Developing the use of diagrammatic representations in primary mathematics through professional development

References

• Ainsworth , S. 1999 . The functions of multiple representations . Computers & Education , 33 : 131 – 152 .
   Web of Science ®Google Scholar
• Back , J. , De Geest , E. , Hirst , C. and Joubert , M. 2009 . Final report: Researching effective CPD in mathematics education (RECME) , Sheffield : NCETM .
   Google Scholar
• 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 .
   Web of Science ®Google Scholar
• Barmby , P. , Bilsborough , L. , Harries , T. and Higgins , S. 2009a . Primary mathematics: Teaching for understanding , Buckingham : Oxford University Press .
   Google Scholar
• Barmby , P. , Harries , T. , Higgins , S. and Suggate , J. 2009b . The array representation and primary children’s understanding and reasoning in multiplication . Educational Studies in Mathematics , 70 : 217 – 241 .
   Web of Science ®Google Scholar
• Barmby, P., and J. Milinkovic. 2011. “Pre-service teachers’ use of visual representations of multiplication.” In Proceedings of the 35th Annual Conference of the International Group for the Psychology of Mathematics Education (Vol. 2), edited by B. Ubuz, 105–112. Ankara, Turkey: PME.
   Google Scholar
• Baturo, A. R., and T. J. Cooper. 1999. “Fractions, reunitisation and the number-line representation.” In Proceedings of the 23rd Annual Conference of the International Group for the Psychology of Mathematics Education (Vol. 2), edited by O. Zaslavsky, 81–88. Haifa, Israel: PME.
   Google Scholar
• Behr , M. J. , Lesh , R. , Post , T. P. and Silver , E. A. 1983 . “ Rational-number concepts ” . In Acquisition of mathematics concepts and processes , Edited by: Lesh , R. and Landau , M. 91 – 126 . New York : Academic Press .
   Google Scholar
• 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 .
   Google Scholar
• Borko , H. 2004 . Professional development and teacher learning: Mapping the terrain . Educational Researcher , 33 ( 8 ) : 3 – 15 .
   Google Scholar
• Boulton-Lewis , G. M. 1998 . Children’s strategy use and interpretations of mathematical representations . Journal of Mathematical Behavior , 17 ( 2 ) : 219 – 237 .
   Google Scholar
• Brophy, J. 1991. “Conclusion.” In Advances in research on teaching (Vol. 2), edited by J. Brophy, 349–364. Greenwich, CT: JAI Press.
   Google Scholar
• Burkhardt , H. and Schoenfeld , A. H. 2003 . Improving educational research: Toward a more useful, more influential, and better-funded enterprise . Educational Researcher , 32 ( 9 ) : 3 – 14 .
   Google Scholar
• Charalambous , C. Y. and Pitta-Pantazi , D. 2007 . Drawing on a theoretical model to study student’s understandings of fractions . Educational Studies in Mathematics , 64 ( 3 ) : 293 – 316 .
   Google Scholar
• Cobb , P. , Yackel , E. and Wood , T. 1992 . A constructivist alternative to the representational view of mind in mathematics education . Journal for Research in Mathematics Education , 23 ( 1 ) : 2 – 33 .
   Web of Science ®Google Scholar
• Cohen , L. , Manion , L. and Morrison , K. 2000 . Research methods in education , 5th ed. , London : RoutledgeFalmer .
   Google Scholar
• Creswell , J. W. 1994 . Research design: Qualitative and quantitative approaches , London : Sage .
   Google Scholar
• Desimone , L. M. , Porter , A. C. , Garet , M. S. , Yoon , K. S. and Birman , B. F. 2002 . Effects of professional development on teachers’ instruction: Results from a three-year longitudinal study . Educational Evaluation and Policy Analysis , 24 ( 2 ) : 81 – 112 .
   Web of Science ®Google Scholar
• Duval, R. 1999. “Representation, vision and visualization: Cognitive functions in mathematical thinking.” In Proceedings of the Twenty-first Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, edited by F. Hitt, and M. Santos, 3–26. Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
   Google Scholar
• Flevares , L. M. and Perry , M. 2001 . How many do you see? The use of nonspoken representations in first-grade mathematics lessons . Journal of Educational Psychology , 93 ( 2 ) : 330 – 345 .
   Web of Science ®Google Scholar
• Flick , U. 2009 . An introduction to qualitative research , 4th ed. , London : Sage .
   Google Scholar
• Franke , M. L. , Carpenter , T. P. , Levi , L. and Fennema , E. 2001 . Capturing teachers’ generative change: A follow-up study of professional development in mathematics . American Educational Research Journal , 38 ( 3 ) : 653 – 689 .
