Maths……No Problem!

National Curriculum in Focus

National Curriculum in Focus is dedicated to unpicking the new curriculum and how to understand and develop the requirements of the new programmes of study for mathematics. You can find previous features in this series here

Planning for misconceptions – Teach, Learn, Confuse

The focus on mastery in the National Curriculum is a focus on understanding. Understanding builds from experiencing a concept in lots of different ways in different contexts. One of the most striking observations about the lessons taught by teachers from Shanghai, as part of the national England-China project, has been that the teachers focus on provoking misconceptions. One of the visiting teachers explained this as: ‘Teach, Learn, Confuse.’

For some teachers, planning to confuse and expecting all children to struggle within maths lessons will be a new experience. The importance of struggle in learning has been highlighted by numerous educationalists (including Stigler, Dweck and Boaler) and can be represented by the Learning Pit (Nottingham) where cognitive conflict or ‘wobble’ is encouraged in order to make learners think deeply.

This focus on provoking misconceptions is part of a focus on reasoning and allows learners to develop sound generalisations. Misconceptions, which can lead to incorrect generalisations, are often the result of limited experience and a limited diet of questions which have provided a one-dimensional view of a concept. Planning for misconceptions can be supported by using the Teaching for Mastery booklets produced by the NCETM. The questions in each domain give a sense of the breadth of understanding expected and many are deliberately shaped to expose misconceptions. For example the following question from the Y2 booklet could expose a number of misconceptions, all arising from a limited experience of finding quarters:

Y2 assessment materials question

Depending on the children’s experiences they may consider that:

  • the first does not have a quarter shaded because they can only see two parts
  • the second does have a quarter shaded because it is one of four parts
  • the third and fourth ones do not have a quarter shaded because they have not seen a square split like this for quarters before (it has always been split into four squares) and they do not think that the rectangle or triangle can be a quarter of the square. They may also think that one of them is correct and that the other cannot be because they look different to each other so can’t both be a quarter of the same sized square
  • the fifth one does not have a quarter shaded because it is not split into four parts or because the parts shaded are not together.

The starting point for any teaching sequence in maths is to be clear about what it is the children need to understand about the concepts included and, therefore, what they should be able to generalise at the end of the teaching sequence. This then leads to considering potential misconceptions; it will be important to plan for children to explore concepts in a variety of contexts and expect them to use a variety of representations in order to expose and address misconceptions and build a more complete understanding. This is explained in the Teaching for Mastery booklets as follows:

“A pupil really understands a mathematical concept, idea or technique if he or she can:

  • describe it in his or her own words;
  • represent it in a variety of ways (e.g. using concrete materials, pictures and symbols – the CPA approach);
  • explain it to someone else;
  • make up his or her own examples (and non-examples) of it;
  • see connections between it and other facts or ideas;
  • recognise it in new situations and contexts;
  • make use of it in various ways, including in new situations.”

This list of ways to demonstrate understanding provides teachers and pupils with different ways to challenge thinking and explore concepts. It is important to note that sometimes teachers will expose misconceptions which they had not anticipated (for an example of this, see the Seen and Heard article this month); we are always interested in unusual and unexpected misconceptions so please email these to us for inclusion in future issues of the magazine.

Ten things the maths team want you to know (1)