474 Teaching Children Mathematics / May 2004

Problem-SolvingStrategies of

First GradersThe significance of problem solving in the K–12

mathematics curriculum has been well docu-mented over the years by professional organi-

zations and individuals. The National Council ofSupervisors of Mathematics (NCSM 1977) and theNational Council of Teachers of Mathematics(NCTM 1980, 1989, 2000) endorse the inclusion ofproblem solving at all grade levels. For example,NCTM (1989) states, “The development of each stu-dent’s ability to solve problems is essential if he orshe is to be a productive citizen” (p. 6). In the samepublication, NCTM asserts that “problem solvingshould be the central focus of the mathematics cur-riculum. As such, it is a primary goal of all mathe-matics instruction and an integral part of all mathe-matical activity. Problem solving is not a distincttopic but a process that should permeate the entireprogram and provide the context in which conceptsand skills can be learned” (p. 23). NCTM reiteratesits position on problem solving in Principles andStandards for School Mathematics (NCTM 2000).

Cognitively GuidedInstruction (CGI)Cognitively Guided Instruction (CGI) is an exam-ple of a research-based, problem-solving approach

to teaching mathematics. As the name implies, anunderstanding of the learner’s thinking guidesinstruction. The idea originated at the University ofWisconsin with Thomas Carpenter and his col-leagues. Although it is intended primarily forgrades K–3, CGI is such a broad philosophy ofteaching that it can be adapted for use at almost anygrade level.

In this approach, the teacher presents studentswith mathematical word problems set in the con-text of their environment and allows students thefreedom to create their own strategies for solvingthe problems using available resources. They canuse physical, pictorial, or symbolic representationsand are encouraged to explore different ways tosolve a particular problem. In the process of solv-ing problems, children invent their own algorithmsand share their problem-solving strategies with therest of the class. The process of obtaining ananswer is more important than the answer itself.

Based on a student’s response to a problem, theteacher prepares appropriate follow-up problems.For example, if a student has not solved a problemsatisfactorily, the teacher may give him or her aless difficult problem, perhaps one with lessernumbers. The CGI approach easily addressesNCTM’s Process Standards (Problem Solving,Reasoning and Proof, Communication, Connec-tions, and Representation); this is one of its great-est benefits. More important, learners becomeadept at these processes.

The CGI teacher is a facilitator of learningrather than a disseminator of knowledge. He or sheavoids imposing adult thinking on students. Theteacher spends a lot of time questioning and listen-ing to students’ explanations in order to determine

Emam Hoosain, ehoosain@aug.edu, teaches mathematics edu-cation at Augusta State University in Augusta, Georgia. He isinterested in problem solving, methods of teaching mathemat-ics, and research in mathematics education. Renee Chance,rhchance@burke.net, teaches first grade at Waynesboro Pri-mary School in Waynesboro, Georgia. She is interested in how

young children think and solve mathematical problems.

By Emam Hoosain and Renee H. Chance

Copyright © 2004 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

how they are thinking. Therefore, the teacher mustpossess good questioning skills. Once the teacherhas determined how the learner is thinking, he orshe tries to build on the learner’s ideas. CGI isbased on the premise that children come to schoolwith a great deal of mathematical knowledge; thetask of the teacher is to uncover this knowledge andbuild on it.

CGI places great emphasis on individualization,conceptual understanding, and higher-order think-ing. More emphasis is placed on formative evalua-tion than on summative evaluation. The approachdoes not use any teaching materials such as text-books, so the teacher must prepare his or her ownproblems. This and other responsibilities such asclassroom management make teaching challengingfor the CGI teacher. For a discussion of the charac-teristics of a CGI classroom, see Chambers andHankes (1994).

Classroom management issues are importantconcerns for the CGI teacher, especially if the classis large, because CGI is an individualizedapproach. If students are properly oriented to CGI,however, they become independent, responsible,

and accountable learners and, therefore, can workon their own while the teacher is attending to onestudent. Independence, responsibility, and account-ability take time to develop. From the inception ofinstruction, the teacher must make students awareof what is required of them and implement policiesconsistently. Allowing students to make their owndecisions facilitates development of these qualities.For example, students choose strategies that aremeaningful to them to solve a problem, and theyselect what materials to use. They do not depend onexplicit instructions from the teacher. Instead ofcorrecting students’ mistakes, the teacher can helpstudents correct their own mistakes by questioning,pointing out contradictions, providing counter-examples, suggesting that a result be checkedagainst the conditions in the problem, or askinganother student to share his or her perspective ofthe task. The teacher can teach students basicresearch skills to answer their own questions; inthis way, the teacher is relieved of the responsibil-ity to answer students’ questions. Students becomeaccountable because they know that the teachermay call on anyone to explain his or her solution to

Teaching Children Mathematics / May 2004 475

a problem. Students can help one another stayfocused by identifying off-task behaviors and deal-ing with them. Students can develop independence,responsibility, and accountability only if theteacher gives them opportunities to do so.

