Children’s interpretations of arithmetic word problems

Do children fail to solve arithmetic word problems because they lack the mathematical knowledge to do so, or because they don’t understand certain words or phrases in the problems? At Yale University, Denise Dellarosa Cummins conducted two experiments to test conflicting explanations for the errors children make on written problems. The results of these experiments indicated that word-problem errors are often caused by misinterpretation of certain verbal expressions commonly used in arithmetic texts, and that children can solve these problems if they are rewritten using unambiguous language.

Learning to solve word problems is an important part of mathematics instruction since real-world problems requiring math seldom come in the form of ready-made equations. Cummins stresses that understanding how children interpret word problems is of great importance to educators because all instruction, ande particularly reteaching, depends on identifying where children’s misunderstandings lie, as well as the nature of those misunderstandings.

Two very different theories

Previous research has led to the development of two very different theories to explain students’ failure on word problems. One of these holds that when children fail to solve a word problem, it is because they lack an understanding of the mathematical concepts underlying the problem. According to this theory, a student’s math knowledge actually enables the student to understand the semantics of word problems. Proponents of this theory claim poor performance reflects insufficient mathematical knowledge. At the elementary level, this often means a lack of understanding of part-to-whole relationships.

According to the second theory, the major source of difficulty children encounter when trying to solve word problems is in interpreting certain words or phrases; students are unable to solve the problem not because they lack the mathematical knowledge to do so, but because they don’t fully understand key verbal expressions in the problem. Those researchers who support the second theory hypothesize that the acquisition of problem-solving skill primarily denotes improvement in a child’s ability to understand the language of written problems. Cummins’ experiments were aimed at determining which theory was more accurate.

Children’s interpretations of language

The purpose of the first experiment was to gather information about children’s interpretations of typical word problems. In this experiment, 24 first grade children were given 13 word problems to solve. They were also required to draw a picture of each problem and then to identify one picture among several that correctly illustrated the situation described in each problem. The problems used in the study included the kinds of verbal expressions children commonly find difficult, such as comparatives, “Mary has five more marbles than John,” and terms like “altogether” and “some”. Afterwards, the children’s drawings and choices of illustrations were compared to their problem solutions. The results showed that children who could not recognize or draw a picture that accurately represented the situation described in a crucial phrase of a problem could not accurately solve that problem.

Children’s interpretations of arithmetic concepts

The purpose of Cummins’ second experiment was to determine whether misunderstandings occurred because the children did not know the arithmetic concepts demanded by the problem. Eleven children from the first experiment, who had consistently failed to solve word problems or correctly represents them in pictures, were given reworded versions of the same problems. Cummins believed that if these children produced incorrect depictions and solutions of standard problems, but correct depictions and solutions to the same part-whole relationships in reworded problems, then the difference in their performance could not be due to insufficient knowledge of the part-whole concepts tested by the problems. An example of a standard problem was: “Mary has 3 marbles. John also has some. Mary and John have 5 marbles together. How many marbles does John Have?” The same problem, reworded, read: “There are five marbles. Three of them belong to Mary. The rest belong to John. How many belong to John?”

Analysis of the second experiment’s results revealed significantly improved performance on the reworded problems. Students found the reworded versions of the problems both easier to represents in pictures and easier to solve. The number of correct responses was 85% on the reworded problems compared to 30% on the standard-language problems.

The results of these experiments demonstrate, Cummins believes, that a significant percentage of young children misinterpret certain words and phrases in standard written problems found in arithmetic texts. The use of easier-to-understand language resulted in dramatically improved performance on these word problems. This demonstrated that linguistic skill had been a significant factor in solving common word problems. By analyzing how children perform on variously worded problems, educators can more accurately measure both their linguistic skills and their mathematical knowledge. Cummins asks that educators take care not to underestimate children’s logical and mathematical knowledge by confusing it with linguistic skill. Both skills need to be developed if students are to perform well on standard arithmetic word problems.

“Children’s Interpretations of Arithmetic Word Problems” Cognition and Instruction Volume 8 Number 3 pp. 261-289.

Published in ERN November/December 1991 Volume 4 Number 5

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