1. 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

2. 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.

3. 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

4. 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.

5. 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

6. Cognitive Theory of Multimedia Learning (Mayer, 2001) CTML framework is based on three assumptions: cognitive theories of dual channels, limited capacity and active processing.

7. Multimedia Learning Principles (Mayer, 2001)  Multimedia Principle It is better to present an explanation in words and pictures than solely in words.  Coherence Principle Exclude extraneous words, pictures, and background sounds.  Spatial Contiguity Principle 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.

8. 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.

9. 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.”

10. Types of Technology Representations Pictorial only Simulations Combined Pictorial and Numeric

11. Types of Technology Representations  Features of Concept Tutorials Provides Directions (words) Includes Numeric Information Includes Pictorial Models Provides Feedback

12. Research Question 1 How does the presentation modality (virtual manipulatives, physical manipulatives) influence students’ learning of fraction and algebra concepts?

13. Research Question 2 Are there differences in students’ responses by the test item mode of representation (pictorial/symbolic)?

14. Research Question 3 How can cognitive load theory be used to interpret the results in these two presentation modalities?

15. Participants and Setting  36 third grade students in two classes at the same school for 12 days of instruction

16. Materials  Physical manipulatives- deluxe fraction circles Hands on Equations  Virtual Manipulatives fraction applets Algebra Balance applet

17. Materials  Virtual Fraction Symbolic representation Iconic representation

18. Materials  Physical Fraction Circles and Fraction CD Mat

19. Materials  Virtual Algebra Balance Scale

20. Materials  Physical Hands-On Equations ®

21. 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.

22. Cross-over design Fraction Unit Posttest #1 Algebra Unit Posttest #2 Data Collection Group 1 Pretest on Fraction and Algebra Physical Manipulative Fraction Circles Fraction Posttest Virtual Manipulative - Algebra Balance Scale Algebra Posttest Interview & Preference Survey Group 2 Pretest on Fraction and Algebra Virtual Manipulative- Fraction Adding Applet Fraction Posttest Physical Manipulative -Hands on Equations Algebra Posttest Interview & Preference Survey

23. Data Sources  Pretest  Posttests: Fraction/ Algebra Achievement tests with 8 pictorial, 8 symbolic items  Field notes, student interviews, and classroom videotapes.

24. Results and Conclusions

25. Results #1: Analysis by manipulatives and mathematics concepts

26. 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 Algebra83.33 (SD = 14.34) Group 1 80.00 (SD = 20.16) Group 2 Fraction75.55(SD = 19.91) Group 2 45.55 (SD = 17.05) Group 1

27. Analysis by manipulative type and mathematics concept Source dfFp Manipulative Types 115.03.000 *** Mathematics Concepts 124.11.000 *** Manipulatives x Concept 19.62.003 **

28. Line plot show an interaction effect

29. 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.

30. 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. ”

31. 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.”

32. Results #2: Analysis of test items

33. Results: Question 2: Are there differences in students’ responses by the test item mode of representation (pictorial/symbolic)? 3/8+1/2= 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? Symbolic items Word problems Pictorial items

34. Results: Question 2: Analysis by test item types Mathematics content and Manipulative condition Pictorial test items (8 problems) Symbolic test items (8 problems) Physical manipulatives- Fraction Circles (Group 1) 58.33 (SD=21.86) 22.22 (SD=31.37) Virtual manipulatives- Fraction Applets (Group2) 86.11 (SD=17.61) 70.83 (SD=29.70)

35. 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 -36.11.001 *** Physical Manipulative – Word Items -50.00.000 *** Virtual Manipulative – Pictorial Items -63.88.000 *** Virtual Manipulative – Symbolic Items -48.61.001 *** Virtual Manipulative – Word Items -55.55.000 ***

36. 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.

37. Results #3: Unique features of the manipulatives that impacted learning

38. 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.

39. 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.

40. 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.

41. Implications for Classroom Instruction and Recommendations 1) Consider different manipulatives for different purposes 2) Professional training 3) Time for reflective discussion to support students’ making mathematics connections

42. 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.

43. Thank you for your attention! jsuh4@gmu.edu pmoyer@gmu.edu