Year 3 and 4 children s progress in mathematics

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1 1 of 23 Year 3 and 4 children s progress in mathematics Year 3 and 4 children s progress in mathematics Introduction and context Primary schools work hard to ensure that as many of their Year 6 pupils as possible leave primary education with a strong understanding of the key principles of mathematics, a broad range of mathematical skills and the ability to apply their knowledge and understanding in a range of contexts. In order to achieve these aims, it is important that children make good progress in mathematics throughout Key Stage 2. However, many schools report that their tracking of pupil progress shows uneven progress through Key Stage 2, with too many children making slow progress in Years 3 and 4. A milestone used by schools to determine progress at the end of Year 4 is the standard set by the National Curriculum s level 3 attainment target for mathematics. The level descriptions provide broad summaries of attainment and making judgements about a child s progress is a matter of determining if an Attainment Target s description provides a best fit for the assessment evidence that has been collected over the year. It is hoped that schools can draw on the findings from this project to inform the assessment process, to support their planning and to target everyday mathematics provision in order help children overcome barriers to learning and ensure that more children make good progress in mathematics throughout Key Stage 2. The project set out to investigate the mathematical attainment and characteristics of pupils in Years 3 and 4. Information was gathered through assessment activities conducted with children in Years 3 and 4 in schools in two local authorities. The project aimed to answer the following questions: For children who entered Key Stage 2 working at level 2c: How does low attainment on entry into Key Stage 2 impact on progress in Years 3 and 4? For children whose slow rate of progress in Years 3 and 4 suggested that, without targeted intervention, they were unlikely to attain level 3 by the end of Year 4: Are these children struggling with the same aspects of mathematics as children who appear stuck at level 2c at the end of Year 2? What are those common areas of difficulty? For children working at all levels in lower Key Stage 2: Are there aspects of mathematics within the Year 3 and 4 curriculum that appear to present general difficulties to pupils in Years 3 and 4? This report aims to provide information that can be used to identify and support children in overcoming barriers to learning and so make good progress in mathematics in early Key Stage 2 and throughout their later education. The report summarises the key findings from the project relating to the questions above. It also includes some recommendations for actions and strategies that schools might incorporate into their mathematics provision for children: who enter Key Stage 2 working at level 2c; whose progress in early Key Stage 2 is slow; and more generally for all children in Years 3 and 4. In this way, it is hoped that teachers and schools may be able to draw on the findings of this study to help them consider and develop appropriate teaching strategies or intervention approaches.

2 2 of 23 Year 3 and 4 children s progress in mathematics Structure of the report Introduction and context... 1 Summary of findings and recommendations... 2 Related National Strategies resources... 6 Project findings Progress in mathematics in lower Key Stage Overview of the approach taken in the project Appendix Summary of findings and recommendations In this study, Year 3 and Year 4 children were observed working on a range of number-based activities involving level 2 and early level 3 learning objectives. These observations were carried out with children who were making slower than expected progress and who did not appear likely, without targeted intervention, to achieve a secure level 3 by the end of Year 4. For contrast, assessment activities were also carried out with children who had made good progress and appeared to be on track in terms of mathematical attainment. Within each of these categories, the project involved children who had entered Key Stage 2 working at a secure or high level 2 and children who had entered Key Stage 2 stuck at level 2c. (This project incorporates some followup to a Year 2 study, Children who get stuck at level 2C in mathematics, downloadable from The information gathered from these observations was collated and analysed in order to identify areas of mathematics where the children in each category commonly demonstrated secure understanding and areas where they commonly exhibited difficulties. In this section of the report, key findings are given, outlining the areas of difficulty commonly demonstrated by: the children who entered Key Stage 2 working at level 2c the children identified as making slow progress in Years 3 and 4 a large proportion of the Year 3 and 4 children, including those making expected progress. It is hoped that teachers and schools might find it useful to look at these identified areas of difficulty alongside the assessment information for their own pupils in Years 3 and 4, and to identify any common difficulties in learning that might be inhibiting their progress. This should support schools in identifying, clarifying and addressing the main barriers to progress in mathematics for these pupils. The summary of findings outlining key areas of difficulty for each group of children is followed by suggested teaching foci that, where appropriate, teachers might build into their planning and teaching or use to inform any intervention provision to help pupils address the identified difficulties. These recommendations are drawn from a wide range of evidence including information from the study, observation of good practice in Key Stage 2 classrooms, and wider experience of consultancy and support undertaken within primary schools.

