using a problem-solving approach in the primary grades

Using a Problem-solving Approach in the Primary GradesAuthor(s): Patricia F. CampbellSource: The Arithmetic Teacher, Vol. 32, No. 4 (December 1984), pp. 11-14Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194008 .

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Using a Problem-solving Approach

in the Primary Grades By Patricia F. Campbell

According to the National Council of Teachers of Mathematics (1980), the focus of school mathematics in the 1980s must be on problem solving. Furthermore, computation is to be a tool for problem solving. The impor- tance of problem solving as a goal in mathematics education cannot be dis- puted; however, the de-emphasis of computation may cause feelings of uneasiness for many primary-level teachers. These feelings can be accen- tuated by such statements as "Pri- mary-level curricula contain practical- ly no mathematical problem-solving experiences" (Greenes 1981). Where does this dilemma leave the typical primary-level teacher, given the existing primary mathematics curriculum and the demands from parents and school administrators that young children develop a mastery of addition and subtraction?

The source of this anxiety over a focus on problem solving may be the lack of a universally accepted defini- tion of problem solving. Krulik and Rudnick (1980) define problem solving as "a process . . . the means by which an individual uses previously acquired knowledge, skills, and understanding to satisfy the demands of an unfamiliar situation" (p. 4). During problem solving, the student must take what has been learned and apply it to a new situation. If the student knows a procedure that can guarantee

Patricia Campbell teaches mathematics education and computer education courses at the University of Maryland, College Park, MD 20742. She is currently doing research on the learning of mathematics by elementary school- children.

a solution and if this technique can be easily applied, then it could be said that there was "no problem." How- ever, if this skill is still being acquired, then deciding what process to use and how to use it may be quite a problem (see Heller and Greeno 1978).

It is often tacitly assumed that children must acquire computational

Students must comprehend the problem, then conceptualize it in a general setting.

skills before they can apply them to the solution of problems. However, preschool children invent strategies for solving mathematical problems in- dependent of instruction (Ginsburg 1977; Groen and Resnick 1977). This behavior suggests that computational skills are not prerequisites for problem solving. Consider the so-called simple, one-step story problem. For example:

Three children were playing in a sandbox. Two more children came and played with them. How many children were in the sandbox?

Primary teachers may be able to use this problem setting as a means of developing an understanding of addition rather than using addition as a means of modeling the problem, as typically done in the middle grades. Rather than emphasizing only the ac- quisition of computational skill for its own sake, teachers may take advan- tage of the young child's problem- solving strategies. In this way, addition and subtraction have meaning

from the beginning and are not simply part of a complex scheme involving the writing of numerals and the state- ment of facts.

This approach is conducive to the development of an understanding of the operations of addition and subtraction as well as practice with the computational skills involved. Recent research (Carpenter, Hiebert, and Moser 1981) supplies evidence sup- porting this method. At the same time, if these one-step problems are used for problem solving (where students need to understand the problem and plan for a solution), then they may serve as a foundation for developing skills needed to solve nonroutine problems (Carpenter et al. 1980).

What Influences Young Children's Interpretation and Solution of Problems? Before using a problem as the setting for developing or clarifying the meaning of addition or subtraction, we should consider what influence the structure of the problem and its mode of presentation may have on the student's ability to solve it. Generally, teachers should consider the structure of the problem, the use of manipulatives or- diagrams to represent the problem, and the choice of pictures or oral statements to communicate the problem setting.

Carpenter, Hiebert, and Moser (1981) classified the structure of addition and subtraction problems as joining/separating, part-part-whole, comparison, and equalizing. The joining/

December 1984 И



separating problem has an initial set and an action (either direct or implied) that causes a change in the quantity of the initial set. The part-part- whole class consists of problems in which two amounts are considered indepen- dently or as part of a whole and action does not take place. Comparison problems involve the comparison of two amounts. Equalizing problems involve comparison, with an implied action on one quantity.

The data collected by Carpenter, Hiebert, and Moser indicate that first- grade children can recognize different problem situations and use that recognition to determine what solution strategy to apply to a given problem.

With respect to the presentation of the problem, research offers some guidelines that teachers should keep in mind. An oral problem with no

We can foster computational skills and problem-solving skills in the same setting.

manipulative is, as expected, the most difficult for young children to understand. An oral problem with concur- rent physical movement of objects to depict the action in the problem generally prompts the greatest number of correct responses (Ibarra and Lind-

it has been suggested that approach- ing the one-step problems in a problem-solving setting may provide a base for the growth of skills in solving nonroutine problems as the student matures. Research has been cited to support the feasibility of such an approach as well as to indicate the varia- bles within problems that seem to influence young children's perception. How can all these factors be combined for effective instruction?

