Presentation Mathmaster 031210
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Use of ICT for acquiring, practicing and assessing algebraic expertise
Christian [email protected]/~christianb
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.
• Freudenthal Institute of Science and Mathematics Education, Utrecht University, the Netherlands
• Supervisors: Paul Drijvers and Jan van Maanen• Educational research
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
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)
Mathematical proficiency
Kilpatrick
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"
Tall: procept
Algebraic expertise
• Arcavi• Kop & Drijvers• Pierce & Stacey• (Structure sense,
e.g. Hoch & Dreyfus)
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
Design research
(Tessmer)
Prelim: criteria for toolsFirst 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)
1st cycle: 1-to-1s
Qual. analysis (video, camtasia, atlas TI)
Symbol SenseQuality of tool Feedback
6 multihour think-aloud1-to-1 sessions with
17/18 year oldsI want to know what’s going
on in their minds
2nd cycle
Jan-Mar 2010, Enkhuizen
Digital Mathematical Environment (DME)www.fi.uu.nl/dwo/voho
Two 6vwo 17/18 yr olds
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)
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)
Digital mathematical environment
http://www.fi.uu.nl/dwo/voho , developer: Peter Boon
CUSTOM CORRECTIVE FEEDBACK
DESCRIPTIVE
BUG-RELATED
HINTS
Authoring tasks
Example student work
In action
Another student example
3rd cycle
• Oct/nov 2010
• 11 schools, around 350 students
• “Algebra met Inzicht” (AmI)
• www.algebrametinzicht.nl
Data collection & analysis
• Scores per module in DME
• Pre- and posttest scores
• Attitude scales
• General characteristics
• Log files
• Log notes
• Audio, video clips
Excel
Analyzing log data
C4.5Decision trees
Future?
http://www.fi.uu.nl/~christianb/downloads/poc_equation.htm
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?
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)
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.