JOURNAL OF ELEMENTARY SCIENCE EDUCATION. Vol. 8. No.2. Pp. 66-87. (1996). © 1996CollegeofEducation. The Universityof WestFlorida

Integrating Math and Science ThroughProblem Centered Learning in Methods

Courses: Effects on Prospective Teachers'Understanding of Problem Solving

by Carol Briscoe & David Stout

AbstractThis study reports our understanding of the views onproblem-solving developed by prospective elementaryteachers as a result ofexperiencing problem centeredlearning in an integrated math-science methodscourse. The course was designed to assist theprospective teachers to develop their problem solvingskills in mathematics and science contexts and todevelop problem-solving activities that integratemathematics and science to use in their classrooms.Small group activities, where students had multipleopportunities to interact with concrete materials andto discuss their ideas, were emphasized. Data sourcesincluded audio taped interviews, field notes, anddocuments such as portfolios and lesson plans studentsproduced during problem solving activities. Fourassertions related to students' attitudes towardproblem solving, their knowledge ofproblem solvingand problem solving processes, and the roleofproblemsolving in elementary classrooms are discussed.Implications for planning and implementingintegrated mathematics and science problem solvingactivities in methods courses and for continuedresearch are presented.

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IntroductionWe begin to accelerate up the rough mountain side.The mountain of problem solving. Since this roadhas been seldom traveled it has many rough edgesthat we attempt to smooth out as we drive along...[Ittakes} perseverance to get to the top ofthe mountain,sometimes there is a detour or two, but we remainheaded in the right direction. The more we drive themore comfortable it gets...Sometimes we drove sofastwe did not realize how much we really saw on ourtrip, but as we look back. we realize what a steep climbwe made...There are many trips in life and each onemakes us better and stronger for having traveled it.(Cindy, Fall 1994).

John Dewey (1916) considered problem solving through themethods of science, coupled with reflective thinking, a primary purposeof science instruction. As Champagne and Klopfer (1977, 1981) note,since the first issue of Science Education was published, improvingstudents' abilities to solve problems using inquiry skills and criticalthinking has been a concern of science educators. Furthermore, inthe most recent calls for change in the teaching of science, andmathematics as well, emphasis is again placed on problem-centeredlearning, as the focus of good teaching (American Association for theAdvancement of Science, 1989; Rutherford and Ahlgren, 1990;National Council of Teacher of Mathematics, 1989; National ResearchCouncil, 1989). Involving individuals in hypothesizing, predicting,observing, measuring, experimenting, collecting and analyzing dataand communicating their ideas immerses them in the activities commonto both mathematics and science.

Problem solving skills can best be developed if a firmfoundation is provided through a child's earliest experiences withinschool. However, elementary science is most often taught from areading rather than experimental approach, and mathematicsinstruction emphasizes the algorithmic approaches to problems

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represented in textbooks (McKnight, 1989; Romberg & Carpenter,1986). Researchers cite as reason for these approaches to teaching,the fact that many elementary teachers lack sufficient personalunderstanding of problem-solving processes to be able to plan lessonsthat enable children to be problem solvers (Funkhouse, 1993;Gonzales,1994; Harty, Kloosterman, & Matkin, 1991; Martens, 1992).

It is well understood that teachers tend to teach as they weretaught. Typically prospective elementary teachers have not hadopportunities in their own science and mathematics experiences tomake sense of the knowledge generated in these two areas as a productof problem solving processes (Cuban, 1982). When these prospectiveteachers enter methods courses their lack of understanding of problemsolving processes is coupled with anxiety towards the subject matter.This creates a context that does not favor their constructing anunderstanding of how to teach mathematics or science in meaningfulways.