Initially the student must understand the problem. This comprehension may be indicated by the child's ability to tell the story of the problem in a paraphrased form when asked, "Now you tell me the story." All that

Fig. 1 Two sketches to describe the same problem

The children in their study did not attempt to use or rely on one solution strategy; rather, their strategy seemed related to problem type. Because the mathematics curriculum of these students had not yet introduced formal instruction in symbolic representa- tions of addition and subtraction, they relied on their natural problem-solving skills, including counting, and their experience in using objects to represent problem settings. The high level of success evidenced by these students supports the use of problem settings as a means of developing addition and subtraction concepts rather than simply as a means of applying previously learned algorithms.

vail 1979). When pictures depict problems rather than oral statements, Campbell (1981) noted a progression in the young child's interpretation of the pictures from set recognition to additional recognition of motion to perception of mathematical relation- ships between the sets.

Problem Solving in the Primary Grades The premise outlined thus far has been that primary-level children may gain an understanding of the concepts of addition and subtraction as well as practice in computational skills through one-step problems. Further,

is required at this stage is that children be able to translate the problem into their own words. To help their comprehension, students could act out the events or watch them being acted out as the problem is read aloud by the teacher. Instead of reading the entire problem aloud and then expecting the children to enact it, the teacher should probably read only one portion of the problem (e.g., "David, Sarah, and Gregory were drawing pictures on the blackboard") and have that portion acted out before reading the next portion ("Then Nancy and Ramon also came up to the blackboard to draw pictures"). The conclusion of the enactment prompts the question for the

12 Arithmetic Teacher



class to answer ("Now how many children are drawing at the blackboard?"). At this point the actors may return to their seats and the teacher may ask the other students to tell what happened in their own words.

Next the child must take from the story the essentials of the problem. The student must conceptualize the problem in a general setting. For example, in the problem above, the child must come to see that "this is a problem where there were some people and more came. So now I have to find how many people there are altogeth- er." Having the students describe, as in a time line, the events of the problem can encourage this conceptualization. For example, the teacher may ask, "What was the very first thing that happened? Then what? . . ."The children may also be led to see the structure of the problem. For example, five other children could enact this problem. There could be four girls and one boy (rather than two girls and three boys). They could draw different pictures on the blackboard, but the enactment would be similar and the solution would be the same. By discussing which parts of the acted- out problem are useful for answering the question, the children begin to learn what part of the enactment, and hence what elements of the problem, deserve their attention and what information or actions may be disregarded. In this way the children are encouraged to see the structure of the problem. Having conceptualized the problem, the students may represent it with fingers, counters, or other manipulatives. Using objects as the model, counting will generally yield a solution. At this point, the students should be encouraged to consider whether the solution makes sense.

The student may then symbolize the general form that has been described (either verbally or with objects) by writing a number sentence and solving it. This activity calls for the student to use numbers to record an event. To emphasize the use of the symbols as a recording device and to aid students in attributing meaning to abstract symbols, the class may practice recording each phase of the action as it occurs or recording each part of the

problem as it is read (e.g., "Three children came up to the blackboard, so we can write a 3 to remember that is what happened first"). If children have difficulty at this stage, it may first be necessary to symbolize the problem by using a diagram to represent the story or enactment (see fig. 1). The diagram provides a more con- crete model related to the enactment or manipulatives used previously and, at the same time, introduces numerals to indicate the quantities.

By acting out problems and discussing them, the children can learn a language of communicating about solving problems.

Additional activities to aid symbolization may call for another skill. Stu- dents may be presented with an abstract number sentence and asked to invent a situation to accompany the symbols. This task requires the children to create a story or problem to go with the number sentence. In this way the children are encouraged to consider the structure of the problem and the meaning of the arithmetic operation. As students compare their problems and see that one number sentence may represent many different events, their conceptualization of the operations of addition and subtraction will be enhanced.

