This should be more accurately titled “How My Children Were Taught Numeracy by Their Primary School”. I know that different schools have different strategies and everything is under constant review. However, I think it is true to say that all primary schools use techniques that were not taught to the parents of today when they were at school. I’ve used my children’s experience to

- Understand some of the new techniques
- Understand why numeracy teaching has changed

## Why Has Numeracy Teaching Changed?

When I was a lad, we were taught how to count, add up and take away. We learned our times tables and then we tackled long multiplication and division. That’s how I seem to remember it — but it was a few years ago. Now there appears to be a bewildering range of additional techniques; number lines, counting on, number squares, number bonds, chunking, the grid method and the lattice method to name a few. When my children were taught using these methods, I couldn’t understand the logic. Why are all these additional techniques necessary?

Fortunately my local primary school sent parents “A guide for parents on how addition, subtraction, multiplication and division are taught”. After reading this guide, I began to understand not only the techniques but also the logic of teaching a range of different methods to achieve the same end. This is how the four rules (addition, subtraction, multiplication and division) are taught:

## Addition

**Counting On**: Hold one number in your head and count on.

**Using a Known Fact**: Rapid response to a fact known by heart, for example

**Number Bonds**: Learn pairs of numbers that make 10 or 20 and recall them immediately.

**Using a Derived Fact**: Using a known fact to work out a new one, for example if 15 + 5 = 20 (number bond) then 15 + 6 = 21.

**Hundred Squares:** Use knowledge of multiples of ten to add, for example 30 + 50 or 32 + 50. Number squares maybe used.

**Adding Several Numbers**: A number of techniques; for example — starting with the largest number or looking for pairs of numbers that make 10.

**Partitioning and Recombining:** Splitting numbers up. Add the ‘tens’ first and then add the ‘units’.

**For example:**

46 + 63 = (40 + 60) + (6 + 3) = 100 + 9 = 109

**Counting on in Multiples of 100, 10 or 1:** Start with the largest number and count on by partitioning the second number and adding the tens and then the units. Even the units can be split to make it easier.

**For example:**

76 + 47 = 76 + 40 + 7

**Adding Significant Digits First:** This is an extension of partitioning; add hundreds, then tens and then units.

**For example:**

Although this is set out formally in columns, children are expected to develop mental strategies to do these sums in their heads.

**Adding Near Multiples of 10:** Using the fact that to add 9 is the same as to add 10 and then subtract 1. It’s a mental strategy, for example:

53 + 9 = (53 + 10) — 1 = 63 — 1 = 62

76 + 39 = (76 + 40) — 1 = 116 — 1 = 115

328 + 199 = (328 + 200) — 1 = 528 — 1 = 527

**Compensation:** This builds on ‘adding near multiples of 10′. Add too much and then subtract the difference.

**For example:**

754 + 86 = (754 + 100) — 14 = 854 — 14 = 840

367 + 92 = (367 + 100) — 8 = 467 — 8 = 459

**Standard Written Methods with Exchanging:** Finally, the standard method I remember from school – carefully writing in columns so that units, tens, hundreds etc. line up and carrying below the line:

## Subtraction

**Counting Out:** To find the answer to 9 — 3, hold up 9 fingers and fold down three.

**Counting Back From**: To find the answer to 9 — 3 count back three numbers from 9.

**Counting Back To**: Count back from the largest number to the lowest.

**Number Bonds**: If the number bonds to 1o and 20 are known (see addition, above), these facts can be used for subtraction.

**For example:**

If 6 + 4 = 10 then 10 – 4 = 6.

**Using Derived Facts**: If it’s known (from number bonds) that 20 — 7 = 13 then 21 — 7 must be 14.

**Partitioning into Tens and Units:** Splitting numbers up into tens and units

**For example:**

47 — 24 = (40 and 7) — (20 and 4)

40 — 20 = 20

7 — 4 = 3

20 + 3 =23

A variation is to only partition the number to be subtracted:

47 — 24 = 47 — 20 — 4 = 27 –4 = 23

Partitioning can be extended to thousands and decimals.

**Counting Up**: Subtraction by addition! Start with lowest of the two numbers and count up to the highest.

Best explained with an example:

132 — 56

Then just add 70 + 4 + 2 = 76.

This method can be used with larger numbers and decimals.

**Subtracting Near Multiples of 10**: Subtracting by 9 is the same as subtracting by 10 and adding 1.

54 — 9 = (54 — 10) + 1 = 44 + 1 = 45

76 — 39 = (76 — 40) + 1 = 36 + 1 = 37

438 — 199 = (438 — 200) + 1 = 238 + 1 = 239

**Same Difference:** Add or subtract an amount to make one of the numbers more manageable *as long as you do the same to both* numbers. Best explained with some examples:

**83 — 47 **– it’s easier to subtract 50 so add three to both numbers – the difference is the same:

83 – 47 = 86 – 50 =36

**Negative Numbers**: A method to avoid “borrowing” (see below) by partitioning the numbers in the sum:

**Borrowing / Traditional Method**: This is the method I remember from school.

This is best explained by seeing all the workings, step by step. This video from the Khan Academy is great. It not only shows how to use the borrowing method — it also starts by breaking it down and showing the logic that lies behind it.

