Mathematics Educatio11 Research Journal 2001/ Vol. 13/ No. 2/74-90

The Role of Fantasy Contexts in Word ProblemsLynda Wiest'

University of Nevada/ Reno

This paper reports on the efficacy of different contexts for word problems given toGrade Four and Grade Six students. In particular, the use of fantasy contexts wereexamined and compared with both adult and children/s real-world contexts. Thestudy found that students expressed an interest in the fantasy contexts, and solvedproblems using these contexts as well as or better than real-world problems.

Mathematical word problems are common school devices intended to helpstudents apply classroom learning to the "real world". Few would disagree that'~how children transfer knowledge between school and the outside world may bethe central problem in education" (Baranes/ Perry, & Stigler, 1989/ p. 287). Standardword problems/ however/ have not lived up to their promise. They often do notspark important mathematical thinking. They may in fact inhibit it, leadingReusser (1988) to this unfortunate finding: "The major result observed in most ofour studies is the extent to which textbook problem solving contexts can impair thequality of comprehension" (p. 334).

Word problems are likely to endure as school mathematics tasks/ and theircontexts-verbal aspects of the problem statement, such as the story line-havemuch to do with the problem-solving experience. Therefore, it is worthwhile toseek to improve word-problem contexts in ways that maximise their potential forfostering thinking, .which is "at the heart of mathematical problem solving"(Willoughby, 1983, p.58). Further/ because various student sUbgroups reactdifferentially to problems based on their context, studying and improving word­problem context also relates to equity issues in mathematics education.

Fantasy contexts interest many children, although whether this leads to betterproblem solving remains an open question. It can be hypothesised, however/ thatfantasy contexts lend themselves to abstract and creative thinking, importantabilities in any field as well as in daily life. The topic of the research reported hereis how solving fantasy~omparedwith real-world-word problems influences (a)children's preferences for solving various problems and (b) children's problem­solving performance.

Related Literature

Impact of Word-Problem Context on Problem SolvingWord problems are "notoriously difficult to solve" (Cummins, Kintsch,

Reusser, & Weimer, 1988/ p. 405)/ particularly nonroutine and multistep types(Lopez & Sullivan, 1991; Murphy & Ross, 1990). One aspect of word problemssome researchers have found to impact student performance is problem context.Word-problem context is defined as the "nonmathematical meanings present in theproblem statement" (Kulm, 1984/ p. 17t for example, the "story" in which themathematics problem is set. Problem context may help problem solvers give

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meaning to the mathematical content in a problem. It is likely to influence, inparticular, the problem-solving stages of understanding a problem and planningits solution (Boaler, 1993a; Kulm, 1984). Bickmore-Brand (1993) says context isfoundational to mathematical activity: "Context is paramount to the c'onstructionof meaning the whole way through. It is the backdrop against which the parts haveto make sense" (p. 3). '

Problem context investigated, in this study was the type in which themathematical aspects of the problem (e.g., mathematical relationships and numbersize) remained constant while the story topics varied, specifically, within thefantasy and real-world genres. Most studies reviewed here took place at the Grade4 to Grade 8 levels. This may be a particularly important point in time formaintaining student interest in mathematics. The degree of interest, and thusmotivation, problem context generates in problem solvers is one way problemcontext influences problem-solving performance. It may inspire problem solvers toengage more fully in a problem and to persevere longer in solution attempts (d.Boaler, 1993b, 1994; Murphy & Ross, 1990). Therefore, affective variables, besidessuccess in achieving correct solution methods and answers, are important toconsider.

Hembree (1992) conducted a meta~analysis of 44 studies, involving Grade 4 toundergraduate students, in which word-problem context differed while themathematical structure remained constant. Better performance was most stronglyassociated with familiar contexts. Concrete (vs. abstract) and imaginative (vs.ordinary) problems, the latter using fantasy or unusual circumstances, showedborderline significance in their positive impact on problem-solving performance.

.Numerous studies have investigated the impact of personalised problems­those using students' names or other personal information-on student interestand problem-solving success. All such studies found positive effects on themeasured variable of interest or achievement (e.g., Davis-Dorsey, Ross, &Morrison, 1991; Hart, 1996; Kloosterman, 1992; L6pez & Sullivan, 1991, 1992).L6pez and Sullivan (1992) found significant differences favouring personalisationon problem-solving scores for two-step but not for one-step problems, althoughstudents also scored higher on the latter, in comparison with nonpersonalisedproblems. They say personalisation may be particularly important for moredemanding (e.g., unfamiliar or mathematically complex) cognitive tasks.Personalised word problems may be more motivationat meaningfut concrete, andfamiliar (Davis-Dorsey et al., 1991; L6pez & Sullivan, 1992). Grouppersonalisation-using dominant interests of a group of students-has also beenshown to benefit students' problem-solving scores compared withnonpersonalisation (L6pez & Sullivan, 1992), but individual personalisation ismore effective in impacting students' attitudes and preferences (L6pez & Sullivan,1992; Murphy and Ross, 1990).

