Blog: Dr. Helen Drury, Director of Mathematics Mastery
Assessment for a world-leading mathematics education?
Posted: 31/07/15The new national curriculum for primary maths is intended to ensure that pupils leave school with the knowledge, skills and ambition to succeed in modern Britain and to compete in the global labour market.
It has been designed to ensure England has the most productive, most creative and best educated young people of any nation. So the Standards and Testing Agency (STA) have a pretty huge job – it falls to them to develop the test that all pupils will take at the end of primary school.
At the end of June this year, the STA published the sample materials for the key stage 2 tests from 2016. There was something nagging at me when I first looked through the two ‘Reasoning’ papers from the sample key stage 2 assessments – paper 2 and paper 3. Having had the privilege of being part of the group that had drafted the new national curriculum for maths, I had a fairly clear idea of what I expected 11 year olds to be achieving with their mathematics. Somehow these sample papers didn’t quite match that. Yet page 5 of the mark scheme clearly maps each test question to the national curriculum. It wasn’t what was being tested that was bothering me, but the way that it was being tested.
In the early development of the ‘mastery’ approach to teaching mathematics, I spent a lot of time studying Singapore’s Primary School Leaving Exam (PSLE). It was clear they set a very high bar. To get my head round what felt funny about the sample key stage 2 papers, I decided to look again to Singapore. As the PSLE is for 12 year olds, I looked at papers written by Singaporean teachers for 9 year olds (P3, our Year 4), 10 year olds (P4, our Year 5) and 11 year olds (P5, our Year 6). You can see the papers here: raffles.guru/singapore-math/ .
Looking back at these Singaporean test papers clarified for me what makes a great test. Yes, some questions on a good maths test might focus solely on recall and computation. But the really important questions are those that assess knowledge and mathematical reasoning. STA’s sample key stage 2 tests include some great questions that do exactly this. For example, question 15 of paper 3, where pupils are asked to join dots on a grid to make a quadrilateral that has 3 acute angles. There’s no getting around the knowledge recall required here – to be successful the pupil will need to be know the properties of a quadrilateral and an acute angle. But the demand is higher than this – the recall is necessary but not sufficient.
How do we really test fluency?
There are three elements that constitute mathematical fluency: efficiency, accuracy and flexibility (Russell, 2000). If test questions are too straightforward, pupils can get full marks with just accuracy – their approach need not be efficient or flexible. For us to become world leaders in mathematics education, we need pupils who do more than just recording carefully and double-checking their answers. It’s important that pupils don’t become bogged down in too many steps (they’re efficient) and use an appropriate method for the numbers involved (and flexible).
As Daniel Willingham explains in his article Is it true that some people just can’t do math?, “for most topics, it does not make sense to teach concepts first or to teach procedures first; both should be taught in concert. Gaining knowledge and understanding of one supports comprehension of the other.” We therefore need our assessments to reflect this, with questions that require students to use efficient strategies, and to understand them.
As procedural fluency and calculation are already being assessed on Paper 1 – the arithmetic paper – papers 2 and 3 are opportunities to assess this deeper type of fluency – efficiency, accuracy and flexibility. However, what’s disappointing with the sample key stage two tests is that there are too many questions that focus on straightforward knowledge recall, rather than conceptual understanding, reasoning or problem solving.
For example, question 14 on Paper 2 is rather repetitive, arguably assessing the same skill three times:
Likewise, question 3 of paper 3 asks:
 Which reminds me – the notation used on these sample tests is inconsistent with the national curriculum! The convention is that we use spaces, not commas, to separate sets of three digits.
Place value understanding is fundamental to mathematical success – it’s important that it features on the test papers, but are all of these questions the best possible ones? I found an equivalent question on a Singaporean paper for 9-10 year olds (paper 4A), but the reasoning demand is perhaps a little higher:
On Paper 4A (9-10 year olds), Singaporean pupils reason using understanding of place value and the subtraction algorithm.
But we don’t have to go all the way to Singapore to find great questions that require real fluency and mathematical reasoning.
The importance of reasoning
In this summer’s key stage 2 assessment, paper 1 question 21, pupils were asked:
This demands strong procedural fluency of place value and rounding to the nearest hundred. Only pupils with strong understanding will be able to reason that they are looking for the numbers furthest from multiples of a hundred (such as 149 and 151).
In Singapore, assessments tend to demand that pupils demonstrate their fluency through reasoning mathematically. A question for 8 year olds (2B) asks pupils to find the total mass of a bag of flour and a bag of rice:
Pupils need to demonstrate proficiency with reasoning. They might begin by deciding that the mass of the rice is more straightforward to find than the mass of the flour, then finding the sum of 3kg + 2kg +2kg to know that the rice is 7kg. To find the mass of the flour, they need to use the information that:
1kg + 1kg + 2kg + 2kg = the mass of the flour + 500g + 500g + 1kg
The pupil who tackled this paper seems to have used the strategy of subtracting 1kg from each side of the equals sign. This involves knowing that there are 1000g in 1kg, so 500g + 500g = 1kg. The flour therefore has a mass of 4kg. The total mass of the rice and flour is 7kg + 4kg = 11kg.
Another question for 8 year olds (2B) shows three boxes labelled A, B and C:
Pupils need to reason that they should subtract 124g from 285g. Given these digits, mental calculation is most efficient: 285g – 124g = 161g. Pupils then need to reason that they need to subtract a further 83g from 161g. They might use the formal written algorithm, or calculate mentally, e.g. 161g – 83g = 160g + 1g – 80g – 3g = 160g – 80g + 1g – 3g = 80g – 2g = 78g.
These questions aren’t just logic questions – they do require skill in reasoning, but not just reasoning. Each question requires pupils to be fluent with key mathematical concepts and skills.
Again, we don’t need to go as far as Singapore to find questions that demand both fluency and reasoning. Here’s question 12 from this summer’s key stage 2 paper 2:
Pupils need to use knowledge that 40 ÷ 10 = 4, that 4 x 5 = 20, and that 40 + 20 = 60 (or 4 x 15 = 60). Just as importantly, the question requires pupils’ fluency with this calculation to be sufficiently robust that they can concentrate on the reasoning without being distracted by the mechanics of the calculation.
And on question 15 of this summer’s key stage 2 paper 1, fluency with place value and subtraction were assessed with this question:
Hope for the future
So, will we have a world leading assessment in 2016 for our new curriculum? I believe we can. For our standards to rival those of higher performing jurisdictions, we need to see fewer straight knowledge recall questions, and a few more of those questions that require pupils to demonstrate fluency through reasoning and application. The evidence shows we’re just as good at designing those questions as the higher performing jurisdictions – better, even. Fingers crossed that’s what we’ve got to look forward to in summer 2016.