Educational Studies in Mathematics

, Volume 73, Issue 3, pp 297–328 | Cite as

Visual templates in pattern generalization activity

  • F. D. RiveraEmail author


In this research article, I present evidence of the existence of visual templates in pattern generalization activity. Such templates initially emerged from a 3-week design-driven classroom teaching experiment on pattern generalization involving linear figural patterns and were assessed for existence in a clinical interview that was conducted four and a half months after the teaching experiment using three tasks (one ambiguous, two well defined). Drawing on the clinical interviews conducted with 11 seventh- and eighth-grade students, I discuss how their visual templates have spawned at least six types of algebraic generalizations. A visual template model is also presented that illustrates the distributed and a dynamically embedded nature of pattern generalization involving the following factors: pattern goodness effect; knowledge/action effects; and the triad of stage-driven grouping, structural unit, and analogy.


Pattern generalization Algebraic thinking Visual templates Additive reasoning Multiplicative reasoning Algebraic generalization Abduction Pattern goodness 

Supplementary material

10649_2009_9222_MOESM1_ESM.doc (98 kb)
ESM 1 Karen’s graphical representation of her thinking in relation to Fig. 9 (DOC 98 kb)
10649_2009_9222_MOESM2_ESM.doc (118 kb)
ESM 2 Graphical illustration of Emma’s direct formula \( s = n + n - 1 \) in relation to Fig. 9 (DOC 118 kb)
10649_2009_9222_MOESM3_ESM.doc (5.1 mb)
ESM 3 Additional examples of linear figural patterns that yield a CSG (DOC 5217 kb)
10649_2009_9222_MOESM4_ESM.doc (66 kb)
ESM 4 Diana’s visual analysis of her pattern in Fig. 9 (DOC 66 kb)
10649_2009_9222_MOESM5_ESM.doc (260 kb)
ESM 5 Cherrie’s written work in relation to the Fig. 3 task (DOC 260 kb)
10649_2009_9222_MOESM6_ESM.doc (268 kb)
ESM 6 Emma’s written work in relation to the Fig. 3 task (DOC 268 kb)
10649_2009_9222_MOESM7_ESM.doc (846 kb)
ESM 7 Karen’s written work on items A and B of the Fig. 3 pattern task (DOC 846 kb)
10649_2009_9222_MOESM8_ESM.doc (68 kb)
ESM 8 Frank’s justification of the formula \( S = 2x - 1 \) in relation to the Fig. 9 pattern (DOC 68 kb)
10649_2009_9222_MOESM9_ESM.doc (72 kb)
ESM 9 Frank’s justification of the formula \( {\text{S}} = 2\left( {x - 1} \right) + 1 \) in relation to the Fig. 9 pattern (DOC 72 kb)
10649_2009_9222_MOESM10_ESM.doc (100 kb)
ESM 10 Cherrie’s justification of her formula \( x = 2n - 1 \) in relation to the Fig. 9 pattern (DOC 100 kb)
10649_2009_9222_MOESM11_ESM.doc (48 kb)
ESM 11 A figural transformation of Tamara’s pattern in Fig. 16 demonstrating the algebraic generalization \( s = n\left( {n - 1 + 1} \right) \) (DOC 48 kb)
10649_2009_9222_MOESM12_ESM.doc (50 kb)
ESM 12 Two numerical approaches in obtaining a direct formula for the pattern in Fig. 16 (DOC 50 kb)


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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Department of MathematicsSan Jose State UniversitySan JoseUSA

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