The Holmes Group (1986) suggests that providingopportunities for prospective teachers to construct knowledge ofproblem solving, of content and of teaching methods in an integratedfashion, within a context that emphasizes processes for generatingknowledge, promotes development of problem solving skills.Accordingly, we designed our courses to establish a learningenvironment where students would work toward solutions of problemsin a manner that modeled the knowledge creating activities of scientistsand mathematicians. As we considered the tasks we would assignstudents, we tried to construct those that would encourage studentsto participate in problem solving as outlined in Good, Mulryan andMcCaslin (1992), "(1) maintaining the intention to learn (2) whileenacting alternative task strategies (3) in the face of uncertainty" (p.173). The tasks we set were designed to involve students in designingexperiments and collecting data, then using mathematical reasoningto reach a solution.

This study undertook to examine whether generatingknowledge of science conceptsthrough small group problem-solvingactivities assisted prospective teachers to develop an understanding

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of the nature of problem solving in mathematics and science teaching.We also investigated how experience with problem-solving influencedprospective teachers' confidence in their problem-solving skills. Finallywe were interested in finding out whether these kinds of experiences,would aid our students in developing their ability to plan and teachproblem solving activities in math and science.

Specific questions that guided the research included:-How does practice in scientific problem solving influence individuals'recognition and application of problem solving skills?-How do group interactions facilitate development of individuals'problem solving abilities?-How does practice in scientific problem solving influence individuals'confidence in themselves as problem-solvers and their views of problemcentered learning in science and mathematics teaching for elementarychildren?

MethodContext and participants

This study involved 62 students, 8 males and 54 females,enrolled in two sections of Mathematics and Science methods courses.All had taken prerequisite courses including one mathematics and onescience content course designed for elementary teachers. Other thanthese two courses most students had completed only 6-7 hours ofscience and 6 hours of mathematics courses required for completingbasic studies.

The methods courses were two three hour courses scheduledback to back on the same day. Students were enrolled incomplementary sections to form cohort groups who attended bothclasses. The instructors, one a mathematics educator, the other ascience educator, worked as a team planning and implementing thecurriculum. Both instructors were present during all class meetings.A key feature of the curriculum was the special time block set asidefor problem solving. The problems presented to the students weregrounded in mathematics and science content and selected to challengethe students to use both science process skills and mathematics skillsas they solved them (Appendix A). The prospective teachers, in self

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selected groups of four, worked toward solution of the problems. Arequirement was that all solutions offered were substantiated by thedata collected. The tasks were completed during class time, althoughsome students also met outside of class to work on them. In additionstudents were also provided with a set of optional "problems of theweek" from Vande Walls (1994), that they could work on individuallyor together and place in their portfolios to demonstrate growth intheir problem solving abilities.

Data SourcesQualitative data regarding students' application of problem

solving skills were obtained from three sources. Transcripts ofinterviews with students served as one data source. Questions posedin the interview focused on students' perspectives on a portfolio theywere keeping that was to represent their growing understanding ofscience and mathematics teaching, problem solving, and how childrenlearn. No questions focused specifically on problem solving. However,30 students brought up the topic of problem solving during theinterviews. The second primary data source was the portfoliosproduced by the students. Although students were required to have asection on problem solving in their portfolios, the quantity and qualityof materials placed in this section were left up to the students.Accordingly, not all students emphasized problem solving in theirportfolios. However, 37 students had substantial sections dealingwith problem solving. Between these two sources of anecdotal data54 students (87%) of the total enrolled in the course were represented.A third source ofdata was the problem solutions with accompanyingexplanations and lesson plans produced by each group of students forthe Tasks. From these data we constructed assertions regardingstudents' attitude toward and developing knowledge of problemsolving. Additional data from field notes served as sources totriangulate assertions constructed from the interviews, portfolios, andproblem solutions.

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AnalysisThe first step of analysis of data consisted of reviewing the

transcripts and portfolios. The interview transcriptions and portfolioentries were read 'separately by each researcher and major themes inthe ideas expressed by students in these records were constructed.The researchers then met and negotiated a classification system forthe data that was consistent with the themes that had been constructedindividually. Where possible categories were divided into sub­categories based on common threads in the comments expressed bystudents. Three major areas of reflection represented in these datasources were: 1) reflections on students' personal attitudes towardproblem solving in mathematics and science; 2) reflections on theirknowledge of problem-solving and problem-solving processes; and3) reflections on application of problem solving to classroom teachingand children's learning of mathematics and science. Subcategoriesfor each major theme were also generated (See Table 1).