Some researchers have used various topics, such as animals, sports, or fashion,for word problems with the idea that they bring about differential problem-solvingresponse based on student interest and background knowledge. Boaler (1994)found gender-related differences in achievement that appeared to relate toproblem-context topic. Renninger (1992) concluded from her work with fifth- andsixth-grade students that individual interests and noninterests embedded in

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contexts influence the way students engage in and complete tasks in terms of, forexample, entry into and perseverance on a task. However, this was much less thecase with better problem solvers.

Boaler (1994) says the situation-specificity of word-problem contexts tends to 'influence students in a way that clouds the deeper mathematical connections'among problems. She attributes this response, in part, to the types ofcontexts usedin mathematics classrooms, which tend to be superficial in meaning and to cuestudents to look for and apply a "correct" procedure. She suggests, "If students canlearn mathematics in such a way that enables them 'to see the underlyingsimilarities between questions set in different contexts they will probably alsodevelop enhanced capability ~n transferring their school learned mathematics to'real world' situations" (1993a, p. 342). Boaler's (1993a) study involving 12- and 13­year-olds in two schools with different instructional orientations provides supportfor her contentions. Students in the more process-oriented school tended to usesimilar procedures across mathematically siinilar word problems with differentcontexts. Students with more traditional mathematics instruction were, moreresponsive to context, using varied procedures with problems of similarmathematical structure but differing contexts.

Word-problem context can have, differential impact on some groups ofstudents in terms of interest and achievement. Sex-stereotyped contexts favouringmales, such as science themes, have been proposed as one possible factor in femaleunderachievement in solving word problems (Jones & Smart, 1995; Murphy &Ross, 1990). However, sex-stereotyped contexts favouring females have also beenassociated with harming females' achievement by engaging them more fully infamiliar or interesting contexts (see Boaler, 1993a, 1994). Sex-stereotyped contextsmay be a source of test bias to consider in designing assessment tasks (Boaler,1993a, 1994; Chipman, Marshall, & Scott, 1991). Word-problem context may alsohave differential impact upon student subgroups formed by age (Davis-Dorsey etal., 1991; Parker & Lepper, 1992), ability level (Marshall, 1995; Renninger, 1992),and community type in which students reside (Murphy & Ross, 1990).

Use ofFantasy Contexts in School MathematicsMany children like fantasy themes and story elements (Gillespie & Naden,

1994; Huck, Hepler, & Hickman, 1993). Nevertheless, the fantasy genre rarelyappears in contemporary mathematics textbooks. To find out why, I surveyedindividuals associated with textbook authoring and publishing, mainly authors ofmathematics textbooks for the elementary and middle grades. Eighteen people-56percent of the 32 contacted-responded to some or all of my questions. All of the15 respondents who commented on genres currently used for word-problemcontexts emphasised use of realistic themes and real-world data. One textbookauthor said, "The themes that are avoided are those that are 'unrealistic.' It is theage of rea~ism: no humour, no fantasy, no fiction."

Survey respondents offered two main reasons for the trend away from fantasyand toward real-world problem contexts. Foremost, they assigned the Curriculumand Evaluation Standards for School Mathematics (NCTM, 1989) a key role in theincreased emphasis on realism. This first version of the mathematics standards

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continually mentions "real-world" mathematics, although it does not define theterm clearly. The second reason given was parental objection, which tends to bereligious in nature and attacks printed verbiage across the school curriculum. Ingeneral, fundamentalist religious groups contend that opening the mind toimagination may allow all sorts of ideas to rush in, creating a potential forquestioning God and being distracted from His Word (DeIFattore,1992).Publishers tend to avoid controversial topics (Webb, 1995).

