How To Teach Mathematics To Children - Counting

Asking the right mathematical questions at the right time to your child can make the difference that enables deep learning to occur and a solid foundation for other concepts to be built upon.

CountingAs a primary school teacher, I am often asked many questions about the best strategies to use when helping a child with mathematics or about what mathematical concepts to focus on with a child to get them to have a deeper understanding of mathematical content. There is a lot of information about teaching mathematics to a young child so I thought I would cover a couple of topics through three blogs in a series titled “How to teach mathematics to young children”. This is the first blog of the series focusing on “Counting”.

Number sense is very important for children to be able to effectively and efficiently work with numbers to solve problems in their everyday life. Counting is one part of number sense and it becomes a very important skill that assists children to solve problems.

Children’s early numeracy (mathematical) development can be described using ‘growth points’. These can be thought of as “steps” on a learning journey. Each “step” is a growth point that a child must be successful at before moving onto the next. They move through the growth points to achieve a deep understanding of that particular domain. (eg counting, place value, additions and subtraction, multiplication and division)

Below is a list of the growth points of counting that children should move through and parent tips of how to support their child at home to achieve a given growth point.

Growth Point 1 - Rote counting to at least 20

This growth point is about the child being able to count forwards independently to at least 20 without making mistakes. (skipping a number or saying the wrong number) It is hoped that most children are able to do this before starting Kindergarten; however most children are unable to do this. Giving children many opportunities to count in the early years gives them a big advantage when they start school.

Parent tips for working with a child to achieve this growth point.

  • Give your child many opportunities of hearing you count forwards in your everyday life (eg counting the fruit you put into a bag at the green grocer or counting the number of cars in a line in the carpark as you walk back to your car)
  • Point out numbers in environmental print (print used in everyday life) eg number plates, house numbers on letter boxes, digital clocks, phone numbers on advertisements)
  • Sing counting songs (5 little ducks, 10 green bottles)
  • Read counting picture books (Ten little ladybugs by Melanie Gerth and Counting kisses by Karen Katz)
  • Play counting games e.g. The parent and child take turns saying a number. You could include more people in the game when the child gets more confident with their knowledge.
  • Don’t always stop counting at a number ending with a zero (10, 20, 30, 40 etc) Start at 0 and finish at 35.

Growth Point 2 - Confidently counts a collection of around 20 objects

This growth point is about the child understanding that each object represents 1 and confidently uses this knowledge to count a collection of objects. A lot of children find it hard to organise their collection in order to count it correctly. Most children usually keep their collection in a pile and try to point to each object while counting; however they will miss counting some objects or count objects twice using this method. Our focus should be getting children to organise their collection to make counting the objects easier. By moving the object to another space when it has been counted, or setting up the objects in rows or lines, it will help the child to successfully count them. To achieve this growth point, the child must be able to count at the same pace as they point to the object. (Many children count slower than their finger pointing to the object) This growth point is usually achieved between 4 - 7 years old.

Parent tips for working with a child to achieve this growth point.

  • Give your child many opportunities to count different objects for you (e.g. Ask the child to get 5 potatoes for you to peel, ask them to count how many necklaces you have, ask them how many toy cars they have)
  • Play simple board games to get your child moving their player piece the amount of spots on the board that the dice tells them eg snakes and ladders, trouble (when your child gets good at this encourage them to try and tell you the number on the dice without counting them. This is called subitising and it is important for ‘number’ understanding)
  • Play guessing games with your child. For example, put some (13 - 25) toy cars in a bucket and ask your child to tell you how many cars they think are in there. Then get your child to check by counting them.
  • Practise arranging objects in different ways and get your child to investigate which ways are the easiest to count. (moving objects, placing them in rows, placing objects in a line) You can demonstrate how you set up the objects.
  • Play matching games (eg Matching the collection of objects to the numeral or matching pictures of objects with their numeral)

Growth Point 3 - Counts forwards and backwards from various starting points between 0 - 120 and knows numbers before and after a given number

This growth point is about the child being able to start from any number between 0 - 120 and count forwards from that number and backwards from that number confidently. They must also be able to identify what number comes before and what number comes after each number. Achieving this growth point helps the child with the “count on” strategy in mathematics. For example if a child had 23 apples and a friend gave them 4 more, they could start with 23 in their head and continue counting four more (24, 25, 26, 27) instead of starting at 1 and counting to 27. Children are exposed to lots of language in mathematics and it is important they understand the terms, more and less and before and after to achieve this growth point. This growth point is usually achieved between 6 - 8 years old.

Parent tips for working with a child to achieve this growth point.

  •  Focus on counting which moves from one decade to another (e.g. Asking your child to start at 45 and count forwards by ones. The focus is to see if they can go from the 40’s to the 50’s confidently. Then ask them to start at 45 and count backwards by ones. The focus is to see if they can go from the 40’s to the 30’s confidently).
  • Constantly use lots of terms to ask the same counting question. Children need to be exposed to as many mathematical terms as possible in order to answer questions. Some children get ‘stuck’ on a growth point as they do not understand a particular mathematical term. (e.g. Tell the child to pick any number between 30 - 120. Once they have picked a number, say 67 ask them what is one less or the number before or what it would be if you took one away, or what it would be if you subtract 1. The child needs to understand all these terms. Then you could ask them what is the number after, or one more, or add one or plus one)
  • The biggest error that a child usually makes to keep them from achieving this growth point is knowing the numbers beyond 109. So many children will say 120 after 109. It is important for children to see the numbers in a line (not a table that is usually on posters) so they realise that numbers increase in size as we count forwards. So many children only have experiences of counting up to 100 and this really limits their abilities. Even though we as adults recognise the patterns of numbers children do not. Encourage your child to start at 105 and count forwards or 116 and count backwards. Get your child to make the number 109 with blocks and 120 with blocks and show them that there are many numbers between.
  • Allow your child many opportunities to make numbers over 100. Children need to know numbers in a triad. They need to know what the numeral is, what the name of the number is and what the amount of the number looks like. A lot of parents and teachers take away the maths equipment too early and this creates a lot of set backs to children’s mathematical development. Encourage your child to prove their understanding with equipment.

