The development of proficiency in the fraction domain : affordances and constraints in the curriculum

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[1]  P. Cobb,et al.  Sociomathematical Norms, Argumentation, and Autonomy in Mathematics. , 1996 .

[2]  B. Pepin,et al.  AN INVESTIGATION OF MATHEMATICS TEXTBOOKS IN ENGLAND, FRANCE AND GERMANY: SOME CHALLENGES FOR ENGLAND , 2002 .

[3]  Knut Neumann,et al.  FROM PISA TO EDUCATIONAL STANDARDS: THE IMPACT OF LARGE-SCALE ASSESSMENTS ON SCIENCE EDUCATION IN GERMANY , 2010 .

[4]  Wolff‐Michael Roth.  What is the meaning of “meaning”? A case study from graphing , 2004 .

[5]  P. M. Hiele.  Structure and Insight: A Theory of Mathematics Education , 1985 .

[6]  Dorianne Almann,et al.  What Research Says to the Science Teacher. , 1978 .

[7]  H. Freudenthal.  Geometry between the devil and the deep sea , 1971 .

[8]  Km Kariene Mittendorff.  Career conversations in senior secondary vocational education , 2010 .

[9]  G Geeke Bruin-Muurling,et al.  The development of proficiency in the fraction domain : affordances and constraints in the curriculum , 2006 .

[10]  Lejla Batina,et al.  Montgomery in Practice: How to Do It More Efficiently in Hardware , 2002, CT-RSA.

[11]  Amy J. Hackenberg.  Units coordination and the construction of improper fractions: A revision of the splitting hypothesis , 2007 .

[12]  Robert P. Hunting,et al.  Context and process in fraction learning , 1990 .

[13]  de Gm Gijs Bakker,et al.  Allocated online reciprocal peer support via instant messaging as a candidate for decreasing the tutoring load of teachers , 2010 .

[14]  Susan Magidson.  Building bridges within mathematics education: Teaching, research, and instructional design☆ , 2005 .

[15]  Hans Freudenthal,et al.  Weeding and Sowing: Preface to a Science of Mathematical Education , 1977 .

[16]  E.W.R. Oemar Said.  De Da Vinci case : een onderzoek naar de relaties tussen vernieuwende leeromgevingen en de motivatie en regulatievoorkeuren van leerlingen in het MBO , 2009 .

[17]  R. Tzur.  Teacher and Students' Joint Production of a Reversible Fraction Conception. , 2003 .

[18]  Jere Confrey,et al.  Splitting Reexamined: Results from a Three-Year Longitudinal Study of Children in Grades Three to Five. , 1995 .

[19]  Susanne Prediger.  The relevance of didactic categories for analysing obstacles in conceptual change: Revisiting the case of multiplication of fractions , 2008 .

[20]  M. Raman.  Coordinating informal and formal aspects of mathematics: student behavior and textbook messages , 2002 .

[21]  P. C. Moreira,et al.  Academic mathematics and mathematical knowledge needed in school teaching practice: some conflicting elements , 2008 .

[22]  P. Brok,et al.  Students' goal orientations, information processing strategies and knowledge development in competence-based pre-vocational secondary education , 2005 .

[23]  William Byers,et al.  How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics , 2007 .

[24]  Lejla Batina,et al.  Flexible Hardware Design for RSA and Elliptic Curve Cryptosystems , 2004, CT-RSA.

[25]  Susan E. B. Pirie,et al.  Growth in mathematical understanding: How can we characterise it and how can we represent it? , 1994 .

[26]  K.P.E. Gravemeijer,et al.  Fractions, percentages, decimals and proportions : a learning-teaching trajectory for grade 4, 5 and 6 , 2008 .

[27]  C.A.C. van Bragt.  Students' educational careers in higher education : a search into key factors regarding study outcome , 2010 .

[28]  L. S. Vygotskiĭ,et al.  Mind in society : the development of higher psychological processes , 1978 .

[29]  Hans Freudenthal,et al.  Revisiting mathematics education : China lectures , 1991 .