3 3 of 23 Year 3 and 4 children s progress in mathematics Summary of findings for children who entered Key Stage 2 working at level 2c One aim of this project was to gather information about how low attainment on entry into Key Stage 2 might impact on children s progress in Years 3 and 4. Evidence from the study showed that the children who entered Key Stage 2 working at level 2c appeared likely, without targeted intervention, to remain working below the level of their peers in certain key areas of mathematics including: counting quickly and accurately in steps other than one understanding place value in whole numbers with three or more digits confidence and accuracy in adding and subtracting 2-digit and 3-digit whole numbers and multiplying and dividing single-digit and 2-digit whole numbers recall and use of number facts recording steps to support calculations involving more than one step. However, they did show age-appropriate attainment in some practical aspects of mathematics and aspects where models and images were used to support understanding, such as: applying understanding of place value to money understanding and using number lines to position and order numbers representing multiplication as an array and using this image to find products. Recommended teaching foci for children who enter Key Stage 2 working at level 2c Plan and implement targeted intervention based on key areas of difficulty identified in the report Children who get stuck at level 2c in mathematics. This could be incorporated into these children s daily mathematics lessons through the use of guided group work, or be delivered as a supplementary programme of support early in Year 3 (or final stage of Year 2). Make the learning and use of number bonds a high priority. Provide a variety of frequent and regular, short and focused activities that involve practice in memorising and recalling number facts. These are organised in ways that help pupils to commit families of related facts to memory. Support children towards becoming more efficient and accurate with calculation. For example, children need to be taught in a structured way how to use number bonds to 10 to bridge across multiples of 10 in order to calculate efficiently rather than relying on counting in ones. Model and teach how to use recording to keep track of the steps to calculations initially using number lines or other jottings and progressing to more efficient methods that develop the skills of recording alongside more formal calculation strategies. Integrate appropriate use of practical resources and contexts, models and images into everyday teaching to support, secure and extend children s understanding of mathematical concepts and processes, gradually removing the scaffolds as children become more confident. Summary of findings for children making slow progress in Years 3 and 4 As part of this project, information was gathered about the mathematical understanding, skills and difficulties demonstrated by children identified as making slow progress in Years 3 and 4. This information was analysed to try to clarify the areas of difficulty commonly experienced by these children and to explore whether these areas of difficulty were similar to those experienced by the children working at a low level in mathematics when entering Key Stage 2.

4 4 of 23 Year 3 and 4 children s progress in mathematics Evidence from the study showed that the children who were making slow progress in mathematics in Years 3 and 4 demonstrated difficulties in many of the same areas of mathematics as those children who entered Year 3 at level 2c. Both groups of children experienced difficulties in using place value in 3-digit numbers, calculating efficiently with 2 and 3-digit numbers and recording steps in their working. In particular, the children making slow progress in Years 3 and 4 commonly demonstrated: an insufficiently robust understanding of place value in 3-digit numbers many children did not fully appreciate the relationships between the place value columns and were unable, for example, to count accurately in tens when moving across a hundreds boundary inefficient calculation methods such as counting in individual steps of ten and steps of one, rather than drawing on knowledge of number bonds, or answering a division calculation by sharing out counters rather than using related multiplication facts limited use of recording to keep track of the steps in their calculations, coupled with inefficient methods, meaning that many children lost track and made errors In addition, the progress of children making slow progress in Years 3 and 4 also appeared to be hampered by difficulty in using the vocabulary and language of mathematics to explain their methods and ideas and to develop their thinking. Recommended teaching foci for children making slow progress in mathematics in Year 3 and 4 Use practical equipment and resources to secure children s understanding of place value in whole numbers with three or more digits, and to secure their understanding of the relationship between the place value columns. For example, children might use bundles of ten and one hundred straws to secure understanding that one hundred is equivalent to 10 tens, two hundreds is equivalent to 20 tens etc and vice versa Use equipment such as base 10 apparatus to support children in developing a broader understanding of partitioning and exchange, for example to explore ways of partitioning the number 234 into , and , and to model the effect on the digits in each column when counting over boundaries in preparation for column arithmetic. Model how to use the value of digits to support explanations, for example when ordering numbers or to explain calculation methods. Think aloud with children, and structure and scaffold activities to give them experience in developing greater precision in the use of the vocabulary of place value. Present children with a blend of practical and mental tasks that are linked to methods of recording to ensure that all children know and can record the key number bonds. Model how to use number bonds to calculate efficiently, moving children from counting in ones to bridging across multiples of 10 and 100 and deriving and using number facts that involve multiples of 10 and a 100. Teach children how to record steps to calculations moving between empty number lines, number sentences and column methods, helping children to explain and compare their recording. Model the use of accurate mathematical vocabulary and associated language, giving children time to listen and repeat what they hear and to rehearse and use mathematical language. Model and promote the accurate use of mathematical language to explain ideas and reasoning and to solve problems, providing children with regular and carefully scaffolded experience in conducting mathematical dialogue with adults and with their peers.

