Problem Posing: A Neglected Component in MathematicsCourses for Prospective Elementary and

Middle School Teachers

Nancy A. GonzalesDepartment of Mathematics and StatisticsUniversity of New Mexico

The purpose of this article is to describe a scheme "which is designed to guide the prospective teacherbeyond the point ofproficiency as a student solving given problems. By experiencing several stages ofinquiry, the scheme provides a mechanismwhereby the prospective teacher advances toward assumingthe role of a teacher posing his or her own problems. Although the scheme has been developedwithinthe context of a standard mathematics course for preservice teachers, the ideas set forth are equallyappropriate for incorporation within a science course for prospective teachers.

The National Council ofTeachers of Math-ematics has called for problem solving to be thecentral focus of the curriculum for school math-ematics (NCTM, 1989). However, the destiny ofany problem-solving efforts lies in the hands of theclassroom teacher; in fact, the success of anycurricular reforms in mathematics "ultimatelydepends on classroom teachers" (Pejouhy, 1990).Hence, problem solving will become the centralfocus only if the teacher recognizes its importanceand fosters a classroom environment which isconducive to exploration, inquiry, reasoning, andcommunication.

Innovation in problem solving does not justhappen. Teachers must possess sufficient contentknowledge to enable them to discover (or recog-nize) potentially rich mathematical situations. Thenit takes skill on the part of the teacher who mustcombine knowledge of content, learning, andteaching in order to transform the mathematicalsituations into problem-solving activities appropri-ate for classroom use.

A Call for Action

Herein lies the challenge for teacher prepara-tion programs in colleges and universities. While itis generally recognized by teacher educators thatprospective teachers require guidance in masteringthe ability to confront and solve problems, what isoften overlooked is the critical fact that, as teachers,they must be able to go beyond the role as problemsolvers. That is, in order to promote a classroomsituation where creative problem solving is thecentral focus, the practitioner must become skillful

in discovering and correctly posing problems thatneed solutions. This idea is not new. In theirwritings over the past twenty years. Brown andWalter have identified important aspects of problemposing in mathematics, and they have describedessential components for a problem-posing course(see, for example. Brown and Walter, 1983).

Based on this author’s experience withpreservice teacher education programs, manyprospective elementary and middle school teacherslack the skills and confidence necessary to gobeyond finding the solution to a given mathematicalproblem. This observation offers one possibleexplanation why the novice teacher might rely soheavily upon the textbook to provide the mathemat-ics curriculum. According to Romberg and Carpen-ter (1986), a review of research on teaching andlearning mathematics reveals the fact that thetextbook is seen as the "authority on knowledge andthe guide to learning.... Ownership of mathematicsrests with the textbook authors and not with theclassroom teacher" (pp. 867-868).

It is imperative that teachers�not the text-book�take control of classroom learning andownership of mathematical knowledge. This isespecially true if problem solving is to become anintegral pan of the mathematics curriculum. Inteacher training programs we must de-emphasizethe authority of the textbook and enhance theprospective teachers’ content knowledge andproblem-posing skills so that they will have confi-dence in determining direction for creative problemsolving.

In his search for understanding how knowledgegrows in teaching. Lee Shulman (1986) found that

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to be a teacher requires a special kind of knowledgethat is neither content nor pedagogy, per se. Hespeaks of pedagogical content knowledge, whichgoes beyond knowledge of the subject matter "tothe dimension of subject matter knowledge/orteaching" (p. 9). Hence, teacher understanding thatcombines content, pedagogy, and learner character-istics guides the teacher in making sense of what isgoing on in the classroom and, subsequently, informulating questions that engage students’ think-ing.

The Development of a Sound Approach

Brown and Walter (1988) recommended thatproblem-posing types of strategies be incorporatedwithin the context of standard mathematics coursesrather than reserved for a previous course here andthere: "We look forward to the time that a specialcourse on problem posing would be redundant" (p.131). Accepting this recommendation as a chal-lenge, during the course of several semesters thisauthor grappled with the notion of developingproblem posing as a specific component within astandard mathematics course for prospectiveelementary and middle school teachers. Such acourse provided the perfect setting and opportunityto begin to formulate a response to the followingquestion: How can college mathematics coursesassist in taking prospective teachers (in this case,elementary and middle school teachers) beyond thepoint of proficiency in solving given problems andprovide a mechanism whereby they will gain asense of the nature and importance of problemposing?

