This work is available here free, so that those who cannot afford it can still have access to it, and so that no one has to pay before they read something that might not be what they really are seeking.  But if you find it meaningful and helpful and would like to contribute whatever easily affordable amount you feel it is worth, please do do.  I will appreciate it. The button to the right will take you to PayPal where you can make any size donation (of 25 cents or more) you wish, using either your PayPal account or a credit card without a PayPal account.

Teaching Children to Count Without Counting:
i.e., to Recognize Numerical Quantities By Patterns and Groupings, Not by Counting the Individual Objects in the Group

Rick Garlikov

First, for Older Kids and Adults: The number of dots on dice or dominoes (or hearts, clubs, spades, diamonds on playing cards, apart from the numeral written in the corners) is often easy to recognize by the patterns of the dots, two pairs for four, two pairs with a dot in the middle between them for five, as in:

And adults sometimes recognize quantities from patterns or groupings of their components.  For example I have eight "Aerobie Orbiter" boomerangs, which are plastic and rubber triangular shape boomerangs.  Four of them are orange and four of them are a sort of magenta  or iridescent pink .  On days in which the wind is erratic in direction or velocity, the boomerangs don't always return to me or near each other, so I have to walk around to pick them up.  With really changing and gusty wind conditions, they can scatter over a fairly large distance, particularly if I aim them in different directions to begin with.  But I need to be sure I have all 8, and counting them each time I pick them up or look to make sure I know where they all are before starting to walk around to pick them up, is faster by groupings than by individual counting.  The following pattern scattered on the ground is easy to see I have all eight because it is fairly easy to see (after having done this a while) there are four orange ones to the left of center and four magenta ones to the right.

It is also fairly easy to see there are all eight when they line up in the following way because you can see at a glance there are four orange-and-pink pairs relatively close to each other:

Or sometimes they can land in the following pattern of 3, 3, and 2:

Or they can land 1, 4, and 3 like this:

When they are in my hand, it is easy to see the various combinations, which might be any of the following or other combinations:


Four pairs of orange and pink ones.


Either 4 and 4 or by color it is two sets of three of one color to make six, and the fourth one of the other color to make either 4 and 4 or two sets of 3 to make 6, plus the two opposite color ones.


Same as just above, but arranged differently in terms of the color positions.


On the left four, it is easy to see 4 by a pair of each color, and on the right it is easy to see four by two pairs of pink/orange or by the two pink and the two orange alternated.


Same as above, but with different grouping of the two pairs on the left -- pinks in the middle between the two orange outer ones.


Either see these as two sets of 3 each on the outsides, with a pair in the middle, for a 6,2 grouping of 8; or
see the left two groups as making 5 from a 3, 2 grouping plus a set of 3 more on the right to make 8 out of the 5,3 grouping.


A 1, 5, 2 grouping to make either 6,2 or 5,3


Now, to Teach Children: It is important that the children first know how to count individual objects at least to ten, and preferably higher.  It is important that they understand what it means in this sense for there to be three things or seven things or ten or twenty things.  The method of counting by grouping is not to teach how to count in general, but how to count faster by recognizing groups.  So the concept of a numerical quantity must already mean something to the child.  None of this will work, as indeed counting individual things itself doesn't work, if children are still at the stage where they count wrong (by skipping numbers or by just naming numbers randomly or by counting two or three numbers as they point to each object instead of giving each item the next, and only one, number).  And notice that even though adults can add the groups or multiply the number of things in a group by the number of groups, what I am talking about here is not necessarily about adding numbers or calculating by multiplication, but combining the groups automatically as when one looks at the grouping of dots on dice as above.  So it is important that children work their way up, as explained below with different groupings of higher and higher numbers.  They can typically see two as a group, then three, then four.  Then they need to see the different groupings that comprise 5 (see below), then 6, 7, etc.  That way they see the quantities at one time as a group rather than having to count and add.  E.g., in seeing the dice above, you can either see the four immediately and the five immediately or you can see the four as being 2 + 2 and adding them, and you can see the five as being 2 plus 1 plus 2 or as being the four plus 1, etc.  We do that either way, particularly with larger and larger groups, but I am talking here about teaching children (through fun practice) to see the whole groups as patterns, with minimal, if any, addition required.  E.g., as an adult your might see the following grouping as being either seven immediately or as four immediately and three immediately that you add together.  The idea is to help the child learn it both ways, but emphasizing here their seeing the seven immediately, through fun practice and familiarity, as just the grouping in this pattern.