5 5 of 23 Year 3 and 4 children s progress in mathematics Summary of findings for children in Year 3 and 4, including those children making good progress Another of the aims of this project was to find out whether there appeared to be any aspects of the Year 3 and 4 mathematics curriculum that presented difficulties to a large proportion of pupils in Year 3 and 4, whatever level they were working at. Information from the project showed that even children generally making good progress in mathematics commonly demonstrated difficulties with the following aspects of the Year 3 and Year 4 curriculum: Choosing appropriate calculation strategies for addition and subtraction based on the particular numbers involved Children tended to be over-reliant on one method that was not necessarily fit for purpose. Carrying out subtraction calculations such as where the unit digit in the number being subtracted is larger than the unit digit of the start number Children commonly tried to partition both numbers and made errors in dealing with the units. Children make ineffective use of counting up to solve appropriate subtraction calculations, particularly when the numbers were close together such as Understanding the relationships between operations Few children were able to generate two linked addition and two linked subtraction calculations using numbers such as 7, 9 and 16. They struggled to find the missing number in calculations as they were not able to draw on an understanding of the relationship between the inverse operations. This difficulty also applied to linking calculations involving multiplication and division. Multiplying and dividing 2-digit and 3-digit numbers by 10. Children were inaccurate, could not relate this to the place value of the digits and were not able to explain the effect. Recommended teaching foci in order to ensure that all children make progress across the Year 3 and Year 4 curriculum Ensure that children have regular opportunities to rehearse, practise and apply calculation strategies and to select strategies for particular calculations, justifying and refining their choices when the method is not fit for purpose. Review progression in the learning and teaching of subtraction. Many of the common subtraction errors occur when children inappropriately partition both numbers in the calculation. Explain to children that when partitioning for subtraction they should partition the second number only, keeping the first number whole e.g. for work out and then subtract 6 from the answer. Ensure that children understand that when two numbers are close together, counting up from the smaller to the bigger number may be a more efficient way than counting back to work out a subtraction calculation mentally. Demonstrate examples when the method would be inefficient and why this is the case. Plan how to draw on children s practical experience in order to secure understanding of the inverse nature of addition and subtraction, and of multiplication and division. For example, multiplying by 2, 5 and 10 and then dividing the answers respectively by 2, 5 and 10 to confirm the undoing of the operation. Build on this understanding and use to find missing numbers in gap calculations, by exploiting inverse operations where it is helpful. Secure children s understanding of the relationships between the place value of digits set out in columns and the effect on the digits of multiplying by 10 and dividing by 10, using structured resources, alongside models and images that children can interpret and visualise.

6 6 of 23 Year 3 and 4 children s progress in mathematics Related National Strategies resources It is hoped that schools and teachers will find the above information that has come out of this project useful. Of course, schools need to draw on a wide range of information and resources to support them in their ongoing assessment of children to help identify any difficulties in mathematics that can inhibit their progress. These resources might also be useful in planning and informing intervention provision to support children in overcoming these difficulties. The following National Strategies resources contain information that is designed to support schools and teachers in ensuring that as many children as possible make good progress in mathematics in early Key Stage 2. Overcoming barriers in mathematics helping children move from level 2 to level 3 (DCSF: PCK-EN) Securing level 2 in mathematics (DCSF: BKT-EN) Securing level 3 in mathematics (Ref: BKT-EN) What I can do in mathematics level 2 (DCSF: DOC-EN-01) What I can do in mathematics level 3 (DCSF: DOC-EN-02) Supporting children with gaps in their mathematical understanding (DCSF: G) Steps to success in mathematics: Securing progress for all children (DCSF: DVD-EN) Moving on in mathematics: Narrowing the Gaps (DCSF: BKT-EN) Children who get stuck at level 2C in mathematics (DCSF: PDF-EN-02) Project findings Progress in mathematics in lower Key Stage 2 The summary of findings in this report have been synthesized from the collated findings from the project. In this section of the report, greater detail is given, drawn from the collated notes of the consultants involved in the project, about the information gleaned in relation to each of the questions: How does low attainment on entry into Key Stage 2 impact on progress in Years 3 and 4? Are children making slow progress in Years 3 and 4 struggling with the same aspects of mathematics as children who appear stuck at level 2c at the end of Year 2? How do the mathematical characteristics of children making good progress in Years 3 and 4 differ from those of children making slow progress? Which aspects of the Year 3 and 4 mathematics curriculum appear to cause difficulty generally to pupils in Year 3 and 4? How does low attainment on entry into Key Stage 2 impacts on progress in Years 3 and 4? The observations carried out as part of this project included observations of children who had entered Key Stage 2 assessed as working at level 2c as well as observations of children assessed as working at a secure level 2. It was therefore possible to do some comparison of the learning outcomes in Years 3 and 4 for pupils who entered Key Stage 2 working below the expected level of attainment and those who entered Key Stage 2 on track.

