Download the module’s lesson plan (pdf, editable format)

We chose cryptography because, nowadays, it is an interdisciplinary domain in itself (Durand-Guerrier, Meyer, & Modeste, 2019). Both mathematical elements (e.g., proofs, number theory) and Informatics elements (e.g., computational complexity, systems design, programming) are fundamental to solving the relevant social, technological, and scientific challenges it poses. Moreover, some cryptography elements encompass intertwined aspects of Informatics and Mathematics (for example, one-way functions are both well-defined mathematical functions and programs that satisfy specific security and efficiency criteria). 

The topic seems halfway through O2 and O3: while cryptography is crucial for today’s societal challenges and its study involves different communities, the two most important disciplines involved (Informatics and Mathematics) are still very recognisable. While, historically, cryptography was in the realm of Mathematics, and while Informatics started as only a tool for faster computations, modern cryptography and its fields of application would not exist without Informatics. The current research in the field of cryptography stimulates a dialectic between the two disciplines, from Informatics to Mathematics (e.g., requiring new research in elliptic curves) and in reverse (e.g., using theorem provers to validate cryptographic properties). As said, cryptography still allows for “disciplinary projections”: elements of the two disciplines are very recognisable inside the field (analogously to what happens in bicultural societies). We choose didactic activities where these projections can shed light on important disciplinary concepts, like computational complexity for Informatics and linear systems for Mathematics. The projections can stimulate exploration and learning of some relevant topics and ideas of each discipline, not only in the specific scope of cryptography but also in other Informatics and Mathematics areas. Moreover, this can foster the exploration of general interactions between the two disciplines.

Consequently, developing learning in both disciplines and their interactions is relevant for secondary teachers who should teach cryptography either in Mathematics courses, Informatics courses, or interdisciplinary courses and projects.

We consider the need to introduce cryptography as a social issue of contemporary society (ENISA, 2016) (following a long-term presence in human history from ancient Greece to nowadays) because  Cryptography is at the core of many  of the economic, social, political interactions that happen in our interconnected, globalised digital society. That is why our module starts with a discussion on end-to-end encryption, the debate around its societal advantages and risks, and stimulating the need to understand how it works to form an informed opinion. Moreover, Informatics education research agrees on the fact that communicating in secret and trying to decrypt messages without knowing the key is engaging and motivating for students (Lindmeier & Mühling, 2020). 

We choose the Didactical Engineering methodology relying on the Theory of Didactical Situations (Brousseau & Warfield, 2020), widely used for decades in Mathematics Education. Learning this methodology can be useful for student teachers as well.

We choose a public-key cryptography activity (based on a computationally hard problem on graphs) for epistemological and didactical reasons. Epistemologically, Informatics and Mathematics are deeply interconnected in the cryptography research field and discipline, and the activity, as we will see, can bring up many topics like algorithms, computational complexity, graphs, matrices, and linear systems. Educationally, informatics and cryptography have an inherent potential for adidacticity, that is, the potential for learning with a strong autonomy left to students’ interactions with the problem. In fact, if one is trying to find the secret key of a cryptosystem (where encryption and decryption algorithms are public), the supposed key can be tested by decrypting the messages that have been encrypted with that key and see if the result is the original plaintext message. Analogously, in computer programming, students can autonomously test their program and check if it works by comparing the desired and actual result of the computation without waiting teacher’s validation.

 Becoming explorers

Aims: Making students aware of the fact that cryptography is crucial today by contextualising it in a social debate about end-to-end encryption and backdoors. Students should realise what disciplines are involved and what knowledge they need to understand and form an opinion in a debate. Moreover, by analysing a historical piece on the birth of modern cryptography, students see how Informatics and Mathematics are intertwined in cryptography, beginning with disciplinary projections. During all the activities, students start encountering basic terminology about cryptography.

Activities: Two individual activities. The first is watching an outreach video about how end-to-end encryption works (the fact that messages are fully encrypted from the sender to the receiver, so no one in the middle can read them) and the opportunity and risks of governments putting a backdoor (a way to decrypt those communications) for security reasons. The second one is reading an outreach text about the birth of asymmetric encryption, highlighting the interaction between mathematicians, computer scientists, and engineers who discovered it. After the two activities, students are given a reflection form about: skills needed to understand the debate; disciplines they recognise in the historical piece and their interactions; disciplinary-specific and crypto-specific terminology they found; initial thoughts about using the crypto interdisciplinary potential to teach crypto, Informatics and Mathematics in schools.

