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Artificial Intelligence Techniques in Software Design for Mathematics Education

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Handbook of Digital Resources in Mathematics Education

Part of the book series: Springer International Handbooks of Education ((SIHE))

Abstract

This chapter presents a state of the art in the design of digital environments for mathematics education, with a particular focus on artificial intelligence techniques. A review of the work done in this area over the last few decades highlights current challenges and distinguishes between symbolic approaches and machine learning. About symbolic approaches, we review automatic reasoning tools in geometry and their potential. We also consider the design and research work around the Casyopée environment and the use of logic programming in the QED-Tutrix intelligent tutoring system. With respect to machine learning, four classes of techniques constitute contemporary AI in computer science. Two examples are discussed: a deep learning system of monument analysis for learning situations in mathematics, technology and art, and a computer classroom simulator that provides a new approach to training teachers.

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Notes

  1. 1.

    This state of the art relies on the current research literature. We draw particularly from Lagrange and Abboud (2018) for sections “Introduction” and “A CAS Kernel in a Digital Learning Environment: The Case of Casyopée” and from Richard et al. (2022) for the remaining sections.

  2. 2.

    The most advanced version is at https://github.com/kovzol/geogebra-discovery/

References

  • Arsac G (1987) L’origine de la démonstration: essai d’épistémologie didactique [the origin of the demonstration: essay on didactic epistemology]. Publications mathématiques et informatique de Rennes 5:1–45

    Google Scholar 

  • Asif R, Merceron A, Ali SA, Haider NG (2017) Analyzing undergraduate students’ performance using educational data mining. Comput Educ 113:177–194. https://doi.org/10.1016/j.compedu.2017.05.007

    Article  Google Scholar 

  • Balacheff N (1994) Didactique et intelligence artificielle. Recherches en didactique des mathématiques 14(1):9–42

    Google Scholar 

  • Barredo Arrieta A, Díaz-Rodríguez N, del Ser J, Bennetot A, Tabik S, Barbado A, Garcia S, Gil-Lopez S, Molina D, Benjamins R, Chatila R, Herrera F (2020) Explainable artificial intelligence (XAI): concepts, taxonomies, opportunities and challenges toward responsible AI. Inf Fusion 58:82–115. https://doi.org/10.1016/j.inffus.2019.12.012

    Article  Google Scholar 

  • Botana F, Recio T (2004) Towards solving the dynamic geometry bottleneck via a symbolic approach. In: International workshop on automated deduction in geometry. Springer, Berlin, Heidelberg, pp 92–110

    Google Scholar 

  • Butler D, Jackiw N, Laborde JM, Lagrange JB, Yerushalmy M (2010) Design for transformative practices. In: Mathematics education and technology-rethinking the terrain. Springer, Boston, MA, pp 425–437

    Google Scholar 

  • Carbonell J (1970) AI in CAI: an artificial-intelligence approach to computer-assisted instruction. IEEE Trans. Man-Machine Systems 11(4):190–202. https://doi.org/10.1109/tmms.1970.299942

    Article  Google Scholar 

  • Carlson M, Jacobs S, Coe E, Larsen S, Hsu E (2002) Applying covariational reasoning while modeling dynamic events: a framework and a study. J Res Math Educ 33(5):352. https://doi.org/10.2307/4149958

    Article  Google Scholar 

  • Emprin F (2022) Modeling practices to design computer simulators for trainees’ and mentors’ education. In: Richard PR, Vélez PM, Vaerenbergh VS (eds) Mathematics education in the age of artificial intelligence: how artificial intelligence can serve mathematical human learning, Mathematics education in the digital era, 17. Springer, pp 319–341. https://doi.org/10.1007/978-3-030-86909-0_14

    Chapter  Google Scholar 

  • Falcade R, Laborde C, Mariotti MA (2007) Approaching functions: Cabri tools as instruments of semiotic mediation. Educ Stud Math 66(3):317–333. https://doi.org/10.1007/s10649-006-9072-y

    Article  Google Scholar 

  • Fayek HM, Lech M, Cavedon L (2017) Evaluating deep learning architectures for speech emotion recognition. Neural Netw 92:60–68. https://doi.org/10.1016/j.neunet.2017.02.013

    Article  Google Scholar 

  • Font L, Gagnon M, Leduc N, Richard PR (2022) Intelligence in QED-Tutrix: balancing the interactions between the natural intelligence of the user and the artificial intelligence of the tutor software. In: Richard PR, Vélez PM, Vaerenbergh VS (eds) Mathematics education in the age of artificial intelligence: how artificial intelligence can serve mathematical human learning, Mathematics education in the digital era, 17. Springer, pp 45–76. https://doi.org/10.1007/978-3-030-86909-0

    Chapter  Google Scholar 

  • Ganesalingam M, Gowers WT (2016) A fully automatic theorem prover with human-style output. J Autom Reason 58(2):253–291. https://doi.org/10.1007/s10817-016-9377-1

    Article  Google Scholar 

  • Howson AG, Wilson B, International Commission on Mathematical Instruction (1990) School mathematics in the 1990s. Cambridge University Press