7 7 of 23 Year 3 and 4 children s progress in mathematics As might be expected, there were some areas of mathematics in which children who entered Key Stage 2 below the expected level of attainment remained below their peers. These areas included: counting quickly and accurately in steps other than one understanding place value in whole numbers with three or more digits confidence and accuracy in adding and subtracting 2 and 3-digit whole numbers and multiplying and dividing single-digit and 2-digit whole numbers recall and use of number facts and facts relating to shape properties, for example that a regular pentagon has 5 equal sides and 5 equal angles recording steps to support calculations involving more than one step. Children who enter Key Stage 2 below age-appropriate expectation and those on track at the end of Key Stage 2 were equally competent in their: use and understanding of place value when it applied to money understanding and use of number lines understanding of multiplication and division, for example using arrays to establish multiplication facts. This implies that the children entering Key Stage 2 working at level 2c were more able to access mathematics when it was presented using models and images such as number lines and practical contexts such as money. Are children making slow progress in Years 3 and 4 struggling with the same aspects of mathematics as children who get stuck at level 2c at the end of Year 2? This section of the report compares evidence gained through this project with information from the DCSF report Children who get stuck at level 2c in mathematics (DCSF: PDF-EN-02). In the 2C report, five key areas of difficulty commonly experienced by children who appear to get stuck at level 2c in mathematics are identified: Place value and the number system Mental calculation Solving problems Recording methods Understanding and using vocabulary For each of these areas of difficulty, a table has been produced that gives common features of the attainment of the pupils making slow progress in Years 3 and 4 in this project alongside common features of the attainment of the Year 2 pupils working at level 2c involved in the Children who get stuck at level 2c in mathematics project. The information in the tables is drawn from collated observation notes of the consultants involved in the projects. It is hoped that comparison of these two sets of information might help teachers to explore progression in understanding as children move from Key Stage 1 to Key Stage 2 and clarify some aspects of mathematics that continue to form barriers to learning for some children.

8 8 of 23 Year 3 and 4 children s progress in mathematics Place value and the number system Children who get stuck at level 2c at the end of Year 2 were able to count aloud in tens from 0 to 100 though there was some confusion when distinguishing between the ty and teen numbers. However, many children were unable to apply the counting to practical contexts. For example once a group of objects had been arranged into groups of 10, almost all children still needed support to in using this knowledge to count how many objects there were altogether. were not able to recognise and state that there are 52 objects altogether in a set containing 50 objects that were arranged in groups of tens alongside another 2 individual objects. struggled to find the total of a small number of coins in a purse; they found it hard to combine several amounts and did not appreciate the power of starting by counting the value of the 10p coins using their skill of counting in 10s. were, when presented with a test question, all unable to identify that 37 has 3 tens while 60% of the children working at level 2b did so accurately Children making slow progress in Years 3 and 4 were less effective than others at continuing sequences; they particularly struggled when required to count backwards for example counting backwards in threes from 25. were generally able to count on and back in tens from any 2-digit start number (up to 100) and were able to draw on this to support counting in tens with 3-digit numbers. However, they commonly struggled to count accurately over hundreds boundaries. were able to combine multiples of ten and units and say quickly the total number of objects this represented. were less able than other children to apply their understanding of place value in contexts such as money. For example, when looking at 126 sweets organised into 1 bag of 100, 2 packs of 10 and 6 loose sweets, they often struggled to give the total cost if sweets cost 1p each. were insecure with place value in 3-digit numbers. They were generally able to partition numbers into their constituent parts based on experience, though they rarely referred explicitly to the value of each digit or demonstrated understanding of the relationship between columns. had a weaker understanding of the relationships between numbers than those making good progress. They could position a number such as 325 between 300 and 400 on a number line, but did not generally use reasoning, for example comparing it to the halfway value of 350, to position it an appropriate proportion of the way along the line. could not read scales accurately and lacked understanding of the relative value of numbers on a number line. They could not work out the value of each interval on a scale and so could not read information accurately from a graph where the scale did not go up in ones. demonstrated limited understanding of simple fractions such as one third.