The fantasy genre has been associated with many proposed benefits, includingintrinsic motivation (Lepper & Hodell, 1989), mental and emotional health(Cullinan & GaIda, 1994; Tunnell, 1994), and moral, social, and intellectualdevelopment (Cullinan & GaIda, 1994; Stewig, 1988). One vital consideration forincluding fantasy in people's lives is its probable impact upon developingcreativity, imagination, and problem-solving abilities (Cullinan & GaIda, 1994;Huck et aI., 1993; Tunnell, 1994). "Fantasy h~lps to develop imagination. Theability to imagine, to conceive of alternative ways of life, to entertain new ideas,. tocreate strange new worlds, to dream dreams are all skills vital to the survival ofhumankind" (Huck et aI., 1993, pp.394-395). Many authors have noted theimportance of fantasy and imagination to invention and problem solving inmathematics and science (Prawat, 1993; Shepard, 1988; Tunnell, 1994). Forexample, Einstein imagined himself travelling alongside a beam of light in his earlythinking, which later led to the theory of relativity; the chemist Kekule's discoveryof the benzene ring stemmed from his dream image of a snake seizing its own tail(Shepard, 1988). These types of incidents may have given rise to Einstein'sfrequently quoted statement, "Imagination is more important than knowledge" (as,cited in Farber, 1985).

Research Methods

SampleStudents in twelve classrooms-six at each of Grade 4 and 6-in a US

midwestern state participated in this research. Six classrooms were located in theonly elementary school in a small town and six were in three (of thirteen)elementary schools in a small city about 50 miles away. The small town is a lower­middle-class area, and the city schools draw from lower- to upper-middle-classareas. The total number of students in the twelve classrooms was 308; however, 35

. did not participate in the study or were excluded from data analysis for variousreasons, mostly due to having very low mathematics or reading ability. Therefore,the study included 273 participants-127 Grade 4 and 146 Grade 6 students,including 138 females and 135 males.

Design and ProceduresWord-problem structure. Word problems were constructed to be parallel in

mathematical structure but different in story content, or context. Mathematicalstructure means the nature of the mathematical relationships expressed in aproblem, or situation. The mathematical structure of the problems in this study is

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one in which smaller groups (subsets) combine to form a larger group (superset).This type of situation has been variously termed group (Marshall, 1995)/ part-part­whole (Carpenter & Moser, 1982)/ and combine/subset-unknown (Riley, Greeno, &Heller, 1983). The problems were two-step translation problems that containedextraneous information. Each had three subsets, one of which was unknown, and asuperset. In abstract form, the problems have the following general structure,where italicised terms represent numerals:

There are a (superset name) and b (extraneous set name). C are (subset name 1)/ dare (subset name 2), and unknown quantity are (subset name 3). How many(superset name) are (subs~t name 3)?

Context categories. The two literary genres used for story content-fantasy andreal-world-were subdivided into two categories each. The resulting contextcategories were low fantasy, high fantasy, children's real-world, and adults' real­world.

For this research, fantasy was defined as "any departure from consensusreality" (Hume, 1984~ p. 21)/ as one might expect the vast majority of people acrosscultures to define such a departure. "It contains some element not found in thenatural world" (Cullinan & Galda, 1994/ p. 199). Low fantasy contexts involve anotherwise normal world that contains an unexplained, nonrational aspect or event.Sample problem:

One day in December, 131 people shopping in a toy store were stunned when 157toys on the shelves suddenly came to life. 49 of the toys danced around the store,46 of them chatted with each other, and the others sang the song "Toyland". Howmany toys in the store sang when they cameto life?

High fantasy contexts involve an imaginary, but internally· consistent, secondaryworld that often includes mythical or unreal characters or creatures. Sampleproblem:

The Secret Forest has 159 redwood trees and is the home of 134 animals that like tokeep to themselves. Of the animals that live there, 37 are unicorns, 54 are fire­breathing dragons, and the others are horses with wings. How many animalsliving in The Secret Forest are horses with wings?

Real-world contexts are those that are "everyday", as in life as it is presumedto be experienced, or potentially experienced, by most people. Children's real-worldcontexts involve everyday situations presumed to be experienced or potentiallyexperienced by most Grade 4 and Grade 6 students. Sample problem:

139 children rushed to their favourite ride at the yearly carnival held by the 137business owners in their town. 34 children hurried to the ferris wheel, 38 went tothe merry-go-round, and the rest chose the roller coaster. How many childrenchose the roller coaster as their favourite ride?

Adults' real-world contexts involve everyday situations presumed to be experiencedor potentially experienced by most adults. The theme or topic of the actual storyline is presumed to be one that is-for the most part-of greater concern, interest,familiarity, or relevance to adults. Sample problem:

;t The Role of Fantasy Contexts in Word Problems 79

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1,.,

lt

159 people visited the Westfield Art Show, held one weekend in August, to see the136 works of art for sale. 58 works of art were sold on Saturday, 33 on Sunday, andthe others were not sold, so they were stored for another art show. How manyworks of art were not sold at the Westfield Art Show?

