kids studying maths

In this blog you will discover how to nurture children with strong numeracy skills. These are skills that all learners must improve on in order to develop a strong sense of numbers and number processes.

But, first let’s clear up the definition of ‘sums’?

Strategies not “SUMS”

“Tell me and I forget, teach me and I may remember, involve me and I learn.” Benjamin Franklin

What are ‘sums’?

The word sum or sums is derived from the word summation, which means to add 2 or more numbers.

Simple.

However, a bit like a lot of things words can lose their original meaning and now ‘sums’ often gets used to describe the following mathematical calculations:

Questions

378 + 567 

415 – 279

135 ÷ 3

146 x 6

 

The ‘Sums’:

Sums

These get called a lot of things: (no name calling)

  • ‘chimney sums’,
  • ‘upstairs downstairs’
  • ‘bus stop division’

They are in fact to give them their proper name ‘algorithms’.

Algorithms allow calculations to be performed in an efficient procedure to get the correct answer.

Why do we teach ‘Algorithms’?

We teach them because …….well ….. we’ve always taught them!! And so we should always keep teaching them – right?

I think they should be banned!!

Yes you heard me right (well read it right).

What is the real need for them in our society? 

If someone asked you what the answer is to £7.89 multiplied by 6 you could do 1 of 2 things.

Possibility 1

  • Round £7.89 up to £8
  • Multiply £8 by 6 to get £48
  • That gives you an estimated answer.

Simple.

You could be clever and take off the 11 pence six times so the answer is exactly = £47.34

Possibility 2

Otherwise you would get out your smart phone type in the calculation and get £47.34

Why do ‘algorithms’ exist?

So why are these taught in schools and why do they exist.

They exist as they are a great way and were a great way to do much bigger calculations like

53,647 x 7

or

345,786 + 968,453

They were great for doing very big addition problems like a Census back in the 1600s!

We had no calculators and we had to add very big numbers efficiently.

Office clerks would perform these very big algorithms, to add very big numbers and with a very big number of numbers.

And they worked.

They were the most efficient method at the time, for big numbers that is.

But now we have computers and spreadsheets that do these instantly.

What’s the problem now?

Now the problem is these are being used to work out simple calculations like

99 + 16

And if a child (or an adult) reverts to a “chimney sum” for this, and they do, then maths education is FAILING our nation, our world!

A lot of young learners could quite easily tell you the answer to this question as they have a strong sense of number and number processes.

Some learners cannot.

Some learners will want to use a “chimney sum”

Why does a 10-year-old choose to use a chimney sum?

Learners want to use a chimney sum for a variety of reasons:

  • Reason 1

They have become the norm for every addition problem since they were taught them in P4.

  • Reason 2

They have not had enough time or developed enough confidence in learning strategies that they understand.

  • Reason 3

They feel safer using this method! People like this strategy, they get the answer correct, they feel happy, they experience success.

Some teachers will argue that then this helps develop certain pupils’ confidence.

They are entitled to their opinion. I am not going to argue with a class teacher as they know their individuals better than I do.

The long-term impact.

There is however a long-term negative impact on teaching algorithms too early.

This becomes the ‘go to method’ for a lot of learners. They learn the rules and procedures to get the ‘right’ answer rather than understanding a strategy.

Algorithms take away the excitement in maths. They take away the investigation, the exploration, the discovery, the engagement, the interactive nature of learning and they deprive learners the opportunity to experience those ‘light bulb’ or the ‘Eureka’ moments in maths. 

…the Euerka! moment is when that sudden flash of inspirisation or realisation happens, as it did to Archimedes.

Learners become scared to try out new strategies in maths as there is a fear of failure, so they then stick to the safe algorithm.

Mindset and Resilience!

Mindset and resilience are key attributes in maths.

I have personally witnessed several pupils coming into secondary school (12 years of age) at the top end of the class but totally scared to lose this ‘tag’ of maths wiz.

They believe they are naturally good at maths and therefore they do not need to try. They do not like it when maths gets challenging or they need to think about math problems.

This is true for all learners. And this is the issue.

A learner who is perceived to be below average development with maths has these same issues. They develop the same fears over maths. They may find success with an algorithm and this becomes a ‘crutch’.

They are reluctant to engage in any new strategy which may help make stronger connections in their neural pathways, but their minds remain partially closed.

Not everyone develops numeracy skills at the same time and pace.

Learners must be allowed to move at their own pace. As parents and teachers need to worry less about comparing your child’s stage of development to other children. As parents we often have immense pride when our child excels in an area of the curriculum.

