Y. Attali and F. Van-der-kleij, Effects of feedback elaboration and feedback timing during computer-based practice in mathematics problem solving, Computers & Education, vol.110, pp.154-169, 2017.

A. Barana and M. Marchisio, Strategies of formative assessment enacted through automatic assessment in blended modality, Proceedings of CERME11, pp.4041-4048, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02430513

J. Choppin, C. Carson, Z. Borys, C. Cerosaletti, and R. Gillis, A typology for analyzing digital curricula in mathematics education, International Journal of Education in Mathematics, vol.2, issue.1, pp.11-25, 2014.

G. Anichini, F. Arzarello, L. Ciarrapico, and O. Robutti, Matematica 2003. Lucca: Matteoni stampatore, 2004.

T. Anderson and J. Shattuck, Design-based research: A decade of progress in education research? Educational Researcher, vol.41, pp.16-25, 2012.

T. Bauer and U. Partheil, Schnittstellenmodule in der Lehramtsausbildung im Fach Mathematik. Mathematische Semesterberichte, vol.56, pp.85-103, 2008.

M. Brandl, The vibrating string -an initial problem for modern mathematics; historical and didactical aspects, 18 th Novembertagung on the History, pp.95-114, 2009.

M. G. Bartolini-bussi and M. A. &mariotti, Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective, Handbook of international research in mathematics education, pp.746-783, 2008.

J. Back, J. Brodie, P. Curzon, C. Myketiak, P. W. Mcowan et al., Making computing interesting to school students: teachers' perspectives, p.13, 2013.

L. Chabanon and J. M. Pastor, L'évolution des performances en calcul des élèves de CM2 à trente ans d'intervalle, Note de la DEPP, vol.19, pp.1-4, 1987.

A. Kuzniak, Teaching and Learning Geometry and Beyond, Mathematics Education and Technology-Rethinking the Terrain. The 17th ICMI Study, 2010.

A. Kuzniak, . Tanguay, and I. Elia, Mathematical working spaces in schooling: An introduction, ZDM-Mathematics Education, issue.6, pp.721-737, 2016.

A. Kuzniak and A. Nechache, Développer un travail géométrique complet et conforme chez les étudiants de première année de master enseignement en France, 2020.

C. Laborde, Integration of technology in the design of geometry tasks with cabrigeometry, International Journal of Computers for Mathematical Learning, vol.6, issue.3, pp.283-317, 2001.

M. Mariotti, Introduction to Proof: The Mediation of a Dynamic Software Environment, Educational Studies in Mathematics, vol.44, pp.25-53, 2000.

. Oecd, Students, Computers and Learning: Making the Connection. PISA, 2015.

M. G. Bartolini-bussi and M. A. Mariotti, Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective, Handbook of international research in mathematics education, pp.746-783, 2008.

A. Clark-wilson, O. Robutti, and N. Sinclair, The mathematics teacher in the digital era, 2014.

E. Faggiano, A. Montone, and M. A. Mariotti, Synergy between manipulative and digital artefacts: a teaching experiment on axial symmetry at primary school, International Journal of Mathematical Education in Science and Technology, vol.49, issue.8, pp.1165-1180, 2018.

M. A. Mariotti, Artifacts and signs after a Vygotskian perspective: the role of the teacher, vol.ZDM, pp.427-440, 2009.

J. Trgalová, A. Clark-wilson, and H. G. Weigand, Technology and resources in mathematics education, Developing Research in Mathematics Education. Routledge, pp.142-161, 2018.

D. L. Ball, M. H. Thames, and G. Phelps, Content knowledge for teaching what makes it special, Journal of teacher education, vol.59, issue.5, pp.389-407, 2008.

C. B. Boyer, The history of the calculus and its conceptual development, 1959.

D. Holland, W. Lachicotte, D. Skinner, and C. Cain, Identity and Agency in Cultural Worlds, 1998.

J. W. Little, Locating learning in teachers ' communities of practice: opening up problems of analysis in records of everyday work. Teaching and Teacher Education, vol.18, pp.917-946, 2002.

C. Loisy, Analyzing Teachers' Work with Resources: Methodological Issues, The 'Resource' Approach to Mathematics Education, pp.257-321, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02480597

B. Pepin, G. Gueudet, and L. Trouche, Refining teacher design capacity: Mathematics teachers' interactions with digital curriculum resources. ZDM -Mathematics Education, vol.49, pp.799-812, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01536244

S. Prediger, A. Bikner-ahsbahs, and F. Arzarello, Networking strategies and methods for connecting theoretical approaches: first steps towards a conceptual framework, ZDM Mathematics Education, vol.40, pp.165-178, 2008.

J. Skott, Changing experiences of being, becoming, and belonging : Teachers' professional identity revisited, ZDM Mathematics Education, vol.51, issue.3, pp.469-480, 2019.

L. Trouche, G. Gueudet, and B. Pepin, Documentational approach to didactics, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01851785

L. Trouche, Evidencing Missing Resources of the Documentational Approach to Didactics, The "Resource" Approach to Mathematics Education, pp.447-489, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02390230

E. Wenger, Communities of Practice: Learning, Meaning, and Identity, 1998.

H. Burkhardt and M. Swan, Task design for systemic improvement: principles and frameworks, Task design in Mathematics Education -Proceedings of ICMI Study, vol.22, pp.431-440, 2013.

M. Joubert, M. Blanchard, and J. Lagrange, Revisiting theory for the design of tasks: Special considerations for digital environments, Digital REFERENCES Abboud, vol.13, pp.183-202, 2007.

K. Ala-mutka, Mapping digital competence: Towards a conceptual understanding, JRC Technical Note, 2011.

S. Hegedus, C. Laborde, C. Brady, S. Dalton, H. S. Siller et al., Uses of technology in upper secondary mathematics education, 2016.

E. Klieme, J. Hartig, and D. Rauch, The concept of competence in educational contexts, pp.3-22, 2008.

M. Kunter, U. Klusman, J. Baumert, D. Richter, T. Vos et al., Professional Competence of Teachers: Effects on Instructional Quality and Student Development, Journal of Educational Psychology, vol.105, issue.3, pp.805-820, 2013.

M. Neubrand, Conceptualizations of professional knowledge for teachers of mathematics, ZDM Mathematics Education, vol.50, issue.4, pp.601-612, 2018.

P. Rabardel, People and technology -a cognitive approach to contemporary instruments, 2002.
URL : https://hal.archives-ouvertes.fr/hal-01020705

C. Redecker, European Framework for the Digital Competence of Educators (DigCompEdu), 2017.

J. Cummins, Cognitive/Academic Language Proficiency, Linguistic Interdependence, the Optimum Age Question and Some Other Matters. Working papers on bilingualism, vol.19, pp.197-205, 1979.

L. Darling-hammond, M. E. Hyler, and M. Gardner, Effective Teacher Professional Development, 2017.

M. Gebhardt, D. Rauch, J. Mang, C. Sälzer, and P. Stanat, Mathematische Kompetenz von Schülerinnen und Schülern mit Zuwanderungshintergrund, PISA 2012: Fortschritte und Herausforderungen in Deutschland, pp.275-308, 2013.

S. Prediger, N. Wilhelm, A. Büchter, C. Benholz, and E. Gürsoy, Sprachkompetenz und Mathematikleistung -Empirische Untersuchung sprachlich bedingter Hürden in den Zentralen Prüfungen 10, vol.36, pp.77-104, 2015.

F. Lipowsky and D. Rzejak, Lehrerinnen und Lehrer als Lerner -Wann gelingt der Rollentausch? Merkmale und, Wirkungen effektiver Lehrerfortbildungen. Schulpädagogik heute, vol.5, issue.3, pp.1-17, 2012.

A. E. Nir and R. Bogler, The antecedents of teacher satisfaction with professional development programs. Teaching and Teacher Education, vol.24, pp.377-386, 2008.

D. Cohen, S. Raudenbush, and D. Ball, Resource, Instruction, and Research. Educational Evaluation and Policy Analysis, vol.25, pp.119-142, 2003.

K. Naguib, The production and reproduction of culture in Egyptian schools, Cultures of Arab Schooling, pp.53-81, 2006.

M. Carlson and I. Bloom, The cyclic nature of problem solving: An emergent problem-solving framework, Educational Studies in Mathematics, vol.58, issue.1, pp.45-75, 2005.

P. Drijvers, Digital Technology in Mathematics Education: Why It Works, vol.8, pp.1-20, 2013.

M. Fahlgren and M. Brunström, A model for task design with focus on exploration, explanation, and generalization in a dynamic geometry environment, Technology, Knowledge and Learning, vol.19, issue.3, pp.287-315, 2014.

M. Haspekian, Teachers' instrumental geneses when integrating spreadsheet software, The Mathematics Teacher in the Digital Era, pp.241-275, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01002961

C. Hoyles and . Lagrange, Mathematics education and technology -Rethinking the terrain, 2010.

M. Misfeldt, A. Szabo, and O. Helenius, Surveying teachers' conception of programming as a mathematics topic following the implementation of a new mathematics curriculum, Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education, pp.2713-2720, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02417074

A. Barbosa and I. Vale, Math trails: meaningful mathematics outside the classroom with preservice teachers, JETEN, vol.12, pp.49-63, 2016.

A. Barbosa and I. Vale, Math Trails: a resource for teaching and learning, Proceedings of the Re(s)sources 2018 International Conference, pp.183-186, 2018.

C. Bonotto, How to connect school mathematics with students' out-of-school knowledge, Zentralblatt für Didaktik der Mathematik, vol.33, issue.3, pp.75-84, 2001.

R. Borromeo-ferri, Estabelecendo conexões com a vida real na prática da aula de Matemática, Matemática, vol.110, pp.19-25, 2010.

A. N. Cahyono and M. Ludwig, Teaching and Learning Mathematics around the City Supported by the Use of Digital Technology, Science and Technology Education, vol.15, issue.1, pp.1-8, 2019.

F. Erickson, Qualitative methods in research on teaching, Handbook of research on teaching, pp.119-161, 1986.

G. Fessakis, P. Karta, and K. Kozas, Designing Math Trails for Enhanced by Mobile Learning Realistic Mathematics Education in Primary Education, iJEP, vol.8, issue.2, pp.49-63, 2018.

M. Ludwig and S. Jablonski, Doing Math Modelling Outdoors -A Special Math Class Activity designed with MathCityMap, 5th International Conference on Higher Education Advances (HEAd'19), pp.901-909, 2019.

K. Richardson, Designing math trails for the elementary school, Teaching Children Mathematics, vol.11, pp.8-14, 2004.

M. Shoaf, H. Pollak, and J. Schneider, Math Trails, 2004.

M. Smith and M. K. Stein, Five practices for orchestrating productive mathematics discussions, 2011.

D. C. Blane and C. Doug, A mathematics trail around the city of Melbourne, 1984.

A. N. Cahyono, Learning Mathematics in a Mobile App-Supported Math Trail Environment, 2018.

R. Cross, Developing mathematics trails. Mathematics Teaching, pp.38-39, 1997.

G. Franke-braun, F. Schmidt-weigand, L. Stäudel, and R. Wodzinski, , 2008.

A. Mit-gestuften-lernhilfen, , pp.27-42

I. Gurjanow, M. Olivera, J. Zender, P. Santos, and M. Ludwig, Mathematics Trails: Shallow and Deep Gamification, International Journal of Serious Games, vol.6, issue.3, pp.65-79, 2019.

A. Lieberoth, Shallow Gamification -Testing Psychological Effects of Framing an Activity as a Game, Games and Culture, issue.10, pp.229-248, 2015.