The participants introduce themselves and “break the ice” on interdisciplinarity and cryptography (pdf, editable format)

The participants watch two videos about end-to-end encryption (video 1, video 2)

The participants read a text about the birth of RSA (pdf, editable format)
 

The participants reflect on the activities they experienced, guided by some open-ended questions (pdf, editable format)

Becoming students

Aims: Students experience an example of a classroom activity designed according to the Theory of Didactical Situation (TDS). They will learn about public-key cryptography, explore the ideas of computational complexity and one-way functions, and manipulate interdisciplinary objects like functions, graphs, matrixes, and linear systems. From teachers’ perspective, they will also experience firsthand how a didactical situation is organised. 

Activities: We start with a brief presentation on the fundamental elements of asymmetric cryptosystems. Then, we present a  cryptosystem based on the perfect dominating set problem on graphs and show how it can be used to encrypt a message. After that, students are split into three groups: each group has to decrypt the same secret message but has different information on cryptographic and mathematical aspects of the system (i.e., in terms of the TDS, each group has a different organisation of the milieu) and an autonomous way to verify if they solved the problem (adidacticity). Afterwards, each group presents and discusses its strategies to decrypt the message. Finally, instructors lead an institutionalisation phase, based on the groups’ presentations, to formalise the disciplinary aspects behind the activity. The different organisation of the three groups leads to different perspectives and approaches to the task and is intended to help student teachers think about the interdisciplinary objects encountered.

The participants participate in a presentation and interactive lecture on fundamental cryptography concepts (pdf, editable format)

The participants are introduced to how to encrypt a message using graphs (pdf, editable format)

The participants in groups try to decrypt a secret message

Group A (pdf, editable format)

Group B (pdf, editable format)

Group C (pdf, editable format)

The participants present their results; instructors discuss the approaches and formalise some aspects of the activity (institutionalisation) (pdf, editable format)

 Becoming analysts

Aims: together with the interdisciplinary analysis explained in the short version, students are also given some TDS methodological tools to analyse the previous learning experience also from a learning design point of view.
Students are given “raw materials” to be analysed both with the same interdisciplinary tools and in light of the TDS in order to design a new didactical situation on a different topic,  which also takes advantage of the interdisciplinarity between Mathematics and Informatics (within cryptography or outside of it)

Activities: First of all, students are given raw teaching materials (es. about the Fibonacci sequence and its computation, and about Primes factorisation and its relationship to computational complexity and cryptography) and asked to analyse it from disciplinary, interdisciplinary and didactical perspectives with guiding questions. After that, students are taught some principles of TDS. Finally, students are asked to design in groups a teaching activity (based on TDS) on one of the analysed topics by leveraging the disciplinary and interdisciplinary aspects they found.

The participants watch divulgation videos about interdisciplinary frameworks (videos)

The participants reflect on interdisciplinarity in activities they experienced, guided by some open-ended questions (pdf, editable format)

The participants explore individually “raw” teaching material that has the potential to be used to design interdisciplinary activities between CS and Math (inside and outside cryptography) (pdf, editable format)

The participants analyse in groups the teaching materials guided by some open questions (pdf, editable format)

The participants follow a lecture about the Theory of Didactical Situations (pdf, editable format)

Becoming designers

Aims: together with the interdisciplinary analysis explained in the section above, students are also given some TDS methodological tools to analyse the previous learning experience also from a learning design point of view.
Students are given “raw materials” to be analysed both with the same interdisciplinary tools and in light of the TDS in order to design a new didactical situation on a different topic,  which also takes advantage of the interdisciplinarity between Mathematics and Informatics (within cryptography or outside of it)

Activities: First of all, students are given raw teaching materials (es. about the Fibonacci sequence and its computation, and about Primes factorisation and its relationship to computational complexity and cryptography) and asked to analyse it from disciplinary, interdisciplinary and didactical perspectives with guiding questions. After that, students are taught some principles of TDS. Finally, students are asked to design in groups a teaching activity (based on TDS) on one of the analysed topics by leveraging the disciplinary and interdisciplinary aspects they found.

The participants design an interdisciplinary learning activity (pdf, editable format)

Download the module’s lesson plan (pdf, editable format)