6.13 or 7.8
Which number is greater?
For teachers, this appears to be a fairly straightforward question but when planning a question such as this, are the responses that the children might give always considered?
Children will often say, with great conviction, that the answer is 6.13 because they apply their understanding that 613 is greater than 78. Let’s not shy away from addressing and tackling mathematical misconceptions such as this, let’s use them to our advantage.
In the guidance report, ‘Improving Mathematics in Key Stages Two and Three’ produced by The Education Endowment Foundation (EEF), addressing misconceptions is stated within recommendation 1. The EEF define a misconception as:
“an understanding that leads to a ‘systematic pattern of errors’. Often misconceptions are formed when knowledge has been applied outside of the context in which it is useful.”
It can be very difficult for a teacher new to teaching a particular year group/Key Stage or an NQT to identify the common misconception, after all we are not mind readers, nor do we have a crystal ball and children will always surprise us! However, by thinking of what we want the children to be able to do at the end of the lesson and by considering what the barriers might be, then we are pre-preparing ourselves to deal effectively with responses in the classroom.
By starting with thinking about the possible misconceptions at the planning stage, teachers can pre-empt the stumbling blocks that the children might face and address it from the beginning of the lesson rather than reacting during, or often after, a task to the misconception. For example, if we take the same question and present it in two ways:
6.13 or 7.8 6.13 or 7.80
Which number is greater? Which number is greater?
This allows the children to explore what is the same and what is different about the two questions as well as allowing them to come to a more accurate conclusion. Showing the redundant zeros, and bringing previous place value learning in, can help children make the connections between previous and new learning. As teachers, we need the children to actively think and explore the learning taking place, arguably even more so if we are addressing a misconception that is grounded in what a child believes to be accurate knowledge. Simply telling the children why something is wrong will not have the best impact on developing their conceptual understanding.
Malcolm Swan (2005) discusses two common ways of teachers reacting to misconceptions:
1. Try to avoid them.
“If I warn learners about the misconceptions as I teach, they are less likely to happen. Prevention is better than cure.”
2. Provoke them and use them as learning opportunities.
“I actively encourage learners to make mistakes and learn from them.”
Things to consider when planning to address misconceptions:
• Focus on specific known difficulties rather than all possible misconceptions associated with a particular area of learning
• Think about how a misconception might have developed and the ‘partial truth’ that it is built on – what inaccurate generalisations have the children previously made and how can this thinking be refined?
• Plan opportunities for collaboration to develop learning and deepen thinking, not only with the teacher but with each other
• Select examples, questions, models and images that allow the children to make accurate generalisations and conclusions
• Compare examples and non-examples of a concept as well as standard and non-standard representations
In my experience, the deep discussion and ‘wow moments’ that occur when unpicking misconception can create a buzz in the classroom. If mathematical misconceptions are appropriately exposed and sensitively handled, children can begin to successfully restructure their understanding and become more successful mathematicians.
Sarah Howlett, Primary Maths Specialist