To be able to extend my understanding of what the children knew in relation to long multiplication, I would strategically encourage the children to solve a long multiplication problem using two, two digit numbers. Here would be an appropriate time to initiate a discussion with the group as to what method the child used and invite the group to give me another method that other members might use, working through the problem step by step. Using open and closed questioning techniques to extend the children’s thinking. Ball (1990 p.5) believes that:
….’From a mathematical standpoint is the fact that opportunities to discuss, share and compare ideas will result in a better understanding of the underlying concepts.’
Moreover, it would give me a clearer insight to any errors and who in the group is not as forthcoming in answering the questions for further probing later in the lesson.
If the children were relevantly confident in using the standard column type of long multiplication, they would have the basic skills necessary to use the grid method (NCITT B6 ii). What would have been needed next, was to clarify their understanding of partitioning, as this is important in the setting out the of the grid methods, along with helping to provide a basis for other topics in the mathematic. Frobisher et al (1999 p.39) comments on, ‘partitioning as a very important tool for mental strategies.’ The same strategy would be used to extract information as to their knowledge of partitioning as before in the previous paragraph.
While planning this lesson I felt it was important to introduce new vocabulary, the term ‘distributive law,’ had previously not been used. (NCITT B6 a vi) On speaking to their teacher, this was appropriate and welcomed. This lesson is planned with an attempt to challenge and provide children with concrete experiences so to build on existing knowledge.
The teaching strategy I engaged in was one of was modelling and demonstrating the grid method. I would explain the step-by-step layers needed to solve a long multiplication problem. Using a white board and coloured pens to emphasise the place value part of the lesson i.e. (tens in one colour, units another colour), I would draw the two types of grids and explain to the children what I was doing and demonstrate how to use these grids to simplify and solve long multiplication problems. It is simpler to represent it in a pictorial way (ELPs framework) for you as a reader to understand. See fig 1
X 30 5
20
4
Answer
Fig 1 the sum 24 X 35 Tabular method
It will be necessary, as you can see, to separate the number similar to partitioning. This will allow the children to carry out multiplication of the tens and units of agreed numbers; the next step would be to gather a total of how many hundreds, ten and units are left. The column can then be added to give a grand total. From this, one can see how it underpins distributive law.
For an extension, those who do not find children this grid challenging enough, I have the gelosia or lattice grid method. This method will again give the children a pictorial strategy when working with the multiplication of decimals later on in the spring term.
See fig 2:
Tens Units
8
Fig 2
- 0
The numbers at the top of the grid are multiplied in turn by those on the side. The answers to each of them will be written in the boxes as shown. The tens value is written in the top of the diagonal boxes, the units in the bottom. The diagonal columns can then are added together and produce an answer.
It is through working with the frameworks of Bruner, Liebeck and the National Numeracy strategy that has enabled me to plan my lesson and help justify my main part of the lesson.
The rational behind teaching this particular aspect of multiplication is to help in the progress of the children’s mathematical thinking and giving them alternative strategies in problem solving.
By using this type of strategy all the children are able to participate in the calculations on the board thus keeping their interest and enthusiasm in helping me to solve problems using different methods. Dunn and Jennings (1998) believe that:
‘There are several different methods to choose from but we select the one that provides continuity with earlier learning….. This tabular form of presenting long multiplication can be used for multiplying brackets in algebra.’
(p.83/84)
The grid method adheres to my key objectives of informal long multiplication using pen and paper. The lesson involved revisiting partition of number, sharpening the children’s applied knowledge for utilization under different circumstances.
My lesson promotes discussion about a range of methods and size of number, how to multiply by tens, discussed about where to put the numbers and in some cases made their own assumptions based on their own experiences. Wells (1992, p. 287) draws your attention to the value of discussion and how through it, can facilitate learning in a collaborative situation:
Discourse is itself a form of action. For in producing and responding to the linked and reciprocally related moves that make upon a sequence of discourse, participants are able to act on each other, guiding and influencing each other’s understanding of, and involvement in, their joint endeavour.
The children during this lesson, learnt to break down a more complex problem or calculation into simpler steps before attempting a solution using different methods. Through discussion the children were able to share their thought about these methods and talk about what it was that they found difficult.
By revisiting the partitioning method and column multiplication algorithms, I was able to cover much of my objectives. In addition it gave me the opportunity to check the children conceptual understanding of place value and multiplication tables, ready for the class teacher to move on to multiplying decimal fractions in future lessons.
Ben offered his hand in the initial stages of the lesson, when we were looking at the column algorithm and, began to give me his approach for this type of sum. His step-by-step account was slow but correct in his thinking. I noticed though, that he had a problem with the carrying over to the tens/hundreds column. I used this opportunity and asked him if he could spot his mistake. Puzzled and searching he eventually saw it and corrected it. In a later conversation with him he confessed,
‘Mrs Ryan, I get a little confused with the carrying bit.’
I hoped that this grid method would help Ben and elevate his problem with his carrying. However, his problems could be spotted throughout the addition of both column and partition algorithms and so although this grid method would help in his long multiplication, a different strategy should be sought to help him in the other algorithms.
For the degree of my lesson I would demonstrate the grid method, I engaged in discussions with the children asking open questions and encouraging them to offer answers to the calculations and predict what would happen next as I worked through the tabular grid.
‘Now I have multiplied these two number together, what should I do next?
Joy this time offered her hand, ‘um, Miss I think you have to multiply it by the next number in that row.’
To which I replied, ‘with which number Joy?’
Louise shouted out, ‘the same number Miss.’
This gave me a clear indication that they had some understanding of what was happening using the grid method.
I gave further example of these grids a number of times. Each time involving them in the calculations until I felt they were confident to try some on their own.
