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Original Articles

Geometric and algebraic approaches in the concept of complex numbers

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Pages 681-706 | Received 06 Jul 2005, Published online: 20 Feb 2007

References

  • Arcavi , A . 2003 . The role of visual representations in the learning of mathematics . Educational Studies in Mathematics , 52 : 215 – 241 .
  • Pimm , D . 1995 . Symbols and Meanings in School Mathematics , London : Routledge .
  • Riding , R and Wigley , S . 1997 . The relationship between cognitive style and personality in further education students . Personality and Individual Differences , 23 ( 3 ) : 379 – 389 .
  • National Council for Teacher of Mathematics . 2000 . Principles and Standards for School Mathematics , VA : Reston .
  • Sfard , A . 1998 . “ Symbolizing mathematical reality into being or how mathematical discourse and mathematical objects create each other ” . In Symbolizing and Communicating: Perspectives on Mathematical Discourse, Tools and Instructional Design , Edited by: Cobb , P , Yackel , KE and McClain , K . 37 – 98 . Mahwah : Erlbaum .
  • Glas , E . 1998 . Fallibilism and the use of history in mathematics education . Science and Education , 7 : 361 – 379 .
  • Thompson , P and Saldanha , L . 2003 . “ Fractions and multiplicative reasoning ” . In Research Companion to the Principles and Standards for School Mathematics , Edited by: Kilpatrick , J. , Martin , G and Schifter , D . 95 – 114 . Reston, VA : National Council of Teacher of Mathematics .
  • Henle , JM . 1986 . An Outline of Set Theory , New York : Springer Verlag .
  • 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 .
  • Harkin , A and Harkin , J . 2004 . Geometry of generalized complex numbers . Mathematics Magazine , 77 ( 2 ) : 118 – 129 .
  • Craig , HV . 1943 . Vector and Tensor Analysis , New York and London : McGraw-Hill .
  • Demetriadou , H and Gagatsis , A . 1996 . “ On the history of the concept of vector and its introduction in element and geometry textbooks ” . In Didactics and History of Mathematics , Edited by: Gagatsis , A and Rogers , L . 371 – 403 . Thessaloniki : Erasmus ICP-95-G-2011/11 .
  • Burton , DM . 1988 . The History of Mathematics – An Introduction , Wm.C. : Brown Publishers .
  • Leitzel , JR . 1989 . “ Critical considerations for the future of algebra or a reaction to Algebra: what should we teach and how shoud we teach it? ” . In Research Issues in the Learning and Teaching of Algebra , Edited by: Wanger , S and Kieran , C . 25 – 32 . NJ, Hillsdale : Erlbaum .
  • Cuoco , A . 1997 . Constructing the complex numbers . International Journal of Computers for Mathematical Learning , 2 : 155 – 186 .
  • Tall , D . 2004 . Introducing three worlds of mathematics . For the Learning of Mathematics , 23 ( 3 ) : 29 – 33 .
  • Riding , R and Cheeman , I . 1991 . Cognitive styles – an overview and integration . Educational Psychology , 11 : 193 – 215 .
  • Rayner , S and Riding , R . 1997 . Towards a categorization of cognitive styles and learning styles . Educational Psychology , 17 : 5 – 28 .
  • Mousoulides , N and Gagatsis , A . 2004 . “ Algebraic and geometric approach in function problem solving ” . In Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education , Edited by: Johnsen Hoines , M and Berit Fuglestad , A . Vol. 3 , 385 – 392 . Bergen, , Norway : Bergen University College .
  • Eliot , J and Smith , IM . 1983 . An International Directory of Spatial Tests , UK : NFER .
  • Toru , I . 2005 . Why some students have trouble with maps and other spatial representation . Journal of Geoscience Education , 2 ( 5 ) : 184 – 197 .
  • Markmann , AB . 1999 . Knowledge Representation , Mahwah, NJ : Erlbaum .
  • Gagatsis , A , Elia , I and Mougi , A . 2002 . The nature of multiple representations in developing mathematical relations . Scientia Paedagogica Experimentalis , 39 ( 1 ) : 9 – 24 .
  • Schnotz , W . 2002 . Towards an integrated view of learning from text and visual displays . Educational Psychology Review , 14 ( 1 ) : 101 – 120 .
  • Elia , I , Gagatsis , A and Deliyianni , E . 2005 . “ A review of the effects of different modes of representations in mathematical problem solving ” . In Proceedings of the 4th Mediterranean Conference on Mathematics Education , Edited by: Gagatsis , A. , Spagnolo , F. , Makrides , Gr. and Farmaki , V. Vol. 1 , 271 – 286 . Palermo, , Italy : University of Palermo, Cyprus Mathematical Society .
  • DeWindt-King , AM and Goldin , GA . 2003 . Children's visual imagery: aspects of cognitive representation in solving problems with fractions . Mediterranean Journal for Research in Mathematics Education , 2 ( 1 ) : 1 – 42 .