The number sentence is the mathematical model of the one-step problem. Lindvall and Ibarra (1979) note that the typical first-grade student does not construct the incomplete number sentence (3 + 2) as a step in the problem solution. Rather, the number sentence is only a way of representing the solution process that has already taken place. The young child cannot determine whether a solution makes sense from the number sentence. That determination should be made before the symbolization occurs. Using a number sentence to solve a problem does not occur until much later when problems are more complex and the conceptualization of

the operations is already developed. Primary teachers may realize that

they use many of these ideas when they work individually with students who are having difficulty solving a computation or workbook problem. Frequently teachers explain 3 + 4 = by stating, "Suppose you had three candies and I gave you four more. Let's use the counters to see how many candies you would have." The problem-solving approach places this technique in the forefront as a means of developing students' understanding of the addition of three units and four units rather than using it as an afterthought following the formal- ization of the number sentence 3 + 4 =

Conclusion Can the primary-level teacher take the existing curriculum, develop students' understanding of arithmetic operations, and teach problem solving? Yes, by using a problem-solving approach from the beginning of the instruction process. This approach does not isolate addition and subtraction concepts from their application in the children's experience. When teachers encourage students to analyze the problem's structure by modeling the essential elements of the problem through physical representation, young children can gain further understanding of the problem and the meaning of arithmetic operations. Symbol- izing the problem and writing the solution fosters their skill in computation. Furthermore, this method may help prevent children from developing the idea that mathematics is a collec- tion of arbitrary rules or meaningless procedures that must be memorized and executed. Although this idea may not be evident at the primary level except in students' emerging depen- dence on the teacher to check the correctness of their answers (Brandau and Easley 1979), it has been verified at the middle school level (Erlwanger 1973; 1974). The problem-solving approach at the primary level will also help students develop the language to communicate about solving problems, a skill that many students lack today (Zweng 1979). In addition, students

December 1984 13



will be acquiring a problem-solving style that may serve as the foundation for developing the skills that are needed to solve nonroutine problems in the future.

References

Brandau, Linda, and Jack Easley. Understand- ing the Realities of Problem Solving in the Elementary School. Columbus, Ohio: ERIC/ SMEAC, 1979.

Campbell, Patricia F. "What Do Children See in Mathematics Textbook Pictures?" Arith- metic Teacher 28 (January 1981): 12-16.

Carpenter, Thomas P., Mary Kay Corbitt, Hen- ry S. Kepner, Jr., Mary Montgomery Lind- quist, and Robert E. Reys. "Solving Verbal Problems: Results and Implications from Na- tional Assessment." Arithmetic Teacher 28 (September 1980):8-12.

Carpenter, Thomas P., James Hiebert, and James M. Moser. "Problem Structure and First-Grade Children's Initial Solution Pro- cesses for Simple Addition and Subtraction Problems." Journal for Research in Mathe- matics Education 12 (January 1981):27-39.

Erlwanger, Stanley H. "Benny's Conception of Rules and Answers in IPI Mathematics." Journal of Children s Mathematical Behavior 1 (Autumn 1973):7-26.

. "Case Studies of Children's Concep- tions of Mathematics." Unpublished doctoral dissertation, University of Illinois at Urbana, 1974.

Ginsburg, Herbert. Children's Arithmetic: The Learning Process. New York: D. Van Nos- trand Co., 1977.

Greenes, Carole E. "Beyond the Textbook." Arithmetic Teacher 28 (March 1981):2.

Groen, Guy, and Lauren B. Resnick. "Can Preschool Children Invent Addition Algo- rithms?" Journal of Educational Psychology 69 (1977):645-52.

Heller, Joan I., and James G. Greeno. "Infor- mation Processing Analyses of Mathematical Problem Solving." Paper presented at the National Science Foundation Seminar on Ap- plied Problem Solving, March 1978, at North- western University.

Ibarra, Cheryl G., and C. Mauritz Lind vail. "An Investigation of Factors Associated with Children's Comprehension of Simple Story Problems Involving Addition and Subtraction Prior to Formal Instruction on These Opera- tions." Paper presented at the 57th Annual Meeting of the National Council of Teachers of Mathematics, April 1979, Boston.

Krulik, Stephen, and Jesse A. Rudnick. Prob- lem Solving: A Handbook for Teachers. Bos- ton: Allyn & Bacon, 1980.

Lindvall, C. Mauritz, and Cheryl G. Ibarra. The Development of Problem-solving Capabili- ties in Kindergarten and First-Grade Chil- dren. Pittsburgh: Learning Research and De- velopment Center, University of Pittsburgh, 1979.

National Council of Teachers of Mathematics. An Agenda for Action: Recommendations for School Mathematics of the 1980s. Reston, Va.: The Council, 1980.

Zweng, Marilyn J. "The Problem of Solving Story Problems." Arithmetic Teacher 27 (September 1979):2-3. m

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