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Math-Towersand
Collaborative Learning Communities
Geoffrey Roulet
Faculty of Education
Queen’s University, Canada
geoff.roulet@queensu.ca
Online Interactive CollaborativeMathematics Explorations & Problem-Solving
for Students Ages 11 to 15 years(Grades 6 to 10)
Mathematics EducationReform
over the past two decades
has been driven by a
Social Constructivist
image of mathematics
Mathematics EducationReform
Social Constructivism
Mathematical
Talk
Student
Investigations
In Classrooms
In Classrooms
Emergence ofCollective Knowledge
Complex Phenomena
Complexity Science decentralized control
all pupils feel free to contribute ideas neighbour interactions
active exchange of ideas redundancy among agents
some overlap of ideas to support exchange internal diversity
divergence of ideas to stimulate thinking
Can Social ConstructivistMathematical Environments
be constructedOnline?
Many websites presenting interesting mathematical problems for student investigation
Some also provide tools (applets) to support investigation
Very few also support online collaboration
Can Social ConstructivistMathematical Environments
be constructedOnline?
Math-Towers
YesBut neither easy nor guaranteed
Active Exchange of Ideas
Adventure/Fantasy Context
Class is presented with a challenge by the
Lord or Lady
of the castle
Active Exchange of Ideas
Workplace & Tools
to support Investigation
Active Exchange of Ideas
Workplace & Tools
to support Investigation
Decentralized Control
Pupil not Computer
sets direction of Investigation
Active Exchange of Ideas
Communication Tool
for sharing ideas
Active Exchange of Ideas
Communication Tool
for sharing ideas
Redundancy Among Agents
Agreement on Initial Shared Ideas
Building ConfidenceGroup 2:
All square numbers have the pocket TR, 0 bounces and with 7x7=7 squares
Group 5:
If your table is a equal square it will do no bounces and go across 10 squares an go to the top right corner.
Group 4:
As long as the numbers are not the same both ways the whole thing will be filled in but if the numbers are the same it will just go in the hole across from where it is going from.
Group 6:
12 by 12 is 0 bounces and 12 squares
Internal Diversity
a Contradiction
promoting Renewed Investigation
Active Exchange of Ideas
Sharing of Revised Ideas
Building to a SolutionGroup 1:
When one side divides into the other you get only one bounce and less squares
Group 5:
6 by 8 doesn't cover the whole square.
Group 7:
Any Table with 10 and a odd number it takes up all squares!!!!!
Group 3:
we found anything by 11 goes over every single square
Group 6:
14 by 10 table is 70 squares and 11 bounces and in tr
Student Activity inMath-Towers
Shows evidence of: decentralized control
all pupils feel free to contribute ideas neighbour interactions
active exchange of ideas redundancy among agents
some overlap of ideas to support exchange internal diversity
divergence of ideas to stimulate thinking
Student Activity inMath-Towers
Shows evidence of the building of:
Collective Knowledge
Successful Progress through theTrial Chamber
Successful Climbingof theTower
Can Social ConstructivistMathematical Environments
be constructedOnline?
Can Social ConstructivistMathematical Environments
be constructedOnline?
Yes
the technology can support
Social ConstructivistMathematical Environments
Math-Towers provides: decentralized control
all pupils feel free to contribute ideas neighbour interactions
active exchange of ideas redundancy among agents
some overlap of ideas to support exchange internal diversity
divergence of ideas to stimulate thinking
When a group of students isinclined and coached towards
such activity
Math-Towers provides: decentralized control
all pupils feel free to contribute ideas neighbour interactions
active exchange of ideas redundancy among agents
some overlap of ideas to support exchange internal diversity
divergence of ideas to stimulate thinking
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