## 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

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.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.English, L.D. and Warren, E.A.: 1998; 'Introducing the variable through pattern exploration',

*Mathematics Teacher*91(2), 166–170.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.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.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.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.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.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.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.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.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.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.National Council of Teachers of Mathematics: 2000,

*Principles and Standards for School Mathematics*, NCTM, Reston, VA.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.Radford, L.: 2000, 'Signs and meanings in students' emergent algebraic thinking: A semiotic analysis',

*Educational Studies in Mathematics*42, 237–268.Schoenfeld, A.H.: 1988, 'When good teaching leads to bad results: The disaster of well taught mathematics courses',

*Educational Psychologist*23(2), 145–166.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.Stacey, K.: 1989, 'Finding and using patterns in linear generalizing problems',

*Educational Studies in Mathematics*20, 147–164.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.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.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.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.

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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