Supporting Numeracy: A Guide for School Support Staff

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Ashley Compton, Helen Fielding & Mike Scott

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    Preface

    This book has been written for people supporting mathematics in primary schools. It is of particular interest to those undertaking a Foundation Degree or pursuing Higher Level Teaching Assistant (HLTA) qualification. The role of the teaching assistant (TA) has grown considerably in recent years and, as a result, much higher levels of skills and knowledge are required to fulfil this role effectively. This book will help to raise your skills and knowledge about supporting mathematics by explaining the underlying concepts and the underpinning research. Common misconceptions are considered, alongside suggestions about how to support children of all abilities. Case studies from practising teaching assistants are used to put this all in context. As well as addressing the primary mathematics curriculum, some wider issues are explored.

    A key feature of this book is the inclusion of research summaries. These should help you to understand the reasoning behind the teaching strategies and help you consider how children learn. However, research articles are often written in very technical language. The summaries are designed to make these more accessible but they are also intended to encourage you to read further about the topic. The references and further reading sections provide suggested texts to deepen your understanding. To use this book effectively it is important to undertake the tasks and reflect on your own experience in the light of what you have read.

    Opportunities for teaching assistants to attain recognition as professionals are increasing as their profile is raised. Part of this process for some is gaining recognition as a Higher Level Teaching Assistant. To do this TAs must meet published standards. This book includes activities for reflection and discussion which are related to these standards. The intention is to provide a framework within which TAs can explore ideas related to these competencies. It should be made clear that fulfilling the activities will hopefully be good preparation for TAs intending to follow the HLTA route, but will not provide a comprehensive route to meeting the standards without other input.

    Regarding the Authors

    The authors have all taught mathematics across the primary age range and currently are senior lecturers in higher education. They teach primary mathematics across a wide range of programmes including undergraduate and postgraduate teaching qualifications, and the Foundation Degree in Educational Studies for Teaching Assistants.

    Exercises, extensions to tasks and other useful information can be found in the appendix section at the back of the book.

    The authors would like to thank the following teaching assistants who so kindly allowed us to include case studies taken from their Foundation Degree assignments:

    • Emma-Jane Bowden
    • Lorna Garfoot
    • Anne-Marie Goode
    • Kerry Hugill
    • Tracy Mountford
    • Lynn Pope
    • Donna Young
    • Patricia Fox
    • Katy Green
    • Samantha Heeley
    • Jacqueline Johnson
    • Tamsin Nash
    • Amanda Wright
  • Appendices

    The appendices are listed by chapter and contain additional information, tables and exercises to supplement the text.

    Appendix 1
    Task 1.1

    Take a few moments to note down where you have used or encountered mathematics over the last 24 hours. This could involve calculations, making estimations, using measurements including time, handling data or working with shapes. Discuss how you have used maths knowledge most with a colleague.

    Some possible areas might include the following:

    • Going to work – reading exact times; estimating time for activities and journey; sorting and matching clothing; packing various shapes into a lunch box or bag; reading scales in the car such as speedometer and fuel gauge.
    • Cooking – measuring capacity, mass and time; diameter; area and volume; estimating; ratio.
    • Shopping – estimating running total; selecting appropriate money; checking change.
    • DIY/sewing – exact measuring of lengths and angles.
    • Decorating – exact measuring; estimating measures; multiplying.
    • Television – extracting data from a table (TV guide).
    • Banking – calculating to check statement.
    Task 1.4

    Reflect on the strategies employed in this activity (see Case study 1.1). Identify those which helped to make this a positive and accessible experience for the children.

