1. Progression in written methods
September 2017
Sally Hewitt

2. Objectives of this seminar
To identify the key steps in progression for the teaching of written methods in KS1 and KS2
To explore the representation of written methods using models and images and concrete resources
To develop trainees ability to apply their subject knowledge to the teaching of written methods
To share problems, contexts and rich tasks to support creativity in the teaching of written methods
To identify areas in which you need to develop your knowledge and understanding of mathematics as a learner yourself and as a teacher.

3. Standards addressed
TS 3 Demonstrate Good Subject and Curriculum Knowledge
Have a secure knowledge of the relevant subject/s and curriculum areas, foster and maintain pupils’ interest in the subject and address misunderstandings
Demonstrate a critical understanding of developments in the subject and curriculum areas and promote the value of scholarship
 If teaching early mathematics, demonstrate a clear understanding of appropriate teaching strategies

4. Research
Haylock, D., & Cockburn, A. (2008). Understanding mathematics for young children
Skemp, 1978; Richard R Skemp; Relational understanding and instrumental understanding. Arithmetic Teacher, 26 (1978), pp. 9–15.
Ofsted (Office for Standards in Education) (2011) Good practice in primary mathematics: evidence from 20 successful schools.
practiceprimary-mathematics-evidence-20-successful- schools

5. Good practice
“ At each stage in developing skills in addition, subtraction, multiplication and division, the schools follow a similar pattern in:
establishing pre-requisite knowledge of the number system such as place value, families of number facts and partitioning
calculating in practical contexts and using hands-on resources such as base-10 materials
developing mental methods supported by jottings and visual images such as number lines
establishing written forms of recording, moving towards more efficient methods over time. ”
Good Practice in Primary Mathematics: Evidence from 20 successful schools (2011)

6. Making sense of mathematics
Pictures
Symbols
Concrete
experiences
Language
Haylock and Cockburn 2008 6

7. Developing mental methods
Explore the resources on your table… thinking back to the lecture, how could you use these resources to provide concrete experiences for children in adding, subtracting, multiplying and dividing?
Some key words….
Chaining, stringing, sharing, grouping, comparing, combining, reducing, partitioning, arrays

8. Representation – use concrete resources to help teach written methods
Remember !
It is good practice to use manipulatives to represent written methods!

9. Using resources to add two digit numbers=
Use diennes to help you solve this addition calculation
Would a place value chart help?

10. =
30+20 = 50
1+4 = 5
= 55
OR VERTICALLY =
30 + 1
20 + 4
= 55
You have partitioned the number into 10s and 1s and recombined

11. Now try
Place value counters
Expanded written method
Compact written method
Partition vertically
40 + 7
70 + 6
= 123 47
1 3
1 1 0
1 2 3
4 7
1 2 3 1
Schools will have their own calculation policies – ask for a copy or where you can find it on day 1!

12. What about subtraction?
Use diennes to work out =
What happens if you try 43 – 25 =
How could you record this?
Now try 425 – 143 using place value counters
and record the expanded and compact methods
diennes v place value counters?

13. Subtraction: decomposition
Common Misconception?
Expanded decomposition
Compact decomposition
Schools will have their own calculation policies –
ask for a copy or where you can find it on day 1!

14. Depth of knowledge
Write these on post its -order them based on depth of knowledge of subtraction formal written methods
762 – – 368
243 –

15. 8 0 6 2 7 4 _________ 2 Common Misconception?
Six subtract four is two. Then you can’t do it, Miss. Seven can’t be taken from nothing!
_________ 2
Children’s Errors in Mathematics, Hansen, 2011

16. Decomposition

17. Try these…
9682
3768 -
2083
1294

18. Written methods for multiplication rely on ability to recall facts
Learning tables… what have you seen?
chanting learning one a week
multiples counting patterns
games trio cards bingo
round the class cards tests 18

19. Importance of knowing facts
Children need to draw on a bank of known facts to derive facts and do written and mental calculation
For example if you know 3 x 7 = 21 you can work out other 'facts' such as 21 ÷ 7 = 3 by using the inverse relationship between x and ÷
3 x 8 = by adding one more 3
6 x 7 = by doubling 3 x 7 = 21
30 x 7 = by using understanding of multiplying by 10
13 x 7 = by partitioning 13 into

20. Using Concrete and Pictorial Representations
Have a go at representing 13 x 4 = ….. Think carefully about how best to organise your counters/diennes 1 10

21. The Grid Method
Can you use manipulatives to represent this multiplication…….diennes or place value counters?

22. Moving from grid to vertical methods
There are 21 spaces on each level of the car park. There are 3 levels. How many spaces are there altogether?

23. What about 25 x 5? Use PV counters Complete the grid method
Complete the compact method
What can you show using the counters?

