# Modelling Dependency Structures Produced by the Introduction of a Flipped Classroom

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. The Flipped Subject

#### 2.2. Variables under Study

- Success rate: fraction of students who manage to succeed in the course out of the students taking the exam.
- Performance rate: fraction of students who manage to succeed in the course out of the students matriculated in it.
- Percentage of students sitting the final exam.
- Grade received by the students in the continuous evaluation.

- Time spent on studying the course, besides the class attendance (question 6).
- Whether the student used to prepare the out-of-class material every week (question 7).
- Students’ preferences for the teaching methodology (question 8).
- Students’ beliefs about whether they would pass the course (question 23).
- Whether each student thought that they would have learnt more with a traditional methodology in the teaching group (question 12).
- Assessment of the students regarding the methodology used in the teaching group (question 15).
- Assessment of the students regarding the methodology used in the working group (question 16).
- Perceptions of the students of the in-class formative assessment (question 19.1).

#### 2.3. Bayesian Networks

**Qualitative component**: a directed acyclic graph (DAG) consisting of one node for each variable in the model and a set of edges linking statistically dependent variables.**Quantitative component**: a conditional distribution $p\left({x}_{i}\right|pa\left({x}_{i}\right))$ for each variable ${X}_{i}$, $i=1,\dots ,n$ given its parents in the graph, denoted as $pa\left({X}_{i}\right)$.

**Serial connections**.- An observation on ${X}_{1}$ will influence the certainty about ${X}_{3}$ and, through this last variable, ${X}_{1}$ will also have an impact on ${X}_{5}$. Likewise, evidence on ${X}_{5}$ will influence ${X}_{1}$ through ${X}_{3}$.
- However, if we know the value of ${X}_{3}$, the path is blocked and the evidence cannot flow from ${X}_{1}$ to ${X}_{5}$ and neither follow the contrary way.

**Diverging connections**.- Variables ${X}_{3}$ and ${X}_{4}$ are dependent on ${X}_{2}$ and so, there is a flow of information from ${X}_{3}$ to ${X}_{4}$ or back again while no information about ${X}_{2}$ is known.
- However if a value for variable ${X}_{2}$ is observed, the flow of information from ${X}_{3}$ to ${X}_{4}$ is stopped and evidence about ${X}_{3}$ has no effect on ${X}_{4}$.

**Converging connections**.- Variable ${X}_{3}$ depends on both variables ${X}_{1}$ and ${X}_{2}$. But ${X}_{1}$ and ${X}_{2}$ are not related unless we have some information about ${X}_{3}$, this is, information can only be transmitted between ${X}_{1}$ and ${X}_{2}$ if we have information about ${X}_{3}$.

**Serial and Diverging connections:**Information flows between variables while the state of the middle variable is unknown.**Converging connections:**Information flows as long as information about the middle variable (or some of its descendants) is known.

