Math Concepts and Activities for Three and Four Year Olds

Parents discussing math readiness with Jason Frand, PhD

May 12, 2009


The Counting Challenge 

Consistently assigning the correct number to a group of objects is a major accomplishment for a pre-kindergarten child, but one which we all achieve.  Let’s look at what is involved when counting so that we can understand why kids make the mistakes they do as they learn to count.

Counting successfully involves**:

Frances Stern has a fabulous web site with video clips illustrating what kids do when learning to count and suggestions for how parents can respond. I recommend you view her site Talk About…Math! 

But, counting is only one destination along a mathematical journey, one which is achieved between ages four and six for most kids.  There are several other math readiness concepts which lay the foundation for counting and kindergarten math.  My goal is to share some information about how your child learns and the three concepts critical for counting and learning mathematics.  My hope is that once you understand these general ideas, then you can select the specific activities that you can do on a day-to-day basis to support your child’s development.  The ideas are organized around three topics:

1.  How kids learn
2.  Critical early childhood math concepts
3.  Activity categories

The most important take-aways from our discussion are these general guidelines:

1.  How Kids Learn

The theory of intellectual development that has very significantly influenced education is that of Jean Piaget.  Piaget developed an extensive model of how our knowledge develops.  At the core, Piaget says we construct our knowledge through interacting with our environment.  We go through specific stages, and we can not progress to the next stage until we go through the previous stage.  Overly simplified, the stages can be described as follows: 

There is a second theory of knowledge development that complements Piaget’s ideas, namely Lev Vygotsky, who (in just a couple of words) says we learn through social interaction and language.  The implication of these theories for your pre-school child is that you should create a rich environment in which your child can play, be a safe explorer, investigating and testing everything, and one in which you provide verbal support, labeling, explanations, discussions. 

The more informal, natural your interactions, the better.  In other words, you don’t need to walk round and point to things and tell your child what it is and tell them to repeat it, you should label things naturally.  For pre-language children, you tell them what you are doing (I’m changing your diaper and wiping your bottom;  label items when you use them: spoon, bowl, book, etc.)   For the three year old its find to say “this toy car is bigger than that one” and with your four year old, after reading some books its fine to say, “so how many books did we read?” and try to count them (but don’t worry about counting errors –). 

Our discussion this morning is geared for pre-kindergarten children, mainly ages three and four, who are in the pre-operational concrete stage.  For these children, direct interaction with the physical environment is their primary learning mode.  For them, seeing written symbols is less important than the physical manipulation of things.

2.  Critical Early Childhood Math Concepts

There are three critical concepts (classifying, ordering, and matching) that your child must internalize before they can be successful with kindergarten math and beyond.

But what do classifying, ordering, and matching have to do with understanding numbers and math?  To answer this question, let’s begin with a non-math example.  We all understand “the concept of chair” – there is a seat, some legs, and a back.  We’ve gain that understanding through countless encounters with hundred’s of different variations of what we mean by a chair. We can think of the abstract notion of a chair as a generalization of all sorts of things that fit the description of a chair.  For example, if the object we were looking at had a seat and legs but no back, we’d call it a stool.  If he object has four legs, a back, and a long rectangular seat, we’d call it a bench.  If the object is small (child size), with a picture of Elmo with Elmo’s lap like a seat and his body as a back, and sitting on the floor, we’d call it a chair.  We use classification (the result of sorting things by some criteria) to put items into the category “chair” or “not chair.”

Numbers have a similar property.  We use classification (the result of sorting things by some criteria) to create a category which we then label with a number.  Let me use the number 4 as an example.  For a child to understand “the concept of four” the child needs countless encounters with categories that share the property of four:  four books, four people, four toys, four animals, four houses, four chairs, four different things, etc.  

But classifying things into a category of four isn’t enough to understand the concept of four.  Size – larger than/smaller than, greater than/less than, etc. – the ordering of things, is also important.  To understand the concept of four the child needs to understand that a category of four is less than a category of five and more than a category of three.  Being able to arrange things in order, by size, amount, weight, loudness, etc., as well as classifying, is needed to understand the concept of four.  

But classification and ordering are still not enough to understand the concept of four.   We also need a way to assign a number to the category.  We use matching to make the assignment.  The way we label a category of four is through matching a memorized sequence of labels – the names we’ve given numbers: one, two, three, four – with the objects in the category.   And, the matching we make is a particular type, a one-to-one matching: one number label for each item in the category.  If you have two children, there is a one-to-two matching between you and your children.  In a marriage, there is a one-to-one matching between spouses.  But children don’t start with doing a one-to-one matching, especially a matching in which they need to use a sequence of words they’ve memorized to be paired with physical objects of various sizes, shapes, colors, etc.  As with classifying and ordering, your child needs lots of opportunities to make single and multiple pairing of things before they can count correctly.  