Assuming the responsibility as teacher-re-searcher (Duckworth, 1986), the author beganexperimenting with a variety of ways in which toguide preservice teachers beyond the familiar roleof "students solving mathematics problems" with aview to acquiring a new role as "teachers posingmathematics problems." The pedagogical task wasone of enabling the prospective teachers to knowand to appreciate the different roles being played asproblem solver and problem poser and to acquireflexibility in moving from one mode to the other.

The remainder of this paper provides a descrip-tion of the scheme that has proved quite successfuland reasonable under the conditions imposed by thestructure of a well-established mathematics course.Even though the ideas have been developed within

the context of a mathematics course, the scheme isequally appropriate for use within a science coursefor teachers.

Step One: Learning to Monitor the ThoughtProcess

Early in the semester, the students are intro-duced to Polya’s four-step method for solvingproblems (Polya, 1973). Using specific verbalproblems for discussion, the students explore theintent of each step in the process. During theseinitial stages, the students also gain experience inthe identification and use of different strategies forsolving verbal problems (e.g., drawing a diagram,making a table, writing an equation, solving asimpler problem, etc.).

Having attained a basic understanding of theprocess, the students are then required to keep anotebook containing details leading to the solutionof a given set of verbal problems�includingquestions posed as a result of clarifying the task of agiven problem. They are instructed to solve theseproblems by traversing each ofPolya’s four steps inthe problem-solving process. They are required towrite (in their own words) a description of each stepas follows:

1. Understanding the ProblemTake note of all thoughts and questions which

cross your mind as you attempt to come to gripswith the problem. Ask yourself questions such as:"What is the problem all about?" "What am Igiven?" "What do I need to find?"

2. Devising a PlanDescribe the strategy or strategies which you

will use. Describe the plan for the chosen strategyor strategies.

3. Carrying out the PlanPerform the necessary computations (or

drawings, tables, etc.) and describe the steps thatyou take.

4. Looking BackCheck the results. If the results do not make

sense, then begin the process all over again. Checkto see if there might be other solutions or otherstrategies which will yield the same solution(s).

The students are required to carry out a self-reflecting inquiry and written discussion. Theymust indicate all thoughts, questions, attempts(correct or incorrect), frustrations, interpretations,and any parameters or restrictions they may haveplaced on a problem.

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This notebook writing exercise provides thefirst opportunity for the students to monitor theirown thoughts and to generate scrutinizing ques-tions�that is, in the spirit of the "managerial"strategy as described by Alan Schoenfeld (1982).Within the context of solving a given set of prob-lems, probing questions are posed such as: Whatam I trying to find? Is all the given data relevant tothe solution? Do any assumptions have to bemade? Arc there different ways of interpreting thegiven information, stated condition(s), or task?Further, how do any different interpretationsinfluence the analysis? As the questions are posedand resolved (if possible), students reach a deeperunderstanding of each of the given problems.

Following is an excerpt taken from onestudent’s notebook. It serves to illustrate how thestudent monitors her thoughts as she strives forunderstanding and searches for a meaningfulsolution.

Example of a textbook problem. Jim has savedsome silver dollars. He wants to divide themamong Tom, Dick, Mary, and Sue so that Tom gets1/2 of the total amount, Dick gets 1/4, Mary gets1/5, and Sue gets 9 of the dollars. How manydollars has Jim saved? (Billstein, Libeskind, & Lott,1990,p. 840)

Excerptfrom Student’s WorkA. Understanding the ProblemJim has some money and he wants to give it

away�what a nut! The problem is to find out howmuch money he has. That is the unknown�the bigone. He tells us only how many coins Sue will getand what fraction of his coins the others will get.The number of coins the others will get are smallerunknowns. I don’t see any excess information�other than a few words.

Figure 1. Circle graph drawn by the student as anaid to devising a plan.

B. Devising a PlanI think I’d draw a picture. Sue gets nine coins,

but what fraction of the whole is that?Then, I’d try to write an equation to get Sue’s

fraction of the whole amount.1/2 + 1/4 + 1/5 + x = 1 The whole amount saved.

10/20 + 5/20 + 4/20 + x = 119/20+;c=l

x= 1/20Now, I know something. I know how many

coins Sue gets and what fraction of Jim’s money the9 coins are.

I don’t think this is "carrying out the plan." Ithink it is part of the strategy!

