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Designing creative electronic books for mathematical creativity


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Designing creative electronic books for mathematical creativity

  1. 1. The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement n° 610467 - project “M C Squared”. This publication reflects only the author’s views and Union is not liable for any use that may be made of the information contained therein. Eirini Geraniou – UCL Institute of Education ( Christian Bokhove – University of Southampton ( Manolis Mavrikis – UCL Knowledge Lab ( Saturday, 12th of November 2016 Designing creative electronic books for mathematical creativity
  2. 2. Overview • 5 mins – Introduce the MC2 project and its aims • 10 mins – Introduce MC-squared platform, the different functionalities, tools, such as widgets • 15 mins – Present the list of current c-books and focus on 3 c-books • 8 mins – Show some data • 2 mins – Share Key messages • Discussion
  3. 3. Aims • Design and develop a new genre of authorable e-book, which we call 'the c- book' (c for creative) • Creative Mathematical Thinking (CMT) • Initiate a ‘Community of Interest’ (CoI) • A community of interest consists of several stakeholders from various ‘Communities of Practice. • England, Spain, Greece, France • Within these, teachers who co-design and use resources for teaching, can contribute to their own professional development. • Social Creativity • UK CoI: learning analytics and feedback (e.g. Fischer, 2001; Wenger, 1998; Jaworski, 2006)
  4. 4. MC Squared Platform JAVA/HTML5 disclaimer • C-books available • Widget list • At the end, we can demo the platform: and login as guest
  5. 5. 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.
  6. 6. Creativity (a) Social creativity (SC) in the design of CMT resources (c-book units) (b) Creative mathematical thinking (CMT) has been drawn on Guilford’s (1950) model of • fluency (the ability to generate a number of solutions to a problem), • flexibility (the ability to create different solutions), • originality (the ability to generate new and unique solutions), and • elaboration (the ability to redefine a problem).
  7. 7. Tryingtooperationalizecreativity Geometry example (object=quadrilaterals) Algebra example (object=expression) Create a certain object with a widget that can produce these objects. If ‘certain’ is not included then (originality) Create a (certain) quadrilateral Create a (certain) number We should be able to ask the widget what classes/types have been produced (fluency) From the 9 types of quadrilaterals 3 were made by the student: square, rhombus and trapezium. Different numbers or expressions were made We should be able to ask the widget how these classes/types were produced (flexibility) The square and rhombus were made once. The trapezium in two distinct ways. The numbers/expressions were made in very distinct ways If the outputs are numbers they can be compared over a group of students (originality) Student was only one who made a rhombus. / Teacher submitted ‘non original’ answers. Student was only one who made this expression. / Teacher submitted ‘non original’ answers. Explain what you created and how you created them in your own words (elaboration) Student supplies an elaboration on the types and processes of quadrilateral making. Student supplies elaboration on the types and processes of making numbers/expressions.
  8. 8.
  9. 9. Fluency
  10. 10. Flexibility
  11. 11. Originality
  12. 12. Feedback as cue for creativity Numbers v7
  13. 13. Also feedback for more open tasks 14
  14. 14. Integrated design
  15. 15. Case Study – Reflections c-book We considered CMT as • (i) the ‘construction’ of math ideas or objects, in accordance to constructionism that sees CMT being expressed through exploration, modification and creation of digital artefacts • (ii) Fluency (as many answers as possible) and Flexibility (different solutions/strategies for the same problem) • (iii) novelty/originality (new/unusual/unexpected ways of applying mathematical knowledge in posing and solving problems). (Daskolia & Kynigos, 2012; Papadopoulos et al. 2015; 2016) • as a thinking ‘process’ in a mathematical activity in order to produce a ‘product’ (e.g. a solution to a mathematical problem).
  16. 16. Case Study - Aims • explore the potential of the Reflection digital book and of automated feedback and reflection • focus on building ‘bridges’ to the maths involved (and may be ‘hidden’) in digital resources. Bridging activities = short tasks or questions used to intervene and encourage students to reflect upon mathematical concepts and problem-solving strategies they use throughout a sequence of activities (or simple interactions) with a digital tool. Such activities could take various forms from questions or prompts within the digital tool to paper-based worksheets or verbal teacher’s interventions.
