Development of Teachers’ Conceptions Through Learning and Teaching: The Meaning and Potential of Multiple-Solution Tasks

REFERENCES

• Ball , D. L. 1993 . “ Halves, pieces, and twoths: Constructing representational contexts in teaching fractions ” . In Rational numbers: An integration of research , Edited by: Carpenter , T. , Fennema , E. and Romberg , T. 157 – 196 . Hillsdale, NJ : Erlbaum .
   Google Scholar
• Barbeau , E. J. and Taylor , P. J. 2009 . ICMI Study-16 Volume: Mathematical challenge in and beyond the classroom. Study Volume of ICMI study 16 , New York, NY : Springer .
   Google Scholar
• Brousseau , G. 1997 . Theory of didactical situations in mathematics , Dordrecht, , The Netherlands : Kluwer .
   Google Scholar
• Brown , J. S. , Collins , A. and Duguid , P. 1989 . Situated cognition and the culture of learning . Educational Researcher , 1 : 32 – 41 .
   Google Scholar
• Cooney , T. and Shealy , B. 1997 . “ On understanding the structure of teachers’ beliefs and their relationship to change ” . In Mathematics teachers in transisition , Edited by: Fennema , E. and Scott-Nelson , B. 87 – 109 . Mahwah, NJ : Lawrence Erlbaum .
   Google Scholar
• Da Ponte , J. P. and Chapman , O. 2006 . “ Mathematics teachers’ knowledge and practices ” . In Handbook of research on the psychology of mathematics education: Past present and future , Edited by: Gutiérrez , A. and Boero , P. 461 – 494 . Rotterdam : Sense .
   Google Scholar
• Davydov , V. V. 1996 . Theory of developing education [Razvivayushee obuchenie] , Moscow : Intor .
   Google Scholar
• Dhombres , J. 1993 . Is one proof enough? Travels with a mathematician of the baroque period . Educational Studies in Mathematics , 24 : 401 – 419 .
   Google Scholar
• Dreyfus , T. and Eisenberg , T. 1986 . On the aesthetics of mathematical thought . For the Learning of Mathematics , 6 ( 1 ) : 2 – 10 .
   Google Scholar
• Ervynck , G. 1991 . “ Mathematical creativity ” . In Advanced mathematical thinking , Edited by: Tall , D. 42 – 53 . Dordrecht, , The Netherlands : Kluwer .
   Google Scholar
• Fennema , E. and Romberg , T. A. 1999 . Classrooms that promote mathematical understanding , Mahwah, NJ : Erlbaum .
   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
• House , P. A. and Coxford , A. F. 1995 . Connecting mathematics across the curriculum: 1995 Yearbook , Reston, VA : National Council of Teachers of Mathematics .
   Google Scholar
• Kennedy , M. M. 2002 . Knowledge and teaching . Teacher and Teaching: Theory and Practice , 8 : 355 – 370 .
   Google Scholar
• Kieren , T. E. 1990 . Understanding for teaching for understanding . The Alberta Journal of Educational Research , 36 : 191 – 201 .
   Google Scholar
• Kilpatrick , J. , Hoyles , C. , Skovsmose , O. and Valero , P. 2005 . Meaning in mathematics education , Dordrecht, , The Netherlands : Kluwer .
   Google Scholar
• Krainer , K. 2003 . Teams, communities & networks . Journal of Mathematics Teacher Education , 6 : 185 – 194 .
   Google Scholar
• Krutetskii , V. A. 1976 . The psychology of mathematical abilities in school children , Edited by: Teller , J. , Kilpatrick , J. and Wirszup , I. Chicago : The University of Chicago Press .
   Google Scholar
• Lave , J. 1996 . Teaching, as learning, in practice . Mind, Culture & Activity , 3 ( 3 ) : 149 – 164 .
   Google Scholar
• Lave , J. and Wenger , E. 1991 . Situated learning: Legitimate peripheral participation , Cambridge, UK : Cambridge University Press .