Each researcher then went back to the original data sourcesand organized the data within the categories and subcategories thathad been constructed. The total number of student entries coded byboth researchers was 144. The initial inter-rater agreement on theclassification of these entries was .70. The difference was found to beattributable to two factors. First, there were two sub-categories thatwere very similar; a) under the first category - group work and itseffect on students' growth in problem solving skills, and b) under thethird category - group work and its effect on students' attitudes towardproblem solving. In several cases the raters distinguished these twoitems differently. The second discrepancy was in the number of itemscoded by each researcher. In some cases an item coded by one wasnot coded by the other. In each case, the discrepancies were discussedand resolved so that final agreement reached 100 percent. The numberof data entries that fit within each category were then grouped, talliedand summarized in Table 1.

The categorized data together with the students' problemsolution reports and field notes supplied the information that formed

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2750

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1653

the bases for constructing the following assertions related to theoriginal research questions.

InterpretationsAssertion I: Lack of prior experience in problem solving led tofrustration in students' early attempts to solve problems; however, asgroups worked together, students' confidence in their problem solvingabilities increased.

The data presented in Table 1 indicate that among the 54students who reflected on problem solving in their portfolios or duringinterviews 27 made one or more statements that related to their attitudetoward problem solving. Most of these students (85%) noted thatproblem solving was a source of frustration for them. A commonthread that ran throughout these data as reason for frustration wasthe lack of prior experiences in science and mathematics contextswhere problem solving was used to promote learning. Their concernalso was related to the fact that several of the problems had more thanone correct answer. Comments included:

I have never before been given a problem inschool and been told that there may be many answersor no answer... How will I know if I'm on the righttrack if I'm not given an example or an expectedoutcome? (L.T, Portfolio).

The group may have been intimidated with atask that required them to find their own scientificsolution to a problem without benefit of specificguidance and without knowledge that there was indeedone correct answer.. ..I too felt this intimidation (D.D.,Portfolio, Task I).When presented the second task, students noted how difficult

it was to depend only on their experimental data to develop and defendan answer. Most had never been asked to work in this way before.These reflections are representative:

Trying to find a formula that would workproved to be very difficult. It seemed that once we

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thought that we were headed in the right direction,things would fall apart.. ..We wanted to look in theencyclopedia so bad that we could taste it. (L.R.,Portfolio, Task ll).

I don't feel confident enough to defend myposition ....I was never asked why I thought my answerwas correct or how I arrived at my answer. (C.v.,Portfolio, Task ll).Our providing students with opportunities to "be scientists,"

learning to trust data they had collected to construct knowledge ofscience concepts, seemed to be the most frustrating feature of problemsolving for the students. The history of their educational experienceswhere science was learned as facts to be memorized and doing sciencewas following directions was difficult to overcome. We presented aprobability task with the intentions that, by collecting data and usingthe rules of probability to develop tentative conclusions regardingrelationships among genetics crosses, students would become moreconscious of the tentative nature of scientific knowledge. We foundthat, rather than viewing data as useful in developing scientific ideas,many students would not commit to any solution as trustworthyregardless of the preponderance of data. They still wanted to knowthe "right" answer.

On the other hand some students did develop confidence intheir ability to apply the processes of problem solving and to acceptuncertainty:

I can make decisions of where I'm going togo, and like the first problem that we did, I could notaccept an answer that I was unsure of. I had to haveconcrete answers. But now I've been able to acceptthat its ok, because this is all the information 1 haverc.v, Portfolio, Task lll).

