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**:

- Memorizing the names of the numbers
- Saying them in a certain order
- Connecting each number-word to an object to be counted
- Recognizing the pattern of the numbers
- Understanding that the final number said is the total quantity in the group
- Knowing that the order in which we count the objects does not affect the total
- Believing that the number of objects in a group stays the same even after they are rearranged

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:

- Whatever you do, make it natural, make it fun.
- Don’t worry about correcting the "wrong-ness" of the child's answer; they will make mistakes and these will diminish over time as they develop conceptually.
- Be sure to stay inside your "comfort zone."
- Kids are not identical; they pick up stuff at their own rate.
- All children go through the same stages, but at different rates and ages, and each child progresses at different rates with different concepts.
- Not all 2 year olds are at the same stage.
- Understanding comes after (sometimes long after) memorization; tragically with math, sometimes understanding never comes.
- Early math needs to be fun, and should occur only in a play environment; we don’t want to turn them off at an early age, give them anxieties because they are not doing it right.
- There is no need to do “formal” math with your child. Keep in informal, natural, and make it part of the set of activities your child does.

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:

- The first stage is from birth to about age two, where our five senses – taste (remember how everything goes into the mouth), touch, sight, hearing, and smell – are the way we makes sense of the world (develop our minds).
- From about age two to about age seven, we are in the pre-operational concrete stage where we need to physically manipulate things to understand them. Oral language develops during this stage, but visual perception overpowers reasoning.
- Next is the operational concrete stage, from about age 7 to about age 11, where we have the ability to think abstractly, but only about concrete or observable things. Also, during this stage, written language and number symbols help us develop our understanding of the world.
- At about age 11 or 12 we enter the formal operational stage where we start using abstract logical reasoning to learn new ideas and further our development.

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.

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.

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

- For all activities, use around the house items in a natural setting.
- Use appropriate vocabulary while doing activity (For example, there are more blue blocks than red blocks, this doll is bigger than that doll, etc.).
- Sing math songs and rhymes to help learn the number sequence through about 20.
- Making math fun is the single most important preparation you can give your child for kindergarten. Don’t worry about correcting the "wrong-ness" of your child's answers; they will make mistakes and these will diminish over time as they develop conceptually.
- Avoid saying, “That’s not right” or “You’re wrong.” Instead, you can say “My turn,” and do the task with your child modeling the “correct” response.
- Also, avoid giving impossible tasks which are beyond your child’s readiness.

- Sort objects by 1 attribute (e.g., color, shape, size, material, pattern, texture, etc.)
- Sort objects by 2 attributes (e.g., red triangles, large cars, soft toys, etc.).
- Sort objects according to function (e.g., things we drink with, things we use to write, things we ride in, etc.).
- Identify items that belong or
don’t belong to a group (e.g., three squares and a circle; a boat, a car, a bicycle, and a
carrot; etc.).

- Compare opposites (e.g., long/short, big/small, etc.).
- Order 3 objects (e.g., small, medium, large; big, bigger, biggest).
- Order 4 to 7 objects (e.g., stacking cups, funnels, blocks, books).
- Do reverse ordering (e.g., biggest to smallest, longest to shortest).
- Do double ordering (e.g., big doll gets big chair, small doll gets small chair, etc).

- Match 2 identical items (e.g., these socks go together, this puzzle piece goes into this shaped slot, etc.).
- Pair items from two groups. There may be the same number of items in each group (e.g., 4 red and 4 blue blocks, 3 spoons and 3 cups), or different number of items in each group (e.,g there are 4 people in your family to eat, but many forks. You pair one fork for each person; you have 3 red blocks and 10 blue blocks, and you pair one blue with each red; etc.). Do not worry about “how many” more or less there are; that comes much later.

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.

An extremely useful website for both concepts and activities is www.pbs.org/parents/ .

* 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, http://www.talkaboutmath.org/

*** 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; http://ecrp.uiuc.edu/v4n1/kirova.html

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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