Remember Me
Or use your Academic/Social account:


Or use your Academic/Social account:


You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.


Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message


Verify Password:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Ainsworth, Shaaron E.
Languages: English
Types: Unknown
This thesis reports the design and evaluation of multi-representational learning environments that teach aspects of number sense. COPPERS is concerned with children's belief that mathematical problems can have only a single correct answer. CENTS addresses the skills and knowledge required for successful computational estimation. Although, there is much multi-representational software and a significant body of research which suggests that learning with multiple external representations (MERs) is beneficial, little is known about the conditions under which MERs promote effective learning. To address this, a framework was proposed for considering MERs. It consists of a set of dimensions along which multi-representational software can be described and specifies learning demands of MERs. This framework was used to generate predictions about the effectiveness of different multi-representational systems. Experiments investigated children's performance in multiple solutions and computational estimation before they received direct teaching and tested whether the learning environments could help children develop these skills. Each experiment examined how specific aspects of the learning environments contributed to learning outcomes. Experiments with COPPERS showed that children's pre-test performance was generally poor. Improved post-test performance on multiple solutions tasks occurred when children gave substantially more answers on the computer than their pre-test base-line. They rarely chose this strategy for themselves. It was found that providing a tabular representation of solutions in addition to the familiar row and column representation improved learning. Estimation is difficult for primary school children, but limited teaching led to substantial improvements in strategies and accuracy of estimates. Three experiments with CENTS addressed the effects of MERs on learning. When representations were too difficult to co-ordinate, then either children did not improve at understanding the accuracy of estimates, or focused their attention upon a single representation. Additionally, varying how information was distributed across representations influenced how representations were used. These experiments show that when considering learning with MERs, it is not sufficient to consider the effects of each representation in isolation. Behaviour with representations changes depending on how they are combined. These findings are discussed in terms of their implications for the design of multi-representational learning environments.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • Chapter 1: Introduction ...................................................................................................
    • Chapter Two: Number Sense- Computational Estimation and Multiple Solutions 3.8 CONCLUSION ............................................................................................................
    • 9.6.2 How much information eachrepresentationshould express..........................265 9.6.3 Similarity between Representations ...............................................................266
    • 9.6.4 How many representations?..........................................................................267 9.6.5 Automatic Translation
    • 9.7 FUTURE WORK.........................................................................................................271 9.7.1 Multiple Solutionsand ComputationalEstimation Dugdale, S. (1982). Green globs: A micro-computer application for graphing of equations. Mathematics Teacher, j, 208-214.
    • Duncker, K. (1945). On problem solving. Psychological Monographs, 5$(5).
    • Ehrlich, K., & Johnson-Laird, P. N. (1982). Spatial descriptions and referential continuity. Journal of Verbal Learning and Verbal Behaviour, 21,296-306.
    • Elsom-Cook, M. (1990). Guided discovery tutoring. London: Paul Chapman Publishing Ltd.
    • Fischbein, E., Drei, M., Nello, M. S., & Merino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, Ik(1), 3-17.
