THE APPLICATION OF DUAL CODING
THEORY IN MULTI-REPRESENTATIONAL
VIRTUAL MATHEMATICS ENVIRONMENTS
PME 31 SEOUL, KOREA
July 8-13, 2007
Jennifer M. Suh, Ph.D.
George Mason University
Patricia S. Moyer-Packenham, Ph.D.
George Mason University
Overview of the Study
This study applies Dual Coding Theory
(DCT) and Cognitive Theory of Multimedia
Learning in a mixed methods study
comparing mathematics achievement in
two third-grade classrooms using two
different representations: virtual
manipulatives and physical manipulatives.
Overview of our presentation
Brief overview of theoretic framework including
cognitive science, multimedia learning, and
mathematics education
Discussion of research in a third grade classroom
using virtual and physical manipulatives
Implication of this study in relation to cognitive
science, multimedia learning and mathematics
education
THEORETICAL FRAMEWORK
Dual Coding Theory
(Clark & Paivio 1991, Paivio, 1986, 2006)
Information for
memory is processed
and stored by two
interconnected
systems and sets of
codes. These sets of
codes include visual
codes and verbal
(symbolic) codes.
THEORETICAL FRAMEWORK
Cognitive Load Theory
(Chandler & Sweller, 1994; Sweller, 1988, 1999)
The premise of Cognitive Load Theory is
that an individual’s working memory has a
limited capacity for attending to and
processing incoming sensory data.
intrinsic cognitive load
 extraneous cognitive load

Cognitive Theory of
Multimedia Learning
(Mayer, 2001)
CTML framework is based on three assumptions: cognitive
theories of dual channels, limited capacity and active
processing.
Multimedia Learning Principles
(Mayer, 2001)

Multimedia Principle


Coherence Principle


Exclude extraneous words, pictures, and background sounds.
Spatial Contiguity Principle


It is better to present an explanation in words and pictures
than solely in words.
Present corresponding words and pictures near rather than far
from each other or simultaneously rather than successively.
Individual Differences Principle

Design principles are more important for low- rather than
high-knowledge learners, for for high- rather than low-spatial
learners.
Physical Manipulatives
When manipulatives are used, the senses
are brought into learning: students can
touch and move objects to make visual
representations of mathematical concepts.
Virtual Manipulatives
Moyer, Bolyard, and Spikell (2002) defined a virtual
manipulative as “an interactive, Web-based visual
representation of a dynamic object that presents
opportunities for constructing mathematical knowledge.”
Types of Technology Representations

Pictorial only
Simulations

Combined Pictorial and Numeric
Types of Technology Representations

Features of Concept Tutorials




Provides
Includes
Includes
Provides
Directions (words)
Numeric Information
Pictorial Models
Feedback
Research Question 1
How does the presentation modality
(virtual manipulatives, physical
manipulatives) influence students’ learning
of fraction and algebra concepts?
Research Question 2
Are there differences in students’
responses by the test item mode of
representation (pictorial/symbolic)?
Research Question 3
How can cognitive load theory be used to
interpret the results in these two
presentation modalities?
Participants and Setting

36 third grade students in two classes at
the same school for 12 days of instruction
Materials

Physical manipulativesdeluxe fraction circles
Hands on Equations

Virtual Manipulatives
fraction applets
Algebra Balance applet
Materials

Virtual Fraction
Iconic
representation
Symbolic
representation
Materials

Physical Fraction Circles and Fraction CD Mat
Materials

Virtual Algebra Balance Scale
Materials

Physical Hands-On Equations®
Research Design
Within-subject crossover repeated measures
(Campbell & Stanley, 1963)
All subjects received both treatments, which
allowed each student to serve as his or her own
comparison control.
Cross-over design
Fraction
Unit
Posttest
#1
Algebra
Unit
Posttest
#2
Data
Collection
Group Pretest
on
1
Physical
Manipulative
Fraction
Circles
Fraction
Posttest
Virtual
Manipulative
- Algebra
Balance
Scale
Algebra
Posttest
Interview &
Preference
Survey
Group Pretest
on
2
Virtual
ManipulativeFraction
Adding Applet
Fraction
Posttest
Physical
Manipulative
-Hands on
Equations
Algebra
Posttest
Interview &
Preference
Survey
Fraction
and
Algebra
Fraction
and
Algebra
Data Sources
Pretest
 Posttests: Fraction/ Algebra Achievement
tests with 8 pictorial, 8 symbolic items
 Field notes, student interviews, and
classroom videotapes.