9 9 of 23 Year 3 and 4 children s progress in mathematics Mental Calculation Children who get stuck at level 2c at the end of Year 2 were hampered by a lack of understanding of place value and therefore used a very restricted range of simple mental calculation strategies such as counting on in ones. were able to identify the new total when they were to add 1 to a given number but very few children could give the new total when asked to add 10. often resorted to using strategies that involved counting in ones. Where they could count small numbers of objects or fingers, they were generally accurate. When asked to handle bigger numbers, they would continue to count in ones using fingers; this often led to inaccuracy through their inability to keep track of the count. were only beginning to understand and use counting on or back in various steps as a calculation strategy, though they generally continued to return to counting in ones. had very limited understanding of the inverse relationship between addition and subtraction. Over half of the children were unable to explain how to undo an addition operation. Only one child was able to make an accurate link between a subtraction calculation and a known addition fact. could halve small even numbers, such as 8 by using objects or fingers but were not able to recall these facts; they could not halve larger numbers. When presented with a test question, only 14% of children could find half of 60. This compared with 43% of those children working at 2b. had little practical understanding of multiplication and could not apply what they knew to solve simple problems: one boy, who explained that doing tables was one of his favourite things in mathematics was not able to give the total amount of money in a purse that contained three 5p coins. Children making slow progress in Years 3 and 4 generally recognised the importance of the number 10 as the next counting unit after one and consequently counted in tens and ones to add and subtract 2-digit numbers. were commonly able to partition 2 and 3-digit numbers into units and powers of 10, but were not clear how they could draw on this understanding to carry out a calculation and improve the efficiency of the method they used. demonstrated weak recall and use of addition facts to 20. They commonly relied on counting on fingers rather than using number facts to inform their calculations, for example to find the difference between two numbers whose difference fell in that range such as 57 and 64. Were reluctant to record steps when adding and subtracting 2-digit numbers; they often tried to retain a count in their heads which commonly led to inaccuracy. commonly carried out addition and subtraction by counting in individual steps of 10 and 1 rather than adding or subtracting a multiple of 10 and 1. were generally unable to identify and record the 2 addition number sentences and 2 subtraction number sentences when given a number family such as 17, 8 and 25. (This was also true of a surprising number of pupils making good progress.) In their subtraction sentences, the order of numbers was often incorrect. were unable to double and halve 2-digit numbers. (This was also true for a surprising large proportion of the children making good progress in Years 3 and 4.) had poorly developed understanding of the operations of multiplication and division. While they were generally able to respond to informal multiplication and division calculations such as What are six tens? and How many twos make 12? by counting up in equal groups, they often using fingers to keep count. They struggled to relate multiplication (and division) to arrays and were insecure when using and interpreting the x and symbols.

10 10 of 23 Year 3 and 4 children s progress in mathematics were not confident when asked to work with 3-digit numbers; they generally chose to answer calculations involving smaller numbers regardless of the difficulty of the calculations involved such as 24 7, rather than Solving problems Children who get stuck at level 2c at the end of Year 2 were able to identify when to use addition to solve a problem, particularly where it involved combining groups but struggled to identify how to solve problems involving subtraction. were unable to identify a way to work out how many more bricks were in one set than another, even though the bricks were in front of them and they had already counted the bricks in each set. struggled to interpret and find methods to solve unusual problems, for example, on a test question only 44% of 2c children were able to find ways of putting counters onto a grid to make two lines of four counters; all 2b children did this successfully. Children making slow progress in Years 3 and 4 struggled to identify the appropriate operation to solve a word problem. found it difficult to interpret phrases such as How many more...?' when set in practical contexts. This was also true for children who were making good progress. were reluctant to try an alternative approach to solving problems where their initial approach was not successful. Recording methods Children who get stuck at level 2c at the end of Year 2 successfully modelled simple problems using equipment or drawings when this was suggested. However, when asked to record what they had done to solve a problem, most children s immediate response was to attempt to write a number sentence. Addition sentences were generally accurately written, but when writing subtraction sentences many children struggled to write the appropriate numbers in the correct order. in their response to test questions, children were insecure when asked to interpret number sentences with missing numbers. For example, only 29% of 2c children correctly completed the equations = 8 and = 9, compared to 86% of children working at level 2b. Children making slow progress in Years 3 and 4 recorded little or nothing to support their mental calculations involving 2-digit numbers, instead holding the steps in their head. This meant that their calculations were not always accurate as they lost track of the stages they had worked through. demonstrated confusion about the relative positions and values of the digits in 2-digit and 3-digit numbers in subtraction and division number sentences. rarely demonstrated good understanding of the role of the = sign, particularly when it was in a non-standard position within a number sentence (for example, presented as 14 = 46 - rather than 46 = 14). found it difficult to find the missing numbers in gap calculations. This was also true for children making good progress.