In summary, for a child to understand the concept of a number, they need lots of experience with sorting things into categories, arranging things in order, and making matches and pairing of things.  Things, real tangible physical things.  For young children, under the age of seven or eight, at the pre-operational concrete stage, working with real physical materials is essential for their mental development.  Playing with real things, blocks and toys, water and sand, is essential for the development of the ideas related to classifying, ordering, and matching.

One of the major developmental milestones Piaget identified for a child to move from the pre-operational concrete stage to the operational concrete stage is the concept of conservation.  Conservation means that the quantity doesn’t change, even though the shape (visual clues) change.  For example, if you show your child two balls of play dough of the same size, and ask if there is the same amount of play dough in each, they would say “yes.”  If you then roll one ball into a hot dog and ask is there the same amount of play dough in each now, the child would say “no,” the longer one has more (or the higher one has more).  Roll it back into a ball, and presto, they are the same again.    Or, if you have two identical glasses of juice, both filled to the same level, and pour one into a wider-shorter glass, and ask your child which has more now,  they would probably say the taller one.  The visual clues overwhelm any “logical reasoning” that the amount of juice hasn’t changed.  Or, if you  have 7 red block and 7 blue block and they are right next to each other and ask if there are the same number of each, they say yes.  Then spread the blue blocks apart and ask again if there are the same number of each.  The pre-operational concrete stage child would say there are more of the blue.    Push the blue into a tight circle, and suddenly there are more of the red.  Again, the visual overpowers reasoning.  (And, now you know why your child like to break a cookie into a few pieces.  That way they always have more cookie.)  But, with experiences over time, the conservation milestone is achieved (along with other developmental changes) and we move into the operational concrete stage.

Conservation is a critical concept which takes countless experiences over years with soils, liquids, foods, toys, shapes, and so many other things, to acquire.  So, even as our children acquire the ability to classify, order and match things, until they conserve, the category, or order, or pairing up is easily modified by simply re-arranging the items, say just spreading them out, even though the essential property used to group, order or match hasn’t change.  So, for our children, mastering number understanding requires mastering some very fundamental ideas, and that’s where you can play a role.

Before we discuss some activities, let me respond to a question asked at the discussion:  Why is water and sand play important?  This ties directly into conservation.  Think about what your child does when playing in the tub with a couple of containers of different sizes.  When they pour water from a larger container into a smaller, they pour until the larger is empty, not concerned about overflow.  If  they pour from the smaller, they refill until and pour until the larger overflows.  It isn’t until much later that they are interested in pouring to match the capacity of the container.  And, if they have a few containers, some that are identical, and some that look different but with the same capacity, they can discover as they pour that the quantity doesn’t overflow even though the containers different in appearance.  And, these experiences need to extend to solids, like sand, and with blocks that fit into the same space in different ways, etc.   

3.  Activities***

Let’s now go through a list of categories of activities you can do with your child.  First some general guidelines:

Concept area:  Classification – Sorting Objects

Concept area:  Ordering 

Concept area: Matching (pairing items) and One-to-One Correspondence

I’ve been reviewing the research on these developmental ideas, and the consensus is that these are ideas which cannot be taught, but will be acquired by the child when ready. For the child, classifying, ordering, and matching, are only learned through countless experiments, trail and error, random play. Kids are not identical; they pick up stuff at their own rate.  So, understanding of sorting, ordering and matching are related to their stages of development, not their age Just as we can’t eat for our children, we can’t learn for them.  And, just as we can’t control how fast or tall they grow, we can’t control how fast or how much they understand.  Understanding comes after (sometimes long after) memorization; tragically with math, sometimes understanding never comes.  HOWEVER, you can and should create an environment where your child can explore these concepts and thus learn them. and facilitated interactions which direct your child toward the concepts. 

Useful Website 

An extremely useful website for both concepts and activities is .

* This paper was written based on a workshop organized for the Culver City Mom’s Club.  The original announcement:  Join Donna & Isaac for a morning of fun math activities and discussion with special guest Jason Frand, PhD (in charge of Math Olympics program at Linwood Howe Elementary).  The little ones can play inside, outside, upside-down while parents discuss early math activities for preschool aged kids. All are welcome (even if you are kidless).

** List copied from “Talk About…Math!  A Guide to  Raising Children Who Can Do Math” by Frances Stern,

*** Adapted from “Learning to Guide Preschool Children's Mathematical Understanding: A Teacher's Professional Growth,” by Anna Kirova and Ambika Bhargava, Journal of Early Childhood Research & Practice, Spring 2002;

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Jason Frand, PhD.
Math Olympiad Coach and
Los Angeles County-wide Math Olympiad Tournament Chairperson
Linwood Howe Elementary School
Culver City Unified School District
Adjunct Assistant Professor, Retired
Assistant Dean and Director, Retired
UCLA Anderson School of Management
prepared May 12, 2009