Designing and using interactive applets for conceptual ...
Transcript of Designing and using interactive applets for conceptual ...
Designing and using interactive applets forconceptual understanding
Anthony Morphett
The University of Melbourne
ANZMC Melbourne
10 December 2014
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Visualisation and conceptual thinking
How can we support our students to develop a solid conceptual
understanding of mathematics & statistics?
I visualisation
visual representations of concepts, relationships
abiding images
I interactivity
students take ownership of visualisation by manipulating it
themselves
−→ interactive applets for conceptual learning
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Visualisation and conceptual thinking
How can we support our students to develop a solid conceptual
understanding of mathematics & statistics?
I visualisation
visual representations of concepts, relationships
abiding images
I interactivity
students take ownership of visualisation by manipulating it
themselves
−→ interactive applets for conceptual learning
2 / 20
Visualisation and conceptual thinking
How can we support our students to develop a solid conceptual
understanding of mathematics & statistics?
I visualisation
visual representations of concepts, relationships
abiding images
I interactivity
students take ownership of visualisation by manipulating it
themselves
−→ interactive applets for conceptual learning
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Interactive applets
Why use applets?
I Visual representations of concepts, relationships
I Targeted conceptual focus
I Tailored to a particular teaching context
I Transferrable across learning/teaching domains
I Flexible – multiple uses, entry points
I Accessible – low barriers to use
I Interactive – telling a story
I Engaging – fun, creative thinking
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Limit of a sequence - ε-M
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Limit of a sequence - ε-M
Visualisation:
I blue/orange regions
I red/green points
Targeted:
I Difficult but important concept
I Compare two sequences – based on teaching need
Flexible:
I convergence
I divergence
I bounding
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Limit of a sequence - ε-M
Coherence:
I Same notation as lectures
I Same colour/layout as related ε-δ applet
Transferrable:
I Use in lectures, one-on-one consultations
I Common ‘visual vocabulary’ for discussions
Interactive:
I Reveal components one-by-one when ready
I Enhances dialogue
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Differentiability
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Differentiability
Visualisation:
I Multiple representations
I Clear image of why/how differentiability fails
Targeted:
I Deep understanding of concept
I Address common misconceptions
I Supports key examples
Interactive:
I Leaves a ‘trace’ of previous actions
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CI’s, hypothesis testing and p-valuesVariation of p with μ0 in a hypothesis test
Drag μ0 (the point on the lower graph) to see how the p -value changes (the point on the upper graph).
The lower graph shows the distribution for X under the null hypothesis. The two tails corresponding to p areshaded.The upper graph shows how the p -value changes as the distance between μ0 and x changes.Drag the point on the lower graph to change μ0 , or click the play button in the bottom-left corner to animate μ0.The sample mean x and the standard error σ
n√ remain fixed.
13 September 2013, Created with GeoGebra
x
x − 1.96nσ
√
X
X
p = 0.45
|x − μ | = 0.75nσ
0 √
x + 1.96nσ
√
Variation of p with μ0 in a hypothesis test - GeoGebra Dynamic Worksheet http://www.ms.unimelb.edu.au/~awmo/demos/p-value-geogebra.html
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CI’s, hypothesis testing and p-values
Visualisation:
I Linking concepts often treated separately
I Multiple visual representations of accept/reject regions
I Challenging viewpoint: x̄ is fixed, µ0 changes
Flexible:
I Simple: accept/reject regions and confidence interval
I More challenging: p vs. µ0
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CI’s, hypothesis testing and p-values
Interactive:
I Question: what would the graph of p vs µ0 look like?
I Think then test
I Reveal components one-by-one when ready
Engaging:
I ‘Drag me!’
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Principles
Minimise technological barriers
I applet ‘just works’ in most browsers, devices
I uses familiar syntax
I hosting taken care of by Geogebratube
I easily distributed via web link, etc
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Principles
Minimise cognitive load
I correspondence between user interface elements (view) and
conceptual elements (model)
I physical interaction - tactile, ‘embodied cognition’
I colour coding of semantically related elements
Reduce extraneous mental effort
Maximise mental resources available for concepts
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What else are we doing?
Applets for
I Calculus: sequences & series, Riemann sums, ODEs
I Statistics: confidence intervals & hypothesis testing, power, random
variables, order stats, MLEs, ...
I Others: eigenvectors, difference equations
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What else are we doing?
Supporting resources
I online tutorial exercises
I teaching notes
I ‘how-to’ guides or similar
Evaulation
I quick surveys immediately after applet use
I collect analytics data
I focus groups, interviews etc
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GeoGebra
The applets are constructed in GeoGebra
I Freely available interactive geometry/graphing/CAS system
I Open source
I Java application, cross-platform (Windows, Mac, iPad ...)
I Developed by educators, for education
I Increasingly popular in secondary education
www.geogebra.org
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GeoGebra
Geogebra is a good platform for such projects.
I Rapid development
I Minimal programming - build by construction
I Extensive documentation & community support
I Exports to HTML5 - no Java, plugins required!
I Host applets publicly (Geogebratube) or privately (Moodle, etc)
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GeoGebra
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Our applets may be found at
http://www.melbapplets.ms.unimelb.edu.au
or at our GeoGebratube profile
http://geogebratube.org/user/profile/id/36916
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Acknowledgements
Project members:
I Sharon Gunn
I Robert Maillardet
I Anthony Morphett
Research assistants:
I Max Flander
I Sabrina Rodrigues
I Simon Villani
Associates:
I Deb King
I Robyn Pierce (MGSE)
I Christine Mangelsdorf
I Liz Bailey
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