Early Numeracy

Assessment for Teaching and Intervention

Robert J Wright, James Martland, Ann K Stafford

• Publisher: SAGE Publications Ltd
• Year: 2006
• Print ISBN: 9781412910200, 9781412910194
• eBook ISBN: 9781446227145
• Edition: 2
• Topic: Education
• Subtopic: Teaching Mathematics

?Unlike many books based on research work this book doesn?t just let teachers know what is going wrong with children?s learning it actually gives some practical advice about what to do next. The whole book is based upon extensive observation and recording of individual children and their difficulties with mathematics. These children are the most difficult to plan for in a busy classroom and the authors appreciate the small steps and the different learning styles approaches needed for these children. This approach fits well with the NNS wave and springboard materials but takes the analysis of the individual?s difficulties to a more detailed level. The book brings together research carried out across a range of countries and therefore shows the versatility of the approaches taken. This will be a very useful book for trainee teachers as it exemplifies how to use assessment to feed into teaching. It will be helpful for class teachers and mathematics co-ordinators as well as SENCOs to assist in improving the teaching and learning for specific children in our schools? - Mary Briggs, Primary and Early Years PGCE Co-ordinator, Institute of Education, University of Warwick

?This is a highly practical resource that will be appreciated by classroom and specialist teachers alike. It will provide teachers new to the Math Recovery program with practical help and support to diagnose errors and misconceptions in early numeracy. Practicing Math Recovery Specialists will be thrilled with the addition of four new highly focused assessments and an elaboration of the Learning Framework in Number.

Early Numeracy is admirably grounded in international research and well-established theory, characteristics that are much sought after in the current data driven educational environment.

Like many others, I was drawn to Math Recovery after reading the first edition of Early Numeracy. This second edition is a treasure - it is exciting to consider the impact it will have on children and teachers, and to the growth of the Math Recovery program? - Audrey Murray, Lead Teacher, Midwest Math Recovery Training Center, Minneapolis

This text has been fully updated to include developments and refinements brought about by widespread international application of the assessment tools in the Mathematics Recovery Programme. The book will help practitioners to identify and provide detailed analyses of all children but especially those who are able and those who underachieve in early numeracy. It will enable teachers, learning support personnel, numeracy consultants and educational psychologists to advise colleagues and parents on children?s number knowledge and strategies for early numeracy.

The Mathematics Recovery Programme has been successfully applied in Australia, the United Kingdom and Ireland, the United States and Canada, both in specialist interventions and classroom settings. The revised version shows how familiarisation with, and understanding of, the diagnostic assessment tools has allowed teachers to become more knowledgeable in understanding children?s difficulties and misconceptions, and more skilled and confident in planning programmes of intervention and monitoring the children?s progress.

This new edition includes:

- Integrated frameworks of useful tasks for assessing children?s number knowledge and strategies;

- Four separate and revised diagnostic assessment interviews;

- Assessments for addition and subtraction strategies, Base Ten Arithmetical strategies, Early Grouping strategies, and Advanced Grouping strategies in the four operations;

- How the assessment process has impacted significantly on teachers? professional development and contributed to the raising of standards in early numeracy.

1

Children, Numeracy and Mathematics Recovery

Summary

We present the stories of three children in different countries who were struggling with early number. The governmental initiatives to raise standards are discussed but questioned as to whether they would meet the needs of children experiencing difficulty in the bottom 20 per cent of attainment. We show all three children have grown mathematically as a result of a detailed assessment leading to a teaching intervention which possessed clearly defined objectives and distinctive teaching approaches. Not only did each child attain, or surpass, the age-related norm but also as a result of the exercise teachers reported changes in their coverage of the curriculum and in their teaching style. The teachers also professed confidence to advise colleagues and to disseminate their learning and results.

We would like you to meet three young children. Judy is Australian and attends school in New South Wales, Denise lives in the North West of England and Michael is from South Carolina in the USA. Their classrooms are thousands of miles apart, in opposite hemispheres and in different time zones. Though they receive schooling under different educational administrations, Judy, Denise and Michael have several features in common. First, they are approximately the same age, between 6 and 7 years. Secondly, at the time of the interviews all three have been in school for almost a year and a half. They are regular attendees and well adjusted to school. However, they also share a problem. They are all experiencing difficulty with basic numeracy and are already falling behind the expected performance level of their year group.