## 3. Results

#### 3.1. Analysis of the Students’ Performances

#### 3.2. Analysis of Student Attitudes

#### 3.3. Analysis of the Dependency between Variables

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- O’Flaherty, J.; Philiphs, C. The use of flipped classrooms in higher education: A scoping review. Internet High. Educ.
**2015**, 25, 85–95. [Google Scholar] [CrossRef] - Dillon, K. Statisticophobia. Teach. Psychol.
**1982**, 9, 117. [Google Scholar] [CrossRef] - Philiphs, L.; Phillips, M. Improved student outcomes in a flipped statistics course. Adm. Issues J.
**2016**, 6, 88–98. [Google Scholar] - McBride, C. Flipping Advice for Beginners: What I Learned Flipping Undergraduate Mathematics and Statistics Classes. PRIMUS
**2015**, 25, 694–712. [Google Scholar] [CrossRef] - Guerrero, S.; Beal, M.; Lamb, C.; Sonderegger, D.; Baumgartel, D. Flipping Undergraduate Finite Mathematics: Findings and Implications. PRIMUS
**2015**, 25, 814–832. [Google Scholar] [CrossRef] - Overmyer, J. Research on Flipping College Algebra: Lessons Learned and Practical Advice for Flipping Multiple Sections. PRIMUS
**2015**, 25, 792–802. [Google Scholar] [CrossRef] - Lo, C.W.; Hew, K.F.; Chen, G. Toward a set of design principles for mathematics flipped classrooms: A synthesis of research in mathematics education. Educ. Res. Rev.
**2017**, 22, 50–73. [Google Scholar] [CrossRef] - Gilboy, M.B.; Heinerichs, S.; Pazzaglia, G. Enhancing Student Engagement Using the Flipped Classroom. J. Nutr. Educ. Behav.
**2015**, 47, 109–114. [Google Scholar] [CrossRef] [PubMed] - Limniou, M.; Schermbrucker, I.; Lyons, M. Traditional and flipped classroom approaches delivered by two different teachers: The student perspective. Educ. Inf. Technol.
**2018**, 23, 797–817. [Google Scholar] [CrossRef][Green Version] - Strayer, J.F. How learning in an inverted classroom influences cooperation, innovation and task orientation. Learn. Environ. Res.
**2012**, 15, 171–193. [Google Scholar] [CrossRef] - Kraut, L. Inverting an Introductory Statistics Classroom. PRIMUS
**2015**, 25, 683–693. [Google Scholar] [CrossRef] - Wilson, S.G. The Flipped Class: A Method to Address the Challenges of an Undergraduate Statistics Course. Teach. Psychol.
**2013**, 40, 193–199. [Google Scholar] [CrossRef] - Castillo, E.; Gutiérrez, J.; Hadi, A. Expert Systems and Probabilistic Network Models; Springer: New York, NY, USA, 1997. [Google Scholar]
- Jensen, F.V.; Nielsen, T.D. Bayesian Networks and Decision Graphs; Springer: New York, NY, USA, 2007. [Google Scholar]
- Fernández, A.; Morales, M.; Rodríguez, C.; Salmerón, A. A system for relevance analysis of performance indicators in higher education using Bayesian networks. Knowl. Inf. Syst.
**2011**, 27, 327–344. [Google Scholar] [CrossRef] - Madsen, A.L.; Jensen, F.V. Lazy propagation: A junction tree inference algorithm based on lazy evaluation. Artif. Intell.
**1999**, 113, 203–245. [Google Scholar] [CrossRef][Green Version] - Shenoy, P.P.; Shafer, G. Axioms for probability and belief function propagation. In Uncertainty in Artificial Intelligence 4; Shachter, R., Levitt, T., Lemmer, J., Kanal, L., Eds.; Springer: Amsterdam, The Netherlands, 1990; pp. 169–198. [Google Scholar]

Group | Dunn’s p-Value |
---|---|

A-B | 0.0102 |

A-C | 0 |

A-D | 0 |

B-C | 0.0111 |

B-D | 0.0016 |

C-D | 0.1730 |

Variable | Description |
---|---|

REP | Whether or not the student was retaken the course. |

CALL | If the students passed the course in June call, September call, they failed or |

they did not sit the final exam. | |

PCE GD | Percentage of grade got in the formative assessment in the teaching group. |

PCE GT | Percentage of grade got in the work group activities. |

PROB | Percentage of mark got by the students in the final exam out of the total mark |

for probability exercises. | |

RV | Percentage of mark got by the students in the final exam out of the total mark |

for random variable and probability distribution exercises. | |

INF | Percentage of mark got by the students in the final exam out of the total mark |

for exercises about inference. |

**Table 3.**Comparison of the success ratio, performance ratio, and percentage of students sitting the exam by teaching group (CI for dif.: Confidence interval for the difference between the means).

Comparison | Performance Rate | Success Rate | % of Students Sitting the Exam | |||
---|---|---|---|---|---|---|

between Groups | p-Value | 95% CI for dif. | p-Value | 95% CI for dif. | p-Value | 95% CI for dif. |

A and C | 1 | (−0.1387, 0.1387) | 0.7465 | (−0.2966, 0.1770) | 0.3040 | (−0.0789, 0.2813) |

B and C | 0.6739 | (−0.1934, 0.1063) | 0.8797 | (−0.2986, 0.2073) | 0.6566 | (−0.2379, 0.1307) |

C and D | 0.1500 | (−0.1114, 0.24) | 1 | (−0.3937, 0.3108) | 0.0778 | (−0.0148, 0.4220) |

**Table 4.**Comparison of the success ratio, performance ratio, and percentage of students sitting the exam by teaching group (CI for dif.: Confidence interval for the difference between the means).

