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Our Espresso 22 looked at the research surrounding the development of graphicacy in young children, with one of the implications for the classroom being that ‘It is important for students to invent and explore their own early representations of data and not impose conventional representations on them’. Reading this took me back to the 1990s when I was raising and home-schooling my children in Canada, and to the wonderfully innovative school through which my children’s education was supervised.
Mossleigh Demonstration School was established by Don and Anne Green, a talented and highly experienced pair of educators who were based in the Center for Gifted Education at the University of Calgary. They established Mossleigh School to demonstrate how their theories about education could work in practice. I will address the detailed practicalities of how the classrooms worked, and how each student chose the way they would be taught, in future blogs. In this blog I shall explore their idea of allowing students to present their work in whatever way they (the students) wished to, and how this could work in mathematics.
A quotation from Disessa et al (1991) in Espresso 22 shows that children may learn how to use mathematical conventions without having any underlying understanding about the conventions they are using. Allowing students to demonstrate their understanding in their own ways – whatever they may be – may give a clearer indication of their actual knowledge and how they are able to apply it, although all of these ideas are of course problematic in various ways. In addition, it fosters each child’s creativity and motivation along with the realisation that mathematics is most likely a part of everything that interests them. As suggested by Gilderdale et al. (2017), ‘Students may come into the Mathematics classroom with a passion for Science, Geography, Music, Art or even skateboarding, and this provides us with an opportunity to show them the Mathematics that underpins what they love’.
At Mossleigh all subjects were presented to students in small modules with lists of objectives which would build on each other, the last of which would always be ‘Demonstrate your ability to…’ This was where each student could present the result of their work on that particular module. An example was ‘Demonstrate your ability to show the circumference, diameter and radius of a circle’. My eldest daughter, a budding artist who received this objective at the end of a module on the properties of circles, created a circular island out of bright orange clay with a narrow blue ring of sea around it (circumference). A trail etched into the island ran straight across it through its centre (diameter) while another led from the centre straight to the shore (radius). Not only was this demonstration accepted as a fulfilment of the objective, it was welcomed by her learning community as an example of thinking outside – or indeed through – the conventions of mathematical presentation. The conventions were present in some way, but so was an important element of creativity.
My colleague Lucy Rycroft-Smith recently showed us an example of a graph she had made when she was around eight years old, and I was so intrigued that I took a scan of it:
Again, some of the conventions, or structural elements, are there – the squared paper, the axes, the labelling – but there is no title to tell us what this graph is about. Do we need one, however? Can we ‘read’ the graph all the better because it has been presented creatively, and even more importantly, does it show some insight into her understanding of graphical methods that might be lost with a more restricted assessment method? I would argue that we know exactly what is being demonstrated here and the child-that-was-Lucy’s ability to demonstrate it clearly and wonderfully.
These are both examples of artistic and thoughtful ways to demonstrate mathematical understanding. Can you think of how the scientists, geographers, musicians or skateboarders in your classroom might choose to demonstrate the properties of a circle or the progression of sunrise and sunset over a period of two weeks? How might you do it, if you had complete carte blanche and a few hours to spare?
DiSessa, A. A., Hammer, D., Sherin, B., & Kolpakowski, T. (1991). Inventing Graphing: Meta-Representational Expertise in Children. Journal of Mathematical Behaviour, 10, 117–160.
Gilderdale, C., Kiddle, A., Lord, E., Warren, B., & Watson, F. (2017) Approaches to Learning and Teaching Mathematics: a toolkit for international teachers. Cambridge: Cambridge University Press.
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