  17. 17. Case Study • METHODOLOGICAL TOOL: • “design experiment” (Collins et al., 2004; Cobb et al., 2003). • SAMPLE: • 21 Grade-7 students and their class teacher • 2 lessons in the school’s computer lab. • DIFFERENT ROLES: • Researchers as ‘participant observers’ • Teacher as COI member to design the Reflections c-book, offered assistance in technical issues and ensured that all students were on task and answered the bridging activities • DATA: • logged answers in the MC2 platform, voice recordings of students’ elaborations on their interaction/answers and a student evaluation questionnaire
  18. 18. Case Study – Reflections c-book (A – F) Excerpts from the Reflection c-book and (G) a sample solution of (F)
  19. 19. Questionnaire A Likert multiple-choice questionnaire consisting of questions such as: • (1) How satisfied were you after completing the c-book activities? • (2) How easy to use do you think the c-book is? • (3) How free did you feel to experiment with the c-book and try out your ideas? • (4) I feel I understand Reflection now. • Another two questions (5 and 6) gave them options to pick on their thoughts on the c-book and their preferred features. • The questionnaire finished with three more questions to request suggestions from students.
  20. 20. Findings (Students) • Language. Moving from informal terminology (“the other shape moves”, “we have to flip the shape” or “count how many down from the mirror line”) to using mathematical terms (“the reflected church” or the “reflection line”). • Superficial responses. • “if you move the green shape, the orange shape moves with it”. • They seemed to have noticed that the 2 shapes (green and orange ‘F’) are linked, but only 2 were able to articulate that they maintain the same distance from the Reflection line. • Bridging activities revealed students’ solving strategies and their CMT. • Competition task. 3 different strategies: (i) counting boxes across and down, (ii) tilt the head so that the reflection line becomes vertical and then find the reflected image and (iii) imagine using tracing paper on the screen to find the reflected image.
  21. 21. QUESTIONS RESULTS (Q4) I feel I understand Reflection now. 85% (18/21) responded on with an answer above 4 in the Likert scale (Q5)-(Q6) Students’ thoughts on the c-book and their preferred features. 43% (9/21) said that it included problems that they would not have tried to solve. 60% (13/21) answered that it helped them see the idea of reflection in different ways. 3 students made comments that showed that they appreciate the advantages of digital technologies: “the digital book help[s] because you could have actually test[ed] out your ideas and improve if it’s wrong or not” Students’ Suggestions They recognized the dynamicity of such resources and how seeing the immediate feedback on their actions helps them validate an answer or solution.
  22. 22. Findings (Teacher) • Teacher’s keenness to use Digital Technologies was re-enforced through • authoring activities, • designing bridging activities and • participating in the creation of the Reflection c-book • Reflection c-book was further developed • The most notable improvement was breaking down the bridging questions to smaller questions with guidance, and using the feedback affordances to encourage the CMT aspect of flexibility in terms of the strategies. KEY MESSAGE The c-book technology can be integrated in the mathematics classroom and promote a positive learning experience through the use of playful activities for students, matched with carefully designed bridging activities, followed by constructionist activities that allow deeper exploration of the subject matter.
  23. 23. Conclusions • Operationalising creativity: challenging • Added value of technology: more than the sum of the parts • Authoring • Widgets with different features • Feedback • Storing student data • Different people can work together in participatory design (Community of Interest) with c-books acting as boundary objects • Creative and interactive activities made by designers (creative process authoring) • Collaboration within CoI between designers, teachers and computer scientists. • Interactivity: feedback design • More than one widget factories used • All student data stored • Sum is more than the parts…
  24. 24. MC Squared Platform JAVA/HTML5 disclaimer •C-books available •Widget list • To demo the platform, go to: and login as guest
  25. 25. Thank you • Eirini Geraniou • • Christian Bokhove • • Twitter: @cbokhove •
  26. 26. REFERENCES Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9-13. Collins, Alan., Diana Joseph & Katerine Bielaczyc (2004). Design Research: Theoretical and Methodological Issues. The Journal Of The Learning Sciences, 13(1), 15-42. Daskolia, M., & Kynigos, C. (2012). Applying a Constructionist Frame to Learning about Sustainability. Creative Education, 3, 818–823. 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. Guilford, J.P. (1950). Creativity. American Psychologist, 5, 444–454. 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. Papadopoulos, I., Barquero, B., Richter, A., Daskolia, M., Barajas, M., & Kynigos, C. (2015). Representations of Creative Mathematical Thinking in Collaborative Designs of C-book units. In K. Krainer & N. Vondrová (Eds.), Proceedings of the Ninth Conference of the European Society for Research in Mathematics Education (4-8 February 2015) (pp. 2381–2387). Prague, Czech Republic. Papadopoulos, I., Diamantidis, D. & Kynigos, C. (2016). Meanings around angle with digital media designed to support creative mathematical thinking. Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education, (Vol. 4, pp. 35–42). Szeged, Hungary. Wenger, E. (1998). Communities of Practice: Learning, Meaning, Identity. Cambridge University Press.