Rate | Value in 2016 | Value in 2017 | p-Value | 95% CI for dif. |
---|---|---|---|---|

Performance | 0.18 | 0.21 | 0.8040 | (−0.2065, 0.1309) |

Success | 0.27 | 0.39 | 0.4793 | (−0.3748, 0.1460) |

% sitting exam | 0.65 | 0.55 | 0.4308 | (−0.11, 0.297) |

**Table 5.**Comparison of the grades in years 2016 and 2017 (N: sample size; SD: standard deviation; CV: coefficient of variation).

2016 | 2017 | |
---|---|---|

N | 33 | 31 |

Min. | 0 | 1.09 |

Max. | 9.6 | 7.35 |

Mean | 2.63 | 4.32 |

Median | 1.50 | 3.9 |

SD | 2.63 | 1.51 |

CV | 1 | 0.35 |

**Table 6.**Conditional probabilities of variable inference module (INF) given CALL using flipped classroom.

Inference | Probability | Conditional Probabilities in 2017 Given CALL | ||
---|---|---|---|---|

Module | in 2016 | June | September | Not Pass |

$[0,{0}^{\prime}25)$ | 0.75 | 0.18 | 1 | 0.67 |

$[{0}^{\prime}25,{0}^{\prime}50)$ | 0.08 | 0.36 | 0 | 0.13 |

$[{0}^{\prime}50,{0}^{\prime}75)$ | 0.14 | 0.18 | 0 | 0.2 |

$[{0}^{\prime}75,1]$ | 0.04 | 0.27 | 0 | 0 |

**Table 7.**Conditional probabilities of variable CALL given the probability module (PROB) using a flipped classroom.

CALL | Prior | PROB | |||
---|---|---|---|---|---|

Probability | [0,0’25) | [0’25,0’50) | [0’50,0’75) | [0’75,1] | |

June | 0.20 | 0 | 0.55 | 0.80 | 1 |

September | 0.02 | 0 | 0 | 0.20 | 0 |

Not pass | 0.78 | 1 | 0.45 | 0 | 0 |

**Table 8.**Conditional probabilities of variable CALL given the random variables module (RV) using flipped classroom.

CALL | Prior | RV | |||
---|---|---|---|---|---|

Probability | [0,0’25) | [0’25,0’50) | [0’50,0’75) | [0’75,1] | |

June | 0.20 | 0 | 0.29 | 0.80 | 1 |

September | 0.02 | 0 | 0.14 | 0 | 0 |

Not pass | 0.78 | 1 | 0.57 | 0.20 | 0 |

CALL | Prior | INF | |||
---|---|---|---|---|---|

Probability | [0,0’25) | [0’25,0’50) | [0’50,0’75) | [0’75,1] | |

June | 0.20 | 0.05 | 0.67 | 0.33 | 1 |

September | 0.02 | 0 | 0 | 0.17 | 0 |

Not pass | 0.78 | 0.95 | 0.33 | 0.50 | 0 |

CALL | Prior | RV | |||
---|---|---|---|---|---|

Probability | [0,0’25) | [0’25,0’50) | [0’50,0’75) | [0’75,1] | |

June | 0.18 | 0 | 0.17 | 0.64 | 0.20 |

September | 0.25 | 0.07 | 0.50 | 0.36 | 0.80 |

Not pass | 0.57 | 0.93 | 0.33 | 0 | 0 |

CALL | Prior | PROB | |||
---|---|---|---|---|---|

Probability | [0,0’25) | [0’25,0’50) | [0’50,0’75) | [0’75,1] | |

June | 0.18 | 0.07 | 0.48 | 0.27 | 0.34 |

September | 0.25 | 0.13 | 0.34 | 0.53 | 0.60 |

Not pass | 0.57 | 0.80 | 0.18 | 0.20 | 0.06 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Maldonado, A.D.; Morales, M.
Modelling Dependency Structures Produced by the Introduction of a Flipped Classroom. *Mathematics* **2020**, *8*, 19.
https://doi.org/10.3390/math8010019

**AMA Style**

Maldonado AD, Morales M.
Modelling Dependency Structures Produced by the Introduction of a Flipped Classroom. *Mathematics*. 2020; 8(1):19.
https://doi.org/10.3390/math8010019

**Chicago/Turabian Style**

Maldonado, Ana D., and María Morales.
2020. "Modelling Dependency Structures Produced by the Introduction of a Flipped Classroom" *Mathematics* 8, no. 1: 19.
https://doi.org/10.3390/math8010019