When we received this problem, we all seemedso much more confident of our ability to solve it. Inthe previous two problems there seemed to be a general

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sense of loss as to where to even begin. (C.P.,Portfolio,Task ill)

Of the 18 students who specifically expressed a positive change intheir personal attitudes toward problem solving, seven attributed theirchange in attitude to the benefits of group interaction:

My own thinking while trying to solve thisproblem was frustration at first; but it became betteras we threw our ideas on the table. One person wouldsay something and that would help spark an answerfrom someone else. (M.R., Portfolio, Task I).

Furthermore, as shown in Table 1, 30 of the 54 students noted thatthey valued problem centered learning as an important and appropriateway to learn and to teach science and mathematics to children (Seealso Assertion 3). As one commented:

1 can see how my attitude toward problem­solving has changed. I thought that it was the mostdifficult thing to do at the beginning of this term. Butnow the ideas for problem-solving lessons [forchildren] just flow. (R.S., Portfolio, Task III).

Overall, 43 of the 54 students (80%) stated that they had developedpositive attitudes toward problem solving and/or they valued problemsolving as a way to teach science and mathematics to children.

Assertion 2: Trial and error methods were thefocus ofearly problemsolving strategies, however, as students worked together they learneda variety ofeffective strategies for seeking solutions to the problemspresented.

Pizzini, Shepardson and Abell (1989) identify three knowledgestates through which students progress as they attempt to solve agiven problem (1) identification of initial information about theproblem, (2) identification and application of operational procedures,and (3) a goal state or solved problem. Our data suggest thatprogression through these states is not a natural process for beginningproblem solvers. The general approach to solution for Task I was to

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begin collecting data without considering whether the data werenecessary to solve the problem. As one student noted,

Without even knowing what the problem wasasking, I wanted to begin some type ofcomputation....1found it very difficult to read and re-read the problemwithout beginning to solve for the answer. (Civ.,Portfolio, Task I).

This approach is common among beginning problem solvers (Woods,1983; Shoenfeld, 1985). Not knowing what to do, yet needing to dosomething they read the problem and decide quickly what to do andpursue that direction without reconsidering.

Frustration arose when students had collected the data andthen did not know what to do with it. It was not until they realizedthat they had nowhere to go with the information that they discussedthe problem and attempted to define what it was they were lookingfor. Once the problem was defined students began using their data inconjunction with data provided by the instructors to solve the problem.However, not all students took the most efficient route to solution.Many were confused by extraneous information on the nutrition chanand attempted to use it in the solution to the problem.

On the other hand, as our students practiced problem solvingin small groups, their approaches to solution became more efficienland resembled more closely a progression through the three statesdescribed above. The urge to "dive right in" and get some data gaveway to students discussing what they knew and defining the problemclearly before manipulating materials. For Task II, most groupsreported beginning the task by discussing what they knew about thecontext and defining the problem. Trial and error was used initially tcinvestigate the problem, but in most group reports there was evidencethat a solution process that involved hypothesis testing was eventuallyplanned, then carried out.

Students' approaches to problem solving were more organizedduring Task III. Portfolio reflections suggest that planning the routeto solution was an important activity:

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We discussed what we felt we needed to do tosolve the problem and gave our reasons why we feltthat way and then set about collecting our data. (C.v.,Portfolio, Task III)

We are as a group, discussing what we arefinding and recording our findings clearly as we go.(A.H., Portfolio, Task III)

As students worked toward solution of problems they also becamemore aware of the various processes they were using to make senseof the problem and the data. Of the 54 students who are represented,43 (80%) reflected on the nature and processes of problem solving.Thirty one students' comments focused on the thinking processes thatare involved in problem solving. Particular emphasis was placed onthe importance of process vs. product and understanding the reasoningbehind a solution as represented in these reflections:

I finally see that the problem does not have tobe hard, but should promote discovery and exploration.I also realized that it is the process that goes on duringthe problem, rather than the solution that is mostimportant because this is where the children areengaging and interacting with each other as they areexploring (M.R. Portfolio).