Now, I have something "real" to work with.1/20 of Jim’s money = 9or (1/20) x == 9

Maybe this should be called a sub-plan?I just realized that this simple equation can get

to the answer. I -was thinking I’d needed to use thecoins that Tom, Dick and Mary were getting. Idon’t�because ifjcis the amount of Jim’s coinsand I know what fraction of the coins (1/20) x = 9,then, I can solve for x.

C. Carrying out the Plan(1/20) x= $9

(20)(1/20) ;c= $9(20)^=$180

D. Looking BackNow, I’d like to see how much the others get to

see if I have the right answer. Set up equation:(1/2)180 + (1/4)180 + (1/5)180 + (1/20)180 = 180Solve: 90+45+36+9=180

180=180I think that since Sue must get $9, and to fit

into the fraction of the total amount, there can beonly one solution.

Author’s ReflectionsIt is particularly interesting to note how the

student justifies carrying out the solution for Sue’sfraction of the whole amount during the devising-a-plan phase. She devises and carries out a "sub-plan," which yields an equation for solving for "thebig one" (that is, the amount of money that Jim hassaved). It is evident that the student was beingquite honest with detailing her thoughts as theyoccurred. Otherwise, she could have summarizedher plan as: draw a picture; write an equation tofind out what fraction of the whole amount Suegets; use the preceding solution and the knowledge

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that Sue has 9 coins to write an equation to find outhow much money Jim has saved.

However, it appears that this account wouldhave been retrospective rather than metacognitive.According to Hunkins (1989), when a learnerfunctions at a metacognitive level, he or she main-tains in consciousness the purpose, the procedures,and the manner of engaging in the meaning-makingprocess. "Such cognizance allows the individual tomodify the approach when required by the results orthe unfolding situation" (p. 9). The student’s workshows evidence of modifying her approach whenshe realizes that her "sub-plan" has yielded a simpleequation which will lead to the solution of theproblem.

Step Two: Posing Related Problems

Following the students’ experience with theproblem-solving process, the notion of a relatedproblem is introduced. For our purposes, a relatedproblem is defined to be any modification (variationor extension) of the given problem statement. For auseful discussion of problem variations�completewith examples�see Charles and Lester (1982, pp.51-56).

To ease the students slowly into taking on therole ofproblem poser, this author has found thatextending Polya’s four-step method to a fifth step�"posing a related problem"�can be an effectivenext step. The students are asked to generatevariations of the problems which they have alreadysolved during the aforementioned notebook assign-ment. They are instructed to add a fifth step to eachof the previously solved problems:

5. Posing a Related ProblemUse the given problem and modify it to obtain

a variation of the given problem.

Related Problems Produced by StudentsUsing the aforementioned Example of a

Textbook Problem, following is a student-generatedvariation of the problem statement. Note that thestudent poses a related problem by changing thevalues of the given data and by changing thecontext of the original problem.

Related problem #1. Lisa is trying to work outher monthly budget. She knows that her monthlyincome is spent in the following way: 1/2 pays forthe rent on her apartment, 1/6 goes to utilities, 1/5pays for food, and she has $80 left for miscella-neous expenses. What is Lisa’s monthly income?

Following is another illustration of a student-generated variation of the given textbook problem.

Related problem Wl. Jesse has 36 dollars in hissavings account. He withdraws money from hisaccount in order to lend money to three of hisfriends. He lends Caria 1/3 of his savings, and helends Phil 1/6 of his savings. After he lends Anna acertain amount of money, he has 1/12 of the origi-nal savings left in his account. How much did helend Anna?

In this case, the original problem statement wasmodified by: (a) changing the values of the givendata, (b) reversing given and wanted informa-tion�since now the total amount is known andAnna’s fraction of the total amount is unknown,(c) changing the names of the persons involved,and (d) changing the money-giving situation to amoney-lending situation.

The list of techniques for posing variations of agiven problem is unlimited, as demonstrated in theexample that follows.

Related Problem #3. Each week Amandareceives an allowance. This past week, the onlymoney she had to spend came from her weeklyallowance, but she ran short of cash by $3.00.Amanda spent her allowance in the followingmanner: 1/10 of her allowance went on bus fare,1/2 went toward the purchase of the latest "NewKids on the Block" tape, and 1/4 was spent on chipsand cokes at lunch. During the weekend (after shehad spent her allowance on the items listed above)Amanda wanted to go to the movies, but shecouldn’t because the movies cost $6.00. How muchdoes Amanda receive for an allowance?

