Math Learning in a Science Museum—Proposal for a Workshop Design Based on STEAM Strategy to Learn Mathematics. The Case of the Cryptography Workshop
Abstract
:1. Introduction
1.1. Mathematics Learning in a Science Museum Based on van Hiele’s Theory with STEAM Strategy
- There are different levels of reasoning in students, referring to mathematics.
- Each level supposes a way of understanding, a particular way of thinking, so that a student can only understand and reason with the mathematical concepts appropriate to his or her level of reasoning.
- Therefore, the teaching process must be adapted to the student’s level of reasoning. Teaching that takes place at a higher level than that of the students will not be understood.
- The teaching process must be oriented towards facilitating progress at the level of reasoning, so that this progress is made quickly and effectively.
1.2. Didactic Use of Cryptography for Mathematics Learning
2. Preliminaries
3. Objectives, Research Design and Methodology
3.1. Objectives
- To introduce a proposal for science workshop design in a museum, based on van Hiele’s mathematics learning model and the STEAM strategy.
- To use this general proposal to design a scientific workshop for learning mathematics about cryptography.
3.2. Research Design and Methodology
- The starting point is a real, complex and open situation, with social implications and very close to the visitors, which acts as a guiding thread for the action. Prior knowledge survey: the language and questions used in the workshops are adapted to the visitors’ prior knowledge.
- Information, directed guidance, explanation, free guidance, and integration (STEAM) activities developing the different communicative, visual, pictorial, logical, applied and digital skills. Presented in a motivational way, through different games or challenges, but scientifically rigorous.
- Paying attention to the gender perspective, development of SDG (sustainable development goal) values, competence and creation of scientific vocation are discussed.
- A final product and reflection are obtained.
4. Proposal for a Cryptography Workshop
4.1. Activity 1: Everyone Has Secrets I
- Small groups are set up and the sign code is provided.
- Opinions on the current importance of cryptography are shared.
- Different news is analysed by the students.
- A recap or outline of the work done is made.
4.2. Activity 2: The Challenge Is Deciphering It I
- Small groups are created, encryption challenges are provided and discussed.
- Simple types of encryptions by substitution and translation, such as scytale, Polybius and Caesar, are discussed. Matrix definition, additive structure and divisibility criteria are explained.
- Message decryption with scytale and Polybius. Construction of a Caesar cipher wheel.
- Opinions are shared on the strategies used to solve the challenges.
- New challenge proposal by the students.
- A recap or outline of the work completed is made.
4.3. Activity 3: Everyone Has Secrets II
- Small groups are set up and the flag code is given out.
- Opinions are shared on the strategies used to solve the challenges.
- New challenge proposal by the students.
- A recap or outline of the work completed is made.
4.4. Activity 4: The Challenge Is Deciphering It II
- Small groups are created, encryption challenges are provided and discussed.
- Commentary on the generalisation of substitution ciphers. Matrix definition, matrices addition and subtraction.
- Educational escape code decryption to open a lock on a four-letter surprise chest.
- Strategies used to solve the challenges are shared.
- A recap or outline of the work completed is made.
4.5. Activity 5: Enigma Machine
- Small groups are created, encryption challenges are provided and discussed.
- Commentary on automatic cryptography and the enigma machine.
- Construction, encryption and decryption of messages with an enigma machine recreation.
- Strategies used to solve the challenges are shared.
- A recap or outline of the work completed is made.
4.6. Activity 6: The Challenge Is Deciphering it III
- Small groups are created, encryption challenges are provided and discussed.
- Commentary on the generalisation of substitution ciphers, row matrix, column matrix, square matrix, matrix multiplication and inverse of a matrix.
- Educational escape code decryption to open a lock on a three-letter surprise chest.
- Strategies used to solve the challenges are shared.
- A recap or outline of the work completed is made.
4.7. Activity 7: My Secrets Are Yours
- Small groups are created, encryption challenges are provided and discussed.