I no longer find value in only the correctanswer, but also in the process by which the answer isfound ...The real importance of problem solving is thatone should actively try to solve problems oneencounters (G.M. Portfolio).Eleven students identified working in groups as a positive

contributor to their development of problem solving skills and morepositive attitude toward problem solving. Statements that indicatethe benefits students found working in groups include:

I've learned that working in groups helps a lotbecause you can get a lot accomplished throughthat...everybody's come up with their own way andwe've had to kind of settle on something and so I think

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that's good and I've learned a lot through the problemsolving (K.T., Interview)

This problem taught me that cooperativelearning does work. I learned how to work thisproblem from my group members. Without their helpI would have been in the dark (K.c., Portfolio, TaskIII).

Because we did not study closely the actual interactions amongmembers of the groups we cannot address what factors of groupinteraction facilitated students to develop enhanced problem solvingskills. We would suggest, based on student comments, that somefactors include: a) the nature of the group itself and whether allmembers shared equally in solving the task; b) the nature of the taskand how much prior knowledge of the context in which it was framedthe students had; c) how comfortable students were in sharing theirideas with others, especially when they were unsure of their ownknowledge; and d) the confidence of the student in entering into thesolution process - whether he or she could at least enter into it atsome level. From a constructivist perspective, each of these factorsinfluence the capacity of the student to merge motivation, emotion,cognitive strategy and metacognitive awareness into an intellectualstructure that favors growth in problem solving skills.

Assertion 3: Experiencing problem centered learning as an effectiveway to facilitate their own meaningful learning, Students came toview problem solving as an appropriate and important method forteaching children science and mathematics.

An important goal in our implementing problem-solving inour methods courses was to facilitate students' development ofattitudes and skills that would help them to become elementary teacherswho would teach science and mathematics effectively. Accordingly,we required our students not only to solve problems, but to designproblem centered learning activities for the children they taught duringa practicum experience connected with the course. Students' makingthe connection between problem solving and their own learning is a

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first step toward developing confidence in the method as an appropriateway to teach children. If prospective teachers come to value theirproblem solving experiences from a personal perspective they canbegin to see the benefits of its application in the classroom.

As Table I shows, 30 of the 54 students reflected on problemsolving in connection with teaching or children's learning. Eight ofthese students noted that problem solving as a method of learningmathematics and science encourages children to be thinkers and sixteenstudents noted that problem solving encourages meaningful learningbecause it is fun, interesting, and relevant. Specific commentsincluded:

I have changed my way of thinking aboutproblem-solving and guided discovery in several wayssince this class. Because this class has provided theopportunities for me to experience problem-solvingand guided discovery lessons as a student, I see howthese types of lessons allow children to begin to thinkand explore in a completely different way (M.R.,Portfolio) .

I liked the way we did the problem solving inclass ....Now I know not to just give the children thefacts. Show them something and let them find outthings on their own, .... If they're thinking about itthemselves and finding out the solutions, it helps them,you know, remember more (A.R., Interview).

Problem solving is definitely a good way to goabout teaching children because I've learned it willstimulate them to be more curious, because it makesme more curious. (K.T., Interview).

The group problem solving situations havedemonstrated how meaningful experiences provideconnections between prior knowledge and newlearning...Since I have benefitted from this type ofteaching and have practiced planning and implementing

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problem-centered lessons, I feel very well prepared touse problem solving in my classroom (M.C., Portfolio).For some students the practicum experience was an important

factor in changing their ideas about problem solving. The followingare examples of how practicum experience influenced their thinking:

I previously thought that problem solvingactivities would not work with younger elementarychildren, but I know that I was wrong. It is just asimportant for younger children to develop criticalthinking skills that will help them later in life (J.Z.,Portfolio).

The students I worked with at the elementaryschool were quite a bit more open-minded about theprocess. They showed an enthusiasm level that seemedto be a little more intense than usual....It was ratherexciting to see them take ownership of the learningprocess (D.D., Portfolio).

Until they presented their own problem-solving activities to children,the prospective teachers tended to project their own difficulty withproblem-solving as something that would be common in the children,However, as the examples given above illustrate, the children's positivereactions to problem-solving activities influenced the prospectiveteachers to reflect on and change this belief.