Word-problem construction. Most story topics and elements used for the fantasyand children's real-world problems were drawn from literature on children'sinterests. Efforts were made to choose story content of relatively equal interest togirls and boys. Topics for the adults' real-world contexts were gleaned from Grade4 and Grade 6 mathematics textbooks. These topics were chosen because they weredeemed to be of greater interest or relevance to adults than to children.

Numerous aspects of the problems were controlled to create mathematicallycomparable word problems. The first sentence of each problem contains thesuperset number and an extraneous number, each of which has a different verballabel (e.g., "toys" and "people"). The second sentence begins with the two knownsubsets and ends" with the unknown subset. The third-and shortest-sentenceposes a question soliciting the quantity of the unknown subset, and it includes theverbal label that should accompany the numerical answer. The number of words inthe problems ranges from 50 to 60. Grade-level-specific terms, such as "fourthgraders", were adapted to match the students working with the problems.

Problem readability was addressed in several ways. Feedback from five Grade4 and Grade 6 teachers who read the problems, as well as information from a pilotstudy, was used to adjust wording and change vocabulary in some problems.Pronunciation and meaning of selected vocabulary were discussed with studentsbefore they completed forms, and reading and comprehension help was providedduring task completion.

Study phases. After a pilot study to refine materials and procedures, three studyphases-followed by individual interviews-were conducted near the end of atraditional school year. Students completed the first three phases individually in awhole-class setting, each phase occurring one week after the previous. Studentsparticipating in interviews did so privately with the researcher. In Phase 1 studentsused a five-item, Likert-type scale ("really dislike", "dislike", "okay", "like","really like") to indicate their preferences for nonquantitative "story ideas" thatformed the basis for word problems solved in Phase 3. For example, the children'sreal-world problem shown earlier appeared in this form:

Many children rushed to their favourite ride at the yearly carnival held by thebusiness owners in their town. Some children hurried to the ferris wheel, somewent to the merry-go-round, and the rest chose the roller coaster.

Students used a similar scale in Phase 2 ("very bad", "bad", "okay", "good","very good") to rate the research problems in terms of their preferences for solvingthe problems, without actually solving them. In these first two phases a commentspace was available after each rated item for student? to provide reasons for theirratings. Most students completed the comment sections. In Phase 3, students wereasked to solve five problems. Four problems were research problems of interest,one in each of the four context categories. The fifth, located third in the problemsequence, appeared on the surface to be similar to the others but called for adifferent solution method (one-step subtraction).

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Each student received 12 of the 24 story ideas in Phase 1 and 8 of the 24problems in Phase 2 to rate so that she or he would work with a manageablenumber of items. To minimise the effects of interest in individual problems, onlythe three most highly rated story ideas in each of the four categories in Phase 1-bygrade level-were used for problems solved in Phase 3. Therefore, Phase 3included 12 of the original 24 problems, with each student solving four. For eachphase, each student received the same number of items in each of the fourcategories: three in each for Phase 1, two in each for Phase 2, and one in each forPhase 3. In ,each phase, several forms that included different story ideas orproblems were constructed so that a portion of the sample addressed all research·items. Because the order in which a problem appears within a problem set has beenshown to affect problem-solving performance (Zollman, 1986/1987), problemorder varied across forms. For each phase, the same number of each version of theform was randomly distributed within each classroom.

Finally, 31 students who had participated in the first three study phases wereinterviewed individually in a private setting. Sessions were audiotaped.Participants consisted of 17 girls and 14 boys. These included 15 Grade 4 studentsand 16 Grade 6 students who volunteered from among those who had been invitedbecause of their varied responses-across students-on earlier forms. Interviewedstudents solved and rated problems as in -Phase 3, explaining their thinkingthroughout this process. They also answered several questions concerning the

"influence of problem context. For example, they reported the types of wordproblems they liked to solve and-if not offered-the types of story topics theyprefer. They also discussed their probable reaction to solving two word problemswith the same length and mathematics requirements but different stories, one likedand one disliked.

Data AnalysisRatings for items in Phases 1 and 2 were assigned a score from 1 to 5, 1 being

the most negative of the five response choices ("really dislike" or "very bad") and 5being the most positive ("really like" or "very good"). Phase 3 problem solutionswere scored 1 if the solution plan was appropriate and 0 if it was inappropriate."Appropriate" meant the plan would lead to a correct answer if implementedcorrectly. I scored all problems, and to establish intercoder reliability, amathematics education doctoral student scored one classroom set of problems:four problems for each of 28 students, a total of 112 problems. Our scores matchedfor 107 problems. After discussing the remaining five items, we agreed on thescores assigned to all 112 problems.