    The following are some of the strategies used:

    • Using the skittles put the mathematics into the context of a game. It also made it a kinaesthetic activity which would have been more accessible to some learners.
    • The children were working in small groups so each got to have a turn rolling the ball. The group was small enough to ensure everybody participated in the discussion. The small groups also allowed shyer children to contribute who would not normally speak in a whole-class situation.
    • The TA asked for explanations as well as answers. This told her more about the children's thinking processes but also provided models for the other children.
    • She used open-ended questions to probe the children's thinking and encourage reasoning skills.
    • Allowing the children to teach her their methods raised their self-esteem and confidence.
    Appendix 2
    Task 2.2

    Keep taking turns drawing and describing different arrangements until they are reasonably accurate.

    Most students find they begin quite generally but they –

    • get more and more precise about orientation of paper/base/surface and shapes;
    • begin to use language of measurement;
    • begin to use language of movement and direction.
    Task 2.3

    Make a list of all the mathematical words you can think of that could be confused with everyday words.

    Some potentially confusing mathematical words:

    take awaydifferenceproduct
    timesprimevolume
    facepropertyplane
    kitenetbase
    regulardegreeodd
    evenunitsproper (fraction)
    formulaconstantmatch
    die (singular of dice)signoperation
    changerelationshipplan
    round (to the nearest …)tablerow
    columnaxesorigin
    meanrangemode
    pair/pearsum/somewhole/hole
    Appendix 3
    Task 3.3

    Try these calculations mentally:

    Analyse your own methods. What known facts did you use and what ideas did you apply? Would the children you work with do things the same way?

    The answers to the questions are given below with some of the ways you might have solved them but many other methods are possible.

    • 66 − 29 = 37

      A common way to think about this is compensating so you subtract 66 − 30 and then add 1. You might also use near doubles since 66 is double 33, so the answer is 33 + 4 (the difference between 33 and 29).

    • 66 + 8 = 74

      You might use the known fact that 6 + 8 = 14 and then added this to 60. Alternatively, you could partition the 8 into 4 + 4 to bridge through the nearest 10, so 66 + 4 = 70; 70 + 4 = 74.

    • 309 − 77 = 232

      You might count up using a number line image, bridging through 10s and 100s. 77 + 3 = 80; 80 + 20 = 100; 100 + 200 = 300; 300 + 9 = 309; so the answer is 3 + 20 + 200 + 9 = 232. You might compensate with 307 − 77 and then add 2 or just partition into 300 − 70 + 9 − 7.

    • 309 + 95 = 404

      You might partition and recombine to make 304 + 100 = 404. You could also compensate with 309 − 100 + 5.

    • 4.08 − 1.3 = 2.78

      It is often easier to think of decimals as money: £4.08 – £1.30. You can then count up.

    • 4.08 + 3.1 = 7.18

      Again, try changing this to money: £4.08 + £3.10. This can he partitioned easily into units, tenths and hundredths.

    • 180 ÷ 12 = 15

      If you think of this as a fraction you can remove common factors gradually. It is probably a known fact that 18 and 12 are both in the 6 × table, so 180 and 12 both have 6 as a factor. Dividing numerator and denominator by 6 gives 180/12 = 30/2. This leaves another known fact: 30 ÷ 2 = 15. You might also use known facts such as (12 × 10) + (12 × 5).

    • 25 × 7 = 175

      Known facts are useful: 25 × 4 = 100. You could double this and compensate: 25 × 8 − 25 = 200 − 25 = 175.

    • 9.6 ÷ 0.4 = 24

      Changing the question by multiplying both numbers by 10 gives 96 ÷ 4 which most people find easier. You could think about halving 96 and then halving that answer. You might also recognise that 80 ÷ 4 = 20 and 16 ÷ 4 = 4.

    • 20 × 19 = 380

      You could compensate the near double to 20 × 20 − 20 = 400 − 20 = 380.

      It is likely that children would not do things in the same way as you because you will be bringing past teaching and experience to bear on solving these problems. However, if you have supported maths lessons, especially at Key Stage 2, you are likely to have developed and changed your approach to include some of the taught strategies.

    Task 3.4

    Solve this calculation in your head: (19 + 18) × (72 − 67). Now solve it using a calculator. Analyse the difference; what knowledge, understanding and skills did you need in each case? Which method is more appropriate for you in this particular instance?

