Enhancing Representations in Linear Algebra Through Virtual Reality

Ferdie Rivera, Plamen Koev, & Yingjie Liu, PIs

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Patrick Stafford, VR Programmer

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November 29, 2023

This material is based upon work supported by the National Science Foundation under Grant No. 2315756. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the foundation.

Advisory Board

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Wasin So, SJSU

Chris Rasmussen, SDSU

Abraham Wolcott, SJSU

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Michelle Zandieh, ASU, Consultant

Current Status

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We just started. We were funded August 2023.

The process is not that easy. We meet one hour a week.

But it’s fun and rewarding.

Representations in Linear Algebra

Algebraic

Vector Equation

Matrix Equation

Graphical

Research on Constructing Representations in Math (Vergnaud)

• Not static but a dynamic process

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• Determined by individual concepts-in-action and theories-in-action

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Documented Problems in Learning Linear Algebra:

Global Findings

Documented Problems in Learning Linear Algebra:

Global Findings

Documented Problems in Learning Linear Algebra:

Global Findings

Documented Problems in Learning Linear Algebra:

Global Findings

Documented Problems in Learning Linear Algebra:

Global Findings

Documented Problems in Learning Linear Algebra:

Local Findings - SJSU Math 129A/39 Data 2015-2022

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2015 - 2020: About 30% Math 129A and/or 39 students received D, F, and W/WU

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2015- 2022: About 21% received D, F, and W/WU

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Documented Problems in Learning Linear Algebra:

Local Findings - Classroom Data (n = 49) Spring 2021

Structural Discernment Tasks

Documented Problems in Learning Linear Algebra:

Local Findings - Classroom Data (n = 49) Spring 2021

Documented Problems in Learning Linear Algebra:

Local Findings - Classroom Data (n = 49) Spring 2021

Documented Problems in Learning Linear Algebra:

Local Findings - Classroom Data (n = 49) Spring 2021

Structural Construction Tasks

subspace

xxxx

X

Documented Problems in Learning Linear Algebra:

Local Findings - Classroom Data (n = 49) Spring 2021

Documented Problems in Learning Linear Algebra:

Local Findings - Classroom Data (n = 49) Spring 2021

Documented Problems in Learning Linear Algebra:

Local Findings - Classroom Data (n = 49) Spring 2021

Transfer Task

(Nonlinear transformation

Margalit & Rabinoff, 2019)

Documented Problems in Learning Linear Algebra:

Local Findings - Classroom Data (n = 49) Spring 2021

Documented Problems in Learning Linear Algebra:

Local Findings - Classroom Data (n = 49) Spring 2021

Documented Problems in Learning Linear Algebra:

Local Findings - Classroom Data (n = 49) Spring 2021

How to Address the Problems: Learn from Research

Algorithmic Thinking

Geometric intuition

Representational fluency

Metamathematical understanding

Formalist structure and language

How to Address the Problems: Use VR to Support 4E Cognition

4Es of Cognition: Learning mathematics is not just in the head. It involves a

complex of brain, body, and environment.

How to Address the Problems: Use VR to Support 4E Cognition

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Body contributes to learning; brain co-evolves with the body

(e.g., hand);

How to Address the Problems: Use VR to Support 4E Cognition

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Body is coupled to an environment (physical, social, cultural)

How to Address the Problems: Use VR to Support 4E Cognition

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The primary relationship between body and environment is geared toward action.

How to Address the Problems: Use VR to Support 4E Cognition

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Role of instruments and objects in the environment and how they mediate in learning.

How to Address the Problems: Use VR to Support Better Long Term/ Semantic Processing of Abstract Concepts

How to Address the Problems: VR Supports Semantic Processing by Concretizing the Abstract

Semantic Memory and Processing

Concrete Concepts

Abstract Concepts

Processed via senses

Processed via language

How to Address the Problems: VR Supports Semantic Processing by Concretizing the Abstract

Abstract Concepts

Processed via language

Our Research Questions

• To what extent does a visual-geometric and action-oriented VR-mediated intervention support and enhance a deep theoretical understanding of linear algebra concepts?

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• What is the nature of student reasoning elicited in a VR-mediated environment for learning linear algebra?

VR Modules

VR Modules

Types of activities

• Explore concepts and relationships
• Interact with simulations

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Advantages of the VR Modules

• Starting point of exploration is R³
• VR modules overcome constraints in other math apps (e.g., Desmos, Geogebra)
• VR modules will be available for use in all platforms

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Underlying conceptual unifying structure

• Spanning (two vector operations and multiplicative structures)
• Matrix multiplication (Ax and AB)

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VR Module 1 Vector Manipulation

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• Explore linear combinations of one or more vectors
• Describe geometrically subspaces of R³

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• Informally understand linear independence and dependence of sets of vectors

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VR Module 1 Vector Manipulation

• Useful in describing solutions sets of linear systems

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VR Module 1 Vector Manipulation

• Useful in describing solutions sets of linear systems

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VR 1 Vector Manipulation Structure

VR 1 Vector Manipulation Structure

Vector panel

Rotation panel

VR 1 Vector Manipulation Structure

Vector panel

Hides a vector

Opposite vector

Activates the vector’s translation function

(see the tip of the vector with the two colored perpendicular segments)

VR Module 2 Vector Operations

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• Explore vector addition of two and three vectors

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• Informally understand coordinate systems based on basis vectors

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Project Website

https://sites.google.com/sjsu.edu/ferdierivera/nsf-vr-grant-project

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