Each participant's ratings within a context category were added to form onescore. This "Created four scores for each participant-one in each category-for eachof the first two phases. For example, in Phase 1 each participant rated three storyideas within each of the four categories. Because the score for each rating rangedfrom 1 to 5, a participant's combined rating for the three low fantasy items rangedfrom 3 to 15, and the same was true for each of the other three categories. Similarly,in Phase 2 participants' individual category scores ranged from 2 to 10 becausestudents rated two problems in each category. Scores for participants who did not

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complete all items within a category were excluded from analysis. In Phase 3 it wasnot necessary to combine scores, because participants only solved one problem ineach category. Data for participants from both grade levels were combined inPhase 1 and 2 analyses. Phase 3 data are reported both separately and together bygrade level, because greater variability appeared in actually solving the problemsdue to the effects of ability level.

Means and standarddeviations were computed by context category for each ofthe three study phases. Pearson correlations were run between Phase 1 preferenceratings and Phase 3 problem-solving scores. Phase 3 context categories werecompared for significant differences among categories using an ANOVA withrepeated measures-for each grade level separately and for both grade levelscombined.

Results

Table 1 shows Phase 1 mean scores and standard deviations for story-idearatings in the four context categories. Scores represent the following ratings: 3­"really dislike", 6-"dislike", 9-"okay", 12-"like", and 15-"really like".

Table 1Ph~se 1 Ratings: Preferences for Story Ideas (N = 257)

Low fantasy High fantasyChildren's Adults'real-world real-world

M 9.76 10.19 10.19 7.77

SD 2.61 2.93 2.30 2.54

Preferences for low fantasy, high fantasy, and children's real-world story ideaswere relatively comparable, with lower preference expressed for adults' real-worldthemes. Standard deviations show that students' ratings varied somewhat more forfantasy than for real-world story ideas, with scores for high fantasy showing thegreatest standard deviation.

Mean scores listed in Table 2 represent the following Phase 2 ratings of wordproblems students imagined they would solve: 2-."very bad", 4-"bad", 6­"okay", 8-"good", and 10-"very good".

Table 2Phase 2 Ratings: Pre-Solution Preferences for Solving Problems (N = 243)

Low fantasy High fantasyChildren's Adults'real-world real-world

M 6.58 6.90 6.98 6.19

SD 1.75 1.89 1.74 1.74

As In Phase I, the low fantasy, high fantasy, and children's real-world

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categories received highest ratings. Although adults' real-world problems rankedlast, scores in this category increased-relative to the other three categories-fromPhase 1. As in Phase 1, the most variable response, as indicated by standarddeviations, occurred for high fantasy problems. .

Mean problem-solving scores from Phase 3 (see Table 3) were obtained fromraw scores of 1 for appropriate solution plans and a for inappropriate solutionplans, reflecting plans that would or would not lead to a correct answer ifimplemented properly. One-way analysis of variance showed significantdifferences (p' < 0.0001) between grade levels in overall performance. No significantdifferences by gender appeared at either grade level.

Table 3Phase 3 Problem-Solving Scores, By Grade

GradeLow High Children's Adults'

fantasy fantasy real-world real-world

Four (N = 104) M 0.63 0.55 0.57 0.57SD 0.49 0.50 0.50 0.50

Six (N = 135) M 0.75 0.79 0.83 0.68SD 0.44 0.41 0.38 0.47

Combined scoreg;".ibr Grade 4 and Grade 6 are relatively similar for the lowfantasy, high fantasy, and children's real-world categories. The lowest scoresoccurred in the adults' real-world category. Results of an ANOVA with repeatedmeasures indicate that students performed significantly better (p < 0.01) onchildren's real-world problems than on adults' real-world problems. However, asignificant interaction by grade level exists. An ANOVA with repeated measuresrun for individual grade levels shows that no significant differences appear amongcategories for Grade 4, although low fantasy problems yielded top performance.On the other hand, Grade 6 students' performance showed significant differencesfor three category pairings. Grade 6 students performed significantly better onchildren's real-world problems than on low fantasy problems (p < 0.05) or adults'real-world problems (p < 0.001), and they attained significantly higher scores onhigh fantasy than on adults' real-world problems (p < 0.05). Thus, the Grade 6students' performance was best on children's real-world problems and worst onadults' real-world problems.

