But developing kids’ ability to see a collection of objects (or a number) as being made up of smaller ‘parts’ is something that traditionally has not been focused on in early maths. It’s important to do this, however, because it leads on to early understanding that ‘bits’ can be moved around from one number to another to make calculations much easier. Kids who begin developing this sort of flexible thinking about numbers from the very beginning of their schooling have a distinct advantage over the rest. Yet it is something that is frequently not taught!

So if you have a 4 or 5 year old, and you are busy teaching them to say the 0-9 number counting pattern to 10 or higher, and teaching them to count up to at least 6 objects (pointing to each object only once and not missing any out), now is the time to teach them to subitize and to think about number partitions.

How to start:
Get a 30cm x 20cm (approx) piece of cardboard or a box lid and five small objects, eg toy animals, pegs, pasta shells, counters. Sit on the floor facing your child, and say:

I’m going to show you some things for just a second and I want to see if you can tell me how many you see. There won’t be time to count, and this is not about counting. It’s about looking. First you need to close your eyes. When they’re closed, I’ll get ready.

Put one thing behind your cardboard barrier. Tell your child to open his eyes. Reveal the object behind your cardboard for just a second or two, and then hide it again.

(It’s one. I just knew.)

Ask:

How many teddies did you see?

(Answer: One)

How did you know it was one?

(Answer: I just knew.)

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Repeat, but this time place two objects behind your barrier.
Ask the same questions:

How many teddies did you see?

(Answer: Two)

How did you know it was two?

(Answer: I saw one and one, and so I knew it was two.)
Strongly encourage this type of response because the child has looked at the two in parts, and is therefore partitioning.

(Other likely answer: I just knew.) This answer is acceptable, but you should point out that you can also look at the two in parts, ie one and one. Note, this is NOT about teaching addition, it is about looking at a number in parts. You can practise using two objects of different colours to help make the partitioning clearer to your child. He’s more likely to say he saw one red one and one yellow one.

(It’s two. I saw one red one and and one yellow one.)

However, you don’t want your child to become dependent on using colour for partitioning, so you can vary your teaching by using two objects of different sizes. Your child is also likely to say he knew it was two because he saw one big one and one small one.

(It’s two. I saw one little bear and one big bear.)

Eventually you want him to be able to say he saw one and one when the objects are exactly the same.

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Repeat the process, with three objects.
Based on what you have previously done, you will probably find that here your child will say he knew it was three because he saw one and one and one. This is fine because it is partitioning. But it’s now time to point out that with three objects, sometimes you can see two objects and one object. You will need to demonstrate a few times using different objects grouped in different arrangements.

(It’s three. I saw two bears over here and one bear over there.)

(It’s three. I saw one red bear and two yellow bears.)

(It’s three. I saw two and one.)

(It’s three. I saw one and two.)

Work on subitizing 1, 2 and 3 until you see that your child has a great understanding and can give you the ‘right sort of answers’ to the two questions.

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Then move on to subitizing with four objects.
Repeat the same process, using a variety of objects and remembering that you will need to point out the different partitions over time, saying:

I saw two and two that time!
I saw three, and one over there that time!
I saw one and three that time!

Work on subitizing 1, 2, 3 & 4. Look around for opportunities to subitise when you’re out and about (eg pictures on posters that you can cover with your hand) or playing a game with involving dice.

(It’s four. I saw two and two.)

Children learn to recognise the numbers on a die very easily. They just know, for example, that a six is a six without needing to count the dots. Take the opportunity to point out that you can look at a six on a die as three and three, or as two and two and two. Don’t be tempted to teach subitizing with just a die, as you want your child to see partitions in all situations, and standard dice always use the same dot arrangements.

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Later on you can move on to subitizing with five and six objects, but it’s best to wait until your child is very comfortable with subitizing one to four objects.

Remember that subitizing is ‘seeing at a glance’ and you show the objects for no more than a second or two so there is no time for counting. The word ”subitize’ comes from the Latin word ‘subitus’, meaning ‘sudden’.

Here are some you many already have that are great for particular areas of Maths:

Number pattern & order:  Bertie and the Bear  by Pamela Allen (K & upwards)

Making comparisons:  Counting on Frank  by Rod Clement (K & upwards)

Subitizing:  The Grouchy Ladybird  by Eric Carle (K & upwards)

Partitioning : One Is a Snail, Ten Is a Crab  by Sayre & Cecil (K & upwards)

Counting large numbers:  Out for the Count  by Cave & Riddell (K & upwards)

Thinking about millions:  The Giant Jam Sandwich  by John Vernon Lord (K & upwards)

Have fun!

Teachers have moved away from this method nowadays, and this is because they understand that children reap much greater benefits from learning to partition numbers to calculate. Simply put, this means breaking bits off one number and adding them to the other, or vice versa. There are many wonderful and varied strategies that teach and reinforce this idea.

Below is an example of a strategy called Bridging, which uses tens grids.  Bridging to 10 is a great way to teach partitioning skills while adding 9 to any single digit number. Kids love this strategy!

In this example (9 + 5), we are in fact partitioning the 5 into 4 + 1, and then moving the 1 over to join the 9 and fill up the ten grid on the left-hand side. 9 + 5 has now been rearranged to become 10 + 4. Why? Because 10 plus anything is very easy to add, so it makes sense to rearrange the numbers. This sort of flexibility in mathematical thinking is based on true understanding of number and is something to encourage.

Now that the grid on the left is full, you can just see the answer is 14.

If you want to teach bridging to your child, print some grids (under the free downloads menu), find counters or any small objects and demonstrate how to solve 9 + 5. Explain that you put 9 counters on one grid for the first number, and 5 on the other grid for the second number. Now tell your child that you’re going to bridge to ten, and move one counter across to fill up the ten. Point out that you can now see the answer …10 + 4. Explain that bridging to 10 is a good idea because adding a number to 10 is much easier than adding a number to 9.

Now give your child some grids and counters, and some problems, eg 9 + 4, 9 + 7, 9 + 6.

Bridging to 10 can easily be extended to bridging to 20, 30 or higher when children fully understand it. and can rewrite addition problems such as 9 + 4 as 10 + 3. Bridging to 20 involves adding a single digit number to 19, eg 19 + 6, 19 + 8. Bridging to 30 involves adding a single digit to 29, eg 29 + 3, 29 + 7.

For example, if a 5 year old can say the numbers 0-9 forwards and backwards in order, it’s fine to use a chart or number line to help them start reading the numbers in order. When they can read them accurately, they’re ready to start learning how to write them. As they practise reading and writing 0-9 over time, they can also learn to say the next group of numbers, 10-19. And when this is mastered, they’ll be ready to learn to read 10-19 and later, write these numbers.

Kids develop a very sound understanding of the 0-9 number pattern when this method is continued right through to the higher numbers.

1-100 number charts are very useful for counting forwards and backwards by ones, twos, tens, fives etc. Often, however, 1-100 is the only number chart children are presented with because it’s assumed that if they’re fine with numbers and patterns up to 100, they’ll be fine with all the higher numbers. This is frequently not the case.

Many children get stuck on saying numbers in order above 100. For example, you’ll often find they hesitate after saying 109 and have to think very hard about what comes next. The same often happens after saying 119.

To help avoid this uncertainty, I recommend you use a 0-119 chart, and when they’re comfortable with the upper end, get them to fill in their own add-on chart below it. You can download a 0-119 and blank chart here.

It’s also a good idea to give them practice grids to fill in when learning to write a section of the 0-9 pattern that they can already say and read. You can download some practice grids here. You’ll notice that some squares are blacked out on these grids. This is because kids often copy the last digit in each column. The black squares on the grids force them to think about the numbers in order because they can’t just copy the pattern.

Number charts can also be useful for demonstrating or reinforcing addition and subtraction strategies, for example, counting on from the larger number in addition (We get to the same point if we add 5 + 13 or 13 + 5. Which is easier to do?), or adding or taking away chunks of a number in addition or subtraction problems (14 + 6 can become 14 + 2 + 2 + 2 – three jumps of 2). However, a child who finds ‘chunking’ too abstract on a number chart will need to have it explained with hand-on materials.

Charts are great for highlighting skip-counting patterns. Kids LOVE using stickers or whiteboard markers to show patterns on laminated charts! Remember that they need to start on odd numbers as well as even numbers when skipcounting by twos, and should also have varied starting numbers when skipcounting in other patterns. This helps develop a much broader understanding of the patterns.

Algorithms can be in any language, from English, Spanish etc to mathematical or computer programming languages.

We use algorithms every day. For example, a cake recipe is an algorithm. A set of instructions for checking the oil level in your car is an algorithm.

Algorithms are designed to be helpful and show you an efficient way of doing something. Are they dangerous? It depends. Certainly not intentionally. But the ability to use an algorithm does not imply that a person understands the underpinning of that algorithm… in other words, they may not understand why they’re doing what they’re being told to do.

In Maths, algorithms are frequently used in calculating the answers to addition, subtraction, multiplication and division problems. (They have many other applications too, of course.) However, they do not help children to understand the mathematical concepts behind addition, subtraction, multiplication and division. And when they are used as the sole method for teaching children how to calculate, many children do not gain the understanding of what is actually going on mathematically. In this way, algorithms CAN be dangerous.

Let’s look at an addition problem and use a standard algorithm to calculate the answer:

The usual procedure is to add the right hand column first (the ‘ones’), and if the result is 10 or above, to ‘carry’ the ‘one’ by writing it somewhere in the ‘tens’ column and write down the second digit (the ‘ones’ number) below in the ‘ones’ column. Then we add all the single digits in the ‘tens’ column.

On the face of it, this looks very easy because you only have to add small numbers at any point in the process, and most children find it quite easy to add small numbers. So where’s the danger?

Algorithms generally don’t correspond to the way people think about numbers. Studies have shown that most children stop thinking logically about the numbers they’re working with when they’re using algorithms. They just think about the numbers column by column. If they make an error, they’re unlikely to notice it because they don’t think about the ‘reasonableness’ of their answer.

Algorithms also tend to ‘unteach’ place value. For example, a 7-8 year old child can usually show that 16 is actually 10 + 6 (and that the 1 in 16 actually means 1 group of 10 ones) if they have materials and are focusing just on this sort of partitioning task. When this child is taught to add using an algorithm, however, they are told to add the numbers in the ‘ones’ column (in this case arriving at 16), and then put the 1 from the 16 into the ‘tens’ column. This is where their fledgling understanding of place value is instantly challenged. Often, in this context, the child will not know that the 1 actually means 10 ones, or understand why they’re putting it in the next column to add to the other numbers.

Check your child’s current understanding of place value! Draw 23 circles quickly on a piece of paper as balloons. Ask your child first to count them and then write down the number of balloons on the piece of paper. Do not say the number ‘twenty-three’ aloud at any point, even to confirm they’re right. Instead, underline the 2 in the 23 they’ve written. Then point to the 2 and say, “Use a pen to show me how many balloons this underlined digit represents.”

So should we teach algorithms to young children if they can actually hinder the development of their mathematical understanding?

Current Maths educational theory tells us that it is not a good idea to introduce addition, subtraction, multiplication and division using algorithms. It is fine to teach algorithms as an alternative method when you are sure that a child fully understands these mathematical concepts and can partition numbers flexibly.

So how should we teach addition, subtraction, multiplication and division then?

Just looking at addition, there are many different strategies that are meaningful AND build the flexibility in thinking about numbers that children should be developing.

Here are a couple to think about:

Near doubles

For example, 6+5 is almost 5+5 or ‘double 5′. You just have to add 1. 6+5 is also nearly 6+6 or ‘double 6′. If you look at it like this, you just have to subtract 1. Your child should be thinking about doubles and halves, and learning doubles facts, if not already doing so. Doubles facts (eg ‘double 1′ or ’1+1′ is 2) should be taught using materials if it’s a new thing.

Bridging to 10 (or 20, 30 etc)

We can work out what 9 plus ‘something’ is very easily. For example, 9+3 can be looked at as 10+2 if we think about it flexibly. By taking 1 away from the 3 and adding it to the 9, 9+3 becomes 10+2. Adding 10+2 (or 10+ anything) is a breeze! This idea is best introduced using two grids with ten squares in each (easy to draw), and counters or small objects. Ask the child to put 9 counters on the first grid (one per square) and 3 counters on the second grid. So 9 is then represented on one grid and 3 on the other. Then show your child how to bridge to 10. Get them to move 1 of the counters away from the 3 and across to fill up the 9 in the other grid to make 10. Tell your child they’re looking at the answer, ie 10+2 (12)!

We could use this strategy to solve our 19+7 problem above. We’d be bridging to 20 in this case, and would end up seeing 19+7 as 20+6. (Children need to be familiar with bridging to 10 before attempting to bridge to 20.)

Why do they do this, when you’re just trying to help?

About kids

Kids know when they don’t understand something, but they often pretend they do understand because they’re afraid their lack of understanding will be exposed. Yes, they do care what their parents think! By creating a storm, they’re trying to protect themselves from this exposure.

Kids who guess a lot in Maths are also attempting to cover up their lack of understanding.

About parents

Parents often think that if they show their child something once or twice, they’ll get it and be fine. When this doesn’t work, parents get frustrated and often blame the child. Whether they voice this or not, the child picks up on it, and feels worse. They don’t want the parent to help any more and so the protests start.

Why kids often don’t understand your explanations

1 The work is currently at too high a level for this child, and so any explanations are going to go over their head. This is a very common issue with Maths homework (and Maths at school). The child needs to be taken back a step (or two or three) and have explanations and experiences at that level. The teacher should adjust the homework. Otherwise it’s a waste of time for everyone involved.

2 The homework is slightly above the correct level for this child. You explain it and the child seems to understand it, but then loses it again. You explain again and the same thing happens. This means the child really doesn’t understand it yet, and needs a lot more modelling from you, and then practice. If you reduce the size of the numbers to below 10 when you’re explaining something, you’ll find it often makes a huge difference. You can slowly increase the size of the numbers when you see that the child does understand properly.

Just a word of warning. If a child doesn’t understand something that you’ve just tried to explain, you need to keep your cool and think about the next logical step you can take to get it across. The following steps go backwards in order from (1) requiring a high level of understanding to (4) requiring a low level of understanding. So start at (1), and continue going down the list until your child gets it.

1 Try explaining a different way. If this doesn’t work…
2 Try reducing the size of the numbers involved. So if you were up in the 100s, go down to the 10s. If this doesn’t work…
3 Get some hands-on materials that you can use to represent the situation and model the maths situation using these materials. Your child obviously needs to SEE it in action. If this doesn’t work…
4 Get the real items you are talking about in the maths situation if you can, AND reduce the size of the numbers even further (below 10). Now your child should get it!

Also, don’t expect a child to remember anything after one explanation. They need lots of modelling.

Let’s look in a little more detail at some of the features just mentioned:

They can choose from a range of games, and the same game can be played multiple times, with different content each time.

Children love games, and the apps are always bright, fun and graphically engaging. But are they mathematically engaging for a child who doesn’t get maths? The answer is, yes they are. Initially. But they are not tutors, and children who don’t understand the concepts behind the games will come unstuck and not want to play after a few failed attempts. For children who do understand the concepts, they’re great, especially as the content always changes.

Some even give parents statistics and progress reports.

Parents naturally want to know that their child is getting on well, and there’s nothing like a progress report generated on the spot. But are these reports really of value? If a child is getting poor results in one area, for example, what then?  The parent can mention it to the child’s teacher, and that may be helpful. However, because maths understanding builds on earlier level understanding, a problem that an app highlights may actually indicate a problem at a much earlier level. If this is the case, hammering away at the game’s level will do no good, and just frustrate everybody.

Games are often leveled, not always in difficulty, but frequently in the time given to answer questions.

We put a lot of pressure on our children to do maths quickly, but is this really a useful thing? Well, yes it’s fine, as long as we are sure that the child understands what’s going on. If a child doesn’t understand, they’ll guess when under pressure. Alarm bells should go off in your head if your child is guessing at maths. It’s a clear indicator that they don’t understand, and you need to take action. Get a proper assessment, find out what level they’re truly at and work on with hands-on materials on the relevant concepts.

Now there’s that phrase…hands-on. There is a significant difference between pictures on a screen and hands-on experiences. A child who doesn’t understand with pictures will often get it if they are given manipulatives because this is an earlier mathematical stage. An even earlier stage is acting situations out.

Number sense, greater than/less than, addition & subtraction are stalwart features in apps for Years 1& 2.

Modern maths has been embraced whole-heartedly by our education systems, and the latest apps are reflecting this. But if modern maths is all that it’s cracked up to be, why then are there so many children failing? The answer is that very few of our educational systems have moved far enough ahead in their understanding about teaching maths. Modern maths is all about children understanding mathematical situations and recognising what to do, but not all educators understand what to do if a child still doesn’t get it after they’ve gone through their palette of strategies. A lot more training is necessary to develop greater understanding of what to assess, how to take a child back far enough and then how to move them forward effectively.

Apps are great in giving practice in maths that a child already understands. If, however, our educational systems are still not up to speed with helping children with maths difficulties, do you really think that the makers of apps are?

Click here for help with 5-12 year olds’ maths