This account describes how number develops in the preschool range, from about three to five years, and provides a more detailed overview of the various skills and understandings that children need in order to achieve competence in counting. A curriculum with intentional teaching should address all of these topics.
Children Develop an Everyday Math
Context and overview
Young children, even infants, develop essentially non-verbal basic concepts of quantity: more/less, order, same, and adding/subtracting. Children learn most of these things on their own, without much adult help. Children often use these concepts in everyday life, for example, to determine who has more or less ice cream. Children’s concepts and procedures are useful under certain conditions but need to be enriched. (Perhaps that’s why number was invented: the shepherd needs to know not only that he has a lot of sheep, but exactly how many.) This is what children know and what they need to learn at roughly ages three, four and five.
Children need to be able to see that there are more objects here than there. They often solve this problem not by counting but by physical appearance. "This flock of geese in the sky must be larger because it covers a greater area than does the other flock." This approach is often adequate but can lead to wrong answers and confusion.
Judgments of more or less are sufficient for many purposes, but sometimes a comparison between more than two things needs to be made. Thus the idea of order, which includes subtle ideas:
- In a group of three objects, the second item is larger than the one preceding it but smaller than the one following it.
- Also, the item that was first can become last under a new order.
Again young children tend to rely too much on appearances to solve the problems.
The idea of same number evolves, even without adult assistance, through several stages:
- The first step is seeing that two groups identical in shape and arrangement are also the same in number. Thus, if a brown bear and a yellow canary are placed directly below another brown bear and yellow canary, both rows are the same in number (as well as in shape, color, and arrangement).
- The second step is seeing that two groups differing in color or shape can still be the same in number. Thus, if a brown bear and a yellow canary are placed directly under a pink pig and blue heron, both rows are the same in number (and arrangement, although they differ in shape and color).
- The third step is seeing that two groups differing only in arrangement are the same in number. Thus, if a brown bear and a yellow canary are not placed directly under a pink pig and blue heron but instead lie elsewhere, both groups are the same in number (although they differ in arrangement, shape and color).
- The fourth is seeing that one group, when rearranged, has the same number as it did before it was moved around. Thus, if the child first sees a brown bear and a yellow canary in one arrangement, which is then transformed, the child realizes that the number did not change from what it was before the rearrangement.
- The fifth is first seeing that two amounts are the same number when they look similar, for example five eggs in a row and five egg cups in a row both have the same number. But then if there is a transformation (for example spreading the eggs apart so that the line of eggs is longer than the line of egg cups), the child has to be able to understand that the eggs and egg cups are the same in number even though the two lines look different.
The idea of adding as resulting in more and subtracting in less
Children learn that:
- When you add something to an existing set, the result is that you have more than you had at first.
- If you start with two groups of the same number, and by magic (while the child is not looking), one set is now more numerous than the other, you must have added to one or subtracted from the other
- You don’t have to count to arrive at these judgments concerning more and concerning addition and subtraction: you can solve the problem by reason alone.
Later instruction needs to build on all of these ideas when written numbers are introduced.
Learning the Counting Words
Context and overview
In everyday life, we use counting words all the time, selecting items from the supermarket (“we need two bananas”) or playing “10, nine, eight, … blast off!” Children love counting as high as they can, like grown-ups. They may even be interested in the name of the highest number. Fluency in the counting words aids later computation.
Rote memory plus
At first, children memorize the counting words from about one to 10 or so. But their learning doesn’t involve only memory. Children learn some ideas and rules about number too, namely that proper order is essential; numbers are different from letters; and you are not supposed to skip or repeat numbers when you count.
Later, children pick up the underlying structure of number: ten is the basic unit (20, 30, etc.) and we tack units onto the tens (twenty-one etc.). The rules for saying the English counting words from eleven to nineteen are especially hard to learn because they are poorly designed. Eleven should be "ten-one," just like twenty-one. Fifteen should be "ten-five," like twenty-five. The East Asian languages get this right, but English and many other languages do not. By contrast, English is fairly well designed for the number words beginning with twenty. Each of the tens words resembles a unit word. Forty is like four; eighty like eight, and so on. Fifty comes before sixty. (A fairly minor problem is that twenty should sound more like two and ideally should be “two-ten;” thirty should be “three-ten” and so on). After saying a tens word, the child appends the unit words, one through nine. Learning to count to 20 and beyond is a child's first experience with base-ten ideas. In this case, teaching needs to stress the base-ten pattern underlying the counting numbers: the structure. We need to “instructure” (teach the structure) not “instruct.”