M. Ludwig and S. Jablonski, MathCityMap -Mit mobilen Mathtrails Mathe draußen entdecken, /2020), pp.29-36, 2020.

M. Ludwig and J. Jesberg, Using Mobile Technology To Provide Outdoor Modelling Tasks -The MathCityMap-Project, Procedia -Social and Behavioral Sciences, vol.191, pp.2776-2781, 2015.

Y. Park, A Pedagogical Framework for Mobile Learning: Categorizing Educational Applications of Mobile Technologies into Four Types, International Review of Research in Open and Distance Learning, vol.12, issue.2, pp.78-102, 2011.

M. M. Shoaf, H. Pollak, and J. Schneider, Math Trails. The Consortium for Mathematics and Its Applications (COMAP), 2004.

Y. Sung, K. Chang, and T. Liu, The effects of integrating mobile devices with teaching and learning on students' learning performance: A metaanalysis and research synthesis, Computers & Education, issue.94, pp.252-275, 2016.

J. Zender, Mathtrails in der Sekundarstufe I. Der Einsatz von MathCityMap bei Zylinderproblemen in der neunten Klasse, 2019.

H. Bauersfeld, G. Krummheuer, and J. Voigt, Interactional theory of learning and teaching mathematics and related microethnographical studies, Foundations and methodology of the discipline mathematics education, pp.174-188, 1988.

E. Baschek, Mit PrimarWebQuests Sprache fördern, vol.3, pp.10-13, 2019.

E. Baschek, Using ICT in Bilingual Mathematics Classes -an Example, International Symposium Elementary Mathematics Teaching. Proceedings. Opportunities in Learning and Teaching Elementary Mathematics, pp.455-457, 2019.

E. Baschek, Content and language Integrated learning with primarwebquests -Using Information and Communication technology in Bilingual Primary Mathematics Classes. ICME-proceedings, 2021.

C. Bescherer, WebQuests und Mathematikdidaktik. WebQuests als Methode für projektorientierten Mathematikunterricht, WebQuest and Mathematics Education, vol.67, pp.18-19, 2007.

D. Coyle, Content and language integrated learning. Motivating learners and teachers, Scottish Languages Review, vol.13, pp.1-18, 2006.

. Eurydice, Content and language integrated learning (CLIL) at school in Europe, 2006.

T. March, The learning power of WebQuests, Educational Leadership, vol.61, issue.4, pp.42-47, 2004.

H. Moser, Abenteuer Internet. Lernen mit WebQuests, 2008.

G. Donauwörth,

C. Schreiber, WebQuests für die Grundschule: Prima(r)WebQuest [WebQuests for primary school: Prima(r)WebQuest]. lehrer-online, 2007.

C. Schreiber and H. Kromm, , 2020.

G. Baltmannsweiler,

F. Alessio, L. Demeio, and A. I. Telloni, A formative path in tertiary education through GeoGebra supporting the students' learning assessment and awareness, International Journal for Technology in Mathematics Education, vol.26, issue.4, pp.191-203, 2019.

M. Baldacci, Personalizzazione o individualizzazione? Roma: Erickson, 2006.

B. Belland, Instructional Scaffolding in STEM Education, 2017.

A. Cusi and A. I. Telloni, The role of formative assessment in fostering individualized teaching at university level, Proceedings of CERME 11, pp.4129-4126, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02430560

A. Cusi and A. I. Telloni, Students' use of digital scaffolding at university level: emergence of utilization schemes, Proceedings of ICTMT 14, pp.271-278, 2019.

J. Hattie and H. Timperley, The power of feedback, Review of Educational Research, vol.77, issue.1, pp.81-112, 2007.

D. Holton and D. Clarke, Scaffolding and metacognition, International Journal of Mathematics Education in Science and Technology, vol.37, issue.2, pp.127-143, 2007.

R. Pea, The social and technological dimensions of scaffolding and related theoretical concepts for learning, education, and human activity, Journal of the Learning Sciences, vol.13, pp.423-451, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00190619

P. Sharma and M. J. Hannafin, Scaffolding in technology-enhanced learning environments, Interactive Learning Environments, vol.15, issue.1, pp.27-46, 2007.

A. Shvarts and B. Bakker, The early history of the scaffolding metaphor: Bernstein, Luria, Vygotsky, and before, Mind, vol.26, pp.4-23, 2019.

D. Wood, J. Bruner, and G. Ross, The role of tutoring in problem solving, The Journal of Child Psychology and Psychiatry, vol.17, pp.89-100, 1976.

, Base Nacional Comum Curricular, BNCC, 2018.

J. Carpenter and S. Gorg, Principles and Standards for School Mathematics, 2000.

A. Donevska-todorova, Raum und Form" in der Sekundarstufe mit Hilfe von DGS und 3D Druck Technologie. Beiträge zum Mathematikunterricht. Vorträge auf den 54. Jahrestagung der Gesellschaft für Didaktik der Mathematik vom 09, 2020.

A. Donevska-todorova and D. Lieban, Fostering Creativity Through Design of Virtual and Tangible Manipulatives, the Proceedings of the tenth ERME Topic Conference Mathematics Education in the Digital Age (MEDA), p.2020, 2020.

N. A. Drickey, A comparison of virtual and physical manipulatives in teaching visualization and spatial reasoning to middle school mathematics students, pp.1-138, 2000.

A. Cusi, F. Morselli, and C. Sabena, Promoting formative assessment in a connected classroom environment: Design and implementation of digital resources, ZDM, vol.49, issue.5, pp.755-767, 2017.

M. Fahlgren and M. Brunström, How the word 'mathematical'influences students' responses to explanation tasks in a dynamic mathematics software environment, Proceedings of the 5th ERME Topic Conference, pp.83-90, 2018.

M. Fahlgren and M. Brunström, Orchestrating whole-class discussions in mathematics using connected classroom technology, Proceedings of the 14th International Conference on Technology in Mathematics Teaching -ICTMT 14, pp.173-182, 2019.

M. Fahlgren and M. Brunström, Connected Classroom Technology to monitor, select and sequence student responses. Paper presented at the twelfth research seminar of the Swedish Society for, Research in Mathematics Education, 2020.

M. L. Franke, E. Kazemi, and D. Battey, Mathematics teaching and classroom practice, Second handbook of research on mathematics teaching and learning, vol.1, pp.225-256, 2007.

C. Kieran, D. Tanguay, and A. Solares, Researcher-designed resources and their adaptation within classroom teaching practice: Shaping both the implicit and the explicit, From Text to 'Lived' Resources, pp.189-213, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00632737

K. Ruthven and R. Hofmann, Chance by design: devising an introductory probability module for implementation at scale in English early-secondary education, ZDM, vol.45, issue.3, pp.409-423, 2013.

M. K. Stein, R. A. Engle, M. S. Smith, and E. K. Hughes, Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell, Mathematical Thinking and Learning, vol.10, issue.4, pp.313-340, 2008.

D. Bates, M. Mächler, B. Bolker, and S. Walker, Fitting Linear Mixed-Effects Models Using lme4, Journal of Statistical Software, vol.67, issue.1, pp.1-48, 2015.

E. De-corte, L. Verschaffel, and C. &masui, The CLIA-model: A framework for designing powerful learning environments for thinking and problem solving, European Journal of Psychology of Education, vol.19, issue.4, pp.365-384, 2004.

F. Arzarello, F. Olivero, D. Paola, and O. Robutti, A cognitive analysis of dragging practises in Cabri environments, Zentralblatt für Didaktik der Mathematik, vol.34, p.66, 2002.

N. Balacheff, The role of the researcher's epistemology in mathematics education: An essay on the case of proof, ZDM -The International Journal on Mathematics Education, vol.40, issue.3, pp.501-513, 2008.
URL : https://hal.archives-ouvertes.fr/hal-01675739

A. Baccaglini-frank and M. A. Mariotti, Generating conjectures in Dynamic Geometry: The maintaining-dragging model, International Journal of Computers for Mathematical Learning, vol.15, issue.3, pp.225-253, 2010.

, BUVM (Børne-og Undervisningsministeriet) (2019) Matematik Faelles Mål (Mathematics common aims

J. Connor, L. Moss, and B. Grover, Student evaluation of mathematical statements using dynamic geometry software, International Journal of Mathematical Education in Science and Technology, vol.38, issue.1, pp.55-63, 2007.

M. De-villiers, Proof in Dynamic geometry: More than Verification, 9th International Conference in Mathematics Education in the Global Community, 2007.

M. T. Edwards, S. R. Harper, D. C. Cox, J. Quinlan, and S. Phelps, Cultivating deductive thinking with angle chasing. The mathematics Teacher, vol.107, pp.426-431, 2014.

I. H. Højsted, Utilizing affordances of dynamic geometry environments to support students' development of reasoning competency, Nordic Studies in Mathematics Education, vol.25, issue.2, pp.71-98, 2020.

I. H. Højsted, Teachers Reporting on Dynamic Geometry Utilization Related to Reasoning Competency in Danish Lower Secondary School, Digital Experiences in Mathematics Education, vol.6, pp.91-105, 2020.

I. H. Højsted, Guidelines for the design of didactic sequences utilizing dynamic geometry affordances related to mathematical reasoning competency, Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education, 2019.

B. E. Jessen, C. Holm, and C. Winsløw, Matematikudredningen -udredning af den gymnasiale matematiks rolle og udviklingsbehov. København: Institut for Naturfagenes Didaktik, 2015.

C. Laborde, Integration of technology in the design of geometry tasks with cabri-geometry, International Journal of Computers for Mathematical Learning, vol.6, pp.283-317, 2001.

R. Lachmy and B. Koichu, The interplay of empirical and deductive reasoning in proving "if" and "only if" statements in a dynamic geometry environment, The Journal of Mathematical Behavior, vol.36, pp.150-165, 2014.

A. Leung, Discernment and Reasoning in Dynamic Geometry Environments, Selected Regular Lectures from the 12th International Congress on Mathematical Education, pp.451-469, 2015.

M. A. Mariotti, Proof and proving in the classroom: Dynamic Geometry Systems as tools of semiotic mediation, Research in Mathematics Education, vol.14, issue.2, pp.163-185, 2012.

R. Marrades and Á. Gutiérrez, Proofs produced by secondary school students learning geometry in a dynamic computer environment, Educational Studies in Mathematics, vol.44, pp.87-125, 2000.

M. Niss and T. Højgaard, Mathematical competencies revisited, Educational Studies in Mathematics, vol.102, issue.1, pp.9-28, 2019.

N. Sinclair and O. Robutti, Technology and the role of proof: The case of dynamic geometry, Third international handbook of mathematics education, pp.571-596, 2013.

A. Trocki, Evaluating and Writing Dynamic Geometry Tasks, The Mathematics Teacher, vol.107, issue.9, pp.701-705, 2014.

R. T. White and R. F. Gunstone, Probing Understanding, 2014.

K. E. Chin, Making sense of mathematics: supportive and problematic conceptions with special reference to trigonometry (Doctoral dissertation, 2013.

A. Collins, D. Joseph, and K. Bielaczyc, Design research: Theoretical and methodological issues, The Journal of the learning sciences, vol.13, issue.1, pp.15-42, 2004.