11 11 of 23 Year 3 and 4 children s progress in mathematics Understanding and using vocabulary Children who get stuck at level 2c at the end of Year 2 could understand rules, follow instructions and interpret problems that used simple mathematical vocabulary. They found it much more difficult to use appropriate mathematical language to explain their methods, understanding or reasoning. For example, one girl described a rectangle as like a square 4 sides a square goes this way, while another girl tried to explain how she knew that 26 was smaller than is in its twos, 61 is in its sixes. Children making slow progress in Years 3 and 4 struggled to use appropriate mathematical vocabulary and to structure sentences clearly to explain methods and reasoning. could not name or describe properties of common shapes. were not able to interpret the mathematical meaning of terms as more than and difference. This was also true of many children making good progress. From the evidence collected, it is clear that the children interviewed who had made slow progress in Years 3 and 4 had, in the main, overcome most of the specific barriers to learning identified in the research report: Children who get stuck at level 2c in mathematics. In particular: they were able to partition and combine 2-digit numbers and could count in tens accurately involving numbers to 100 most of the children understood the importance of 10s in the addition and subtraction of 2-digit numbers and used this to calculate by counting in tens and ones However, although children making slow progress in Years 3 and 4 had overcome many of the specific barriers to learning that children who appeared stuck at 2c in Year 2 had demonstrated, from the above tables it can be seen that they continued to demonstrate difficulties in the same aspects of mathematical learning, just at a higher level. How do the mathematical characteristics of children making good progress in Years 3 and 4 differ from those of children making slow progress? The researchers analysed the actions of Year 3 and Year 4 children working on a range of number activities, together with their responses to questions and the strategies they drew on to solve problems. The greatest differences between the attainment of children identified as making slow progress in mathematics and the attainment of children identified as making good progress was evident in their understanding of the number system and use of place value. Difficulties in manipulating numbers, using relevant properties and number facts together with place value, limited the ability of children making slow progress to calculate efficiently and to develop and a range of mental calculation strategies. In addition, the evidence collected showed that the children making good progress in mathematics in Years 3 and 4 had acquired particular characteristics and an understanding of mathematics rarely observed in pupils making slow progress. The children making good progress in mathematics in Years 3 and 4 commonly: interpreted and checked answers, applying reasoning to solve unfamiliar problems and working methodically demonstrated persistence they continued to focus on a problem until it was solved, even if this took some time tried different approaches to a problem if their first approach was not successful

12 12 of 23 Year 3 and 4 children s progress in mathematics explained their methods in well-structured, ordered sentences. Which aspects of the Year 3 and 4 mathematics curriculum appear to cause difficulty generally to pupils in Year 3 and 4? There were aspects of mathematics where the attainment of many of the children interviewed was surprisingly low. This was true for children identified as generally making good progress in mathematics in Years 3 and 4 as well as for those children making slow progress. The main areas of difficulty experienced by children involved in the project across different levels or attainment are detailed below. Some details are given to illustrate the particular difficulties children commonly exhibited. These details are from the collated notes of the researchers involved in the project. Choosing appropriate calculation methods Many children relied heavily on one calculation strategy. For some children, this was the use of counting in tens then ones, for others it was partitioning, while some children relied heavily on column written methods. Few children adjusted their strategy to take account of the particular numbers involved. For example, children who tended to use counting in tens then ones to add counted on nine ones when asked to add 9 rather than say adding 10 and then adjusting by one. Children often took longer than necessary to answer a calculation, for example, one able child used a protracted partitioning method to answer : 200 add 100 = 300, add 99 = 399, add 5 = 404, add 30 = 434. While this demonstrated a good understanding of place value it was an inefficient method of calculation. In some cases, it was evident that children were using a mental strategy that they had learnt as a procedure. Using this strategy mechanistically, led to errors when they forgot a step in the procedure and did not have the security of understanding of place value to sort out the method or to find an alternative strategy. For example: to answer , one children recorded = = 70 They gave the answer 78 and even when prompted to check their calculation, the child could not correct the error. Using appropriate calculation methods for subtraction The difficulty some children had in addition, selecting a strategy to take account of the numbers involved, was more wide-scale for subtraction. Interestingly, comparison of the Year 3 and year 4 pupils interviewed showed that Year 3 children were slightly more adept in choosing appropriate subtraction methods than children in Year 4. Even children identified as making good progress in mathematics in Years 3 and 4 were commonly inaccurate when asked to carry out a 2-digit subtraction.