    Google Scholar 

  • Hoyles C, Lagrange JB (2010) Mathematics education and technology-rethinking the terrain: the 17th ICMI study, New ICMI study series, 13. Springer

    Book  Google Scholar 

  • Jankvist UT, Misfeldt M, Aguilar MS (2019) What happens when CAS procedures are objectified ? —the case of “solve” and “desolve”. Educ Stud Math 101(1):67–81. https://doi.org/10.1007/s10649-019-09888-5

    Article  Google Scholar 

  • Jarvis D., Dreise K., Buteau C., LaForm-Csordas S., Doran C., & Novoseltsev A. (2022) CAS use in university mathematics teaching and assessment: applying Oates’ taxonomy for integrated technology. In: Richard, P. R., Vélez, P. M., & Vaerenbergh, V. S.Mathematics education in the age of artificial intelligence: how artificial intelligence can serve mathematical human learning (Mathematics education in the digital era, 17) (pp. 283–317). Springer https://doi.org/10.1007/978-3-030-86909-0_13

  • Kafetzopoulos G, Psycharis G (2022) Conceptualization of function as a covariational relationship between two quantities through modeling tasks. J Math Behav 67. https://doi.org/10.1016/j.jmathb.2022.100993

  • Koncel-Kedziorski R, Hajishirzi H, Sabharwal A, Etzioni O, Dumas Ang S (2015) Parsing algebraic word problems into equations. Trans Assoc Comput Linguist 3:585–597

    Article  Google Scholar 

  • Kovács Z, Parisse B (2015) Giac GeoGebra -Improved Gröbner Basis Computations. In: Computer algebra and polynomials, Gutierrez J, et. al. (Eds.), Lect Notes Comput Sci 8942:126–138. https://doi.org/10.1007/978-3-319-15081-97

  • Kovács Z, Recio T, Vélez MP (2022) Automated reasoning tools with GeoGebra: what are they? What are they good for? In: Richard PR, Vélez PM, Vaerenbergh VS (eds) Mathematics education in the age of artificial intelligence: how artificial intelligence can serve mathematical human learning, Mathematics education in the digital era, 17. Springer, pp 23–44

    Chapter  Google Scholar 

  • Kynigos C, Lagrange JB (2013) Cross-analysis as a tool to forge connections amongst theoretical frames in using digital technologies in mathematical learning. Educ Stud Math 85(3):321–327. https://doi.org/10.1007/s10649-013-9521-3

    Article  Google Scholar 

  • Lagrange JB (2005a) Using symbolic calculators to study mathematics. The case of tasks and techniques. In: Guin D, Ruthven K, Trouche L (eds) The didactical challenge of symbolic calculators. Turning a computational device into a mathematical instrument. Springer, pp 113–135

    Chapter  Google Scholar 

  • Lagrange JB (2005b) Curriculum, classroom practices, and tool design in the learning of functions through technology-aided experimental approaches. Int J Comput Math Learn 10(2):143–189

    Article  Google Scholar 

  • Lagrange JB, Abboud M (2018) Environnements numériques pour l’apprentissage, l’enseignement et la formation: perspectives didactiques sur la conception et le développement. IREM de Paris, Cahiers du Laboratoire de Didactique André Revuz

    Google Scholar 

  • Leduc N (2016) PhD thesis, École polytechnique de Montréal

    Google Scholar 

  • Mariotti MA (2013) Le potentiel sémiotique de Casyopée. In: Postface à Halbert R, Lagrange JB, Le Bihan C, Le Feuvre B, Manens MC, Meyrier X (eds) Les fonctions: comprendre et résoudre des problèmes de la 3ème à la Terminale. L’apport d’un logiciel dédié. IREM de Rennes, pp 78–82

    Google Scholar 

  • Martínez-Sevilla Á, Alonso S (2022) AI and mathematics interaction for a new learning paradigm on monumental heritage. In: Richard PR, Vélez PM, Vaerenbergh VS (eds) Mathematics education in the age of artificial intelligence: how artificial intelligence can serve mathematical human learning, Mathematics education in the digital era, 17. Springer, pp 107–136

    Chapter  Google Scholar 

  • Minh TK (2012) Les fonctions dans un environnement numérique d’apprentissage: étude des apprentissages des élèves sur deux ans. Can J Sci Math Technol Educ 12(3):233–258. https://doi.org/10.1080/14926156.2012.704127

    Article  Google Scholar 

  • Mitchell TM (1997) Machine learning. McGraw-Hill, New York

    Google Scholar 

  • Recio T, Vélez MP (1999) Automatic discovery of theorems in elementary geometry. J Autom Reason 23:63–82

    Article  Google Scholar 

  • Recio T, Richard PR, Vélez MP (2019) Designing tasks supported by GeoGebra automated reasoning tools for the development of mathematical skills. Int J Technol Math Educ 26(2):81–89

    Google Scholar 

  • Richard PR (2004) L’inférence figurale: Un pas de raisonnement discursivo-graphique. Educ Stud Math 57(2):229–263. https://doi.org/10.1023/b:educ.0000049272.75852.c4

