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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