Use of ICT for acquiring, practicing and assessing algebraic expertise

Christian Bokhovec.bokhove@fi.uu.nlwww.fi.uu.nl/~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.