The presence of a paraprofessional in the class-room helps alleviate management problems. Thefirst author of this article observed many CGIclassrooms in North Carolina and did not recog-nize any serious management problems. Theteacher in the investigation described in this articledid not have any problems managing the class.From the beginning, she made students aware ofher expectations and the students cooperated. Shealso used the services of the paraprofessional.

In the mid- to late nineties, workshops to trainteachers to use CGI were conducted in North Car-olina; previously, workshops were offered in Wis-consin. Two national conferences were devotedentirely to CGI. Presentations on CGI often aremade at national, regional, and state conferences.We are not aware of a single organization that coor-dinates CGI workshops and conferences. Readerswho are interested in CGI, however, may want tosearch online for “Cognitively Guided Instruction”

to get information about possible CGI workshopsand the names of contacts. The first author of thisarticle is fully trained to conduct CGI workshopsand was a co-leader of some of the North Carolinaworkshops.

CGI Problem TypesFour broad CGI problem types are related to addi-tion and subtraction: join, separate, part-part-whole, and compare. Each of these has sub-types.For a detailed description of the problem types andchildren’s strategies, see Carpenter et al. (1999),which includes a CD that illustrates children work-ing with these problem types. Knowledge of theproblem types helps the teacher present a balancedcurriculum to students and move students to higherlevels of thinking. Some types of problems aremore difficult than others.

“Separate” problems have three sub-types:result unknown, change unknown, and startunknown. The following is an example of a “sepa-rate change unknown” problem: “Mary had 9cookies. After she gave some of these cookies toJon, she had 5 cookies left. How many cookies didshe give to Jon?”

This article reports the results of an investiga-tion into the problem-solving strategies that agroup of first graders used when solving “separatechange unknown” problems. The children hadalready done some work with “separate resultunknown” problems, and the teacher thought thatthey were ready to move on to “change unknown”problems. Knowledge of these strategies will helpteachers plan more appropriately for students.

ProcedureThe teacher, the second author of this article, readthe problem to each child and reread it if necessary.Sometimes, the teacher and the child read the prob-lem together. The teacher asked the child to solvethe problem using any method and to write anexplanation of the solution. The child could usematerials such as counters, number lines, markers,and drawing paper. After the child’s initial solu-tion, the child gave an oral explanation of the solu-tion process during a dialogue between the teacherand the child. The usefulness of dialogue withlearners is well documented; see, for example,Hoosain (2000) and Buschman (2001). The datawere collected through the teacher’s observationsand dialogues with the students. The students’

476 Teaching Children Mathematics / May 2004

Figure 1John changed his equation after explaining his

thinking.

responses follow; their names have been changed. The teacher gave John the following problem:

There were twenty orange pumpkins on thebackyard fence. Some of them fell down; nowthere are twelve. How many pumpkins felldown?

The teacher read the problem with John andobserved how he solved it. He carefully countedtwenty counters; then he took some away and saidthat the answer was 9. When the teacher asked himto explain how he obtained that answer, herepeated the procedure with the counters. Then hesaid, “I started with twenty, then separated twelvebecause that was what was left, and I got eight.” Hediscovered that his mistake was in counting.

The teacher asked John to write an explanationof how he thought about the problem. He wrote,“20 – 12 = 8.” Recognizing that the equation didnot reflect the information in the problem, theteacher read the problem again with the studentand asked him to explain how he got his subtrac-tion sentence. He smiled as he crossed through thefirst sentence and replied, “It is wrong.” In his sec-ond attempt, he wrote, “20 – 8 = 12.” He explained,“The word some gets a box because I do not knowwhat it is.” His explanation could be a reflection ofhis prior knowledge of how to deal with key wordsin a problem. Perhaps the teacher, who is transi-tioning from a traditional approach to a problem-solving approach, taught him that he must write thesemantic equation, 20 – 8 = 12, instead of the com-putational equation, 20 – 12 = 8. (See Van DeWalle 2001, p. 111, for a discussion of semanticand computational equations.) John was never dis-couraged. He was motivated by his active involve-ment and curiosity. Figure 1 shows his work.