    This is a frightening calculation at first sight. Solve the brackets first.

    An easy way to think about this is 37 × 10 ÷ 2 = 370 ÷ 2 = 185. You could also partition the 37 to get (30 × 5) + (7 × 5) = 150 + 35 = 185.

    The way you solve this using a calculator will depend on whether your calculator has brackets or not. If it does then you can just key in the calculation as shown. Otherwise, you need to know that the calculations in the brackets must be done first and either recorded, put into the calculator's memory or put into your own memory before trying the final multiplication.

    Task 3.5

    Ask some Key Stage 2 children to choose whether to do the following questions mentally or with a calculator:

    Look at the way the children use calculators. Do they try mental methods first? Are they always happy to accept the displayed answer? Do they estimate first to ensure the answer is sensible? Are they able to interpret the calculator display to give the answer?

    Most of these questions could be done mentally with a bit of thought and possibly some jottings. Nevertheless, many people would prefer the security of using a calculator for many of them, especially since some of them look frightening initially, like the question in Task 3.4. Was there a pattern to which questions the children were willing to attempt mentally?

    The children may have had some problems interpreting the calculator display with £12.95 + £17.95 and £1.65 × 6 because only one decimal place shows on the calculator. Another problem is £4.25 = 8 which needs to be rounded to 53p or 54p depending on the context. For £5.00 − 95p and £500 ÷ 25p they needed to convert so that both were pounds or pence. With 50p × 3 they may not have recognised that 150p is £1.50.

    Appendix 4
    Task 4.1

    Can you work out how many different multiplication facts a child will learn in their primary school career, if learning their tables up to 10 × 10, including zeros?

    There are several possible answers depending on which facts you count.

    • You could count all of the statements from 0 × 0 to 10 × 10 in a multiplication square or in individual multiplication tables which gives 121 facts (see Figure A4.1).

      Figure A4.1 Full multiplication square.

    • However, because of commutivity 2 × 5 = 5 × 2. Instead of having to learn the whole multiplication square you only need to learn a multiplication right-angled triangle (see Figure A4.2). This reduces the number of facts by nearly half to 66.

      Figure A4.2 Shaded multiplication square accounting for commutative facts.

    • You might reduce this even further by thinking that 0 × anything = 0 and count this as one fact (see Figure A4.3). That brings the total down to 56.

      Figure A4.3 Shaded multiplication square accounting for commutative facts and 0 × anything = 0 as one fact.

    • You could also consider that 1 × anything = itself and count this as one fact (see Figure A4.4). That makes the total 47.

      Figure A4.4 Shaded multiplication square accounting for commutative facts and 0 × anything = 0 and 1 × anything = itself as single facts.

    You can see that by learning a few rules about multiplication you can greatly reduce the number of facts you need to learn. This is particularly useful for children who have difficulty remembering things. Knowing the multiplication facts is important because it makes calculations much simpler. Looking for patterns in the tables (such as the 5 × table always ends in 5 or 0) is another way of helping children learn them.

    Appendix 5
    Task 5.2

    Think about how you show shapes to children. Do you always show them in the same orientation? Do you always show the same kind? Are your triangles always equilateral? Do you ever show very long, thin oblongs? Think of ways in which you can enrich children's experience of shape.

    Ways in which you can enrich children's experience of shape might include:

    • going for a ‘;shape walk’ around the school or outside;
    • devising a maths trail;
    • initiating cross-curricular activities such as origami in art, Rangoli patterns in RE, constructing with shapes in design & technology;
    • playing games such as ‘guess the shape’ by giving properties one by one.
    Task 5.4

    Try this activity with a colleague. Imagine a square (all four sides the same length, all angles 90°). Now imagine a second square, the same size as the first, next to the first so that they share a side (no overlap). What shape do you have now? Does it matter where you put the second square? Bring on a third square the same size so that it shares a common side with one of the others. What shape do you have now? Does it matter where you put it? Now imagine a fourth square the same size as the others and join it to one of the sides of your shape. What shape do you have now?