C. Kynigos, N. Essonnier, and J. Trgalova, Social creativity in the education sector: The case of collaborative design of resources in mathematics, Creativity Research Journal, vol.32, issue.1, pp.17-29, 2020.
URL : https://hal.archives-ouvertes.fr/hal-02489909

A. Disessa, Changing Minds: Computers, Learning and Literacy, 2001.
URL : https://hal.archives-ouvertes.fr/hal-00702974

I. E. Harel and S. E. Papert, , 1991.

B. Harvey, Computer Science Logo Style, 1997.

L. Healy and C. Kynigos, Charting the microworld territory over time: design and construction in mathematics education, ZDM, vol.42, issue.1, pp.63-76, 2010.

C. Hoyles and R. Noss, What can digital technologies take from and bring to research in mathematics education, Second international handbook of mathematics education, pp.323-349, 2003.

Y. B. Kafai, Playing and making games for learning: Instructionist and constructionist perspectives for game studies, Games and culture, vol.1, issue.1, pp.36-40, 2006.

Y. Kafai and M. Resnick, Constructionism in practice: designing, thinking and learning in a digital world, 1996.

C. Kynigos, Constructionism: Theory of learning or theory of design, Selected regular lectures from the 12th International Congress on Mathematical Education, pp.417-438, 2015.

C. Kynigos and M. Grizioti, Programming Approaches to Computational Thinking: Integrating Turtle Geometry, Dynamic Manipulation and 3D Space, Informatics in Education, vol.17, issue.2, pp.321-340, 2018.

S. Papert, Mindstorms: Children, computers, and powerful ideas, 1980.

H. C. Reggini, Ideas y formas: Explorando el espacio con Logo, 1985.

R. Akkus, B. Hand, and J. Seymour, Understanding students' understanding of functions, Mathematics Teaching, vol.207, pp.10-13, 2008.

F. Arzarello and C. Sabena, Analytic-structural functions of gestures in mathematical argumentation processes, Emerging perspectives on gesture and embodiment, pp.75-103, 2014.

F. Arzarello and C. Sabena, Introduction to the Approach of Action, Production, and Communication (APC), Networking of Theories as a Research Practice in Mathematics Education, 2014.

K. R. Bujak, I. Radu, R. Catrambone, B. Macintyre, R. Zheng et al., A psychological perspective on augmented reality in the mathematics classroom, Computers & Education, vol.68, pp.536-544, 2013.

L. D. Edwards, Gestures and conceptual integration in mathematical talk, Educational Studies in Mathematics, vol.70, pp.127-141, 2009.

A. Hoffkamp, The use of interactive visualizations to foster the understanding of concepts of calculus: Design principles and empirical results, ZDM-Mathematics Education, vol.43, pp.359-372, 2011.

H. Kaufmann, K. Steinbügl, A. Dünser, and J. Glück, General training of spatial abilities by geometry education in augmented reality, Annual Review of Cyber Therapy and Telemedicine: A Decade of VR, vol.3, pp.65-76, 2005.

M. Q. Patton, Qualitative research & evaluation methods, 2002.

L. Radford, L. Edwards, and F. Arzarello, Beyond words, Educational Studies in Mathematics, vol.70, pp.91-95, 2009.

P. W. Thompson and M. P. Carlson, Variation, covariation, and functions: Foundational ways of thinking mathematically, Compendium for research in mathematics education, pp.421-456, 2017.

L. Trouche, From artifact to instrument: mathematics teaching mediated by symbolic calculators, Interacting with Computers, vol.15, issue.6, pp.783-800, 2003.
URL : https://hal.archives-ouvertes.fr/hal-01540327

L. S. Vygotsky, Mind in society, 1978.

J. C. Yen, C. H. Tsai, and M. Wu, Augmented reality in the higher education: Students' science concept learning and academic achievement in astronomy, Procedia-social and behavioral sciences, vol.103, pp.165-173, 2013.

P. Barmby, T. Harries, S. Higgins, and J. Suggate, How can we assess mathematical understanding?, Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, vol.2, pp.41-48, 2007.

K. Charmaz, Constructing grounded theory: A practical guide through qualitative analysis, 2006.

J. Clement, The concept of variation and misconceptions in Cartesian graphing, Focus on Learning Problems in Mathematics, vol.11, issue.1-2, pp.77-87, 1989.

H. L. Johnson, E. Mcclintock, and P. Hornbein, Ferris wheels and filling bottles: A case of a student's transfer of covariational reasoning across tasks with different backgrounds and features, ZDM Mathematics Education, vol.49, issue.6, pp.851-864, 2017.

M. Lichti and J. Roth, How to foster functional thinking in learning environments using Computer-Based Simulations or Real Materials, Journal for STEM Education Research, vol.1, issue.1-2, pp.148-172, 2018.

F. Schlöglhofer, Vom Foto-Graph zum Funktions-Graph, Mathematik lehren, vol.103, pp.16-17, 2000.

H. Vollrath, Funktionales Denken. Journal für Mathematik-Didaktik, vol.10, issue.1, pp.3-37, 1989.

S. Vosniadou and X. Vamvakoussi, Examining mathematics learning from a conceptual change point of view, Instructional psychology, pp.55-70, 2006.

J. Zöchbauer and M. Hohenwarter, Developoing a collaboration tool to give every students a voice in a classroom discussion, Paper presented at the ERME Topic Conference ETC 7, 2020.

M. Ainley, Hands on with teaching Scratchmaths: Examining ScratchMaths as a vehicle for developing computational thinking -and making lessons effective. helloworld cc, vol.10, pp.19-21, 2019.

G. Z. Bereday, Reflections on comparative methodology in education, Comparative Education, vol.3, issue.3, pp.169-287, 1967.

S. Bocconi, A. Chioccariello, G. Dettori, A. Ferrari, and K. Engelhardt, Developing computational thinking in compulsory education -Implications for policy and practice, 2016.

K. Brennan and M. Resnick, New frameworks for studying and assessing the development of computational thinking, Proceedings of the 2012 annual meeting of the American Educational Research Association, 2012.

K. Bråting, C. Kilhamn, and L. Rolandsson, Presented at MADIF12, the twelfth research seminar of the Swedish Society for Research in Mathematics Education, 2020.

A. Mee, Computing in the school curriculum: a survey of 100 teachers, 2020.

M. Misfeldt and S. Ejsing-duun, Learning Mathematics through Programming: An Instrumental Approach to Potentials and Pitfalls, Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education, pp.2524-2530, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01289367

M. Niss and T. Højgaard, Mathematical competencies revisited, Educational Studies in Mathematics, vol.102, pp.9-28, 2019.

S. Papert, Mindstorms: Computers, children, and powerful ideas, 1980.

, Läroplan för grundskolan, förskoleklassen och fritidshemmet, Swedish National Agency for Education, 2018.

J. M. Wing, Computational thinking, Communications of the ACM, vol.49, issue.3, pp.33-35, 2006.

R. K. Yin, Applications of case study research, 2011.

I. Attorps, K. Björk, and M. Radic, Generating the patterns of variation with GeoGebra: the case of polynomial approximations, Int. J. Math. Educ. Sci. Technol, vol.47, issue.1, pp.45-57, 2016.

A. M. Breda and J. Santos, Complex functions with GeoGebra, Proceedings of ICTMT 12, pp.277-284, 2015.

D. Bressoud, I. Ghedamsi, V. Martinez-luaces, and G. Törner, Teaching and Learning of Calculus, 2016.

P. Demarois and D. Tall, Facets and Layers of the Function Concept, Proceedings of PME 20, pp.297-304, 1996.

R. Duval, A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics, Educational Studies in Mathematics, vol.61, pp.103-131, 2006.

A. S. González-martín, Introduction to the papers of TWG14: University mathematics education, Proceedings of CERME 10, pp.1953-1960, 2017.

J. Hattie and H. Timperley, The Power of Feedback, Review of Educational Research, vol.77, issue.1, pp.81-112, 2007.

M. Hohenwarter, , 2018.

A. Hoppenbrock, R. Biehler, and R. Hochmuth, Lehren und Lernen in der Studieneingangsphase, 2016.

I. U. Machromah, M. E. Purnomo, and C. K. Sari, Learning calculus with Geogebra at college, IOP Conf. Series: Journal of Physics: Conf. Series 1180, 2018.

F. Marton and S. Booth, Learning and Awareness, 1997.
URL : https://hal.archives-ouvertes.fr/hal-02092238

L. Moreno-armella, An essential tension in mathematics education, ZDM Mathematics Education, vol.46, pp.621-633, 2014.

C. N. Nobre, M. R. Meireles, N. Vieria-junior, M. N. Resende, L. E. Da-costa et al., The Use of Geogebra Software as a Calculus Teaching and Learning Tool, Informatics in Education, vol.15, issue.2, pp.253-267, 2016.

A. H. O'neil and H. M. Doerr, Using variation theory to design tasks to support students' understanding of logarithms, Proceedings of CERME 9, 2015.

J. Selden and A. Selden, Unpacking the logic of mathematical statements, Studies in Math, vol.29, pp.123-151, 1995.

S. Stavrou, Common errors and misconceptions in mathematical proving by education undergraduates, Issues Undergrad. Math. Prep. Sch. Teach, vol.1, 2014.

D. Tall, Using technology to support an embodied approach to learning concepts in mathematics, Historia e tecnologia no ensino da matematica, vol.1, pp.1-28, 2003.

D. Tall, D. Smith, and C. Piez, Technology and Calculus, Research on Technology and the Teaching and Learning of Mathematics, pp.207-258, 2008.

C. Winsløw, Analysis Teaching and Learning, Encyclopedia of Mathematics Education, 2018.

M. Anastasakis, C. L. Robinson, and S. Lerman, Links between students' goals and their choice of educational resources in undergraduate mathematics, Teaching Mathematics and Its Applications, vol.36, pp.67-80, 2017.

I. Biza, V. Giraldo, R. Hochmuth, A. Khakbaz, and C. Rasmussen, Research on teaching and learning mathematics at the tertiary level: State-of-the-art and looking ahead, 2016.

J. Malmqvist, K. K. Rådberg, U. ;. Lundqvist, and . China, Comparative analysis of challenge-based learning experiences, Proceedings of the 11th International CDIO Conference, 2015.

B. Pepin and G. Gueudet, Curriculum Resources and Textbooks in Mathematics Education, Encyclopedia of Mathematics Education, pp.1-5, 2018.
URL : https://hal.archives-ouvertes.fr/hal-02547059

B. Pepin and Z. J. Kock, Towards a better understanding of engineering students' use and orchestration of resources: Actual Student Study Paths, Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02422663

C. Rasmussen and O. Kwon, An inquiry-oriented approach to undergraduate mathematics, Journal of Mathematical Behavior, vol.26, pp.189-194, 2007.

M. A. Simon and R. Tzur, Explicating the role of mathematical tasks in conceptual learning: An elaboration of the hypothetical learning trajectory, Mathematical Thinking & Learning, vol.6, pp.91-104, 2004.

M. W. Apple, Do the standards go far enough? Power, policy, and practice in mathematics education, J. Res. Math. Educ, vol.23, issue.5, pp.412-431, 1992.

M. L. Balinski and H. P. Young, Fair Representation. Meeting the Ideal of One Man, One Vote, 2001.

C. Bokhove, Supporting Variation in Task Design Through the Use of Technology, Digital Technologies in Designing Mathematics Education Tasks. Potentials and Pitfalls, pp.239-257, 2017.

