Almost all teachers in the primary grades teach children to use algorithms. However, we believe that algorithms are harmful to young children because they "unteach" place value and hinder children's development of number sense. Our belief is based on the research and theory of Jean Piaget and our own research done at Hall-Kent Elementary School in Homewood, Alabama.

The theory of Jean Piaget, constructivism, states that children acquire logical- mathematical knowledge by constructing (making) it from the inside, in interaction with the environment. This theory is in sharp contrast with the traditional assumption that children learn arithmetic by internalizing it from the enviroment.

In the constructivist program at Hall-Kent School, children are not taught any algorithms, but are, instead, encouraged to invent their own procedures for each of the four operations. When children are thus encouraged to do their own thinking, they invariably proceed from left to right in addition, subtraction and multiplication. For example, if they are given a problem such as

876
+359
-------

children in the constructivist program think and say, "Eight hundred and three hundred is one thousand one hundred. Seventy and fifty is one hundred and twenty. And fifteen more is 1,235."

In the constructivist program, children strengthen their knowledge of place value by using it.

They think of the "8" as eight hundreds,

the "7" as seven tens and

the "6" as six ones.

In contrast, children who use algorithms think and say, "Six and nine is fifteen. Put the five down and carry the one (or ten). One and eight and three is twelve."

The algorithm is convenient for adults, who already know place value. For primary age children, who have a tendency to think about every digit as ones, however, the algorithm serves toreinforce this weakness by making them treat every column as ones.

Since space does not permit us to give many research findings about the harmful effects of algorithms, we'll limit ourselves here to what we found out by working with a fourth grade teacher at Hall-Kent School. The teacher, whose name is Cheryl Ingram, taught algorithms for ten years. By September, 1991 however, she had become convinced that there was something wrong with this traditional way of teaching.

She had observed that the children who came to her class seemed to be much better math students after one, two or three years of constructivist teaching than traditionally instructed students.

One of the ways in which Cheryl began to wean children away from algorithms was to write problems such as 876 + 359 horizontally on the chalkboard and to ask children to solve them without using a pencil. As the children volunteered to explain how they got the answer of 1,235 by using the algorithm in their heads, she followed their statements and wrote numbers such as the following for each column:

15
13
+ 12
-------------
40

After the children finished explaining how they got the answer of 1,235, Cheryl said, "But I followed your way and got 40 as the answer. How did you get 1,235?" Most of the children were stumped and became silent. However, someone soon pointed out that the teacher's 13 was really 130, and that her 12 stood for 1,200.

Cheryl used the constructivist approach throughout the 1991-92 school year. In May, 1992, we decided to compare the performance of this group (n=20) with that of the fourth graders to whom she had taught algorithms the preceding year. ( 1990-91, n = 18 ). Since the principal randomly assigns students to heterogeneous groups at the beginning of each school year, the performance of the two groups was comparable. All except two or three of the children in each group had been taught algorithms in grades, 1, 2 and / or 3.

13x11

Each child was asked to solve each problem without paper and pencil, give the answer and then explain how he or she got the answer.

In May, 1991, only 17 percent of the fourth graders in Cheryl's class got the correct answer of 244 for the addition problem. In May, 1992, the percentage increased to 75.
However, the incorrect answers the children gave were much more informative. The wrong answers given in 1991 were 225, 234, 234, 234, 234, 234, 234, 274, 745, 835, 838, 838, 10,099, "May I pass?" and "I wish I could write down ways. . . ."

By contrast, the wrong answers given in 1992 were more reasonable. They were 28, 202, 234, 238, and 243.

The five answer (28%) in the 700s, 800s, and 10,000s clearly demonstrated the poor number sense of the children who used algorithms.
The answers in the 700s or 800s were obtained by adding 6 to the 1 of 185 and sometimes carrying 1 from the tens column. ( Children who have never been taught any algorithms usually begin by adding 180 and 50, They are, therefore, more likely to think that the answer has to be 230 + more than 10.)

As for the subtraction problem, 504 - 306 written vertically, the percentage of children giving the correct answer increased from 39 (in 1991) to 80 (in 1992).
The incorrect answers of the children who were taught algorithms were 108, 108, 108, 108, 192, 196, 202, 202, 208, 408, and "Pass." The four children (22%) who gave the answer of 108 borrowed 10 from the 5 of 504 and then subtracted 3 from 4.
The errors produced in 1992 were 90, 108, 200 and 202. ( Children who have never been taught algorithms usually know that the answer has to be about 200 because they deal with the hundreds first. )

The percentage of children getting the correct answer to 13x11 increased from 6 (in 1991) to 55 (in 1992). In May, 1991, all the children could get the answer by using the algorithm. When they were allowed to use only their heads, however, only one of the 18 children produced the correct answer.
The wrong answers in given in May, 1991, were 11, 13, 42, 64, 113, 133, 133, 141, and 144, and eight children (44%) said, "I donÕt know," "I want to skip it," or "I need a pencil to do it."
By contrast, the wrong answers given in 1992 were 113, 133, 144, 233, and 300.

The detrimental effects of algorithms were even more evident in CherylÕs classroom than in the interviews. The first author regularly observed her math class and was amazed that, throughout the year, the children continued to treat separate columns as if they were isolated digits. By making children treat isolated columns, algorithms prevent the development of number sense.

On October 28, 1991, Cheryl wrote one problem after another on the chalkboard for the students to solve in their own ways. For an hour, she gave only addition problems which had 99 in one of the addends (e.g., 366 + 199; 493 + 99; and 601 + 199). The purpose of this experiment was to find out how long it would take for the class to think about whole numbers instead of focusing on columns.

Almost all of the children in the class used the algorithm during the entire hour and added the ones first, carried 10, added the tens and then carried the 100. However, one of the children - let's call him Joe - had been in constructivist classes since first grade and volunteered solutions such as the following for every single problem: "I changed 366 + 199 to 200+365, and my answer is 565." After an entire hour of this kind of " interaction," only three children caught on to what Joe was doing and imitated him ! The rest of the class did not seem to have "heard" Joe and continued blindly to use the algorithm.

606
+ 149
--------------

600 - 100 = 500,
6 - 49= -43,
500 - 43 = 457.

She was finally thinking about whole numbers ( 606 and 149 ) and no longer thinking about isolated columns. Most of the rest of the class continued to treat each column separately, but Cheryl remarked that in her ten years of teaching fourth grade, she had never seen such excitement and enthusiasm for math.

There were many ups and downs over the year.

December 20 brought a disappointment. Cheryl told the class that she had $50 to spend on Christmas presents and wanted to know if she had enough money to buy the following items: 3 battleships (a game everyone knew) @ $7.99, 2 sweaters @ $11.99, 1 wallet @ $15.00, and 2 dolls @ $8.95. The first volunteer started her answer by saying, "9+9+5=23."

On January 29, Cheryl put the misaligned numbers shown below on the board, and one of her most advanced students said, "20 + 30 = 50, 40 + 60 = 100, 150 + 10 = 160.

25
03
04
+ 65
-------------

When Cheryl asked how many people agreed with this answer, five hands went up. By May, as noted earlier, the majority of children were able to give correct or reasonable answers. We should point out, however, that it was impossible to get all of the children to do their own thinking. The comfortable routines they had learned interfered with their ability to think for themselves.

We hope this article will inspire other teachers to experiment in their classrooms and to write to us about what they did. (School of Education, The University of Alabama at Birmingham, Birmingham, AL 35294-1250)