Judy, Denise and Michael had been assessed in mathematical attainment and placed in the bottom 10 to 25 per cent bracket within their year group.

We know there is ample evidence (Aubrey, 1993; Wright, 1991b; 1994; Young-Loveridge, 1989; 1991) to show that there are significant differences in the numerical knowledge of children when they begin school and that these differences increase as they progress through school. In other words, children who are low attainers at the beginning of schooling tend to remain so, and the gap between them and the average to high attainers tends to increase. Thus the question arises: what will happen to these children if they do not receive additional support in numeracy other than that given by the class teacher as part of their normal duties? Further, since the vast majority of primary and elementary teachers are not specialist teachers of mathematics one might also ask whether the teachers are in a position to remediate the children’s problems even if they know what these are. Since neither Judy, Denise nor Michael is disruptive, rather classified as ‘cooperative’, there is a strong possibility that their difficulties may be overlooked. Let us look at the actual attainment of each child and, though they are in different school systems, see if they experience common difficulties in numeracy lessons.

JUDY

Judy has completed one and a half year’s schooling. Her teacher assessed her knowledge of the number word sequence, the ability to name numerals, and strategies for solving addition and subtraction involving two small collections of counters both of which were covered from view.

Judy showed she had a sound knowledge of the number word sequence in the range of one to ten. She could ‘count’ from one to ten and backwards from ten to one. She could also say the number before or after any number word in the range one to ten. However, she was not able to do this for the numbers in the teens or twenties. When numeral cards were shown to Judy she was able to identify the numerals in the range 1 to 20 but was not able to name many beyond 20.

Judy was given two verbal addition tasks, 5 + 4 and 8 + 3. First, she was presented with five red counters as a group, which were covered, and then four green counters which were placed under a separate screen. She was asked how many counters were there altogether. She successfully solved these tasks but it is revealing to consider the strategy she used in doing so. She invariably counted from one using her fingers or making pointing actions over the screened collections. Her strategy could be described as counting one of the screened collections from one and then continuing her count to include the other screened collection. ‘One, two, three, four, five … six, seven, eight, nine … Nine!’

Judy demonstrates that she possesses one-to-one correspondence and has good counting skills but these are limited to the range one to ten. She identifies numerals up to twenty. She understands the operation of addition but not subtraction. It is important to note that her current number knowledge does not include number facts to ten. She uses a strategy of counting from one to solve additive tasks. Mathematically this is a low-level strategy. Given that she has difficulty with counting in the teens, her strategy is likely to fail her when larger quantities are introduced such as 9 + 6 for example.

DENISE

Denise was 7 years and 1 month at the time of assessment and had been in school for four terms. She could say the number words up to 27 but she had difficulty when saying the number that came after a number when it was above twelve. She also had a lack of fluency when crossing the decade number twenty. Her counting backwards skills were limited to saying the words from ten. When she was asked to start at fifteen and count backwards she said ‘15, 51, 52, 53’. She could say the number word that comes before a number but only if the number was less than ten. When asked to produce the number before a certain number between ten and twenty she always gave the number one more than, even when the task was repeated. For example, she would respond ‘eighteen’ when asked for the number before 17 and ‘twenty-one’ for 20.

Her numeral identification skills were limited to numbers less than 10. She said ‘fourteen’ for 47, ‘fifty-five’ for 15 and ‘thirty-three’ for 13. When Denise was given the same addition problem with the screened counters as Judy, that is, 5 + 4, her strategy was to guess ‘seven’. The task was represented and this time she stated ‘six’. Other than guessing Denise did not have a strategy for solving the addition tasks. The teacher reduced the complexity of the problem by showing her five counters which were then placed under a screen. Two more counters of a different colour were displayed alongside but this time they were not covered. She was asked, ‘How many are there altogether?’ The teacher indicated the total collection by a circular motion of the hand. Denise failed to answer three tasks, 5 + 2, 4 + 4 and 7 + 5, correctly. The level of the tasks was reduced even further by checking if Denise had one-to-one correspondence. First, a linear collection of 13 counters was presented and, secondly, one of 18 counters. Denise successfully demonstrated that she had good one-to-one correspondence in this range.

