If you have taught upper KS2 before, you know how much of a slog it can be to teach the long multiplication method.
Despite best intentions, there will always be a few pupils who are either unsure of the simpler 4 by 1-digit approach or who are not secure on their times tables.
If this academic year will be your first time in year 6, you have all of this to look forward to, but don’t despair – it happens every year.
A quick look through the 2019 SATs arithmetic paper shows there are 4 marks up for grabs for getting the long multiplication questions correct, along with many examples of pupils needing to select this method within the two reasoning papers as it would be the most time efficient method to select in order to get through the paper.
Therefore, it is crucial that pupils become fluent in the method. When I say fluent this is what I mean:
‘Fluency is the process of retrieving information from out of long-term memory with no effort on our working memory, freeing up valuable space in our working memory to give attention to other things.
Read more: Fluency, Reasoning and Problem Solving
The Long Multiplication Method In The KS2 National Curriculum
In the maths National Curriculum for England it mentions long multiplication both in year 5 and year 6. In the year 5 objectives for multiplication and division it states that ‘pupils should be taught to multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication for two-digit numbers.’
While in year 6 it says that, ‘pupils should be multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication.’
The appendix at the end of the year group objectives gives us an insight into what this looks like:
This nicely sets out a progression model for teachers after the class are comfortable with multiplying 3 or 4-digit numbers by a 1-digit number.
How Cognitive Science Has Affected My Teaching of Long Multiplication
Long and Short Term Memory
Two lessons from cognitive science have massively changed the way I approach teaching. The first has been understanding that we have a long-term memory that is near limitless in the information that it can store; and working memory, where we make our conscious thoughts.
Important to note is that the space in our working memory is limited, many researchers put it at between 4 or 7 items. Oliver Caviglioli has graciously sketched a wonderful poster that show this process.
From the model we can see that the person uses their attention to take things from the environment into the working memory. We then attempt to encode this information into our long-term memory, but some information may be forgotten for a myriad of reasons.
When that information is in our long-term memory we can bring it back to the forefront of our working memory to use it. If those memories remain dormant for too long however (that is we don’t recall those memories for a long period of time) they too can be forgotten.
Cognitive Load Theory And The Long Multiplication Method
The other lesson from cognitive science that has impacted on my teaching has been that of cognitive load theory. Cognitive load theory attempts to explain why it is that we fail to encode new information from our working memory into our long-term memory.
This could be down to many reasons, such as: the work being too complicated; being covered too quickly; too many distractors in the environment; not having prior knowledge of topic (we will come back to this later) etc.
How does this help us teach the long multiplication method? Well let’s be clear about something first. My outcome for the first lesson or two will be to give my pupils confidence in learning the method. When we look at the second example from the National Curriculum appendix:
These are the steps that will be required to complete it:
- Set the question in the formal method
- Remember to start the process of multiplication with the units
- Multiply 6 by 4
- Write the answer down correctly – including any carrying
- Multiply 6 by 2
- Add anything that you have carried from the previous multiplication.
- Multiply 6 by 2
- Write the answer down correctly
- Drop a zero as we are now multiplying with 10s
- Multiply 2 by 4
- Write the answer correctly
- Multiply 2 by 2
- Write the answer down
- Multiply 2 by 1
- Write the answer correctly
- Add the two answers up together correctly
So that’s a total of 16 steps that children need to become fluent in when learning this new process to get to the final answer. Bearing in mind the limits of our working memory, this is a lot to take on and can quite easily overwhelm it and prevent this information from being encoded.
It has certainly been my experience that those pupils who know their multiplication facts are are fluent in 3 or 4 digit by 1-digit multiplication have an easier time working with these larger numbers.
This makes sense, as if they are fluent in these areas they are effectively reducing what their working memory needs to attend to. Assuming fluency in these two things, what they need to learn is reduced from 16 to 4-6 things.
A child who is not secure in multiplication is likely to use so much of their working memory on solving the multiplication part of the question that all the other steps, as we saw in the model earlier, are forgotten.
This is an important point for teachers to recognise: it’s not that one child has an innate ability to do long multiplication and one child does not. It’s that one child has simply retained the crucial knowledge needed to be successful and therefore can make the connection to prior knowledge to drastically reduce what they need to actively work out.