13 13 of 23 Year 3 and 4 children s progress in mathematics Children commonly used partitioning to answer subtraction calculations. Where the unit digit in the number being taken away was larger than the unit digit in the start number, this frequently led to errors. Some examples of the children s responses and the errors children made are included in the table below I did and that equals 40 And then I took away 2 from 4 is 2 so that makes 42 (Applied a method they only partially understood.) I knew that = 30...then = 3 so it s 33 Is it right? (When encouraged, couldn t think of a way to check.) = 10-8 = 2 and 2-2 = 0 (Knew it could not be 0 but did not know how to correct the mistake.) (Had confused addition with subtraction and confidently applied a partitioning approach and took no account of the size of the answer.) Children were more likely to use counting back to answer calculations such as 72 6, than a counting on method to answer calculations such as In some cases, children were over-reliant on column written methods, for example, one child used a vertical decomposition method to work out and wrote the answer as 04. When asked to explain the steps in their method, children could describe the steps in the procedure rather than explain how the processes used the values of the digits in the numbers. Understanding the relationships between the operations Children did not demonstrate good understanding of the inverse relationships between addition and subtraction and between multiplication and division. Very few children were able to give the two addition and two subtraction facts linking three numbers in an addition/subtraction family, for example the numbers 7, 8 and 15. When given a calculation such as , almost all children chose to work it out without looking carefully at the numbers first. Even those children, who arrived at the answer 39, rarely noticed or appreciated that the answer was the same as the initial number. When prompted with Do you notice anything? under half of the children commented that they had got back to the same start number. Only a few children were able to explain why that might be so. Few children were able to identify the multiplication fact they could use to help them to answer a given division question, such as Children often struggled to answer missing number calculations by trial and improvement; they were not able to recognise how an inverse operation could be applied. Surprisingly few children understood that doubling and halving were inverse operations. Using place value to multiply numbers by 10 All children could use multiplication facts to multiply a single-digit number by 10. Few, however, could accurately multiply a larger number by 10. Some errors included: 10 x 25 = sweets at 10p each would cost 20.32

14 14 of 23 Year 3 and 4 children s progress in mathematics Few children could explain the effect of multiplying a number by 10. Most who did get the answer right explained their methods in words such as You just add a zero. Children even made errors when using this rule for example saying: 35 x 10 = 305 When asked to multiply a 3-digit number by 10, an able child partitioned the number, correctly multiplied each part by 10, and then recombined the parts, stating the correct answer without comment. Overview of the approach taken in the project The study was carried out to gather information on the following questions: For children who entered Key Stage 2 working at level 2c: How does low attainment on entry into Key Stage 2 impact on progress in Years 3 and 4? For children whose slow rate of progress in Years 3 and 4 suggested that, without targeted intervention, they were unlikely to attain level 3 by the end of Year 4: Are these children struggling with the same aspects of mathematics as children who appear stuck at level 2c at the end of Year 2? What are the common areas of difficulty? For children working at all levels in lower Key Stage 2: Are there any aspects of mathematics within the Year 3 and 4 curriculum that appear to present general difficulties to pupils in Years 3 and 4? This study was carried out by two consultants. Between them the consultants spent time in 6 schools, across 2 local authorities. The consultants selected schools that were known to them and invited them to take part in the project. The schools were selected so that there was variety in terms of size, catchment and setting. Each school involved was asked to use their tracking and assessment information to identify 8 pupils for observation as part of the study. There were 4 pupils from each of Year 3 and Year 4. The 4 Year 3 pupils represented pupils who had entered: Year 3 working at level 2c and had since made good progress Year 3 working at level 2c and had since made slow progress Year 3 working at level 2b/2a and had since made good progress Year 3 working at level 2b/2a and had since made slow progress The 4 Year 4 pupils represented pupils who had entered: Year 3 working at level 2c and had since made good progress Year 3 working at level 2c and had since made slow progress Year 3 working at level 2b/2a and had since made good progress Year 3 working at level 2b/2a and had since made slow progress Feedback from the observations was discussed with teachers in order to gain wider information about pupils attainment in mathematics and to support the schools involved in their ongoing development of teaching and learning of mathematics in Key Stage 2. Each school was given detailed feedback from observations of their pupils as it was hoped that involvement in the project would inform the ongoing work of their teachers and teaching assistants.

15 15 of 23 Year 3 and 4 children s progress in mathematics The project was based on observations of children working on a range of, mainly number-based, activities. Documentation and guidance was produced in order to support the consultants engaged in carrying out these observations and to ensure quality and consistency of approach, see the Appendix below. The document: Progress in mathematics in lower Key Stage 2 observation notes sheet, takes Assessing Pupil Progress statements for Levels 2 and 3, gives a range of related activities that children might engage with, and some questions for consultants to draw on as they interacted with children during the observations. The sheets contained space for consultants to jot down key observation points that demonstrated children s understanding or difficulties with particular elements of the learning objectives. A copy of this document is included in this report so that teachers might draw on it to support their own observation of children in Year 3 or Year 4, and might incorporate such observation-based activities into their own assessment practices. In total, consultants observed 48 children working on mathematics. As the children worked on the tasks, consultants noted the responses of the children and later engaged them in a structured discussion about what they had been doing, in order to assess the children s understanding and application of number skills. They went on to analyse children s responses and assess the underlying understanding that was revealed for each child. Consultants then collated assessment points identifying common areas of understanding, gaps in understanding, difficulties children had experienced and barriers to learning they had identified. Common elements of the feedback from the consultants involved in the project were identified and form the findings presented in this report. This report is written so that teachers and schools might reflect on commonalities between the findings from the project and the mathematical development of their own pupils in Years 3 and 4. In this way, teachers and schools may be able to draw on these findings to help them consider and develop appropriate teaching strategies or intervention approaches to support children in overcoming key barriers to learning and so make good progress in mathematics in lower Key Stage 2.

