problem solving in the primary grades

Problem Solving in the Primary GradesAuthor(s): Charlotte L. Wheatley and Grayson H. WheatleySource: The Arithmetic Teacher, Vol. 31, No. 8 (April 1984), pp. 22-25Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41192375 .

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Problem Solving in the Primary Grades

By Charlotte L. Wheatley and Grayson H. Wheatley

Rationale Problem solving has clearly become the focus of curricular reform in mathematics in the eighties. The recommendations of the NCTM in An Agen- da for Action (NCTM 1980) are being taken seriously by teachers, princi- pals, and textbook writers. The re- sults of the most recent national mathematics assessment reveal a drop in problem-solving skills, thereby docu- menting a problem in our schools. Funding for research in problem solving by the National Science Founda- tion is evidence of the attention being given to these skills. Over the past few years, numerous projects de- signed to study problem solving in children have been supported (e.g., Wheatley and Wheatley 1979ft; Zweng 1979).

However, problem solving can mean different things to different people. In this article, problem solving means "What do you do when you don't know what to do?" As Krulik and Rudnick (1980, p. 3) define problem solving, "the individual is con- fronted by something he or she does not recognize, and to which he or she cannot apply a model."

While attention is being focused on teaching problem solving in the inter- mediate grades, we must not neglect

Charlotte Wheatley is employed in administra- tion at Purdue University, West Lafayette, IN 47906. She is actively involved in the prepara- tion of curriculum materials in mathematics. Grayson Wheatley is also at Purdue University. He is presently developing curricular materials and conducting research on problem solving.

the earlier grades. Recent studies have shown that young children can solve problems, sometimes better than children who have been exposed to rule-oriented learning (Carpenter, Hiebert, and Moser 1981; Ginsburg 1980; Hebbeler 1977). Early childhood is a critical time to build a readiness for problem solving. Young children can profit from activities that encourage them to explore their environment mathematically and build a network of relationships that can form a basis for problem solving. The recommendations in this article are based on the Piagetian assumption that children construct knowledge for them- selves and that a curriculum that presents mathematics as a set of rules to be learned is antithetical to the way children learn.

Problem-solving Readiness Activities Problem-solving readiness can be achieved within the framework of early childhood education. The sugges- tions that follow give the primary school teacher assistance in building a mathematics problem-solving program. The specific problems and activities that follow have been found to be effective with children in the primary grades.

1 . Create an atmosphere conducive to exploration and learning. Children respond favorably to a warm, anxiety- free, success-oriented atmosphere. An environment that reflects trust and a sincere interest in children is an important aspect of learning. Devel-

oping a climate for problem solving is a necessary first step in teaching problem solving.

2. Pose interesting and challenging oral problems for exploration. Chil- dren in the primary grades often respond more favorably, especially in the beginning, to the oral presentation of problems. With this approach, the teacher, working with the entire class or small groups of students, presents the problem using pictures of objects.

The type of problems selected plays a major role in the success of the problem-solving program. The following problems are examples of some that stimulate problem-solving development by focusing on learning a process rather than getting an answer.

Example A

Marc put five puppies in a box. Sally, took out three puppies. Tom put one puppy in the box. How many puppies are in the box? (See fig. 1.)

Example В

How many blocks in figure 2 are not red?

Example С

Put the same number of lions in each cage in figure 3. Move some lions from cage A to cage B. With Unifix cubes, show how many lions you have.

3. Present a variety of mathematical problems. Problems should be posed to encourage the child to per-

22 Arithmetic Teacher



form higher level mental processes rather than just use a rule to get an answer. Situations that allow the child an opportunity to observe, describe, classify, order, compare, conjecture, question, and find a pattern should form the basis of problem-solving development.

4. Encourage children to develop problem-solving strategies. Ginsburg (1980) reports that young children ex- hibit surprising intellectual strength in the area of invented strategies for school arithmetic. kThe invented strategy reflects the child's contribu- tion to the work of understanding" (p.

44). For example, an invented strategy might be finger counting combined with saying the names of numbers in order.

5. Introduce specific problem-solving strategies, for example, model it, act it out, guess and test, make a list, and look for a pattern. Informally guide the children to consider and reflect on these problem-solving strategies. The problems presented in sug- gestions 2 and 9 can be explored and solved by using one or more of these strategies.

6. Emphasize counting activities

and exercises. Both forward and backward counting become powerful problem-solving tools. These counting activities allow the child to develop and build patterns. Houlihan and Ginsburg (1981) report that first- and second-grade students use counting heavily to solve problems. The use of a calculator with a constant addend can assist in the development of counting and pattern-search skills. Many counting activities can be found in Keystrokes: Calculator Activities for Young Students, Counting and Place Value (Rey s et al. 1980).

7. Supply children with manipula-

Fig. 1 Fig. 2

Fig. 3-

April 1984 23



tive materials, for example, Unifix cubes, attribute pieces, Cuisenaire rods, and counting frames. The materials provide models that help children make the problems concrete. From the manipulations and associated mental activities comes the abstrac- tion of ideas.

8. Encourage the interaction of pupils. Learning is enhanced by the ex- change of ideas in a group setting. As Piaget has shown, social interaction facilitates the shift from egocentrism to a broader view of the world.

9. Have children write, illustrate, role play, and create stories about numbers. Ways of implementing this suggestion include the following:

a. Present a picture, and ask the children to write a number story about the picture. Discuss the picture with the class. Sample pictures for discus- sion are shown in figure 4.

b. Have students work on suggest- ed problems in small groups. Prob- lems can be prepared on cards for selection by the pupils. A problem- solving interest center can also be established and used by groups of students.

