Children's Constructions of Knowledge for Fraction Division After Solving Realistic Problems

references

• Bates , T. and Rousseau , L. 1986 . Will the real division algorithm please stand up? . The Arithmetic Teacher , 33 ( 7 ) : 42 – 46 .
   Google Scholar
• Bergen , P. M. 1966 . Action research on division of fractions . The Arithmetic Teacher , 13 ( 4 ) : 293 – 295 .
   Google Scholar
• Bezuk , N. S. and Armstrong , B. E. 1993 . Understanding division of fractions . The Mathematics Teacher , 86 ( 1 ) : 43 – 46 .
   Google Scholar
• Borko , H. , Eisenhart , M. , Brown , C. A. , Underhill , R. G. , Jones , D. and Agard , P. C. 1992 . Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily? . Journal for Research in Mathematics Education , 23 : 194 – 222 .
   Web of Science ®Google Scholar
• Bray , C. J. 1963 . To invert or not to invert . The Arithmetic Teacher , 70 ( 5 ) : 274 – 276 .
   Google Scholar
• Brickman , B. 1955 . More rationalizing division of fractions . The Arithmetic Teacher , 2 ( 1 ) : 25 – 26 .
   Google Scholar
• Brownell , W. A. 1938 . Two kinds of learning in arithmetic . The Journal of Educational Research , 31 : 656 – 664 .
   Google Scholar
• Bruner , J. 1986 . Actual minds, possible worlds Cambridge, MA : Harvard University Press .
   Google Scholar
• Burns , M. 1991 . Introducing division through problem solving experiences . The Arithmetic Teacher , 38 ( 8 ) : 14 – 18 .
   Google Scholar
• Capps , L. R. 1962 . Division of fractions . The Arithmetic Teacher , 9 ( 1 ) : 10 – 16 .
   Google Scholar
• Carpenter , T. P. 1986 . “ Conceptual knowledge as a foundation for procedural knowledge. ” . In Conceptual and procedural knowledge: The case of mathematics Edited by: Hiebert , J. 113 – 132 . Hillsdale, NJ : Erlbaum .
   Google Scholar
• Carpenter , T. P. and Moser , J. M. 1982 . “ The development of addition and subtraction problem-solving skills. ” . In Addition and subtraction: A cognitive perspective Edited by: Carpenter , T. P. , Moser , J. M. and Romberg , T. A. 9 – 24 . Hillsdale, NJ : Erlbaum .
   Google Scholar
• Carroll , W. M. and Porter , D. 1997 . Invented strategies can develop meaningful mathematical procedures . Teaching Children Mathematics , 3 ( 1 ) : 370 – 374 .
   Google Scholar
• Chabe , A. M. 1963 . Rationalizing “inverting and multiplying.” . The Arithmetic Teacher , 10 ( 5 ) : 272 – 273 .
   Google Scholar
• Cobb , P. 2000 . “ Conducting teaching experiments in collaboration with teachers. ” . In Handbook of research design in mathematics and science education Edited by: Kelly , A. E. and Lesh , R. A. 307 – 333 . London : Erlbaum .
   Google Scholar
• Cobb , P. , Yackel , E. and Wood , T. 1991 . “ Curriculum and teacher development: Psychological and anthropological perspectives. ” . In Integrating research on teaching and learning mathematics Edited by: Fennema , E. , Carpenter , T. P. and Lamon , S. J. 83 – 119 . Albany : State University of New York Press .
   Google Scholar
• D'Ambrosio , U. 1985 . Ethnomathematics and its place in the history and pedagogy of mathematics . For the Learning of Mathematics , 5 ( 1 ) : 44 – 48 .
   Google Scholar
• D'Ambrosio , U. 1990 . The history of mathematics and ethnomathematics: How a native culture intervenes in the process of learning science . Impact of Science on Society , 160 : 369 – 377 .
   Google Scholar
• Elashhab , G. A. 1978 . A division of fractions—Discovery and verification . School Science and Mathematics , 78 ( 2 ) : 159 – 162 .
   Google Scholar
• Huinker , D. 1998 . “ Letting fraction algorithms emerge through problem solving. ” . In The teaching and learning of algorithms in school mathematics, NCTM 1998 yearbook Edited by: Morrow , L. J. and Kenney , M. J. 170 – 182 . Reston, VA : NCTM .
   Google Scholar
• Johnson , H. C. 1965 . Division with fractions—Levels of meaning . The Arithmetic Teacher , 12 ( 5 ) : 362 – 368 .
   Google Scholar
• Kamii , C. 1985 . Young children reinvent arithmetic New York : Teachers College Press .
   Google Scholar
• Kamii , C. and Clark , F. B. 1995 . Equivalent fractions: Their difficulty and educational implications . Journal of Mathematical Behavior , 14 ( 4 ) : 365 – 378 .
   Google Scholar
• Kamii , C. and Warrington , M. A. 1995 . Division with fractions: A Piagetian, constructivist approach . Hiroshima Journal of Mathematics Education , 3 : 53 – 62 .