Apart from categorical similarities and differences in performance, it isimportant to note that considerable variation exists among individual problemswithin and across categories. Individual problem scores ranged from 0.41 to 0.83for Grade 4 and 0.57 to 0.89 for Grade 6. Another prominent finding appears bygender. Gender differences in scores for the same items are not large for mostproblems. However, the discrepancy is quite large for a few items. The mean scoredifference exceeds 0.20 for three problems, two favouring Grade 4 males and onefavouring Grade 6 females. All three of these most extreme cases were fantasyproblems, two high fantasy (Grade 4) and one low fantasy (Grade 6).

Pearson correlations between Phase 1 story-idea preference ratings and Phase3 problem-solving scores show that only one of the 24 problems solved in Phase 3(12 at each grade level) significantly correlates with its Phase 1 preference rating.This result may be attributable to chance. A comparison of group means ratherthan individual pairs of scores also indicates that little consistency exists betweenPhase 1 preference ratings and Phase 3 problem-solving scores, and that nopatterns of association between these two phases emerge by context category.

Influence of Word-Problem Context on Problem SolvingMany students' written and oral comments illustrate their belief that word­

problem context affects their interest in and manner of solving a problem.Numerous students wrote that they had liked or would like so!ving particular

problems because the story was interesting. Some students claimed to try harder or"think worse" based on a like or dislike for specified problem contexts.Unpredictability of content impacted at least one student, who said a problem"made him think" because he was expecting a different ending. Some studentsmaintained that particular problems were harder or easier than others on a form,which implies context effects given the mathematical equivalence of the problems.

Three-quarters of the 31 interviewed students thought different storiesaccompanying the same mathematics problem would affect the way they solved it.Half of these students said they would exert differential effort on the problems.'Almost half said they would perceive the difficulty level of the problemsdifferently. About one-fourth said contexts of differing interest would affect theirdesire to do the problems, their success in solving them, their attentiveness, or howmuch they like the problems or like solving them. Preferred contexts alsoencourage students to look back at or "check" problems, develop greater interest inmathematics and doing problems, and "get more out of" the solved problem,whereas disliked contexts distract, confuse, or bore them. Fantasy contexts evokedstronger responses than real-world contexts and were typicaiIy associated withgreater interest and engagement or distraction and obscurity.

The following students' oral and written comments demonstrate some beliefsabout the effects of story context:

You'd like, "Aw, this [disliked problemJ is dull. It's boring. I can't do it". It's gonnabe harder, because it's just boring. But once you get into, like, fantasy, and it reallyjumps at ya, you're gonna wanna solve it, instead of thinking about how big it is,how much the numbers are, and if it's too hard. You're gonna wanna think, "Wow,I wanna read this. I wanna try to get, solve this, see what happens". (Grade 4female)

The Role of Fantasy Contexts ill Word Problems

Relation of Story-Idea Preference Ratings to Problem-Solving Scores

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If there's, like, fairy tale or something, I just do it. If it's, like, science fiction, I liketo read it, too. If it's a fairy tale, I don't try to read it all. I just skim it, 'n get what Isuppose-, that's what we're supposed to do. (Grade 6 male)

It [fantasy problem] kept me amused. Made me think harder. We should getproblems like this and the one about the braces [another fantasy problemJ. (Grade 6male)

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Stupid stuff like rain coming up from the sidewalk tends to kinda throw us .... Itthrows me off when they do that, cuz I concentrate on-"Well, that's stupid"­and things like that, and think, "Well, could that really happen?" 'n stuff. Thatkinda distracts me. I think something real unrealist~c throws me off. That's why Idon't like 'em much, cuz I skim. I don't really read the problem. I just look fornumbers. And, so, I have to go back and read the whole sentence. (Grade 6 female)

Mainly when I'~ working hard and I hear something creative, weird, or funny, itwakes me up. It sort of just gives you a laugh and helps you keep from beingbored. (Grade 6 male)

I like fantasy, but not really too unrealistic. And I like realistic stuff, 'n ... stuff thatcould happen to you, instead of stuff that couldn't. I work better on math problemsthat I can relate to, like something that could happen to me. And I do more workon those that I can relate to than I do the ones I can't. (Grade 6 female)

Discussion

Phase 1 Ratings: Preferences for Story IdeasStudents expressed differential preferences for various story ideas or verbal

contexts. By category, they showed roughly equivalent preferences that tendedl toward the "like" side of the rating scale for low fantasy,' high fantasy, and

children's real-world story ideas. Preferences for adults' real-world story ideascontrast with the other three types, falling somewhat on the "dislike" si~e of therating' scale, perhaps due to use of topics that are unfamiliar, irrelevant, oruninteresting to children. Conversely, the appeal of children's real-world storyideas might be grounded in familiarity, relevance, and interest. Preference for thetwo fantasy contexts might be based on familiarity due to fantasy being common tochildren's manner of thinking and to the large body of fantasy literature, movies,video games, and so forth geared toward children. The novelty aspect of fantasyalso evoked interest in many children, although this was also a disliked element forsome.

