a langauge of patterns for mathematical learning
DESCRIPTION
Talk at the Technion, Haifa, 13 April,2008TRANSCRIPT
A language of patterns for
mathematical learning
Yishay Mor,
London Knowledge Lab
10 April, 2008,
Department of Education in Technology and Science, Technion, Haifa
ProblemKeep the rain out
ContextCold, wet, poor.
Method of solutionThatched roof
RelatedTimber frame, Slanted roof,Chimney
Design pattern: problem + context +
method of solution
Construction, communication,
collaboration => mathematical learning
patterns for mathematical learning
The Problem
Learning mathematics is a complex business.
Building technology is a complex business.
Building technology for learning mathematics is
complex2.
Even when someone gets it right, success is
hard to replicate.
WebLabs (http://www.weblabs.eu.com)
• EU funded, Sept. 2002 – Sept. 2005, directed by professors Richard Noss and Celia Hoyles. Grant # IST-2001-32200.
• Students & researchers from UK, Italy, Sweden, Bulgaria, Portugal, Cyprus.
creating new ways of representing and expressing mathematical and scientific knowledge in European
communities of young learners (10 – 14).
Our aim was to transform the web into a medium in which European students collaboratively construct and critique
each others' evolving knowledge and working models.
patterns for mathematical learning
Learning patterns
Kaleidoscope JEIRP: 1 year,
15-20 members, 7 institutes, 6 countries
~24 case studies, ~150 patterns
patterns for mathematical learning
Today's talk
Design science and design patterns (the short
version).
Context – number sequences, construction,
collaboration.
A bit of theory.
Some patterns.
Future work.
patterns for mathematical learning
design …
“everyone designs who devises courses of action aimed at changing existing situations into desired ones” (Simon, 1969, p 129).
patterns for mathematical learning
… based research
Design based research is a methodology for the study of function. Often referred to as design research or design experiments. Concerned with the design of learning processes, taking account of the
involved complexities, multiple levels and contexts of educational settings. The primary aim is to develop domain-specific theories in order to
understand the learning process.(Mor & Winters, 2006)
patterns for mathematical learning
Design patterns
[describe] a problem which occurs over and over again in our environment, and then describes the core of the solution to that problem, in such a way that you can use this solution a million times over, without ever doing it the same way twice(Alexander et al., 1977)
patterns for mathematical learning
example: activity nodes
Design problemCommunity facilities scattered individually through the city do nothing for the life of the city.
Design solutionCreate nodes of activity throughout the community, spread about 300 yards apart.
http://www.uni-weimar.de/architektur/InfAR/lehre/Entwurf/Patterns/030/ca_030.html
patterns for mathematical learning
pattern structure
• Problem / intent
• Context
• Solution
• Examples
• Related patterns
• Notes
The theory (ies)
The patterns
Mathematical game pieces
Mathematical content is often injected
artificially into games or other activities, as
sugar-coating. This has a dual effect of
ruining the game and alienating the
mathematics. By contrast, for many
mathematicians, mathematics is the game.
Problem / Intent
Context
Games for mathematical learning.
Mathematical game pieces (II)
Identify an element of the mathematical content you wish
to address in this game.
Find a visual, animated or tangible representation of this
element which is consistent with the game metaphors.
Design your game so that these objects have clear
purpose and utility as game elements in the gameplay
structure.
Mathematical game pieces: examples
Soft scaffolding
Technology should be designed
to scaffold learners' progress, but
an interface that is too rigid
impedes individual expression,
exploration and innovation.
Problem / Intent
Context
Interactive learning interface
Soft scaffolding (II)
Provide scaffolding which can easily be overridden by the learner
or by the instructor. Let the scaffolding be a guideline, a
recommendation which is easier to follow than not, but leave the
choice in the hands of the learner. For example:
When providing a multiple-selection interface, always include an
open choice, which the user can specify (select 'other' and fill in
text box).
When the user is about to stray off the desired path of activity,
warn her, ask for confirmation, but do not block her.
Soft scaffolding: examples
Narrative spaces
Constructing narrative is a fundamental mechanism for making
sense of events and observations. To leverage it, we must give
learners opportunities to express themselves in narrative form.
Problem / Intent
Context
Digital environments for
collaborative learning.
Narrative spaces (II)
Provide learners with a narrative space: a medium, integrated with
the activity design, which allows learners to express and explore
ideas in a narrative form:
Allow for free-form text, e.g. by supporting soft scaffolding.
Choose narrative representations when possible.
Mark narrative elements in the medium:
Clearly mark the speaker / author, to support a sense of voice.
Date contributions to support temporal sequentiality ('plot').
Use semi-automated meta-data to provide context.
Narrative spaces: examples
Narrative spaces: examples
Objects to talk with
Natural discourse makes extensive use of
artefacts: we gesture towards objects that
mediate the activity to which the
discussion refers. This dimension of
human interaction is often lost in
computerized interfaces.
Problem / Intent
interfaces which allow learners to
converse about a common activity.
Context
Objects to talk with (II)
Learning activities often involve the use or construction of
artefacts. When providing tools for learners to discuss their
experience, allow them to easily include these artefacts in the
discussion.
If the activity is mediated by or aims to produce digital artefacts,
then the discussion medium should allow embedding of these
artefacts. The medium should support a visual (graphical,
symbolic, animated or simulated) 1:1 representation of these
objects.
Objects to talk with: Example
This is the real graph that was produced by
the cumulate total of the halving-a-number
robot. It looks like the top of my graph but
I made the fatal mistake
of thinking it started at
zero. I also said it
wouldn’t go over 100,
which was very wrong.
Streams
EP-Streams
How do you represent an infinite object in a finite medium? How do
you model number sequences in a way which is consistent both
with intuition and with mathematical formalism?
Construction activities / Microworlds where learners use
programming to explore complex, dynamic or infinite structures.
Problem / Intent
Context
Streams in ToonTalk
Add-a-number Add-up Nest
a1, a
2, a
3 …
1
1ai,
12a
i, 1
3ai …
Guess my X
Sustaining a mathematical discussion is vital to the establishment
of socio-mathematical norms and to the collaborative construction
of knowledge in the community. This goal is especially difficult to
achieve in geographically distributed communities.
We address this by A challenge exchange game of build this
puzzles, using a league chart to orchestrate sustained social
interaction.
Problem / Intent
A Challenge exchange of Build this puzzles, using a League chart to orchestrate social interaction.
GmX: Example
The patterns
http://patternlanguagenetwork.org
stickmen: a visual language for design patterns?
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