Low-attaining students’ representational strategies: tasks, time, efficiency, and economy

Abstract

There are many potential ways to represent arithmetical tasks, but students' choices may be limited by beliefs that only certain standardised representations are 'legitimate' in school mathematics. Furthermore, concern for the quantity and speed of 'work done' can override opportunities for meaningful engagement with the content. This paper draws on a sample of the informal representational strategies observed during a microanalytic study of 11-15 year old students with low prior attainment in mathematics. In absence of pressure to provide quick answers, or to obtain them in a prescribed manner, students worked flexibly, participating in arithmetical reasoning, attempting and succeeding in tasks they were previously unable to engage with. The relationships between representational strategies, economy and efficiency are discussed in relation to multiplicative thinking. These have pedagogical implications for the representational expectations placed on students with difficulties in mathematics, particularly in learning support and intervention contexts.

An exploration of arts teachers' beliefs and judgements concerning students with SEND

“…Furthermore, as authors, we recognise that all subjects in school have the capacity to be creative; it is not the sole provenance of the arts. For instance, Finesilver (2017) observed students employing a wide variety of creative visuospatial representations in their mathematical problem‐solving, with colourful imagery, modelling and metaphor mixed with conventional symbolic forms. Meanwhile, Jones (2013) highlights that collaboration amongst peers is promoted in MFL lessons, where there is also scope to develop applied skills, such as filmmaking in a real‐life linguistic context, which, coincidently, ‘bends the rules’ of classroom formalities.…”

Beyond categories: dynamic qualitative analysis of visuospatial representation in arithmetic

“…This is unsurprising in educational cultures in which formal symbolic notations are perceived as 'the almost sole desired and valued outcomes of mathematics learning' (Karsenty et al, 2007), and Mason (1980) addresses some of the educational problems that result from using symbols without due consideration. While few would argue that symbolic notation and standard calculation procedures are not highly useful mathematical tools, an education system that rushes learners into exclusively symbolic notation can result in the performance of conceptually empty 'maths-like behaviours' made of fragments of poorly understood symbolic representation, and the unfortunate acceptance (or even misguided encouragement) of this by teachers (Finesilver, 2017b). In fact, analysis of individuals' visuospatial representational strategies would ideally be considered not only in terms of individual cognitive development, but in relation to the educational contexts in which they learn.…”

“…The data derived from a series of six interactive problem-solving interviews methodologically situated at a point between clinical interview and naturalistic pedagogic activity, with the dynamic assessment principle of first gauging unassisted capability on a task, then calibrating support (if any) in situ to the needs of the individual (Elliott et al 2018). The guiding principles regarding representation were (1) strong encouragement of students' freedom to follow their own preferences, ideas, and strategies and (2) absence of time pressure on tasks (see Finesilver (2017b) for further discussion of this point).…”

ERME column

“…Another recurring theme in many of these discussions is the advocacy of working with different forms of representation, and the valuing of non-standard forms. Group discussions have pointed to pressures that exist across many educational contexts for teachers to privilege particular standardized forms of representation over alternatives, in order to push students to acquire as swiftly as possible selected so-called "efficient" ways to produce answers [7]. Since these pressures may prematurely curtail students' creativity and intuitive approaches when engaging in problem solving [4], discussions in our working group have focused on how to support teachers' use of more diverse representational forms and formats that enable wider inclusivity, providing all learners with opportunities to engage more meaningfully with mathematical activity and knowledge.…”