   Web of Science ®Google Scholar
• Garet , M. S. , Porter , A. C. , Desimore , L. , Birman , B. F. and Yoon , K. S. 2001 . What makes professional development effective? Results from a national sample of teachers. . American Educational Research Journal , 38 ( 4 ) : 915 – 945 .
   Web of Science ®Google Scholar
• Gersten , R. , Chard , D. J. , Jayanthi , M. , Baker , S. K. , Morphy , P. and Flojo , J. 2009 . Mathematics instruction for students with learning disabilities: A meta-analysis of instructional components . Review of Educational Research , 79 ( 3 ) : 1202 – 1242 .
   Web of Science ®Google Scholar
• Goldin , G. A. 1998 . Representational systems, learning and problem solving in mathematics . Journal of Mathematical Behavior , 17 ( 2 ) : 137 – 165 .
   Google Scholar
• Greeno , J. G. and Hall , R.P. 1997 . Practicing representation: Learning with and about representational forms . The Phi Delta Kappan , 78 ( 5 ) : 361 – 367 .
   Web of Science ®Google Scholar
• Greer , B. 1992 . “ Multiplication and division as models of situations ” . In Handbook of research on mathematics teaching and learning , Edited by: Grouws , D. A. 276 – 295 . New York : Macmillan .
   Google Scholar
• Hall , N. 1998 . Concrete representations and procedural analogy theory . Journal of Mathematical Behavior , 17 ( 1 ) : 33 – 51 .
   Google Scholar
• Hannula, M. S. 2003. “Locating fraction on a number line.” In Proceedings of the 2003 Joint Meeting of the PME and PMENA (Vol. 3), edited by N. A. Pateman, B. J. Dougherty, and J. Zilliox, 17–24. CRDG, College of Education, University of Hawaii, Hawaii.
   Google Scholar
• Hart , K. M. , ed. 1981 . Children’s understanding of mathematics: 11–16 , London : John Murray .
   Google Scholar
• Hiebert , J. and Carpenter , T. P. 1992 . “ Learning and teaching with understanding ” . In Handbook of research on mathematics teaching and learning , Edited by: Grouws , D. A. 65 – 97 . New York : Macmillan .
   Google Scholar
• Hiebert , J. and Wearne , D. 1992 . Links between teaching and learning place value with understanding in first grade . Journal for Research in Mathematics Education , 23 ( 2 ) : 98 – 122 .
   Web of Science ®Google Scholar
• Hill , H. C. and Ball , D. L. 2004 . Learning mathematics for teaching: Results from California’s mathematics professional development institutes . Journal for Research in Mathematics Education , 35 ( 5 ) : 330 – 351 .
   Web of Science ®Google Scholar
• Kaestle , C.F. 1993 . The awful reputation of education research . Educational Researcher , 22 ( 1 ) 23 and 26–31
   Google Scholar
• Kamii , C. and Clark , F. B. 1995 . Equivalent fractions: Their difficulty and educational implications . Journal of Mathematical Behavior , 14 : 365 – 378 .
   Google Scholar
• Kaput , J. J. 1991 . “ Notations and representations as mediators of constructive processes ” . In Radical constructivism in mathematics education , Edited by: von Glasersfeld , E. 53 – 74 . Dordrecht : Kluwer .
   Google Scholar
• Kerslake , D. 1986 . Fractions: Children’s strategies and errors , London : NFER Nelson .
   Google Scholar
• Kieren , T. 1976 . “ On the mathematical, cognitive, and instructional foundations of rational numbers ” . In Number and measurement: Papers from a research workshop , Edited by: Lesh , R. 101 – 144 . Columbus , OH : ERIC/SMEAC .
   Google Scholar
• Korthagen , F. A. J. and Kessels , J. P. A. M. 1999 . Linking theory and practice: Changing the pedagogy of teacher education . Educational Researcher , 28 ( 4 ) : 4 – 17 .
   Google Scholar
• Kvale , S. 2007 . Doing interviews , London : Sage .
   Google Scholar
• Lampert , M. 1986 . Knowing, doing, and teaching multiplication . Cognition and Instruction , 3 : 305 – 342 .
   Google Scholar
• Larkin , J. H. and Simon , H. A. 1987 . Why a diagram is (sometimes) worth ten thousand words . Cognitive Science , 11 : 65 – 69 .
   Web of Science ®Google Scholar
• Lean , G. and Clements , K. 1981 . Spatial ability, visual imagery, and mathematical performance . Educational Studies in Mathematics , 12 ( 3 ) : 267 – 299 .
