Students' Mathematical Noticing

Abstract

Even in simple mathematical situations, there is an array of different mathematical features that students can attend to or notice. What students notice mathematically has consequences for their subsequent reasoning. By adapting work from both cognitive science and applied linguistics anthropology, we present a focusing framework, which treats noticing as a complex phenomenon that is distributed across individual cognition, social interactions, material resources, and normed practices. Specifically, this research demonstrates that different centers of focus emerged in two middle grades mathematics classes addressing the same content goals, which, in turn, were related conceptually to differences in student reasoning on subsequent interview tasks. Furthermore, differences in the discourse practices, features of the mathematical tasks, and the nature of the mathematical activity in the two classrooms were related to the different mathematical features that students appeared to notice.

Testing students ability to use derivatives, integrals, and vectors in a purely mathematical context and in a physical context

“…However, this definition does not account for all the phenomena associated with knowledge building and the current theories of knowledge [21,22]. Rather than a mere application of previous knowledge, modern perspectives view transfer as an active, student-centered dynamic process, governed by the students' epistemic frames [5] and "noticing" of relevant problem features [23,24], the visual attributes and "affordances" of the problem [25][26][27][28], and dependent on the disciplinary context [29]. For example, in the preparation for future learning (PFL) perspective [30] evidence of transfer is not sought in "one-shot" students' performances, but rather in the whole process of learning.…”

Rethinking Learning Trajectories in Light of Student Linguistic Diversity

“…We focus on student reasoning on two of these tasks, the Pool Problem (adapted from Lobato et al, 2013) and the Roof Problem (from Cuoco & Kerins, 2013). The written tasks and our planned followup questions appear in Appendix A.…”

“…The interview protocol was developed and sequenced using prior research and the school's existing curriculum resources as starting points. Four tasks were adapted from the school curriculum (Cuoco & Kerins, 2013) and one task was from Lobato et al (2013). Each of the interview tasks, together with the planned follow-up questions, afforded opportunities for students to use forms of reasoning across the trajectory in Figure 1, with the exception of the Slope Calculations task.…”

Valid and valuable: lower attaining pupils’ contributions to mixed attainment mathematics in primary schools

“…Mason comments that "what you do not notice you cannot act upon" (2002, p. 7) and in relation specifically to classroom mathematical activity, Lobato, Hohensee, and Rhodehamel (2013, p. 844) conclude that "what students notice mathematically has consequences for their subsequent reasoning". Lobato et al (2013) find that the specific nature of teacher pedagogies, such as the implicit or explicit focus of, and vocabulary used in, questions and prompts, as well as the nature of board annotations and gestures, impact on what is noticed at the time, and on pupils' subsequent noticing behaviour. Importantly, Hohensee (2016) further suggests that pedagogies aimed at promoting noticing appear to have particular potential to support pupils identified as lower attaining.…”

“…I define mathematical noticing as "the detection of mathematical features, relations, or processes", and focus on noticing as an essential precursor to mathematical reasoning. Whilst noticing could be considered to be part of the reasoning process rather than a precursor to it, following Lobato et al (2013), I separate the perceptual or analytic discernment of features, specifically in this study, mathematical pattern, structure and property, from the use of them in constructing reasoning and position noticing as a necessary but insufficient condition for mathematical reasoning.…”