메뉴 건너뛰기




Volumn 40, Issue 1, 2008, Pages 111-129

"Rising to the challenge": Using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students

Author keywords

[No Author keywords available]

Indexed keywords


EID: 59049088002     PISSN: 18639690     EISSN: 18639704     Source Type: Journal    
DOI: 10.1007/s11858-007-0069-5     Document Type: Article
Times cited : (50)

References (47)
  • 2
    • 84867389553 scopus 로고    scopus 로고
    • Why do we learn mathematics? Because it organizes our minds An encounter between former Soviet Union and Israeli cultures of mathematics education
    • Palermo, Italy: Palermo University
    • Amit, M., & Burda, I. (2005). "Why do we learn mathematics? Because it organizes our minds" An encounter between former Soviet Union and Israeli cultures of mathematics education. Proceedings of the 57th conference of the international commission for the study and improvement of mathematical education (pp. 16-22). Palermo, Italy: Palermo University.
    • (2005) Proceedings of the 57th conference of the international commission for the study and improvement of mathematical education , pp. 16-22
    • Amit, M.1    Burda, I.2
  • 4
    • 84867389156 scopus 로고    scopus 로고
    • Paper presented at the 13th conference of the International Community of Teachers of Mathematical Modeling and Applications, IN, USA
    • Amit, M., & Neria, D. (2007). Assessing a modeling process of a linear pattern task. Paper presented at the 13th conference of the International Community of Teachers of Mathematical Modeling and Applications, IN, USA.
    • (2007) Assessing a modeling process of a linear pattern task
    • Amit, M.1    Neria, D.2
  • 5
    • 64949157815 scopus 로고    scopus 로고
    • Struggling with variables, parameters, and indeterminate objects or how to go insane in mathematics
    • In: H. L. Chick, J. L. Vincent (Eds.),). Melbourne, Australia: University of Melbourne
    • Bardini, C., Radford, L., & Sabena, C. (2005). Struggling with variables, parameters, and indeterminate objects or how to go insane in mathematics. In: H. L. Chick, J. L. Vincent (Eds.), Proceedings of the 29th conference of the international group for the psychology of mathematics education (Vol. 2, pp. 129-136). Melbourne, Australia: University of Melbourne.
    • (2005) Proceedings of the 29th conference of the international group for the psychology of mathematics education , vol.2 , pp. 129-136
    • Bardini, C.1    Radford, L.2    Sabena, C.3
  • 6
    • 84859268295 scopus 로고    scopus 로고
    • Aninvestigation of beginning algebra students' ability to generalize linear patterns
    • In: M. J. Hoins, & A. B. Fuglestad (Eds.), Bergen, Norway: Bergen University College
    • Becker, J. R., &Rivera, F. (2004). Aninvestigation of beginning algebra students' ability to generalize linear patterns. In: M. J. Hoins, & A. B. Fuglestad (Eds.), Proceedings of the 28th conference of the international group for the psychology of mathematics education (Vol. 1, p. 286). Bergen, Norway: Bergen University College.
    • (2004) Proceedings of the 28th conference of the international group for the psychology of mathematics education , vol.1 , pp. 286
    • Becker, J.R.1    Rivera, F.2
  • 7
    • 4544328910 scopus 로고    scopus 로고
    • Analysing cognitive learning process through group interviews of successful high school pupils: Development and use of a model
    • Dahl, B. (2004). Analysing cognitive learning process through group interviews of successful high school pupils: Development and use of a model. Educational Studies in Mathematics, 56, 129-155.
    • (2004) Educational Studies in Mathematics , vol.56 , pp. 129-155
    • Dahl, B.1
  • 8
    • 17444368354 scopus 로고
    • Soviet studies in mathematics education
    • In: J. Kilpatrick (Ed.), & J. Teller (Trans.), Reston, VA: National Council of Teachers of Mathematics (original work published in 1972)
    • Davydov, V. V. (1990). Soviet studies in mathematics education. In: J. Kilpatrick (Ed.), & J. Teller (Trans.), Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula. Reston, VA: National Council of Teachers of Mathematics (original work published in 1972).
    • (1990) Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula
    • Davydov, V.V.1
  • 10
    • 3843147658 scopus 로고    scopus 로고
    • Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students' errors
    • De Bock, D., van Dooren, W., Jannsens, D., & Verschaffel, L. (2002). Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students' errors. Educational Studies in Mathematics, 50, 311-334.
    • (2002) Educational Studies in Mathematics , vol.50 , pp. 311-334