The teacher gave the same problem to Mark. Theteacher and the student read the problem together.Initially, Mark struggled with the problem; then heslowly wrote the number sentence “20 – __ = 12.”He started to work with the counters but abandonedthem and resorted to the number line. He placed onefinger on 20 and the other on 12; then he countedthe numbers between 12 and 20 and obtained 8,which he wrote in the blank space. The number sen-tence he wrote was “20 – 8 = 12.” He seemed satis-fied with his solution and very proud of his method.He explained his method step by step to the teacher.Figure 2 shows his solution.

To introduce some variety, the teacher gave Joana slightly different problem:

There were seventeen black cats in a tree. Someof the cats jumped down. Now there are eightcats in the tree. How many cats jumped down?

Joan carefully drew seventeen cats and crossed outeight of them. The way she wrote her number sen-tence, “17 – 9 = 8,” was consistent with the writtenproblem; that is, she wrote the semantic equation.When asked, she clearly explained her procedureto the teacher. She said, “I started with seventeencats and some went away. There are eight now, soI will mark them out and find out what went away.”She carefully wrote her explanation on the problemsheet. Figure 3 shows her solution.

ConclusionThis investigation has shown that children can bevery creative when they are allowed to choosetheir methods for solving mathematical problems.These first graders of different ability levelssolved problems that many people would considertoo advanced for them. They used different strate-gies (modeling and counting) as well as differentrepresentations (concrete, pictorial, and sym-bolic). The different strategies they used could bean indication of their levels of thinking. Markseemed to be the most advanced student because

Teaching Children Mathematics / May 2004 477

Figure 2Mark used a number line to solve the problem.

he counted on the number line; the number line ismore abstract than counters. Joan’s level of think-ing seemed to be between those of Mark and John,and her explanation was the most fluent. Perhapsher explanation was indicative of her confidenceand oral skills. John struggled a little before heobtained the solution. In his quest for the semanticequation, he appeared to use a trial-and-errorapproach, which he tried to reconcile with his pre-vious knowledge.

Based on the teacher’s observations, we con-cluded that the students benefited immensely fromthe experiences. The students demonstrated a will-ingness to pursue the solutions and to share themwith the teacher and the rest of the class, because itallowed them to be the teacher for a while. Theylearned from one another when they shared theirsolutions; they were able to see multiple solutionstrategies for the same problem. Some studentslearned to correct their own mistakes after seeingother solution strategies. John corrected what hethought was a mistake.

The students were actively engaged in the tasks;they were motivated and interested; they developedconceptual understanding, judging from their

explanations; and they demonstrated improvedsocial (collaboration) and communication (writing,reading, listening, and speaking) skills. They artic-ulated their thinking clearly enough for the teacherto understand, and they did not appear inhibited indoing so. Their confidence was boosted; Mark, forexample, appeared satisfied with and proud of hissolution. We believe that the students’ reasoningand problem-solving skills improved; however,time and more of such problem-solving experi-ences may be necessary to see explicit evidence ofthe improvement of these skills.

The teacher’s reflections indicated that she, too,benefited from the experience. Most important, herapproach to teaching young children and her inter-actions with them underwent significant changes.She has recognized the need to listen carefully towhat students are saying, and she plans more thor-oughly. She is using her knowledge of students’thinking to plan appropriate tasks and questions.She has become more reflective about her practicewith a view toward improving it. She is more tol-erant of and open-minded to students’ solutionstrategies.

Occasionally, the teacher might have attemptedto impose her thinking on the students. For exam-ple, in the case of John, the teacher might haveinsisted on the semantic equation. Although this isinconsistent with the philosophy of CGI, it isunderstandable in someone who is transitioningfrom a traditional teaching approach to a CGIapproach. The teacher was at Level 2, Transitional,according to the levels in becoming a CGI teacheridentified by Carpenter et al. (1999). At this level,she would use traditional and CGI approaches; soshe would allow students to use their own strate-gies and also inject her strategies.

The results of this investigation lend supportfor a problem-solving approach to the teaching ofmathematics. They are consistent with the findingsof Riordan and Noyce (2001) and Reys et al.(2003), who concluded that children who aretaught through a standards-based (usually prob-lem-based) curriculum do better on standardizedtests than do children who are taught through a tra-ditional curriculum. The results also are consistentwith the findings of the Treisman study cited bySingham (1998). Treisman found that a problem-solving approach to the teaching of mathematicsresulted in improved grades for college students. Astrong case for the use of a problem-solvingapproach in teaching mathematics is rapidlydeveloping.

478 Teaching Children Mathematics / May 2004

Figure 3Joan drew cats to find the answer.

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