Use of ICT for acquiring, practicing and assessing algebraic expertise Christian Bokhove [email_address] www.fi.uu.nl/~chr...
Context <ul><li>Christian Bokhove </li></ul><ul><li>12 yr Teacher maths/ict secondary school </li></ul><ul><li>St. Michaël...
Problem statement <ul><li>Transition secondary    higher education </li></ul><ul><ul><li>Lack of Algebraic expertise? </l...
Overview  In what way can the use of ICT support acquiring, practicing and assessing relevant mathematical skills? Assessm...
Mathematical proficiency <ul><li>Kilpatrick </li></ul>
Sfard <ul><li>&quot;First there must be a process performed on the already familiar objects, then the idea of turning this...
Tall: procept
Algebraic expertise <ul><li>Arcavi </li></ul><ul><li>Kop & Drijvers </li></ul><ul><li>Pierce & Stacey </li></ul><ul><li>(S...
Extension of Gestalt <ul><li>Gestalt </li></ul><ul><li>Visual salience </li></ul>Bokhove, C., & Drijvers, P. (2010). Symbo...
Design research (Tessmer)
Prelim: criteria for tools <ul><li>First choose a tool </li></ul><ul><li>Evaluation instrument, externally validated, f ir...
1st cycle: 1-to-1s Qual. analysis (video, camtasia, atlas TI)  Symbol Sense Quality of tool Feedback 6 multihour think-alo...
2nd cycle <ul><li>Jan-Mar 2010, Enkhuizen </li></ul><ul><li>Digital Mathematical Environment (DME) www.fi.uu.nl/dwo/voho  ...
Design choices <ul><li>Follow from 1-to-1 sessions prototype </li></ul><ul><li>4 activities in 4 categories </li></ul><ul>...
Crises <ul><li>“ Failure is, in a sense, the highway to success, inasmuch as every discovery of what is false leads us to ...
Digital mathematical environment http://www.fi.uu.nl/dwo/voho   , developer: Peter Boon
Authoring tasks CUSTOM CORRECTIVE FEEDBACK DESCRIPTIVE BUG-RELATED HINTS
Example student work
In action
Another student example
3rd cycle <ul><li>Oct/nov 2010 </li></ul><ul><li>11 schools,  around 350  students </li></ul><ul><li>“ Algebra met  Inzich...
Data collection & analysis <ul><li>Scores per module in DME </li></ul><ul><li>Pre- and posttest scores </li></ul><ul><li>A...
 
Analyzing log data C4.5 Decision trees
Future? <ul><li>http://www.fi.uu.nl/~christianb/downloads/poc_equation.htm   </li></ul>
Discussion <ul><ul><li>Questions? </li></ul></ul><ul><ul><li>Almost every time discussion understandibly ends with the cut...
Selected references <ul><li>Bokhove, C., & Drijvers, P. (2010). Digital tools for algebra education: criteria and evaluati...
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Presentation Mathmaster 031210

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Presentation for Mathmaster students of Utrecht University, december 3rd 2010, by Christian Bokhove

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Presentation Mathmaster 031210

