Impact case study database

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By adding new capabilities for computing with semigroups, new and faster algorithms for key problems, new databases and new links to other systems, our research insights have enhanced software widely used in teaching and learning of advanced mathematics. Through this, our research has changed the practices of advanced mathematics teaching at more than 27 institutions in 23 countries, affecting courses taken by over 5,700 people (students), improving the students' learning and understanding. Specifically, more teachers are using computational algebra software to teach advanced mathematics, enabling them to make their courses more investigative and student-centred, leading to improved learning and understanding in their students. Our research has also improved the experience, learning and understanding of the approximately 20,000 weekly users of the CoCalc web-based cloud computing and course management platform for computational mathematics.

Pathway from Research to Impact

The pathway from our research to this impact runs through an open-source software system called GAP (Groups, Algorithms, Programming) [S1]. GAP is a software package which allows the user to calculate interactively with complex mathematical objects, such as groups, semigroups, permutations, matrices and so on. It is highly extensible and, with appropriate extension packages, has functionality covering much of abstract algebra, combinatorics, representation theory and other key disciplines in pure mathematics. Our research insights are realised as algorithms, databases and software technologies, which are distributed as part of GAP, which is itself used for teaching.

Our development of new algorithms for computing with semigroups has expanded GAPs capabilities in semigroup theory from almost nothing, making it possible to use GAP for teaching (as well as researching) this key topic. As one measure of the scale of this expansion, since 2012 (the oldest version available, but already including some of our work) the number of documentation entries containing “semigroup” in their names increased from 700 to over 2,000 [S2]. This expanded capability provided the basis for subsequent changes in teaching practice and student learning.

Our new algorithms for permutation groups and linear algebra increased the speed of key computations across the whole system. This allows students to work flexibly and interactively with more and larger examples of the mathematical objects they are studying. This is especially important as the smallest examples in these fields (the only ones accessible to hand computation or with previous generations of software) are often atypical and can mislead students. A professor of mathematics at Colorado State University (Fort Collins) emphasised this in his letter, writing " The use of computation makes it feasible to [study examples] in cases that are significantly larger than what could be done by hand, and allows [students] to study not only individual phenomena, but also to avoid misguided intuition, based on the availability of only small examples" [S3, p. 5]. A professor of algebra from the University of Granada wrote, " fast and responsive performance benefits their educational use, for example by allowing students to experiment with larger examples gaining deeper understanding" [S3, p. 4].

Adding databases to the system builds still further on the ability of students to work with examples, allowing them to systematically study all examples and to look for patterns. It also increases the efficiency of teaching by making it trivial for the teacher to identify and supply examples for the students. A lecturer at Vrije Universiteit, Brussels, wrote " Through the study of examples using the databases available in GAP... a natural approach to the formulation of general theorems can be developed." [S3, p. 6].

Finally, thanks to our research on linking mathematical software, GAP is now also part of the broader SageMath mathematics software system [S4] and the CoCalc on-line scientific collaboration and teaching system [S5], enabling thousands of additional students to benefit from its capabilities. As a direct output of our research [R6], GAP can now be used though the Jupyter notebook user interface, which many students and teachers are familiar with from the popular Python programming language and which is widely used in data science and scientific data analysis.

Reach of Impact

Both GAP and SageMath are open-source software systems, which anyone is free to download and use, without informing the authors. Evidence for the reach of the impact through their direct use is thus unavoidably incomplete, based on surveys, requests for support and publications. Our most recent survey of the use of GAP, SageMath and CoCalc in teaching [S6] received 46 responses between 22 September 2020 and 01 December 2020, describing teaching in 23 countries to a total of 5,757 people (students) between August 2013 and December 2020. Of particular note are courses delivered to over 2,500 people (students) in at least seven African countries. Teaching included undergraduate (26 reports), postgraduate (26 reports) and professional development courses (10 reports). Among these are 740 people (students) taking computer supported courses in semigroup theory, something that would have been impossible without our research. Earlier data collections through the GAP website [S7] and GitHub [S8] definitively identify 7 additional sites, in 7 additional countries where GAP has been used in teaching between August 2013 and December 2020.

GAP and SageMath are key components of the CoCalc online service [S5] which provides additional tools for teaching, course management, assessment, etc. As of February 2020, CoCalc had over 20,000 unique weekly logins, growing at about 20% per year [S9].

Significance of impact

Evidence for the significance of the specific changes flowing from our insights is found in our survey [S6]. As well as the 740 students whose semigroup theory courses would simply have been impossible without our algorithms and software, 88% of responses agreed that the ability to explore larger examples was an important or very important benefit of using the systems, and 72% said that the performance and responsiveness of the software was a significant factor in delivering the benefits of the software for their teaching. These benefits flow from our new and faster algorithms. Mathematical databases were rated “very” or “extremely” important by 50%. The significance of our work on linking software is seen in the extent of use of GAP via SageMath and CoCalc (52% of those who responded to our survey use it this way, and we know [S9] that the majority of CoCalc users use SageMath).

Evidence of widespread use and high value placed on computational tools for teaching in this area is found in the publication by independent authors, of textbooks for courses relying on the use of GAP and SageMath. We know of 8 editions of textbooks published between August 2013 and December 2020 which rely on GAP [S7] and 24 which rely on SageMath [S10]. In France, SageMath (under its previous name “Sage”) is on the national list of software permitted for the “Aggregation”, the qualifying examination for secondary school teachers [S11].

A pedagogical research project in 2018 at the Universitat d’Alacant and Universitat Politècnica de València studied the use of GAP in courses on Group Theory and Coding Theory. It reported a positive experience and included a recommendation that the University " Integrate GAP classes into the subject from the beginning, thus giving time to ... see more examples and applications of the program" [S12, p. 8, our translation].

Further evidence is found in the letters [S3] submitted as part of the follow-up to the survey:

"I have made the experience that by giving the students the possibility to work concretely with the new algebraic objects they have seen in the courses the motivation to confront abstract algebraic problems hugely increases. .....

The interplay of theory with the calculations of more complex examples, for which manual computations are overly technical or simply not feasible, allows a much deeper insight into the underlying algebra." [S3, p. 6]

“ … the possibility of using structures as groups, rings and fields either finite or infinite allows the student to implement cryptographic protocols in platforms which the students could not construct themselves in a computer without an extraordinary effort.” [S3, p. 3] *(*describing teaching a cryptography course).

"The combination of Sagemath with Jupyterlab is a useful tool that makes possible to combine GAP with other mathematical tools, for both the use of students (for teamwork and labs) and for the use of teachers in the class room. It offers a learning by experimentation which would be hardly achieved without these tools. ..... the inclusion of GAP in Sagemath has allowed to interact groups with other mathematical objects " [S3, p. 2].