Counting Things: How Many Are There?
Context and overview
Children’s ideas about same, more, less, and order are heavily influenced by perception and by their own imperfect logic (for example, that what looks like more is more). These are good ideas but lack precision, so children need help in taking the next step. The counting words that children learn early on can be used for enumeration; in determining the exact number of a collection, it is the cardinal number that tells how many. Accurate enumeration and understanding of cardinal number are fundamental for all arithmetic (and measurement) and are not as simple as they seem. Rather they involve key mathematical ideas and strategic thinking.
Principles needed to understand enumeration
Enumeration refers to using the counting words to figure out the number of objects. (This includes any object, from imaginary monsters to marbles.) Children must learn to follow several rules and principles to enumerate accurately. This set of rules is fundamental:
- Say number words in their proper order.
- Match one number word with only one thing (one-to-one correspondence between number word and thing).
- Count each thing once and only once.
Given these rules and principles, there are several ways to enumerate with accuracy. Children need to be able to:
- "See” small numbers (up to four or so) without counting. This is subitizing, which can reduce the drudgery of counting.
- Count one object at a time.
- Point at objects.
- Push objects aside to keep track of which ones have been counted.
- Put objects in a line or other orderly arrangement.
- Count on the fingers.
- Group objects into convenient groups that can be subitized or counted.
- Group by 10s.
- Check the answer.
Children need to learn to use these approaches in appropriate situations. For example, if there are only two objects, subitizing may useful, but if there are nine, then pushing objects aside may be indicated.
Children who enumerate accurately also need to understand the result achieved. Suppose a child accurately counts five things. Correct enumeration alone does not necessarily mean that the child understands cardinality. Asked how many there are, the child may simply count the objects another time. For that child, answering the question of how many simply activates the counting routine but does not provide an understanding of the result. Children need to learn several things about cardinal number. The core idea is that correct enumeration yields the cardinal value of set. The last number word does not refer to the last object counted but to the set as a whole. When we count, the number one refers to the first object; two refers not to the second object counted but to the two objects in the new group, and so on. Furthermore, once the child has determined that there are five objects in the set, it does not matter if they are hidden, or if the objects are simply rearranged (say from a straight line to a circle). There are still five objects. This is conservation of number.
Common mistakes or misconceptions
When counting, children often rely too heavily on physical appearance, just as they did in determining more or less. One goal of teaching should be to help children learn that reason must trump appearance. Children need to think abstractly about tangible things. Eventually, they need to embed understanding of cardinal number (for example, the abstract idea that there are five objects here) within the larger system of number, for example, that five comes after four and is half of 10.
Everyday Numerical Addition and Subtraction
Context and overview
Next we need to understand how concepts of more/less, order, same, adding and subtracting without exact number (knowing that adding means making a set larger even if you don't know of the exact number), and enumeration get elaborated to create numerical addition and subtraction. Children learn some of this on their own, but adults can and should help.
These concepts need to be learned to understand addition (subtraction is similar):
- Addition can be thought of in several ways, including combining two sets, increasing the size of one set, and jumping forward on a number line.
- Simple counting is also adding, one at a time.
- The order of addition makes no difference (the commutative property).
- Adding zero changes nothing.
- Different combinations of numbers can yield the same sum.
- Addition is the inverse of subtraction.
Strategies used to add (or subtract)
Children often begin by using concrete objects and fingers to add but gradually learn mental calculation and remember some of the sums.
- Using concrete objects, children may do the following to solve a simple problem like 3 + 2: They may count all ("I have three here and two there and now I push them together and count all to get five") or they may count on from the larger ("I can start with three and then say, four, five.")
- Approaching the problem mentally, children may solve the problem by derived facts, building on what is known ("I know that two and two is four, so I just add one to get five") and by memory ("I just know it!").
More features of numerical addition and subtraction
- It’s always useful to have backup strategies in case one doesn’t work. For example, if unsure about memory, the child can always count to get the answer.
- It’s important for the child to be able to check the answer.
- It’s important for the child to explain why 3 + 2 gives five as the answer, since proof is a social act requiring language.
- The child needs to learn different strategies for different set sizes. (Counting one by one is good for adding small sets but tedious and inefficient for larger sets.)
- The child should also be able to describe how he got the answer. (Self-awareness is one aspect of metacognition. Of course, remembering what you just did is essential for describing it in words.)
- Language is vital for describing one’s work and thinking, and to convince others; children need to learn mathematical vocabulary.