  1. 1. Use of ICT for acquiring, practicing and assessing algebraic expertise Christian Bokhove [email_address] www.fi.uu.nl/~christianb
  2. 2. Context <ul><li>Christian Bokhove </li></ul><ul><li>12 yr Teacher maths/ict secondary school </li></ul><ul><li>St. Michaël College, Zaandam, the Netherlands, tradition math/ict projects </li></ul><ul><li>Phd research. ( www.dudocprogramma.nl ) aimed at math curriculum. </li></ul><ul><li>Freudenthal Institute of Science and Mathematics Education, Utrecht University, the Netherlands </li></ul><ul><li>Supervisors: Paul Drijvers and Jan van Maanen </li></ul><ul><li>Educational research </li></ul>
  3. 3. Problem statement <ul><li>Transition secondary  higher education </li></ul><ul><ul><li>Lack of Algebraic expertise? </li></ul></ul><ul><ul><li>Entry exams </li></ul></ul><ul><li>Use of ICT </li></ul><ul><ul><li>“ Use to learn” vs. “Learn to use” </li></ul></ul><ul><ul><li>Position statement NCTM (2008): ICT can be a valuable asset </li></ul></ul>
  4. 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. 5. Mathematical proficiency <ul><li>Kilpatrick </li></ul>
  6. 6. Sfard <ul><li>&quot;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) </li></ul><ul><li>  </li></ul><ul><li>&quot;interiorization&quot;, </li></ul><ul><li>&quot;condensation&quot;, and </li></ul><ul><li>&quot;reification&quot; </li></ul>
  7. 7. Tall: procept
  8. 8. Algebraic expertise <ul><li>Arcavi </li></ul><ul><li>Kop & Drijvers </li></ul><ul><li>Pierce & Stacey </li></ul><ul><li>(Structure sense, e.g. Hoch & Dreyfus) </li></ul>
  9. 9. Extension of Gestalt <ul><li>Gestalt </li></ul><ul><li>Visual salience </li></ul>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. 10. Design research (Tessmer)
  11. 11. Prelim: criteria for tools <ul><li>First choose a tool </li></ul><ul><li>Evaluation instrument, externally validated, f irst formulate want we want, then see what there is. A selection: </li></ul><ul><ul><li>Stores both answers & solutions students; </li></ul></ul><ul><ul><li>Steps & freedom to choose own strategy; </li></ul></ul><ul><ul><li>Authoring tool for own questions; </li></ul></ul><ul><ul><li>Intuitive interface incl. equation editor (‘use to learn’ vs. ‘learn to use’) </li></ul></ul><ul><ul><li>60+ tools evaluated; </li></ul></ul>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 )
  12. 12. 1st cycle: 1-to-1s Qual. analysis (video, camtasia, atlas TI) Symbol Sense Quality of tool Feedback 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. 13. 2nd cycle <ul><li>Jan-Mar 2010, Enkhuizen </li></ul><ul><li>Digital Mathematical Environment (DME) www.fi.uu.nl/dwo/voho </li></ul><ul><li>Two 6vwo 17/18 yr olds </li></ul>
  14. 14. Design choices <ul><li>Follow from 1-to-1 sessions prototype </li></ul><ul><li>4 activities in 4 categories </li></ul><ul><li>Randomization (note “strange values”) </li></ul><ul><li>Crises </li></ul><ul><li>Feedback (many types, Hattie & Timperley) </li></ul><ul><li>Formative scenario’s: first a lot of feedback then gradually less </li></ul>Bokhove, C. (2010). Implementing feedback in a digital tool for symbol sense. . International Journal for Technology in Mathematics Education . 17 (3)
  15. 15. Crises <ul><li>“ 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. </li></ul><ul><ul><li>Van Hiele: crisis of learning </li></ul></ul><ul><ul><li>Productive failure (Kapur) </li></ul></ul><ul><ul><li>Impasse (VanLehn et al) </li></ul></ul><ul><ul><li>Perturbation (Doll) </li></ul></ul><ul><ul><li>Disequilibrium (Piaget) </li></ul></ul>
  16. 16. Digital mathematical environment http://www.fi.uu.nl/dwo/voho , developer: Peter Boon
  17. 17. Authoring tasks CUSTOM CORRECTIVE FEEDBACK DESCRIPTIVE BUG-RELATED HINTS
  18. 18. Example student work
  19. 19. In action
  20. 20. Another student example
  21. 21. 3rd cycle <ul><li>Oct/nov 2010 </li></ul><ul><li>11 schools, around 350 students </li></ul><ul><li>“ Algebra met Inzicht” (AmI) </li></ul><ul><li>www.algebrametinzicht.nl </li></ul>
  22. 22. Data collection & analysis <ul><li>Scores per module in DME </li></ul><ul><li>Pre- and posttest scores </li></ul><ul><li>Attitude scales </li></ul><ul><li>General characteristics </li></ul><ul><li>Log files </li></ul><ul><li>Log notes </li></ul><ul><li>Audio, video clips </li></ul>Excel
  23. 24. Analyzing log data C4.5 Decision trees
  24. 25. Future? <ul><li>http://www.fi.uu.nl/~christianb/downloads/poc_equation.htm </li></ul>
  25. 26. Discussion <ul><ul><li>Questions? </li></ul></ul><ul><ul><li>Almost every time discussion understandibly ends with the cut Skills vs. Understanding </li></ul></ul><ul><ul><li>Is one is more dominant? </li></ul></ul><ul><ul><li>Does symbol sense exist apart from skills and/or vice versa </li></ul></ul><ul><ul><li>Methodology: what about distance learning? </li></ul></ul>
  26. 27. Selected references <ul><li>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 ) </li></ul><ul><li>Bokhove, C., & Drijvers, P. (2010). Symbol sense behavior in digital activities. For the Learning of Mathematics , 30 (3), 43-49. </li></ul><ul><li>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. </li></ul><ul><li>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. </li></ul><ul><li>Tall, D. (2008). The Transition to Formal Thinking in Mathematics. Mathematics Education Research Journal, 20(2), 5-24. </li></ul>
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