B. A. Bradberry, A Geometric View of Some Apportionment Paradoxes, Math. Mag, vol.65, issue.1, pp.3-17, 1992.

A. Brüning, Lernen mit einer digitalen Lernumgebung, MU, vol.54, issue.6, pp.19-26, 2008.

P. Daume, Finanz-und Wirtschaftsmathematik im Unterricht. Band 1 Zinsen, Steuern und Aktien, 2016.

H. Freudenthal, Weeding and Sowing. Preface to a Science of Mathematical Education, 1978.

M. Gauglhofer, Analyse der Sitzverteilungsverfahren bei Proportionalwahlen, 1988.

G. Heintz, Werkzeugkompetenzen. Kompetent mit digitalen Werkzeugen Mathematik betreiben. Menden: medienstatt, 2017.

H. Henn, Die Mathematik der Einkommenssteuer, Neue Materialien für einen realitätsbezogenen Mathematikunterricht, vol.3, pp.95-105, 2017.

J. R. Hitchner, Financial Valuation. Application and Models, 2017.

. Oecd, PISA 2015 Assessment and Analytical Framework: Science, Reading, Mathematic, Financial Literacy and Collaborative Problem Solving (rev, 2017.

J. Roozenbeek and S. Van-der-linden, Fake news game confers psychological resistance against online misinformation, Palgrave Communications, vol.5, 2019.

O. Skovsmose, Linking Mathematics Education and Democracy: Citizenship, Mathematical Archaeology, Mathemacy and Deliberative Interaction, ZDM, vol.30, issue.6, pp.195-203, 1998.

J. Sweller, Cognitive load theory, learning difficulty, and instructional design, Learn. Instr, vol.4, issue.4, pp.295-312, 1994.

A. Vohns, Einkommensteuertarife mit Schulmathematik erkunden -Ein Zugang aus bildungstheoretischer Perspektive, MU, vol.63, issue.3, pp.3-13, 2017.

H. Weigand, Competencies and Digital Technologies-Reflections on a complex Relationship, Proceedings of the 13th International Conference on Technology in Mathematics Teaching, pp.40-47, 2017.

H. Weigand and E. Bichler, Towards a competence model for the use of symbolic calculators, ZDM, vol.42, issue.7, pp.697-713, 2010.

H. Winter, Bürger und Mathematik, ZDM, vol.22, issue.4, pp.131-147, 1990.

H. Winter, Mathematikunterricht und Allgemeinbildung. M. GDM, vol.61, pp.37-46, 1995.

E. Wittmann and . Ch, Developing mathematics education in a systemic process, Educ. Stud. Math, vol.48, issue.1, pp.1-20, 2001.

F. Arzarello and C. Soldano, Approaching Proof in the Classroom Through the Logic of Inquiry, Compendium for Early Career Researchers in Mathematics Education. ICME-13 Monographs, 2019.

B. Barzel, T. Leuders, S. Prediger, and S. Hußmann, Designing tasks for engaging students in active knowledge organization, ICMI Study 22 on Task Design -Proceedings of Study Conference, pp.285-294, 2013.

J. Dewey, Logic: The theory of inquiry, The Later Works of John Dewey, vol.12, pp.1925-1953, 1938.

R. Duval, A cognitive analysis of problems of comprehension in a learning of mathematics, Educational studies in mathematics, vol.61, issue.1-2, pp.103-131, 2006.

H. Freudenthal, Mathematics as an educational task, 1973.

J. Hintikka, The principles of mathematics revisited, 1998.

J. Hintikka, Inquiry as inquiry: a logic of scientific discovery, 1999.

F. Schacht and O. Swidan, Exploring pre-calculus with augmented reality. A design-based-research approach, Proceedings of CERME 11-Eleventh Congress of the European Society for Research in Mathematics Education, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02428775

C. Soldano and F. Arzarello, Approaching secondary school geometry through the logic of inquiry within technological environments, Using Mobile Technology in the Teaching and Learning of Mathematics, Mathematics Education in the Digital Era, vol.12, pp.247-261, 2018.

C. Soldano and C. Sabena, Semiotic potential of inquiring-game activities, Proceedings of PME 43-International Group for the Psychology of Mathematics Education, Pretoria (Sudafrica), pp.7-12, 2019.

O. Swidan, F. Schacht, C. Sabena, M. Fried, J. El-sana et al., Engaging Students in Covariational Reasoning within an Augmented Reality Environment, Augmented Reality in Educational Settings, pp.147-167, 2019.

N. Balacheff, Aspects of proof in pupils' practice of school mathematics, Mathematics, teachers and children, pp.216-235, 1988.

C. Benedicto, C. Acosta, A. Gutiérrez, E. Hoyos, and A. Jaime, Improvement of gifted abilities in a 3d computer environment, 12th International Conference on Technology in Mathematics Teaching, pp.24-28, 2015.

D. H. Clements and J. Sarama, Learning trajectories in mathematics education, Mathematical Thinking and Learning, vol.6, issue.2, pp.81-89, 2004.

C. Dimitriadis, Developing mathematical giftedness within primary schools: A study of strategies for educating children who are gifted in mathematics (Doctoral dissertation), Brunel University School of Sport and Education, 2010.

G. Harel and L. Sowder, Students' proof schemes: Results from exploratory studies, Research in collegiate mathematics education, vol.7, pp.234-283, 1998.

A. Jaime and A. Gutiérrez, Investigación sobre estudiantes con alta capacidad matemática, Investigación en Educación Matemática XXI, pp.71-89, 2017.

A. Jaime, A. Gutiérrez, and C. Benedicto, Problemas con extensiones. Propuesta para estudiantes con alta capacidad matemática, vol.79, pp.7-14, 2018.

R. Marrades and A. Gutiérrez, Proofs produced by secondary school students learning geometry in a dynamic computer environment, Educational Studies in Mathematics, vol.44, issue.1-3, pp.87-125, 2000.

A. I. Sacristán, N. Calder, T. Rojano, M. Santos-trigo, A. Friedlander et al., The influence and shaping of digital technologies on the learning -and learning trajectories -of mathematical concepts, Mathematics education and technology, pp.179-226, 2010.

M. Simon, Hypothetical learning trajectories in mathematics education, Encyclopedia of mathematics education, pp.272-275, 2014.

M. Simon and R. Tzur, Explicating the role of mathematical tasks in conceptual learning: an elaboration of the hypothetical learning trajectory, Mathematical Thinking and Learning, vol.6, issue.2, pp.91-104, 2004.

N. Sinclair and O. Robutti, Technology and the Role of Proof: The Case of Dynamic Geometry, Third International Handbook of Mathematics Education, pp.571-596, 2013.

A. Bodin, What does to assess mean? The case of assessing mathematical knowledge, In Investigations into assessment in mathematics education, pp.113-141, 1993.

D. Checa and A. Bustillo, A review of immersive virtual reality serious games to enhance learning and training, Multimedia Tools and Applications, pp.1-27, 2019.

N. Herscovics and L. Linchevski, A cognitive gap between arithmetic and algebra, Educational Studies in Mathematics, vol.27, issue.1, pp.59-78, 1994.

U. T. Jankvist and M. Niss, A framework for designing a research-based "maths counsellor" teacher programme, Educational Studies in Mathematics, vol.90, issue.3, pp.259-284, 2015.

C. Kieran, Learning and teaching of algebra at the middle school through college levels: Building meaning for symbols and their manipulation, Second Handbook of Research on Mathematics Teaching and Learning, vol.2, pp.707-762, 2007.

C. Linsell, A Hierarchy of Strategies for Solving Linear Equations, Crossing divides: Proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia, vol.1, pp.331-338, 2009.

M. Otten, M. Van-den-heuvel-panhuizen, and M. Veldhuis, The balance model for teaching linear equations: a systematic literature review, International Journal of STEM Education, vol.6, issue.1, p.30, 2019.

S. E. Pirie and L. Martin, The equation, the whole equation and nothing but the equation! One approach to the teaching of linear equations, Educational Studies in Mathematics, vol.34, issue.2, pp.159-181, 1997.

S. Rhine, R. Harrington, and C. Starr, How Students Think When Doing Algebra, 2018.

M. Slater, Place illusion and plausibility can lead to realistic behaviour in immersive virtual environments, Philosophical Transactions of the Royal Society B: Biological Sciences, vol.364, pp.3549-3557, 1535.

J. Suh and P. S. Moyer, Developing students' representational fluency using virtual and physical algebra balances, Journal of Computers in Mathematics and Science Teaching, vol.26, p.155, 2007.

J. Vlassis, The balance model: Hindrance or support for the solving of linear equations with one unknown, Educational Studies in Mathematics, vol.49, issue.3, pp.341-359, 2002.

A. Bakker, Great Britain: British Library Cataloguing-in-Publication Data, 2019.

L. Cohen, L. Manion, and K. Morrison, Research Methods in Education, 2018.

P. Drijvers, Integrating Technology in Mathematics Education: Theoretical Perspectives, Mathematics Education and Technology-Rethinking the Terrain, vol.13, pp.89-132, 2009.

E. Faggiano, A. Montone, and A. Mariotti, Synergy between manipulative and digital artefacts: A teaching experiment on axial symmetry at primary school, International Journal of Mathematical Education in Science an Technology, vol.49, issue.8, pp.1165-1180, 2018.

A. Gutiérrez, Visualization in 3-dimensional geometry: In search of a framework, Proceedings of the 20th PME International Conference, vol.1, pp.3-19, 1996.

A. Gutiérrez, La investigación sobre enseñanza y aprendizaje de la geometría, Badajos, España: FESPM y SAEM Thales, pp.13-58, 2006.

S. Hershkovitz, Concrete manipulatives and Virtual Manipulatives: Learning Fractiones in 4 th Grade, vol.13, 2016.

D. Lieban, Exploring opportunities for connecting physical and digital resources for mathematics teaching and learning, 2019.

P. Rabardel, People and Technology -A Cognitive Approach to Contemporary Instruments. Francia: Université Paris, vol.12, 2002.
URL : https://hal.archives-ouvertes.fr/hal-01020705

N. Sinclair, Recent research on geometry education: an ICME-13 survey team report. ZDM Mathematics Education, vol.48, pp.691-719, 2016.

R. Vágová, Plane, physical or dynamic representations of 3D objects. Which do you prefer when teaching solid geometry?, Aplimat 2019: Proceedings of the 18 th conference on applied mathematics, pp.1237-1251, 2019.

R. Vágová, M. Kmetová, and D. Lieban, The interplay of physical and digital resources to explore cube cross-section, Aplimat 2020: Proceedings of the 19 th conference on applied mathematics, pp.1067-1081, 2020.

A. Arcavi, The role of visual representations in the learning of mathematics, Educational Studies in Mathematics, vol.52, issue.3, pp.215-241, 2003.

A. Barbosa and I. Vale, Math trails: a resource for teaching and learning, Proceedings of the Re(s)sources 2018 International Conference, pp.183-186, 2018.

B. Barnbaum, The art of photography: An approach to personal expression, 2010.

L. Bragg and C. Nicol, Seeing mathematics through a new lens: using photos in the mathematics classroom, Australian Mathematics Teacher, vol.67, issue.3, pp.3-9, 2011.

S. Brown and M. Walter, The art of problem posing, 2005.

F. Erickson, Qualitative methods in research on teaching, Handbook of research on teaching, pp.119-161, 1986.

C. Godfrey, The Board of Education Circular on the Teaching of Geometry, Mathematical Gazette, vol.5, pp.195-200, 1910.

E. Gutstein, Reading and Writing the World with Mathematics: Toward a Pedagogy for Social Justice, 2006.

P. Kenderov, A. Rejali, M. Bartolini-bussi, V. Pandelieva, K. Richter et al., Challenges Beyond the Classroom-Sources and Organizational Issues, Challenging Mathematics In and Beyond the Classroom -New ICMI Study Series, vol.12, pp.53-96, 2009.

