Generalization of patterns: the tension between algebraic thinking and algebraic notation

Abstract

This study explores the attempts of a group of preservice elementary school teachers to generalize a repeating visual number pattern. We discuss students' emergent algebraic thinking and the variety of ways in which they generalize and symbolize their generalizations. Our results indicate that students' ability to express generality verbally was not accompanied by, and did not depend on, algebraic notation. However, participants often perceived their complete and accurate solutions that did not involve algebraic symbolism as inadequate.

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REFERENCES

  1. Charbonneau, L.: 1996, 'From Euclid to Descartes: Algebra and its relation to geometry', in N. Bednarz, C. Kieran and Lee, L. (eds.), Approaches to Algebra: Perspectives for Research and Teaching, Kluwer Academic Publishers, Dordrecht, pp. 15–37.

    Google Scholar 

  2. Dörfler, W.: 1991, 'Forms and means of generalization in mathematics', in A.J. Bishop (ed.), Mathematical Knowledge: Its Growth through Teaching, Kluwer Academic Publishers, Dordrecht, pp. 63–85.

    Google Scholar 

  3. English, L.D. and Warren, E.A.: 1998; 'Introducing the variable through pattern exploration', Mathematics Teacher 91(2), 166–170.

    Google Scholar 

  4. Harel, G. and Tall, D.: 1991, 'The general, the abstract and the generic in advanced mathematics', For the Learning of Mathematics 11(1), 38–42.

    Google Scholar 

  5. Ishida, J.: 1997, 'The teaching of general solution methods to pattern finding problems through focusing on a evaluation and improvement process', School Science and Mathematics 97(3), 155–162.

    Article  Google Scholar 

  6. Iwasaki, H. and Yamaguchi, T.: 1997, 'The cognitive and symbolic analysis of the generalization process: The comparison of algebraic signs with geometric figures', in E. Pehkonen (ed.), Proceedings of the 21st International Conference for Psychology of Mathematics Education. Lahti, Finland, pp. 105–112.

  7. Kaput, J.: 1999, 'Teaching and learning a new algebra', in E. Fennema and T. Romberg (eds.), Mathematics Classrooms that Promote Understanding, Erlbaum, Mahwah, NJ, pp. 133–155.

    Google Scholar 

  8. Kaput, J. and Blanton, M.: 2001, 'Algebrafying the elementary mathematics experience', in H. Chick, K. Stacey, J. Vincent and J. Vincent (eds.), Proceedings of the 12th ICMI Study Conference on the Future of the Teaching and Learning of Algebra (Vol. 1), Melbourne, Australia, pp. 344–351.

  9. Kieran, C: 1989, 'A perspective on algebraic thinking', in G. Vergnaud, J. Rogalski and M. Artigue (eds.), Proceedings of the 13th Annual Conference of the International Group for the Psychology of Mathematics Education, July 9–13, Paris, France, pp. 163–171.

  10. Lee, L.: 1996, 'An initiation into algebraic culture through generalization activities', in N. Bednarz, C. Kieran and Lee, L. (eds.), Approaches to Algebra: Perspectives for Research and Teaching, Kluwer Academic Publishers, Dordrecht, pp. 87–106.

    Google Scholar 

  11. Mason, J.: 1996, 'Expressing generality and roots of algebra', in N. Bednarz, C. Kieran and Lee, L. (eds.), Approaches to Algebra: Perspectives for Research and Teaching, Kluwer Academic Publishers, Dordrecht, pp. 65–86.

    Google Scholar 

  12. Mason, J.: 1998, 'Researching from the inside in mathematics education', in A. Sierpinska and J. Kilpatrick (eds.), Mathematics Education as a Research Domain: A Search for Identity, Kluwer Academic Publishers, Dortrecht, pp. 357–377.

    Google Scholar 

  13. Mason, J.: 2002, 'What makes an example exemplary? Pedagogical and research issues in transitions from numbers to number theory', in J. Ainley (ed.), Proceedings of the 26th Annual Conference of the International Group for the Psychology of Mathematics Education, July 21–26, 2002, Norwich, UK, Volume 1, pp. 224–229.

  14. National Council of Teachers of Mathematics: 2000, Principles and Standards for School Mathematics, NCTM, Reston, VA.

    Google Scholar 

  15. Orton, A. and Orton, J.: 1999, 'Pattern and the approach to algebra', in A. Orton (ed.), Pattern in the Teaching and Learning of Mathematics. Cassell, London, pp. 104–120.

    Google Scholar 

  16. Radford, L.: 2000, 'Signs and meanings in students' emergent algebraic thinking: A semiotic analysis', Educational Studies in Mathematics 42, 237–268.

    Article  Google Scholar 

  17. Schoenfeld, A.H.: 1988, 'When good teaching leads to bad results: The disaster of well taught mathematics courses', Educational Psychologist 23(2), 145–166.

    Article  Google Scholar 

  18. Sfard, A.: 1995, 'The development of algebra - Confronting historical and psychological perspectives', in C. Kieran (ed.), New Perspectives on School Algebra: Papers and Discussions of the ICME-7 Algebra Working Group (Special Issue), Journal of Mathematical Behavior 14, 15–39.

  19. Stacey, K.: 1989, 'Finding and using patterns in linear generalizing problems', Educational Studies in Mathematics 20, 147–164.

    Article  Google Scholar 

  20. Threlfall, J.: 1999, 'Repeating patterns in the primary years', in A. Orton (ed.), Pattern in the Teaching and Learning of Mathematics, Cassell, London, pp. 18–30.

    Google Scholar 

  21. Zazkis, R. and Campbell, S.: 1996, 'Divisibility and multiplicative structure of natural numbers: Preservice teachers' understanding', Journal for Research in Mathematics Education 27(5), 540–563.

    Article  Google Scholar 

  22. Zazkis, R. and Liljedahl, P.: 2001, 'Exploring multiplicative and additive structure of arithmetic sequence', in M. van den Heuvel-Panhuizen (ed.), Proceedings of the 25th International Conference for Psychology of Mathematics Education, Utrecht, Netherlands, pp. 439–446.

  23. Zazkis, R. and Liljedahl, P.: 2002, 'Arithmetic sequence as a bridge between conceptual fields', Canadian Journal of Mathematics, Science and Technology Education 2(1), 93–120.

    Article  Google Scholar 

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Zazkis, R., Liljedahl, P. Generalization of patterns: the tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics 49, 379–402 (2002). https://doi.org/10.1023/A:1020291317178

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Keywords

  • Elementary School
  • School Teacher
  • Accurate Solution
  • Elementary School Teacher
  • Algebraic Thinking