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Christian received his PhD in 2011 at the Freudenthal Institute,Utrecht University, the Netherlands and is lecturer at the University of Southampton, United Kingdom. In this talk, with space for some hands-on activities, Christian will present a wide spectrum of research initiatives that all involve the use of technology to support mathematics education itself and research into mathematics education. It will cover (i) his involvement in the development of educational software at the Freudenthal Institute , (ii) the evolution from fragmented technology to coherent digital books and their most important features, (iii) numerous examples of software modules/books, including from STEM and the enGasia project;

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- 1. Using technology to support mathematics education and research Dr. Christian Bokhove 13 July 2017 Hong Kong
- 2. Who am I • Dr. Christian Bokhove • From 1998-2012 teacher maths, computer science, head of ICT secondary school Netherlands • National projects Maths & ICT at Freudenthal Instituut, Utrecht University • PhD 2011 under Prof. Jan van Maanen and prof. Paul Drijvers • Lecturer at University of Southampton – Maths education – Technology use – Large-scale assessment – Computer Science stuff
- 3. Contents WORKING WITH THE FREUDENTHAL INSTITUTE ALGEBRA AND TECHNOLOGY DIGITAL MATHEMATICAL BOOKS ENGASIA CONCLUSION
- 4. Contents WORKING WITH THE FREUDENTHAL INSTITUTE ALGEBRA AND TECHNOLOGY DIGITAL MATHEMATICAL BOOKS ENGASIA CONCLUSION
- 5. Wisweb and WELP • Wisweb: collections of (Java) applets • WELP: integrate the use of the applets in lessons
- 6. Galois and sage projects • Government grant programme for teacher innovations • Make integrated version • First version of the ‘Digital Mathematics Environment’ (Peter Boon) • Sage: prize money
- 7. More on FI With thanks to Prof. Paul Drijvers
- 8. Hans Freudenthal (1905-1990) „Mathematics as human activity“ • construct content from reality • organize phenomena with mathematical means
- 10. RME Key Characteristics • Meaningful contexts as starting point for learning • Progressive mathematization from informal strategies and (horizontal and vertical) • Intertwinement of content strands • Interaction • Room for students’ own constructions (Treffers, 1987)
- 11. What do we mean by “Realistic”? “Realistic” may have different meanings: • Realistic in the sense of feasible in educational practice • Realistic in the sense of related to real life (real world, phantasy world, math world) • Realistic in the sense of meaningful, sense making for students • Realistic in the sense of “zich realiseren” = to realize, to be aware of, to imagine • HF: “How real the concepts are depends on the conceiver” 11
- 12. Key RME design heuristics A. Guided reinvention B. Didactical phenomenology C. Horizontal and Vertical Mathematization D. Emergent Modeling
- 13. A. Guided reinvention • Reinvention: Reconstructing and developing a mathematical concept in a natural way in a given problem situation. • Guidance: Students need guidance (from books, peers, teacher) to ascertain convergence towards common mathematical standards • Reinvention <-> guidance: a balancing act
- 14. B. Didactical phenomenology The art to find phenomena, contexts, problem situations that … • … beg to be organized by mathematical means • … invites students to develop the targeted mathematical concepts These phenomena can come from real life or can be ‘experientially real’
- 15. ‘Realistic’ context Mathematical model Mathematical objects, structures, methods Horizontal mathematization Translate Vertical mathematization Abstract C. Mathematization
- 16. D. Emergent modeling • View on mathematics education which aims at the development of models • Models of informal mathematical activity develop into models for mathematical reasoning • Level structure by Gravemeijer situational referential general formal
- 17. Some debate • Implementation ok? (Gravemeijer, Bruin-Muurling, Kraemer, & van Stiphout, 2016) • Influence of contexts? (Hickendorff, 2013) • Procedural skills and conceptual understanding go hand in hand (Rittle-Johnson, Schneider, & Star, 2015) • Not enough emphasis on procedural skills e.g. algorithms (Fan & Bokhove, 2014) I don’t see a contradiction doing both. Combined in subsequent PhD work
- 18. Contents WORKING WITH THE FREUDENTHAL INSTITUTE ALGEBRA AND TECHNOLOGY DIGITAL MATHEMATICAL BOOKS ENGASIA CONCLUSION
- 19. Part of PhD Algebraic skills Year 12 students Netherlands often disappointing
- 20. But even when rewriting skills are OK… …many things can go wrong
- 21. So conceptual understanding and pattern recognition are important!
- 22. Use of ICT
- 23. So can’t we use ICT for acquiring, practicing and assessing algebraic expertise?
- 24. Equations: in-between steps, multiple strategies allowed
- 25. Store student results, and use these as a teacher to study misconceptions and for starting classroom discussions students
- 26. Design principles (i) students learn a lot from what goes wrong, (ii) but students will not always overcome these if no feedback is provided, and (iii) that too much of a dependency on feedback needs to be avoided, as summative assessment typically does not provide feedback. These three challenges are addressed by principles for crises, feedback and fading, respectively.
- 27. Crisis-tasks “students learn a lot from what goes wrong”
- 28. Feedback: worked examples and hints IDEAS feedback, webservice with Jeuring et al
- 29. Fading “too much of a dependency on feedback needs to be avoided”
- 30. Hands-on: Equations http://is.gd/hkeng1 These are HTML5 versions of those applets.