R. Leikin, Exploring mathematical creativity using multiple solution tasks, Creativity in mathematics and the education of gifted students, pp.129-145, 2009.

A. Meier, M. Hannula, and M. Toivanen, Mathematics and outdoor photography experience -exploration of an approach to mathematical education, based on the theory of Dewey's aesthetics, LUMAT, International Journal on Math, Science and Technology Education, vol.8, issue.2, pp.146-166, 2018.

M. Munakata and A. Vaidya, Encouraging creativity in mathematics and science through photography, Teaching Mathematics and its Applications, vol.31, pp.121-132, 2012.

, Principles To Actions: Ensuring Mathematical Success for All, 2014.

E. Silver, Fostering creativity through instruction rich in mathematical problem solving and problem posing, ZDM, vol.3, pp.75-80, 1997.

M. Smith and M. K. Stein, Five practices for orchestrating productive mathematics discussions, 2011.

E. Stoyanova, Problem posing in mathematics classrooms, Research in Mathematics Education: a contemporary perspective, pp.164-185, 1998.

P. Sullivan and P. Liburn, Good questions for math teaching, 2002.

I. Vale and A. Barbosa, A fotografia na aula de matemática: uma experiência promotora de conexões, 2019.

I. N. Amado, A. P. Canavarro, S. Carreira, and R. T. Ferreira, , pp.183-186

I. Vale, T. Pimentel, and A. Barbosa, The power of seeing in problem solving and creativity: an issue under discussion, Broadening the scope of research on mathematical problem solving: A focus on technology, creativity and affect, pp.243-272, 2018.

C. Batanero, V. Navarro-pelayo, and J. D. Godino, Effect of the Implicit Combinatorial Model on Combinatorial Reasoning in Secondary School Pupils, Educational Studies in Mathematics, vol.32, issue.2, pp.181-199, 1997.

D. H. Clements and S. Mcmillen, Rethinking concrete manipulatives, Teaching Children Mathematics, vol.2, issue.5, pp.270-279, 1996.

J. Dinkelaker and M. Herrle, Erziehungswissenschaftliche Videographie: Eine Einführung, 2009.

L. D. English, Children's strategies for solving two-and three-dimensional combinatorial problems, Stepping Stones for the 21st Century. Australasian Mathematics Education Research, pp.139-156, 2007.

K. Höveler, Das Lösen kombinatorischer Anzahlbestimmungsprobleme: Eine Untersuchung zu den Strukturierungs-und Zählstrategien von Drittklässlern (Doctoral Dissertation, 2014.

K. Höveler, Children's Combinatorial Counting Strategies and their Relationship to Conventional Mathematical Counting Principles, Teaching and Learning Discrete Mathematics Worldwide: Curriculum and Research, pp.81-92, 2018.

K. Höveler and J. Winzen, Design-Prinzipien zur Entwicklung eines digitalen Arbeitsmittels zur Kombinatorik, Beiträge zum Mathematikunterricht 2020

T. Huhmann, . K. Höveler, and K. Eilerts, Counteracting representational volatility in geometry class through use of a digital aid: results form a qualitative comparative study on the use of pentomino learning environments in primary school, International Symposium Elementary Maths Teaching: SEMT'19: Opportunities in Learning and Teaching Elementary Mathematics, pp.211-220, 2019.

R. P. Hunting, Clinical Interview Methods in Mathematics Education Research and Practice, Journal of Mathematical Behavior, vol.16, issue.2, pp.145-165, 1997.

E. Lockwood, A Set-Oriented Perspective on Solving Counting Problems, For the Learning of Mathematics, vol.34, issue.2, pp.31-37, 2014.

W. A. Parnell and J. Bartlett, iDocument: How Smartphones and Tablets are Changing Documentation in Preschool and Primary Classrooms, Young Children, vol.67, issue.3, pp.50-59, 2012.

S. Prediger and L. Zwetzschler, Topic-specific design research with a focus on learning processes: The case of understanding algebraic equivalence in grade 8, Educational design research. Part B: Illustrative cases, pp.407-424, 2013.

W. Schipper, Handbuch für den Mathematikunterricht an Grundschulen, 2009.

A. Schulz and D. Walter, Practicing place value': How children interpret and use virtual representations and features, Proceedings of the CERME 11, pp.2941-2948, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02428823

J. Winzen and K. Höveler, Die iOS App "Notizen, Ein digitales Arbeitsmittel zum Lösen von Kombinatorikaufgaben. Die Grundschulzeitschrift, vol.320, pp.28-32, 2020.

B. Wollring, Zur Kennzeichnung von Lernumgebungen für den Mathematikunterricht in der Grundschule, Lernumgebungen auf dem Prüfstand. Zwischenergebnisse aus den Forschungsprojekten, pp.9-26, 2006.

A. Bakker, An Introduction to Design-Based Research with an Example From Statistics Education, 2014.

, Mathematics Education and Technology -Rethinking the Terrain. The 17 th ICMI Study, 2010.

H. Weigand, Looking back and ahead-didactical implications for the use of digital technologies in the next decade, Teaching Mathematics and Its Applications, vol.33, issue.1, pp.3-5, 2014.

A. Donevska-todorova and D. Lieban, Fostering Heuristic Strategies in Mathematical Problem Solving with Virtual and Tangible Manipulatives, the Proceedings of the tenth ERME TC Mathematica Education in the Digital Age (MEDA), p.2020, 2020.

A. Donevska-todorova, Problemorientiertes Lernen zur Leitidee Raum und Form mit Hilfe von DGS und 3D Druck Technologie. Beiträge zum Mathematikunterricht. Vorträge auf den 54, 2020.

A. Flores, J. Park, and S. A. Bernhardt, Interactive Technology to Foster Creativity in Future Mathematics Teachers, Creativity and Technology in Mathematics Education. Mathematics Education in the Digital Era, vol.10, 2018.

S. Lee and R. Carpenter, Creative Thinking for 21st Century Composing Practices, Across the Disciplines, vol.12, 2015.

R. Leikin and B. Sriraman, Creativity and giftedness: interdisciplinary perspectives from mathematics and beyond, 2017.

A. E. Kelly, R. A. Lesh, and J. Y. Baek, Handbook of design research methods in education, Technology, Engineering and Mathematics Learning and Teaching, 2008.

A. Sánchez, V. Font, and A. Breda, Secondary school preservice teachers' references to the promotion of creativity in their master's degree final projects, Proceedings of CERME 11, vol.8, 2019.

F. M. Singer and C. Voica, When Mathematics Meets Real Objects: How Does Creativity Interact with Expertise in Problem Solving and Posing, Creativity and Giftedness, pp.75-103, 2017.

V. Gopalan, J. A. Bakar, A. N. Zulkifli, and A. Alwi, A review of augmented reality elements in science learning, Journal of Telecommunication, Electronic and Computer Engineering, vol.10, issue.1-10, pp.87-92, 2018.

Á. Martínez-sevilla, C. Ureña, and T. Recio, Augmented Reality , Maths Walks and GeoGebra, 2018.

C. Rossi, Egyptian Architecture and Mathematics, Handbook of the Mathematics of the Arts and Sciences, pp.1-12, 2018.

J. Szul?yk-cieplak, A. Duda, and B. Sidor, 3D Printers -New Possibilities in Education, Advances in Science and Technology Research Journal, vol.8, issue.24, pp.96-101, 2014.

. References-geogebra,

E. V. Laski, J. R. Jor'dan, C. Daoust, and A. K. Murray, What makes mathematics manipulatives effective? Lessons from cognitive science and Montessori education, SAGE Open, vol.5, issue.2, p.2158244015589588, 2015.

K. Orton, D. Weintrop, E. Beheshti, M. Horn, K. Jona et al., Bringing Computational Thinking Into High School Mathematics, Proceedings of the 12 th International Conference of the Learning Sciences, pp.705-712, 2016.

. Ris, Rechtsinformationssystem des Bundes. Bundesrecht konsolidiert: Lehrplan neue Mittelschule, 2020.

D. Weintrop, E. Behesti, M. Horn, K. Orton, J. Kemi et al., Defining Computational Thinking for Mathematcis and Science Classrooms, 2015.

R. Hershkowitz, B. B. Schwarz, and T. Dreyfus, Abstraction in context: Epistemic actions, Journal for Research in Mathematics Education, vol.32, pp.195-222, 2001.

I. Kohanová and T. Solstad, Linear figural patterns as a teaching tool for preservice elementary teachers -the role of symbolic expressions. 11th CERME, 2019.

R. Olenburg, Die Mathematik der Bildverarbeitung, Materialien für einen realitätsbezogenen Mathematikunterricht, vol.9, 2006.

P. Sari, A. Hadiyan, A. , and D. , Exploring derivatives by means of GeoGebra, International Journal on Emerging Mathematics Education, vol.2, issue.1, pp.65-78, 2018.

J. R. Stahley, Students' Qualitative Understanding of the Relationship Between the Graph of a Function and the Graphs of Its Derivatives (Doctoral dissertation, 2011.

G. Van-der-hoek, Hellinggrafiek schetsen. Wiswise, 2019.

P. Vos, N. D. Braber, G. Roorda, and M. J. Goedhart, Hoe begrijpen en gebruiken docenten van de schoolvakken natuurkunde, scheikunde en economie het wiskundige concept 'afgeleide'. Tijdschrift voor Didactiek der Bètawetenschappen, vol.27, pp.37-62, 2010.

C. M. Chituc, M. Herrmann, D. Schiffner, and M. Rittberger, Towards the Design and Deployment of an Item Bank: An Analysis of the Requirements Elicited, ICWL 2019. LNCS 11841, 2019.

B. Csapó, G. Molnár, and K. Toth, Comparing Paper-and-Pencil and Online Assessment of Reasoning Skills. A Pilot Study for Introducing Electronic Testing in Large-scale Assessment in Hungary. The Transition to Computer-Based Assessment, 2009.

P. Drijvers, Digital assessment of mathematics: Opportunities, issues and criteria. Mesure et Évaluation En Éducation, vol.41, pp.41-66, 2018.

P. Drijvers, L. Ball, B. Barzel, M. K. Heid, Y. Cao et al., Uses of Technology in Lower Secondary Mathematics Education: A Concise Topical Survey, 2016.

J. Fraillon, J. Ainley, W. Schulz, T. Friedman, and D. Duckworth, Preparing for Life in a Digital World: IEA International Computer and Information Literacy Study, 2018.

E. Geraniou and U. T. Jankvist, Towards a definition of "mathematical digital competency, Educational Studies in Mathematics, vol.102, issue.1, pp.29-45, 2019.

F. Goldhammer and F. Zehner, What to Make Of and How to Interpret Process Data, Measurement: Interdisciplinary Research and Perspectives, vol.15, issue.3-4, pp.128-132, 2017.

K. Hoogland and D. Tout, Computer-based assessment of mathematics into the twenty-first century: pressures and tensions, ZDM Mathematics Education, vol.50, pp.675-686, 2018.

, VERA -An overview. Institute for Educational Quality Improvement, 2020.

V. Kaptelinin, I. Affordances, and . Lowgren, The Encyclopedia of Human-Computer Interaction, vol.2020

. Kmk, Bildungsstandards im Fach Mathematik für den Mittleren Schulabschluss, 2003.

U. Kortenkamp and J. Richter-gebert, Using Automatic Theorem Proving to Improve the Usability of Geometry Software, Proceedings of MathUI, p.13, 2004.

