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by Constance Kamii and Linda Joseph
Taken from ENC Online
Most second graders can write correct answers to double-column addition problems by adding the ones first and proceeding as follows:
When the mathematics hour is finished and we interviewed these children individually, however, most of them say that the 1 in 16 means one. We have interviewed hundreds of children in grades 1-3, and their teachers are always incredulous when they find out that the children said the 1 in 16 means one. We will describe our interview so that teachers will be able to test each other's children in the same way. (We say "each other's children" because children sometimes think about the answer the teacher expects when they are tested by the person who taught the same material.)
The Interview
When a child comes in to begin the interview, the interviewer first shows him or her a 3 x 5 card with "16" written on it. After the child says that the card indicates "sixteen," he or she is asked to count out sixteen chips. The interviewer then circles the 6 of 16 with the back of a pen and asks, "What does this part (the 6) mean? Could you show me with the chips what this part (the 6) means?" First and second graders do not have any trouble up to this moment in the interview.
The interviewer then circles the 1 of 16 and asks, "What about this part (the 1)? Could you show me with the chips what this part (the 1) means?" (Note the use of the term "this part"; avoid using any other word.) Almost all first and second graders respond by showing only one chip, in spite of the hours of place-value instruction they have received.
The interviewer continues to probe: "You showed me all these chips (pointing out the sixteen chips) for this number (circling the 16 on the card), . . . and these (pointing to six chips) for this part (circling the 6 on the card), . . . and this chip (pointing) for this part (circling the 1 on the card). What about the rest of the chips (pointing to the nine or ten chips that were not used to show the two parts on the card)? Is this how it's supposed to be, or is there something strange here?" A few children reply that something is strange, but most of them say that they do not see anything wrong with what they have said.
The proportion of children who say that the 1 means ten is generally 0 percent at the end of first grade, 33 percent at the end of the third grade, and 50 percent at the end of fourth grade. In showing ten chips to indicate that the 1 means ten, however, many children include the six chips they used to explain the meaning of 6.
Why don't children in the primary grades understand place value? The answer to this question is complicated, and it will be easier to explain why after presenting a new way of teaching place value and double-column addition in second grade based on the theory of Jean Piaget. This method, developed at Hall-Kent School near Birmingham, Alabama, resulted in better understanding of place value at the end of second grade than the traditional, textbook method. Sixty-six percent of the children said the 1 in 16 meant ten at the end of the school year in 1987, and 74 percent said the 5 in 54 meant fifty. These percentages are higher than what is usually found at the end of fourth grade with traditional instruction. (Needless to say, teaching the test item was not included, and the children were never instructed to make groups of ten objects. These second graders' entire mathematics education consisted of games in kindergarten and first grade and of games and discussions in second grade, without any textbooks or workbooks.)
A Way of Teaching Place Value and Double-Column Addition
The essence of our way of teaching based on Piaget's theory is to foster the children's own natural thinking and to encourage them to exchange points of view. But two preliminary remarks are in order.
First, our second graders were never taught place value or double column addition when they were in first grade. Research has shown that first graders are unlikely to understand place value (Bednarz and Janvier 1982; Brun, Giossi, and Henriques 1984; C. Kamii 1985; M. Kamii 1980, 1982; Ross 1986). First graders understand "16" as 16 ones (and not as 1 ten and 6 ones).
Second, we never taught place value apart from addition or another operation. For reasons that will be clarified shortly, we never used objects such as straws bundled together in sets of ten and never asked children to circle pictures on a page to show how many tens and ones are represented.
We were heavily influenced by Madell (1985), who said that when children are encouraged to do double-column addition in their own natural way, "they universally proceed from left to right." With respect to
for example, he stated the following:
Invariably, in a problem like this, the seven and eight-year-olds first compute the tens. The details vary:
a) Some will actually record a 7 in the tens column before looking at the ones. These children then come back and erase.
b) Others, having arrived at 7 as the sum of 3 and 4, do not record that 7 before checking the ones column to see if it contains another ten.
c) A few of the most sophisticated students check the ones first. Noting (often by estimation) that there are more than 10 ones in the ones column, they come back to sum the tens and record 8 before returning to the ones and the last detail of the computation.
This last process is the closest that the children get on their own to the standard right-to-left procedure. Even for the addition of 3- and 4-digit numbers where a right-to-left process would seem more efficient, the children uniformly prefer the other direction.