MICHAEL

Michael could count forwards up to 29 but halted at twenty. He could not say the number that came after a number within the range one to 29. He was not able to count backwards and as a consequence could not give the number which comes before a number. He was shown numeral cards to identify and was proficient in identifying the numbers up to 10. However, he had difficulties with the larger numbers especially the teens. For example, he said ‘twenty’ for 12 and ‘fifty’ for 15. He also failed to identify decade numbers.

When assessed on the additive tasks 5 + 4 and 8 + 3, he portrayed the same behaviours as Denise. He guessed. He was not able to solve tasks involving partially covered collections and he did not appear to be interested in trying to find a solution. He was successful, however, in demonstrating one-to-one correspondence with collections of 13 and then 18 items.

WHAT DO THE THREE HAVE IN COMMON?

We have presented the individual characteristics of the children but what attainments do they have in common? The number word and numerals knowledge and the strategies used for addition give us the clues. Collectively the three children have some proficiency in saying the forward number word sequence into the high twenties but all experience problems with teen and decade numbers. Judy and Denise can count backwards from ten but Michael is lost here. The lack of facility with number word sequences is also shown in the poor skills at saying the number which comes after or before a number word. Judy can identify numerals up to 20 but Denise and Michael’s numeral identification skills are good only in the 1–10 range. Of the three children, Judy shows the most advanced strategy for the addition of two screened collections by the fact that she counts from one to arrive at the total. However, this is still a low-level strategy. Denise and Michael could not solve these problems, not even when they were posed using partially screened collections. But, they did have one-to-one correspondence. Overall, it is clear that they are performing at a low level of numeracy given the standard that is expected for their year group.

Our experience in assessing children in the three countries has led us to believe that Judy, Denise and Michael are not unusual in their level of mathematical knowledge. We are sure that teachers everywhere will readily be able to substitute the name of one, or more, of their children for the above.

The problem is not being neglected. The desire to raise standards of numeracy is a stated national education priority for many nations. Certainly Australia, the USA and England are currently undertaking systemic initiatives in early mathematics and numeracy.

In Australia, for example, the Commonwealth Government worked jointly with the states and territories to establish agreed National Numeracy Benchmarks for 8-year-olds. Commonwealth, state and territory ministers responsible for education have agreed the following literacy and numeracy goals: ‘That every child leaving primary school should be numerate, and be able to read, write and spell at an appropriate level. That every child commencing school from 1998 will achieve a minimum acceptable literacy and numeracy standard within four years’. (Numeracy = Everyone’s Business, p. 3, 1997).

In the past eight years many of the state and territory education systems in Australia, as well as the national education system in New Zealand, have developed and implemented major new programmes in early numeracy. A detailed description of some of the most significant examples of these can be found in Bobis et al. (2005).

In the USA the Standards 2000 Project released the National Council of Teachers of Mathematics updated Standards entitled Principles and Standards for School Mathematics. The publication includes ten standards for children’s learning which span pre-school through to grade 12. The Standards emphasize how learning should grow across the four grade bands – K–2, 3–5, 6–8 and 9–12 – and are seen as statements of criteria for excellence in school mathematics. Standard 1: Number and Operation, for example, states that mathematics instructional programmes should foster the development of number and operation sense so that all students:

• understand numbers, ways of representing numbers, relationships among numbers, and number systems.
• understand the meaning of operations and how they relate to one another.
• use computational tools and strategies fluently and estimate appropriately.

In the UK the government launched the National Numeracy Strategy which set out a Framework for Teaching Mathematics (DfEE, 1999a). The Framework provides a year-on-year list of key objectives that are expected to be attained by the ‘great majority’ of children.

Examination of the published documents for each country reveals a commonality in describing numerate children. The publications indicate the need for children to be confident and competent in working with numbers, money and measurement. Statements, listing the skills, knowledge and understanding to be attained, show that children should be able to calculate accurately and efficiently, both mentally and on paper, and have a sense of the size of a number and where it fits in the number system. They should have a knowledge of number facts and have strategies for solving problems. They should also be able to check the reasonableness of an answer and be able to explain their methods. They should recognize when it is appropriate to use calculators and have the skills to use them effectively.