- The child should be able to apply the math in real situations or stories about real situations (such as word problems).
Context and overview
Children need to develop number sense, a concept that is notoriously difficult to define in a simple and exclusive way. I like to think of it as mathematical street smarts, which can be used in just about any area of number, including those discussed above. Number sense, which helps the child to make sense of the world, has several components, each of which undergoes a process of development.
Thinking instead of calculating
Number sense involves using basic ideas to avoid computational drudgery, as when the child knows that if you add two and three and get five, then you don’t have to calculate to get the answer to three-and-two.
Use what is convenient
Number sense involves breaking numbers into convenient parts that make calculation easier, as when we mentally add 5 + 5 + 1 instead of 5 + 6.
Knowing what’s plausible or impossible
Number sense may involve a “feel” for numbers in the sense of knowing whether certain numbers are plausible answers to certain problems (if you are adding two and three you know that the answer must be higher than three; anything lower is not only implausible but impossible).
Number sense involves intuitions about relationships among numbers. (For example, "this is 'way bigger' than that.")
Number sense involves fluency with numbers, as when the child knows immediately that eight is bigger than four, or sees that there are three animals without having to count.
This involves figuring out the approximate number of a group of objects and is related to the notion of plausible answers.
The Transition to Written, Symbolic Math
Context and overview
Formal, symbolic mathematics can provide children with more powerful tools and ideas than those provided through their informal everyday math. Formal math (and its use of symbols) developed in several cultures and is now virtually universal. Children need to learn it.
Everyday origins and formal math
Children encounter math symbols in everyday life: elevator numbers, bus numbers, television channels and street signs are among the many. Often parents, television, and software activities introduce some simple symbolic math, such as reading the written numbers on the television or on playing cards.
Schools certainly have to teach formal math. But doing so is not easy. Even if they are competent in everyday math, children may have trouble making sense of and connecting their informal knowledge to what is taught in school. Teachers often do not teach symbolism effectively. If children get off on the wrong symbolic foot, the result may be a nasty fall down the educational stairs. So the goal for teachers is to help children, even beginning in preschool, to understand why symbols are used, and to use them in a meaningful way to connect already-known informal mathematics to formal symbolic mathematics. The teacher needs to “mathematize” children’s everyday, personal math; that is, help children connect their informal system with the formal mathematics taught in school. It’s not ill-advised or developmentally inappropriate to introduce symbols to young children, if the activity is motivating and meaningful. On the contrary, it is crucial for the teaching of symbols to begin early on, but again, if and only if it is done in a meaningful way.
Here are key issues surrounding the introduction of formal math to young children:
Young children have a hard time connecting numerals and the symbols of arithmetic (+ and -) to their own everyday math
They may add well but be confounded by the expression 3 + 2. It is as if the child is living in alternate realities: the everyday world and the “academic” (in the pejorative sense) world. The everyday world makes sense and the world of school does not. You think for yourself in the former and do what you are told in the latter.
The equals sign (=) is a daunting challenge
The teacher intends to teach the equals sign as "equivalent," and thinks she has, but the child learns it as “makes” (e.g., 3 + 2 makes 5). This is a tale of how child egocentrism meets teacher egocentrism but neither talks with the other.
We should not avoid teaching symbols but need to introduce them in a meaningful way. This means taking account of what children already know and relating the introduction of symbols to that prior knowledge. It also means motivating the use of symbols. Thus if you want to tell a friend how many dolls you have at home, you need to have counted them with number words (symbols) and then use spoken words (“I have five dolls”), written words (“I have five dolls” written on a piece of paper or a computer screen), or written symbols (5) to communicate the result.
Manipulatives can help
Use of manipulatives can be effective in teaching symbolism and formal math, but they are often utilized badly. The goal is not to have the child play with concrete objects but to use these objects to help the child learn abstract ideas. The goal of manipulatives is to get rid of them by putting them in the child’s head to use as needed in thought. For example, suppose the child learns to represent tens and ones with base-ten blocks. Given the mental addition problem 13 plus 25, the child may understand that each number is composed of 10s (the 10 by 10 squares) and some units (the individual blocks), and that solving the problem involves adding one 10 and two more, which is easy, and then figuring out the number of units. The mental images of the 10s and ones provide the basis for her calculation, part of which may be done by memory (one plus two is three) and part of which may be done by counting on her fingers (five fingers and three more give eight).
The basics of number are interesting and deep. Although young children develop a surprisingly competent everyday mathematics, they have a lot to learn and teachers can help.