An early algebra approach to pattern generalisation: Actualising the virtual through words, gestures and toilet paper

Abstract

This paper focuses on pattern generalisation as a way to introduce young students to early algebra. We build on research on patterning activities that feature, in their work with algebraic thinking, both looking for sameness recursively in a pattern (especially figural patterns, but also numerical ones) and conjecturing about function-based relationships that relate variables. We propose a new approach to pattern generalisation that seeks to help children (grades 2 and 3) work both recursively and functionally, and to see how these two modes are connected through the notion of variable. We argue that a crucial change must occur in order for young learners to develop a flexible algebraic discourse. We draw on Sfard's (2008) communication approach and on Châtelet's (2000) notion of the virtual in order to pursue this argument. We also root our analyses within a new materialist perspective that seeks to describe phenomena in terms of material entanglement, which include, in our classroom research context, not just the children and the teacher, but also words, gestures, physical objects and arrangements, as well as numbers, operations and variables.

Agency and assemblage in pattern generalisation: a materialist approach to learning

“…Researchers have investigated how pattern search allows for the successful introduction of basic algebraic thinking in the early years (e.g., Becker & Rivera, 2008;Carraher, Martinez, & Schliemann, 2008;Ferrara & Sinclair, 2016;Moss & Beatty, 2006Radford, 2010a, b;Rivera, 2011). Although drawing on different theoretical perspectives, we highlight some of the relevant findings here.…”

“…Others have drawn attention to the forms of algebraic thinking that can be made accessible to 7-8-year-old learners before any conventional use of alphanumeric symbolism (Radford, 2010a, b;Ferrara & Sinclair, 2016), with pedagogical implications Bto revisit the genetic relationship between arithmetic and algebraic thinking^ (Radford, 2010a, p. 80).…”

“…The tasks used numerical and figural patterns to provide opportunities for students to reason about relations between variables and formulas. The tasks follow the recent lines of mathematics education research that see patterns as a basis for introducing early algebra and for studying processes of generalising (e.g., Ferrara & Sinclair, 2016;Moss & Beatty, 2010;Radford 2010aRadford , 2010bRivera, 2011).…”

Ortaokul Öğrencilerinin Farklı Temsil Biçimlerini Kullanarak Matematiksel Genelleme Yapma Becerileri

“…Beyond being a language in which algebraic letters are used, anticipation emerges as a way of thinking and involves algebraic thinking which requires analyzing, interpreting, verifying, problem solving, predicting, and generalization of values (Kieran, 1996). Research on the development of algebraic thinking seems to focus particularly on generalizations and patterns (eg., Moss & Beatty, 2010;Rivera, 2011;Ferrara & Sinclair, 2016). This is because the relationship between students' ability to understand and generalize patterns (Warren, 2000) or, more clearly, pattern searching is one of the basic and necessary steps to generalize (Jones, 1993).…”

“…yapılan araştırmalarda özellikle genelleme yapma ve örüntüler üzerinde durulduğu görülmektedir (ör., Ferrara & Sinclair, 2016;Moss & Beatty, 2010;Rivera, 2011). Bunun temel nedeni öğrencilerin örüntüleri anlamlandırma ve genelleme yapma becerileri arasında ilişki bulunması (Warren, 2000), daha açık bir ifadeyle örüntü aramanın genelleme yapmanın temel ve gerekli adımlarından biri olmasıdır (Jones, 1993).…”