    See Figure A5.1 for possible arrangements at each stage.

    Figure A5.1 Visualisation of squares.
    Task 5.5

    How would you describe your location at this moment? How many different ways can you find to describe it? Ask some children in the classroom to describe their position. Can you encourage them to think of different ways?

    You might describe your location geographically (at a specific address, in a city, in a country); using prepositions (beside, next to, in front of, behind); using latitude and longitude; using compass points; using measurements.

    Children's understanding of location and position is affected by:

    • not understanding that to describe position you must have a reference point to start with;
    • not having developed an understanding of how to represent 3D in a 2D perspective;
    • representing things as they recognise them rather than as they are actually seeing them, making maps and plans difficult to appreciate.
    Task 5.7

    What are the standard units for these aspects of measures: length, capacity, weight, time, temperature? Find items which can serve as benchmarks for each of these units. What sort of clothes would you wear for 0°C, 10°C, 20°C and 30°C?

    The main metric measures are:

    • Length: millimetres (mm); centimetres (cm); metres (m); kilometres (km) Benchmark: doorhandles are about 1 m above the ground
    • Capacity: millilitres (ml); centilitres (cl); litres (1) Benchmark: a bottle of wine is usually 75 cl or 750 ml
    • Weight: milligrams (mg); grams (g); kilograms (kg); tonnes (t) Benchmark: a bag of sugar is 1 kg
    • Temperature: degrees Celsius; 0°C = freezing point of water; 100°C = boiling point of water
      • Clothing for 0°C: winter coat, scarf, gloves, jumper, hat
      • Clothing for 10°C: jacket, long sleeves or sweatshirt
      • Clothing for 20°C: shirtsleeves
      • Clothing for 30° C: shorts and t-shirt
    • Time: seconds (s); minutes (min); hours (hr); days; decades; years; centuries Time is not metric (which is base 10) but is partially base 60.
    Appendix 6
    Task 6.3

    Think about data handling you have been involved in with children. Which parts of the data cycle were included? Which parts took up the most time? How could the higher-order thinking, the interpreting and drawing conclusions, be emphasised?

    Collecting the data and drawing the graphs take up the most time. Using ICT to present the data can speed things up if the children are familiar with the technology. To ensure time for higher-order thinking you can start the data cycle at the interpreting stage by providing data such as Children's Census information, television guides, train timetables, results from science experiments. The children can then draw conclusions which may lead to new data collection.

    Task 6.4

    Explore some of the difficulties children might have when drawing graphs. Make a checklist of all the things you need to think about when designing a graph, e.g. type of graph, labels, scale. How could you help children to remember these?

    Graph questions:

    • Is the data continuous or discrete?
    • What is the most appropriate type of graph (pictogram, block graph, bar chart, line graph, scattergraph or pie chart)? Why?
    • What are the biggest and smallest numbers? Should you start from 0? What would be an appropriate scale?
    • What is the data about? What should the label on the x-axis/bottom be?
    • What units of measurement are you using? Have you marked this on the y-axis?
    • Does the order of the data matter?
    • What title should your graph have?
    • Would someone understand what this is about if they couldn't talk to you about it?

    Asking these questions of the children as they are working should help them remember all of the steps. You might be able to create a mnemonic with the children for the steps they tend to forget. Notating a completed graph with the important features can also help.

    Appendix 7
    Task 7.5

    Plan some questions from the higher level of the hierarchy for your next mathematics lesson.

    Here are some questions based on the magic square activity in Task 7.6.