   Google Scholar
• Leinhardt, G., R. T. Putnam, M. K. Stein, and J. Baxter. 1991. “Where subject knowledge matters.” In Advances in research on teaching: Vol. 2. Teachers’ knowledge of subject matter as it relates to their teaching practice, edited by J. Brophy, 87–113. Greenwich, CT: JAI Press.
   Google Scholar
• Lesh , R. , Landau , M. and Hamilton , E. 1983 . “ Conceptual models and applied mathematical problem-solving research ” . In Acquisition of mathematics concepts and processes , Edited by: Lesh , R. and Landau , M. 263 – 343 . Orlando , FL : Academic Press .
   Google Scholar
• Mack , N. K. 1990 . Learning fractions with understanding: Building on informal knowledge . Journal for Research in Mathematics Education , 21 ( 1 ) : 16 – 32 .
   Web of Science ®Google Scholar
• Muijs , D. 2004 . Doing quantitative research in education with SPSS , London : Sage .
   Google Scholar
• Ni , Y. 2001 . Semantic domains of rational numbers and the acquisition of fraction equivalence . Contemporary Educational Psychology , 26 : 400 – 417 .
   PubMed Web of Science ®Google Scholar
• Outhred, L., and M. Mitchelmore. 2004. “Student’s structuring of rectangular arrays.” In Proceedings 28th Annual Conference of the International Group for the Psychology of Mathematics Education (Vol. 3), edited by M. J. Høines, and A. Fuglestad, 465–472. Bergen, Norway: PME.
   Google Scholar
• Pape , S. J. and Tchoshanov , M. A. 2001 . The role of representation(s) in developing mathematical understanding . Theory into Practice , 40 ( 2 ) : 118 – 127 .
   Web of Science ®Google Scholar
• Paivio , A. 1969 . Mental imagery in associative learning and memory . Psychological Review , 76 ( 3 ) : 241 – 263 .
   Web of Science ®Google Scholar
• Peck , D. M. and Jencks , S. M. 1981 . Conceptual issues in the teaching and learning of fractions . Journal for Research in Mathematics Education , 12 ( 5 ) : 339 – 348 .
   Google Scholar
• Perkins , D. N. and Unger , C. 1994 . A new look in representations for mathematics and science learning . Instructional Science , 22 : 1 – 37 .
   Web of Science ®Google Scholar
• Pitkethley , A. and Hunting , R. 1996 . A review of recent research in the area of initial fraction concepts . Educational Studies in Mathematics , 30 ( 1 ) : 5 – 38 .
   Google Scholar
• Post , T. R. and Cramer , K. A. 1989 . “ Knowledge, representation, and quantitative thinking ” . In Knowledge base for the beginning teacher , Edited by: Reynolds , M. C. 221 – 232 . New York : Pergamon .
   Google Scholar
• Presmeg , N. C. 1986 . Visualisation and mathematical giftedness . Educational Studies in Mathematics , 17 ( 3 ) : 297 – 311 .
   Google Scholar
• Punch , K. F. 2009 . Introduction to research methods in education , London : Sage .
   Google Scholar
• Rhodes , C. and Houghton-Hill , S. 2000 . The linkage of continuing professional development and the classroom experience of pupils: Barriers perceived by senior managers in some secondary schools . Journal of In-Service Education , 26 ( 3 ) : 423 – 435 .
   Google Scholar
• Robson , C. 2002 . Real world research , 2nd ed. , Oxford : Blackwell Publishing .
   Google Scholar
• Skemp , R. R. 1986 . The psychology of learning mathematics , Harmondsworth : Penguin Books .
   Google Scholar
• Shulman , L. S. 1986 . Those who understand: Knowledge growth in teaching . Educational Researcher , 15 ( 2 ) : 4 – 14 .
   Google Scholar
• Sowell , E. J. 1989 . Effects of manipulative materials in mathematics education . Journal for Research in Mathematics Education , 20 ( 5 ) : 498 – 505 .
   Web of Science ®Google Scholar
• Tashakkori , A. and Teddlie , C. 1998 . Mixed methodology , London : Sage .
   Google Scholar
• Teacher Training Agency . 1998 . National standards for subject leaders , London : TTA .
   Google Scholar
• Webb, N. M., K. Gold, and S. Qi. 1990. Mathematical problem-solving processes and performance: Translations among symbolic representations (CSE No. 316). Los Angeles, CA: UCLA Center for Research on Evaluation, Standards, and Student Testing.
   Google Scholar
• Wilson , S. M. and Berne , J. 1999 . Learning and the acquisition of professional knowledge: An examination of research on contemporary professional development . Review of Research in Education , 24 : 173 – 209 .
   Web of Science ®Google Scholar