24. Another example – using concrete resources for regrouping
H T s x
H T s x 1

25. Short written multiplication 38 x 7 =30 8 7
210 56
266 38
× 7
x 7
×7= 56
266
Leads to short multiplication
× 7
2 6 6 5

26. Long Written Multiplication 286 x 29
Estimating
286x29 is approx 300x30=9000 x
200 80 6 20
4000
1600
120 9
1800
720 54
5720
2574
8294
286 x x20= x20= x20= x9= x9= x9=
286
X 29
x 9
7 5
x 20
1 1
8294 1

27. Some vocabulary for division
Divisor
Quotient
Dividend
Remainder

28. Informal recording of division
Concrete resources use and grouped to calculate answers.
Blank number lines created to use as a pictorial resource.
Larger ‘chunks’ of the multiples of the divisor are used on a number line.

29. ‘Chunking’ – Let’s have a go…!
Potential pitfalls…?…

30. 5 6 7 8 The array is an image for division too
Models and images for 56  7
Either:
How many 7s can I see? (grouping)
Or:
If I put these into 7 groups how many in each group? (sharing)
5 6 7 8

31. Use of Representation 1 2 1
363 ÷ 3 =
3 6 3 3

32. Use of Representation
364 ÷ 3 = 1 2
rem 1
3 6 4 3

33. Use of Representation
345 ÷ 3 = 1 1 5
3 4 5 3 1

35. Suggested Methods for Division
Appendix of National Curriculum suggests this method for short division:
What skills and concepts do the children need to have developed in order to carry out short division?

36. Prior Skills and Concepts
Recall fluently multiplication facts to 12 x 12
Recognise multiples
Visualise and understand how a four-digit number can be partitioned and recombined into multiples of 1000, 100, 10 and 1 with both concrete and abstract representations.
Visualise the relative quantity of the numbers.
Know the value of a digit because of its position in a number.
Understand the effect of multiplying by 10, 100 and 1000.
Understand that multiplication and division are inverses and use this relationship to estimate and check answers.
Understand the concept of a remainder after division.
it is more efficient to calculate mentally.

37. Let’s have a go at short division!
81 ÷ 3 =
445 ÷ 4 =
561 ÷ 5 =
723 ÷ 3 =

38. Mathematics Appendix 1, National Curriculum
Long Division
Mathematics Appendix 1, National Curriculum

39. Let’s have a go at long!
195 ÷ 12 =
495 ÷ 11 =
555 ÷ 15 =

40. Pause for thought …
“Ofsted have questioned the validity of ‘standard’ written methods in a number of documents (Ofsted, 1983, 202, 2008, 2011).”
ATM (2014)
“Chunking is more holistic, focusing throughout on the whole calculation and its meaning, rather than just rules for generating successive digits. As a result, rather than just the ability to follow a ritualised procedure. However, children need to have an established, ideally compact, method for written subtraction before using chunking efficiently.”

41. Written or mental methods?
Write each of these on a post it – would you use a written or mental method? Why? Explain to others in your group.
÷ 12 = x 101
220 ÷

42. Reflection… Do you understand the maths involved?
Do you know the prior learning needed for each strategy?
Do you know where children might struggle?
Could you ‘scaffold’ in order to support? If not, this might be something for your Action Plan.

43. Further Reading
Hansen A (2010) Children's Errors in Mathematics: Understanding Common Misconceptions in Primary Schools Exeter: Learning Matters
Haylock, D. & Cockburn, A. (2013) Understanding Mathematics for Young Children (Fourth Edition) London: Sage.
Haylock, D. (2010) Mathematics Explained for Primary Teachers (Fourth Edition) London: Sage
Anghileri, J.(1991) The language of multiplication and division. In Durkin K. and Shire B. (Eds.)Language in Mathematical Education Buckingham: Open University Press
Anghileri, J. (1997) Using counting in multiplication and division. In Ian Thompson (ed.) Teaching and Learning Early Number Open University Press
Anghileri, J. (1999) Issues in teaching multiplication and division. In I. Thompson (ed.) Issues in Teaching Numeracy In Primary Schools. Buckingham: Open University Press
Anghileri, J. (2000) Intuitive approaches, mental strategies and standard algorithms. In J. Anghileri (ed.) Principles and Practices in Arithmetic Teaching. Buckingham: Open University Press
Anghileri, J. (2000) Teaching Number Sense. London: Continuum
Brown, M Number operations. In K. Hart, (ed.) Children’s understanding of mathematics: Windsor: NFER-Nelson 28
Greer, B. (1992) Multiplication and division as models of situations. In D. Grouws Handbook of Research on Mathematics Teaching and Learning. New York: Macmillan Publishing Company