    • Hypothesising and predicting:
      • What do you think the total will be?
      • What number do you think should go in the middle? Why?
      • Do you think there will be any patterns?
      • Do you think this will be easy or hard? Why?
    • Designing and comparing procedures:
      • What should we do first?
      • Will any resources help us?
      • How will you keep track of the solutions you've tried already?
    • Interpreting results:
      • What patterns have you found?
      • What does that tell us about adding even and odd numbers?
      • Why is 5 in the middle?
      • How is the total related to the middle number? Why?
    • Applying reasoning:
      • Is it possible to have a different arrangement for a 3 × 3 square?
      • Could you use a different set of numbers in a 3 × 3 square?
      • How can the patterns from the 3 × 3 square help us solve a 5 × 5 square?
      • Will a 4 × 4 square have the same patterns?
    Task 7.6

    Try to complete the magic square (Figure 7.2) using the numbers 1 to 9. If you find this too easy, try some of the extensions suggested [in the text]. Once you have found the solution, think about how you solved it. What questions could you ask the children to help them with the problem-solving process? What resources or models could you use to help them? What would be appropriate questions or clues for children who are getting frustrated?

    See Figure A7.1 for a completed magic square.

    Figure A7.1 Completed magic square.

    Suggested resources:

    • A spreadsheet version of the magic square (see Figure A7.2 for formulae) will allow the children to try out numbers and see immediately whether the totals are all the same. This lets the child focus on the problem-solving rather than the arithmetic.

      Figure A7.2 Formulae for a spreadsheet magic square.

    • Number cards 1 − 9 will let the children try out different solutions without having to record and erase.
    • Cuisenaire rods for 1 to 9, several of each, will help the children see which totals you can make with three different rods.
    • Provide a partially completed square.
    • Give the children a completed 3 × 3 square and ask them to analyse the patterns, then try to make a new arrangement for 3 × 3 or try a 5 × 5 square.

    Suggested questions and clues:

    • Which position in the square is in the most sums?
    • Which positions are in the fewest sums?
    • What should the biggest number be paired with?
    • What do you think the total might be?
    • Why couldn't the total be 6? Or 24?
    • Do you notice anything about the odd and even numbers?
    • 5 goes in the middle.
    • The total is 15.
    • What ways do you know to make 10 with two numbers?

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    Further Reading

    For more information about teaching mathematics and learning mathematics the following titles will be helpful.
    Anghileri, J. (2001) Principles and Practices in Arithmetic Teaching. Buckingham: Open University Press.
    Berger, A., Morris, D. and Portman, J. (2000) Implementing the Numeracy Strategy for Pupils with Learning Difficulties. London: David Fulton.
    Cooke, H. (2001) Passport to Professional Numeracy. London: David Fulton.
    DfES (2004a) Mathematics Module: Induction training for teaching assistants in primary schools. London: DfES. (Ref: DfES/0572/2004)
    Frobisher, L., Monaghan, J., Orton, A., Orton, J., Roper, T. and Threlfall, J. (1999) Learning to Teach Number. Cheltenham: Stanley Thornes.
    Gates, P. (ed.) (2001) Issues in Mathematics Teaching. London: Routledge/Farmer. http://dx.doi.org/10.4324/9780203469934
    Goulding, M. (2004) Learning to Teach Mathematics in the Secondary School. London: Fulton.
    Haylock, D. (2006) Mathematics Explained for Primary Teachers. London: Sage.
    Headington, R. (1997) Supporting Numeracy. London: David Fulton.
    Koshy, V. (1999) Effective Teaching of Numeracy. London: Hodder & Stoughton.
    Mooney, C., Ferrie, L., Fox, S., Hansen, A. and Wrathmell, R. (2000) Primary Mathematics: Knowledge and understanding. Exeter: Learning Matters.
    Nunes, T. and Bryant, P. (1996) Children Learning Mathematics. Oxford: Blackwell.
    Suggate, J., Davis, A. and Goulding, M. (2001) Mathematical Knowledge for Primary Teachers. London: David Fulton.
    Williams, S. and Goodman, S. (2000) Helping Young Children with Maths. London: Hodder & Stoughton.

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