Sharpening the Focus on Numbers and Counting: Preschool Educators Differentiating Aspects of Mathematical Knowledge for Teaching

The purpose of this study is to examine how preschool educators (teachers and childminders) perceive numbers and counting as learning content in preschool, and how their perceptions develop during a prolonged period of repetitive, reflective group discussions. The authors seek knowledge about how the staff educators learn to differentiate numbers and counting as learning content in preschool practices and activities they themselves plan, enact, and then discuss and evaluate in a common forum. This means that the study does not focus on the educators' knowledge about mathematics in general.

The authors find that during the further-education programme, there is a general development in the participants' perceptions from a more common perception of numbers and counting to enhanced awareness and increased focus on numbers and counting as independent content for learning in preschool. This is evident in that the documented practices and activities of the participants change and become more focused during the programme.

Many of the initial documentations in the study show activities and play situations where participants assume that numbers and counting are happening. However, the participants place more emphasis on other aspects of mathematics, such as the children knowing which cup is bigger and sorting sticks in a long row to encourage discussions of location (in front, behind). According to the authors, the initial documentations generally visualise situations where the children are to use counting as a strategy to solve tasks where they need to know "how many items". This results in the fact that the children’s knowledge of numbers and counting is actually expressed and used, but not made an object for inquiry of deeper meaning behind numbers and counting.

The analysis shows that the participants' documentations change focus later in the programme. According to the authors, in the later documentations, there are many more opportunities for learning since focus is more directed towards the children’s process of making meaning from the learning content. Numbers and counting are made independent learning content, for example arranging sets of objects in order of quantity and relating them to numbers. A game was also documented: "The worm in the box", the purpose of which is for the children to find out how many pearls (out of ten) are hidden in a box by counting the ones outside the box. According to the authors, the later documentations thereby have a clear goal for learning, as they are directed towards the process of making meaning from numbers orchestrated in different ways to challenge the children's conceptions of numbers and use of counting strategies.

According to the authors, this shift in the participants' perceptions is an important result since it implies that, during a prolonged period of reflective group discussions, the participants have managed to differentiate aspects of numbers and counting as content for learning that was not present in their perceptions at the beginning of the programme. On this basis, the study concludes that a collaborative approach seems beneficial in that new aspects are brought to the fore, influencing the educators' reflective practice.

Focus group discussions were carried out with eight preschool educators from four preschools in one municipality. The focus groups were part of a further-education programme aiming promote preschool educators' collective learning about preschool mathematics. A total of six group discussions were held over a period of eight months. Each group discussion lasted 2½ hours and was based on the educators' own documentation of preschool practices and activities that involved numbers and counting. A total of 40 documentations, such as photographs, video recordings and practice narratives, were presented and discussed. These documentations constitute the primary data material in this study. Based on learning perspectives inspired by Ference Marton's Variation Theory, the study analysed the learning content as expressed in the educators' documentations, and how the educators' perceptions of this learning content develop over time.

Björklund, C., & Alkhede, M. (2017). Sharpening the Focus on Numbers and Counting: Preschool Educators Differentiating Aspects of Mathematical Knowledge for Teaching. Mathematics Teacher Education and Development, 19(3), 117-134.