Between counting and multiplication: low-attaining students’ spatial structuring, enumeration and errors in concretely-presented 3D array tasks

TY - JOUR

T1 - Between counting and multiplication

T2 - low-attaining students’ spatial structuring, enumeration and errors in concretely-presented 3D array tasks

AU - Finesilver, Carla

PY - 2017/6

Y1 - 2017/6

N2 - The move from additive thinking to multiplicative thinking requires significant change in children's comprehension and manipulation of number relationships. This change is not a single cognitive leap, but involves various different conceptual components, and can be a slow, multi-stage process for some learners. Unit arrays are a key visuospatial representation for teaching/learning about multiplicative relationships, but most research focuses on 2D (rectangular) arrays, and that which does address 3D (cuboid) arrays still frequently uses 2D representations of these. This paper documents low-attaining children’s partially-developed multiplicative thinking as they work on concretely-presented 3D array tasks; it also presents a framework for microanalysis of learners’ early multiplicative thinking in array tasks. Data derives from a small but cognitively diverse set of participants, arithmetically low-attaining for their ages, and relying heavily on counting-based strategies: this enabled detailed analysis of small but significant differences in their arithmetical engagement with the arrays. The analytical framework combines and builds on previous structural and enumerative categorizations, and is appropriate for use with a variety of array representations including but not limited to the blocks of unit cubes in the task sequence described here.

AB - The move from additive thinking to multiplicative thinking requires significant change in children's comprehension and manipulation of number relationships. This change is not a single cognitive leap, but involves various different conceptual components, and can be a slow, multi-stage process for some learners. Unit arrays are a key visuospatial representation for teaching/learning about multiplicative relationships, but most research focuses on 2D (rectangular) arrays, and that which does address 3D (cuboid) arrays still frequently uses 2D representations of these. This paper documents low-attaining children’s partially-developed multiplicative thinking as they work on concretely-presented 3D array tasks; it also presents a framework for microanalysis of learners’ early multiplicative thinking in array tasks. Data derives from a small but cognitively diverse set of participants, arithmetically low-attaining for their ages, and relying heavily on counting-based strategies: this enabled detailed analysis of small but significant differences in their arithmetical engagement with the arrays. The analytical framework combines and builds on previous structural and enumerative categorizations, and is appropriate for use with a variety of array representations including but not limited to the blocks of unit cubes in the task sequence described here.

KW - numeracy

KW - counting

KW - arithmetical strategies

KW - multiplicative thinking

KW - low attainment

KW - qualitative analysis

KW - microgenetic methods

U2 - 10.1080/10986065.2017.1295418

DO - 10.1080/10986065.2017.1295418

M3 - Article

SN - 1098-6065

VL - 19

SP - 95

EP - 114

JO - Mathematical Thinking and Learning

JF - Mathematical Thinking and Learning

IS - 2

ER -