  • Stylianou , D . 2001 . “ On the reluctance to visualize in mathematics: Is the picture changing? ” . In Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education , Edited by: van den Heurel–Panhuizen , M. Vol. 4 , 225 – 232 . Utrecht, , The Netherlands : Freudenthal Institute .
  • Janvier , C . 1987 . “ Representation and understanding: The notion of functions as an example ” . In Problems of Representation in the Teaching and Learning of Mathematics , Edited by: Janvier , C. 67 – 72 . Hillsdale, NJ : Lawrence Erlbaum Associates .
  • Duval , R and á paraître . 2005 . Langage, symboles, images, schémas … De quelle manière interviennent-ils dans la compréhension, en mathématiques et en dehors des mathématiques . Bollettino dei Docenti di Matematica , 50 in press
  • Leinhardt , G , Zaslavsky , O and Stein , MK . 1990 . Functions, graphs and graphing: tasks, learning and teaching . Review of Educational Research , 60 : 1 – 64 .
  • Even , R . 1998 . Factors involved in linking representations of functions . Journal of Mathematical Behavior , 17 ( 1 ) : 105 – 121 .
  • Cheng , PCH . 2000 . Unlocking conceptual learning in mathematics and science with effective representational systems . Computers and Education , 33 : 109 – 130 .
  • Greeno , JG and Hall , RP . 1997 . Practicing representation: learning with and about representational forms . Phi Delta Kappan , 78 : 361 – 367 .
  • Kalathil , RR and Sheril , MG . 2000 . “ Role of students’ representations in the mathematics classroom ” . In Fourth International Conference of the Learning of Sciences , Edited by: Fishman , B. and O’Connor-Divelbiss , S. 27 – 28 . Mahwah, NJ : Erlbaum .
  • 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 .
  • Sierpinska , A . 1992 . “ On understanding the notion of function ” . In The Concept of Function: Aspects of Epistemology and Pedagogy , Edited by: Dubinsky , E and Harel , G . 25 – 58 . United States : Mathematical Association of America .
  • Lesh , R , Post , T and Behr , M . 1987 . “ Representations and translations among representations in mathematics learning and problem solving ” . In Problems of Representation in the Teaching and Learning of Mathematics , Edited by: Janvier , C. 33 – 40 . Hillsdale, NJ : Lawrence Erlbaum Associates .
  • Duval , R . Figures’ representational function in geometry and figure's multiple and parallel entries . Twentieth- first annual meeting of the north American chapter of the International Group for the Psychology of Mathematics Education . Cuervanaca, Mexico. A preliminary version for the working group on representations and mathematics visualization ,
  • Duval , R . 2002 . The cognitive analysis of problems of comprehension in the learning of mathematics . Mediterranean Journal for Research in Mathematics Education , 1 ( 2 ) : 1 – 16 .
  • Duval , R . 2001 . “ Pourquoi les representations semiotiques doivent-elles etre placees on centre des apprentissages in mathematiques? ” . In Learning in Mathematics and Science and Educational Technology , Edited by: Gagatsis , A . Vol. 1 , 67 – 90 . Nicosia : Intercollege Press-University of Cyprus .
  • Vinner , S . 1989 . The avoidance of visual considerations in calculus students . Focus on Learning Problems in Mathematics , 11 ( 2 ) : 149 – 156 .
  • Gagatsis , A , Elia , I and Andreou , S . 2003 . Representations and mathematics learning: functions and number line . Euclides γ , 59 : 5 – 34 . (in Greek)
  • National Council of Teacher of Mathematics . 1989 . Curriculum and evaluation standards for school mathematics , Reston, VA : Author .
  • Lerman , IC . 1981 . Classification et analyse ordinale des données , Paris : Dunod .
  • Gras , R , Peter , P , Briand , H and Philippe , J . Implicative statistical analysis . Proceedings of the 5th Conference of the International Federation of Classification Societies . Edited by: Hayashi , C , Ohsumi , N , Yajima , N , Tanaka , Y , Bock , H and Baba , Y . Vol. 2 , pp. 412 – 419 . New York : Springer-Verlag .
  • Bodin , A , Coutourier , R and Gras , R . 2000 . CHIC: Classification Hiérarchique Implicative et Cohésitive-Version sous Windows – CHIC 1.2 , Rennes : Association pour la Recherche en Didactique des Mathématiques .
  • Pantziara , M , Gagatsis , A and Pitta-Pantazi , D . The use of diagrams in solving non-routine problems . Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education . Edited by: Johnsen Hoines , M and Berit Fuglestad , A . Vol. 3 , pp. 489 – 496 . Bergen, , Norway : Bergen University College .

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