- 31. Contents WORKING WITH THE FREUDENTHAL INSTITUTE ALGEBRA AND TECHNOLOGY DIGITAL MATHEMATICAL BOOKS ENGASIA CONCLUSION
- 32. Towards digital textbooks • Digital textbook: theory, examples, explanations • Interactive content (in MC-squared widgets) • Interactive quizzes (formative assessment, feedback) • Integrated workbook
- 33. FP7 EU project Designing creative electronic books for mathematical creativity
- 34. The environment stores student work. Separate ‘schools’ can have several classes. This is the ‘edit’ mode of the environment : this c-book is about planets c-books can have several pages: each circle indicates a page. Other options are available as well C-book pages can have random elements, like random values. Pages consist of ‘widgets’, which can range from simple text to simulations (here: Cinderella). Some widgets can give automatic feedback. The MC-squared project aims aims to design and develop a new genre of creative, authorable e-book, which the project calls 'the c-book MC-squared platform based on Utrecht University’s ‘Digital Mathematics Environment’ (now Numworx). https://app.dwo.nl/en/student/
- 35. Authorable
- 36. Feedback
- 37. Also geometry
- 39. Bokhove, C., & Redhead, E. (2017). Training mental rotation skills to improve spatial ability. Online proceedings of the BSRLM, 36(3)
- 42. Hands-on: Cube Buildings http://is.gd/hkeng2: Cube Buildings http://is.gd/hkeng3: Planets These are HTML5 versions of those applets.
- 43. Contents WORKING WITH THE FREUDENTHAL INSTITUTE ALGEBRA AND TECHNOLOGY DIGITAL MATHEMATICAL BOOKS ENGASIA CONCLUSION
- 44. enGasia project 1. Compare geometry education in England, Japan and Hong Kong → some shown now. 2. two digital resources (electronic books) will be designed. They are then implemented in classrooms in those countries. 3. The methodology will include a more qualitative approach based on lesson observations and a quasi- experimental element.
- 45. This could be a geogebra widget but perhaps not necessary. More important is feedback.
- 46. Challenges • Differences in curriculum regarding geometry • School and teacher participation • Software: Java Now writing the findings in several articles.
- 47. Flowchart • Prof. Miyazaki and team http://engasia.soton.ac.uk
- 48. Contents WORKING WITH THE FREUDENTHAL INSTITUTE ALGEBRA AND TECHNOLOGY DIGITAL MATHEMATICAL BOOKS ENGASIA CONCLUSION
- 49. Technology-added value of the c-books • Creative and interactive activities made by designers (creative process authoring) • Collaboration within CoI between designers, teachers and computer scientists. Feeds into DA component (see later section) • Interactivity: feedback design • More than one widget factories used • All student data stored • Sum is more than the parts… Bokhove, C., (in press). Using technology for digital maths textbooks: More than the sum of the parts. International Journal for Technology in Mathematics Education.
- 50. Thank you • Contact: –C.Bokhove@soton.ac.uk –Twitter: @cbokhove –www.bokhove.net • Most papers available somewhere; if can’t get access just ask. • I’ll add the references and post on Slideshare
- 51. Bokhove, C., (in press). Using technology for digital maths textbooks: More than the sum of the parts. International Journal for Technology in Mathematics Education. Bokhove, C., &Drijvers, P. (2010). Digital tools for algebra education: criteria and evaluation. International Journal of Computers for Mathematical Learning, 15(1), 45-62. Online first. Bokhove, C., & Drijvers, P. (2012). Effects of a digital intervention on the development of algebraic expertise. Computers & Education, 58(1), 197-208. doi:10.1016/j.compedu.2011.08.010 Bokhove, C., & Redhead, E. (2017). Training mental rotation skills to improve spatial ability. Online proceedings of the BSRLM, 36(3) Fan, L., & Bokhove, C. (2014). Rethinking the role of algorithms in school mathematics: a conceptual model with focus on cognitive development. ZDM-International Journal on Mathematics Education, 46(3), doi:10.1007/s11858- 014-0590-2 Fischer, G. (2001). Communities of interest: learning through the interaction of multiple knowledge systems. In the Proceedings of the 24th IRIS Conference S. Bjornestad, R. Moe, A. Morch, A. Opdahl (Eds.) (pp. 1-14). August 2001, Ulvik, Department of Information Science, Bergen, Norway. Freudenthal, H. (1991). Revisiting Mathematics Education. China Lectures. Dordrecht: Kluwer Academic Publishers. Gravemeijer, K., Bruin-Muurling, G., Kraemer, J-M. & van Stiphout, I. (2016). Shortcomings of mathematics education reform in the Netherlands: A paradigm case?, Mathematical Thinking and Learning, 18(1), 25-44, doi:10.1080/10986065.2016.1107821 Hickendorff M. (2013), The effects of presenting multidigit mathematics problems in a realistic context on sixth graders' problem solving, Cognition and Instruction 31(3), 314-344. Jaworksi, B. (2006). Theory and practice in mathematics teaching development: critical inquiry as a mode of learning in teaching. Journal of Mathematics Teacher Education, 9(2), 187-211. Rittle-Johnson, B. Schneider, M. & Star, J. (2015). Not a one-way street: Bi-directional relations between procedural and conceptual knowledge of mathematics. Educational Psychology Review, 27. doi:10.1007/s10648-015-9302-x Treffers, A. (1987). Three dimensions. A model of goal and theory description in mathematics instruction – The Wiskobas project. Dordrecht: D. Reidel Publishing Company. Wenger, E. (1998). Communities of Practice: Learning, Meaning, Identity. Cambridge University Press.