2,226
Views
129
CrossRef citations to date
0
Altmetric
Original Articles

How Many Decimals Are There Between Two Fractions? Aspects of Secondary School Students’ Understanding of Rational Numbers and Their Notation

&
Pages 181-209 | Published online: 14 Apr 2010

REFERENCES

  • Andres , M. , Di Luca , S. and Pesenti , M. 2008 . Finger counting: The missing tool? . Behavioral Sciences , 31 : 642 – 643 .
  • Carey , S. 1985 . Conceptual change in childhood , Cambridge, MA : MIT Press .
  • Carey , S. 1991 . “ Knowledge acquisition: Enrichment or conceptual change? ” . In The epigenisis of mind: Essays on biology and cognition (pp. 257–291) , Edited by: Carey , S. and Gelman , R. Hillsdale, NJ : Lawrence Erlbaum .
  • Carey , S. 2004 . Bootstrapping and the origin of concepts . Daedalus , 133 : 59 – 68 .
  • Carey , S. and Gelman , R. 1991 . The epigenesis of mind: Essays on biology and cognition , Hillsdale, NJ : Lawrence Erlbaum .
  • Carey , S. and Spelke , E. 1994 . “ Domain-specific knowledge and conceptual change ” . In Mapping the mind: Domain specificity in cognition and culture , Edited by: Hirschfeld , L. A. and Gelman , S. A. 169 – 200 . Cambridge, MA : Cambridge University Press .
  • diSessa , A. A. 1993 . Toward an epistemology of physics . Cognition and Instruction , 10 : 105 – 225 .
  • diSessa , A. A. 2006 . “ A history of conceptual change research ” . In Cambridge handbook of the learning sciences , Edited by: Sawyer , R. K. 265 – 281 . New York : Cambridge University Press .
  • diSessa , A. A. 2008 . “ A bird's eye view of the ‘pieces’ vs. ‘coherence’ controversy (from the ‘pieces’ side of the fence) ” . In International handbook of research on conceptual change , Edited by: Vosniadou , S. 35 – 60 . Mahwah, NJ : Lawrence Erlbaum .
  • Dörfler , W. 1995 . “ Mathematical objects, representations and imagery ” . In Exploiting mental imagery with computers in mathematics education , NATO ASI Series F: Advanced Educational Technology Edited by: Sutherland , R. and Mason , J. 82 – 94 . Berlin, , Germany : Springer-Verlag .
  • Duval , R. 2006 . A cognitive analysis of problems of comprehension in a learning of mathematics . Educational Studies in Mathematics , 63 : 103 – 131 .
  • Fischbein , E. , Deri , M. , Nello , M. and Marino , M. 1985 . The role of implicit models in solving problems in multiplication and division . Journal for Research in Mathematics Education , 16 : 3 – 17 .
  • Gelman , R. 1990 . First principles organize attention to and learning about relevant data: Number and animate-inanimate distinction as examples . Cognitive Science , 14 : 79 – 106 .
  • Gelman , R. 1994 . “ Constructivism and supporting environments ” . In Implicit and explicit knowledge: An educational approach , Edited by: Tirosh , D. 55 – 82 . New York : Ablex .
  • Gelman , R. 2000 . The epigenesis of mathematical thinking . Journal of Applied Developmental Psychology , 21 : 27 – 37 .
  • Giannakoulias , E. , Souyoul , A. and Zachariades , T. 2007 . “ Students’ thinking about fundamental real numbers properties ” . In Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education , Edited by: Pitta-Pantazi , D. and Philippou , G. 426 – 425 . Cyprus : ERME, Department of Education, University of Cyprus .
  • Greer , B. 2004 . The growth of mathematics through conceptual restructuring . Learning & Instruction , 14 : 541 – 548 .
  • Greer , B. 2006 . “ Designing for conceptual change ” . In Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education: Vol. 1. , Edited by: Novotná , J. , Moraová , H. , Krátká , M. and Stehlíková , N. 175 – 178 . Prague : PME .
  • Hartnett , P. M. and Gelman , R. 1998 . Early understandings of number: Paths or barriers to the construction of new understandings? . Learning and Instruction , 8 : 341 – 374 .
  • Keil , F. C. 1994 . Concepts, kinds, and cognitive development , Cambridge, MA : MIT Press .