    • de Bock, D.1    van Dooren, W.2    Jannsens, D.3    Verschaffel, L.4
  • 11
    • 3843111596 scopus 로고
    • Forms and means of generalization in mathematics
    • In: A. Bishop (Ed.),. Mahwah, NJ: Erlbaum
    • Doerfler, W. (1991). Forms and means of generalization in mathematics. In: A. Bishop (Ed.), Mathematical knowledge: Its growth through teaching (pp. 63-85). Mahwah, NJ: Erlbaum.
    • (1991) Mathematical knowledge: Its growth through teaching , pp. 63-85
    • Doerfler, W.1
  • 12
    • 0001940455 scopus 로고
    • Advanced mathematical thinking processes
    • In: D. Tall (Ed.),. Dordrecht, The Netherlands: Kluwer
    • Dreyfus, T. (1991). Advanced mathematical thinking processes. In: D. Tall (Ed.), Advanced mathematical thinking (pp. 25-41). Dordrecht, The Netherlands: Kluwer.
    • (1991) Advanced mathematical thinking , pp. 25-41
    • Dreyfus, T.1
  • 13
    • 0002573795 scopus 로고
    • Reflective abstraction in advanced mathematical thinking
    • In: D. Tall (Ed.),. Dordrecht, The Netherlands: Kluwer
    • Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In: D. Tall (Ed.), Advanced mathematical thinking (pp. 95-123). Dordrecht, The Netherlands: Kluwer.
    • (1991) Advanced mathematical thinking , pp. 95-123
    • Dubinsky, E.1
  • 14
    • 33847379311 scopus 로고    scopus 로고
    • Children's intuitive mathematics: The development of knowledge about nonlinear growth
    • Ebersbach, M., & Wilkening, F. (2007). Children's intuitive mathematics: The development of knowledge about nonlinear growth. Child Development, 78, 296-308.
    • (2007) Child Development , vol.78 , pp. 296-308
    • Ebersbach, M.1    Wilkening, F.2
  • 15
    • 34248512314 scopus 로고    scopus 로고
    • Connections between generalizing and justifying: Students reasoning with linear relationships
    • Ellis, A. B. (2007). Connections between generalizing and justifying: Students reasoning with linear relationships. Journal for Research in Mathematics Education, 38, 194-229.
    • (2007) Journal for Research in Mathematics Education , vol.38 , pp. 194-229
    • Ellis, A.B.1
  • 16
    • 33845507055 scopus 로고    scopus 로고
    • Introducing the variable through pattern exploration
    • English, L. D., & Warren, E. A. (1998). Introducing the variable through pattern exploration. The Mathematics teacher, 91, 166-170.
    • (1998) The Mathematics teacher , vol.91 , pp. 166-170
    • English, L.D.1    Warren, E.A.2
  • 17
    • 3843056901 scopus 로고    scopus 로고
    • Some reflections on mathematics classroom notebooks and their relationship to the public and private nature of student practices
    • Fried, M. N., & Amit, M. (2003). Some reflections on mathematics classroom notebooks and their relationship to the public and private nature of student practices. Educational Studies in Mathematics, 53, 91-112.
    • (2003) Educational Studies in Mathematics , vol.53 , pp. 91-112
    • Fried, M.N.1    Amit, M.2
  • 18
    • 13544264781 scopus 로고
    • Identifying the gifted student in mathematics
    • Greenes, C. (1981). Identifying the gifted student in mathematics. Arithmetic Teacher, 28(6), 14-17.
    • (1981) Arithmetic Teacher , vol.28 , Issue.6 , pp. 14-17
    • Greenes, C.1
  • 19
    • 0010129062 scopus 로고
    • The general, the abstract and the generic in advanced mathematics
    • Retrieved from, Accepted 29 September 2007
    • Harel, G., & Tall, D. (1991). The general, the abstract and the generic in advanced mathematics. For the Learning of Mathematics, 11, 38-42. Retrieved from http://www.warwick.ac.uk/staff/David. Tall/pdfs/dot1991b-harel-flm.pdf. Accepted 29 September 2007.
    • (1991) For the Learning of Mathematics , vol.11 , pp. 38-42
    • Harel, G.1    Tall, D.2
  • 20
    • 29444445492 scopus 로고    scopus 로고
    • The teaching of general solution methods to pattern finding problems through focusing on an evaluation and improvement process
    • Ishida, J. (1997). The teaching of general solution methods to pattern finding problems through focusing on an evaluation and improvement process. School Science and Mathematics, 97, 155-162.
    • (1997) School Science and Mathematics , vol.97 , pp. 155-162
    • Ishida, J.1
  • 21
    • 0343376045 scopus 로고    scopus 로고
    • Teaching and learning a new algebra
    • In E. Fennema, & T. A. Romberg (Eds.),. Mahwah, NJ: Erlbaum
    • Kaput, J. J. (1999). Teaching and learning a new algebra. In E. Fennema, & T. A. Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 133-155). Mahwah, NJ: Erlbaum.
    • (1999) Mathematics classrooms that promote understanding , pp. 133-155
    • Kaput, J.J.1