So first start out with, say four, blue poker chips and let the child count them, if necessary, to see how many there are.  And suppose the child counts them 1, 2, 3, 4 and says there are four chips.  Without adding or removing any, just clearly push them all to some place and ask if there are still four.  Even if the child has to count them again, that is fine.  But if you keep doing that, at some point the child will see that there are still four, no matter where you put them all or if you separate them into groups of two pairs or three and one, etc.  As long as you don't take any away or add any, the child will see eventually they are always four, no matter where or how they are arranged.  You can even stack them together, and let the child see that doesn't change the total number of them.  It is really important they see that the total number doesn't change, so that they can become comfortable seeing the different patterns there are for that number of things.

Then particularly divide them into two and two, and then into three and one, and into one, two, and one.  You might even want to put a divider of some sort, like a ruler between the two pairs or the group of three and one, or two dividers between a left one, two middle ones, and a right one, or between two individual separate ones and the group of two together.  Then move all four to one side of a divider and show them there are none on the other side, and yet you still have four chips.

Then tell them you are going to change out two of the blue ones for two red ones, and then ask them how many chips there are in the group.  Still four.  So color doesn't matter.  Or you may get some other objects and exchange them one for one with the chips to show them that the quantities don't depend on the color, the size, the shape, what the object is, etc.  You don't have to say a lot or go into an explanation; just let them see it and make sure they see there are still four things.

Then do the same thing for larger groups; five things. But be sure they can see five either as 2, 1, 2 (as in dice) or as 3,2, and 4,1, or 3, 1, 1 no matter where they are; e.g., 2, 1, 2 might be in front of them as 1, 2, 2 or 2, 2, 1. 
2, 1, 2                

or as            


1, 2, 2              

or as             


2, 2, 1

or as


Again, the idea is to help the child, and give the child fun practice, seeing any of these as 5 immediately, not having to either count or add.  That is the point of this.  As adults we do this both ways, particularly with larger groups, but children can go ahead and learn the pattern way before learning to add numbers, and this might help them learn to add and subtract (and later to multiply) without just having to memorize tables or flash cards, etc.

Then six; then seven; then eight.  Group them all in lots of different ways and let the kid play with them and group them him/herself in whatever combinations s/he wants to, each time saying these are still eight (or seven or however many is in the group).  Let the child play with as large a group as s/he wants to.  They may learn to recognize even larger groups by patterns than you can.  I have a large set of (91) dominoes that has dominoes up to 'double-twelves'.   Playing dominoes with a child is a good way to help them learn quantities by pattern recognition; also dice games. The more they play, the faster and more easily they will recognize the quantities and start to see patterns of various sorts.

Or you can use poker chips and stack them and show the child that if you have a stack of 10 chips, you can tell a stack of equal height will also have 10 chips without having to count them.  And then let him/her count them to verify it.  And if the child can count by tens, s/he can quickly count a large quantity of poker chips by stacking them in groups of ten and then make equal stacks and count up the chips by tens (ten, twenty, thirty, forty) etc.  Or if the chips are lying down in stacks in rows in a tray, your might be able to count them by 50's once you figure out there are 50 chips (five stacks of 10) per row in the tray.

Or you can ask the child who knows that five and five are ten to show you five chips the quickest possible way from a stack of ten to see whether s/he can figure out to divide the stack of ten into two equal stacks.   All of this gives practice with numbers in a lot of different ways, and is generally fun for a child.  If you later start with stacks that are exponents of 2, you can take a stack of, say, 16, divide it in half to get two stacks of 8; divide those in half to get four stacks of 4 each (each stack being 1/4 of the original stack of 16), then in halves again to get 1/8 the original stack, etc.  Or you can start with a stack or grouping of 12 and have them make three equal stacks (or groupings) to see there are four in each, and that those are each thirds of the original; or a stack or grouping of ten and make five equal groupings (each 1/5 of the original) etc.  (I said use 12 for the thirds grouping instead of 9, because they might confuse it being thirds because you end up with 3 in the group, which is accidental -- the three in each group or stack is not what makes each group be called a third of the original group or stack.) It is an easy way to begin to teach the concept of fractional amounts, though there is more to fractions than this, and fractions in general can be far more complicated, even conceptually (not just in doing calculations), than most adults realize (see More About Fractions Than Anyone Needs to Know and also Teaching "Quantity" Fractions).

If a child does this, they will also notice all kinds of things themselves about patterns or quantities you might not know yourself or that you and they will just know intuitively, such as order, arrangement, and manner of grouping not mattering for the combined total rather than through having to memorize anything.

This work is available here free, so that those who cannot afford it can still have access to it, and so that no one has to pay before they read something that might not be what they really are seeking.  But if you find it meaningful and helpful and would like to contribute whatever easily affordable amount you feel it is worth, please do do.  I will appreciate it. The button to the right will take you to PayPal where you can make any size donation (of 25 cents or more) you wish, using either your PayPal account or a credit card without a PayPal account.