  • Khoury , H. A. and Zazkis , R. 1994 . On fractions and non-standard representations: Preservice teachers’ concepts . Educational Studies in Mathematics , 27 : 191 – 204 .
  • Kilpatrick , J. , Swafford , J. and Findell , B. 2001 . Adding it up. Helping children learn mathematics , Washington, DC : National Academy Press .
  • Lamon , S. J. 1999 . Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers , Mahwah, NJ : Lawrence Erlbaum .
  • Lucas , J. R. 2000 . The conceptual roots of mathematics , London : Routledge .
  • Malara , N. 2001 . “ From fractions to rational numbers in their structure: Outlines for an innovative didactical strategy and the question of density ” . In Proceedings of the 2nd Conference of the European Society for Research Mathematics Education II , Edited by: Novotná , J. 35 – 46 . Praga : Univerzita Karlova v Praze, Pedagogická Faculta .
  • Mamede , E. , Nunes , T. and Bryant , P. 2005 . “ The equivalence and ordering of fractions in part-whole and quotient situations ” . In Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education: Vol. 3 , Edited by: Chick , H. L. and Vincent , J. L. 281 – 288 . Melbourne : PME .
  • Markovits , Z. and Sowder , J. 1991 . Students’ understanding of the relationship between fractions and decimals . Focus on Learning Problems in Mathematics , 13 ( 1 ) : 3 – 11 .
  • Merenluoto , K. and Lehtinen , E. 2002 . “ Conceptual change in mathematics: Understanding the real numbers ” . In Reconsidering conceptual change: Issues in theory and practice , Edited by: Limon , M. and Mason , L. 233 – 258 . Dordrecht, , The Netherlands : Kluwer .
  • Merenluoto , K. and Palonen , T. 2007 . “ When we clashed with the real numbers: Complexity of conceptual change in number concept ” . In Reframing the conceptual change approach in learning and instruction , Edited by: Vosniadou , S. , Baltas , A. and Vamvakoussi , X. 247 – 263 . Oxford, UK : Elsevier .
  • Moskal , B. M. and Magone , M. E. 2000 . Making sense of what students know: Examining the referents, relationships and modes students displayed in response to a decimal task . Educational Studies in Mathematics , 43 : 313 – 335 .
  • Mitchell , A. 2005 . “ Measuring fractions ” . In Proceedings of the 28th Conference of the Mathematics Research Group of Australasia , Edited by: Clarkson , P. , Downton , A. , Gronn , D. , Horne , M. , McDonough , A. , Pierce , R. and Roche , A. 545 – 552 . Melbourne : MERGA . Building connections: Research, theory, and practice
  • Moss , J. 2005 . “ Pipes, tubes, and beakers: New approaches to teaching the rational-number system ” . In How students learn: Mathematics in the classroom , Edited by: Donovan , M. S. and Bransford , J. D. 121 – 162 . Washington, DC : National Academic Press .
  • Moss , J. and Case , R. 1999 . Developing children's understanding of the rational numbers: A new model and an experimental curriculum . Journal for Research in Mathematics Education , 30 ( 2 ) : 122 – 147 .
  • Neumann , R. 2001 . “ Students’ ideas on the density of fractions ” . In Developments in mathematics education in German-speaking countries: Selected papers from the Annual Conference on Didactics of Mathematics, Munich, 1998 , Edited by: Weigand , H. G. , Peter-Koop , A. , Neil , N. , Reiss , K. , Torner , G. and Wollring , B. 97 – 104 . Hildesheim, , Germany : Gesellschaft für Didaktik der Mathematik .
  • Ni , Y. and Zhou , Y-D. 2005 . Teaching and learning fraction and rational numbers: The origins and implications of whole number bias . Educational Psychologist , 40 ( 1 ) : 27 – 52 .
  • Nunes , T. and Bryant , P. 2008 . Rational numbers and intensive quantities: Challenges and insights to pupils’ implicit knowledge . Anales de Psicologia , 24 ( 2 ) : 262 – 270 .
  • O’Connor , M. C. 2001 . “Can any fraction be turned into a decimal?” A case study of a mathematical group discussion . Educational Studies in Mathematics , 46 : 143 – 185 .