16 16 of 23 Year 3 and 4 children s progress in mathematics Appendix Progress in mathematics in lower Key Stage 2 observation notes sheet Notes on using these questions as starting points: questions have been included in this document to assess children s understanding in some important areas of mathematics. You will need to prioritise key areas to assess for each child/pair of children the questions suggested only provide starting points for assessing aspects of mathematics. You will need to adapt questions and add in further questions in the light of children s responses in particular, you should incorporate probing questions throughout the assessment in order to assess the depth and breadth of children s understanding. Main attainment focus: Numbers and the number system Possible activity: Sweet shop (using sweets which come in packs of 10) APP statement(s) Possible questions Child: Child: (Establishing facts) Use mental recall of 10 x table (L3) Begin to understand the place value of each digit (L2) Understand the place value of numbers to 1000 (L3) count sets of objects reliably (L2) begin to use decimal notation in contexts such as money (L3) How many sweets do you think might be in a pack? How many sweets do you think will be in the bag? (each bag to have 10 packs of 10) How could you check? Choose a number card (range of 2 and 3- digit number cards) where you think you could count the right number of sweets to match your number into this bowl. Why did you choose that number of bags...? If each of your sweets costs 1p, how much would you have to pay for them altogether?

17 17 of 23 Year 3 and 4 children s progress in mathematics APP statement(s) Possible questions Child: Child: discuss their work using mathematical language, e.g. with support (L2) discuss their mathematical work and begin to explain their thinking, (L3) Begin to understand the place value of each digit (L2) Understand the place value of numbers to 1000 (L3) Focus on partitioning and recombining I am going to give you a bowl with some sweets in. Count how many sweets there are and write the number on this card. Which of your two numbers is smaller? How do you know? Can you place these two numbers into your order? Try to place the numbers onto this number line. Roughly where would it go? How did you decide? What if I added ten more sweets to your pile? How many sweets would you have now? How did you know that? Keep adding 10 to your number how far can you go? If a shop sold this number of sweets every day for 10 days, how many sweets is this altogether? How do you know? If I have 7 sweets in this hand and 50 in the other, how many sweets have I got altogether? What about 120 and 14? 90 and 52? Can you choose the place value cards you need to make the number 347? What about 208? Can you tell me what number is missing: 729 = 700 +? How much would I have to add to 30 to get the number 52?

18 18 of 23 Year 3 and 4 children s progress in mathematics APP statement(s) Possible questions Child: Child: begin to use halves and quarters (L2) use simple fractions that are several parts of a whole and recognise when two simple fractions are equivalent, (L3) use mental calculation strategies to solve number problems including those involving money and measures (L2) Here are some packs of sweets (can differentiate here). Can you give me half/quarter of them? How did you work this out? What fraction of the sweets have you kept? Here are some packs of sweets. If I take one pack, what fraction of the sweets have I taken? What about if I take two packs? Possible activity: sequences (need two sets of sequence cards including some two missing numbers are at the end, two where in the middle, two standard sequences and two involving negative numbers and one pattern sequence). Also use to assess use of number facts. APP statement(s) Possible questions Child: Child: recognise sequences of numbers, including odd and even numbers (L2) recognise negative numbers in contexts such as temperature Ask each child to pick a sequence where they think they can identify the missing numbers. What numbers do you predict will come next in this sequence? How do you know?

19 19 of 23 Year 3 and 4 children s progress in mathematics APP statement(s) Possible questions Child: Child: (L3) recognise a wider range of sequences (L3) What numbers are missing from this sequence? How do you know? Prompt to particular examples if necessary. predict what comes next in a simple number, shape or spatial pattern or sequence and give reasons for their opinions (L2) review their work and reasoning (L3) Pattern sequence How many squares do you think will be in the next pattern in the sequence? How could you check? Mental calculation (drawing on ideas from: Securing level 3 section) Provide range of calculation problems and ask children to choose examples they can do but will have to think about to work out and explain to you/each other. Encourage to record their methods onto whiteboards. Have resources such as number lines/hundred squares available. APP statement(s) Possible questions Child: Child: choose the appropriate operation when solving addition and subtraction problems (L2) use the knowledge that subtraction is the inverse of addition (L2) use mental recall of addition and subtraction facts to 10 (L2) record their work in writing, (L2) What number is 27 more than 45? Find the difference between 18 and 31 What number sentences can you make using the numbers 7, 15 and 8? Can you give me three other numbers that you can use to make addition and subtraction sentences in this way? Range of additions including 3-digit Roughly how big do you expect the answer to be?