Growth Point 4 - Can count from 0 by 2’s, 5’s and 10’s

This growth point is about the child recognising simple number patterns by counting by 2’s, 5’s and 10’s. We need to help the child realise that we count by these patterns to make counting quicker and more efficient. We need children to learn how to choose the the most efficient type of counting for each situation. For example if we had 11 pairs of socks, we could count the socks by 2’s or if there were 6 bunches of 10 flowers, we could count the flowers by 10’s) This growth point is usually achieved between 7 - 9 years old.

Parent tips for working with a child to achieve this growth point.

  • Get your child to watch you count objects during your everyday life by 2’s, 5’s and 10’s. You could also get your child to do it with you. (pairs of socks when doing the laundry, counting how much money you have in coins - 10 cent pieces or 5 cent pieces)
  • Rote count by 2’s, 5,s and 10,s in the car, or on a walk and any other time you have quiet time with your child. Make sure you always go past 109.
  • Write out a number line and get your child to colour in all the numbers that you say when you count by 2’s. You can do the same with 5’s and 10’s. Try not to use a table (where numbers are in rows and after each decade the numbers start a new row) as children need to see that numbers are linear (increase in size as a pattern) We have a lot of children who say that the number before 39 is 29 as it is the number on top of it in a table.

Growth Point 5 - Being given a non zero starting point, count by 2’s, 5’s and 10’s

This growth point is about the child using their knowledge of number patterns to count by 2’s, 5’s and 10’s starting from a number that is not usually in the rote learnt pattern. For example the child can start at 4 and count by 5’s. (4, 9, 14, 19, 24, 29 etc) They need to be able to count by these patterns in both a forward and backward direction. This growth point is usually achieved between 8 - 10 years old.

Parent tips for working with a child to achieve this growth point.

  • Practise saying number patterns with your child by saying one number each. Make sure you start at a number that is not normally in their rote learnt pattern. You could take turns at saying numbers in the pattern.
  • Write out a number line and get your child to colour in all the numbers that you say when you count by 2’s from a number that is not normally in the rote learn pattern. You can do the same with 5’s and 10’s.
  • Get the child to continue patterns that you have written down. Eg 86, 76, 66, 56, 46, _____,  ________, ________
  • You can expose your child to the concept that there are numbers that are smaller than 0. Talk about temperatures in cold countries that can get to minus degrees. A thermometer is a number line that goes vertical instead of horizontal.

Growth Point 6 - Can count from a non zero starting point by any single digit number and can apply counting skills in a practical task.

This growth point is about the child being able to count by 1’s, 2’s, 3’s, 4’s, 5’s, 6’s, 7’s, 8’s and 9’s from any number. They can confidently count forwards and backwards by any of these number patterns. They also need to be able to use this skill and apply it to solving mathematical problems. It is important that children are able to communicate how they solved a problem in order for them to have a deep understanding of it. This growth point is usually achieved between 9 - 12 years old.

Parent tips for working with a child to achieve this growth point.

  • Practise saying number patterns with your child by saying one number each. Make sure you start at a number that is not normally in their rote learnt pattern. You could take turns at saying numbers in the pattern. (Count forwards and backwards by 3’s, 4’s, 6’s, 7’s, 8’s, 9’s)
  • Ask your child questions about multiple numbers before and after a given number. (e.g. 3 more than 124, 6 less than 133, 8 after 89, 9 before 121)
  • Practise counting negative numbers. (For example start at 3 and count backwards by 4’s. : 3, -1, -5, -9, -13, -17)
  • Make up mathematical problems that require counting to solve it. (Eg There were 6 tables at a wedding. One table had 9 people on it and the other tables had 8. How many people altogether?)

Growth Point 7 - Counting using fractions and decimals

This growth point is about the child counting by numbers that are less than a whole. They can confidently count using fractions and decimals. For example they can count by halves, quarters, fifths, sixths, tenths etc. They can successfully count forwards and backwards using fractions and decimals. This growth point is usually achieved from 10 years old to the beginning years of high school.

Parent tips for working with a child to achieve this growth point.

  • Practise counting by fractions with your child. Talk about pizzas or cakes to give them a context (a real life situation) You could take turns at saying numbers in the pattern. (half, one, one and a half, two, two and a half, three) or (by fifths - one fifth, two fifths, three fifths, four fifths, one, one and one fifth etc)
  • Practise counting by decimals with your child. You could take turns at saying numbers in the pattern. (count forwards by 0.6 starting at 0.7 : 0.7, 1.3, 1.9, 2.5)

The main focus of teaching your child mathematics is having lots and lots of conversation about it. Mathematics is very literacy (English) based and in order for your child to succeed they need to have a deep understanding of many mathematical terms. We have moved on from textbooks and rote learning that we experienced when we went to school to having a focus on children learning concepts for a deeper knowledge and understanding of mathematics.

I hope you have found this first blog in the series “How To Teach Mathematics To Young Children” useful and informative. The next blog will be about The growth points of place value, which is another domain that has a large impact on a child’s number sense. I hope you have been able to work out what growth point your child is at in counting and have found the strategies that you can do with your child useful to help them move onto the next growth point.

Thank you for taking the time to read my blog about counting. I look forward to hearing any thoughts or comments that you have.

Until next time …….

Kelly Pisani

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