Young children's spatial structuring ability and emerging number sense

Abstract

This thesis documents research into the role of young children’s spatial structuring ability in the development of number sense, particularly in terms of insight into numerical relations. We take Battista and Clements’ (1996, p. 503) definition to define the act of spatial structuring as “the mental operation of constructing an organization or form for an object or set of objects”. Insight into numerical relations involves the structuring (e.g., (de)composing) of quantities (e.g., understanding six to be three and three but also five and one or four and two), which is essential for the development of higher-order mathematical abilities. Through exploring and comparing, for example, symmetrical double-structures (as represented by egg cartons) to dot configurations (as represented by dice configurations), children can come to recognize an underlying structure such as two rows of three. Such insights can help to establish children’s awareness of spatial structures, to support children’s ability to recognize and manipulate such structures in various contexts and settings, and to use spatial structuring to abbreviate numerical procedures. The research questions are defined as follows: 1. What strategies for solving spatial and numerical problems characterize young children’s spatial structuring abilities? 2a. How can young children be supported in learning to recognize and make use of spatial structures for abbreviating numerical procedures? 2b. What characterizes a learning ecology that can facilitate the development of children’s spatial structuring ability? The research followed the guidelines of Realistic Mathematics Education (RME) and socio-constructivism. It was conducted at a local elementary school with an intervention group that was a combined grade 1 and grade 2 with four- to six-year old children (N = 21). A comparable non-intervention group (N = 17) did not participate in the instruction experiment, but was included to provide additional data for developing and analyzing the interviews and the instruction experiment. To answer the first research question, a set of tasks was designed for gauging children’s spatial structuring and numerical ability in a one-to-one clinical (pre- and post-) interview setting. These interviews contributed to creating a strategy inventory for classifying children’s strategies into one of four phases in a developmental trajectory, which describes children’s spatial structuring ability regarding this particular interview setting. Cumulative cyclic, classroom-based, design research was used to answer the second research question. This involved designing, testing and refining a hypothetical learning trajectory (HLT) with five corresponding instruction activities for an instruction experiment. The retrospective analysis of the data followed the principles of constant comparison. Qualitative analyses of the instruction experiment and comparisons between the phase classifications for the pre- and post-interview, reflected the benefit of an instructional setting that supports awareness of spatial structuring for fostering young children’s insight into numerical relations. This also highlighted the role of the learning ecology in a kindergarten setting regarding an overarching context, shared vocabulary, children’s own productions and constructions, interactive learning, and the socio-mathematical norm of spatial structuring. The teachers who participated had gained an increased awareness of the role of spatial structuring ability in young children’s mathematical development. As such, the local instruction theory contributes to the need for educational practice that stimulates spatial structuring.