However quite often parents can feel disappointed, stressed or perhaps even embarrassed about our Childs’ development if it is below average, especially with numeracy or literacy.

Let us look at the development of a child from infancy.

Some learn to walk at 10 months some learn to walk when they are 2 years old.

Some say their first word at 10 months some children don’t speak until they are 3 years old.

Although there are exceptional circumstances where expert intervention may be needed most children learn to crawl, talk and walk when they are ready.

How can we improve standards of all?

Unless you understand the Arabic number system then this will statement below will mean nothing to you:

arabic number addition

The problem with maths is that it can really be an abstract subject.

It is up to teachers and educators to bring maths alive using concrete materials and pictorial/visual resources.

Allow and provide ample opportunities for learner to make connections rather than going too quickly on to the recital of abstract maths facts such as:

7 + 3 = 10

The abstract part ( 7 + 3 = 10 ) should be the final piece in the jigsaw only after investigating, playing and understanding is established.

Appropriate tasks for age, stage and development.

missing addend question

A friend of mine called me a while ago to see if I could help his daughter in P3 with this question set for homework.

I asked the girl what the answer to the question below was:      

After a quick discussion it was clear the young girl did not have a strategy that she was confident in using.

I then asked the young girl to answer this question:

Uncertainty and still a lack of confidence here. She got the correct answer using her fingers.

However, she did not have confidence or security with single digit addition.

 

Yet her homework task was to find the missing number in an addition question with two 2-digit numbers.

That is how maths anxiety, stress and mental blocks with maths begin to manifest.

What supports and scaffolding can we do to support development of this type of question

missing addend question

We could represent the numbers 38 and 63 using cubes or counters.

I am going to represent it visually within 10 frames and use this to help me determine the missing number in the addition problem.

38 is made up of 3 TENS and 8 ONES

63 is made up of 6 TENS and 3 ONES

To get from 38 to 63 we had to add 2 TENS and 5 ONES in total.

This type of learning can really help a learner who struggles to grasp the abstract nature of the problem and bring it alive.

Seeing is believing.

In this case seeing in understanding.

Why did I say ban algorithms earlier?

Maybe ban is a bit strong. Maybe they can still serve a purpose. Maybe!

But I will need convincing that they can help develop mathematical thinking.

My point is if you had never taught an algorithm, I do not think any learner would come up with a chimney sum to do any of the following calculations:

35 + 24

58 + 27

98 + 17

ALGORITHMS are not a natural way of thinking!

Let us focus on the question 98 + 17

Here are a few popular strategies and methods:

All these strategies require thought!

They require your brain to be active!

They ensure your brain is making connections, it is firing up, it is developing and strengthening connections each time it does a calculation like this.

A quick look at division strategies to help convince you about strategy vs procedures

Ask a 10-year-old how they would do:

52 ÷ 4

If they do not know the answer a series of question like this may help them understand the answer.

40 ÷ 4

44 ÷ 4

48 ÷ 4

52 ÷ 4

I like to focus on this strategy of building on from what we do know.

Let’s talk about key numbers when dividing.

  • 30 is a key number when dividing by 3
  • 40 is a key number when dividing by 4
  • 70 is a key number when dividing by 7
  • 130 is a key number when dividing by 13

What strategy do I investigate with my 9 year old son?

I focus on this key number strategy stated above.

51 ÷ 3

We would say 30 is a key number so let us split the:

51 up into 30 and 21 and divide both separately

30 ÷ 3 = 10

21 ÷ 3 = 7

SO

51 ÷ 3

= 10 + 7 

= 17

In conclusion: Strategies that develop understanding will stand the test of time

“Tell me and I forget, teach me and I may remember, involve me and I learn.” Benjamin Franklin

That quote sums up the whole point of this blog – that is why I opened the blog with it.

Learning is about discovery. It is about those light bulb moments. It is about engaging children in the fun, in the investigation and using the resources that allow that understanding to take place.

We need to allow time for this to take place on an individual basis.

Never stop using the concrete and pictorial/visual resources to aid understanding.

There are 10 years olds out there who still have not developed confidence in strategy for a question like:

9 + 6 

Are they still getting the opportunity to use counters and visual resources?

We may never get it right for every single learner, but we must always believe we can, and we must always strive to achieve this.

Sometime in education we might not see the penny drop. We may not see that light bulb moment happen. But rest assured if you are teaching strategies that develop that conceptual understanding there is an extremely high probability the learning will click.

Next year’s teacher might see that light bulb moment happen live, but your efforts in the previous year will have laid the foundation.

I will leave you with this quote.

“Plant a tree you will never see”