Growing Prior Knowings of Zero: Growth of Mathematical Understanding of Learners with Difficulties in Mathematics

Currently, there is a large proportion of learners experiencing difficulties in mathematics. Much of the intervention research for children with great difficulty learning mathematics has focused on accommodations to the peripheral supports of mathematics, like creating step by step plans, and not on strategies to help children conceptualize mathematics or enable them to mathematize. We know very little about the conceptual development and how to effect change in conceptual understanding of mathematics for children who have great difficulty learning mathematics. At the same time, in mathematics education research, zero is a known area of difficulty for many students and misconceptions regarding zero can persist into university and adulthood. This dissertation explores growth in understanding with three learners experiencing difficulties in mathematics and their growing conceptions of zero. Utilizing the Pirie Kieren Theory for the Dynamical Growth of Mathematical Understanding and its model for tracking growth on a small scale, I ask the questions, (i) What is the process of change, the growth of understanding, that each child passes through? and (ii) What are the images and prior knowings that children experiencing difficulties in mathematics have about zero, and how do they thicken? The analysis presented here is mainly of the task-based clinical interviews in which each learner participated. Data from parental surveys, task-based interventions and classroom observations are used to support this analysis.

Results of my research indicate how learners may be thickening and revisiting their prior knowings. Thickening occurs either as a foundation to anchor growth, or as a comparative for new growth. Results of my research also indicate that on the small-scale of tracking growth there is a juncture between expectation and result where growth has the potential to occur or not occur. This research provides descriptive evidence of intervention specifically for growth in understanding that takes into account the juncture between expectation and result. Finally, because zero is a paradox, understandings in Primitive Knowing around zero require multiple revisitings.