[30]  William J. Boone,et al.  Rigour in quantitative analysis: The promise of Rasch analysis techniques , 2005 .

[31]  L. Brueckner.  Analysis of Errors in Fractions , 1928, The Elementary School Journal.

[32]  E. Fischbein,et al.  THE ROLE OF IMPLICIT MODELS IN SOLVING VERBAL PROBLEMS IN MULTIPLICATION AND DIVISION , 1985 .

[33]  Leslie P. Steffe,et al.  A New Hypothesis Concerning Children ’ s Fractional Knowledge , 2002 .

[34]  Kpe Koeno Gravemeijer.  How Emergent Models May Foster the Constitution of Formal Mathematics , 1999 .

[35]  Richard Lesh,et al.  ORDER AND EQUIVALENCE OF RATIONAL NUMBERS: A CLINICAL TEACHING EXPERIMENT , 1984 .

[36]  Carol F. Novillis.  An Analysis of the Fraction Concept into a Hierarchy of Selected Subconcepts and the Testing of the Hierarchical Dependencies. , 1976 .

[37]  T. Kuhn,et al.  The Structure of Scientific Revolutions. , 1964 .

[38]  Jeremy Kilpatrick,et al.  Understanding mathematical literacy: The contribution of research , 2001 .

[39]  Wiebe E. Bijker,et al.  Science in action : how to follow scientists and engineers through society , 1989 .

[40]  G Geeke Bruin-Muurling,et al.  Error correction for combinational logic circuits , 2000 .

[41]  Thomas J. Cooper,et al.  Mathematical Analogs and the Teaching of Fractions. , 1999 .

[42]  M. R. Espejo.  Applying the Rasch Model: Fundamental Measurement in the Human Sciences , 2004 .

[43]  M. Raman.  Epistemological messages conveyed by three high-school and college mathematics textbooks , 2004 .

[44]  Leslie P. Steffe,et al.  The construction of an iterative fractional scheme: the case of Joe , 2001 .

[45]  Nico F. Benschop,et al.  Associative Digital Network Theory: An Associative Algebra Approach to Logic, Arithmetic and State Machines , 2009 .

[46]  Wolff‐Michael Roth,et al.  “Vygotsky’s Neglected Legacy”: Cultural-Historical Activity Theory , 2007 .

[47]  K.P.E. Gravemeijer,et al.  Context Problems in Realistic Mathematics Education: A Calculus Course as an Example , 1999 .

[48]  David Tall,et al.  Ambiguity and flexibility: A proceptual view of simple arithmetic , 1983 .

[49]  C. Larson.  Locating Proper Fractions On Number Lines: Effect of Length and Equivalence , 1980 .

[50]  S. Goodchild,et al.  An Exploratory Study of Mathematics Test Results: What is the Gender Effect? , 2009 .

[51]  Kenneth R. Koedinger,et al.  Using Cognitive Models to Guide Instructional Design: The Case of Fraction Division , 2001 .

[52]  G. Masters.  A rasch model for partial credit scoring , 1982 .

[53]  H. Freudenthal.  Didactical Phenomenology of Mathematical Structures , 1983 .

[54]  E. Rohaan,et al.  Testing teacher knowledge for technology teaching in primary schools , 2006 .

[55]  David Tall,et al.  Advanced Mathematical Thinking , 1994 .

[56]  John Olive,et al.  Making sense of instruction on fractions when a student lacks necessary fractional schemes: The case of Tim , 2006 .

[57]  R. Guy.  The strong law of small numbers , 1988 .

[58]  David F. Treagust,et al.  Using Large-scale Assessment Datasets for Research in Science and Mathematics Education: Programme for International Student Assessment (PISA) , 2007 .

[59]  H. Freudenthal.  Mathematics as an Educational Task , 1972 .

[60]  Susan J. Lamon,et al.  The Development of Unitizing: Its Role in Children's Partitioning Strategies. , 1996 .