Z. Kovács, T. Recio, and M. P. Vélez, Using Automated Reasoning Tools in GeoGebra in the Teaching and Learning of Proving in Geometry, International Journal for Technology in Mathematics Education, vol.25, issue.2, pp.33-50, 2018.

T. Pelkola, A. Rasila, and C. Sangwin, Investigating Bloom's Learning for Mastery in Mathematics with Online Assessment, Informatics in Education, vol.17, issue.2, pp.363-380, 2018.

H. Rölke, National educational standards in Germany: Methodological challenges for developing and calibrating standards-based tests, Proceedings of world conference on E-Learning in corporate, government, healthcar, and higher education 2012, pp.173-198, 2007.

C. Sangwin, C. Cazes, A. Lee, and K. L. Wong, Mathematics Education and Technology-Rethinking the Terrain: The 17th ICMI Study, 2010.

C. J. Sangwin and I. Jones, Asymmetry in student achievement on multiple-choice and constructed-response items in reversible mathematics processes, Educational Studies in Mathematics Education, vol.94, pp.205-222, 2017.

K. Stacey and D. Wiliam, Technology and Assessment in Mathematics, Third International Handbook of Mathematics Education, pp.721-754, 2013.

M. Artigue, Connecting and integrating theoretical frames: The TELMA contribution, International Journal of Computers for Mathematical Learning, vol.14, pp.217-240, 2009.

K. Bokhove and P. Drijvers, Digital tools for algebra education: Criteria and evaluation, International Journal of Mathematics Learning, vol.15, pp.45-62, 2010.

P. Filho and C. Mercat, Teaching computational thinking in classroom environments: A case for unplugged scenario. Conference: Re(s)sources 2018 -Understanding Teachers' Work Through Their Interactions with Resources for Teaching, 2018.

S. Hadjerrouit and H. H. Gautestad, Evaluating the usefulness of the visualization tool SimReal+ for learning mathematics: A case study at the undergraduate level, Learning Technologies for Transforming Large-Scale Teaching, Learning, and Assessment, pp.71-89, 2019.

A. G. Springer-nature-switzerland,

M. Misfeldt and S. Ejsing-duun, Learning mathematics through programming: An instrumental approach to potentials and pitfalls, Proceedings of CERME9, vol.25, pp.27-35, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01289367

M. Q. Patton, D. Pérez-marín, R. Hijón-neira, A. Bacelo, and C. Pizarro, Can computational thinking be improved by using a methodology based on metaphors and scratch to teach computer programming to children?, Computers in Human Behavior, pp.1-10, 2002.

D. Topalli and N. E. Cagiltay, Improving programming skills in engineering education through problem-based game projects with Scratch, Computers & Education, vol.120, pp.64-74, 2018.

V. J. Shute, C. Sun, and J. Clarke, Demystifying computational thinking, Educational Research Review, pp.1-10, 2017.

J. M. Wing, Computational thinking and thinking about computing, Philosophical Transactions of the Royal Society A, vol.366, pp.3717-3725, 1881.

J. M. Wing, Computational thinking benefits society. Social Issues in Computing, 40th Anniversary Blog, 2009.

L. Alcock, D. Ansari, S. Batchelor, M. Bisson, B. De-smedt et al., Challenges in mathematical cognition: A collaboratively-derived research agenda, Journal of Numerical Cognition, vol.2, issue.1, pp.20-41, 2016.

Y. Attali and F. Van-der-kleij, Effects of feedback elaboration and feedback timing during computer-based practice in mathematics problem solving, Computers and Education, vol.110, pp.154-169, 2017.

E. Fischbein, Tacit models and mathematical reasoning, For the Learning of Mathematics, vol.9, pp.9-14, 1989.

P. Iannone, M. Inglis, J. P. Mejía-ramos, A. Simpson, and K. Weber, Does generating examples aid proof production, Educational Studies in Mathematics, vol.77, issue.1, pp.1-14, 2011.

P. Iannone and A. Simpson, Mapping University Mathematics Assessment Practices, 2012.

G. Kinnear, Delivering an online course using STACK, Contributions to the 1st International STACK, 2018.

F. K. Lester, On the theoretical, conceptual, and philosophical foundations for research in mathematics education, ZDM -International Journal on Mathematics Education, vol.37, issue.6, pp.457-467, 2005.

J. P. Mejia-ramos, E. Fuller, K. Weber, K. Rhoads, and A. Samkoff, An assessment model for proof comprehension in undergraduate mathematics, Educational Studies in Mathematics, vol.79, issue.1, pp.3-18, 2012.

C. J. Sangwin, New opportunities for encouraging higher level mathematical learning by creative use of emerging computer aided assessment, International Journal of Mathematical Education in Science and Technology, vol.34, issue.6, pp.813-829, 2003.

C. J. Sangwin, Computer aided assessment of mathematics, 2013.

C. Sangwin, Reasoning by Equivalence: The Potential Contribution of an Automatic Proof Checker, Proof Technology in Mathematics Research and Teaching. Mathematics Education in the Digital Era, vol.14, pp.313-330, 2019.

C. J. Sangwin and N. Köcher, Automation of mathematics examinations, Computers & Education, vol.94, pp.215-227, 2016.

,

V. J. Shute, Focus on Formative Feedback, Review of Educational Research, vol.78, issue.1, pp.153-189, 2008.

W. J. Sutherland, E. Fleishman, M. B. Mascia, J. Pretty, and M. A. Rudd, Methods for collaboratively identifying research priorities and emerging issues in science and policy, Methods in Ecology and Evolution, 2011.

A. Watson and J. Mason, Mathematics as a Constructive Activity, 2006.

F. References-drasgow, Technology and Testing: Improving Educational and Psychological Measurement, 2015.

A. Lemmo, M. Mariotti, and A. , From paper and pencil-to Computer-based assessment: some issues raising in the comparison, Proceedings of the 10th Conference of European Research in Mathematics Education, pp.3540-3547, 2017.

P. Nichols and R. Kirkpatrick, Comparability of the computer-administered tests with existing paper-and-pencil tests in reading and mathematics tests, Annual Meeting of the American Educational Research Association, 2005.

M. Ripley, The transition to computer-based assessment, pp.92-98, 2009.

M. Russell and W. Haney, Testing writing on computers: An experiment comparing student performance on tests conducted via computer and via paper-andpencil. Education policy analysis archives, vol.5, pp.1-18, 1997.

A. H. Schoenfeld, Mathematical problem solving, 1985.

W. D. Way, L. L. Davis, and S. Fitzpatrick, Practical questions in introducing computerized adaptive testing for K-12 assessments, pp.1-20, 2005.

F. D. Davis, Perceived Usefulness, Perceived Ease of Use, and User Acceptance of Information Technology, MIS Quarterly, vol.13, issue.3, pp.319-340, 1989.

L. S. Feldt, Estimating the Reliability of a Test Battery Composite or a Test Score Based on Weighted Item Scoring. Measurement and Evaluation in Counseling and Development, vol.37, pp.184-191, 2004.

J. Hattie and H. Timperley, The Power of Feedback, Review of Educational Research, vol.77, pp.81-112, 2007.

K. Hoogland and D. Tout, Computer-based assessment of mathematics into the twenty-first century: pressures and tensions, ZDM, vol.50, issue.4, pp.675-686, 2018.

N. Kocher and C. Sangwin, Automation of mathematics examinations, Computers & Education, vol.94, pp.215-227, 2016.

M. Mellone, M. Ribeiro, A. Jakobsen, G. Carotenuto, P. Romano et al., Mathematics teachers' interpretative knowledge of students' errors and nonstandard reasoning, pp.1479-4802, 2020.

N. Movshovitz-hadar, O. Zaslavsky, and S. Inbar, An Empirical Classification Model for Errors in High School Mathematics, Journal for Research in Mathematics Education, vol.18, issue.1, pp.3-14, 1987.

J. Parkes, Reliability in Classroom Assessment, SAGE Handbook of Research on Classroom Assessment. SZAGE, pp.107-124, 2012.

K. Rakoczy, B. Harks, E. Klieme, W. Blum, and J. Hochweber, Written feedback in mathematics: Mediated by students' perception, moderated by goal orientation. Learning and Instruction, vol.27, pp.63-73, 2013.

C. Sangwin, , 2013.

J. Threlfall, P. Pool, M. Homer, and B. Swinnerton, Implicit aspects of paper and pencil mathematics assessment that come to light through the use of the computer, Educational Studies in Mathematics, vol.66, issue.3, pp.335-348, 2007.

A. Singh, S. Karayev, K. Gutowski, P. ;. Abbeel, R. C. Feskens et al., Effects of Feedback in a Computer-Based Learning Environment on Students' Learning Outcomes: A Meta-Analysis, Proceedings L@S '17. van der Kleij, vol.85, pp.475-511, 2015.

J. R. Anderson, Language, memory and thought, 1976.

D. Boud and E. Molloy, Rethinking models of feedback for learning: The challenge of design, Assessment & Evaluation in Higher Education, vol.38, issue.6, pp.698-712, 2013.

P. Brusilovsky and E. Millán, User Models for Adaptive Hypermedia and Adaptive Educational Systems, The Adaptive Web (Bd. 4321, pp.3-53, 2007.

T. De-jong and M. G. Ferguson-hessler, Types and qualities of knowledge, Educational Psychologist, vol.31, issue.2, pp.105-113, 1996.

R. Duval, Representation, Vision and Visualization: Cognitive Functions in Mathematical Thinking. Basic Issues for Learning, Proceedings of the 21th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education: Bd. I, pp.3-26, 1999.

M. Fahlgren, M. Brunström, F. Dilling, B. Kristinsdóttir, .. G. Pinkernell et al., Technology-rich assessment in mathematics, 2019.

A. Fricke, Operative Lernprinzipien im Mathematikunterricht der Grundschule, 1970.

J. Hattie, H. Timperley, G. Kent, and C. Foster, Re-conceptualising conceptual understanding in mathematics, Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education, vol.77, pp.2656-2661, 2007.

J. Hiebert and P. Lefevre, Conceptual and procedural knowledge in mathematics: An introductory analysis, Conceptual and procedural knowledge: The case of mathematics, pp.1-27, 1986.

G. Kent and C. Foster, Re-conceptualising conceptual understanding in mathematics, Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education, pp.2656-2661, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01289445

C. I. Johnson and H. A. Priest, The feedback principle in multimedia learning, pp.449-463, 2005.

A. N. Kluger and A. Denisi, The Effects of Feedback Interventions on Performance: A Historical Review, a Meta-Analysis, and a Preliminary Feedback Intervention Theory, Psychological Bulletin, vol.119, pp.254-284, 1996.

E. H. Mory, Feedback research revisited, Handbook of research on educational communications and technology, pp.745-784, 2004.

L. Erlbaum,

S. Narciss, Feedback Strategies for Interactive Learning Tasks, Handbook of Research on Educational Communications and Technology, pp.125-144, 2008.

G. Pinkernell, Conceptualising knowledge of mathematical concepts or procedures for diagnostic and supporting measures at university entry level, Proceedings of the 11th Congress of European Research in Mathematics Education, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02422666

G. Pinkernell, L. Gulden, and M. Kalz, Automated Feedback at task level: Error analysis or worked out examples-Which type is more effective?, Proceedings of the 14th International Conference on Technology in Mathematics Teaching

E. Ras, D. Whitelock, and M. Kalz, Measuring and visualizing learning in the information-rich classroom, pp.21-40, 2016.

A. Renkl, Worked-out examples: Instructional explanations support learning by self-explanations, Learning and Instruction, vol.12, issue.5, pp.529-556, 2002.