Students' preferences appear to relate not only to the major category type (e.g.,adults' real-world), but also to individual item content (e.g., animals as a topic orcollecting cans to earn money as a story line). Written comments indicate thatstudents sometimes based their ratings on major categories, as in "Because it has todo with things that are make-believe" or "This is too imaginary". The low ratings

.of adults' real-world story ideas also suggest a categorical response, that is, ageneral response to a context category. Somewhat more often, however, students'comments addressed context specifics, such as rap music or pollution, suggesting aneed for a more in-depth look at individual contexts. Therefore, the content ofindividual items used specifically for this research probably influenced results bycategory, as an interaction between these micro and macro factors seems apparentfrom these research findings.

The greater variability of students' response to fantasy compared with real­world story ideas aligns with other research evidence that children tend to either

The Role of Fantasy Contexts in Word Problems 85

like or dislike fantasy and to respond to it categorically (Cullinan & GaIda, 1994;Huck et al., 1993). Interest in real-world contexts more commonly falls along a less­extreme continuum and more often relates to specific story content. It makes sensethat real-world contexts are familiar to more students, making response to themless subject to individual variation.

Phase 2 Ratings: Pre-Solution Preferences for Solving ProblemsThe main categorical difference between Phases 1 and 2 is the relative increase

in scores in the adults' real-world category. This is not surprising. Children havebeen enculturated into often less-than-thoughtful, but sometimes successful,approaches to solving word problems, most of which they expect to have real-

. world contexts. Students may be more concerned about ~ problem's perceivedsolvability than the interest value of its context. Grade 7 students in Lester,Garofalo, .and Kroll's (1989) research wrote word problems they considered to beinteresting, which the class subsequently rated from 1 (very boring) to 5 (veryinteresting). They then solved a problem of their choice from those they had rated.In hopes of improved likelihood of getting a right answer, three-quarters chose aproblem they had assigned a 1 or 2. The researchers attribute this phenomenon toclassroom grading practices.

Perhaps students perceive more boring problems, in this research the adults'real-world problems, as being easier because they are less distracting. It may alsobe that students perceive them as what "real" (school) word problems should looklike or that they are more familiar with and have had more experience solving real­world problems, which dominate school mathematics materials. Moreover,students may hold different standards and expectations for stories read for theirown sake versus those tied to mathematics problems they will solve. In any case,these ingrained perceptions might affect students' problem-solving attitudes andperformance.

Most likely, ratings of children's real-world problems did not increase fromPhase 1 to Phase 2 as much as the adults' real-world ratings because the formerwere already fairly high in Phase 1. However, mean ratings for adults' real-worldproblems did not rise to the level of any other category, hinting that story contentfor these problems still influenced student response.

Phase 3: Problem-Solving ScoresGeneral context category does not appear to influence-by itself-problem­

solving success as measured by an ability to devise an appropriate solution plan,because problem-solving scores within each category show considerable variation.Nevertheless, context category appears to have some effect that interacts withindividual story content, because three pairings of context categories for Grade 6differ significantly in mean scores. As noted earlier, story-idea preferences do notcorrelate with problem-solving scores, negating the sole influence of interest inaccounting for these differences. Appeals to familiarity as "the" contributing factorprobably cannot be made, either, because the adults' real-world category for Grade

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4 does not differ significantly from the other categories as it does for Grade 6. Onewould expect that Grade 6 students would be more familiar with the adult worldthan Grade 4 students, yet adults' real-world problems appeared to disadvantagethose in Grade 6 more than those in Grade 4 in this study. The difference in scoresbetween the two real-world categories for. Grade 6 is interesting, because those

. students have had several years' experience solving adults' real-world problems inschool mathematics. Perhaps this is the very reason for the lower scores. Those inGrade 6 might have given less attention to overly familiar problem types that theyassumed they knew how to solve. Of course/ their minimal interest in these typesof problems undoubtedly is a contributing factor. Interest level may be particularlyimportant as students progress through schoot because motivation decreases withadvanced schooling (Anderman & Maehr/ 1994).

Scores on some individual items in this research are rather discrepant betweenfemales and males. This suggests a need to examine data for various studentgroupings in addition to doing global analyses, and it again implicates problemcontext as a significant factor in students' problem-solving performance. Types ofproblem context in which we embed problems may thus raise issues of educationalequity.