- Commentary on the division of secrets and Parakh secret-sharing scheme [50]. Polynomial interpolation, linear equation systems, Cramer’s method, Newton’s method for quadratic interpolation and modular arithmetic.
- Educational escape code decryption to open a lock on a four-letter surprise chest.
- Strategies used to solve the challenges are shared.
- A recap or outline of the work completed is made.
4.8. Activity 8: Prime Numbers and Their Significance
- Small groups are created, activity objectives are provided and discussed.
- Commentary on RSA encryption, public and private key, divisibility criteria, prime numbers, prime factor decomposition, integer powers, congruences, Euclid’s algorithm and Bexout’s theorem.
- Investigation of the need to use very large prime numbers.
- The products obtained in the different studies are shared through presentations, panels and exhibitions.
- A recap or outline of the work completed is made.
4.9. Activity 9: Everyone Is Encrypted
- Small groups are created, activity objectives are provided and discussed.
- Commentary on DNA, living being code, nitrogenous basis, chromosomes, cell nucleus and genetics [60].
- Vegetal DNA extraction and analysis of plant hybridisation analysis.
- The strategies used for DNA extraction and Mendel’s mathematical reasoning are shared, in addition to polynomials and Newton’s binomial.
- A recap or outline of the work completed is made.
4.10. Activity 10: Steganography
- Small groups are created, activity objectives are provided and discussed.
- Commentary on algorithmics, mobile applications, digital photography and scratch.
- Programming encryption and decryption algorithms for Polybius and Caesar with scratch [61], as well as the use of steganography to hide messages in photographs.
- Strategies used to program algorithms are shared.
- A recap or outline of the work completed is made.
5. Discussion
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Science Workshop at the Museum | STEAM Project [30,31] | Learning Mathematics Phases (according to van Hiele’s Model) |
---|---|---|
Several pre-set activities with a topic that relates and contextualises them. | The starting point is a real, complex, and open situation, with social implications and very close to the visitors, which acts as a guiding thread for the action. | Type of activities: information, directed guidance, explicit, free guidance, and integration. |
Low versatility to adapt the diversity of previous knowledge and short duration. | Well-structured activities are proposed, justifying each action and the relationship between them. These activities are addressed from the different perspectives of STEAM subjects, in order to favour their integration. | Skills to be developed in the activities: communicative, visual, pictorial-manipulative, logical, applied and digital. |
No interaction related to the contents of workshop with visitors before or after the visit is planned at first. | Questions are posed in order to develop a research process with an important experimental part. | |
The workshop contents may or may not be related to those studied at school. | Visitors participate in the process, starting from the very definition of the problem, in the evaluation and in the production of a final product. | |
Funny (motivating), game and mathematics, learning by doing, active interaction and manipulation. | Encourages the visitors to develop creativity, critical thinking, scientific communication and peer-to-peer collaboration. | |
Instructor–visitor communication becomes essential. | As transversal: gender perspective, development of SDG (sustainable development goal) values, digital competence and creation of scientific vocation are discussed. | |
Scientifically rigorous. |
Descriptors of the van Hiele Levels | |
---|---|
Level 1 Visualization | Distinguishes between encoding and encryption. Recognizes the different types of encryptions and the agents involved. Associates the names of the encryption devices with their images. Interprets sentences that describe classic encryption methods. Draws different codes and encryption devices, accurately labelling their parts. Understands the form and meaning of the elements of a flowchart. |
Level 2 Analysis | Recognizes different variants in each type of encryption. Properly describes the elements of each classical and modern encryption system. Translates verbal information about encryption method properties to draw flowcharts. Understands the classification of encryption types according to their characteristics. Properly identifies the use of public and private keys. Recognizes the use of prime numbers and cryptography in different areas of everyday life. Understands the steps of an algorithm and relates them appropriately to the flowchart. |
