5,308
Views
16
CrossRef citations to date
0
Altmetric
Articles

Mathematics Teachers' Reasoning About Fractions and Decimals Using Drawn Representations

, &
Pages 198-220 | Published online: 13 Jul 2011

REFERENCES

  • Ball , D. , Lubienski , S. and Mewborn , D. 2001 . “ Research on teaching mathematics: The unsolved problem of teachers' mathematical knowledge ” . In Handbook of research on teaching , 4th , Edited by: Richardson , V. 433 – 456 . New York, NY : MacMillan .
  • Ball , D. L. , Thames , M. H. and Phelps , G. 2008 . Content knowledge for teaching: What makes it special? . Journal of Teacher Education , 59 ( 5 ) : 389 – 407 .
  • Behr , M. J. , Harel , G. , Post , T. R. and Lesh , R. 1992 . “ Rational number, ratio, and proportion ” . In Handbook of research on mathematics teaching and learning , Edited by: Grouws , D. 296 – 333 . New York, NY : MacMillan Publishing .
  • Behr , M. J. , Harel , G. , Post , T. R. and Lesh , R. 1994 . “ Units of quantity: A conceptual basis common to additive and multiplicative structures ” . In The development of multiplicative reasoning in the learning of mathematics , Edited by: Harel , G. and Confrey , J. 121 – 176 . Albany, NY : State University Press of New York .
  • Behr , M. , Helen , K. , Harel , G. , Post , T. and Lesh , R. 1997 . Conceptual units analysis of preservice elementary school teachers' strategies on a rational-number-as-operator task . Journal for Research in Mathematics Education , 28 ( 1 ) : 48 – 69 .
  • Behr , M. J. , Wachsmuth , I. , Post , T. R. and Lesh , R. 1984 . Order and equivalence of rational numbers: A clinical teaching experiment . Journal for Research in Mathematics Education , 15 ( 5 ) : 323 – 341 .
  • 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 .
  • Borko , H. and Putnam , R. 1996 . “ Learning to teach ” . In Handbook of educational psychology , Edited by: Berliner , D. and Calfee , R. 673 – 708 . New York, NY : MacMillan .
  • Coffey , A. and Atkinson , P. 1996 . Making sense of qualitative data , Thousand Oaks, CA : Sage .
  • D'Ambrosio , B. S. and Mewborn , D. S. 1994 . Children's constructions of fractions and their implications for classroom instruction . Journal for Research in Mathematics Education , 8 ( 2 ) : 150 – 161 .
  • Denzin , N. K. 1989 . The research act: A theoretical introduction to sociological methods , Englewood Cliffs, NJ : Prentice Hall .
  • diSessa , A. A. 1988 . “ Knowledge in pieces ” . In Constructivism in the computer age , Edited by: Forman , G. and Pufall , P. 49 – 70 . Hillsdale, NJ : Lawrence Erlbaum .
  • Dreyfus , T. and Eisenberg , T. 1996 . “ On different facets of mathematical thinking ” . In The nature of mathematical thinking , Edited by: Sternberg , R. J. and Ben-Zeev , T. 253 – 284 . Mahwah, NJ : Lawrence Erlbaum .
  • Fischbein , E. , Deri , M. , Nello , M. S. and Marino , M. S. 1985 . The role of implicit models in solving verbal problems in multiplication and division . Journal for Research in Mathematics Education , 16 ( 1 ) : 3 – 17 .
  • Freudenthal , H. 1983 . Didactical phenomenology of mathematical structures , Boston, MA : D. Reidel .
  • Gagatsis , A. and Shiakalli , M. 2004 . Ability to translate from one representation of the concept of function to another and mathematical problem solving . Educational Psychology , 24 ( 5 ) : 645 – 657 .
  • Goldin , G. 2002 . “ Representation in mathematical learning and problem solving ” . In Handbook of international research in mathematics education , Edited by: English , L. 197 – 218 . Mahwah, NJ : Lawrence Erlbaum .
  • Hall , R. 2000 . “ Videorecording as theory ” . In Handbook of research design in mathematics and science education , Edited by: Kelly , A. E. and Lesh , R. A. 647 – 664 . Mahwah, NJ : Lawrence Erlbaum .
  • Harel , G. and Behr , M. 1995 . Teachers' solutions for multiplicative problems . Hiroshima Journal of Mathematics Education , 3 : 31 – 51 .
  • Harel , G. and Confrey , J. 1994 . The development of multiplicative reasoning in the learning of mathematics , Albany, NY : SUNY Press .
  • Hiebert , J. and Behr , M. 1988 . Number concepts and operations in the middle grades , Reston, VA : National Council of Teachers of Mathematics .
  • Hill , H. C. , Rowan , B. and Ball , D. L. 2005 . Effects of teachers' mathematical knowledge for teaching on student achievement . American Educational Research Journal , 42 ( 2 ) : 371 – 406 .
  • Izsák , A. 2005 . “You have to count the squares”: Applying knowledge in pieces to learning rectangular area . The Journal of the Learning Sciences , 14 ( 3 ) : 361 – 403 .