Results and Conclusions
Results #1:
Analysis by manipulatives and
mathematics concepts
Results: Question 1: How does the presentation modality
(virtual manipulatives, physical manipulatives) influence
students’ learning of fraction and algebra concepts?
Mathematics
Content
Virtual Manipulative
Treatment
Physical Manipulative
Treatment
Algebra
83.33 (SD = 14.34)
Group 1
80.00 (SD = 20.16)
Group 2
Fraction
75.55(SD = 19.91)
Group 2
45.55 (SD = 17.05)
Group 1
Analysis by manipulative type and
mathematics concept
Source
df
F
p
Manipulative Types
1
15.03
.000
***
Mathematics Concepts
1
24.11
.000
***
Manipulatives x Concept
1
9.62
.003
**
Line plot show an interaction effect
Assessment Mean for Group 1 & 2
90%
83%
80%
80%
76%
70%
60%
50%
46%
40%
30%
20%
10%
0%
Fraction
Algebra
Group 1/ Physical Manipulative Fraction -Virtual Algebra Applet
Group 2/Virtual Fraction Applet-Physical Algebra Balance
Summary Results for Question 1
For Fractions, the virtual fraction treatment
group performed statistically better than
the physical fraction circles group on the
posttest.
For Algebra, there was no significant
difference between the two groups.
Students’ voices: Fraction applet
“I like working on the computer with the
tools because it helps me learn more.”
“It shows you pictures.”
“The magic arrow button helped me
because I can see when they (the two
fractions) are equal.”
“You can do it step by step.”
Students’ voices: Fraction circles
“ I liked using the fraction circles and the
fraction mat, but it was hard to solve
problems when you couldn’t find the
common denominator on the fraction
mat.”
“It was hard to find the right fraction
pieces because they were not labeled”
“Some pieces got lost and fell on the floor.”
Results #2:
Analysis of test items
Results: Question 2: Are there differences in students’
responses by the test item mode of representation
(pictorial/symbolic)?
Pictorial items
Symbolic items
3/8+1/2=
Word problems
Mr. Mahlio bought 1/2 pound of ham and 1/3 pounds of turkey for his
sandwich. How much meat did he buy for his big lunch?
Results: Question 2: Analysis by test item types
Mathematics content
and Manipulative
condition
Pictorial test
items
(8 problems)
Symbolic test
items
(8 problems)
Physical
manipulativesFraction Circles
(Group 1)
58.33
(SD=21.86)
22.22
(SD=31.37)
Virtual manipulativesFraction Applets
(Group2)
86.11
(SD=17.61)
70.83
(SD=29.70)
The physical fraction group performed
significantly lower on the symbolic items
compared to all the other test item scores.
Physical Manipulative
Symbolic Items
Physical Manipulative –
Pictorial Items
Physical Manipulative –
Word Items
Virtual Manipulative –
Pictorial Items
Virtual Manipulative –
Symbolic Items
Virtual Manipulative –
Word Items
-36.11
-50.00
-63.88
-48.61
-55.55
.001
***
.000
***
.000
***
.001
***
.000
***
Summary Results: Question 2:

The virtual fraction group performed
significantly better on the pictorial and
symbolic items than the physical fractions
group.

For both treatment groups, the pictorial
test item scores were higher than the
symbolic test items.
Results #3:
Unique features of the
manipulatives
that impacted learning
Results #3: How can cognitive load theory be used to
interpret the results in these two presentation modalities?
Unique features of Physical manipulatives
1) Tactile feature;
2) Physical representation of the symbolic
expression;
3) Encouraged inventive strategies and
mental mathematics; and
4) Over reliance on the manipulatives.
Results #3: How can cognitive load theory be used to
interpret the results in these two presentation modalities?
Unique features of virtual manipulatives:
1) Explicit link between the visual mode
and the symbolic mode;
2) Unique dynamic features;
3) Guided step by step support in
algorithmic process; and
4) Immediate feedback and self- checking
system.
Summary Results: Question 3
There were more features in the virtual
manipulative environment that guided
algorithmic thinking.
Students in the physical manipulative
environment showed inventive strategies
and mental mathematics to solve
problems.
Implications for Classroom
Instruction and Recommendations
1)
2)
3)
Consider different manipulatives for
different purposes
Professional training
Time for reflective discussion to support
students’ making mathematics
connections
Implication for Research
Research should be conducted on other
concept tutorials to see if they can help
facilitate learning of complex algorithms
while providing conceptual understanding.
Thank you for your
attention!
jsuh4@gmu.edu
pmoyer@gmu.edu