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:

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 22

35 = 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?