  • Izsák , A. 2008 . Mathematical knowledge for teaching fraction multiplication . Cognition and Instruction , 26 ( 1 ) : 95 – 143 .
  • Izsák , A. , Orrill , C. H. , Cohen , A. and Brown , R. E. 2010 . Measuring middle grades teachers' understanding of rational numbers with the mixture Rasch model . Elementary School Journal , 110 ( 3 ) : 279 – 300 .
  • Izsák , A. , Tillema , E. and Tunç-Pekkan , Z. 2008 . Teaching and learning fraction addition on number lines . Journal for Research in Mathematics Education , 39 ( 1 ) : 33 – 62 .
  • Kaput , J. J. 1985 . Multiplicative word problems and intensive quantities: An integrated software response , Cambridge, MA : Harvard Graduate School of Education, Educational Technology Center .
  • Kilpatrick , J. , Swafford , J. and Findell , B. 2001 . Adding it up: Helping children learn mathematics , Washington, DC : National Academy Press .
  • Lamon , S. 1994 . “ Ratio and proportion: Cognitive foundations in unitizing and norming ” . In The development of multiplicative reasoning in the learning of mathematics , Edited by: Harel , G. and Confrey , J. 89 – 120 . Albany, NY : State University Press of New York .
  • Lappan , G. , Fey , J. T. , Fitzgerald , W. M. , Friel , S. N. and Phillips , E. D. 2002 . Connected mathematics program , Glenview, IL : Prentice Hall .
  • Lesh , R. , Behr , M. and Post , T. 1987 . “ Rational number relations and proportions ” . In Problems of representation in the teaching and learning of mathematics , Edited by: Janvier , C. 41 – 58 . Hillsdale, NJ : Lawrence Erlbaum .
  • 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 .
  • National Council of Teachers of Mathematics (NCTM) . 2000 . Principles and standards for school mathematics , Reston, VA : National Council of Teachers of Mathematics .
  • Olive , J. 1999 . From fractions to rational numbers of arithmetic: a reorganization hypothesis . Mathematical Thinking and Learning , 1 ( 4 ) : 279 – 314 .
  • Olive , J. 2001 . Children's number sequences: An explanation of Steffe's constructs and an extrapolation to rational numbers of arithmetic . The Mathematics Educator , 11 ( 1 ) : 4 – 9 .
  • Olive , J. and Steffe , L. P. 2002 . The construction of an iterative fractional scheme: The case of Joe . Journal of Mathematical Behavior , 20 : 413 – 437 .
  • Orrill , C.H. , Sexton , S. , Lee , S.-J. and Gerde , C. Mathematics teachers' abilities to use and make sense of drawn representations . The International Conference of the Learning Sciences 2008: Proceedings of ICLS 2008 . Mahwah, NJ. International Society of the Learning Sciences .
  • Post , T. R. , Wachsmuth , I. , Lesh , R. and Behr , M. J. 1985 . Order and equivalence of rational number: A cognitive analysis . Journal for Research in Mathematics Education , 16 ( 1 ) : 18 – 36 .
  • Schwartz , J. 1988 . “ Intensive quantity and referent transforming arithmetic operations ” . In Number concepts and operations in the middle grades , Edited by: Hiebert , J. and Behr , M. 119 – 140 . Reston, VA : National Council of Teachers of Mathematics .
  • Seaman , C. E. and Szydlik , J. E. 2007 . Mathematical sophistication among preservice elementary teachers . Journal of Mathematics Teacher Education , 10 ( 3 ) : 167 – 182 .
  • Shulman , L. S. 1986 . Those who understand: Knowledge growth in teaching . Educational Researcher , 15 ( 2 ) : 4 – 14 .
  • SII/LMT (Study of Instructional Improvement/Learning Mathematics for Teaching) . 2004 . Learning mathematics for teaching , Ann Arbor, MI : Consortium for Policy Research in Education .
  • Smith , J. P. 1995 . Competent reasoning with rational numbers . Cognition and Instruction , 13 ( 1 ) : 3 – 50 .
  • Smith , J. P. , diSessa , A. A. and Roschelle , J. 1993 . Misconceptions reconceived: A constructivist analysis of knowledge in transition . The Journal of the Learning Sciences , 3 ( 2 ) : 115 – 163 .
  • Smith , S. 2003 . “ Representation in school mathematics: Children's representations of problems ” . In A research companion to principles and standards for school mathematics , Edited by: Kilpatrick , J. , Martin , W. G. and Schifter , D. Reston, VA : National Council of Teachers of Mathematics .
  • Steffe , L. P. 2002 . A new hypothesis concerning children's fractional knowledge . Journal of Mathematical Behavior , 102 : 1 – 41 .
  • Steffe , L. P. and Olive , J. 2010 . Children's fractional knowledge , New York, NY : Springer .
  • Strauss , A. and Corbin , J. 1998 . Basics of qualitative research: Techniques and procedures for developing grounded theory , 2nd , Thousand Oaks, CA : Sage .

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.