Assertion 4: Students' initial views ofproblem solving as "gettingthe right answer to math questions" broadened to include "creatingsolutions" as a goal ofproblem solving.

Traditionally, activities that have been tenned mathematics ancscience problem solving have been grounded in contexts that requirecstudents to use given data describing a natural phenomenon or episodefrom everyday life, and calculate relationships among the data usingmathematical expressions. Shoenfe1d (1992) describes these kinds 01

activities as "routine exercises, often organized to provide practiceon a particular mathematical technique that, typically, has just beerdemonstrated to the student" (p. 337).

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Similarly, in science teaching, laboratory work that follows"cookbook" plans has been reported as widely used in schools in theU.S. and Australia (Gallagher & Tobin, 1987; Tobin & Gallagher,1987). These kinds of activities often have low cognitive demandand provide contexts that require little reflective thought orconcentration.

Because most of our prospective teachers described schoolexperiences that reflected these traditional approaches to "problemsolving," their understanding of what scientists and mathematiciansdescribe as problem solving was limited when they entered our course.Resnick (1988) points out that a fundamental component of thinkingmathematically or scientifically is seeing the world in the way scientistsand mathematician do. Thus, one of our goals was to provide theseprospective teachers with experiences that, at least in some facets,resembled the work of scientists and mathematicians. Although wewere unable to provide sufficient experiences to assist our students toconstruct the values and habits ofmind of professional mathematiciansand scientists, our data indicate that these prospective teachers'perspectives of what it means to learn and teach through a problem­centered approach did evolve to some degree. Eighteen of the 43students (40%) who addressed the nature and processes of problemsolving in their reflections, described a changing definition of problem­solving. Representative of students' changing ideas are the followingcomments:

When I first knew what I thought of problemsolving, it was just, to give a problem and solve it. Ialways thought of just one way to solve it.. ..Youweren't supposed to show how you got it. ... I'velearned that there's different ways, and not just oneway (C.P, Portfolio).

It was something I never really thought aboutteaching. I mean, well, you've got the algorithm, thereyou go. You teach the algorithm, you've taught themath...so it was a whole new world going, "why doyou have the algorithm?" (M.G., Interview)

8\

Ijust figured teachers just have the book...Gohome and do your math problems. Its more than that.You have to help the children, not just the process,you have to teach them why they're doing it (M.C.,Interview).A concern for us is that, based on the data, students still connect

problem solving as a process that is associated primarily withmathematics. None of the students placedtheir discussion of problemsolving in the context of solving scientific problems. Although eachstudent was required to develop a unit plan that included four problem­centered activities grounded in the science they intended to teach,these activities were never discussed in relation to problem-solvingprocesses.

Conclusions & ImplicationsOur study is limited by the fact that we had a small number of

student participants and our data was based on anecdotal referencesstudents made within portfolios and interviews designed for otherpurposes than this project. However, we believe that our data supportsan argument that practice in problem solving during the methods coursecan have a positive effect on prospective teachers' attitudes toward,and valuation of, problem centered learning. Also, we found thatconfidence in and use of appropriate problem-solving strategiesincreased as our students progressed through the various tasks.

Research on small group problem solving has suggested thatprocesses of interaction, feedback and discussion within small groupspromotes the acquisition of reasoning skills. As individuals elaboratetheir understanding of a problem through discussion with otherproblem-solvers they develop a better understanding of their ownpersonal problem solving processes (Andre & Phye, 1996; Good,Mulryan, & McCaslin, 1992). Our data support these findings. Theintegration of affective and intellectual spheres of mind is mediatedby the interactions occurring within the group (Vygotsky, 1962; 1978).Formation of links between interpersonal and intrapersonal spheresthrough language is an especially important characteristic of learning

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in social environments. However, more research is needed to clarifyhow these links are constructed as students engage in group problemsolving and how these linkages foster growth in problem solving skills.