We cannot disagree with these aims though clearly Judy, Denise and Michael are considerably behind in meeting the targets. Although the aims are laudable it is less clear how the improvement in attainment is to be achieved. How will the aims be translated into action to help Judy, Denise and Michael? The UK government’s National Numeracy Strategy embarked on a rigorous training programme and published guidance on teaching mental calculation strategies (QCA, 1999a), exemplification of key learning objectives (QCA, 1999b), sample lessons (NNP, 1999) and lists of associated mathematical vocabulary (DfEE, 1999b). Let us now examine this initiative in greater detail.

The National Numeracy Strategy’s Framework for Teaching Mathematics (DfEE, 1999a) provided a year-on-year list of key objectives which the ‘great majority’ of children were expected to attain. Each local education authority (LEA) has been given a target for improvement. The LEAs in turn have set targets for individual schools.

The National Numeracy Strategy has also provided schools with a mechanism for attaining the objectives based upon best practice from European and Pacific-rim countries. The mechanism is a structured three-part daily mathematics lesson. The sections are referred to as ‘Introduction’, ‘Main Teaching Activity’ and ‘Plenary’.

A typical lesson commences with five to ten minutes’ oral work and mental calculation with the whole class, which is conducted at a brisk pace. The aim here is to rehearse, sharpen and develop mental and oral skills that may be called upon in the lesson. This is followed by the main teaching activity lasting 30–40 minutes where the teacher works directly on new input with the whole class. They then have the opportunity to split into group work or individual and paired work, thus providing consolidation and differentiation. The lesson concludes with a plenary session where the children explain their work and discuss the efficiency of different solution strategies. It is also the opportunity for the teacher to help the children to assess their developing skills against the targets they have been set and to record their progress, to make links with other work and to set homework.

Even if the target is reached by 75 per cent of the children, by implication 25 per cent are seen to be failing. Initially, prominence in the strategy was placed on whole-class introductions and summaries, with additional emphases on pace, mental agility and articulation of solutions and method. However, by 2004, despite some improvement being evident with underachieving pupils, greater recognition was being given to the fact that a significant number of children needed small-group intervention to accelerate progress and to allow the children to reach age-related expectations. Further some children would need individual provision to tackle fundamental errors and misconceptions that are preventing progress. Judy, Denise and Michael clearly fall into these categories and, if they do not receive help, will continue to experience failure and begin to feel that they are not part of the mathematics community in the classroom.

The National Numeracy standards indicate that children at Year 1 should have the f...

Table of contents

• Cover Page
• Title
• Copyright
• Dedication
• Contents
• List of Figures
• List of Photographs
• List of Tables
• Contributors
• Acknowledgments
• Series preface
• Preface
• Introduction
• 1 Children, Numeracy and Mathematics Recovery
• 2 The Learning Framework in Number
• 3 The LFIN and the Assessment Interview Schedule 1.1
• 4 The Stages of Early Arithmetical Learning (SEAL)
• 5 Identifying the Stages of Early Arithmetical Learning
• 6 Assessment Interview Schedule 1.2
• 7 Part C of the LFIN: Assessment Interview Schedules 2.1 and 2.2
• 8 Assessment Interview Schedules 3.1 and 3.2
• 9 Recording, Coding and Analyzing the Assessment Interview Schedules
• 10 Linking the Assessment to Teaching
• Appendix 1: Assessment Interview Schedule 1.1
• Appendix 2: Assessment Interview Schedule 1.2
• Appendix 3: Assessment Interview Schedule 2.1
• Appendix 4: Assessment Interview Schedule 2.2
• Appendix 5: Assessment Interview Schedule 3.1
• Appendix 6: Assessment Interview Schedule 3.2
• Appendix 7: Learning Framework in Number
• Appendix 8: Instructional Framework for Early Number
• Appendix 9: Linking the Diagnostic Assessment Interviews to the Learning Framework in Number
• Appendix 10: Materials Required
• Appendix 11: The Mathematics Recovery Coding Schedule
• Glossary
• Bibliography
• Index