As Ausubel said, “The most important single factor influencing learning is what the learner already knows. Ascertain this and teach them accordingly”
It goes without saying that if you know a child is not secure in their multiplication facts then you need to stage an intervention to get them up to speed – contrary to opinion learning multiplication facts is important, and while you may be able to teach times tables for instant recall at earlier ages, by upper KS2 it’s very difficult to find the time.
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Building Pupil Confidence In The Long Multiplication Method
That said, there are still things we can do in the classroom to help pupils get to grips with the procedure. As I mentioned earlier, my aim for the first couple of lessons is to build confidence in the method.
To do that, I ensure that the multiplier is 11. By making the second factor 11 all that is required here is to multiply by one. I have yet to come across a child, who may struggle with their multiplication, who doesn’t know the 1 times table.
This significantly reduces the cognitive load on and helps free up all their working memory to learn the procedure of long multiplication. Of course, these pupils will still have to learn their multiplication facts, but this just helps break down those barriers and helps them become successful.
Now all of sudden, the procedure looks like this:
The step-by-step process to solve the problem is the same as the example above but we have dramatically cut down the strain on working memory.
This makes it far more likely that the procedure will be remembered, as pupils can focus all their attention on understanding the procedure and not on the multiplication. Again, I would like to stress that the purpose of this is so pupils can get to grips with the procedure so it can be internalised.
How To Teach The Long Multiplication Method
Step 1 – Establishing what they know already
To start the lesson I would have several 4 by 1-digit questions on the board for the class to make their way through independently, making sure I get around to all the pupils who I believe may struggle at this and ascertain what they are struggling with – is it the multiplication or the procedure? If it was the former I would assist them with their multiplication tables and if it was the latter, I would go through an example with them.
After sufficient time has passed, I would go through the questions on the board to check for understanding both of the procedure and their knowledge of ‘multiplication’:
- What is the multiplicand and multiplier? (i.e. ‘the top number’ and ‘the bottom number’)
- How do I write this in the column method?
- What is the result of ___ multiplied by____?
- What happens if the product is greater than a single digit?
- What place value do I start multiplying at?
Pupils’ responses to these questions will help plan future interventions. In my experience, I have not come across many pupils whose prior-attainment means they cannot set out the column method calculation correctly.
Third Space’s ‘Maths Bootcamp’ series includes a post on how to teach short multiplication at Key Stage 2, for those pupils who may need additional help.
Multiplication Lessons Resource Pack
Plug gaps and help conquer common KS2 multiplication misconceptions with this set of multiplication activities
Step 2 – Introducing the new idea
During this next part of the lesson, I would show an example of the type of question they would be expected to answer by the end of the unit – in this case it would be a 4 by 2-digit multiplication with any digit, using the long multiplication method.
I would very quickly ask them to spend 30 seconds discussing with each other to see what is different about this question than the one that they did at the start of the lesson.
Once they have picked up that there is a double digit number as the multiplier, I would then solve this silently at a normal pace – the reason for this is to show how effortless it can be and give them the confidence that is something that they do not need to struggle with.
I would then show them another example, this time with 11 as the multiplier – this would be on the same slide as the previous example.
I would then ask: ‘Thumbs up for yes, thumbs down for no. Has the way I have set out the calculation in the column method changed when the multiplier has two digits?”
I would then hope to see all thumbs down. If a child has put their thumbs up, I would engage in a whole-class dialogue to see why this is the case and refer to the example that is on the board.
My next step is to write the calculation out in the column method.
My next instruction to the class would be: ‘For the starter, we looked at examples where the multiplier was a one-digit number. That number would be in the ‘ones’ place value. So with the number that is in the ‘ones’ in this two-digit number, we do exactly the same.’
To ensure everyone is participating I would ask them to show me using fingers or mini-whiteboard the answer to the multiplication questions – not because I think they don’t know it but to keep their working memory firmly on the maths at hand.
On the board I now have:
Now we are onto the new piece of information we want pupils to learn, so I would slow down and explain what is happening here, using this moment again to go reinforce place value.
“So far everything that has happened before is not new to us. Now we have a brand new step. To understand what happens we need to activate our knowledge of place value. The first digit in the multiplier is in the ones and it is worth one.
The second digit is in the tens place so it is worth 10. This means we have 10 multiplied by 3. To show that we are multiplying by 10, we can place a zero in the ones place to act as a place holder.”
Then I would write the zero in the correct place.
“We can then multiply the numbers in the multiplicand as if we were multiplying them by 1.”