After several demonstrations I chose pupils whose contribution to the discussion was less frequent than others and who struggled with the column multiplication. I asked them to solve one, using the grid method, to ensure they had a good understanding.
Some of the concepts under consideration (particularly the gelosia/lattice grid method) did at first seem too abstract and beyond the immediate experience of the children. However, the abstract is made concrete using Liebeck's ELPS (hierarchical approach (Liebeck, 1984). Their understanding may be refined and extended by the use of pictorial or diagrammatic representations. Such cognitive mapping - converting the verbal into the visually memorable - helps children organise what they know and create new patterns of understanding. For this reason much use is made of pictorial worksheets to encourage sequencing and understanding.
One part of the children’s mathematical thinking, involved making a connection between the facts and skills to the concepts needed to use the grid. The children had to use mathematical thinking by how they organised their numbers when placing them in the grid. They had to understand partitioning as away of breaking down the number but still being able to recognise the number has the same value, i.e. 25 x 24= (20 + 5) x (20 + 4). The digits needed to be underneath one another in an orderly fashion. This would help them to minimise the errors when totalling the number together when reaching a final answer.
Whenever possible, children should be taught how, to achieve something as well as what is necessary to achieve it, to learn. Consequently, children are encouraged (via discussion) to think for themselves (via cognitive coaching) and to learn from each other (co-operative learning) during lessons.
The first child I chose to assess was in the process of starting her first grid sheet, (appendix 4). She and her friend had rolled the dice and generated a number 25 x14. Her first error on the sheet can be seen as a simple mistake, she had copied down the number incorrectly and had written 24 x 14. Her second error was a misconception; she multiplied 10x4, 4 x 4 and 4x20 and placed the answers in the wrong boxes.
‘That’s interesting Louise, show me how you did that?
‘Did what Miss?’
‘How you got 16 in that box?’
Well I multiplied that 4 by that 4 there see!’
‘Ah I see!’
‘Well how did you get the 8 in that box?’
‘Oh yeah?’
Louise was confused and needed a further explanation on how to work the grid method and how the distributive law works in this instance. I rectified this with a thorough explanation working the grid with her so she had a better understanding. Further grids on the worksheet could be seen, with the correct thinking and methods used.
The lattice method on the worksheet 1 Louise found the diagonal addition too confusing as you can see he attempted it but left it only half finished. She was also unclear as to what goes where despite explaining this method many times.
The second child Jack, made some simple mistakes, where in the first grid he neglected to write 30 in the box and wrote only 3 yet in the left hand column he correctly wrote the two digit 10 and did not see his mistake. He has misconceptions with place value and multiplying by 10 and lacks factual skill in multiplication tables. I saw this grid while checking on the children and decided that a few might have the same problem. So I recapped the tabular method again on the board to give him a prompt. I was pleased to see that he at least corrected his original error. But was disappointed that he still had not grasped the multiplying by tens, hundreds and thousands. I would plan extra activities with Jack to work on simpler place value, tables. I am aware of his ability through observing him in other activities and think it is a case of revisiting these areas to make them concrete in his thinking.
In Jacks assessment, the pictorial representation of the ELPs framework was not what he needed. I would have liked further time with him to diagnose his preferred learning style and used evidence from that, to plan an effective lesson for him regarding this.
Word count 2729
Askew, M. (1998) Teaching Primary Mathematics A guide for newly qualified and student teachers, London, Hodder & Stoughton Educational.
Ball, G., (1990) Mathematical discussion (Chapters1-3, pages 3-21), Talking & Learning: Primary Maths for the National Curriculum, Blackwell Education
Devereux, K (1982) Understanding learning Difficulties, Milton Keynes, Open University Press.
Durkin, K., Shire, B., Language in Mathematical Education Open University Press, Buckingham
Fisher, R. (1995). Teaching children to learn. Cheltenham: Stanley Thornes (Publishers) Ltd
Frobisher, L. Monaghan, J. Orton J. Roper, T. Threlfall, J (1999) Learing to Teach Number, Cheltenham, Stanley Thorne publishers Ltd.
Hayes, D. (1998). Foundations of Primary Teaching, 2nd Ed, London, David Fulton Publishers Limited.
Jennings, S & Dunne, R. (1997). Mathematic for Primary Teacher, London, Letts Educational.
Liebeck, P. (1984). How children learn mathematics; a guide for parents and teachers. London: Penguin Books Ltd
Littledyke, M & Huxford, L (1998) Teaching the Primary Curriculum for Constructive Learning. London, David Fulton Publishers Ltd.
QCA (1999). The National Curriculum, The Department of Education.
QCA (1999). The National Numeracy Strategy, The Department of Education.
Anghileri, J., (1995) Primary Mathematic Education Reader, University of Gloucestershire.
HMI, (1985) Primary Mathematic Education Reader, University of Gloucestershire. (2002/3)
Appendix 1
-
Using and applying number
1) Pupils should be taught to:
Problem solving
- Break down a more complex problem or calculation into simpler steps before attempting a solution; identify the information needed to carry out the tasks
- Find different ways of approaching a problem in order to overcome any difficulties.
f) Organise work and refine ways of recording
h) Present and interpret solutions in the context of the problem
i) communicate mathematically, including the use of precise mathematical language
- Search for pattern in their results; develop logical thinking and explain their reasoning.
-
Solving numerical problems
4) Pupils should be taught to:
- Choose and use an appropriate way to calculate and explain their methods and reasoning
National Numeracy Strategy
Key Objectives
Year 5
Section 2 page 5
Bullet point 10
Carry out long multiplication of a two-digit by a two-digit integer
Section 3
Pages 26 66-69 section 6 supplement of examples pg 67 outcome year six
