1. Research and practice on tool use for mathematics in secondary education Christian Bokhove 19/9/08

2. Inhoud  Introduction  From a classroom perspective  From a research perspective  Conclusion  Questions/ discussion Feel free to ask questions during the presentation!

3. Introduction

4. Context  Teacher mathematics and computer studies, also ICT coordinator  St. Michael College, a secondary school in Zaandam, near de Zaanse Schans  A lot of “tool” experience in the classroom (classroom perspective)  Now doing research on the use of tools for acquiring, practising and assessing algebraic skills (research perspective) In the Dudoc programme Programme for “allowing” research in secondary school classroom.

5. Practice  Wisweb project  Galois project  Sage project  ACTIVEMATH-EU Many projects rooted in practice. “These seem to work, these don’t” But….these are not scientifically grounded

6. Theory  A lot of research in the field  Yielding many tools  Case studies often indicate successes But….how do these tools do in actual classroom situations

7. From a classroom perspective …but putting it in the context of research

8. Several projects  Wisweb project Aimed at creating applets for various topics in the dutch curriculum Students enthousiastic, some applets “seem to work” But…no coherence  WELP project aimed at grouping applets together for a “leerlijn” (core curriculum) But…no integrated environment Link to wisweb From there on: from project to project…

9. Galois project  Geïntegreerde Algebraïsche Leer Omgeving In School  Aims Inventarisation of tools / “state of the art” Using tools in the classroom AUthoring SCORM And more…  Tools we “dabbled” in: Link to Galois project

10. Maple TA  Approached CAN diensten  In the USA Maplesoft was released  I wanted to see whether it could supplement our math education  Students and teachers are quick to give their opinion. A valuable source for the developer or researcher. Link to maple TA 4

11. STACK  Classroom experiment: accessing the server crashed it  Stability and performance was deemed most important of all points (research)  Hard to install, access to server  But a novel approach to feedback and multi-step exercises. Link to STACK (Invitation: on october 9th I have organized a lecture by Chris Sangwin at the Freudenthal Institute)

12. The practice of computer aided assessment of mathematics raises some fundamental questions. What do to the conventions of traditional mathematical notation mean? How can we enter a mathematical expression into a machine in a simple but unambiguous way? If we generate random problems in mathematics how to we identify and preserve their "essence"? When assessing problems, what mathematical properties are we trying to establish and which can we establish at a technical level? Having established properties of a student's answer, what outcomes should we assign, and how do we generate feedback which is helpful? The STACK project proposes answers to some of these questions in limited contexts.

13. Webwork  Wanted to use this at our school  Moodle integration  But…very hard to install  Now I reverted to the server of the maintainers Link to WebWork

14. Activemath  Classroom experiment with wiskunde D material on differentiation  Group of about 15 students worked with AM on the St. Michael College server  One paragraph on the product rule  Use of registration Link to Activemath site

15. This…

16. Becomes…

17. …

18. Plus other sources (animation)

19. applet Outside AM for registration by Moodle

20. Wiris  Integrated in our Virtual Learning Environment  Input editor and CAS  Demo in our VLE moodle Demo in our VLE moodle But…we as teachers want to be able to edit course material

21. Sage project  Reinvested prize money from Galois project  Aimed at improving the Scorm Applet Generator (See Peters talk)  And providing 20 SCORM packages (creative commons) with algebra.  But also improving Digital Mathematical Environment (developer Peter Boon)  Demo of material Demo of material  Demo of student material (together with Activemath material) Demo of student material

22. II. From a research perspective

23. Problem statement  Bridging the gap Secondary-Higher education Algebraic skills ▼ ? Entry exams e.d. (NKBW, 2007; Tempelaar, 2007; Vos, 2007)  Tool use ▲ Vision of cTWO (2007): (Commission future mathematics education) “Use to learn” vs. “Learn to use” Position statement NCTM (2008)

24. Research question  In what way can ICT be used for acquiring, practising and assessing algebraic skills?  This involves three key topics 1. ICT tools Acquiring, practising and assessing > 2. Assessment 3. Algebraïc skills

25. 1 ICT tools  Anthropological approach (Lagrange, 1999) Task Technology Theory  Instrumental approach (Drijvers, 2007) tool + mental scheme = instrument  Focus on (online) software

26. (ICT) Tool use Examples 10th january 200826

27. 2. Assessment  Practising and testing...  Assessment for learning (Formative) More qualitative and  Assessment of learning (Summative) More quantitative/performance (Black and Wiliam, 1998)  Feedback: several types (Hints!!)  Assessment framework

28. 3. Algebraic skills  Basic skills  Symbol sense (Arcavi, 1994)  Assessment framework > pyramid

29. Simple example  Equation  Students could do this: (conditioned to expand) etc…  Or the student “sees” (Gestalt): or But…are practice and related?

30.  Advocates of insightful learning are often accused of being soft on training. Rather than against training, my objection to drill is that it endangers retention of insight. There is, however a way of training — including memorisation — where every little step adds something to the treasure of insight: training integrated with insightful learning. Freudenthal, H. (1991). Revisiting mathematics education.

31. Design research  Conceptual framework, grounded in existing research  Design cycles  Aimed at making an intervention, but also improving the body of research on a certain topic (e.g. here tool use, symbol sense etc.)  Qualitative analysis (first and second cycle), what feedback, misconceptions etc.  Quantitative (second and third), including high- stakes performance

32. Combining research and practice  Use research to give a theoretical basis for, for example: Criteria on which to rate tools. 27 criteria in 4 categories. Examples  Stability. See example STACK  Authoring. Hardcoding as in Webwork or Wysiwyg  Modes/strategies. (based on literature on formative assessment and feedback)  Process/Answer. Leading to “Tool choice” Focus of the research

33. Conclusion  All tools have their particular strengths and weaknesses  Practice and theory should go hand in hand  Sharing knowledge and more collaboration between seondary education, higher education and researchers could create a win-win situation.

34. Questions and discussion

35. Selected references Arcavi, A. (1994). Symbol Sense: Informal Sense-Making in Formal Mathematics. For the Learning of Mathematics, 14(3), 24-35. Black, P., & Wiliam, D. (1998). Assessment and classroom learning. Assessment in Education: Principles, Policy & Practice, 5(1), 7-73. cTWO. (2007). Rijk aan betekenis, visie op vernieuwend wiskundeonderwijs. Drijvers, P. (2007). Instrument, orkest en dirigent: een theoretisch kader voor ICT-gebruik in het wiskundeonderwijs. Pedagogische Studiën, 84(5), 358-374. Lagrange, J.-B. (1999). Complex calculators in the classroom: Theoretical and practical reflections on teaching pre-calculus. International Journal of Computers for Mathematical Learning, 4, 51-81. NKBW. (2007). Eindrapport Nationale Kennisbank Basisvaardigheden Wiskunde. Tempelaar, D. (2007). Onderwijzen of bijspijkeren? Nieuw Archief voor Wiskunde, 8, 55-59. Vos, P. (2007). Algebra-prestaties van tweedeklassers. Euclides, 82, 129-132.