The Process of Probability Problem Solving: Use of External Visual Representations

REFERENCES

• Anderson , J. 1996 . ACT: A simple theory of complex cognition . American Psychologist , 51 ( 4 ) : 355 – 365 .
   Web of Science ®Google Scholar
• Campbell , K. J. , Collis , K. F. and Watson , J. M. 1995 . Visual processing during mathematical problem solving . Educational Studies in Mathematics , 28 ( 2 ) : 177 – 194 .
   Google Scholar
• Casey , D. P. 1978 . “ Failing students: A strategy of error analysis ” . In Aspects of motivation , Edited by: Costello , P. 295 – 306 . Melbourne : Melbourne Mathematical Association of Victoria .
   Google Scholar
• Clement , J. , Lochhead , J. and Monk , G. S. 1981 . Translation difficulties in learning mathematics . The American Mathematical Monthly , 88 ( 4 ) : 286 – 290 .
   Web of Science ®Google Scholar
• Clements , M. A. 1980 . Analyzing children's errors on written mathematical tasks . Educational Studies in Mathematics , 11 ( 1 ) : 1 – 21 .
   Google Scholar
• Cochran , W. G. 1950 . The comparison of percentages in matched samples . Biometrika , 37 : 256 – 266 .
   PubMed Web of Science ®Google Scholar
• Corter , J. E. and Zahner , D. 2007 . Use of external visual representations in probability problem solving . Statistics Education Research Journal , 6 ( 1 ) : 22 – 50 .
   Google Scholar
• Dean , M. J. 2006 . Explaining performance in the Third International Math and Science Study (TIMSS) 195 Advanced Mathematics Test. (Doctoral Dissertation, Columbia University, 2006) . Dissertation Abstracts International , 67 ( 4 ) : 1305
   Google Scholar
• de Hevia , M.-D. and Spelke , E. S. 2009 . Spontaneous mapping of number and space in adults and young children . Cognition , 110 ( 2 ) : 189 – 207 .
   Google Scholar
• Diezmann , C. M. . Evaluating the effectiveness of the strategy ‘Draw a diagram’ as a cognitive tool for problem solving . Proceedings of the 18th Annual Conference of Mathematics Education Research Group of Australasia . Darwin, Australia. Edited by: Atweh , B. and Flavel , S. pp. 223 – 228 . MERGA .
   Google Scholar
• Douville , P. and Pugalee , D. K. . Investigating the relationship between mental imaging and mathematical problem solving . Proceedings of the International Conference of the Mathematics Education into the 21st Century Project . Brno, Czech Republic. Edited by: Rogerson , A. pp. 62 – 67 . Mathematics Education into the 21st Century Project .
   Google Scholar
• Edens , K. and Potter , E. 2008 . How students “unpack” the structure of a word problem: Graphic representations and problem solving . School Science and Mathematics , 108 ( 5 ) : 184 – 196 .
   Google Scholar
• English , L. D. 1997 . Mathematical reasoning: Analogies, metaphors, and images , Mahwah, NJ : Lawrence Erlbaum .
   Google Scholar
• Ericsson , K. A. and Simon , H. A. 1993 . Protocol analysis: Verbal reports as data , Cambridge, MA : MIT Press .
   Google Scholar
• Ginsburg , H. 1997 . Entering the child's mind: The clinical interview in psychological research and practice , Cambridge, , UK : Cambridge University Press .
   Google Scholar
• Goodman , N. 1976 . Languages of art , Indianapolis, IN : Hackett .
   Google Scholar
• Hadamard , J. 1945 . The psychology of invention in the mathematical field , New York : Oxford University Press .
   Google Scholar
• Hall , V. , Bailey , J. and Tillman , C. 1997 . Can student-generated illustrations be worth ten thousand words? . Journal of Educational Psychology , 89 ( 4 ) : 677 – 681 .
   Google Scholar
• Hannafin , R. D. , Burruss , J. D. and Little , C. 2001 . Learning with dynamic geometry programs: Perspectives of teachers and learners . The Journal of Educational Research , 94 ( 3 ) : 132 – 144 .
   Web of Science ®Google Scholar
• Hannafin , R. D. and Scott , B. 1998 . Indentifying critical learner traits in a dynamic computer-based geometry program . The Journal of Educational Research , 92 ( 1 ) : 3 – 12 .
