1. Use of ICT for acquiring, practicing and assessing algebraic expertise Christian Bokhove c.bokhove@fi.uu.nl www.fi.uu.nl/~christianb

2. Context Christian Bokhove 12 yr Teacher maths/ict secondary school St. Michaël College, Zaandam, the Netherlands, tradition math/ict projects Phd research. (www.dudocprogramma.nl) aimed at math curriculum.www.dudocprogramma.nl Freudenthal Institute of Science and Mathematics Education, Utrecht University, the Netherlands Supervisors: Paul Drijvers and Jan van Maanen Educational research

3. Problem statement Transition secondary  higher education –Lack of Algebraic expertise? –Entry exams Use of ICT –“Use to learn” vs. “Learn to use” –Position statement NCTM (2008): ICT can be a valuable asset

4. Overview In what way can the use of ICT support acquiring, practicing and assessing relevant mathematical skills? Assessment - Formative (for) v Summative (of) - Feedback (Black & Wiliam, 1998) ICT tool use - Instrumentation - Task, technology, theory (Chevallard, 1991) Algebraic expertise - Basic skills - Symbol Sense: gestalt, pattern salience, local salience, circularity (Arcavi, 1994)

5. Mathematical proficiency Kilpatrick

6. Sfard "First there must be a process performed on the already familiar objects, then the idea of turning this process into a more compact, self-contained whole should emerge, and finally an ability to view this new entity as a permanent object in its own right must be acquired.“ (Sfard, 1992) 1."interiorization", 2."condensation", and 3."reification"

7. Tall: procept

8. Algebraic expertise Arcavi Kop & Drijvers Pierce & Stacey (Structure sense, e.g. Hoch & Dreyfus)

9. Extension of Gestalt Gestalt Visual salience Bokhove, C., & Drijvers, P. (2010). Symbol sense behavior in digital activities. In press. For the Learning of Mathematics. Gestalt view: pattern salience, local salience and strategic decision

10. Design research (Tessmer)

11. Prelim: criteria for tools First choose a tool Evaluation instrument, externally validated, first formulate want we want, then see what there is. A selection: –Stores both answers & solutions students; –Steps & freedom to choose own strategy; –Authoring tool for own questions; –Intuitive interface incl. equation editor (‘use to learn’ vs. ‘learn to use’) –60+ tools evaluated; Bokhove, C., & Drijvers, P. (2010). Digital tools for algebra education: criteria and evaluation. International Journal of Computers for Mathematical Learning, 15(1), 45-62. (link)link

12. 1st cycle: 1-to-1s Qual. analysis (video, camtasia, atlas TI) Symbol SenseQuality of toolFeedback 6 multihour think-aloud 1-to-1 sessions with 17/18 year olds I want to know what’s going on in their minds

13. 2nd cycle Jan-Mar 2010, Enkhuizen Digital Mathematical Environment (DME) www.fi.uu.nl/dwo/voho www.fi.uu.nl/dwo/voho Two 6vwo 17/18 yr olds

14. Design choices Follow from 1-to-1 sessions prototype 4 activities in 4 categories Randomization (note “strange values”) Crises Feedback (many types, Hattie & Timperley) Formative scenario’s: first a lot of feedback then gradually less Bokhove, C. (2010). Implementing feedback in a digital tool for symbol sense.. International Journal for Technology in Mathematics Education. 17(3)

15. Crises “Failure is, in a sense, the highway to success, inasmuch as every discovery of what is false leads us to seek earnestly after what is true, and every fresh experience points out some form of error which we shall afterwards carefully avoid.” Keats. Van Hiele: crisis of learning Productive failure (Kapur) Impasse (VanLehn et al) Perturbation (Doll) Disequilibrium (Piaget)

16. Digital mathematical environment http://www.fi.uu.nl/dwo/vohohttp://www.fi.uu.nl/dwo/voho, developer: Peter Boon

17. CUSTOM CORRECTIVE FEEDBACK DESCRIPTIVE BUG-RELATED HINTS Authoring tasks

18. Example student work

19. In action

20. Another student example

21. 3rd cycle Oct/nov 2010 11 schools, around 350 students “Algebra met Inzicht” (AmI) www.algebrametinzicht.nl

22. Data collection & analysis Scores per module in DME Pre- and posttest scores Attitude scales General characteristics Log files Log notes Audio, video clips Excel

24. Analyzing log data C4.5 Decision trees

25. Future? http://www.fi.uu.nl/~christianb/downloads/poc_equation.htm

26. Discussion –Questions? –Almost every time discussion understandibly ends with the cut Skills vs. Understanding –Is one is more dominant? –Does symbol sense exist apart from skills and/or vice versa –Methodology: what about distance learning?

27. Selected references Bokhove, C., & Drijvers, P. (2010). Digital tools for algebra education: criteria and evaluation. International Journal of Computers for Mathematical Learning, 15(1), 45-62. (link)link Bokhove, C., & Drijvers, P. (2010). Symbol sense behavior in digital activities. For the Learning of Mathematics, 30(3), 43-49. Kilpatrick, J., Swafford, J. & Findell, B. (2001). The Strands of Mathematical Proficiency. In J. Kilpatrick, J. Swafford & B. Findell (Eds.), Adding It Up: Helping Children Learn Mathematics (pp 115-155). Washington: National Research Council. Sfard, A. (1991). On The Dual Nature Of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin. Educational Studies in Mathematics 22, 1-36. Tall, D. (2008). The Transition to Formal Thinking in Mathematics. Mathematics Education Research Journal, 20(2), 5-24.