# Teaching Mathematics at Distance: A Challenge for Universities

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## Abstract

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## 1. Introduction

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- As they moved to distance teaching, what artifacts did university professors adopt in their didactic transposition of mathematics?
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- What effect does the number of students and the university degree course have on how a professor explains mathematics?
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- Does distance teaching have disadvantages for teaching mathematics, and if so what?
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- Do university mathematics professors gain any advantages in terms of professional skills from a potential crisis? Do opportunities arise from difficulty?

## 2. Theoretical Background

## 3. Methodology

#### 3.1. Participants

#### 3.2. Data Collection and Method of Analysis

- (1)
- Courses as part of mathematics (and possibly physics) degree courses, in which mathematics is quite appreciated by students, generally offered in small classes.
- (2)
- Mathematics courses (analysis, geometry, physical mathematics) in engineering degree courses, in which mathematics is on average appreciated by students, generally offered in large classes (even more than 100 students).
- (3)
- Basic mathematics courses offered for science degree courses (e.g., biology, geology, natural sciences, agriculture, computer science, architecture, economics, etc.), where mathematics is (often) little appreciated by students, generally offered to classes with many students.

## 4. Data and Analysis

^{2}(4) = 22.86, p < 0.001, allows us to confirm that the passage to distance teaching has pushed the professors in a non-random way to better prepare their lessons.

I do not have a writing tablet, blackboard, or anything else, so the only way is to write by hand before the lesson, scan the many sheets and insert them on the PC to share during the lesson: hard work.

It is easier to show the progress of an exercise.I believe that the professor’s personal writing makes the text “less cold” and more effective from a didactic point of view.

To have more contact with students.Students follow better.I like the idea of eye contact.It is important that students see me, both to create a “classroom” relationship and because I often show objects or “help” myself with gestures.

I do not think it is necessary and I prefer students to focus on the content of the presentation.[…] since my class consists of 180 students, I prefer not to show myself on video to avoid any unpleasant episodes.

Sharing is necessary to show what I have prepared, but also to show further examples, to do exercises, to focus on key steps… using a writing tablet.I think it is necessary to share the screen in a presentation, especially for mathematical formulas. I couldn’t do without it.During a three-hour lesson in front of the class, I fill four blackboards at least ten times, because I always have to write, now with no other way I show the many papers I write before each lesson, so I always talk and share.I use Mathematica software. Students see either my notebook where I explain the topic, or my notebook where I solve a problem.

^{2}(2) = 0.53, p > 0.50). Almost all the professors (95%) used gestures less than before or in the same way as before. In particular, the median test, χ

^{2}(2) = 8.67, p = 0.013, shows that professors who had more students (percentage of attendants between 75% and 100%) reported using gestures in the same way as before, while those who have few students (percentage of attendants between 0% and 50%) reported having used fewer gestures.

^{2}(3) = 27.05, p < 0.001).

With distance learning, having already prepared the material/presentation to share during the lesson (which then becomes teaching material available to students), it is possible to explain more quickly and therefore explain more topics than in traditional lessons.The use of the blackboard meant more time to explain the topics. Students don’t have to take notes. I provide them at the end of each lesson.Actually, with distance learning you do more with the same amount of time.I spend lot of time handwriting what I would write on the blackboard while I explain, and because I can’t see the eyes of the students I’m not inclined to repeat concepts or demonstrations, and so I do everything I set out to do.

^{2}(3) = 6.42, p < 0.10. A Kruskal–Wallis test showed that this inability depends on neither the number of students attending the course, χ

^{2}(2) = 0.699, p = 0.70, nor on the degree course, χ

^{2}(2) = 0.960, p = 0.62. The double entry table (Table 9) correlates the strategies adopted by professors with the previous answer. We can see that those who claimed they were able to perceive that the students were keeping up with the lesson did so by asking the students directly whether they were following the explanation (30%), by asking the students questions about the content they had explained (27%), or by deducing it from the interventions made by students (24%).

^{2}(1) = 4.35, p < 0.10.

^{2}(12) = 15.72, p = 0.21, meaning that the association between the two thematic variables were no different from chance. However, considering each variable individually (see the total column in the Table 12), the frequencies of the identified categories were significantly different for the loss in distance modality, χ

^{2}(4) = 22.26, p < 0.001, with the “Human exchange” category more frequent compared to the other categories. No differences were revealed for the gain in distance modality variable.

I discovered other ways of teaching, through the use of technology.I learned how to use the iPad board and the Microsoft Teams platform.

The ability to produce a PDF file that reproduces exactly what I wrote on the board. In this way students will find faithful notes and avoid transcribing inaccuracies that often cause confusion on important issues such as definitions or various observations.A greater sense of responsibility on the part of the students, who have understood that to follow well they must “study” punctually and in advance.The blackboards are published, the lesson is recorded, I have the ability to use software and also to show a video, which in traditional lessons I did not do because it was too laborious.

