Charlie Harber is Deputy Lead for the Primary Mathematics Advisory team at Herts for Learning.
Over the last few weeks I have had the opportunity to engage at a deeper level with this report, through attending a very informative session led by Peter Henderson (EEF Programme Manager) and hosted by Sandringham school; but mainly when I was preparing to lead a session on how these recommendation might be translated to real life classroom practice. It was during this workshop session that I again encountered that old bug bear of teachers i.e. being aware of what some of the research says, particularly all the amazing new perceptions coming out of cognitive science, (it really is an exciting time to be in teaching), agreeing with outcomes BUT not having the time to actually crystallise what does that mean for my classroom? How can I maximise this research to ensure that my pupils benefit? And from this lack of time, comes the guilt.
Recommendation 1: Use assessment to build on pupils' existing knowledge and understanding.
‘It is important that misconceptions are uncovered and addressed rather than side-stepped or ignored’ (EEF Improving Mathematics in Key Stages Two and Three, page 9)
Challenges with this for a classroom teacher:
- What are the possible misconceptions?
- How do you expose them without negative repercussions?
- Once exposed, how do you successfully alter or deepen pupils’ conceptual understanding?
Within ESSENTIALmaths we provide developmental subject knowledge support for both the specialist and non-specialist maths primary teacher.
Within the Key Concepts paragraphs for the teachers we aim to improve teachers’ subject knowledge, explain the rationale behind the models, representations and language.
Within the learning sequences we provide deliberate and informed (Rowland, 2008) examples of:-
Teacher questions; supporting teachers to clearly identify what the key focus of the learning is at this stage.
Modelled pupil responses; raising awareness of what pupils can and should be expected, scaffolded and challenged into verbalising to show their emerging understanding.
‘Teachers with knowledge of the common misconceptions can plans lessons to address potential misconceptions before they arise, for example, by comparing examples and non-examples when teaching a new concept’
(EEF Improving Mathematics in Key Stages Two and Three, page 9)
Conversation cartoons allow teachers to expose misconceptions head on and build them into an effective teaching tool. The power lies in that they draw attention to the targeted aspect by depicting common misconceptions within a safe environment, revealing and challenging pupils’ ideas, in some cases causing cognitive conflict. Pupils feel more willing to share their current beliefs if they feel that others (even fictional characters) share a similar understanding. By having more than one explanation or strategy presented at the same time, there is a heightened opportunity for the pupils to notice, compare and contrast (Kullberg, 2017). Depth can be added by asking pupils to consider why someone would think this – can they identify the gap in their understanding? The teacher is supported in developing mathematical conversations with the pupils, nurturing an environment where they collaborate to construct and deepen understanding.
This example from Year 4 addresses a common misconception frequently experienced by pupils.
In both of the children's responses, the language and sentence structure are carefully constructed so that they appear equally feasible. The incorrect response reflects what many pupils believe; treating the numbers to the right of the decimal point as whole numbers. The teacher has the freedom to focus on assessment and facilitating social construction of concept.
Provoking and proving
Teaching conceptual understanding of fractions multiplied by whole numbers can be challenging. Many less experienced teachers may inadvertently teach a trick in order to avoid questions which they may not be confident in answering.
In this cartoon, two differing views are not presented, instead it is modelling how to successfully continue and challenge someone else's response in a constructive way.
This scenario naturally leads pupils' own thoughts and investigations – how can we find out? Again, social constructionism and collaboration used to ensure deeper understanding.
Some pupils are locked onto single pathways, and will continue to use a previously successfully strategy for calculations beyond the point of efficiency. In this cartoon, pupils are provided with the opportunity to discuss three differing but correct strategies. It is only through this direct comparison that pupils begin to explore efficiency.
The calculating has been done for the pupils, thus relieving the cognitive load on mental maths and allowing the focus to stay on key learning outcome – which strategy and why?
Teachers are encouraged to deepen and sustain discussion by asking additional questions such as;
Which one is the pupil's default method? Which one is the most efficient? Are these different? What could the calculation be that would mean that the 'find 1% strategy’ becomes the most efficient?
Improving Mathematics in Key Stages Two and Three – Guidance report (2018) Educational Endowment Foundation (online). Available at https://educationendowmentfoundation.org.uk/tools/guidance-reports/math…;
Rowland, T (2008) The purpose, design and use of examples in the teaching of elementary mathematics. Educational Studies in Mathematics, 69(2), 149-163
Kullberg, A., Runesson Kempe, U. & Marton, F. What is made possible to learn when using the variation theory of learning in teaching mathematics? ZDM Mathematics Education (2017) 49: 559.