Teaching Boolean Logic with augmented reality and boundary logic

IE 543 – Virtual Interface DesignTrond Nilsen

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Introduction

The Problem Background

Augmented Reality Boundary Logic

The Application Visualization Interaction

Justifications Conclusion & Future Work

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The Problem

Propositional / Boolean logic is fundamental to reason Necessary for rational argument Not well practiced or understood Abstract and verbal Normally taught formally in university

Intuitively understood even by children Should be taught earlier Important for decision making Can be understood visually

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Augmented Reality

Overlaying virtual imagery on real world Tangible User Interface

Physical objects mapped to virtual objects Particularly suitable for augmented reality

Caveat : Today’s AR often is not perfect The system described could be implemented with AR today But it would likely face significant difficulties in deployment Assumes hypothetical ‘ideal’ head mounted AR

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Background – Boundary Logic 1

Symbolic algebra based on division of space Fundamental symbol – enclosure ()

An enclosure divides space into ‘inside’ and ‘outside’ No cardinality or uniqueness.

Somewhat counter-intuitive when read symbolically

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()() = () Calling

(()) = <void> Crossing

((a)) = a Involution

(() a) = <void> Occlusion

a (b a) = a (b) Pervasion

Background – Boundary Logic 2

Expressions in Boolean logic map to expressions in boundary logic

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<void> = FALSE

() = TRUE

(a) = NOT a

a b = a OR b

((a) (b)) = a AND b

(a) b = IF a THEN b

((a) b) (a (b)) = a XOR b

(a b) ((a) (b)) = a IFF b

Visualizing Boundary Logic

Due to strong nesting rules, boundary logic is hierarchical in form and can be visualized as a tree of ‘pipes’ Truth of expression at left A, B, C are truth variables <void> is still false

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TRUE = () =

A OR B = () () =

A AND B = ((A) (B)) =

IF (A OR B) THEN (B AND C) = (A B) ((B) (C)) =

Visualizing Boundary Logic – AR

One marker per element Marker for each variable Marker for root Marker for crossing Markers for branch base and branch

Reference orientation from root Linking determined by placement

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Visualizing Boundary Logic – AR

Display symbolic representations alongside AR Valid marker placement shown through highlighting Activities:

Manual truth resolution Walk through a proof (inductive) Interactive proofs (deductive)

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Justification – Motivation

Novelty Novelty may break despondency ‘Wow!’ effect Varied content and learning activity

Less Formal Assume: Similarity to fun tasks = easier to motivate Less de-motivating than classroom teaching

Flow Requires dynamic activity with cycle of action & feedback Intensely motivating

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Justification – Inductive Learning

Two forms of reasoning: Deductive: learn by taking rules & facts, then extending Inductive: learn by generalizing rules from examples

Most mathematical teaching is deductive Can lead to a focus on syntactic application

Most students learn best with combo Inductive reasoning is often under-developed

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Justification – Experiential Anchoring

Correlation between strong experience and memory Richness, intensity, meaning Knowledge contextualize with experience is better recalled

Novelty of activity ‘Wow!’ effect

Interactivity supports student participation and ownership Social context Increased motivation

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Justification – Active & Spatial Learning

Direct mapped interaction Movement of marker tags directly affects expressions Reduces cognitive load, freeing attention

Supports active exploration of system Explore symbol combinations / expression configuration by

moving tags (similar to jigsaw puzzle) Particularly important for learning of spatial / configurational

knowledge

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Justification – Sensory Integration

Mayer’s Multimedia theory The more senses that are employed, the stronger the learning Applies for combinations of all ‘five’ senses

Tangible UI affords greater sensory integration Learning is stronger when multiple senses are engaged Particularly important for spatial learning. Supports kinaesthetic learners

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Justification – Learning Styles

Students utilize different learning styles Generally not exclusive The more learning styles supported, the better Many schemes

Verbal / Visual Global / Sequential Active / Passive Intuitive / Sensing

Particular teaching styles map to particular learning styles Traditional classroom mathematics tends to be visual,

sequential, passive, and intuitive System supports visual, global, active, sensing learners.

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Conclusion

AR system for teaching Boolean logic using visualization and manipulation of boundary logic

Justifications: Motivation Inductive Learning Experiential Anchoring Active & Spatial Learning Sensory Integration Learning Styles

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Future Work

Implement Hoped to, but unable to fit this into time available Suitable for implementation with ARToolkit

Evaluate vs traditional classroom teaching vs visual classroom teaching vs desktop PC equivalent

Improved theoretical basis for AR in education Apply to more complex algebras

Predicate logic, tense logic, etc Number theory

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