  • 22
    • 84867373374 scopus 로고    scopus 로고
    • Middle grade students' algebraic understanding in a reform curriculum
    • Kerbs, A. S. (2003). Middle grade students' algebraic understanding in a reform curriculum. School Science and Mathematics, 103, 233-243.
    • (2003) School Science and Mathematics , vol.103 , pp. 233-243
    • Kerbs, A.S.1
  • 23
    • 77951263997 scopus 로고    scopus 로고
    • When do gifted high school students use geometry to solve geometry problems?
    • Koichu, B., & Berman, A. (2005), When do gifted high school students use geometry to solve geometry problems? Journal of Secondary Gifted Education, 16(4), 168-179.
    • (2005) Journal of Secondary Gifted Education , vol.16 , Issue.4 , pp. 168-179
    • Koichu, B.1    Berman, A.2
  • 25
    • 33846165073 scopus 로고    scopus 로고
    • Recursive and explicit rules: How can we build student algebraic understanding?
    • Lanin, J. K., Barker, D. D., & Townsend, B. E. (2006). Recursive and explicit rules: How can we build student algebraic understanding? Journal of Mathematical Behavior, 25, 299-317.
    • (2006) Journal of Mathematical Behavior , vol.25 , pp. 299-317
    • Lanin, J.K.1    Barker, D.D.2    Townsend, B.E.3
  • 26
    • 17144392686 scopus 로고    scopus 로고
    • An initiation into algebraic culture through generalization activities
    • In N. Bednarz, C. Kieran, & L. Lee (Eds.),. Dordrecht, The Netherlands: Kluwer
    • Lee, L. (1996). An initiation into algebraic culture through generalization activities. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 87-106). Dordrecht, The Netherlands: Kluwer.
    • (1996) Approaches to algebra: Perspectives for research and teaching , pp. 87-106
    • Lee, L.1
  • 27
    • 0037550195 scopus 로고    scopus 로고
    • How students structure their own investigations and educate us: What we've learned from a fourteen year study
    • In: A. D. Cockburn, & E. Nardi (Eds.),). Norwich, UK: University of East Anglia
    • Maher, C. A. (2002). How students structure their own investigations and educate us: What we've learned from a fourteen year study. In: A. D. Cockburn, & E. Nardi (Eds.), Proceedings of the 26th international group for the psychology of mathematics education (Vol. 1, pp. 31-46). Norwich, UK: University of East Anglia.
    • (2002) Proceedings of the 26th international group for the psychology of mathematics education , vol.1 , pp. 31-46
    • Maher, C.A.1
  • 28
    • 3843109453 scopus 로고    scopus 로고
    • Expressing generality and roots of algebra
    • In: N. Bednarz, C. Kieran, & L. Lee (Eds.),. Dordrecht, The Netherlands: Kluwer
    • Mason, J. (1996). Expressing generality and roots of algebra. In: N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp. 65-86). Dordrecht, The Netherlands: Kluwer.
    • (1996) Approaches to algebra: Perspectives for research and teaching , pp. 65-86
    • Mason, J.1
  • 30
    • 0003865802 scopus 로고    scopus 로고
    • National Council of Teachers of Mathematics, Reston, VA: National Council of Teachers of Mathematics
    • National Council of Teachers of Mathematics. (2000). Principals and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
    • (2000) Principals and standards for school mathematics
  • 31
    • 84892332194 scopus 로고    scopus 로고
    • When the wrong answer is the "good" answer: Problem-solving as a means for identifying mathematical promise
    • In: J. Novotna, H. Moraova, M. Kratka, & N. Stehilkova (Eds.),). Prague, the Czech Republic: Charles University
    • Neria, D., & Amit, M. (2006). When the wrong answer is the "good" answer: Problem-solving as a means for identifying mathematical promise. In: J. Novotna, H. Moraova, M. Kratka, & N. Stehilkova (Eds.), Proceedings of the 30th conference of the international group for the psychology of mathematics education (Vol. 4, pp. 225-232). Prague, the Czech Republic: Charles University.
    • (2006) Proceedings of the 30th conference of the international group for the psychology of mathematics education , vol.4 , pp. 225-232
    • Neria, D.1    Amit, M.2
  • 32
    • 85186303333 scopus 로고    scopus 로고
    • Patterns and the approach to algebra
    • In: A. Orton (Ed.),. London: Cassell
    • Orton, A., & Orton, J. (1999). Patterns and the approach to algebra. In: A. Orton (Ed.), Pattern in the teaching and learning of mathematics (pp. 104-120). London: Cassell.
    • (1999) Pattern in the teaching and learning of mathematics , pp. 104-120
    • Orton, A.1    Orton, J.2
  • 33
    • 0004235832 scopus 로고
    • (2nd ed.). Princeton: Princeton University Press
    • Polya, G. (1957). How to solve it (2nd ed.). Princeton: Princeton University Press.