   Google Scholar
• Kieren , T. E. 1988 . “ Personal knowledge of rational numbers: Its intuitive and formal development. ” . In Number concepts and operations in the middle grades Edited by: Hieben , J. and Behr , M. J. 162 – 181 . Hillsdale, NJ : Erlbaum .
   Google Scholar
• Krich , P. 1964 . Meaningful vs. mechanical method, teaching division of fractions by fractions . School Science and Mathematics , 64 ( 8 ) : 697 – 708 .
   Google Scholar
• Mack , N. K. 1995 . Confounding whole-number and fraction concepts when building on informal knowledge . Journal for Research in Mathematics Education , 26 ( 5 ) : 422 – 441 .
   Web of Science ®Google Scholar
• McMeen , G. H. 1962 . Division by a fraction—A new method . The Arithmetic Teacher , 9 ( 3 ) : 122 – 126 .
   Google Scholar
• Merriam , S. B. 1988 . Case study research in education: A qualitative approach San Francisco : Jossey Bass .
   Google Scholar
• Miller , G. H. 1957 . How effective is the meaning method? . The Arithmetic Teacher , 4 ( 2 ) : 45 – 49 .
   Google Scholar
• National Council of Teachers of Mathematics. 1989 . Curriculum and evaluation standards for school mathematics Reston, VA : Author .
   Google Scholar
• National Council of Teacher of Mathematics. 1991 . Professional standards for teaching mathematics Reston, VA : Author .
   Google Scholar
• National Council of Teachers of Mathematics. 2000 . Principles and standards for school mathematics Reston, VA : Author .
   Google Scholar
• Novillis , C. F. 1979 . Why teach elementary school students the division meaning of fractions? . School Science and Mathematics , 79 ( 8 ) : 705 – 708 .
   Google Scholar
• Nowlin , D. 1996 . Division with fractions . Mathematics Teaching in the Middle School , 2 ( 2 ) : 116 – 119 .
   Google Scholar
• Pothier , Y and Sawada , D. 1983 . Partitioning: The emergence of rational number ideas in young children . Journal for Research in Mathematics Education , 14 ( 4 ) : 307 – 317 .
   Google Scholar
• Schoenfeld , A. 1985 . Mathematical problem solving New York : Academic Press .
   Google Scholar
• Sharp , J. 1998 . “ A constructed algorithm for the division of fractions. ” . In The teaching and learning of algorithms in school mathematics, NCTM 1998 yearbook Edited by: Morrow , L. and Kenney , M. 204 – 207 . Reston, VA : NCTM .
   Google Scholar
• Sharp , J. , Garofalo , J. and Adams , B. 2002 . “ Children's development of meaningful fraction algorithms: A kid's cookies and a puppy's pills. ” . In Making sense of fractions, ratios and proportions: 2002 NCTM yearbook Edited by: Litwiller , B. and Bright , G. 18 – 28 . Reston, VA : National Council of Teachers of Mathematics .
   Google Scholar
• Silvia , E. M. 1983 . A look at division with fractions . The Arithmetic Teacher , 30 ( 5 ) : 3S – 41 .
   Google Scholar
• Slesnick , T. 1982 . Algorithmic skill vs. conceptual understanding . Educational Studies in Mathematics , 13 ( 2 ) : 143 – 154 .
   Google Scholar
• Streefland , L. 1978 . Some observational results concerning the mental constitution of the concept of fraction . Educational Studies in Mathematics , 9 ( 1 ) : 51 – 73 .
   Google Scholar
• Streefland , L. 1991 . Fractions in realistic mathematics education Dordrecht/Boston/London : Kluwer .
   Google Scholar
• Streefland , L. 1993 . “ Fractions: A realistic approach. ” . In Rational numbers: An integration of research Edited by: Carpenter , T. P. , Fennema , E. and Romberg , T. A. 289 – 325 . Hillsdale NJ : Erlbaum .
   Google Scholar
• von Glasersfeld , E. 1990 . “ Constructivism in mathematics education: Why some like it radical. ” . In Constructivist views on the teaching and learning of mathematics. Journal for Research in Mathematics Education Monograph Edited by: Davis , R. B. , Maher , C. A. and Noddings , N. 19 – 30 . Reston, VA : National Council of Teachers of Mathematics . (No. 4)
   Google Scholar
• Warrington , M. A. 1997 . How children think about division with fractions . Mathematics Teaching in the Middle School , 2 ( 6 ) : 390 – 394 .
   Google Scholar
• Warrington , M. A. and Kamii , C. 1998 . Multiplication with fractions: A Piagetian constructivist approach . Teaching Children Mathematics , 3 ( 5 ) : 339 – 343 .
   Google Scholar
• Wearne , D. and Hiebert , J. 1988 . A cognitive approach to meaningful mathematics instruction: Testing a local theory using decimal numbers . Journal for Research in Mathematics Education , 19 ( 5 ) : 371 – 384 .
   Web of Science ®Google Scholar