Making Associativity Operational

Abstract

The purpose of this paper is to propose an operational idea for developing algebraic thinking in the absence of alphanumeric symbols. The paper reports on a design experiment encouraging preschool children to use the associative property algebraically. We describe the theoretical basis of the design, the tasks used, and examples of algebraic thinking in 5-6-year-old children. Theoretically, the paper makes a critical distinction between operational and structural meanings of the notion of equality. We argue that mathematical thinking involving equality among young learners can comprise both an operational and a structural conception and that the operational conception has a side that is productively linked to the structural conception. Using carefully designed hands-on tasks, the crux of the paper is the realization of algebraic thinking (in verbal mathematics) as operationally experienced in the ability to transform one number structure, with a quantity that is subject to change, into another through equality-preserving transformations.

Conceptual knowledge of the associativity principle: A review of the literature and an agenda for future research

“…No studies have quantitatively compared performance on concrete and abstract associativity problems, although Klein & Bisanz (2000) anecdotally reported that on at least one occasion, 25% of preschoolers used associativity shortcuts on problems presented with poker chips. Asghari & Khosroshahi (2016) observed similar behaviours in 5 − 6 year olds. Thus, it can be argued that adults are likely to have some form of understanding of associativity given that young children do, and that difficulties may arise when applying that knowledge to more abstract 'a + b − c' or 'a × b ÷ c' digitbased problems.…”

Increasing the use of conceptually-derived strategies in arithmetic: using inversion problems to promote the use of associativity shortcuts

“…with words or objects), but not in an abstract context with digits (Gilmore & Bryant, 2006). Given that some pre-schoolers make correct judgements of associativity with concrete objects (Klein & Bisanz, 2000;Asghari & Khosroshahi, 2016), a complete absence of knowledge is unlikely. It may be more likely that they do not know that the shortcut is valid and efficient on digit-based problems.…”

Η Αντιμεταθετική Ιδιότητα Ως Μέσο Ενίσχυσης Των Νοερών Υπολογισμών Πρόσθεσης Των Μικρών Παιδιών

“…Η χρήση της προσεταιριστικής ιδιότητας σε προσθαφαιρέσεις (a+b-c) έχει παρατηρηθεί σε μικρά παιδιά (Klein, & Bisanz, 2000. Asghari, & Khosroshahi, 2017 που χρησιμοποιούν αντικείμενα.…”