   Google Scholar
• Leikin , R. 2003 . Problem-solving preferences of mathematics teachers . Journal of Mathematics Teacher Education , 6 : 297 – 329 .
   Google Scholar
• Leikin , R. 2006a . About four types of mathematical connections and solving problems in different ways . Aleh—The (Israeli) Senior School Mathematics Journal , 36 : 8 – 14 .
   Google Scholar
• Leikin , R. 2006b . “ Learning by teaching: The case of Sieve of Eratosthenes and one elementary school teacher ” . In Number theory in mathematics education: Perspectives and prospects , Edited by: Zazkis , R. and Campbell , S. 115 – 140 . Mahwah, NJ : Erlbaum .
   Google Scholar
• Leikin , R. 2007 . “ Habits of mind associated with advanced mathematical thinking and solution spaces of mathematical tasks ” . In The Fifth Conference of the European Society for Research in Mathematics Education—CERME-5 2330 – 2339 . Retrieved October, 3, 2009, from http://ermeweb.free.fr/CERME5b/
   Google Scholar
• Leikin , R. 2009 . “ Multiple proof tasks: Teacher practice and teacher education ” . In Proceeding of the 19th ICMI Study conference: Proofs and proving in mathematics education , Edited by: Lin , F.-L. , Hsieh , F.-J. , Hana , G. and De Villiers , M. Vol. 2 , 31 – 36 . Taiwan : National Taipei University .
   Google Scholar
• Leikin , R. and Lev , M. 2007 . “ Multiple solution tasks as a magnifying glass for observation of mathematical creativity ” . In Proceedings of the 31st International Conference for the Psychology of Mathematics Education , Edited by: Woo , J.-H. , Lew , H.-C. , Park , K.-S. and Seo , D.-Y. 161 – 168 . Korea : The Korea Society of Educational Studies in Mathematics .
   Google Scholar
• Leikin , R. and Levav-Waynberg , A. 2007 . Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks . Educational Studies in Mathematics , 66 : 349 – 371 .
   Google Scholar
• Leikin , R. and Levav-Waynberg , A. 2008 . Solution spaces of multiple-solution connecting tasks as a mirror of the development of mathematics teachers’ knowledge . Canadian Journal of Science, Mathematics and Technology Education , 8 ( 3 ) : 233 – 251 .
   Google Scholar
• Leikin , R. , Levav-Waynberg , A. , Gurevich , I. and Mednikov , L. 2006 . Implementation of multiple solution connecting tasks: Do students’ attitudes support teachers’ reluctance? . FOCUS on Learning Problems in Mathematics , 28 : 1 – 22 .
   Google Scholar
• Leikin , R. and Zazkis , R. Learning through teaching mathematics: Development of teachers’ knowledge and expertise in practice , Springer . (in press).
   Google Scholar
• Lloyd , G. M. 2002 . “ Mathematics teachers’ beliefs and experiences with innovative curriculum materials. The role of curriculum in teacher development ” . In Beliefs: A hidden variable in mathematics education? , Edited by: Leder , G. C. , Pehkonen , E. and Torner , G. 149 – 159 . Utrecht, , The Netherlands : Kluwer .
   Google Scholar
• National Council of Teachers of Mathematics . 2000 . Principles and standards for school mathematics , Reston, VA : Author .
   Google Scholar
• Piaget , J. 1985 . The equilibration of cognitive structures: The central problem of intellectual development , Chicago : University of Chicago Press .
   Google Scholar
• Polya , G. 1973 . How to solve it. A new aspect of mathematical method , Princeton, NJ : Princeton University Press .
   Google Scholar
• Roth , W.-M. 1998 . Designing communities , Boston : Kluwer Academic .
   Google Scholar
• Scheffler , I. 1965 . Conditions of knowledge. An introduction to epistemology and education , Glenview, IL : Scott, Foresman & Company .
