Mathematics is an abstract subject, representations have the potential to provide access and develop understanding. Consider the model below:
Maths is built on relatively few structures, that can be applied in an infinite number of ways, to an infinite number of problems. People who enjoy maths, see the bottom of the diagram and know that from relatively little knowledge, they are able to solve a huge variety of problems. Often those who do not like maths, see the top, and are overwhelmed by the vast sea of problems they need to learn to solve. In the teaching for mastery approach, well chosen representations, such as part-part whole models, and the Singapore bar model, allow all pupils to access the structure of the mathematics.
It is often said that the teaching for mastery approach is ‘just good teaching’. This is certainly true around the way we use manipulatives, part whole models and bar modelling. One of the keys to mastery, however, is exposing structure to stop artificial early success – we often assume that our ‘rapid-graspers’ can already do, so don’t need exposure to such things. In fact it is only through such exposure that they are able to build a deep understanding.
“Mathematical tools should be seen as supports for learning. But using tools as supports does not happen automatically. Students must construct meaning for them. This requires more than watching demonstrations; it requires working with tools over extended periods of time, trying them out, and watching what happens. Meaning does not reside in tools; it is constructed by students as they use tools.”
– (Hiebert 1997 p 10) Cited in Russell (May, 2000). Developing Computational Fluency with Whole Numbers in the Elementary Grades
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