The authors are members of the research project “Promoting success in mathematics” (PTDC/CPE-CED/121774/2010), funded

by Fundação para a Ciência e Tecnologia, the Portuguese national funding agency for science, research and technology.

Mathematics in Pre-service Teacher Education and the quality of learning: The Monty Hall problem

Fernando Luís SantosPiaget Institute, Universidade Nova de Lisboa, UIED

António DomingosUniversidade Nova de Lisboa, UIED

context

Theoretical framing

Theoretical framing

Eddie Gray and David Tall

Advanced mathematical thinking, procept, proceptual divide

Theoretical framing

John Biggs and Kevin Collis

SOLO (Structure of the Observed Learning Outcome) taxonomy

Theoretical framing

Yrjö Engeström

Third generation of Activity Theory

setting

In a TV contest, a contestant chooses one of three doors; behind one of the doors there is a prize and behind the other two there is nothing. After the

competitor choose a door, the host opens one of the other and reveals that there is no prize. The host then asks the competitor's choice whether to keep or

want to switch. It is advantageous, in statistical terms, to switch or keep?

data

Two students (Raquel and Mariana)

Raquel from Business, Mariana from Education

Both correct answers

Two different kinds of answers

data

Raquel (Business student)

Raquel (Business student)

Raquel (Business student)

For her the problem is solved.

First outcome (decision tree) evaluated as possibly relational.

Second outcome (algorithm) evaluated as multi-structural.

Raquel (Business student)

Mariana (Education student)

=INT(RAND()*3)+1

On the two first cells, generates a random number from 1 to 3.Doors that the contestant could choose.

=IF(C4=B4;IF(B4=1;IF(RAND()<0,5;2;3);IF(B4=3;IF(RAND()<0,5;1;2);IF(RAND()<0,5;1;3)));IF(C4=1;IF(B4

=2;3;2);IF(C4=2;IF(B4=1;3;1);IF(B4=2;1;2))))

Generates one of three numbers avoiding the numbers on the firsts two cells, it's the door that the host will open.

=IF(D4=1;IF(C4=2;3;2);IF(D4=2;IF(C4=1;3;1);IF(C4=1;2;1)))

=IF(E4=B4;1;0)

Check for victory. 1 win, 0 lose.

Mariana (Education student)

This empirical experiment allow her to simulate the outcome.

This outcome was evaluated as relational close related to

extended abstract.

Mariana (Education student)

Final remarks

Evidences of proceptual divide: Raquel clearly with a procedural thinking taking refuge on algorithms and Mariana proceptual thinking with some meaningful combinations of procepts.

The model of analysis used to characterize the outcomes appears to work in identifying the different outcomes.

The authors are members of the research project “Promoting success in mathematics” (PTDC/CPE-CED/121774/2010), funded

by Fundação para a Ciência e Tecnologia, the Portuguese national funding agency for science, research and technology.

Mathematics in Pre-service Teacher Education and the quality of learning: The Monty Hall problem

Fernando Luís SantosPiaget Institute, Universidade Nova de Lisboa, UIED

António DomingosUniversidade Nova de Lisboa, UIED