   Web of Science ®Google Scholar
• Hegarty , M. and Kozhevnikov , M. 1999 . Types of visual-spatial representations and mathematical problem solving . Journal of Educational Psychology , 91 ( 4 ) : 684 – 689 .
   Web of Science ®Google Scholar
• Hollebrands , K. 2003 . High school students’ understandings of geometric transformations in the context of a technological environment . Journal of Mathematical Behavior , 22 ( 1 ) : 55 – 72 .
   Google Scholar
• Hurley , S. M. and Novick , L. R. 2010 . Solving problems using matrix, network, and hierarchy diagrams: The consequences of violating construction conventions . The Quarterly Journal of Experimental Psychology , 63 ( 2 ) : 275 – 290 .
   Web of Science ®Google Scholar
• Kaufmann , G. 1990 . “ Imagery effects on problem solving ” . In Imagery: Current developments , Edited by: Hampson , P. J. , Marks , D. E. and Richardson , J. T. E. 169 – 197 . New York : Routledge .
   Google Scholar
• Kintsch , W. and Greeno , J. G. 1985 . Understanding and solving word arithmetic problems . Psychological Review , 92 ( 1 ) : 109 – 129 .
   PubMed Web of Science ®Google Scholar
• Kirshner , D. and Awtry , T. 2004 . Visual salience of algebraic transformations . Journal for Research in Mathematics Education , 35 ( 4 ) : 224 – 257 .
   Web of Science ®Google Scholar
• Konold , C. 1989 . Informal conceptions of probability . Cognition and Instruction , 6 ( 1 ) : 59 – 98 .
   Web of Science ®Google Scholar
• Koedinger , K. R. and Anderson , R. 1997 . Intelligent tutoring goes to school in the big city . International Journal of Artificial Intelligence in Education , 8 ( 1 ) : 30 – 43 .
   Google Scholar
• Landy , D. and Goldstone , R. 2007 . Formal notations are diagrams: Evidence from a production task . Memory & Cognition , 35 ( 8 ) : 2033 – 2040 .
   PubMed Web of Science ®Google Scholar
• Latour , B. and Woolgar , S. 1986 . Laboratory life: The social construction of scientific facts , Princeton, NJ : Princeton University Press .
   Google Scholar
• Lean , G. and Clements , M. A. 1981 . Spatial ability, visual imagery, and mathematical performance . Educational Studies in Mathematics , 12 ( 3 ) : 267 – 299 .
   Google Scholar
• Lehrer , R. , Schauble , L. , Carpenter , S. and Penner , D. E. 2000 . “ The inter-related development of inscriptions and conceptual understanding ” . In Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design , Edited by: Cobb , P. , Yackel , E. and McClain , K. 325 – 360 . Mahwah, NJ : Lawrence Erlbaum .
   Google Scholar
• Mayer , R. 1989 . Systemic thinking fostered by illustrations in scientific text . Journal of Educational Psychology , 81 ( 2 ) : 240 – 246 .
   Web of Science ®Google Scholar
• Mayer , R. 1992 . “ Mathematical problem solving: Thinking as based on domain specific knowledge ” . In Thinking, problem solving, and cognition , Edited by: Mayer , R. 455 – 489 . New York, NY : W. H. Freeman & Co .
   Google Scholar
• Mayer , R. and Gallini , J. 1990 . When is an illustration worth ten thousand words? . Journal of Educational Psychology , 82 ( 4 ) : 715 – 726 .
   Web of Science ®Google Scholar
• Mendenhall , W. , Beaver , R. J. and Beaver , B. M. 2003 . Introduction to probability and statistics , 11th , Belmont, CA : Duxbury .
   Google Scholar
• Molitor , S. , Ballstaedt , S. P. and Mandl , H. 1989 . “ Problems in knowledge acquisition from text and pictures ” . In Knowledge acquisition from text and pictures , Edited by: Mandl , H. and Levin , J. 3 – 35 . North-Holland : Elsevier Science .
   Google Scholar
• Mosteller , F. 1980 . Classroom and platform performance . The American Statistician , 34 ( 1 ) : 11 – 17 .
   Web of Science ®Google Scholar
• National Council of Teachers of Mathematics . 2003 . Principles and standards for school mathematics , Reston, VA : NCTM .
