A Langauge of Patterns for Mathematical Learning

A language of patterns for

mathematical learning

Yishay Mor,

London Knowledge Lab

[email protected]

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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