[61]  Paul Cobb,et al.  Information-processing psychology and mathematics education: A constructivist perspective , 1987 .

[62]  Robert P. Hunting,et al.  A review of recent research in the area of initial fraction concepts , 1996 .

[63]  Rod Nason,et al.  Young children's partitioning strategies , 2000 .

[64]  Debra L. Junk,et al.  Fractions as the Coordination of Multiplicatively Related Quantities: A Cross-Sectional Study of Children's Thinking , 2006 .

[65]  Douglas A. Grouws,et al.  Handbook of research on mathematics teaching and learning , 1992 .

[66]  Maarten Dolk,et al.  Reforming mathematics learning in Indonesian classrooms through RME , 2008 .

[67]  K.P.E. Gravemeijer,et al.  Developing realistic mathematics education , 1994 .

[68]  M. Rajuan,et al.  Student teachers' perceptions of learning to teach as a basis for supervision of the mentoring relationship , 2008 .

[69]  A. Treffers.  Three Dimensions: A Model of Goal and Theory Description in Mathematics Instruction ― The Wiskobas Project , 1986 .

[70]  Abraham Arcavi,et al.  The role of visual representations in the learning of mathematics , 2003 .

[71]  A. Sfard.  On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin , 1991 .

[72]  Georg Rasch,et al.  Probabilistic Models for Some Intelligence and Attainment Tests , 1981, The SAGE Encyclopedia of Research Design.

[73]  L. Verschaffel,et al.  Add? Or Multiply? A study on the development of primary school students' proportional reasoning skills; , 2009 .

[74]  Y. Engeström,et al.  Polycontextuality and Boundary Crossing in Expert Cognition: Learning and Problem Solving in Complex Work Activities. , 1995 .

[75]  Nancy K. Mack.  Confounding whole-number and fraction concepts when building on informal knowledge. , 1995 .

[76]  Wolff‐Michael Roth,et al.  Representations of Scientists in Canadian High School and College Textbooks. , 2008 .

[77]  J. Lave.  Cognition in Practice: Outdoors: a social anthropology of cognition in practice , 1988 .

[78]  Koeno Gravemeijer,et al.  Methoden in het reken-wiskundeonderwijs, een rijke context voor vergelijkend onderzoek , 1993 .

[79]  H. Freudenthal.  Why to teach mathematics so as to be useful , 1968 .

[80]  R. Lesh,et al.  Acquisition of mathematics concepts and processes , 1983 .

[81]  Leen Streefland,et al.  Fractions in Realistic Mathematics Education: A Paradigm of Developmental Research , 1991 .

[82]  RW Ralph Hooreman.  Synchronous coaching of trainee teachers : an experimental approach , 2008 .

[83]  Leen Streefland,et al.  Learning from history for teaching in the future , 2003 .

[84]  Susan J. Lamon,et al.  Ratio and Proportion: Connecting Content and Children's Thinking. , 1993 .

[85]  Larry D. Yore,et al.  FIRST CYCLE OF PISA (2000–2006)—INTERNATIONAL PERSPECTIVES ON SUCCESSES AND CHALLENGES: RESEARCH AND POLICY DIRECTIONS , 2010 .

[86]  Cees A. W. Glas,et al.  DETECTION OF DIFFERENTIAL ITEM FUNCTIONING USING LAGRANGE MULTIPLIER TESTS , 1996 .

[87]  H. Poincaré.  La science et l'hypothèse , 1968 .

[88]  Guy Brousseau,et al.  Le contrat didactique : le milieu , 1990 .

[89]  Michael P. Clough,et al.  The Nature of Science in Science Education: An Introduction , 1998 .

[90]  S. Vosniadou,et al.  The development of students’ understanding of the numerical value of fractions , 2004 .

[91]  D. Russell.  Rethinking Genre in School and Society , 1997 .

[92]  van de,et al.  Competentiegerichtheid en scheikunde leren: over metacognitieve opvattingen, leerresultaten en leeractiviteiten , 2007 .

[93]  Anne Roche,et al.  Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction , 2009 .