   Google Scholar
• Schoenfeld , A. H. 1983 . Problem solving in the mathematics curriculum: A report, recommendations’ and an annotated bibliography , Washington, DC : The Mathematical Association of America .
   Google Scholar
• Sfard , A. 1991 . On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin . Educational Studies in Mathematics , 22 : 1 – 36 .
   Google Scholar
• Shulman , L. S. 1986 . Those who understand: Knowing growth in teaching . Educational Researcher , 5 ( 2 ) : 4 – 14 .
   Google Scholar
• Sierpinska , A. 1990 . Some remarks on understanding in mathematics . For the Learning of Mathematics , 10 : 24 – 36 .
   Google Scholar
• Silver , E. A. , Ghousseini , H. , Gosen , D. , Charalambous , C. and Font Strawhun , B. T. 2005 . Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom . Journal of Mathematical Behavior , 24 : 287 – 301 .
   Google Scholar
• Silver , E. A. and Marshall , S. P. 1990 . “ Mathematical and scientific problem solving: Findings, issues, and instructional implications ” . In Dimensions of thinking and cognitive instruction , Edited by: Jones , B. F. and Idol , L. Vol. 1 , 265 – 290 . Hillsdale, NJ : Erlbaum .
   Google Scholar
• Simon , A. M. 1997 . “ Developing new models of mathematics teaching: An imperative for research on mathematics teacher development ” . In Mathematics teachers in transition , Edited by: Fennema , E. and Scott-Nelson , B. 55 – 86 . Mahwah, NJ : Erlbaum .
   Google Scholar
• Simon , M. A. 2000 . “ Research on the development of mathematics teachers: The Teacher Development Experiment ” . In Research design in mathematics and science education , Edited by: Kelly , A. E. and Lesh , R. A. 335 – 359 . Mahwah, NJ : Erlbaum .
   Google Scholar
• Simon , M. A. 2007 . “ Constraints on what teachers can learn from their practice: Teachers’ assimilatory schemes ” . In Proceedings of the 31st International Conference for the Psychology of Mathematics Education , Edited by: Woo , J-H , Lew , H.-C. , Park , K.-S. and Seo , D.-Y. 137 – 142 . Korea : The Korea Society of Educational Studies in Mathematics .
   Google Scholar
• Skemp , R. R. 1987 . The psychology of learning mathematics , Hillsdale, NJ : Erlbaum .
   Google Scholar
• Steinbring , H. 1998 . Elements of epistemological knowledge for mathematics teachers . Journal of Mathematics Teacher Education , 1 : 157 – 189 .
   Google Scholar
• Stigler , J. W. and Hiebert , J. 1999 . The teaching gap: Best ideas from the world's teachers for improving education in the classroom , New York : The Free Press .
   Google Scholar
• Strauss , A. and Corbin , J. 1990 . Basics of qualitative research: Grounded theory procedures and techniques , Newbury Park, CA : Sage .
   Google Scholar
• Thompson , A. G. 1992 . “ Teachers’ beliefs and conceptions: A synthesis of the research ” . In Handbook of research on mathematics teaching and learning , Edited by: Grouws , D. A. 127 – 146 . New York : Macmillan .
   Google Scholar
• Vygotsky , L. S. 1978 . Mind in society: The development of higher psychological processes , Cambridge, MA : Harvard University Press .
   Google Scholar
• Yerushalmy , M. , Chazan , D. and Gordon , M. 1990 . Mathematical problem posing: Implications for facilitating student inquiry in classrooms . Instructional Science , 19 : 219 – 245 .
   Google Scholar
• Zaslavsky , O. , Chapman , O. and Leikin , R. 2003 . “ Professional development of mathematics educators: Trends and tasks ” . In The second international handbook of mathematics education , Edited by: Bishop , A. J. , Clements , M. A. , Brunei , D. , Keitel , C. , Kilpatrick , J. and Leung , F. K. S. 875 – 915 . Dordrecht, , The Netherlands : Kluwer .
   Google Scholar