## Multiplication

**Repeated Addition**: This method shows that 4 x 3 = 3 + 3 + 3 + 3

**Doubling**: First double the tens, then the units and add the two answers together.

**For example:**

47 x 2 = (40 x 2) + (7 x 2) = 80 + 14 = 94

**Multiplying by Multiples of 10**: Digits move one place to the left when multiplying by 10, two places to the left when multiplying by 100 and three places to the left when multiplying by 1000.

**For example:**

9 x 10 = 90

9 x 100 = 900

0.3 x 1000 = 300

6 x 30 = 6 x 3 x 10 = 18 x 10 = 180

**Multiply by 4, 5, 8, 20 and 25**

To multiply by 4, double and double again.

To multiply by 5, multiply by 10 and then halve.

To multiply by 20, multiply by 10 and then double (or vice versa)

To multiply by 8, multiply by 4 and double.

To multiply by 25, multiply by 100, halve and then halve again.

**Multiplying by Near Multiples of 10**: To multiply by 9 is the same as multiplying by 10 and then subtracting the number you multiplied.

**For example:**

15 x 9 = (15 x 10) — 15 = 150 –15 = 135

13 x 29 = (13 x 30) — 13 = 390 — 13 = 377

**Multiplying using Factors**: Numbers that appear complicated can often be broken down into more manageable numbers by finding their factors.

**For example:**

35 x 2235 = 7 x 5

22 = 2 x 11

so 35 x 22 = 7 x 5 x 2 x 11

Then look for easy multiplications within the new sum:

7 x 5 x 2 x 11 = 7 x 11 x 5 x 2 = 77 x 10 = 770

so 35 x 22 = 770

**Multiplying by Doubling and Halving**: In a multiplication problem, it’s possible to achieve the same answer by doubling on of the numbers and halving the other.

**For example:**

24 x 25 = 12 x 50 = 6 x 100 = 600

**Written Methods**

**The Grid Method**

I explained this in detail here — The Grid Method

There’s also:- The Lattice Method

### Long Multiplication

The traditional method that I remember:

43 x 36

## Division

### Sharing

If you have 6 sweets and you are sharing them a friend, how many do you get each? Often done with the actual objects– in this case– sweets!

### Grouping

To solve 8 ÷ 2 put 8 objects into groups of 2 and see how many groups there are.

Alternatively use a number line.

**For example**: 18 ÷ 3 could be read as “How many groups of 3 are needed to reach 18?”

Dividing by 10, 100, 1000:

To divide by 10, move the digits one place to the right, the decimal point remains fixed. To divide by 100, move the digits two places to the right and to divide by 1,000 move the digits three places to the right.

90 ÷ 10 = 9

90 ÷ 100 = 0.9

90 ÷ 1000 = 0.09

#### Finding Quarters

To divide by 4, half and half again.

### Finding Fractions of Amounts

To find a fraction of an amount is to perform a division. For example:

¼ of 32 = 32 ÷ 4 = 8

To find vulgar fractions, divide by the denominator (the bottom number) and multiply by the numerator (the top number).

**For example:**

¾ of 72 = (72 ÷ 4) x 3 = 18 x 3 = 54

#### Using Factors with Division

A division can be made more manageable by finding the factors of the number you are dividing by.

**For example:**

90 ÷ 6 6 = 3 x 2 so 2 and 3 are factors of 6

90 ÷ 3 = 30 and 30 ÷ 2 = 15 therefore 90 ÷ 6 = 15

## Written Methods

**Chunking**

Another case where a video is worth a million words:

### The Bus Stop Method/Long Division

This is the traditional, formal method I remember from school. This is certainly best explained with a video, here’s a link to the Khan Academy’s introduction to long division.

## So, Why Has Numeracy Teaching Changed in Primary Schools

The above details how my children’s primary school approaches numeracy but it doesn’t explain why methods have changed. I must admit, before I went through this information, I didn’t understand why it was necessary to teach anything other than the formal methods I remembered such as carrying over, borrowing, long multiplication and long division. I thought that these traditional methods were the most robust and teaching other methods was confusing and counterproductive.

I now see it differently. I can see that once a child has mastered counting and is starting to learn times tables, it’s a huge jump to these formal methods. The intermediate methods outlined above act as stepping stones to help children fully understand numbers. They give all children the best chance of making progress, not just the minority that just find numeracy easy. It’s difficult, when you know how to add up, to fully appreciate the benefit of this step-by-step approach. Your natural inclination is to be impatient and say why not just teach the final method. I now understand how these steps should allow children of all abilities to build their numeracy skills in a relatively consistent and reliable manner.

I can see some challenges with these methods. Firstly, it must be painful for some children to go through all these steps– especially if they have been taught the fundamentals at home. I suppose this makes the case for some “flipped learning” even at primary school– so all children can learn at their own pace. Secondly, when teaching methods change from generation to generation, it’s very helpful for schools to provide as much information as possible to parents. In addition, parents may have to “go back to school” if they want to best support their children.

These are just my views based on a sample size of just two — my children! What do you think? Do you think that these new methods are better than the traditional, formal methods?