Errors and Misconceptions in Maths at Key Stage 2

Working Towards Success in SATS

Mike Spooner

• Publisher: David Fulton Publishers
• Year: 2012
• ISBN:
  9781136611292
• Edition: 1
• Topic: Éducation
• Subtopic: Éducation générale

The activities in this book are designed both to help children to learn and to understand math concepts, and also to prepare them for taking SATS papers at KS2. There is plenty of research evidence to suggest that children are helped in their understanding of math problems if teachers focus on common misconceptions, and if children are given opportunities for discussion and explanation of their own understanding. Children can also feel stigmatized by being made to focus on their own errors.

Mike Spooner has developed activities that present already completed work which children then assess, correct and discuss - in this way they can analyze mistakes without damage to their own self-esteem. The activities are presented in the same format as the SATS papers, to give children practice in coping with that format. The book also contains writing frames that children can use to structure their discussions of math problems.

Using the Exercises

1. Explain to the Children that:
   • they will be working with exercises that have been completed by other children;
   • in all the exercises some or all of the answers are incorrect;
   • the number of errors they should find is written at the start of each separate section of the exercise.
2. Ask the children to:
   • mark the work and establish which examples are incorrect;
   • suggest why the ‘unknown child’ might have made the errors;
   • suggest ways in which they would help the child overcome the problems exhibited (what would they say, is there any equipment or diagram they might use, etc.?);
   • prepare to report back to the rest of the class in a plenary session.
3. Organising the class. In the trials the exercises were used within the standard structure of the daily mathematics lesson. Discussion was achieved either with the whole class looking at some examples or with small groups working alone before reporting back to the class in the plenary session.
   In some classes children worked in groups on different topics and took responsibility for reporting back to the rest of the class on specific errors to avoid. They also made posters to support and reinforce their advice.
4. Setting up the tasks for children working in unsupported groups. Writing frames will help to structure the report (Appendices B and C). Copies of the briefing sheet can be placed on the desk for children and learning support assistants to refer to (Appendix A).
5. Teacher’s notes. The teacher’s notes summarise aspects of the problem and refer to relevant comments from the Standards reports (QCA 1999 and 2000).

Learners’ Health Warning

The value of these exercises can only be accessed through discussion and opportunities to explain thinking.

The Exercises

Numbers and the Number System

Rounding numbers

Decimals

Calculation

Multiplying and dividing by 10 and 100

Solving numerical problems

Interpreting calculator displays

Co-ordinates

Using protractors and rulers

Reading scales

Numbers and the Number System

Some Problem Areas

• The examples in this section will check on the definition of the terms prime, multiple and factor.
• Dealing with intersections and membership of more than one set in Carroll and Venn diagrams.

Evidence from SATs

From QCA (2000):

Two questions assessed children’s recognition of prime numbers. In Question A12 children were asked to classify three numbers according to whether they were even or not even, and prime or not prime. Although half the children working at level 4 were able to classify all three numbers correctly, a substantial number of children thought that nine was prime and that 17 was not prime. This suggests that they were more confident in classifying odd or even numbers than they were in their understanding of prime numbers.

Teaching Points

1. The error exercises will be useful in getting children to provide clear definitions of the terms prime, multiple and factor.
2. Venn and Carroll diagrams are often used in SATs. It can be very useful to describe what each area of the diagram will contain, e.g.
   
3. The final exercise in this section looks at the effects of adding 1,10,100 and 1000. Children who find this difficult would benefit from seeing concrete models of these operations, e.g. ‘If 990 were modelled with Diene’s apparatus and 10 were add...