Next, I would call upon all pupils to solve the multiplication, again showing me on their fingers to ensure participation.
Finally, I would ask pupils to look at the other worked example on the board and to tell their partner what the final step would be –the addition of the two products. The class would do this with me, showing the answers with their fingers/mini-whiteboard.
That will leave us with the finished product of:
Repeat the above process with 2 more examples.
As you go through each example, get the pupils to do more of the explaining, particularly when it comes to the dropping of the zero and reminding one another to add the two products together. If you find children struggling, stop and rehearse this to ensure the correct language is being embedded.
Insist on correct answers in full sentences and correct language. When pupils are unable to do this, I ask for a volunteer who I have picked out who can do this to give a model answer, and then get the original pupils who were unable to answer at first to repeat what was said.
Step 3 – Pupil’s turn
I would then provide two long multiplication questions that I would ask the pupils to complete independently. During this time, I will observe and support as required.
In previous blogs, I have mentioned being aware of learning vs performing and this is no different. Despite hearing pupils give really articulate answers during step 2 or getting both questions right in step 3, I am still very much aware that although these pupils are performing well, nothing has changed in their long-term memory as they are merely repeating what has been shown to them.
Depending on the outcome of step 3, I will either need to: go over more examples and alter my explanations, or continue onto step 4.
Step 4 – Pupil Practice
Happy that pupils are able to copy the process and understand it, I would now provide a worksheet for them to complete. I will not differentiate the worksheet; every child will have equal access to the work.
To differentiate the work sheet would only lead to an increase in the attainment gap. The differentiation will come from additional instruction that I may give during this time.
The worksheet that I would give would not be 20 questions of the same topic. Here I would make use of interleaving. 10 questions of what I have taught would be on the sheet in random order, the other 10 questions would be made up from previous taught content.
Read more about interleaving: 8 Differentiation Strategies You Can Use Across The Attainment Gap
Again, these would be allocated in a random order so that pupils have to switch between what has been taught in this moment and strengthening the retrieval of previously learnt content. This continuous switching helps the encoding process.
Where possible make the content relatable to what has been taught; for example, as I have taught multiplication I would have some division questions from the previous year’s objectives in there to reinforce that division is the inverse of multiplication.
When revising for SATs you may want to interleave long multiplication problems with long division problems to further reinforce the relationship between the two.
The last multiplication question would also have a different multiplier than 11 to see if pupils could apply the process when the demand on working memory is greater.
As this happens, I would be circulating the room to gauge how pupils are doing – not only on the questions from this lesson but previous content too. Pupils are free to skip over questions that they are not sure of.
Step 5 – Shared marking
In this step, pupils will be called on to give answers and the whole class can mark as they hear the answer. If some of them disagree with an answer we can discuss it as a class until the correct answer is found.
Step 6 – Diagnostic question
Diagnostic assessments are an incredibly effective way to see into pupils’ understanding of a concept. They work by posing a question and giving 4 possible answers.
While one answer is correct, the other three distractors will be carefully planned to show a specific misconception. An example of the one I would use in this lesson is below.
Which question shows the correct answer?
In this example A is correct but you can see how each other answer could be an error a child could make:
- In B they have dropped a zero when multiplying by the ones.
- In C they forgot to drop the zero when multiplying by the tens column
- In D they forgot to add on the one that had been carried over when the added 8 to 6.
It is having this selection of incorrect answers that makes diagnostic questions so powerful; they clearly identify what the pupil is thinking, and can provide you with immediate feedback on performance which you can correct based on the answer given.
When doing this in lessons, I assign each letter a number so A=1, B=2 etc which corresponds with the number of fingers I want them to hold up. I then give the command ‘think’. Pupils will think about what the correct answer is.
I will then say ‘hide’ and they will cover the fingers they wish to show on one hand with the other. Finally, I will say ‘show’ and the pupils show me the corresponding finger and I can quickly look around the classroom to see the answers they have given.
The other benefit of diagnostic questions is to discuss through the wrong answers and get the bottom of why they are wrong. These make for fantastic discussion points and really get the class thinking and looking to find the errors.
The Third Space Learning Maths Hub has a large collection of diagnostic assessments available for download.
It worth repeating again that the main aims for this lesson are to build pupil confidence and begin to learn this method of multiplication.
As their confidence grows and the process is embedded further, the multiplier can be changed and reasoning and problem solving questions can be introduced and answered with greater independence.