    • Forrester, M. A., Latham, J., & Shire, B. (1990). Exploring estimation in young primary school children. Educational Psychology, IQ(4), 283-300.
    • Fox, M. (1988) Theory and design for a visual calculator for arithmetic. CITE technical report no. 32, Institute of Educational Technology,Open University.
    • Fuson, K. C. (1992). Researchon whole number addition and subtraction. In D. Grouws (Eds.), Handbook of teaching and learning mathematics(pp. 243-275). New York: Macmillan.
    • Genter, D. (1989). The mechanismsof analogical learning. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp 199-242). Cambridge: CambridgeUniversity Press.
    • Gilmore, D. J., & Green, T. R. G. (1984). Comprehensionand recall of miniature programs.International Journal of Man-Machine Studies,21,31-48.
    • Gilmore, D. J. (1996). The relevance of HCI guidelines for educational interfaces. Machine-Mediated Learning, .(2), 119-133.
    • Green, T. R. G. (1989). Cognitive dimensions of notations. In A. Sutcliffe & L. Macaulay (Ed.), People and Computers . (pp 443-460). Cambridge: Cambridge University Press.
    • Green, T. R. G. (1990). The cognitive dimension of viscosity: A sticky problem for HCI. In D. Diaper, D. Gilmore, G. Cockton, & B. Shackel (Ed.), HumanComputer Interaction - INTERACT 90. (pp 79-86). Amsterdam: Elsevier Science.
    • Green, T. R. G., Petre, M., & Bellamy, R. K. E. (1991). Comprehensibility of visual and textual programs: a test of superlativism against the 'match-mismatch' conjecture. In Proceedings of Empirical Studies of Programmers: Fourth Workshop (pp. 121-146).
    • Green, T. R. G., & Petre, P. (1996). Usability analysis of visual programming environments: a 'cognitive dimensions' framework. Journal of Visual Languages and Visual Computing, 2,134-174.
    • Greer, B. (1992). Multiplication and division as models of situations. In D. Grouws (Eds. ), Handbook of teaching and learning mathematics (pp 276-295) New York: Macmillan PublishingCompany.
    • Hennessy, S., O'Shea, T., Evertsz, R., & Floyd, A. (1989). An intelligent tutoring system approach to teaching p ismary mathematics (CITE Report No. 93). Institute of Educational Technology, Open University.
    • Hennessy, S., Twigger, D., Driver, R., O'Shea, T., O'Malley, C., Byard, M., Draper, S., Hartley, R., Mohamed, R. & Scanlon, E. (1995) Design of a computeraugmented curriculum for mechanics. International Journal of Science Education. 17(11,75-92.
    • Hofstadter, D. R. (1985). Metamagical themas: Questing for the essence of mind and pattern. London: Penguin Books.
    • Howe, C.J., Rodgers, C., and Tolmie, A. (1990) Physics in the primary school; peer interaction and the understanding of floating and sinking. European Journal of Psychology of Education. 459-75.
    • Hoz, R., & Harel, G. (1989). The facilitating role of table forms in solving algebra speed problems: Real or imaginary. In Proceedings of the 13th International Conference for the Psychology of Mathematics Education, vol. 2 (pp. 123- Kaput, J. J. (1989). Linking representations in the symbol systems of algebra. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of Algebra (pp. 167-194).Hillsdale, NJ: LEA.
    • Kaput, J. J. (1992). Technology and mathematics education. In D. A. Grouws (Ed. ), Handbook of teaching and learning mathematics (pp. 515-556) New York: Macmillan Publishing Company.
    • Kaput, J. (1994). Democratizing accessto calculus: New routes using old roots. In A. Schoenfeld (Eds. ), Mathematical thinking and problem solving (pp. 77-156). Hillsdale, NJ: LEA.
    • Kaput, J., & Maxwell-West, M. (1994). Missing-value proportional reasoning problems: Factors affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 237-292). Albany, New York: State University of New York Press.
    • Koedinger,K. R., & Anderson,J. R. (1990). Abstractplanning and perceptualchunks: Elementsof expertisein Geometry.Cognitive Science,14,511-550.
    • Laborde, C. (1996). Towards a new role of diagrams in dynamic geometry? In Proceedings of the European Conference of Artificial Intelligence in Education, (pp. 350-356). Lisbon.
    • Lampert, M. (1986a). Knowing, doing, and teaching multiplication. Cognition and Instruction, .(4), 305-342.
    • Lampert, M. (1986b). Teaching multiplication. Journal of Mathematical Behaviour, (3), 241-280.
    • Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 21(1), 29-63.