B. Rittle-johnson and J. R. Star, Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations, Journal of Educational Psychology, vol.99, issue.3, pp.561-574, 2007.

E. Rosch, Prototype Classification and Logical Classification: The Two Systemp, New trends in conceptual representation: Challenges to Piaget's theory? (p, pp.73-86, 1983.

C. J. Sangwin, Computer Aided Assessment of Mathematics, 2013.

V. J. Shute, Focus on Formative Feedback, Review of Educational Research, vol.78, issue.1, pp.153-189, 2008.

K. Stacey and D. Wiliam, Technology and Assessment in Mathematics, Third International Handbook of Mathematics Education, pp.721-751, 2012.

S. Tacoma, S. Sosnovsky, P. Boon, J. Jeuring, and P. Drijvers, The Interplay between Inspectable Student Models and Didactics of Statistics, Digital Experiences in Mathematics Education, vol.4, issue.2-3, pp.139-162, 2018.

D. Tall and M. Bakar, Students' mental prototypes for functions and graphs, International Journal of Mathematical Education in Science and Technology, vol.23, issue.1, pp.39-50, 1992.

F. M. Van-der-kleij, R. C. Feskens, and T. J. Eggen, Effects of Feedback in a Computer-Based Learning Environment on Students' Learning Outcomes: A Meta-Analysis, Review of Educational Research, vol.85, issue.4, pp.475-511, 2015.

R. Vom-hofe and W. Blum, Grundvorstellungen" as a Category of Subject-Matter Didactics, Journal Für Mathematik-Didaktik, vol.37, issue.S1, pp.225-254, 2016.

M. Fahlgren, Designing for the integration of dynamic software environments in the teaching of mathematics, Dissertation, 2015.

S. Olsher, M. Yerushalmy, and D. Chazan, How Might the Use of Technology in Formative Assessment Support Changes in Mathematics Teaching? For the learning of mathematics, vol.36, pp.11-18, 2016.

F. Rønning, Influence of computer-aided assessment on ways of working with mathematics. Teaching Mathematics and its Applications, vol.36, pp.94-107, 2017.

C. Sangwin, Computer aided assessment of mathematics using STACK, Selected regular lectures from the 12th international congress on mathematical education, pp.695-713, 2015.

H. U. Hoppe, R. De-groot, and R. Hever, Implementing technology-facilitated collaboration and awareness in the classroom, Transformation of Knowledge through Classroom Interaction, pp.130-142, 2009.

B. Kristinsdóttir, F. Hreinsdóttir, and Z. Lavicza, Using silent video tasks for formative assessment, Proceedings of the 14th International Conference on Technology in Mathematics Teaching-ICTMT 14, pp.189-196, 2020.

D. Wiliam, Embedded Formative Assessment, 2011.

D. Wright, J. Clark, and L. Tiplady, Designing for formative assessment: A toolkit for teachers, Classroom assessment in mathematics: Perspectives from around the globe, pp.207-228, 2018.

, Mathematics Education, pp.2042-2049, 2007.

I. &. Berget and O. Bolstad, Perspektiv på matematisk modellering i Kunnskapsløftet og Fagfornyinga. Nordisk tidsskrift for utdanning og praksis, 2019.

P. Blanchard, R. L. Devaney, and G. R. Hall, Differential equations, 2002.

M. Blomhøj and T. H. Jensen, Developing mathematical modelling competence: Conceptual clarification and educational planning, Teaching Mathematics and its Appl, vol.22, issue.3, pp.123-139, 2003.

P. ;. Cobb, R. Lehrer, and . Van-oers, Modeling, symbolizing and tool use in statistical data analysis, Symbolizing, modelling and tool use in mathematics education, pp.171-195, 2002.

P. Drijvers and K. Gravemeijer, Computer Algebra as an Instrument, The Didactical Challenge of Symbolic Calculators, pp.163-196, 2005.

K. Gravemeijer, M. ;. Stephan, R. Lehrer, and . Van-oers, Emergent models as an instructional design heuristic, Symbolizing, modelling and tool use in mathematics education, pp.145-169, 2002.

M. Ju and O. N. Kwon, Ways of talking and ways of positioning: Students' beliefs in an inquiry-oriented differential equations class, Journal of Mathematical Behavior, vol.26, pp.267-280, 2007.

S. Kvale, Interview. En introduktion til det kvalitative forskningsinterview. Hans Reitzels Forlag, 2001.

J. Lithner, A research framework for creative and imitative reasoning, Educational Studies of Mathematics, vol.67, pp.255-276, 2008.

M. Niss, ICME-13 Monographs, Springer We can see that it increases with 3 cm 2 for every cm AB increases with 1. That is the relationship. Height of AB is 1, then the area of the rectangle is 3, The impact of Dutch mathematics education on Danish mathematics education. In: Heuvel-Panhuizen, M. van den, pp.317-324, 2020.

, Clara But what if it changes all the time

, This is also wrong, she quits. Finally, she writes x=ab3. Clara writes f(x)=, then she googles 'function equation', turns back to GeoGebra and writes f(x)=1x+3, looks at the graph done by GeoGebra and deletes it, Dea begins to write in the input field, x=, then x=3, instead of an equation

P. Drijvers, L. Ball, B. Barzel, M. K. Heid, Y. Cao et al., Uses of Technology in Lower Secondary Mathematics Education -A Concise Topical Survey, 2016.

R. Duval, Understanding the Mathematical Way of Thinking -The Registers of Semiotic Representation, 2017.

D. Guin and L. Trouche, The complex process of converting tools into mathematical instruments: The case of calculators, International Journal of computers for Mathematical Learning, vol.3, issue.3, pp.195-227, 1998.
URL : https://hal.archives-ouvertes.fr/hal-01538696

H. L. Johnson and E. Mcclintock, A link between students' discernment of variation in unidirectional change and their use of quantitative variational reasoning, Educational Studies in Mathematics, vol.97, pp.299-316, 2018.

H. Jungwirth, Everyday computer-based maths teaching: The predominance of practical activities, Proceedings30th conference of the International Gorup for the Psychology of Mathematics Education, vol.3, pp.377-384, 2006.

M. Niss and T. Højgaard, Mathematical competencies revisited, Educational Studies in Mathematics, vol.102, issue.1, pp.9-28, 2019.

C. Buteau, G. Gueudet, E. Muller, J. Mgombelo, and A. I. Sacristán, University students turning computer programming into an instrument for 'authentic' mathematical work, International Journal of Mathematical Education in Science and Technology, pp.1-22, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02528599

P. Drijvers and K. Gravemeijer, The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument, pp.163-196, 2005.

C. Hoyles and R. Noss, A computational lens on design research, ZDM, vol.47, issue.6, pp.1039-1045, 2015.

M. Mladenovi?, I. Boljat, and ?. ?anko, Comparing loops misconceptions in block-based and text-based programming languages at the K-12 level. Education Information Technologies, vol.23, pp.1483-1500, 2018.

R. Noss, Constructing a conceptual framework for elementary algebra through Logo programming, Educational Studies in Mathematics, vol.17, issue.4, pp.335-357, 1986.

S. Papert, Mindstorms: children, computers, and powerful ideas, 1980.

P. Rabardel, People and technology -a cognitive approach to contemporary instruments, 2002.
URL : https://hal.archives-ouvertes.fr/hal-01020705

M. Romero, A. Lepage, and B. Lille, Computational thinking development through creative programming in higher education, International Journal of Educational Technology in Higher Education, vol.14, issue.42, pp.1-15, 2017.

G. Vergnaud, A comprehensive theory of representation for mathematics education, The Journal of Mathematical Behavior, vol.17, issue.2, pp.167-181, 1998.

L. Benton, P. Saunders, I. Kalas, C. Hoyles, and R. Noss, Designing for learning mathematics through programming: A case study of pupils engaging with place value, International journal of child-computer interaction, vol.16, pp.68-76, 2018.

C. Buteau, G. Gueudet, E. Muller, J. Mgombelo, and A. Sacristán, University Students Turning Computer Programming into an Instrument for 'Authentic' Mathematical Work, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02528599

C. Buteau, E. Muller, and B. Ralph, Integration of programming in the undergraduate math program at Brock University, Proceedings of Math+Coding Symposium, 2015.

C. Buteau, A. I. Sacristán, and E. Muller, Roles and Demands in Constructionist Teaching of, Computational Thinking in University Mathematics. Constructivist Foundations, vol.14, issue.3, pp.294-309, 2019.

P. Drijvers, M. Doorman, P. Boon, H. Reed, and K. Gravemeijer, The teacher and the tool: instrumental orchestrations in the technology-rich mathematics classroom, Educational Studies in Mathematics, vol.75, issue.2, pp.213-234, 2010.

C. Hoyles and R. Noss, Learning Mathematics & Logo, 1992.

S. Papert, Mindstorms: Children, computers, and powerful ideas, 1980.

P. Rabardel, Les hommes et les technologies; approche cognitive des instruments contemporains, 1995.
URL : https://hal.archives-ouvertes.fr/hal-01017462

L. Trouche, Managing complexity of human/machine interactions in computerized learning environments: Guiding students' command process through instrumental orchestrations, vol.IJCML, pp.281-307, 2004.

G. Vergnaud, Toward a cognitive theory of practice, Mathematics education as a research domain: A search for identity, pp.227-241, 1998.

A. Baccaglini-frank, Dragging, instrumented abduction and evidence, in processes of conjecture generation in a dynamic geometry environment, ZDM, vol.51, pp.779-791, 2019.

E. Geraniou and U. T. Jankvist, Towards a definition of "mathematical digital competency, Educ Stud Math, vol.102, pp.29-45, 2019.

G. Hanna, Mathematical Proof, Argumentation, and Reasoning, Encyclopedia of Mathematics Education, 2014.

K. Hollebrands, A. Conner, and R. C. Smith, The nature of arguments provided by college geometry students with access to technology solving problems, Journal for Research in Mathematics Education, vol.41, issue.4, pp.324-350, 2010.

C. Hoyles and J. Lagrange, Introduction, Mathematics Education and Technology-Rethinking the Terrain: The 17th ICMI Study, pp.1-11, 2010.

M. Misfeldt and U. T. Jankvist, Instrumental genesis and proof: understanding the use of computer algebra systems in proofs in textbook, Uses of Technology in Primary and Secondary Mathematics Education, p.13, 2018.

. Monographs, , pp.375-385

D. Jeannotte and C. Kieran, A conceptual model of mathematical reasoning for school mathematics, Educ Stud Math, vol.96, pp.1-16, 2017.

U. Jankvist and M. Misfeldt, CAS assisted proofs in upper secondary school mathematics textbooks, REDIMAT -Journal of Research in Mathematics Education, vol.8, issue.3, pp.232-266, 2019.

M. Niss and T. Højgaard, Mathematical competencies revisited, Educ Stud Math, vol.102, pp.9-26, 2019.

B. Pedemonte, Argumentation and algebraic proof, ZDM, vol.40, pp.385-400, 2008.

P. Rabardel, People and technology: A cognitive approach to contemporary instruments, 2002.
URL : https://hal.archives-ouvertes.fr/hal-01020705

P. Rabardel and G. Bourmaud, From computer to instrument system: A developmental perspective, Interacting with Computers, vol.15, issue.5, pp.665-691, 2003.

A. Simpson, The anatomy of a mathematical proof: Implications for analyses with Toulmin's scheme, Educ Stud Math, vol.90, issue.1, pp.1-17, 2015.