Conclusions

Traditionally, the textbook has been the most frequently used->in some cases/the only-instructional material in the classroom (Ornstein/ 1992). It is likely tocontinue to wield great influence upon teaching and learning. Most textbooks andother instructional materials will probably continue to rely upon word problems asone type of application- and thinking-oriented medium for mathematics learningat all age levels. Similarly/ standardised and other types of test measures will mostlikely continue to include word problems in their assessments of mathematicsknowledge and abilities. Well-constructed word problems can have an importantplace in t.lle mathematics curriculum, provided effective instructional m~thods

accompany them.Despite their confinement to an artificial setting/ word problems can serve as

sources of mathematical thinking, which is central to transfer of learning, oruseability of mathematic~l knowledge. As Hiebert et al. (1996) and Renninger(1992) say, the critical factor is not so much the problem itself as the problematisingof it, or engaging it as a genuine mental challenge. Kloosterman (1992) found in hisresearch that "students enjoyed the non-routine problems because they wereunique and challenging, not because they saw extensive real-life applications inthem" (p. 36). Lave (1993) echoes these sentiments thus:

Authentic dilemmas that furnish opportunities to improvise new practice may beeither imaginary or real . .. . The real trick may not be one of finding acorrespondence between everyday problems and school problems, but makingword problems truly problematic for children in school-that is, part of a practicefor which the children are practitioners. Given lively imaginations, it does notmatter whether the problems conform to life experience, but it is important thatthey engage the imagination, that they become really problematic. (pp. 85,88)

The Role of Fantasy Contexts in Word Problems 87

We must continue to investigate the many aspects of word-problem contextthat impact students' problem-solving performance and mathematics learning,including differential effects upon student subgroups whO might be particularlyadvantaged or disadvantaged by certain types of contexts. Developers ofinstructional materials, as well as teachers who use these materials and constructtheir own, stand to benefit much from these research efforts, as do-mostimportantly-students. However, information about problem-context effects aloneis not ·sufficient. As noted, teaching method is crucial to using word problemssuccessfully as a learning medium. We must continue to find ways to improveinstructional use of word problems and to institute effective measures for staffdevelopment in this area.

I suggest that fantasy contexts be included among those we use for wordproblems in mathematics teaching. Many children like fantasy, and therelationsh~p of this interest to problem-solving performance merits furtherinvestigation. Among other benefits, fantasy has been associated with abstract andcreative thinking. Because fantasy contexts are abstract and allow students to workwith more varied and less familiar problem types, they may increase cognitivedemands in mathematical problem solving. Their greater obscurity may mean thatstudents cannot jump too quickly to a perceived familiar format based onsuperficial clues. This may force students to engage contexts more fully, processingproblems more deeply to establish underlying mathematical relationships andidentify questions posed. Attending to the mathematical structure of problems inthis way enhances an ability to make connections among problems of differentcontexts, in other words, to transfer learning to similar situations. Learning toprocess problems more attentively and meaningfully will most likely proveespecially beneficial for problems that demand greater effort, such as thosecontaining more than one step or extraneous information. The variety fantasycontexts add can yield affective as well as cognitive benefits. Inclusion of fantasyproblems might influence some students to enjoy solving word problems and,therefore, to want to do more or to develop a more favourable attitude towardword problems or mathematics in general.

Students in this research expressed interest in fantasy contexts and solvedfantasy problems as well as or better than real-world problems. Nevertheless, itmay take time and special effort to teach students how to visualise and representfantasy problems. These practices support the goals of good problem-solvinginstruction in general. It is more important to seek longer-term benefits than short­term "feel-good factors", such as student performance with unrealistically high"success" rates based on an ability to recognise standard word-problem forms thatminimise real mathematical thinking.

Grade 6 students have shown that, given a choice of word problems to solve,they will select problems with a variety of contexts and not merely thosedemonstrating their expressed interest areas (Morrison, Ross, & Baldwin, 1992).Therefore, we might provide students with a variety of parallel problems andallow them some choice in the problems they solve. This would offer alternativesfor students who object to or have a low preference for particular problems, as wellas increase students' investment in their problem-solving efforts. .

88 Wiest

Acknowledgments

The author would like to thank Frank K. Lester, Jr. for directing the doctoraldissertation (Wiest, 1996/1997) on which this article is based.

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AuthorLynda R. Wiest, Department of Curriculum and Instruction, University of Nevada, Reno,NV 89557. E-mail: <wiest@unr.edu>.