Thanks to John Dabell for this post which addresses key misconceptions in maths and suggests an effective activity to overcome them. John is a teacher with over 20 years teaching experience across all key stages. He has worked as a national in-service provider and is a trained OfSTED inspector.
Maths, like any subject, is messy and getting to grips with it is often chaotic, unruly and disordered.
When it comes to learning new concepts then children don’t always ‘get it’ straight away. It might take hours, days, weeks, months or years; conceptual development and ‘mastery’ can be hard-won. Sometimes concepts remain out of reach.
There are lots of errors children make on their maths journeys that we tend to call misconceptions, although a more generous and forgiving phrase is ‘ideas in embryo’.
It’s important to collect and compile these misconstructions so we can analyse them, discuss them and, crucially, use them as teaching, learning an assessment opportunities.
Paying close attention to misconceptions, being forensic with them and building them into teaching episodes is a valuable teaching technique.
One of the recommendations in the recent EEF report Improving Mathematics at Key Stages 2 &3 is to tackle misconceptions head-on, and not to side-step them but exploit them as formative assessment opportunities in order to help pupils “develop richer and more robust conceptions”.
Misconceptions that are commonly made acquire the label ‘classic mistakes’, however they aren’t silly or careless mistakes or a sign of ‘wrong thinking’ but a natural stage in conceptual development. You only know what you know! If you think 1/10 is double 1/5 then there is a logic at work that needs unpacking not ridiculing.
There are many examples of maths misconceptions, but some ‘classics’ include:
1. 62 = 12
2. 7 x 0 = 7
3. Four hundred and eight is written as 4008
4. 0.10 = point ten
5. 0.5 x 10 = 0.50
6. 6 -:- ½ = 3
7. -5 + 3 = -8
8. 4% is 0.4 as a decimal
9. 1/3 + 1/2 = 2/5
10. ¼ -:- 1/8 = 1/2
11. There are no numbers between 2.2 and 2.3
12. 0.2 x 0.4 = 0.8
13. 0.625 > 0.9
14. 0.4 is smaller than 0.400
15. 5 -:- 20 = 4
16. 5/16 is smaller than ¼
17. 2.1 hours = 2 hours 10 minutes
18. A rectangle has two lines of symmetry
19. Shapes with bigger areas have bigger perimeters
20. The largest acute angle is 89⁰
What do we do?
We have to make specific efforts to uncover misconceptions and use them to our advantage for promoting cognitive conflict.
Misconceptions shouldn’t be swept under the carpet but milked as mathematical moments and used as opportunities to refine and upgrade knowledge and understanding. These can be explored inside talk-rich contexts such as concept cartoons, true/false/sometimes statements or balloon debate odd-one-out activities. Counter-examples are particularly well-suited to deep discussions.
Talking about misconceptions in dialogue-rich activities encourages children to review their thinking; it helps make their thinking transparent and leads to far more sophisticated reasoning and reflection.
It shows pupils that misconceptions are often the result of over-generalising from simple cases they have met before.
When discussion is centred on misconceptions as a natural part of learning pupils can feel safer and more secure to open up about their own ideas. As Mackle (2017) says in his book,
“The number one rule in my own classroom is that I’m interested in how you got there, not where you are and that it is okay to make mistakes.”
Start to keep a glossary of the maths misconceptions that you encounter in your day-to-day teaching as this will act as a very valuable teaching resource - especially when you pool your collections together within a professional learning network.
You can download some free Classic Mistake Maths Posters at www.calculatorsoftware.co.uk/classicmistake/gallery.htm
The NCETM document ‘Misconceptions with the Key Objectives’ is a valuable document to support teachers with developing their practice.
Count On contains lots of PDFs explaining some of the popular misconceptions in mathematics.
Hansen, A. (ed) (2005) Children’s Errors in Mathematics. Learning Matters Ltd: Exeter
Keogh, B., Dabell, J. and Naylor, S., (2008) Concept Cartoons in Mathematics. Millgate House
Mackle, K (2017) Tackling Misconceptions in Primary Mathematics: Preventing, identifying and addressing children’s errors. Routledge: Oxon