# EFFECT OF THE PRESENCE OF EXTERNAL REPRESENTATIONS ON ACCURACY AND REACTION TIME IN SOLVING MATHEMATICAL DOUBLE-CHOICE PROBLEMS BY STUDENTS OF DIFFERENT LEVELS OF INSTRUCTION

- 535 Downloads
- 12 Citations

## Abstract

This study explores the effects of the *presence of external representations of a mathematical object* (ERs) on problem solving performance associated with short double-choice problems. The problems were borrowed from secondary school algebra and geometry, and the ERs were either formulas, graphs of functions, or drawings of geometric figures. Performance was evaluated according to the reaction time (RT) required for solving the problem and the accuracy of the answer. Thirty high school students studying at high and regular levels of instruction in mathematics (HL and RL) were asked to solve half of the problems with ERs and half of the problems without ERs. Each task was solved by half of the students with ERs and by half of the students without ERs. We found main effects of the representation mode with particular effect on the RT and the main effects of the level of mathematical instruction and mathematical subject with particular influence on the accuracy of students’ responses. We explain our findings using the cognitive load theory and hypothesize that these findings are associated with the different cognitive processes related to geometry and algebra.

## Key words

accuracy external representations internal representations level of instruction reaction time solving mathematical problems split attention effect## Preview

Unable to display preview. Download preview PDF.