The in-depth planning of the lessons.The lesson is smoother, and the material I have prepared will still help me in the years to come.Comforts.Time.

A part of human contact, useful to transmit the passion for discipline.Eye contact with students.Human contact with students.The ability to see positive or negative reactions to what is proposed not explicitly manifested through interventions.When I saw the empty eyes of so many students I understood that they hadn’t understood me and so I repeated it all with other words until I saw their eyes full: I lost the joy of having been able to fill those eyes.

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Trend of COVID-19 cases in Italy until June 2020 (taken from https://lab24.ilScheme24.ore.com/coronavirus/ accessed on 30/06/2020).

**Table 1.**Cross-tabulation between the number of students who were members of the Teams and the percentage of students who attend distance learning, compared to the degree courses (n = 38).

% of Distance Learning Students | Total | |||||
---|---|---|---|---|---|---|

From 0 to 50 | From 51 to 75 | From 76 to 100 | ||||

Mathematics or physics | # students members of Teams | From 1 to 10 | 0.0% | 4.0% | 28.0% | 32.0% |

From 11 to 40 | 4.0% | 24.0% | 28.0% | 56.0% | ||

From 41 to 100 | 8.0% | 0.0% | 4.0% | 12.0% | ||

More than 100 | 0.0% | 0.0% | 0.0% | 0.0% | ||

Total | 12.0% | 28.0% | 60.0% | 100.0% | ||

Engineering | # students members of Teams | From 1 to 10 | 0.0% | 0.0% | 0.0% | 0.0% |

From 11 to 40 | 0.0% | 0.0% | 14.3% | 14.3% | ||

From 41 to 100 | 0.0% | 14.3% | 14.3% | 28.6% | ||

More than 100 | 28.6% | 14.3% | 14.3% | 57.1% | ||

Total | 28.6% | 28.6% | 42.9% | 100.0% | ||

Other | # students members of Teams | From 1 to 10 | 0.0% | 0.0% | 0.0% | 0.0% |

From 11 to 40 | 16.7% | 0.0% | 0.0% | 16.7% | ||

From 41 to 100 | 0.0% | 16.7% | 33.3% | 50.0% | ||

More than 100 | 33.3% | 0.0% | 0.0% | 33.3% | ||

Total | 50.0% | 16.7% | 33.3% | 100.0% | ||

Total | # students members of Teams | From 1 to 10 | 0.0% | 2.6% | 18.4% | 21.1% |

From 11 to 40 | 5.3% | 15.8% | 21.1% | 42.1% | ||

From 41 to 100 | 5.3% | 5.3% | 10.5% | 21.1% | ||

More than 100 | 10.5% | 2.6% | 2.6% | 15.8% | ||

Total | 21.1% | 26.3% | 52.6% | 100.0% |

Before COVID-19 Pandemic | During COVID-19 Pandemic | |
---|---|---|

Percent | Percent | |

Everything in detail | 39.5 | 68.4 |

Basic step ladder | 47.4 | 26.3 |

No preparation | 13.2 | 5.3 |

Total | 100.0 | 100.0 |

Percent | |
---|---|

Not online; in front of the class with blackboard and chalk | 81.6 |

Not online; in front of the class with blackboard and chalk; slide | 7.9 |

Not online; in front of the class with blackboard and chalk; slide; mathematical software | 5.3 |

Not online; in front of the class with blackboard and chalk; mathematical software | 2.6 |

Not online; in front of the class with blackboard and chalk; mathematical software; laboratory activities in groups | 2.6 |

Total | 100.0 |

**Table 4.**What do you use to digitize the formal and symbolic writings of mathematics? (multiple answer options)—(n = 38).

Percent | |
---|---|

Writing tablet | 26.3 |

Writing tablet; slides | 18.4 |

Mathematical software | 18.4 |

Slides | 18.4 |

Mathematical software; writing tablet | 13.2 |

Mathematical software; slides | 2.6 |

Mathematical software; writing tablet; slides | 2.6 |

Total | 100.0 |

**Table 5.**Explain how much you use your writing (writing tablet, sheets shown to students, etc.)—(n = 38).

Percent | |
---|---|

To improve teaching effectiveness and content exposure | 58.1 |

To recreate the traditional atmosphere of class | 19.4 |

I prefer to use other modalities | 12.9 |

I cannot use it | 9.7 |

Total | 100.0 |

**Table 6.**Frequency with which a certain action takes place on the PC during the distance lesson (n = 38).

Activating the Camera | Sharing the Screen | |
---|---|---|

Percent | Percent | |

Never | 21.1 | 5.3 |

Sometimes | 18.4 | 7.9 |

Often | 0.0 | 10.5 |

Always | 60.5 | 76.3 |

Total | 100.0 | 100.0 |

**Table 7.**Frequency of formats during distance learning compared to the usual presentation of topics? (n = 38).

Natural Language in an Informal Way | Mathematical Language | Iconic Representations | Gestures | |
---|---|---|---|---|