   Google Scholar
• Nemirovsky , R. 1994 . On ways of symbolizing: The case of Laura and the velocity sign . Journal of Mathematical Behavior , 13 ( 4 ) : 389 – 422 .
   Google Scholar
• Novick , L. 1990 . Representational transfer in problem solving . Psychological Science , 1 ( 2 ) : 128 – 132 .
   Web of Science ®Google Scholar
• Novick , L. 2002 . “ Spatial diagrams: Key instruments in the toolbox for thought ” . In The psychology of learning and motivation: Advances in research and theory, 40 , Edited by: Medin , D. 279 – 325 . San Diego, CA : Academic .
   Google Scholar
• Novick , L. and Hmelo , C. 1994 . Transferring symbolic representations across nonisomorphic problems . Journal of Experimental Psychology: Learning, Memory, and Cognition , 20 ( 6 ) : 1296 – 1321 .
   Web of Science ®Google Scholar
• Novick , L. and Hurley , S. 2001 . To matrix, network, or hierarchy: That is the question . Cognitive Psychology , 42 ( 2 ) : 158 – 216 .
   PubMed Web of Science ®Google Scholar
• Novick , L. , Hurley , S. and Francis , M. 1999 . Evidence for abstract, schematic knowledge of three spatial diagram representations . Memory & Cognition , 27 ( 2 ) : 288 – 308 .
   Google Scholar
• O'Connell , A. A. and Corter , J. E. . Student misconceptions in probability problem-solving . Paper presented at annual meeting of the American Educational Research Association . Atlanta, GA. April ,
   Google Scholar
• Olive , J. 1998 . “ Opportunities to explore and integrate mathematics with the Geometer's Sketchpad ” . In Designing learning environments for developing understanding of geometry and space , Edited by: Lehrer , R. and Chazan , D. 395 – 418 . Mahwah, NJ : Lawrence Erlbaum .
   Google Scholar
• Pantziara , M. , Gagatsis , A. and Elia , I. 2009 . Using diagrams as tools for the solution of non-routine mathematical problems . Educational Studies in Mathematics , 72 ( 1 ) : 39 – 60 .
   Google Scholar
• Penner , D. E. , Giles , N. D. , Lehrer , R. and Schauble , L. 1996 . Building functional models: Designing an elbow . Journal of Research in Science Teaching , 34 ( 2 ) : 125 – 143 .
   Google Scholar
• Polya , G. 1957 . How to solve it , Princeton, NJ : Princeton University Press .
   Google Scholar
• Presmeg , N. C. 1986 . Visualization in high school mathematics . For Learning of Mathematics , 63 ( 3 ) : 42 – 46 .
   Google Scholar
• Presmeg , N. 2006 . “ Research on visualization in learning and teaching mathematics ” . In Handbook of research on the psychology of mathematics education: Past, present and future , Edited by: Gutiérrez , Á. and Boero , P. 205 – 236 . Rotterdam : Sense .
   Google Scholar
• Reusser , K. 1996 . “ From cognitive modeling to the design of pedagogical tools ” . In International perspectives on the design of technology-supported learning environments , Edited by: Vosniadou , S. , De Corte , E. , Glaser , R. and Mandl , H. 81 – 103 . Mahwah, NJ : Lawrence Erlbaum .
   Google Scholar
• Rival , I. 1987 . Picture puzzling: mathematicians are rediscovering the power of pictorial reasoning . The Sciences , 27 : 40 – 46 .
   Google Scholar
• Robinson , K. M. 2001 . The validity of verbal reports in children's subtraction . Journal of Educational Psychology , 93 ( 1 ) : 211 – 222 .
   Web of Science ®Google Scholar
• Ross , B. H. 1984 . Remindings and their effects in learning a cognitive skill . Cognitive Psychology , 16 ( 3 ) : 371 – 416 .
   PubMedGoogle Scholar
• Ross , B. H. 1989 . “ Remindings in learning and instruction ” . In Similarity in analogical reasoning , Edited by: Vosniadou , S. and Ortony , A. 438 – 469 . Cambridge, , UK : Cambridge University Press .
   Google Scholar
• Ross , B. H. and Kennedy , P. T. 1990 . Generalizing from the use of earlier examples in problem solving . Journal of Experimental Psychology: Learning, Memory, & Cognition , 16 ( 1 ) : 42 – 55 .