Instead of using base-ten blocks as Madell did (for reasons that will be given shortly), we use the chalkboard to write problems and to facilitate the exchange of ideas among children. The teacher writes one problem after another on the chalkboard, such as the following:
The children raise their hands when they have an answer. The entire class can work together, or the teacher can work with small groups.
We do not teach any procedure for double-column addition but do encourage students to invent many different ways.
When most of the hands are up, the teacher calls on individual children and writes all the answers that are given by them. Being careful not to say that an answer is right or wrong, the teacher then asks for an explanation of each answer or procedure invented by the children. For
for example, one child may say, "I took two from seven to make ten," and the teacher writes "2" as follows:
As the child speaks, the teacher then writes "+ 2 = 10" next to the 8, and "=5" next to 7 - 2. Sometimes the teacher crosses out the 8 and writes "10" next to it and crosses out the 7 and writes "5" next to it, as follows:
The next child may say, "Seven plus seven equals fourteen; so one more has to be fifteen." The teacher goes on to ask, "Did anybody do it a different way?" until no one can think of any other way.
The children are thus encouraged to invent all kinds of ways and to agree or disagree with each other. We do not say that one way is better than another, as long as it makes sense, and each child is free to adopt the way that is best for him or her.
With problems such as
all our children prefer to start with the tens as Madell stated. Most say something like the following:
"Thirty and forty are seventy."
(The teacher writes "70" on the board, to the side, anywhere, as a memory tool.)
"Six and six are twelve."
(The teacher writes "12," again to the side.)
"Take ten from the twelve and put it with the seventy, and that's eighty."
(The teacher erases the 1 of 12 and changes the 7 (of 70) to 8.)
"And two more are eighty-two."
(The teacher writes "82" under the problem.)
Some children say the following:
"Thirty and forty are seventy."
(The teacher writes "70," to the side somewhere.)
"Six and four is another ten; so that's eighty."
(The teacher changes the 7 (of 70) to 8.)
"And two more are eighty-two."
(The teacher changes the 0 (of 80) to 2.)
When a child says, "Three and four are seventy," the children either raise their hands or shout, "Disagree!" depending on the rule they have agreed to. At the beginning of the year, however, the teacher is the only one who says, "I thought three and four were seven. How did you get seventy?" (The teacher is careful not to be the omniscient authority who decrees what is right or wrong. She or he only agrees or disagrees and tries to present another point of view on an equal footing.)
Differences with Traditional Instruction
Our way of teaching place value and double-column addition is different from traditional instruction in the following three ways. The differences are listed first and elaborated on later.
a) We do not teach place value as a separate activity to get children ready for double-column addition.
b) We do not teach any procedure or algorithm for double-column addition and, instead, encourage children to invent many different ways.
c) We encourage children to agree or disagree with each other and to adopt the ideas that make sense to them.
We encourage students to discuss procedures and to adopt ideas that make sense to them.
1. We do not teach place value separately because by the end of first grade all the children know that 10 + 10 = 20, and it seems best to encourage them to use this knowledge in such problems as 10 + 12, 20 + 20, 15 + 5, and 15 + 7. Traditional place-value instruction with bundles of straws, worksheets, and so on, is not helpful because it assumes that number and place value can be transmitted to the child from the outside with materials and pictures such as the ones shown in figure 1.
An understanding of "tens and ones" requires the construction in one's head of two systems that function simultaneously: a system of ones and a system of tens. These systems have to be created by each child, through his or her own mental activity, from the inside. Figure 2 illustrates the system of tens created mentally by the child on the system of ones he created before.
The system of ones is a synthesis of two kinds of relationships created by the child: order and hierarchical inclusion. The relationship of order is shown in figure 2 by the line connecting the thirty-two elements, and hierarchical inclusion is represented by the ovals indicating the mental inclusion of one in two, two in three, and so on. This system of ones is the natural numbers that by first grade almost all children have built on their own.
The system of ones is completely overlooked if we try to teach about "tens and ones" with materials such as those illustrated in figure 1. Figure 3 is an attempt to clarify the inadequacy of the conceptualization shown in figure l(b). The spatial arrangement of the thirty-two objects is the same in both figures. Figure 3, however, includes the system of ones, which consists of order and hierarchical inclusion. Order is represented with straight lines connecting the thirty-two elements, and hierarchical inclusion is shown with oval-like shapes, up to twelve. The system of tens is indicated with heavier lines. The system of tens also consists of two kinds of relationships, namely, order and hierarchical inclusion. Note that "tens and ones" in this conceptualization is far more complicated than the organization shown in figure l(b).
