Exploring the efficacy of a spatial reasoning approach to teaching and learning fractions in the early years

Fractions are widely recognised as one of the most difficult areas of the school mathematics curriculum to teach and learn. A deep understanding of fractions supports further learning in areas such as algebra, proportional reasoning, and statistics. In Australia, research has established that many students in the primary and middle years of schooling experience considerable difficulty in recognising, naming, and renaming common fractions. This difficulty is consistently reflected in Australian students’ results on international assessments of mathematical literacy.  One issue contributing to these difficulties, is the way in which fractions are represented in the Australian Mathematics Curriculum content descriptors. Within the early years, children are expected to recognise equal parts of a whole, primarily emphasising the part-whole meaning of fractions. This focus promotes an additive, counting-based approach to the teaching and learning of fractions. Such an approach overlooks the critical foundations fractions have in fair sharing contexts and masks the multiplicative nature of fractions. The foundation of early fraction understanding needs to be based on fair sharing or equal partitioning. Fair sharing is an intuitive idea that very young children experience in a range of contexts before they begin school. Research on children’s early number learning suggests that fair sharing is a highly spatial activity. This is particularly evident when children share collections of items, or partition continuous objects fairly based on attributes such as geometric similarities and spatial arrangements. Spatial reasoning and its association with mathematics achievement is well established in the literature. Although there is increasing interest in examining young children's spatial reasoning and mathematical development in general, there is very little research that has examined the role spatial reasoning may play in relation to early fraction learning. This thesis aims to address this gap in the research.  A Design-Based Research (DBR) methodology was employed, to explore the extent to which young children could demonstrate an understanding of a broader range of fraction meanings, experienced through a spatial reasoning approach. Consistent with DBR, a local instruction theory was developed to frame the design and implementation of an iterative teaching experiment that introduced children to the fraction as an operator, fraction as a measure and fraction as a relation meanings. The local instruction theory was represented by a series of conjectured key indicators that proposed a pathway for developing the three meanings of fractions through a spatial reasoning approach. The key indicators informed the design of an intervention program comprising of 13 consecutive 60-minute lessons, which were implemented with 70, Year 1 and 2 children at three primary schools in regional South Australia. In its final form, the four key indicators of the local instruction theory were: creating and justifying equal shares; reinitialising the unit; recognising proportional equivalence and connecting multiplicative relations. The results of the study revealed that young children developed rich understandings of the three meanings of fractions, which were directly supported by spatial reasoning. The children demonstrated an understanding of the fraction as an operator meaning, through a focus on spatial visualisation to predict and justify the outcome of sharing geometric shapes, and small collections of objects. The children developed the fraction as a measure meaning specifically in the way they were able to work with unit fractions, composite units, and their ability to name different fraction parts. Spatial structuring was a critical construct that supported the children’s ability to make connections about the associated fraction as a measure ideas, and fraction magnitude. Further, the children demonstrated an ability to recognise fraction and proportional equivalence, through contexts that developed their spatial proportional reasoning. The children were able to work with early fraction as a relation ideas, specifically simple ratio, which was supported by spatial structuring. Additionally, the influence of spatial structuring within fraction as a relation contexts, positively influenced children’s part-part relations for whole number quantities. Another critical finding of this study was the way in which children used gesture to communicate both spatial and mathematical knowledge. The children's spontaneous use of gesture throughout the intervention provided greater insights into their use of spatial reasoning strategies, and how these were connected to their understanding of fractions. This is an important finding that makes a significant contribution to what is currently known about how young children use self-initiated gestures to explain and communicate their mathematical thinking.  As the three meanings of fractions are not typically taught or considered to be entirely appropriate for children in the early years, this study challenges the foundations of the current curriculum expectations that promote additive approaches to learning fractions. The findings provide powerful insights into what is possible in the early years of schooling. Specifically, the ability for children to develop a multiplicative foundation for an extended range of fraction meanings, though a spatial reasoning approach. Given the persistent difficulties children in the middle and upper years of primary education experience with fractions, an explicit emphasis on spatial reasoning in the early years of school provides an important alternative approach to the teaching and learning of factions.