The student used the following techniques inwriting anew, related problem: (a) change thevalues of the given data, (b) change the context,and (c) change the number of conditions�there isan extra condition that Amanda ran short of cash by$3.00 (to pay for the movies).

Step Three: Creating Verbal Problems

After gaining some experience in posingproblems based on a given textbook problem, theparameters of the role of problem poser arestretched in a more creative direction by asking thestudents to discover and pose novel problemsappropriate for grades K-8. In order to assist themin developing their problems, a set of "Guidelinesfor Assessment of the Discovered Problem"evolved over several semesters (see Table 1), and

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seemed to be a useful tool for both the student (asthe problem writer) and the instructor (as theproblem evaluator).

Following is one student’s discovered problem,which offers a good example of an open-ended realworld application problem for the upper elementaryor middle school grades.

Example of a discovered problem. Mrs.Garcia’s class is trying to earn enough money tobuy a new computer and printer for their classroom.The students have decided to recycle aluminum andglass to earn the money to buy this equipment.How much aluminum and glass will the studentsneed to collect in order to have enough money tobuy the computer and the printer?

This is an excellent laboratory activity in whichstudents must investigate the current: (a) recyclingprices for glass and aluminum, and (b) prices on anew computer and printer. Based on these values,they must make decisions on the composition of the

quantities to be collected. That is, how manypounds of glass and how many pounds of aluminumshould they set as a goal in order to reach the sumof money required to buy the computer and printer?This is a wonderful activity which can placeresponsibility for learning and ownership of knowl-edge within the hands of the young investigators.

Step Four: Developing a Lesson Plan

The final step in the transition from "problemsolver" to "problem poser" is to give the preserviceteachers an opportunity to assume the role of"teacher" by requiring them to prepare a lesson planspecifically designed to facilitate the solution totheir own discovered problem. The structure of thelesson plan is based on the five-step extendedproblem-solving process: Polya’s four steps, plus"posing a related problem" (see Table 2).

Table 1.

Guidelines for Assessment of the Discovered Problem

Characteristic Rating8

1. Problem is clearly stated and concise.2. Vocabulary used is appropriate for the suggested student population.3. Mathematical level is appropriate for the students.4. Problem is realistic or practical.5. Problem is creative1’.6. Context of the problem is interesting.7. Problem lends itself to more than one method of solution.8. Mathematical task prompts further exploratory questions0.9. Mathematical situation provides an opportunity for the student to conjecture,

investigate, and analyze.10. Problem facilitates an understanding of some basic mathematical concept.11. Problem stimulates the use of mathematical skills.12. Solution involves strategy, rather than a mere answer.13. Problem has the potential to develop mathematical reasoning.14. Problem solution provides an opportunity for students to label, verbalize, and

define concepts.15. Solution process promotes the use of models, diagrams, and symbols to represent

concepts.16. Solution process provides experiences in translating from one mode of representation

to another.

a! = poor, 2 = mediocre, 3 = good, and 4 = excellent.assessment is based on the knowledge level of the students.Tor example, "What if...?" types of questions.

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It can be seen that extensive writing is requiredof the students as they imagine themselves facilitat-ing a lesson with a classroom of elementary ormiddle school students. They anticipate questionsthat might be posed by the students, and they recordpossible strategies that may be offered by thestudents.

The preservice teachers are then given anopportunity to utilize their lesson plans in a mockclassroom situation, where their colleagues play therole of elementary or middle school students.Observations made during the mock classroomsessions appear to indicate that the preserviceteacher gains: (a) a perspective on the importantrole that language (choice of words) plays in theunderstanding and interpretation of a word problem;(b) knowledge of mathematical levels appropriatefor different grades (K-8) and types of students(remedial to accelerated); and (c) insight into therole of a teacher as a facilitator of knowledge ratherthan as a deliverer of knowledge.

Further, the mock classroom presentationsserve to reveal the tentative nature of a lesson planwhich is designed to promote inquiry as a means offacilitating the solution to a problem solvingsituation. That is, teachers can try to anticipate asmany questions and strategies as possible, but thestudents may pose unanticipated questions and maysuggest a strategy which the teacher did not eventhink of! This is what problem solving and goodproblem posing are all about.