    • (1957) How to solve it
    • Polya, G.1
  • 34
    • 64949126314 scopus 로고    scopus 로고
    • Algebraic thinking and the generalization of patterns: A semiotic perspective
    • In: S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.).,). Mérida, México: Universidad Pedagógica Nacional
    • Radford, L. (2006). Algebraic thinking and the generalization of patterns: A semiotic perspective. In: S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.). Proceedings of the 28th annual meeting of the North American chapter of the international group for the psychology of mathematics education (Vol. 1, pp. 2-21). Mérida, México: Universidad Pedagógica Nacional.
    • (2006) Proceedings of the 28th annual meeting of the North American chapter of the international group for the psychology of mathematics education , vol.1 , pp. 2-21
    • Radford, L.1
  • 35
    • 77951878394 scopus 로고    scopus 로고
    • Visualizing as a mathematical way of knowing: Understanding figural generalization
    • Rivera, F. (2007). Visualizing as a mathematical way of knowing: Understanding figural generalization. Mathematics Teacher, 101(1), 69-75.
    • (2007) Mathematics Teacher , vol.101 , Issue.1 , pp. 69-75
    • Rivera, F.1
  • 36
  • 38
    • 2942591342 scopus 로고    scopus 로고
    • Mathematical giftedness, problem solving, and the ability to formulate generalizations: The problem-solving experiences of four gifted students
    • Sriraman, B. (2003). Mathematical giftedness, problem solving, and the ability to formulate generalizations: The problem-solving experiences of four gifted students. Journal of Secondary Gifted Education, 14, 151-165.
    • (2003) Journal of Secondary Gifted Education , vol.14 , pp. 151-165
    • Sriraman, B.1
  • 39
    • 2942530299 scopus 로고    scopus 로고
    • Reflective abstraction, uniframes and the formulation of generalizations
    • Sriraman, B. (2004). Reflective abstraction, uniframes and the formulation of generalizations. Journal of Mathematical Behavior, 23, 205-222.
    • (2004) Journal of Mathematical Behavior , vol.23 , pp. 205-222
    • Sriraman, B.1
  • 40
    • 1842768042 scopus 로고
    • Finding and using patterns in linear generalizing problems
    • Stacey, K. (1989). Finding and using patterns in linear generalizing problems. Educational Studies in Mathematics, 20, 147-164.
    • (1989) Educational Studies in Mathematics , vol.20 , pp. 147-164
    • Stacey, K.1
  • 41
    • 33845483078 scopus 로고    scopus 로고
    • Using writing to access students' schemata knowledge for algebraic thinking
    • Steele, D. (2005). Using writing to access students' schemata knowledge for algebraic thinking. School Science and Mathematics, 105, 142-154.
    • (2005) School Science and Mathematics , vol.105 , pp. 142-154
    • Steele, D.1
  • 43
    • 22844453992 scopus 로고    scopus 로고
    • Grade 6 students' preinstructional use of equations to describe and represent problem situations
    • Swaford, J. O., & Langrall, C. W. (2000). Grade 6 students' preinstructional use of equations to describe and represent problem situations. Educational Studies in Mathematics, 31, 89-112.
    • (2000) Educational Studies in Mathematics , vol.31 , pp. 89-112
    • Swaford, J.O.1    Langrall, C.W.2
  • 44
    • 0010864840 scopus 로고
    • Conceptions of school algebra and uses of variables
    • In: A. Coxford, & A. P. Schulte (Eds.),. Reston, VA: National Council of Teachers of Mathematics
    • Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In: A. Coxford, & A. P. Schulte (Eds.), Ideas of algebra, K-12, 1988 Yearbook (pp. 8-19). Reston, VA: National Council of Teachers of Mathematics.
    • (1988) Ideas of algebra, K-12, 1988 Yearbook , pp. 8-19
    • Usiskin, Z.1
  • 45
    • 33751318004 scopus 로고    scopus 로고
    • Investigating functional thinking in the elementary classroom: Foundations of early algebraic reasoning
    • Warren, E. A., Cooper, T. J., & Lamb, J. T. (2006). Investigating functional thinking in the elementary classroom: Foundations of early algebraic reasoning. Journal of Mathematical Behavior, 25, 208-223.
    • (2006) Journal of Mathematical Behavior , vol.25 , pp. 208-223
    • Warren, E.A.1    Cooper, T.J.2    Lamb, J.T.3
  • 46
    • 52649152332 scopus 로고    scopus 로고
    • Generalization of patterns: The tension between algebraic thinking and algebraic notation
    • Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49, 379-402.
    • (2002) Educational Studies in Mathematics , vol.49 , pp. 379-402
    • Zazkis, R.1    Liljedahl, P.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.