End of KS1 and KS2 tests
One of the key observations emerging from examination of teaching practices in Shanghai is that there is a focus on teaching relational understanding, ‘knowing both what to do and why’ (Skemp 1976). This leads to an expectation that, when working on mathematics, children will ‘notice’ things and make decisions based on what they notice.
Looking at the test papers for the end of both KS1 and KS2, it is clear that children who look to notice things and use what they notice, and what they know, to make decisions will have an advantage over children with an instrumental understanding who have memorised what to do and follow this route regardless of the numbers and the context involved.
Paper 1: arithmetic (KS1 and KS2)
Despite their title, demonstrating fluency in these papers is dependent on reasoning. In order to provide space to reason children need to step back from the questions before engaging with finding an answer, so that they allow themselves the opportunity to notice things. Including decision-making as a key element of all mathematics lessons will support children with taking this approach.
It is worth considering, as a whole staff, for which of the questions on the KS2 arithmetic paper you would expect children to use a written method and why. This can be linked to examining particular questions from both the KS1 and KS2 arithmetic papers, discussing how you would expect children to be tackling them if they are demonstrating fluency, and then looking at different responses from children in your school in order to consider the adjustments to teaching that might be needed in order to support children to use what they know and understand.
- Do the children look at the whole calculation before starting to calculate?
- Do they notice that they know 4 + 6 = 10 and that it is easy to add 5 to 10?
- Do the children notice that the two numbers shown in the calculation are close together?
- Do they notice that knowing 1 + 5 = 6 will help them here?
- Do the children notice that 69 is only one away from 70?
- Do they notice that adding one and adding ten is the same as adding eleven?
89 994 + 7 643 =
- Do the children notice that 89 994 is only six away from 90 000?
- Do they notice that adding 7 643 (or 7 637) to 90 000 is easy?
122 456 - 11 999 =
- Do the children notice that 11 999 is only one away from 12 000?
- Do they notice that subtracting 12 000 from 122 456 is easy?
- Do the children notice that 3/5 can be thought of as ‘three lots of one fifth’?
- Do they notice that dividing three of something by three is straightforward?
Papers 2 and 3: reasoning
For some KS2 children, the organisation of papers 2 and 3 was a challenge because the content that was being tested did not get progressively harder; instead content from years three to six was dotted around the papers. This is reflected in the mark scheme (page 4) with the table showing the content domain covered by each question. The start of the table for the two reasoning papers (below) shows that in paper 2 the second question covers content from year 5 and in paper 3 the second question covers content from year 6, whilst question 3 on each paper tests content from lower KS2.
Again, at both KS1 and KS2, the numbers used in the questions on the reasoning papers invite the children to notice things. For example:
- Do the children notice that they can share the boxes between Kemi and Ben; they don’t need to find out how many pencils there are in total?
KS2 Paper 2 Q9
- Do the children notice that if they know the cost of six pencils then they know the cost of three pencils by halving, they don’t need to find the cost of one pencil?
Supporting children to develop relational understanding and expecting them to notice things, related to what they know and understand, and then make decisions based on what they notice is at the heart of teaching for fluency, reasoning and problem-solving, the aims of the National Curriculum. The result will be children who understand the mathematics and can demonstrate this understanding in a test situation.
Skemp, R Relational Understanding and Instrumental Understanding. First published in Mathematics Teaching, 77, 20–26, (1976)
Next time in Digging Deeper we will explore the new teaching for mastery videos.
Page header by Suzette (adapted), some rights reserved