We believeteacher-educators can prepare prospective teachersmore likely to engage in teaching through problem-solving by providingexperiences in problem-solving where the processes and content formathematics and science are fully integrated and necessary to solvethe problems. The problem-centered learning activities providedshould allow opportunities for prospective teachers to construct andcommunicate explicit links between processes and outcomes ofproblem solving. If they are provided with significant opportunitiesto link their own problem solving processes with the development ofproblem-centered lesson plans for children, prospective teachers canbe assisted to view the teaching of problem solving as effective practice.Accordingly, making integrated problem-solving an integral part ofmethods class experiences is an important factor that may lead to thepreparation of more effective elementary teachers who will providethe kinds of science and mathematics experiences for children that areadvocated in the reform literature.

However, whether our students will become teachers who areable to implement problem-solving in their classrooms will depend ona number of factors including the culture of the school in which theybegin their careers, the contexts of their own classrooms and thestrengths of their own beliefs about problem solving in relation toother beliefs that will affect their practice. Perhaps because many ofthese prospective teachers have enjoyed their experiences in problemsolving in the integrated methods course and learned to solve problemsthat connected science and mathematics in a meaningful way, theywill be able to overcome the constraints of traditional schooling andcreate classroom environments consistent with the calls for reform.

Future ResearchFurther research is needed to examine problem-solving

processes of prospective elementary teachers. First, research on thetransference of problem solving experience in mathematics to science

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and vice versa is needed to clarify whether and how students view thetwo areas as related. Secondly, more research that compares transferof problem-solving skills taught through an integrated curriculum totransfer of the same skills taught in a single content area would clarifywhether learning science and mathematics in an integrated fashionhas value in developing more universal problem solving abilities.Thirdly, the links between teachers' knowledge of problem-solvingand their ability to plan and implement problem-centered activitiesneeds to be clarified. Finally the effects of problem-centered instructionin methods courses must be examined by following teachers as theybegin their careers. We must understand how learning throughproblem-solving transfers to teaching children science and mathematicsif we are to promote lasting change.

ReferencesAmerican Association for the Advancement of Science.

(1989). Project 2061: Science for all Americans: Literacy goals inscience, mathematics, and technology. Washington, DC: Author.

Andre, T., & Phye, G. (1986). Cognition, learning, andeducation. In G. Phye & T. Andre (Eds.) Cognitive classroomlearning: Understanding thinking and problem solving (pp.I-19).Orlando: Academic Press.

Champagne, A. B., & Klopfer, L. E. (1981). Problem solvingas outcome and method in science teaching: insights from 60 yearsof experience. School Science and Mathematics. 81(1), 3-8.

Champagne, A. 8., & Klopfer, L. E. (1977). A sixty yearperspective on three issues in science education: I Whose ideas aredominant? II Representation of women III. Reflective thinkingand problem solving. Science Education, 61(4),431-452.

Cuban, L. (1982). Persistent instruction: Another look atconstancy in the classroom. Phi Delta Kappan Oct. 113-118.

Dewey, John. (1916). Democracy and education. In J. A.Boydston (Ed.) The middle works ofJohn Dewey 1899-1924, Vol9. Carbondale, IL.; Southern Illinois University.

84

Funkhouse, C. (1993). An examination of problem-solvingconceptualizations of inservice teachers. School Science andMathematics, 93(2), 81-85.

Gallagher, J. J., & Tobin, K. (1987). Teacher managementand student engagement in high school science. Science Education,71(4),535-555.

Gonzales, N. A. (1994). Problem posing: A neglectedcomponent in mathematics courses for elementary and middleschool teachers. School Science and Mathematics, 94(2), 78-84.

Good, T. L., Mulryan, c., McCaslin, M. (1992). Groupingfor instruction in mathematics: A call for programmatic research onsmall group processes. In D. A. Grouws (Ed.) Handbook ofresearch on mathematics teaching and learning (pp. 165-196),New York: Macmillan.

Harty, H., Kloosterman, P., & Matkin, 1. (1991). Scienceproblem solving approaches in elementary school classrooms.School science and mathematics, 91(1), 10-14.

