Category Archives: Math


I am going to give you 2 different pathways for different approaches to learning arithmetic. I will use 2 different colors to separate the approaches. Neither approach is better or worse than the other. They are just different and both will be covered by following one pathway then coming back and doing the secondary pathway. Or you could simultaneously use both approaches.

The first approach is number fact mastery by counting numerals. When many people think about mastering numbers facts they fear the dreaded flash cards, rote memorization and endless drills. However there are some simple strategies that can be used to teach fact mastery that are fun and enjoyable.

Although this is the fastest and most direct way to master number facts there is also a downside. A child might well learn that 6+3=9, however they may not have a concrete idea of what this really means.

The alternative pathway to reach the goal of fact mastery relies heavily on manipulatives. This is the most common way that arithmetic is taught to children in the early school years. It can involve a lot of play and group work, which often equals a lot of fun, and better learning outcomes.

However I have found that there are several concepts that very young children struggle with. These included bijection and cardinality. Although some children seem to grasp these concepts easily and at a young age the majority of children need a lot of practice and time.

My recommendation is to consider the age and the ability of the child. Pre-school and Kindergarten aged children will benefit mostly from the second pathway because it strengthens their number sense.

Now that a child is able to count by reading numerals proficiently to 20 both forwards and backwards we are going to move onto the next step.

This is an essential step that separates reciting numbers or reading numbers type of counting from counting an actual amount.

Step Three

Bijection and Cardinality

Bijection, also known as one-to-one correspondence is the ability to point at an individual object and know that the object represents 1 counting word.

For a child that has been using a number line you can easily check their proficiency by removing the numerals written on a number line and replacing them with a dash or preferably a dot. If a child is struggling you can remove one or two numbers at a time and have them continue reading some of the numbers as they count up and down.

When using manipulatives (objects), while counting it is best to lay out small amounts of objects. Start with 3, and then 5, and then 10. Only move to larger amounts when there is mastery. Have the child first practice touching each object and saying their corresponding number words. Try and encourage them not to point to an object twice or skip an object. You may need to do the pointing yourself, or guide their fingers. You can also practice bijection by moving the objects from one side the table to the other. Or have them line up the objects.

If the child is still struggling have them start with a group of small objects and have them move the objects one by one to a corresponding numeral on a number line. Make sure this is done in order and no numeral is missed.

Cardinality is the ability to recognize that the last number counted is the answer. Or when using manipulatives it is the total amount.

It sometimes takes children a while to stop at the end of a counting sequence because they are still reciting numbers. Each time you count, either using a number line or objects, make sure to ask your child what the last number they counted was. If they are unsure, as is common, tell them. Explain that the last number is like the destination and it is important to stop there and take note of what it is. 


Step Two
“Counting on”

Once a child has mastered counting forwards and backwards from 0-20 it is time to introduce “counting on.” Continue playing the same activities that were done along the large gross motor number line and the smaller fine motor number line. However instead of starting at 1 the child is to start at any number between 2 and 19. To us it seems an easy task to require a child to demonstrate. However to a young child it is a complex request. Work first on counting forward, then backwards.
Ensure that the number line has numerals written so they can be used as an aid. Once the child starts to garner some confidence you can remove the numerals.

You can also play a counting game where you say the first few numbers in a sequence and then have the child continue counting on. You can also go back and forth saying a number or even several numbers each. All while counting forwards and backwards. Also ask questions such as what comes before or after a certain number.

You can also set out counters, or put some in a bowl. Then have your child continue adding counters. Try to discourage them from counting any that are already set out.

Please hop on over to Push Play Learning for some more information and ideas about counting on. She has a great App review and a YouTube video to give you more ideas.


Step One

Counting Forwards and Backwards


  • Chalk or tape to create a number line.

  • A drawn or printed number line.

  • Counters (cheerios, m&ms, blueberries, little erasers, poker chips etc. NOT coins)


  • Count forwards and backwards from 0-10
  • Count forwards and backwards from 0-20

Sing and count the numerals from 1-20 and from 20-0 using the tune “Mary had a Little Lamb”.

Create a number line on the floor with chalk or painters tape and have the child jump up and down the number line, jumping on each numeral. Start with a few numbers at a time and increase to 20 as the child shows proficiency.

Use a small number line on a piece of paper and have the child count up and down the numerals with their finger.

Lay out a small edible treat, counting up as the items are laid out, then count down as the items are eaten.

You could also use non edible items and remove them as you count down by putting them in a bowl, a dump truck or by feeding them to a toy.

You could lay the items on the numerals for the number line and eventually remove the number line completely.

By using this method you are showing the child that the amount that corresponds with a number (and a numeral) is complete. If the items are already laid out and you are asking a child to count them they may be confused when you point to one item and say this is 1, this is 2, this is 3, this is 4 etc. 4 is never a single item, it is 4 complete items.

Still yearning for some more ideas and fun ways to learning counting backwards? I highly recommend that you hop on over to Push Play Learning. She has YouTube playlist of math songs about counting backwards.
Make sure you check out her other playlists too. She makes it so easy to get a little free time while still creating a learning opportunity for your child.



I know that many people understand the value of teaching their child to read before they enter formal education. Why not? Reading is crucial for every school subject and everyday life in out modern literate society.
Initially teaching a very young child to read appears to be an intimidating and momentous task. But it is journey that many parents are willing to take.
I have written this post about how I taught my own toddler to read.

However it isn’t only early reading literacy that I wish to address. I also want to write about early math literacy. According to to this article about a study from the University of Missouri; the math ability of a child during their formative preschool years is a pretty reasonable predictor or their future math success.

In a nutshell the more confident a young child is with math, the better they will do in math. They might even enjoy it. I know I do! It might even be possible that earlier math literacy will be enough to thwart that dreaded math phobia or anxiety. More children may even consider or pursue careers in mathematical fields.

Even though I love math, and I have always “got it”, I know that there are many parents that are overwhelmed with math and don’t know where to start with their young child. Many parents start by teaching their child to count, which more often that not, is just recitation of numbers. Not counting at all. They fail to implement bijection (one-to-one correspondence) and create awareness that the last number counted is the total amount.
I know parents that have started with a subtitizing or dot program such as Doman Math or Brill Kids Little Math. At the end of the program, or when their child ages past subtitizing age, they become lost and discouraged and don’t know where to turn next.
I also know parents that are unaware that early math literacy is much more than counting and math facts.

My goal is to create a series of simple posts that cover basic steps that can be used to increase a young child’s math literacy. I don’t claim to be an expert. However I have amalgamated information from what I have studied, others experiences, and my own experience. I am currently working with my 3.5 year old son who is doing 1st grade math at a slow, steady and enjoyable pace.

My Math beyond counting series will cover such thing as:
Number sense and numeracy. Including but not limited to addition, subtraction, multiplication and division.
Spatial relationships