    • Larkin, J. H., & Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science, II, 65-99.
    • LeFevre, J., Greenham,S. L., & Waheed,N. (1993). The developmentof procedural and conceptual knowledge in computational estimation. Cognition and Instruction 11(2), 95-132.
    • Lepper, M. R., Woolverton, M., Mumme, D. L., & Gurtner, J. (1993). Motivational techniques of expert human tutors: Lessonsfor the design of computer-based tutors. In S. J. Derry & S. P. Lajoi (Eds.), Computersas cognitive tools (pp. 75- 105).Hillsdale, NJ: LEA.
    • Levine, D. R. (1982). Strategy use and estimation ability of college students. Journal for Research in Mathematics Education, 11,350-359.
    • Lohse, G. L., Biolsi, K., Walker, N., & Rueler, H. (1994). A classification of visual representations. Communications of the A. C.M., fl(l2), 36-49.
    • Luchins, A. S., & Luchins, E. H. (1950). New experiemental attempts at preventing mechanization in problem solving. Journal of General Psychology. 4.2.279- Markovits, Z. (1989). Reactions to the number sense conference. In J. T. Sowder & B. P. Schappelle (Eds.), Establishing foundations for research on number sense Palmer,S. E. (1978). Fundamentalaspectsof cognitive representation.In E. Rosch& B. B. Lloyd (Eds.), Cognition and categorization (pp 259-303). Hillsdale, NJ: Plötzner, R. (1995). The construction and coordination of complementary problem representations in physics. Journal of Artificial Intelligence in Education, 6(2/3), 203-238.
    • Petre (1.993).Using graphical representationsrequires skill, and graphical readership is an acquired skill. In R. Cox, M. Petre,J. Lee, & P. Brna (Eds.), Proceedings of AI-ED93 workshop Graphical Representations. Reasoning and Communication. (pp. 55-58). Edinburgh.
    • Petre,M., & Green, T. R. G. (1993). Learning to read graphics: Some evidencethat 'seeing'an information display is an acquired skill. Journal of Visual Languages and Computing, 4,55-70.
    • Philipp, R. A., Flores, A., Sowder,J. T., & Schappelle,B. P. (1994). Conceptionsand practices of extraordinary mathematics. Journal of Mathematical Behaviour, f3(2), 155-180.
    • Pimm, D. (1995). Symbolsand meaningsin school mathematics.London: Routledge Price, M., & Foreman, J. (1989). A prIME experience. Mathematics in School Resnick, L. B. (1983). A developmental theory of number understanding. In H. P. Ginsburg (Eds. ), The development of mathematical thinking (pp. 109-151).
    • Resnick, L. B. (1989). Defining, assessingand teaching number sense. In J. T. Sowder & B. P. Schappelle (Ed.), Establishing foundations for research on number sense Roberts, M. J., Wood, D. J., & Gilmore, D. J. (1994). The sentence-picture verification task: Methodological and theoretical difficulties. British Journal of Rubenstein, R. N. (1985). Computational estimation and related mathematical skills. Journal for Research in Mathematics Education, I¢(2), 106-119.
    • Santos, M. S. (1994). Students approaches to solve three problems that involve various methods of solution. In Proceedings of the 18th International Conference for the Psychology of Mathematics Education, vol. 4 (pp. 193- Scaife, M., & Rogers, Y. (1996). External cognition: how do graphical representationswork? International Journal of Human-Computer Studies, 5, 185-213.
    • Schoen, H. L., Frisen, C. D., Jarrett, J. A., & Urbatsh, T. D. (1981). Instruction in Schwartz, D. L. (1995). The emergence of abstract representations in dyad problem solving. The Journal of the Learning Sciences, 4(3), 321-354.
    • Self, J. A. (1990). Bypassing the intractable problem of student modelling. In C. Frasson & G. Gauthier (Eds. ), Intelligent tutoring systems (pp. 107-123). Norwood, New Jersey: Ablex Publishing Company.
    • Snow, R. E., & Yalow, E. (1982). Education and Intelligence. In R. J. Sternberg (Eds.), A handbook of human intelligence (pp 493-586). Cambridge: Cambridge Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science,,12257-285.
    • Tabachneck, H. J. M., Koedinger, K. R., & Nathan, M. J. (1994). Towards a Tabachneck, H. J. M., Leonardo, A. M., & Simon, H. A. (1994). How does an expert Xploratorium (1991). Xploratorium.
    • 30% to 20% less 20% to 10% less 10% to 0% less 20% to 10% to 10%less 0% less 30% to 20% less 20% to 10% less 20% to 10% to 10%less 0% less
  • No related research data.
  • Discovered through pilot similarity algorithms. Send us your feedback.

Share - Bookmark

Cite this article