The Holmes Group. (1986). Tomorrow's Teachers: A Reportof the Holmes Group. East Lansing, MI: Author.

Martins, M. L. (1992). Inhibitors to implementing a problem­solving approach to teaching elementary science; Case study of ateacher in change. School Science and Mathematics, 92(3), 150­156.

McKnight, B. J. (1989). Problem solving in elementaryschool science. In D. Gabel (Ed.) What research says to thescience teacher, 5. Washington D.C.: National Association forResearch in Science Teaching.

National Council of Teachers of Mathematics. (1989).Curriculum and evaluation standards for school mathematics.Reston, VA: Author.

National Research Council. (1989). Everybody counts: Areport to the nation on the future ofmathematics education.Washington, D.C.: National Academy Press.

R5

Pizzini, E., Shepardson, D., & Abell, S. (1989). A rationalefor the development of a problem solving model of instruction inscience education. Science Education, 73(5), 523-534.

Resnick, L. (1988). Treating mathematics as an ill-structureddiscipline. In R. Charles & E. Silver (Eds) The teaching andassessing ofmathematics problem solving: Multiple researchperspectives, (pp. 32-60) Reston, VA: NCTM.

Romberg, T. A, & Carpenter, T. P. (1986). Research onteaching and learning mathematics: Two disciplines of scientificinquiry. In M.e. Wittrock (Ed.) Handbook of research on teaching(3rd Ed.) New York: MacMillan.

Rutherford, J. F., & Ahlgren, A (1990). Science for allAmericans. New york: Oxford University.

Shoenfeld, A (1992) learning to think mathematically:problem solving, metacognition and sense making in mathematics.In D. A. Grouws (Ed.) Handbook of research on mathematicsteaching and learning (pp. 334-370), New York: Macmillan.

Shoenfeld, A. (1985). Mathematical problem solving. NewYork: Academic Press.

Tobin K., & Gallagher, J. J. (1987). What happens in highschool science classrooms? Journal ofCurriculum Studies, 19,549-560.

Van de Walle, J. A (1994). Elementary school mathematics;Teaching developmentally. New York: Longman.

Vygotsky, L. S. (1962). Thought and language. Cambridge,MA: MIT Press

Vygotsky, L. S. (1978). Mind in society: The development ofhigher psychological processes. Cambridge, MA: HarvardUniversity Press.

Woods, D. R. (1989). Problem solving in practice. In D.Gabel (Ed.) What research says to the science teacher (Vol 2)problem Solving (pp. 97-121). Washington D.e.: NSTA.

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Appendix A

Problematic Tasks:

Task I - The Fruit Problem. Each group was provided with a fruitpurchased at a local market and were told the purchase price of theitem. Also provided was a chart listing vitamins found in each fruit,and the quantity of vitamin per 100 grams of the fruit. The servingsize of the fruit was also given in the chart. They were challenged tofind the fruit that provided the most vitamin for the money expended.Materials available for student use included balances.

Task II - The Lever Problem. Students were provided with a meterstick, the balance bar from old SAPA balances, and several weightsfrom 1 g to 100 g. They were challenged to develop a "Law of theLever" that would allow them to decide the exact placement of asecond mass of given weight in order to achieve balance of a levergiven the weight and placement of a first mass. The "Law" had to besupported by experimental evidence. Students were requested not touse outside resources in obtaining a solution.

Task III - The Probability Problem. Students were provided withseveral sets of data from simple, single allele genetics crosses (i.e.Two parent peas produce offspring, 650 are tall and 206 are short,what are the gene combinations of the parents) and asked to use pokerchips of various colors to design a model that represented the crosses.Then using the model, collect data that supported their hypothesizedparentage and develop a written argument that the data did indeedsupport their hypothesis.

Carol Briscoe, Division of Teacher Education, University ofWest Florida Pensacola, FL 32514.

David Stout, Division of Teacher Education, University ofWest Florida Pensacola, FL 32514

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