Mathematics Education in Singapore pp 217-240 | Cite as

# Patterns Across the Years—Singapore Learners’ Epistemology

## Abstract

Pattern has a prominent position in the Singapore mathematics curriculum. This chapter reports how learners across the grades thought about patterns, how they recognised patterns, and how they constructed rules to describe the structure underpinning specific patterns. The corpus of data came from four studies. Primary children participated in the first three studies: Age and Individual Differences, Forward and Backward Rule, Colour Contrast whilst Secondary 2 students participated in the fourth, Strategies and Justifications in Mathematical Generalization. All these studies used the mathematics curriculum to design grade-specific mathematical tasks. In general, two types of pattern tasks were used, number patterns presented in tandem with figures and figural patterns. Data with primary children were collected using paper-and-pencil task and clinical interviews were used to collaborate their responses. The fourth study analysed the written responses of the secondary students to paper-and-pencil task. These studies found that learners focused on the surface features to arrive at a rule to describe these number patterns. In the colour-contrast study, compared with monochromatic presentation, those using two colours encouraged learners to present possible general rules. The more able academic stream secondary students were able to arrive at general rules for linear figural patterns. However, all students across the academic spectrum were challenged by quadratic patterns. Findings from the four suggest that it important for teachers to know how to move learners to look for the structure underpinning patterns, numerical and figural, and to construct the all-important general rule.

## Keywords

Colour contrast Linear figural Recursive rule Predictive rule Structure Number patterns Quadratic patterns## References

- Amit, M., & Neria, D. (2008). Rising to the challenge: Using generalization in pattern problems to unearth the algebraic skills of talented pre-algebra students.
*Zentralblatt für Didaktik der Mathematik,**40,*111–129.CrossRefGoogle Scholar - Blanton, M. L., & Kaput, J. J. (2004). Elementary grades students’ capacity for functional thinking. In M. J. Hoines & A. B. Fuglestad (Eds.),
*Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 135–142). Bergen, Norway.Google Scholar - Cai, J., Lew, H. C., Morris, A., Moyer, J. C., Ng, F. F., & Schmittau, J. (2005). The development of students’ algebraic thinking in earlier grades: A cross-cultural comparative perspective.
*Zentralblatt für Didaktik der Mathematik,**37*(1), 5–15.CrossRefGoogle Scholar - Chua, B. L., & Hoyles, C. (2009). Generalisation and perceptual agility: How teachers fared in a generalising problem. In
*Proceedings of the British Society for Research in Learning Mathematics (BSRLM) Conference*(pp. 13–18). Bristol, The United Kingdom: BSRLM.Google Scholar - Chua, B. L., & Hoyles, C (2011). The interplay between format of pattern display and expressing generality. In B. Ubuz (Ed.)
*Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education*(p. 281). Ankara, Turkey: PME.Google Scholar - Chua, B. L., & Hoyles, C. (2012). The effect of different pattern formats on secondary two students’ ability to generalise. In T. Y. Tso (Ed.),
*Proceedings of the 36th Conference of the International Group for the Psychology of Mathematics Education*(Vol 2, pp. 155–162). Taipei, Taiwan: PME.Google Scholar - Chua, B. L., & Hoyles, C. (2013). Rethinking and researching task design in pattern generalisation. In A. M. Lindmeier & A. Heinze (Eds.),
*Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education*(Vol 2, pp. 193–200). Kiel, Germany: PME.Google Scholar - Chua, B. L., & Hoyles, C. (2014). Generalisation of linear figural patterns in Secondary School.
*The Mathematics Educator,**15*(2), 1–30.Google Scholar - Driscoll, M. (1999).
*Fostering algebraic thinking: A guide for teachers grades 6–10.*Portsmouth: NH: Heinemann.Google Scholar - Fosnot, C. T., & Jacob, B. (2010).
*Young mathematicians at work: Constructing algebra*. Reston, Virginia: NCTM.Google Scholar - Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.),
*Algebra in the early grades*(p. 5–17). Reston, VA: NCTM.Google Scholar - Küchemann, D. (2010). Using patterns generically to see structure.
*Pedagogies: An International Journal, 5*(3), 233–250.CrossRefGoogle Scholar - Lee, K., Ng, S. F., & Bull, R. (2017). Learning and solving more complex problems: The roles of working memory, updating, and prior skills for general mathematical achievement and algebra. In D. C. Geary, D. B. Berch, R. Ochsendorf, & K. M. Koepke (Eds.),
*Acquisition of complex arithmetic skills and higher-order Mathematics concepts*(pp. 197–220). http://dx.doi.org/10.1016/B978-0-12-805086-6.0009-6.CrossRefGoogle Scholar - Mason, J. (1990).
*Supporting primary Mathematics: Algebra.*Milton Keynes, UK Open University: The Open University.Google Scholar - 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 Academic Publishers.CrossRefGoogle Scholar - Mason, J. (2008). Making use of children’s powers to produce algebraic thinking. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.),
*Algebra in the early grades*(pp. 57–94). New York: Lawrence Erlbaum Associates.Google Scholar - Mason, J., Stephens, M., & Watson, A. (2009). Appreciating Mathematical structures for all.
*Mathematics Education Research Journal,**21*(2), 10–32.CrossRefGoogle Scholar - Moses, B. (Ed.). (1999).
*Algebraic thinking. Grades K-12.*Reston, VA: NCTM.Google Scholar - Ng, S. F. (2018). Function tasks, input, output and the predictive rule is: How some Singapore primary children construct the rule. In C. Kieran (Ed.),
*Teaching and learning algebraic thinking with 5- to 12-year-olds the global evolution of an emerging field of research and practice.*Springer International Publishing AG.Google Scholar - Radford, L. G. (2001). The historical origins of algebraic thinking. In R. Sutherland, T. Rojano, A. Bell, & R. Lins (Eds.),
*Perspectives in school algebra*(pp. 13–36). Dordrecht: The Netherlands: Kluwer Academic Publishers.Google Scholar - Rivera, F. D., & Becker, J. R. (2005). Figural and numerical modes of generalising in algebra.
*Mathematical Teaching in the Middle School,**11*(4), 198–203.Google Scholar - Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. In A. Coxford (Ed.),
*Ideas of algebra: K-12*(pp. 8–19). Reston, VA: NCTM.Google Scholar - Van De Walle, J., & Bay-Williams, J. M. (Eds.). (2014).
*Elementary and middle school Mathematics: Teaching developmentally*(8th ed.). Essex, UK: Pearson.Google Scholar