design, the results suggest that the positive outcome of the Scaffolded Curriculum group—both in the posttest and in the follow-up test—should be ascribed to the implementation of digital support principles, and not the evidence-based fractions curriculum, in the present sample.

Yet, the finding that in all three groups students maintained their respective knowledge level from the posttest to the follow-up test, but differed in the absolute knowledge level gained during the intervention, suggests a more fine-grained analysis of the exact content successfully learned in each condition. Indeed, the low achievement level in the Traditional group suggests that these students only learned most basic content during classroom practice—yet they could maintain this knowledge after eight weeks without additional instruction. When interpreting this result, it should be kept in mind that students in the Scaffolded Curriculum group did also keep their knowledge level without further instruction, yet on a substantially higher level than students from the Control group—which is arguably a more difficult task, underpinning the positive effect of digital support principles for disadvantaged students.

While students with more favorable prerequisites also profit from evidence-based instruction [

While there is broad empirical evidence that all of our suggested digital support principles mentioned in

Our operationalization of disadvantaged students as students from the lowest school track of the German public-school system could be considered a potential limitation of this study. However, attendance of this track is not only confounded with lower performance on standardized assessments of mathematics competence [

It might also be questioned whether a time period of eight weeks between posttest and follow-up test are enough to assess the sustainability of learning effects. Additional follow-up tests after several months could corroborate our findings. Yet, there is long-established evidence that only eight weeks after instruction about two thirds of successfully learnt information is forgotten [

The finding that the students described in [

Due to specific circumstances of the present study, students did only have access to the iPads—and therefore to the e-textbook providing digital support for the learning of fraction concepts—in school and during their regular mathematics lessons. It is of specific interest, whether providing access to these support principles for learning outside of school may have additional positive effects. Within these rather informal learning scenarios, students face different boundary conditions when engaging in learning tasks (i.e., learning from home during the global COVID-19 pandemic), so that different effects than in regular school classrooms may be expected—which could be focused on in subsequent studies.

Please also refer to [

(PDF)

Please also refer to [

(PDF)

studID (anonymized student id, numbers from 401 to 660); claID (anonymized classroom id, numbers from 1 to 15); grp (factor representing group, i.e., Scaffolded Curriculum, Curriculum, Traditional); sexC (student’s gender, (-0.5 = female; 0.5 = male); delayZ (standardized delay between posttest and follow-up, original

(CSV)

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