  • Pehkonen , E. , Hannula , M. , Maijala , H. and Soro , R. 2006 . “ Infinity of numbers: How students understand it ” . In Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education: Vol. 4 , Edited by: Novotná , J. , Moraová , H. , Krátká , M. and Stehlíková , N. 345 – 352 . Prague : PME .
  • Resnick , L. B. 1989 . Developing mathematical knowledge . American Psychologist , 44 ( 2 ) : 162 – 169 .
  • Resnick , L. B. 2006 . “ The dilemma of mathematical intuition in learning ” . In Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education: Vol 1 , Edited by: Novotná , J. , Moraová , H. , Krátká , M. and Stehlíková , N. 173 – 175 . Prague : PME .
  • Sfard , A. 1991 . On the dual nature of mathematical conceptions. Reflections on processes and objects as different sizes of the same coin . Educational Studies in Mathematics , 22 : 1 – 36 .
  • Sfard , A. 2007 . “ Reconceptualizing conceptual change ” . In Reframing the conceptual change approach in learning and instruction , Edited by: Vosniadou , S. , Baltas , A. and Vamvakoussi , X. 329 – 334 . Oxford, , UK : Elsevier .
  • Siegler , R. S. 2006 . “ Microgenetic analyses of learning ” . In Handbook of child psychology: Volume 2: Cognition, perception, and language , 6th ed. , Edited by: Damon , W. , Lerner , R. M. , Kuhn , D. and Siegler , R. S. 464 – 510 . Hoboken, NJ : Wiley .
  • Smith , C. L. , Solomon , G. E. A. and Carey , S. 2005 . Never getting to zero: Elementary school students’ understanding of the infinite divisibility of number and matter . Cognitive Psychology , 51 : 101 – 140 .
  • Smith , J. P. 1995 . Competent reasoning with rational numbers . Cognition and Instruction , 13 : 3 – 50 .
  • Smith , J. P. , diSessa , A. A. and Roschelle , J. 1993 . Misconceptions reconceived: A constructivist analysis of knowledge in transition . Journal of the Learning Sciences , 3 ( 2 ) : 115 – 163 .
  • Sophian , C. 1997 . Beyond competence: The significance of performance for conceptual development . Cognitive Development , 12 : 281 – 303 .
  • Stafylidou , S. and Vosniadou , S. 2004 . The development of students’ understanding of the numerical value of fractions . Learning and Instruction , 14 : 503 – 518 .
  • Tirosh , D. , Fischbein , E. , Graeber , A. O. and Wilson , J. W. 1999 . “ Prospective elementary teachers’ conceptions of rational numbers ” . Retrieved May 05, 2005 from http://jwilson.coe.uga.edu/Texts.Folder/Tirosh/Pros.El.Tchrs.html
  • Vamvakoussi , X. and Vosniadou , S. 2004 . Understanding the structure of the set of rational numbers: A conceptual change approach . Learning and Instruction , 14 : 453 – 467 .
  • Vamvakoussi , X. and Vosniadou , S. 2007 . “ How many numbers are there in an interval? Presuppositions, synthetic models and the effect of the number line ” . In Reframing the conceptual change approach in learning and instruction , Edited by: Vosniadou , S. , Baltas , A. and Vamvakoussi , X. 267 – 283 . Oxford, , UK : Elsevier .
  • Vosniadou , S. 2007 . “ The conceptual change approach and its reframing ” . In Reframing the conceptual change approach in learning and instruction , Edited by: Vosniadou , S. , Baltas , A. and Vamvakoussi , X. 1 – 15 . Oxford, , UK : Elsevier .
  • Vosniadou , S. , Skopeliti , I. and Ikospentaki , K. 2004 . Modes of knowing and ways of reasoning in elementary astronomy . Cognitive Development , 19 : 203 – 222 .
  • Vosniadou , S. , Vamvakoussi , X. and Skopeliti , I. 2008 . “ The framework theory approach to conceptual change ” . In International handbook of research on conceptual change , Edited by: Vosniadou , S. 3 – 34 . Mahwah, NJ : Lawrence Erlbaum .
  • Vosniadou , S. and Verschaffel , L. 2004 . Extending the conceptual change approach to mathematics learning and teaching . Learning and Instruction , 14 : 445 – 451 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.