с Have children model a problem with concrete materials.

Example 1

If thirty-two pencils are available and eight children are each to receive the same number of pencils, how many should each receive?

For a fifth grader this division problem would be an exercise, but for a first grader it is a problem. Chips or cubes can be used to solve the problem in a concrete, meaningful way. Such experiences can actually develop readiness for the concept of division.

Example 2

I took a picture of some children and some dogs. The picture showed seven heads and twenty- two legs. How many children were in the picture?

Fig. 4

Fig. 5

Although the problem in the ab- stract is very difficult, young children can solve it by using sticks for legs and bottle caps for heads and trying various combinations until the problem's conditions are satisfied. Learn- ing to use manipulatives as thinking tools is an important achievement.

с Acting out the problem is an excellent strategy for young children. Present a problem and ask children to act it out.

Example 1

Two boys are ahead of Bill in a line. Three girls are behind Bill. How many children are in line?

While certain designated children form a line, the other students can

reason toward a solution.

Example 2

Katrina's parents have a big van with six seats. Three people can sit on each seat. If eleven people are in the van already, how many more people can sit in the van?

d. Let children write, illustrate, and role play original problems. The children enjoy solving each other's problems. Discuss and display the children's work. Samples of first graders' problems and illustrations are shown in figure 5.

10. Offer spatial experiences. Spa- tial activities promote the development of imagery, which is critically important in problem solving (Wheat-

24 Arithmetic Teacher



ley and Wheatley 1979a). The stimula- tion of right-hemispheric activity can help to develop creative thinking.

An initial activity for developing visualization builds on memory for images. Present a drawing like the one in figure 6 on an overhead transparen- cy for a few seconds and, from memory, have pupils draw what they saw. After a few moments, give them an- other peek at the picture and encourage them to complete the drawing. Any type of drawing can be created for this activity. You may wish to increase the complexity of the drawing as the children improve with prac- tice.

Other examples of spatial activities can be found in Wheatley and Wheat- ley (1979a).

1 1 . Use games and gamelike situations. By making a game of the problem-solving experience, the child can actively participate in a way that is associated with play. Motivation is high for such activities. Many pre- school mathematics games are available commercially, or they can be constructed by a resourceful teacher. A Mathematics Activity Curriculum for Early Childhood and Special Edu- cation (Richardson et al. 1980) is a rich source of gamelike activities.

12. Enjoy problem solving with your students. The success of problem- solving activities depends heavily on the atmosphere you establish in the classroom. "Let's explore together" and "I enjoy problem solving" are attitudes that must prevail.

Summary Whereas in later grades pupils may refine certain problem-solving strategies (e.g., guess and test, make a list, look for a pattern, or draw a diagram), in early childhood they should be developing a readiness for problem solving through explorations with concrete materials in real-life situations.

Teachers of young children should be aware of problem-solving processes and should provide numerous experiences that stimulate creative thinking. If young children (1) have a keen sense of observation; (2) can

classify, compare, and sequence objects; (3) can recognize, record, and create patterns with objects; (4) can create problems, make predictions, and draw conclusions; (5) can use spatial relations; and (6) can commu- nicate mathematical relationships, they have the "tools" for success in mathematics. The challenge for teachers of young children is to build an interesting and meaningful program for problem-solving readiness around the child's intuitive and informal knowledge of mathematical ideas.

References

Carpenter, Thomas P., James Hiebert, and James M. Moser. "First-Grade Children's Initial Solution Processes for Simple Addi- tion and Subtraction Problems." Journal for Research in Mathematics Education 12 (Jan- uary 1981):27-39.

Ginsburg, Herbert P. "Children's Surprising Knowledge of Arithmetic." Arithmetic Teacher 28 (September 1980):42-44.

Hebbeler, K. "Young Children's Addition." Journal of Children's Mathematical Behavior 1 (1977): 108-12.

Houlihan, Dorothy M., and Herbert P. Gins- burg. "The Addition Methods of First- and Second-Grade Children." Journal for Re- search in Mathematics Education 12 (March 1980:95-106.

Krulik, Stephen, and Jesse Rudnick. Problem Solving: A Handbook for Teachers. Boston: Allyn & Bacon, 1980.

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

. The Agenda in Action: 1983 Yearbook. Edited by Gwen Shufelt. Reston, Va.: The Council, 1983.

Reys, Robert E., Barbara J. Bestgen, Terrence B. Coburn, Robert Marcucci, Harold L. Schoen, Richard J. Shumway, Charlotte L. Wheatley, Gray son H. Wheatley, and A. L. White. Keystrokes: Calculator Activities for Young Students, Counting and Place Value. Palo Alto, Calif.: Creative Publications, 1980.

Richardson, Lloyd I., N. Goodman, N. Hart- man, and H. LePique. A Mathematics Activi- ty Curriculum for Early Childhood and Spe- cial Education. New York: Macmillan Publishing Co., 1980.

Wheatley, Charlotte, and Gray son Wheatley. "Developing Spatial Ability." Mathematics in Schools 8 (1979a): 10-11.

Wheatley, Charlotte L., and Grayson H. Wheatley. Calculator Use and Problem-solving Strategies of Early Adolescents. Wash- ington, D. C: National Science Foundation, 197%. (SED-7919614)

Zweng, Marilyn J. Children's Strategies of Solving Verbal Problems. Washington, D.C.: National Institute of Education, 1979. (NIE- G-78-0094) m

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