This year the Key Stage 2 mathematics test has undergone some big changes to reflect the new National Curriculum. One was the removal of the Mental Mathematics paper, given for the last time in 2015. It involved a 10-minute assessment, administered by playing a CD, in which 11-year-old pupils were expected to carry out 20 calculations in their heads, given 5 seconds, 10 seconds or 15 seconds for each one, and asked to write down the answer to each question, without access to paper to make jottings for working out.
Instead, last May, in addition to two papers testing reasoning, children sat an Arithmetic paper lasting 30 minutes. It asked 36 questions covering context-free calculations for all four operations, including the use of fractions, percentages and decimals. Squared paper was provided in the answer area, for children to show their working. ‘Working out’ is necessary in some cases, as the arithmetic questions now include much more complicated calculations which cannot be answered by mental calculation alone. In some questions, credit may be given for using a traditional algorithm, such as long multiplication, seemingly favoured by Michael Gove when he was Education Secretary, and praised by former Education Minister Elizabeth Truss for their efficiency.
The change in types of questions seems to have triggered a major overhaul in the planning of a teaching week. Teachers interviewed as part of a Nuffield-funded small scale study talk about how they have moved away from mental maths sessions, which took would place, say, every Friday, with children rehearsing for the Mental Mathematics test using one of the many past papers. Instead teachers now have a much greater focus on rehearsal for the Arithmetic paper each week, such as practising eight arithmetic questions every day between January and May. Teachers tell us about the need for children to work at speed to complete the 36 questions in 30 minutes and the huge amount of practice required to achieve this, as some of the types of calculations require much more working out. Teachers also tell us about how they have taught traditional algorithms to some children where they might previously have allowed them to use a less formal but slower method with which they were more comfortable.
The change in types of arithmetic questions, which clearly favour traditional methods to carry out complex calculations more quickly, begs the question of the kind of mathematics we want our children to master and how appropriate these complex context-free calculations are for children in Year 6 of primary school. The question is not a new one and has been discussed throughout the years.
A discussion paper from the University of Chicago on the changing place of algorithms in school mathematics offers some interesting viewpoints. This paper argues that there was a particular need to carry out complex computations by hand before the widespread availability of calculators and computers. However, today employers do not need their workers to be able to mimic a $5 calculator but require people who can think mathematically.
The paper also argues that traditional methods fail with a large number of students, who never become fully proficient in carrying out algorithms for the basic operations because of an overemphasis on procedural proficiency with insufficient attention to the conceptual basis for the procedures. This unbalanced approach produces, according to colleagues at the University of Chicago, students who are plagued by ‘bugs’, such as always deducting the smaller digit from the larger in subtraction.
It is too early to tell whether the removal of the mental maths paper creates similar issues in England, but teachers in our small-scale study believe that a real everyday life skill of quickly adding and subtracting numbers in your head, used when shopping or to work out how long you have to wait for the next bus, is now not being promoted in primary school teaching. Similarly, the Association of Teachers of Mathematics argues that the focus on column methods of division and multiplication for multi-digit numbers skews the curriculum against enjoyment and understanding of mathematics and towards following procedures, and that column methods are unnecessary as a foundation for secondary mathematics. Given the important role of the Key Stage 2 test in prioritising what is taught: do we really want to encourage teachers and schools to focus on written methods rather than developing mental and flexible problem solving?
Photo by JD creative commons