    Article  Google Scholar 

  • Richard PR, Vélez PM, Vaerenbergh VS (2022) Mathematics education in the age of artificial intelligence: how artificial intelligence can serve mathematical human learning, Mathematics education in the digital era, 17, 1st edn. Springer

    Book  Google Scholar 

  • Roanes-Lozano E, Roanes-Macías E, Villar-Mena M (2003) A bridge between dynamic geometry and computer algebra. Math Comput Model 37(9–10):1005–1028. https://doi.org/10.1016/s0895-7177(03)00115-8

    Article  Google Scholar 

  • Robert A, Rogalski J (2002) Le système complexe et cohérent des pratiques des enseignants de mathématiques: Une double approche. Can J Sci Math Technol Educ 2(4):505–528. https://doi.org/10.1080/14926150209556538

    Article  Google Scholar 

  • Robert A, Vandebrouck F (2014) Proximités-en-acte mises en jeu en classe par les enseignants du secondaire et ZPD des élèves: analyses de séances sur des tâches complexes. Recherches en didactique des mathématiques 43(2–3):239–285

    Google Scholar 

  • Roozemond DA (2004) Automated proofs using bracket algebra with Cinderella and OpenMath. In Proceedings of 9th Rhine workshop on computer algebra (RWCA 2004)

    Google Scholar 

  • Saxton D, Grefenstette E, Hill F, Kohli P (2019) Analysing mathematical reasoning abilities of neural models. In: Proceedings of ICLR. In international conference on machine learning

    Google Scholar 

  • Smith A, Min W, Mott BW, Lester JC (2015) Diagrammatic student models: modeling student drawing performance with deep learning. In: International conference on user modeling, adaptation, and personalization. Springer, pp 216–227

    Chapter  Google Scholar 

  • Tessier-Baillargeon M, Richard PR, Leduc N, Gagnon M (2017) Étude comparative de systèmes tutoriels pour l’exercice de la démonstration en géométrie. Annales de didactique et de sciences cognitives 22:93–117. https://doi.org/10.4000/adsc.716

    Article  Google Scholar 

  • Thompson PW (2011) Quantitative reasoning and mathematical modeling. In: Hatfield LL, Chamberlain S, Belbase S (eds) New perspectives and directions for collaborative research in mathematics education, WISDOMe Mongraphs, vol 1. University of Wyoming, pp 33–57

    Google Scholar 

  • Turing AM (1950) Computing machinery and intelligence. Mind 49:433–460

    Article  Google Scholar 

  • Van Vaerenbergh S, Pérez-Suay A (2022) A classification of artificial intelligence systems for mathematics education. In: Richard PR, Vélez PM, Vaerenbergh VS (eds) Mathematics education in the age of artificial intelligence: how artificial intelligence can serve mathematical human learning, Mathematics education in the digital era, 17. Springer, pp 89–106

    Chapter  Google Scholar 

  • Vapnik V (1999) The nature of statistical learning theory. Springer

    Google Scholar 

  • Wu WT (1978) On the decision problem and the mechanization of theorem-proving in elementary geometry. Sci Sinica 21:159–172

    Google Scholar 

  • Zehavi N, Mann G (2011) Development process of a praxeology for supporting the teaching of proofs in a cas environment based on teachers’ experience in a professional development course. Tech Know Learn 16:153–181. https://doi.org/10.1007/s10758-011-9181-2

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Acknowledgments

Third author is partially supported by the grant PID2020-113192GB-I00 (Mathematical Visualization: Foundations, Algorithms and Applications) from the Spanish MCIN/AEI

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Authors and Affiliations

  1. LDAR, Université Paris-Cité, Paris, France

    Jean-Baptiste Lagrange

  2. Université de Montréal, Montréal, QC, Canada

    Philippe R. Richard

  3. Universidad Antonio de Nebrija, Madrid, Spain

    María Pilar Vélez

  4. Universidad de Cantabria, Santander, Spain

    Steven Van Vaerenbergh

Authors
  1. Jean-Baptiste Lagrange
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  2. Philippe R. Richard
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  3. María Pilar Vélez
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  4. Steven Van Vaerenbergh
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Correspondence to Jean-Baptiste Lagrange .

Editor information

Editors and Affiliations

  1. Eindhoven School of Education, Eindhoven University of Technology, Eindhoven, The Netherlands

    Birgit Pepin

  2. University of Paris-Saclay, Orsay, France

    Ghislaine Gueudet

  3. University of Rochester, Rochester, NY, USA

    Jeff Choppin

Section Editor information

  1. No affiliation provided

    Despina Potari

  2. No affiliation provided

    Giorgos Psycharis

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Lagrange, JB., Richard, P.R., Vélez, M.P., Van Vaerenbergh, S. (2023). Artificial Intelligence Techniques in Software Design for Mathematics Education. In: Pepin, B., Gueudet, G., Choppin, J. (eds) Handbook of Digital Resources in Mathematics Education. Springer International Handbooks of Education. Springer, Cham. https://doi.org/10.1007/978-3-030-95060-6_37-1

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  • DOI: https://doi.org/10.1007/978-3-030-95060-6_37-1

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  • Print ISBN: 978-3-030-95060-6

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