All the straight and curved lines in figure 3 represent relationships, which the children create in their heads and impose on the objects. Contrary to the empiricist-associationist assumptions on which traditional mathematics instruction is based, relationships cannot be put into children's heads from sources external to them. Relationships must be created by children through their own mental activity. Therefore, demonstrating figure 3 to children will not contribute to their learning about the system of tens. The reader wishing to know more about children's construction of the system of tens on the system of ones is referred to C. Kamii (1986).
2. The teaching of the right-to-left procedure interferes with children's possibility of learning about tens and ones. Even adults think about thirty-two as "thirty and two," rather than as "two and thirty." When children are encouraged to invent their own procedure of doing
therefore, it is not surprising that they deal with the tens first before going on to the ones.
In traditional instruction, children are made to add the ones first and to treat the tens column as two 3s. They therefore do not think about the two 3s as "thirty and thirty"; yet this thinking helps them construct the ideas of tens and ones.
3. When children are free to accept or reject other people's ideas, they do their own thinking to decide what makes sense to them. If the teacher follows a child's reasoning and writes "70" and "12" to do 36 + 46, for example, and then erases the 1 of 12 and changes the 7 of 70 to 8, children try hard to evaluate each other's line of reasoning. Children who are free to choose the way that is best for them think critically for themselves rather than blindly following the rules that are imposed from the outside.
We do not teach anything in the traditional sense of telling children how to do something, reinforcing "right" answers and correcting "wrong" ones. Piaget's research and theory have proved that neither number nor place value nor operations can be taught by direct transmission from the outside. Young children have to build, or construct, their own logico-mathematical knowledge from the inside, through their own thinking, if they are to reinvent standard procedures for themselves.
Visitors to our school often ask how we start at the beginning of the year. Essentially, the teacher makes up problems so that children can use what they already know to invent ways of dealing with new situations. For example, if they change
to (8 + 2) + 5, the teacher might put
on the chalkboard. Some children will think about the problem as 30 + 5, whereas others may change it to 20 + (9 + 1) + 5. The games they play in first grade, such as "Tens" and "Tens with Playing Cards" (C. Kamii 1985, 151-53), strengthen their natural tendency to think in terms of tens.
It is difficult to understand students' thinking.
It is difficult and exhausting for teachers to try to understand children's thinking. But it is also gratifying because children who have been encouraged to add in ways that make sense to them have a foundation for going on to invent their own ways of solving such problems as 33 - 15 and 8 x 23.
References:
- Bednarz, Nadine, and Bernadette Janvier. "The Understanding of Numeration in Primary School." Educational Studies in Mathematics 13 (February 1982):33-57.
- Brun, Jean, J.-M. Giossi, and Androula Henriques. ''A Propos de l'Ecriture Decimale (Concerning decimal writing)." MATH-ECOLE (published in Geneva, Switzerland) 23, no. 112 (1984):2-11.
- Kamii, Constance. "Place Value: An Explanation of Its Difficulty and Educational Implications for the Primary Grades." Journal of Research in Childhood Education 1 (August 1986):75-86.
- Young Children Continue to Reinvent Arithmetic, 2nd Grade. New York: Teachers College Press, forthcoming.
- Young Children Reinvent Arithmetic. New York: Teachers College Press, 1985.
- Kamii, Mieko. Children's Graphic Represenfation of Numerical Concepts: A Developmental Study. Ed.D. diss., Harvard University, 1982. Dissertation Abstracts International 43 (November 1982): 1478A.
- "Place Value: Children's Efforts to Find a Correspondence between Digits and Numbers of Objects." Paper presented at the Tenth Annual Symposium of the Jean Piaget Society, Philadelphia, 1980.
- Madell, Rob. "Children's Natural Processes." Arithmetic Teacher 32 (March 1985):20-22.
- Ross, Sharon. "The Development of Children's Place-Value Numeration Concepts in Grades Two through Five." Paper presented at the annual meeting of the American Educational Research Association, San Francisco, 1986.