A closer focus: Common misconceptions in maths
By Susan Rose, Senior Research Manager
Friday 28 August 2020
The question-level analyses of the key stage 2 national curriculum mathematics tests provide evidence of the areas of the curriculum which pupils found more difficult than others. This article aims to focus on those areas which caused the most difficulty in both 2018 and 2019, as well as in the standardisation trial for our own year 6 NFER Tests.
- Long multiplication and long division
Long multiplication and long division skills are tested in two-mark questions, contributing one fifth of the total marks for the arithmetic paper. Despite the questions being set out to show pupils that these methods were required, each year approximately 20% of pupils were unable to gain the marks when asked to multiply a three digit number by a two digit number. Furthermore, approximately 25% are unable to do so when multiplying a four digit number by a two digit number. Results from the NFER autumn tests show that this skill is particularly difficult for lower achieving pupils with very few (8%) being able to gain both marks.
Two errors were particularly common in these questions. The first was related to place value. Pupils had a clear misunderstanding when multiplying by the tens digit – interpreting it incorrectly as if it were in the ones column. These pupils require a focus on the long multiplication approach and revisiting what each calculation means within the method. The second issue was errors in arithmetic: pupils showed an understanding of the method required but made more than one arithmetic error. This may be a result of more focus being placed on the method rather than care being taken in the calculations. Therefore, pupils must develop approaches in systematically checking their work in order to avoid such errors.
For long division a majority of pupils (76%) were able to gain marks when asked to divide a three digit number by a two digit number but significantly fewer (50%) were able to do so when asked to divide a four digit number by a two digit number. This is a skill from the year 6 programme of study which may account for pupils having had fewer opportunities to practise this skill. Interestingly, the NFER Test data showed that almost three times as many pupils (29%) chose to answer the question using a short division method as those who opted for long division (10%) which may indicate less familiarity with the method. Arithmetic errors were also very common, again suggesting that pupils may be more focused on the method rather than taking care over the calculation.
These skills also feature in the reasoning papers, usually as part of a three mark problem solving question. Where it is clear that multiplication is required, a similar number of pupils were able to gain the marks as when the skill was tested discretely suggesting that, once pupils are more secure in the long multiplication method, they are more likely to perform better on the three mark question.
- Order of operations
This is an area of the year 6 programme of study that can prove difficult for pupils of all levels of achievement. Evidence suggests that where a question involves the use of brackets, such as in 2019 where they were asked to calculate ‘60 – (30 – 24)’, pupils are clear that the calculation within the bracket needs to be worked out first and the majority (90%) were able to obtain the correct answer. However, pupils are less successful in answering questions where brackets are not used, such as in 2019 when asked to complete ‘6 + 2 x 2 - ? = 6’, which was answered correctly by 55% of pupils. Also, in 2018, when asked to evaluate ‘92 – 36 ÷ 9’, only 51% of pupils managed to gain the mark. This was also the case in the NFER Tests where pupils were asked to complete a very similar question to the 2018 example used here. The most common error across all levels of ability involved pupils correctly evaluating the square, but continuing to complete the calculation from left to right rather than considering the order of operations.
- Conversion of units
Pupils found several questions in both 2018 and 2019 involving the conversion from one unit to another challenging. However, as many of these questions require multiple steps, it is not immediately clear as to whether the difficulty is related to the problem solving nature of the question or the conversion of units. In 2019, a question requiring pupils to calculate the number of litres of juice which could be poured by a machine in a minute given a particular rate of pouring, was answered correctly by less than half (43%) of pupils. In 2018, two similar questions involving a calculation using both kilograms and grams and a second question involving litres and millilitres were answered slightly more successfully (58% and 55% of pupils respectively).
Findings from the NFER Tests suggest that pupils do struggle with converting units and so this is likely to have caused difficulty for some pupils who were unable to gain full marks for these questions. A question in the NFER Tests required pupils to calculate the number of drinks that could be made from a large bottle. Many pupils - particularly those with middle achievement - gave an answer that suggested they recognised the need to divide one value by the other and that they should then multiply to convert from litres to millilitres. However, many gave an answer that was ten times smaller than required, suggesting that they either used an incorrect conversion by multiplying by 100 rather than 1000, or that they made a place value error during their division.
Questions involving time caused pupils a range of different issues. The most simple of these appeared in the 2019 reasoning paper where pupils were required to recall a fact (from the Year 3 programme of study) regarding how many days were in the months of September, October and November. This was answered correctly by only 53% of pupils.
When calculating with units of time the majority of pupils across the different tests were unable to gain the available marks. In 2018, students were given the length of time taken by a child to complete a race and then information about two other children who had finished earlier and later. They were asked to calculate the finishing times of the other children, both of which involved minutes and seconds being added or subtracted. Only 54% of pupils were able to gain the first mark and 22% were able to gain the second.
A similar question in the NFER Tests also asked pupils to subtract a number of minutes from a given time and, despite having a picture of a clock as a visual aid, only 31% of pupils answered correctly. Although there were a variety of incorrect responses, there was evidence that a common misconception was that there are 100 minutes in an hour, which would be particularly evident if pupils were using the column method. An additional issue arose when pupils were able to carry out a time calculation correctly but did so in the 12-hour format, forfeiting the mark as they did not give the answer in the required 24-hour format. These findings show it is important that pupils have opportunities to revisit areas of the curriculum that are in earlier years’ programmes of study both in order to target misconceptions that may never have been addressed, as well as to maintain a secure understanding of that curriculum area.
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