Percent | Percent | Percent | Percent | |

Less than before | 5.3 | 0.0 | 10.5 | 63.2 |

Same as before | 86.8 | 86.8 | 86.8 | 31.6 |

More than before | 7.9 | 13.2 | 2.6 | 5.3 |

Total | 100.0 | 100.0 | 100.0 | 100.0 |

**Table 8.**How much you agree with these statements: “with distance learning you can only explain half the things you have prepared”? (n = 38).

Percent | |
---|---|

Definitely disagree | 60.5 |

Quite disagree | 21.1 |

Quite agree | 7.9 |

Definitely agree | 10.5 |

Total | 100.0 |

**Table 9.**Cross-tabulation between degree of agreement and strategies adopted to perceive whether students are keeping up with the lesson (n = 38).

Categories of Answers for Professors’ Strategies | Total | ||||||
---|---|---|---|---|---|---|---|

Asking the Students If They Are Following the Explanation | Understanding via Student Interventions | Asking Questions about the Lesson Contents | Not Verifiable | ||||

Likert scale items | Definitely not | % of Total | 3.1% | 0.0% | 0.0% | 12.1% | 15.2% |

More no than yes | % of Total | 12.1% | 12.1% | 12.1% | 6.1% | 42.4% | |

More yes than no | % of Total | 12.1% | 12.1% | 6.1% | 0.0% | 30.3% | |

Definitely yes | % of Total | 3.0% | 0.0% | 9.1% | 0.0% | 12.1% | |

Total | % of Total | 30.3% | 24.2% | 27.3% | 18.2% | 100.0% |

**Table 10.**Likert-scale response percentages about concerns regarding distance learning before and during COVID-19 pandemic (n = 38).

Likert-Item Items | Concern before COVID-19 | Concern during COVID-19 |
---|---|---|

Not at all concerned | 7.9% | 36.8% |

Little concerned | 18.4% | 23.7% |

Pretty concerned | 42.1% | 31.6% |

Very concerned | 31.6% | 7.9% |

Total | 100.0% | 100.0% |

**Table 11.**Which of these e-learning platforms do you plan to continue using after the end of COVID-19 pandemic? (multiple response options)—(n = 38).

Percent | |
---|---|

Teams | 28.9 |

No platform | 26.3 |

Teams and Studium | 18.4 |

Studium | 10.5 |

Teams and Moodle | 10.5 |

I do not know | 5.3 |

Total | 100.0 |

**Table 12.**Cross-tabulation between thematic categories of “Gain in distance modality” and “Loss in distance modality” (n = 38).

Thematic Categories for “Loss in Distance Modality” | Total | ||||||
---|---|---|---|---|---|---|---|

Human Exchange | Interaction for Learning Purposes | Both (the Previous Ones) | No Loss | ||||

Thematic categories for “Gain in distance modality” | No profit | % of Total | 13.2% | 5.3% | 7.9% | 0.0% | 26.4% |

Better learning/involvement of students | % of Total | 13.2% | 7.9% | 0.0% | 2.6% | 23.7% | |

Improvement of didactical/technological skills | % of Total | 10.5% | 0.0% | 7.9% | 0.0% | 18.4% | |

Less organizational/physical stress | % of Total | 5.3% | 2.6% | 7.9% | 0.0% | 15.8% | |

Improved lesson planning | % of Total | 2.6% | 10.5% | 2.6% | 0.0% | 15.7% | |

Total | % of Total | 44.8% | 26.3% | 26.3% | 2.6% | 100.0% |

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**MDPI and ACS Style**

Cassibba, R.; Ferrarello, D.; Mammana, M.F.; Musso, P.; Pennisi, M.; Taranto, E.
Teaching Mathematics at Distance: A Challenge for Universities. *Educ. Sci.* **2021**, *11*, 1.
https://doi.org/10.3390/educsci11010001

**AMA Style**

Cassibba R, Ferrarello D, Mammana MF, Musso P, Pennisi M, Taranto E.
Teaching Mathematics at Distance: A Challenge for Universities. *Education Sciences*. 2021; 11(1):1.
https://doi.org/10.3390/educsci11010001

**Chicago/Turabian Style**

Cassibba, Rosalinda, Daniela Ferrarello, Maria Flavia Mammana, Pasquale Musso, Mario Pennisi, and Eugenia Taranto.
2021. "Teaching Mathematics at Distance: A Challenge for Universities" *Education Sciences* 11, no. 1: 1.
https://doi.org/10.3390/educsci11010001