Summary

The work discussed in this paper was con-ducted at the University ofNew Mexico withstudents enrolled in a mathematics course forelementary and middle school teachers. Thescheme is designed to guide the prospective teach-ers through several stages of inquiry: (a) posingquestions�possibly generating new problems�asa result of clarifying the task of a given problem,(b) posing problems which are variations of a givenproblem, (c) discovering and posing novel prob-lems appropriate for grades K-8, and (d) assumingthe role of "teacher" by preparing a lesson planspecifically designed to facilitate the solution to adiscovered problem. As noted earlier, this scheme isequally functional within a course for prospectivescience teachers.

The proposed scheme has evolved over aperiod of about four years. Modifications havebeen made based on sources of data consisting ofstudents’ detailed notebooks (including descriptionsof thoughts, interpretations, questions, and calcula-tions produced in response to each stage as outlinedin the scheme) and the author’s classroom observa-tion notes. An assessment of the students* workappears to indicate that preservice teachers can beguided through a transition from "problem solver"to "problem poser." In addition, flexibility inmovement between the roles can be attained.

One note of warning: the scheme might beapproached by some instructors in an overly

Table 2.

The Lesson Plan Format

StageKeyQuestion

Understanding the problem

Devising a plan

Carrying out the plan

Looking back

Posing a related problem

How will you guide the students into an understanding of the problem?

What strategies might be suggested by the students?

How will you assist the students in carrying out the plan which theyhave devised?

How can you guide the students into recognizing another possible planor another possible solution (if such exists)?

What possible variations for the problem might be suggested by you orthe students?

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Problem Posing

mechanistic manner. There is a danger to outlininga process such as this because it gives the appear-ance of fixed or routine stages. The intent of theproposed scheme is not to mechanize the steps but,rather, to present a suggested guide through severalpossible stages of inquiry. In the spirit of trueproblem solving, flexibility and the stimulation ofcreative questioning must always be present.

hi conclusion, a quotation from an articlewritten by Walter and Brown (1977) is appropriate:"For a long time there has been general interestamong researchers, teachers, and curriculum writersin the area of problem solving in the mathematicscurriculum. The other side of that coin�problemposing�has been a neglected subject, however" (p.4). It would seem that by now, during the presentdecade, we would have already given seriousconsideration to this neglected component.

References

Billstein, R., Libeskind, S., & Lott, J. W. (1990). Aproblem solving approach to mathematics/orelementary school teachers (4th ed.). Red-wood City, CA: Benjamin/Cummings.

Brown, S. I., & Walter. M. I. (1983). The art ofproblem posing. Philadelphia, PA: TheFranklin Press.

Brown. S. I., & Walter, M. I. (1988). Problemposing in mathematics education. QuestioningExchange, 2(2). 123-131.

Charles, R., & Lester, R. (1982). Teaching problemsolving: What, why, & how. Palo Alto, CA:Dale Seymour.

Duckworth, E. (1986, November). Teaching asresearch. Harvard Educational Review, 56(4),481-495.

Hunkins. F. P. (1989). Teaching thinking througheffective questioning. Boston: Christopher-Gordon Publishers.

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

Pejouhy, N. H. (1990, September). Teaching mathfor the 21st century. Phi Delta Kappan, 72(1),76-78.

Polya, G. (1973). How to solve it: A new aspect ofmathematical method (2nd ed.). Princeton, NJ:Princeton University Press.

Romberg, T. A., & Carpenter, T. P. (1986). Re-search on teaching and learning mathematics:Two disciplines of scientific inquiry. In M. C.Wittrock (Ed.), Handbook of research onteaching (3rd ed.). New York: MacmillanPublishing.

Schoenfeld, A. H. (1982). Measures of problem-solving performance and of problem-solvinginstruction. Journal of Research in Mathemat-ics Education, 13, 31-49.

Shulman, L. S. (1986). Those who understand:Knowledge growth in teaching. EducationalResearcher, 15(2), 4-14.

Walter, M. I., & Brown, S. I. (1977). Problemposing and problem solving: An illustration oftheir interdependence. Mathematics Teacher,70(1), 4-13.

Note: The author’s address is: Nancy Gonzales,Department of Mathematics and Statistics, University ofNew Mexico, Albuquerque. New Mexico 87131.

Errata

For more information on USA TODAY’S educational programs or to receive a cata-

log, call 1-800-USA-OOOL This information was inadvertently omitted from the invitedcomment in January’s SSM entitled "USA TODAY: Connecting Science and MathematicsPrinciples to the Daily News."

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