Your new Year 7s and their arithmetic, and some arithmetic for you

Nationally, 70% of Yr 7s achieved the ‘expected standard’ in KS2 Maths SATs tests. This ‘expected standard’ was decided by a panel of teachers looking at the scores from completed test papers and comparing them to the national curriculum, as well as by the testing agency. Perhaps eventually there will be more detailed information about how individual pupils reached the standard, but meanwhile you will have your new intake with little useful information. I have recently been given access to the actual scripts of the KS2 SATs tests for a whole school, where 73% of pupils achieved the ‘expected standard’.

I have looked in detail at the test papers, and also at one particular set of scripts that might suggest where to look for useful information. You will probably be more interested in the strengths and weaknesses shown in these scripts, rather than actual marks, so I hope this blog is useful to you. Perhaps I should also tell you that for this school, 73% was a particularly outstanding achievement because they have been seriously struggling in other areas of school life. You are seeing what happens when a school makes a supreme effort to help their children succeed.

You will know that the test involves three papers. The first paper is purely calculation, apart from a couple of questions at the end that require a little bit of numerical reasoning. The other two papers focus on reasoning mainly in terms of reading a situation and deciding how to apply arithmetical knowledge to it. This setup means that there were no straightforward factual or procedural questions about angles, area, volume, or measurements. So the overall test scores won’t tell you whether your Yr 7s know about these things in straightforward cases.

Some of the students ‘passed’ on the basis of doing formal calculations throughout all three papers, so did very well on the first paper and occasionally succeeded on the other two papers. Other students ‘passed’ on the basis of doing reasonably well across all three papers. So the overall test scores won’t tell you whether students are strong or weak in problem-solving tasks or calculation skills, not even students from the same school.

All of the students in the school used formal calculation methods throughout the first paper.

Here are some of the questions, each worth one mark, that were done with formal column methods on most of the scripts:

987 + 100 =

326 ÷ 1 =

468 – 9 =

? = 435 – 30

50 × 70 =

122,456 – 11,999 =

The few students who merely wrote down answers to these questions, showing no working, did not do well in the test overall. They did not seem to be able to do any formal procedures so had problems when the numbers required more than mental methods could provide.

I wanted to write a more jokey blog, but this is too sad to joke about.

When I last taught Yr 7s three years ago, I was overwhelmed with their confidence in mental methods and their familiarity with numbers. Maybe these students also have that, but they did not show it in the test, opting instead for the safety of showing everything you know how to do. So the overall test scores will not tell you whether they have confidence with numbers, or whether they depend on procedural accuracy. In those ‘high-performing jurisdictions’ we are all a bit fed up with hearing about, procedural fluency includes being able to choose appropriate methods. Hmmm.

So here is something to cheer you up:

Choose five whole numbers from 1 to 10 and list them in ascending order. Next to them in one-to-one correspondence list the rest of the whole numbers in descending order. What is the sum of the absolute differences between each pair? Compare your answer to someone who has chosen different numbers. What happens and why?


Choice Others Absolute difference
2 10 8
4 7 3
6 5 1
8 3
9 1

Try other sets of consecutive or non-consecutive numbers.

All the best,

Anne Watson

Anne WatsonAnne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications.  Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent.