Learning to count numbers

How Children Learn to Count

Education
Maths Tips
5 and under

Counting is easily taken for granted but there's a lot of fascinating research into how we learn to count - and there's more to it than you may think.

The mathematical brain

It’s first worth considering where our capacity to do mathematics comes from.

Neuropsychologist Brian Butterworth in his book “The Mathematical Brain” suggests we’re born with an innate sense of number hard-wired into our brain and he attributes this to a small region of the brain behind the left ear he calls "the number module". He compares this idea to colour – in the same way we perceive the “greenness” of a leaf we can also perceive the “twoness” or “threeness” of a group of objects.

Take counting. Like times tables and algebra, we tend to think it's something kids have to be taught. Wrong, says Butterworth - it's an instinct. Sure, we have to learn the names and symbols of numbers to develop that instinct, but, because the number module is hardwired into the brain, basic counting comes naturally.

Remote tribes can count even when they have no words for numbers. In maths as in language he believes, "kids start off with little starter kits" And their maths starter kit is the number module.

There are other theories too - such as maths being an extension of our spatial awareness – but there’s something nice in the idea of a “little maths starter kit”.

A Word of warning - All this doesn't mean a child is predestined to be either good at maths or not. Far from it, we’re all born ready to learn maths – and it’s what happens in the first 10 years or so that sets us up.

Counting with toddlers

Research suggest that toddlers - even as young as 12 months - have a sense of how many there are in a set - up to around three objects. This comes from their innate sense of number.

Counting is learned when the toddler starts making the connection between this innate sense of "how many there are" and the language we use to count "one, two, buckle my shoe". This is the first stage in learning maths and it's the building block for many early concepts.

Should parents count with their toddlers? Absolutely, using a variety of real objects. And since counting and language are interlinked reading to your toddlers is equally, if not more, important.

Counting - early learning milestones

Here are some stages of learning to count that you may notice your child going through at ages 3 to 5:

Recognising how many objects are in a small set without counting. So if you show your child four apples they won't have to count them to tell you there's four.
Knowing the "number words" from one to ten and their order.
Know the sequence regardless of which number they start on. So if you say "start counting at four" they will count "four, five . ." as opposed to always counting from one.
Conservation of quantity - This is where children realise that the number of objects in a set stays the same unless any are added or removed. So if they count six cans of beans in a straight line, then you rearrange the beans ( in front of their eyes ) into say two stacks of three - they will realise there's still six without recounting.

Counting non-visible objects - your child will realise they can count things they can't touch or even see - such as sounds, members of someone else's family, or even ideas.
Cardinality, not to be confused with carnality - This is knowing that the last number counted is equal to the quantity of the set. If your child counts six oranges 1,2,3,4,5,6 and then you ask "how many oranges are there"? and they count them again then they haven't grasped "cardinality".

Counting on - as a step towards adding

Learning to add comes as an extension of counting. Here are some stages a child goes through to make this connection:

Counting all - For 3 + 5, children will count "one, two, three" and then "one, two, three, four, five" to establish the quantity of the sets to be added – for example, three fingers on one hand and five fingers on the other. The child will then count all the objects "one, two, three, four, five, six, seven, eight"
Counting on from the first number - Some children come to realise that it is not necessary to count the first number to add. They can start with three, and then count on another five to get the solution. Using finger counting, the child will no longer count out the first set, but start with the word ‘Three’, and then use a hand to count on the second added: ‘Four, five, six, seven, eight’.
Counting on from the larger number - It's more efficient when the smaller of the two numbers is counted. The child now selects the biggest number to start with which is "five", and then counts on "six, seven, eight".
The final stage isn't really counting - it's where learners know their number facts and skip the time-consuming counting altogether.

Number lines are great visual tools for making this connection between "counting on" and addition or subtraction - we use them in Komodo a lot. Here's an earlier blog article all about number lines.

Beyond basic counting

Counting is the first mathematical pattern learners encounter. From here they soon begin to count backwards which is a step towards subtraction and they'll also count in twos, fives and tens which are a foundation for multiplication.

The next big step is the idea of place value and counting to base 10. Learners often make this leap simply because it's an obvious and efficient way to count large numbers. In Komodo, we use practice examples like this to help learners make the connection to counting in tens and ones.

It's easy to forget that counting is a key concept in maths with many stages before it's mastered. There's certainly a lot more to it than one, two, three!

I'm Ged, Co-founder of Komodo, ex-maths teacher and dad. If you have any questions please get in touch.

About Komodo - Komodo is a fun and effective way to boost primary maths skills. Designed for 5 to 11 year olds to use in the home, Komodo uses a little and often approach to learning maths (15 minutes, three to five times per week) that fits into the busy routine. Komodo users develop fluency and confidence in maths - without keeping them at the screen for long.

Find out more about Komodo and how it helps thousands of children each year do better at maths - you can even try Komodo for free.

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Help Children Understand the Meaning of Counting — Better Kid Care — Penn State Extension

Topics include: learning about measurement, learning about patterns and sorting, and learning about shapes and space.

Teaching children to count involves more than helping them learn the numbers one to ten. It involves helping children understand the meaning of numbers.

Children learn the meaning of numbers when they are developmentally ready. For instance, children ages two to three might move things as they count, but they might count to three while moving four things. This shows that while they might know the numbers in sequence, they are not able to use them to count. At this age, it’s great to count together with a child, just for fun. You might count the steps as you walk up them or the buses as they go by. This helps children begin to move towards matching one thing at a time with the number as they say it.

Three- to four-year-olds are still learning to understand quantity. While they can count up to five, they are growing in their understanding of what numbers really mean. By age four to six, children can match the numbers one to ten with ten items; this means they are really counting with meaning. They can solve simple problems, such as how many cookies you will need for each person to have one. By the time children reach the ages of five to seven, they can count items and match them; for example, putting five stamps on five letters.

You can help your child learn to count by making counting a fun part of your day. Count socks as you sort them; count the juice boxes in your refrigerator; count the cars and buses going by. The more experience children have with counting, the more they will learn the meaning of numbers. Understanding the meaning of numbers takes experience with counting lots of things, and you can help by giving your child this experience regularly.

Learning about Measurement

Giving your young child a chance to measure things can help her understand both how and why people measure things. Find real measuring jobs for children to work on. Will this table fit here in this space? How tall are you? How much bigger is the plant than it was a month ago?

Inches, feet, and other units of measurement don’t make much sense to a young child. Teach your child to measure with a simple object, such as a shoe. Check how long the rug is with the shoe, or measure the height of a plant with blocks. Then give her a ruler to work on measurement problems. How big a block do we need to fit this space?

Learning about Patterns and Sorting

You can help your child get ready for math by giving him chances to think about patterns and to sort things. This skill is both fun and important in learning math. You can use things that you can find outside: pinecones, leaves, sticks, etc. Lay out a pattern with these objects (leaf, leaf, stick, stick, stone) and then repeat the pattern and ask your child to make the same pattern. Most children enjoy this simple game.

You can help your child develop sorting skills when you do your household chores. Ask your child to put the blue towels in one pile and the white in another. As children grow you can challenge them to use two different ways to sort. You might say, “Can you sort these towels by color and then put all the large towels in this basket and all the small ones here on the shelf?”

Learning about Shapes and Space

Take time to notice shapes and talk about them with your kids. Shapes are everywhere. A simple walk is a great time to talk about them — you might see the rectangles made by the fence posts, or the circular manhole covers. As you pick out shapes, your children will learn how to do this as well.

Children learn best about sizes, shapes, and how things fit together by playing. Blocks are a great toy to teach these spatial relationships. Building structures helps children figure out how things can fit together. They learn about what it means to double things in a practical way, when they discover that they need two blocks to fill a space. When children understand the real-life meaning of these things, the idea of multiplying by 2 makes sense.

Parent Count December 2007

Learning to count up to 20

At the age of 4-5, children are ready to learn counting up to 20. Start with forward counting, gradually moving on to learning backward counting.

Developing online math tasks for children from 2 to 11 years old

Direct quantitative counting

Learn with your child the names of numbers from one to twenty and their spelling. Your child probably already knows how to count to ten. For better memorization of numbers from 11 to 19explain how they are formed: they consist of 10 and one more of the numbers 1, 2, 3,4, 5, 6, 7, 8, 9. Put ten identical objects or counting sticks in a row. This is "twenty" - Old Russian ten. Place one other object or stick of a different color on top. It turned out eleven - "one" to "twenty". Show how the other numbers of the second ten turned out. There are two tens in the number 20.

The next step is to remember the numbers. An excellent simulator for learning numbers is the number line. You can use a regular ruler. Looking at the ruler, the child will not only remember the spelling of numbers and their arrangement in order, but also learn to distinguish: the numbers that are located to the right - more, to the left - less. Encourage such exercises with a ruler. Call the child numbers in random order, let him write down the numbers under your dictation. As an assistant at first, he can use a ruler.

Do simple game exercises:

Name the neighbors. You name a number, the child - neighboring numbers located on both sides of the number line. This exercise not only helps to remember numbers, but also teaches the child to increase and decrease by one.

Quantitative account

Repeat the formation of numbers from eleven to twenty with your child, tell them that numbers up to one hundred are formed in the same way: a number indicating the number of tens + "twenty" and a number indicating the number of units. The only exception is the number "forty", it must be remembered.

Counting skills within the first hundred are well developed by the Connect by Dots exercises. You can also perform simple game exercises with your child:

"Name the neighbors of round numbers." The greatest difficulty for children is the transition to round numbers. This exercise is designed to reinforce this skill. Name a round number, let the child name the neighboring numbers that are on both sides of it on the number line. This exercise also teaches the child to increase and decrease by one.

Perform developmental exercises from Aikyusha

“We pronounce the numbers one by one”. The essence of the exercise: counting up to one hundred, say the numbers in turn with the child. Explain to him that you need to pronounce the number that comes after the one named. You can connect a few more people to the game to make the game more fun.

In addition to counting straight, practice counting backwards, paying special attention to naming numbers in the forward and backward order with the transition through the round number.

Practice writing numbers. Pay attention to the fact that the child writes and “reads” the number from left to right. Children often make mistakes in recognizing numbers, for example, instead of the number 21 they say 12 . Written numbering, like oral numbering, is associated with the number of digits, remind the child that tens are written first, then units. Pay special attention to writing round numbers, in which 0 is written in place of units (they are not in this number). If you miss this moment, you may encounter the fact that the child will write the number 45 as 405.

"Counting and counting". In every possible situation, enlist the help of the child in counting and counting objects. When counting, the child needs to determine the number of items. Make sure that you count according to the rules: in the process of counting, point to objects in order, touching them with your finger. If you count out of order, you can get confused and miss one or more items. Count everything in a row - steps, steps, toys. When counting, the child needs to count a certain number of items, for example, bring 7 pencils or 2 sweets.

"Forbidden number". An adult calls a number up to 10, a child shows it on his fingers. Before that, a number is selected that is forbidden to be shown. When an adult calls this number, the child simply clap his hands. The game consolidates the understanding of the relationship between the name of the number and the quantity that it denotes.

Countdown

After your child has learned counting up to 20, you can start learning backward counting up to 20. Do this in stages - first from 5, then from 10, and only then from 20. To practice skills countdown do exercises:

"How much is left." Going down the stairs, count together with the child how many steps are left, seeing the approaching bus, count the seconds before the doors open, count the seconds before the green traffic light.

Walking games. Play board games with two dice. During the game, the child will need to calculate how many steps he will need to make a move, as well as “walk”, forward or backward.

"Count in range". The adult names the tasks for the child, and the child completes them. For example, count up to 9 to 5, from 17 to 11. The exercise teaches the child to vary numbers, count in a certain range, trains attention.

Do these simple exercises throughout the day, at home or on your way home, and your child will be great at counting from 1 to 20 and back!

Educational online math exercises for children from 2 to 11 years old

How to learn to quickly count in your mind? - Meduza

Why in the mind when you can on a calculator or in a column?

Minimum counting skills, a sense of numbers are the same element of human culture as literate writing and speech, knowledge of a foreign language, a basic understanding of art and the world around.

In addition, when you easily count without improvised means, you feel a completely different level of reality control - you know in advance how much change you will be given in the store or whether it is worth cramming seven into an elevator with a carrying capacity of 400 kilograms.

Think about the fact that the calculator and the actions in a column are a kind of magic. Most likely, you do not understand how it works, and you are forced to simply trust them. And when you have a good understanding of how mathematical operations work and can reproduce them "by hand", your sense of control (and self-confidence) gets a serious bonus.

And finally, mental counting develops your mental abilities: attention, memory, concentration, switching between several streams of thought, and can also serve as a means for meditation or distraction from sad thoughts.

But where can I get tasks for training? Think of examples for yourself?

Of course not. The network is full of mobile applications that will offer you the training of math skills for every taste.

When choosing, keep in mind that a good application, at least, must have sufficiently flexible difficulty settings and keep statistics of the tasks you have solved.

Try these apps for iOS and Android, or look for alternatives on the App Store and Google Play.

How exactly should you train?

There are only four basic mathematical operations - addition, subtraction, multiplication and division. Each action has its own characteristics, but they are not complex. You need to figure it out once, and then train for 5-10 minutes a day, and very soon you will feel that you think better. Most likely, in two or three months you will reach a fairly decent level that can be maintained by episodic training.

Where to start?

Start with the easiest level, single digit addition, and work it up to perfection: 99% correct, 1-2 seconds for each answer. For solving "with the transition through 10" examples, try using the following technique - "Reliance on the ten".

Let's say you need to add 8 and 7.

1) Ask yourself how much 8 is missing from 10 (that's 2).

2) Imagine 7 as the sum of 2 and some second piece (that's 5).

3) First add to 8 that part of the number 7, which was missing to 10, and then that second piece - you get 10 and 5, and this, of course, is 15.

How to add multi-digit numbers?

Here the most important principle is the addition of identical digits to each other. Having broken both numbers into "bit parts", start adding from the highest digits - thousands with thousands, hundreds with hundreds, tens with tens, ones with ones. If necessary, enlarge what happens and count everything together again.

For example, how do you add 456 and 789?

1) 456 consists of three bit parts - 400, 50 and 6.

789 is also divided into three bit parts - these are 700, 80 and 9.

2) Add hundreds with hundreds: 400 + 700 = 1100, tens with tens: 50 + 80 = 130, ones with units: 6 + 9 = 15.

10+5, that is, 1200+40+5 = 1245.

Amendment . When adding the digits, we mixed up the units and added 8 instead of 9 to 6. As a result, the sum also turned out to be incorrect - 1244 instead of 1245. We apologize for the mistake, and do not repeat it - carefully follow the numbers, especially in mental counting!

What about subtraction?

And here you need to start from the basic level - subtracting a single digit from the numbers of the first and second ten - and bring this skill to perfection. As with addition, problems usually arise with "going through 10" subtraction. And here a similar method of "reliance on a dozen" will help.

Let's say we need to subtract 8 from 12.

1) Let's ask ourselves how much we need to subtract from 12 to get 10 (that's 2).

2) We will subtract 8 from 12 in parts - first we subtract this 2, and then everything else. And the rest is how much? (that's 6).

3) After subtracting 2 from 12, we got 10, and we need to subtract 6 more, we get 4. Done!

What about multi-digit numbers? Is everything difficult with them?

Not particularly. It is only important not to confuse the technique of subtraction with the technique of addition. When adding, it was convenient for us to break each of the numbers into digit parts, and here we break only the number that we subtract.

So let's say we need to subtract 512−259.

1) The number 259, which we subtract, consists of three digit parts - 200, 50 and 9. We will subtract them in turn.

2) 512−200 - subtracting hundreds does not affect the tens and units of the number 512 in any way, it only affects hundreds, so the result will be this - 312.

3) From what happened after subtracting hundreds, now subtract tens, 312 -50.

This is similar to subtracting by ten. First subtract 10 from 312 to whole hundreds (units will not be affected), we get 302. And then subtract everything else (50 had to be subtracted in total, 10 has already been subtracted, it remains to subtract 40), we get 262.

4) It remains to subtract the units: 262−9.

Clean transition through ten - first subtract 2, we get 260, and then subtract the rest, 7, we get 260−7 = 253. Here is the answer.

How does multiplication work?

Let's start with single-digit multiplication. First you need to remember that multiplication is when you add the same thing several times. For example, multiplying 4 by 7 means adding four sevens. Using the addition technique, we can easily calculate - two sevens, 7 and 7, will be 14, if we add the third 7, we get 21, and adding the last, fourth seven, we get 28 as a result.0003

Gradually, as a result of training, you will remember the reference multiplication values that are convenient for you and with their help you will be able to quickly calculate neighboring ones. For example, if you need to multiply 6 by 7 (that is, add six sevens), and you remember that 5 times 7 (that is, add five sevens) will be 35, then to get the final result, you just need to add the sixth seven - you get 42.

The most difficult example in the multiplication table is 7∙8. To remember it, there is a good mnemonic rule “five six seven eight”, which means 56 = 7∙8.

How to multiply a multi-digit number by a single-digit number?

Let's look at an example. Let's say we need to multiply 468 by 6.

1) 468 consists of 400, 60 and 8, and all this needs to be multiplied by 6. Well, taken separately, these problems are no more difficult than multiplying single-digit numbers.

2) Going from the most significant digit to the least significant: 400∙6 = 2400 (since 400 is 100 times greater than 4, then the result of 400∙6 will be 100 times greater than the result of 4∙6).

Accordingly, 60∙6 = 360, and 8∙6 = 48.

3) And now, as in addition, we add it all together, grouping the same digits: 400+300)+(60+40)+8 = [add the same digits] =

= 2000+700+100+8 = [group and add the same digits] =

= 2000+800+8 = [further enlarge nothing, we get the answer] = 2808.

How to multiply two-digit numbers?

For an ordinary person, this is already aerobatics! If you have mastered double-digit multiplication, consider that you have been accepted into the world of the mental counting elite. But in fact, there is nothing fundamentally complicated here, just a higher load on short-term memory (at the same time we will train it).

So, for example, let's multiply 78 by 56. This means that we need to add (take) the number 78 56 times.

1) These 56 times can be divided into stages - first add 78 50 times, then 6 times, and then combine the results.

2) Adding the number 78 50 times is not difficult - it is 10 times more than adding it 5 times. 78∙5 = 70∙5+8∙5 = 350+40 = 390. So, 78∙50 = 3900, remember this number.

3) Now let's calculate 78∙6 = 70∙6+8∙6 = 420+48 = 468.

4) Now let's add both results together: 3900+468 = 3000+900+400+60+8 = 3000+1300+60+8 = 4368. Voila!

Amendment . At the final stage, when adding 3900 and 468, we incorrectly split the second number into digits - we forgot about 60. As a result, the total turned out to be 4308. We apologize for the mistake, and do not repeat it - you cannot lose terms in the mental calculation.

Wow, the last thing left is action, division?

Yes, we are at the finish line. And again, let's start from the simplest level: dividing by a single-digit number of those numbers that are familiar to us from multiplying single-digit numbers.

So what is division? In fact, this is the "reverse" operation to multiplication.

For example, dividing 56 by 7 means picking up such a number that if you multiply it by 7, you get 56. Since you are already well versed in the multiplication table by this point, you will surely remember that exactly 8 times 7 gives 56. So, the desired number is 8, 56:7 = 8.

And so always - remember which number, when multiplied, gives the desired result - this is the number that you need.

How to divide multidigit numbers by single digits?

Let's divide 6144 by 8. Our way is to "cut off" the maximum "round" parts from the original number, each of which is guaranteed to be divisible by 8 according to the multiplication table.

1) Select from 6144 as large a part as possible, which is divisible by 8 according to the multiplication table. This will be 5600, because 56 is divisible by 8, and the next number that is divisible by 8 is already 64, which does not suit us, since 6400 is greater than 6144. Great, 6144 is 5600 and 544 (here we needed the subtraction skill ).

We will divide along the way:

6144:8 = [select the maximum convenient round part] =

= (5600+544):8 = [divide the selected part by 8, and work on the second part in the next step] =

= 700+544:8. We will remember

700 as a partial result, and we ourselves will deal with the division 544:8.

2) Similarly, out of 544, the largest part that can be conveniently divided by 8 according to the multiplication table is 480 (after all, 48 is divisible by 8, and the next number - 56 - does not suit us, because 560\u003e 544) . So 544 = 480+64.

We continue the division:

544:8 = [select the maximum convenient round part] =

= (480+64):8 = [divide the selected part by 8, and work on the second part in the next step] =

= 60 +64:8.

60 add to 700, 700+60 = 760 - remember this as the second part of the result and move on to the last division, 64:8.

3) The remaining piece, 64, is also divisible by 8 according to the multiplication table, 64:8 = 8.

Accordingly, the total result of division is 760+8=768. All!

How do you divide by a two-digit number?

The technique of dividing by a two-digit number is the most diverse, unlike anything, exquisite. Let's get acquainted with it on the example of 5148:66.

1) Guess what ten is our result. Recall that 5148:66 means: we are looking for a number that, when multiplied by 66, will give 5148. We will use the "shooting" technique.

Just randomly try the number 20 as a possible candidate. 20∙66 = 1320, which is 4 times less than 5148, which we need.

4 times more than 20 is 80, let's try it. 80∙66 = 5280, it turned out more than the desired 5148, but a little, most likely, this is the “top” ten.

Let's check for reliability 70, the previous before 80 tens. 70∙66 = 4620, which is just less than 5148, great! This means that the number we are looking for lies between 70 and 80.

2) Let's use the mathematical law about the last digit of the result of multiplying two numbers.

It turns out that it always coincides with the last digit of the result of multiplying the last digits of these numbers (try to think why this is so). For example, what number will 123 9 end with0011 4 ∙567 8 ? To the same as 4 ∙ 8 , that is, to 2 (4∙8 = 3 2 ).

Therefore, if we are looking for a number that, when multiplied by 6 6 , will give 5148, then to guarantee this 8 in last place, the desired number can only end in either 3 or 8 (3∙ 6 = 18, 8 6 = 48).

3) With these endings between 70 and 80, we have two total candidates - 73 and 78.

5148 is clearly closer to 5280, so let's check 78 first.

78∙66 = 78∙60+78∙6 = 4680+468 = 5000+148 = 5148, hooray!

(Well, if the result did not converge, then we would check the second number, and it would already fit exactly).

What are the final recommendations?

Here, in general, and all the ways that it is enough to know to train confident counting within 10,000 (and the ability to work mentally with large numbers, perhaps, is already beyond the scope of the necessary general development).

Surely you will also come across other techniques, the so-called. "tricks" of quick counting, but do not rush to get carried away with them. In addition, remember that regularity is more important than intensity - try to exercise on the simulator every day for 5-10 minutes, no more is needed, otherwise there is a high risk of "burning out" and giving up.