Level 3 Classification | Appropriately interprets the visual representation of an algorithm through its flowchart. Formulates precise definitions of the different methods presented. Is capable of building other encryptions based on the presented models. Understands the successive steps to encrypt or decrypt a message in classic and modern cryptography. Use the appropriate statements to develop an encryption or decryption algorithm. Is capable of solving problems from other areas of science and everyday life by applying cryptography. |
Situation | Activity | Type | Concepts | Skills | Sessions |
---|---|---|---|---|---|
Pre-visit (classroom) | Everyone has secrets I | Information and directed guidance | Cryptology, coding, encryption | Communicative and visual | 1 session |
Pre-visit (classroom) | The challenge is deciphering it I | Information, directed guidance and explanation | Encryption and decryption by substitution and transposition, matrix definition, additive structure, divisibility criteria. | Communicative, visual, pictorial-manipulative, logical | 1 session |
Workshop at the museum | Everyone has secrets II | Information and directed guidance | Cryptology, coding, encryption | Communicative and visual | 1 session |
Workshop at the museum | The challenge is deciphering it II | Information, directed guidance and explanation | Encryption and decryption by substitution and transposition, matrix definition, addition and subtraction of matrices. | Communicative, visual, pictorial-manipulative, logical, and applied | |
Workshop at the museum | Enigma machine | Information, directed guidance and explanation | Automatic encryption and decryption, additive structure | Communicative, visual, pictorial-manipulative, logical, and applied | |
Post-visit (classroom) | The challenge is deciphering it III | Information, directed guidance, explanation, and free guidance | Row matrix, column matrix, square matrix, matrix multiplication, and inverse of a matrix. | Communicative, visual, and applied | 1 session |
Post-visit (classroom) | My secrets are yours | Information, directed guidance, explanation, and free guidance | Polynomial interpolation, linear equation systems, Cramer’s method, Newton’s method for quadratic interpolation and modular arithmetic. | Communicative, visual, and applied | 1 session |
Post-visit (classroom) | Prime numbers and their significance | Information, directed guidance, explanation, and free guidance | Public and private key, divisibility criteria, prime numbers, prime factor decomposition, integer powers, congruences, Euclid’s algorithm, and Bexout’s theorem. | Communicative, visual, and applied | 1 session |
Post-visit (classroom) | Everyone is encrypted | Information, targeted guidance and integration | DNA, living being code, nitrogenous basis, chromosomes, cell nucleus and genetics. Polynomials and Newton’s binomial. | Communicative, visual, pictorial-manipulative, logical, and applied | 1 session |
Post-visit (classroom) | Steganography | Information, directed guidance, explanation, free guidance and integration | Algorithmics, mobile applications, digital photography and scratch | Communicative, visual, pictorial-manipulative, logical, applied and digital | 2 sessions |
A | B | C | D | E | F | G | H | I | J | K | L | M |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |
N | O | P | Q | R | S | T | U | V | W | X | Y | Z |
14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 |
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Roldán-Zafra, J.; Perea, C. Math Learning in a Science Museum—Proposal for a Workshop Design Based on STEAM Strategy to Learn Mathematics. The Case of the Cryptography Workshop. Mathematics 2022, 10, 4335. https://doi.org/10.3390/math10224335
Roldán-Zafra J, Perea C. Math Learning in a Science Museum—Proposal for a Workshop Design Based on STEAM Strategy to Learn Mathematics. The Case of the Cryptography Workshop. Mathematics. 2022; 10(22):4335. https://doi.org/10.3390/math10224335
Chicago/Turabian StyleRoldán-Zafra, Juan, and Carmen Perea. 2022. "Math Learning in a Science Museum—Proposal for a Workshop Design Based on STEAM Strategy to Learn Mathematics. The Case of the Cryptography Workshop" Mathematics 10, no. 22: 4335. https://doi.org/10.3390/math10224335
APA StyleRoldán-Zafra, J., & Perea, C. (2022). Math Learning in a Science Museum—Proposal for a Workshop Design Based on STEAM Strategy to Learn Mathematics. The Case of the Cryptography Workshop. Mathematics, 10(22), 4335. https://doi.org/10.3390/math10224335