## References

- Ashcraft, M. H. (1982). The development of mental arithmetic: A chronometric approach.
*Developmental Review, 2*, 213–236.CrossRefGoogle Scholar - Babai, R., Brecher, T., Stavy, R. & Tirosh, D. (2006a). Intuitive interference in probabilistic reasoning.
*International Journal of Science and Mathematics Education, 4*, 627–639.CrossRefGoogle Scholar - Babai, R., Levyadun, T., Stavy, R. & Tirosh, D. (2006b). Intuitive rules in science and mathematics: A reaction time study.
*International Journal of Mathematical Education in Science and Technology, 37*, 913–924.CrossRefGoogle Scholar - Bobis, J., Sweller, J. & Cooper, M. (1993). Cognitive load effects in a primary-school geometry task.
*Learning and Instruction, 3*, 1–21.CrossRefGoogle Scholar - Cohen Kadosh, R. & Walsh, V. (2009). Numerical representation in the parietal lobes: Abstract or not abstract.
*The Behavioral and Brain Sciences, 32*, 313–328.CrossRefGoogle Scholar - Cox, R. (1999). Representation construction, externalised cognition and individual differences.
*Learning and Instruction, 9*, 343–363.CrossRefGoogle Scholar - Funahashi, S. (Ed.). (2007).
*Representation and brain*. Tokyo, Japan: Springer.Google Scholar - Goldin, G. & Steingold, N. (2001). Systems of representations and the development of mathematical concepts. In A. A. Cuoco & F. R. Curcio (Eds.),
*The roles of representation in school mathematics. NCTM 2001 Yearbook*(pp. 1–23). Reston, VA: NCTM.Google Scholar - Goldin, G. A. & Kaput, J. J. (1996). A joint perspective on the idea of representation in learning and doing mathematics. In L. P. Steffe, P. Nesher, P. Cobb, G. A. Goldin & B. Greer (Eds.),
*Theories of mathematical learning*(pp. 397–430). Hillsdale, NJ: Erlbaum.Google Scholar - Goldin, G. A. (2003). Representation in school mathematics: A unifying research perspective. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.),
*A research companion to principles and standards for school mathematics*(pp. 275–285). Reston, VA: NCTM.Google Scholar - Groen, G. J. & Parkman, J. M. (1972). A chronometric analysis of simple addition.
*Psychological Review, 79*, 329–343.CrossRefGoogle Scholar - Hegarty, M. (2004). Diagrams in the mind and in the world: Relations between internal and external visualizations. In A. Blackwell, K. Mariott & A. Shimojima (Eds.),
*Diagrammatic representation and inference: Lecture notes in artificial intelligence 2980*(pp. 1–13). Berlin: Springer.Google Scholar - Hiebert, J. & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 65–97). New York, NY: Macmillan.Google Scholar - Janvier, C. (1987). Translations processes in mathematics education. In C. Janvier (Ed.),
*Problems of representation in the teaching and learning of mathematics*(pp. 27–32). Hillsdale, NJ: Erlbaum.Google Scholar - Janvier, C., Girardon, C. & Morand, J. (1993). Mathematical symbols and representations. In P. S. Wilson (Ed.),
*Research ideas for the classroom: High school mathematics*(pp. 79–102). Reston, VA: NCTM.Google Scholar - Jensen, A. R. (2006).
*Clocking the mind: Mental chronometry and individual differences*. Amsterdam, the Netherlands: Elsevier.Google Scholar - Kaput, J. (1989). Linking representations in the symbol systems of algebra. In S. Wagner (Ed.),
*Research issues in the learning and teaching of algebra*. NCTM: Reston, VA.Google Scholar - Kaput, J. (1998). Representations, inscriptions, descriptions and learning: A kaleidoscope of windows.
*The Journal of Mathematical Behavior, 17*(2), 265–281.CrossRefGoogle Scholar - Kilpatrick, J., Swafford, J. & Findell, B. (Eds.). (2001).
*Adding it up: Helping children learn mathematics*. Washington, DC: National Academy Press.Google Scholar - Larkin, J. H. & Simon, H. A. (1987). Why a diagram is (sometimes) worth 10,000 words.
*Cognitive Science, 11*, 65–100.CrossRefGoogle Scholar - Lesh, R., Landau, M. & Hamilton, E. (1983). Conceptual models in applied mathematical problem solving research. In R. Lesh & M. Landau (Eds.),
*Acquisition of mathematics concepts and processes*(pp. 263–343). New York, NY: Academic.Google Scholar - Lesh, R., Post, T. & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.),
*Problems of representation in the teaching and learning of mathematics*(pp. 33–40). Hillsdale, NJ: Erlbaum.Google Scholar - Luce, R. D. (1986).
*Reaction times: Their role in inferring elementary mental organization*. New York, NY: Oxford University Press.Google Scholar - Mayer, R. E. (2009).
*Multimedia learning*(2nd ed.). Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar - Mayer, R. E. & Anderson, R. (1991). Animations need narrations: An experimental test of a dual-coding hypothesis.
*Journal of Educational Psychology, 83*, 484–490.CrossRefGoogle Scholar - Mayer, R. E. & Anderson, R. (1992). The instructive animation: Helping students build connections between words and pictures in multimedia learning.
*Journal of Educational Psychology, 84*, 444–452.CrossRefGoogle Scholar - Miller, C. A. & Poll, G. H. (2009). Response time in adults with a history of language difficulties.
*Journal of Communication Disorders, 42*(5), 365–379.CrossRefGoogle Scholar - Moreno, R. & Mayer, R. E. (1999). Cognitive principles of multimedia learning: the role of modality and contiguity.
*Journal of Educational Psychology, 91*, 358–368.CrossRefGoogle Scholar - National Council of Teachers of Mathematics (NCTM) (1989).
*Curriculum and evaluation standards for school mathematics*. Reston, VA: The Council.Google Scholar - National Council of Teachers of Mathematics (2000). Standards for school mathematics. In
*Principles and standards for school mathematics*. Reston, VA: NCTM.Google Scholar - Nieder, A. & Dehaene, S. (2009). Representation of number in the brain.
*Annual Review of Neuroscience, 32*, 185–200.CrossRefGoogle Scholar - Pachella, R. G. (1974). The interpretation of reaction time in information processing research. In B. Kantowitz (Ed.),
*Human information processing: Tutorials in performance and cognition*(pp. 41–82). New York: Wiley.Google Scholar - Pape, S. J. & Tchoshanov, M. A. (2001). The role of representation(s) in developing mathematical understanding.
*Theory in Practice, 40*(2), 118–127.CrossRefGoogle Scholar - Posner, M. I. & McCleod, P. (1982). Information processing models—In search of elementary operations.
*Annual Review of Psychology, 33*, 477–514.CrossRefGoogle Scholar - Presmeg, N. (2006). Research on visualization in learning and teaching mathematics. In A. Gutirrez & P. Boero (Eds.),
*Handbook of research on the psychology of mathematics education: Past, present and future*. Sense: Rotterdam.Google Scholar - Schneider, W., Eschman, A. & Zuccolotto, A. (2002).
*E-prime computer software (version 1.0)*. Pittsburgh, PA: Psychology Software Tools.Google Scholar - Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin.
*Educational Studies in Mathematics, 22*, 1–36.CrossRefGoogle Scholar - Stavy, R. & Babai, R. (2008). Complexity of shapes and quantitative reasoning in geometry.
*Mind, Brain, and Education, 2*, 170–176.CrossRefGoogle Scholar - Sternberg, S. (1969). Memory scanning: Mental processes revealed by reaction time experiments.
*American Scientist, 57*, 421–457.Google Scholar - Sweller, J. (1994). Cognitive load theory, learning difficulty and instructional design.
*Learning and Instruction, 4*, 295–312.CrossRefGoogle Scholar - Sweller, J., Ayres, P. & Kalyuga, S. (2011).
*Cognitive load theory*. New York, NY: Springer.CrossRefGoogle Scholar - Sweller, J., van Merriënboer, J. J. G. & Paas, F. G. W. C. (1998). Cognitive architecture and instructional design.
*Educational Psychology Review, 10*, 251–296.CrossRefGoogle Scholar - Tarmizi, R. & Sweller, J. (1988). Guidance during mathematical problem solving.
*Journal of Educational Psychology, 80*, 424–436.CrossRefGoogle Scholar - Waisman, I., Shaul, S., Leikin, M. and Leikin, R. (2012). General ability vs. expertise in mathematics: An ERP study with male adolescents who answer geometry questions. In
*The electronic proceedings of the 12th International Congress on Mathematics Education*(*Topic Study Group*-*3*:*Activities and Programs for Gifted Students*), (pp. 3107–3116). Seoul, Korea: Coex.Google Scholar - Shaul, S., Leikin, M. Waisman, I., and Leikin, R. (2012). Visual processing in algebra and geometry in mathematically excelling students: an ERP study. In
*The electronic proceedings of the 12th International Congress on Mathematics Education*(*Topic Study Group*-*16*:*Visualization in mathematics education*) (pp. 1460–1469). Seoul, Korea: Coex.Google Scholar - Zhang, J. & Norman, D. A. (1994). Representations in distributed cognitive tasks.
*Cognitive Science, 18*, 87–122.CrossRefGoogle Scholar - Zhang, J. J. (1997). The nature of external representations in problem solving.
*Cognitive Science, 21*, 179–217.CrossRefGoogle Scholar