S. E. Toulmin, The uses of argument, 1958.

G. Vergnaud, The theory of conceptual fields, Human Development, vol.52, issue.2, pp.83-94, 2009.

G. Boulet, On the essence of multiplication, For the Learning of Mathematics, vol.18, issue.3, pp.12-19, 1998.

M. G. Bussi and M. Mariotti, Semiotic mediation in the mathematics classroom: Artefacts and signs after a Vygotskian perspective, Handbook of international research in mathematics education, pp.746-783, 2008.

J. Confrey, Splitting, similarity, and rate of change: A new approach to multiplication and exponential functions, pp.291-330, 1994.

N. Jackiw and N. Sinclair, TouchTimes, 2019.

B. C. Burnaby,

A. Maffia and M. A. Mariotti, Intuitive and formal models of whole number multiplication: Relations and emerging structures, For the Learning of Matheamtics, issue.38, p.3, 2018.

M. A. Mariotti, Ict As Opportunities For Teaching-Learning In A Mathematics Classroom: The Semiotic Potential Of Artefacts, Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education, pp.25-40, 2012.

M. Maschietto and M. G. Bussi, Working with artefacts : Gestures , drawings and speech in the construction of the mathematical meaning of the visual pyramid, Educ Stud Math, vol.70, pp.143-157, 2009.

R. Tzur, H. L. Johnson, E. Mcclintock, R. H. Kenney, P. Yan et al., Distinguishing schemes and tasks in children's development of multiplicative reasoning, PNA, vol.7, issue.3, pp.85-101, 2013.

J. L. Schwartz, Intensive quantities and referent transforming arithmetic operations, Number concepts and operations in the middle grades, pp.41-52, 1988.

S. Squire, C. Davies, and P. Bryant, Does the cue help? Children's understanding of multiplicative concepts in different problem contexts, British Journal of Educational Psychology, vol.74, issue.4, pp.515-532, 2004.

G. Vergnaud, Multiplicative structures, Number concepts and operations in the middle grades, pp.141-161, 1988.

P. Verillon and P. Rabardel, Cognition and artifacts: A contribution to the study of though in relation to instrumented activity, European Journal of Psychology of Education, vol.10, issue.1, pp.77-101, 1995.

C. Dedé and J. Richards, Digital Teaching Platforms: Customizing Classroom Learning for Each Student, 2012.

S. Downes, Collective Intelligence and E-Learning 2.0: Implications of Web-Based Communities and Networking, 2010.

N. Heffernan, C. Heffernan, M. Bennett, and M. Militello, Effective and meaningful use of educational technology: Three cases from the classroom, C, 2012.

, Digital Teaching Platforms: Customizing Classroom Learning for Each Student

V. Hoyos, Distance technologies and the teaching and learning of mathematics in the era of MOOC, Handbook of Research on Transforming Mathematics Teacher Education in the Digital Age, 2016.

R. Kop and A. Hill, Connectivism: Learning theory of the future or vestige of the past? IRRODL, vol.9, 2008.

F. Marton, J. Routledge-olive, K. Makar, V. Hoyos, L. Kee-kor et al., Mathematical Knowledge and Practices Resulting from Access to Digital Technologies, Mathematics Education and Technology-Rethinking the Terrain (17 ICMI Study Book), pp.133-178, 2010.

K. Ruthven, Frameworks for analyzing the expertise that underpins successful integration of digital technologies into everyday teaching practice, The Mathematics Teacher in the Digital Era. Mathematics Education in the Digital Era, vol.2, 2014.

G. Siemens, Connectivism: Learning as network-creation. Retrieved, 2005.

R. Sutherland and N. Balacheff, Didactical complexity of computational environments for the learning of mathematics, International Journal of Computers for Mathematical Learning, issue.4, pp.1-26, 1999.
URL : https://hal.archives-ouvertes.fr/hal-00190717

Z. Wang, T. Anderson, C. , and L. , How learners participate in connectivist learning: An analysis of interaction traces from a cMOOC, International Review of Research in Open and Distributed Learning, vol.19, issue.1, 2018.

R. Zbiek and K. Hollebrands, A research-informed view of the process of incorporation of mathematics technology into classroom practice by in-service and prospective teachers, Research on Technology and the Teaching and Learning of Mathematics, vol.1, pp.287-344, 2008.

J. Fish and S. Scrivener, Amplifying the mind's eye: sketching and visual cognition, vol.23, pp.117-126, 1990.

J. Mason, Exploring the Sketch Metaphor for Presenting Mathematics Using Boxer, Computers and exploratory learning, pp.383-398, 1995.

E. Naftaliev and M. Yerushalmy, Solving algebra problems with interactive diagrams: Demonstration and construction of examples, Journal of Mathematical Behavior, vol.30, issue.1, pp.48-61, 2011.

E. Naftaliev and M. Yerushalmy, Design digital tasks: Interactive Diagrams as resource and constraint, The role and potential of using digital technologies in designing mathematics education tasks, pp.153-173, 2017.

E. Naftaliev, Prospective Teachers' Interactions with Interactive Diagrams: Semiotic Tools, Challenges and Well-Trodden Paths, Research on Mathematics Textbooks and Teachers' Resources: Advances and Issues, pp.297-314, 2018.

R. Netz, The shaping of deduction in Greek mathematics, 1999.

G. B. Saxe, Children's developing mathematics in collective practices: A framework for analysis, Journal of the Learning Sciences, vol.11, issue.2-3, pp.275-300, 2002.

C. U. Balsløv, The mutual benefits of using CAS and original sources in the teaching of mathematics, 2018.

B. Buchberger, Computer algebra: The end of mathematics?, ACM SIGSAM Bulletin, vol.36, issue.1, pp.3-9, 2002.

R. Chorlay, Making (more) sense of the derivative by combining historical sources and ICT, History and Epistemology in Mathematics Education -Proceedings of the Seventh European Summer University, pp.485-498, 2015.

G. Hanna, Some pedagogical aspects of proof, Interchange, vol.21, issue.1, pp.6-13, 1990.

G. Harel and L. Sowder, Toward comprehensive perspectives on the learning and teaching of proof, Second Handbook of Research on Mathematics Teaching and Learning, pp.805-842, 2007.

U. T. Jankvist and E. Geraniou, ICT as a way of making original sources more accessible to students, Tzanakis: Proceedings of the Eight European Summer University on the History and Epistemology in Mathematics Education, 2018.

U. T. Jankvist and M. Misfeldt, CAS assisted proofs in upper secondary school mathematics textbooks, REDIMAT -Journal of Research in Mathematics Education, vol.8, issue.3, pp.232-266, 2019.

U. T. Jankvist, M. Misfeldt, and M. S. Aguilar, Tschirnhaus' transformation: mathematical proof, history and CAS, Proceedings of the Eight European Summer University on the History and Epistemology in Mathematics Education, 2018.

J. Lagrange, The Didactical Challenge of Symbolic Calculators: Turning a Computational Device into a Mathematical Instrument, pp.113-135, 2005.

M. Misfeldt and U. T. Jankvist, Instrumental genesis and proof: Understanding the use of computer algebra systems in proofs in textbooks, 2018.

L. Siller, P. Ball, and S. Drijvers, Uses of technology in primary and secondary mathematics education: Tools, topics and trends, pp.375-385

I. M. Olsen and M. Thomsen, Matematikhistorie og it i matematikundervisningen i grundskolen, 2017.

P. Rabardel and G. Bourmaud, From computer to instrument system: a developmental perspective, Interacting with Computers, vol.15, issue.5, pp.665-691, 2003.

A. Sfard, Thinking as communicating: Human development, the growth of discourse, and mathematizing, 2008.

M. Thomsen, Working with Euclid's geometry in GeoGebraexperiencing embedded discourses, vol.20

M. Thomsen and I. M. Olsen, Original sources, ICT and mathemacy, Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education, pp.2060-2061, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02421925

J. Adler, Conceptualising Resources as a Theme for Teacher Education, Journal of Mathematics Teacher Education, vol.3, issue.3, pp.205-224, 2000.

S. F. Akkerman and A. Bakker, Boundary crossing and boundary objects, Review of Educational Research, vol.81, issue.2, pp.132-169, 2011.

J. Bezemer and G. Kress, Writing in Multimodal texts -A social semiotic account of designs for learning, vol.25, pp.166-195, 2008.

J. Bezemer and G. Kress, Multimodality, learning and communication: A social semiotic frame: Routledge, 2015.

G. Gueudet, B. Pepin, A. Restrepo, H. Sabra, and L. Trouche, E-textbooks and connectivity: proposing an analytical framework, International Journal of Science and Mathematics Education, vol.16, issue.3, pp.539-558, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01412819

K. O'halloran, R. A. Beezer, and D. W. Farmer, A new generation of mathematics textbook research and development, ZDM, vol.50, issue.5, pp.863-879, 2018.

C. Oswick and M. Robertson, Boundary objects reconsidered: From bridges and anchors to barricades and mazes, Journal of Change Management, vol.9, issue.2, pp.179-193, 2009.

B. Pepin, G. Gueudet, M. Yerushalmy, L. Trouche, and D. Chazan, Etextbooks in/for teaching and learning mathematics: A disruptive and potentially transformative educational technology, Handbook of International Research in Mathematics Education, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01207678

S. Rezat and R. Sträßer, From the didactical triangle to the socio-didactical tetrahedron: artifacts as fundamental constituents of the didactical situation, ZDM, vol.44, issue.5, pp.641-651, 2012.

S. L. Star, The structure of ill-structured solutions: Boundary objects and heterogeneous distributed problem solving, Distributed artificial intelligence, pp.37-54, 1989.

Z. Usiskin, Studying textbooks in an information age-a United States perspective, ZDM, vol.45, issue.5, pp.713-723, 2013.

J. Williams and G. Wake, Black boxes in workplace mathematics, Educational Studies in Mathematics, vol.64, issue.3, pp.317-343, 2007.

M. Yerushalmy, Challenging the authoritorian role of textbooks, Proceedings of International Conference on Mathematics Textbook Research and Development, vol.191, p.255, 2014.

, Gilbert Greefrath, vol.361

, Rikke Maagaard Gregersen, p.451

, Ghislaine Gueudet, vol.443

, Canan Güne?, p.459

, Angel Gutiérrez, vol.303

. H-ben-haas,

, Said Hadjerrouit, vol.369

, Julia Handl, vol.351

, Nils Kristian Hansen, vol.369

, Mariam Haspekian, vol.3

, Lulu Healy, vol.69

, Ingi Heinesen Højsted 215

, Johanna Heitzer, vol.287

, Petra Hendrikse, p.207

, Rune Herheim, vol.45

, Sara Hinterplattner, vol.353

, Karina Höveler, vol.335

, Veronica Hoyos, vol.467

, Adela Jaime, vol.303

, Ellen Jameson, p.61

, Uffe Thomas Jankvist, vol.255, p.483

, Myrto Karavakou, vol.223

, Camilla Finsterbach Kaup, vol.53

, George Kinnear, vol.377

, Mária Kmetová, p.319

M. J. Koch, , p.61

-. Zeger,

, Ulrich Kortenkamp, vol.231

Y. Kreis,

, Bjarnheiður (Bea) Kristinsdóttir, p.415

, Corinna Kröhn, vol.353

, Jens Krummenauer, vol.69

S. Kuntze, , p.69

, Alain Kuzniak, vol.77

, Chronis Kynigos 223 L Silke Ladel, vol.127, p.231

, Kevin Larkin, vol.231

, Zsolt Lavicza, vol.199, p.415