   Web of Science ®Google Scholar
• Roth , W. M. and McGinn , M. K. 1998 . Inscriptions: Toward a theory of representing social practice . Review of Educational Research , 68 ( 1 ) : 35 – 59 .
   Web of Science ®Google Scholar
• Russell , W. E. 2000 . The use of visual devices in probability problem solving. (Doctoral Dissertation, Columbia University, 2000) . Dissertation Abstracts International , 61 : 1333
   Google Scholar
• Santos-Trigo , M. 1996 . An exploration of strategies used by students to solve problems with multiple ways of solution . Journal of Mathematical Behavior , 15 ( 3 ) : 263 – 284 .
   Google Scholar
• Schoenfeld , A. 1994 . Mathematical thinking and problem solving , Mahwah, NJ : Lawrence Erlbaum .
   Google Scholar
• Schreiber , C. 2004 . The interactive development of mathematical inscriptions – A semiotic perspective on pupils’ externalisation in an internet chat about mathematical problems . ZDM , 36 ( 6 ) : 185 – 194 .
   Google Scholar
• Schwartz , D. L. and Martin , T. 2004 . Inventing to prepare for future learning: The hidden efficiency of encouraging original student production in statistics instruction . Cognition and Instruction , 22 ( 2 ) : 129 – 184 .
   Web of Science ®Google Scholar
• Sedlmeier , P. and Gigerenzer , G. 2001 . Teaching Bayesian reasoning in less than two hours . Journal of Experimental Psychology: General , 130 ( 3 ) : 380 – 400 .
   PubMed Web of Science ®Google Scholar
• Sfard , A. 1994 . Reification as the birth of metaphor . For the Learning of Mathematics , 14 ( 1 ) : 44 – 55 .
   Google Scholar
• Stylianou , D. A. 2002 . On the interaction of visualization and analysis: The negotiation of a visual representation in expert problem solving . Journal of Mathematical Behavior , 21 ( 3 ) : 303 – 317 .
   Google Scholar
• Stylianou , D. A. and Silver , E. A. 2004 . The role of visual representations in advanced mathematical problem solving: An examination of expert-novice similarities and differences . Mathematical Thinking and Learning , 6 ( 4 ) : 353 – 387 .
   Google Scholar
• Tatsuoka , K. K. , Corter , J. E. and Tatsuoka , C. 2004 . Patterns of diagnosed mathematical content and process skills in TIMSS-R across a sample of twenty countries . American Educational Research Journal , 41 ( 4 ) : 901 – 926 .
   Web of Science ®Google Scholar
• Tukey , J. 1977 . Exploratory data analysis , Boston : Addison-Wesley .
   Google Scholar
• Tversky , B. 2001 . “ Spatial schemas in depictions ” . In Spatial schemas and abstract thought , Edited by: Gattis , M. 79 – 112 . Cambridge, MA : MIT Press .
   Google Scholar
• Uesaka , Y. , Manalo , E. and Ichikawa , S. 2007 . What kinds of perceptions and daily learning behaviors promote students’ use of diagrams in mathematics problem solving? . Learning and Instruction , 17 ( 3 ) : 322 – 335 .
   Web of Science ®Google Scholar
• van Garderen , D. and Montague , M. 2003 . Visual-spatial representation, mathematical problem solving, and students of varying abilities . Learning Disabilities: Research & Practice , 18 ( 4 ) : 246 – 254 .
   Google Scholar
• Vygotsky , L. S. 1978 . Mind and society: The development of higher psychological processes , Cambridge, MA : Harvard University Press .
   Google Scholar
• Weaver , J. L. and Quinn , R. L. 1999 . Geometer's sketchpad in secondary geometry . Computers in the Schools , 15 ( 2 ) : 83 – 95 .
   Google Scholar
• Zahner , D. 2005 . Clinical interviewing and problem-solving protocols to uncover the cognitive processes of probability problem solvers. Doctoral dissertation, Columbia University . Dissertation Abstracts International , 66 : 2851
   Google Scholar
• Zahner , D. and Corter , J. E. . Clinical interviewing to uncover the cognitive processes of probability problem solvers . Paper presented at the annual meeting of the American Educational Research Association . April , New Orleans, LA.
   Google Scholar