Aspects of word-problem context that influence children's problem-solving performance.

Students tend to deem word problems one of the most distasteful and anxiety-inducing tasks in the mathematics classroom. They--at least those in the United States--perform poorly on solving word problems (e.g., Kouba, Brown, Carpenter, Lindquist, Silver, Swafford, 1988). Educational-equity issues also surface in relation to this important ability. Girls perform more poorly than boys in solving word problems (Fennema, 1989), particularly the less routine types, and other student subgroups may respond differentially to word problems as well.

Because word problems are difficult for students and play a key role in mathematics instruction, it is worthwhile to seek improved ways of constructing them. The result can lead to enhanced student thinking and insight into which problem aspects impact students' problem-solving performance and in what manner. One aspect of word problems that some researchers have found to impact student performance is problem context, or "non-mathematical meanings present in the problem statement" (Kulm, 1984, p. 17), for example, the "story" in which the mathematics problem is set. The influence of problem context is an important area that has not been studied extensively (Chipman, Marshall, & Scott, 1991; Harvey, 1987). Context as used here refers to the verbal aspect of word problems only (including numerals). It does not encompass other modes of problem presentation, such as concrete or pictorial formats, or the environment in which problems are solved (e.g., classroom climate).

In this article, I discuss aspects of word-problem context that appear to influence students' problem-solving performance. These aspects are gleaned mainly from a study conducted with fourth-and sixth-grade students (Wiest, 1996/97), but also from literature detailing similar work by other researchers.

Kuhn (1984) says problem context may help problem solvers give meaning to the mathematical content in a problem and that it is likely to influence, in particular, the problem-solving stages of understanding a problem and planning its solution. In addressing the potential influence of word-problem context, Ross, McCormick, and Krisak (1986) say, "Learners bring different knowledge structures to a task ... what comprises a meaningful presentation for one student may not be very meaningful to another" (p. 245). Bickmore-Brand (1990/1993) says context is foundational to mathematical activity: "Context is paramount to the construction of meaning the whole way through. It is the backdrop against which the parts have to make sense" (p. 3).

One way problem context may influence problem-solving performance is through the degree of interest and, hence, motivation it sparks. It may inspire problem solvers to engage to a greater degree in a problem and to persevere longer in solution attempts (cf. Boaler, 1993b, 1994; Murphy & Ross, 1990). Thus, affective variables, in addition to problem-solving success measured by correct solutions and answers, are worth considering.

Hembree (1992) conducted a meta-analysis of 44 studies, including study samples ranging from students at the fourth-grade through undergraduate levels, in which problem context was varied while mathematical structure was held constant. He classified the context manipulations as abstract vs. concrete, factual vs. hypothetical, familiar vs. unfamiliar, imaginative vs. ordinary, personalized vs. impersonal, and preferred vs. nonpreferred (indicating whether or not students chose or were assigned problems they solved). Hembree concluded that better performance was most strongly associated with familiar contexts, a finding corraborated by Kouba et al. (1988) in examining the results of the Fourth National Assessment of Educational Progress. Concrete and imaginative problems in Hembree's meta-analysis, the latter using fantasy or unusual circumstances, showed borderline significance in having positive effects upon students' problem-solving performance. No other obvious links surfaced between other types of problem context he reviewed and student problem-solving performance. However, such global analysis "smooths out" the findings of individual studies, some of which have shown that other types of problem context--for example, personalized contexts--benefit student performance.

Numerous studies have addressed the impact of personalizing problems--inserting individual students' names or other personal information into the problems they solve--on student interest/motivation and problem-solving success. In some cases these personalized problems were computer-generated. All studies surveyed found positive effects on the measured variable of either interest or achievement (Davis-Dorsey, Ross, & Morrison, 1991; Hart, 1996; Kloosterman, 1992; Lopez & Sullivan, 1991, 1992; Ross & Anand, 1987; Ross, McCormick, Krisak, & Anand, 1985; Wright & Wright, 1986). Lopez and Sullivan (1992) found significant differences favoring personalization on problem-solving scores for two-step but not for one-step problems (although students also scored higher on the latter) in comparison with nonpersonalized problems. They say personalization may be particularly important for more demanding (e.g., unfamiliar or mathematically complex) cognitive tasks.

Some research has centered about using various topics (viz., animals, sports, clothing/fashion, etc.) for problems with the idea that student interests and background experiences impact performance. Boaler (1994) found achievement differences based on story-content variations in problems that were otherwise alike mathematically. Renninger (1992) concluded from her work with fifth- and sixth-grade students that individually identified interests and noninterests embedded in contexts influence the way students engage in and complete tasks (e.g., in terms of entry into and perseverance on a task). However, this was much less the case with better problem solvers, which seems to support Lopez and Sullivan's (1992) contention that personally interesting or engaging contexts might be most important for more challenging problems.

In general, researchers have found personalization and use of preferred story content (e.g., topic and gender of characters) to be effective on a group basis - using dominant characteristics and interests of a group of students - as well as on an individual basis, but individual personalization may yield greater benefits overall in measures ofproblem-solving success and affective response (cf. Lopez & Sullivan, 1992; Murphy & Ross, 1990). Preferred contexts may contain content that is more personally familiar (Murphy & Ross, 1990), which - as noted earlier - has been associated with greater success in solving word problems. It also is interesting to consider Murphy and Ross's (1990) finding that students had more difficulty with sets of problems that had story contexts of mixed themes than they did with those that had a consistent story theme across all problems. This may, again, illustrate the importance of context in setting a mental framework within which students work to solve problems.

Caldwell and Goldin (1987) studied another type of problem context in their work with junior and senior high school students. They compared the difficulties of concrete versus abstract and factual versus hypothetical contexts for word problems. Overall, concrete problems were easier than abstract, and factual were easier than hypothetical. These results show that differing levels of abstraction seem to influence problem-solving performance.

Boaler (1993a) notes that individual reactions to context and the resulting personal constructions make the effects of context a complex one. Problem context can- for example - interact with the specific numbers used in a problem (Baranes, Perry, & Stigler, 1989). Because of its possible association with dollars and quarters, 100+4 can take on different meaning in money contexts compared with other context types.

Some groups of students respond differently to a word problem's context. Differences have been noted by gender. Sex-stereotyped contexts favoring males (e.g., science themes) have been proposed as one factor that contributes to females' underachievement in solving word problems (Jones & Jones, 1989; Jones & Smart, 1995; Murphy & Ross, 1990). However, sex-stereotyped contexts favoring females also have been associated with harming females' achievement by drawing them into problems in distracting ways (see Boaler, 1993 a, 1994). Sex-stereotyped problem contexts may advantage one sex over another in terms of familiarity of content and in type of affective response elicited (which in turn impacts cognitive response, such as mental investment in a problem and perseverance). Other student subgroups formed by age, ability level, community background, and other such factors can also affect individual response to a problem's context.

Carefully selected problem contexts can increase student interest as well as understanding, and hence problem-solving success, through enhanced ability to represent problems in computational form and to evaluate reasonableness of answers. The potential benefits provide an important message to designers of instructional mathematics materials (Murphy & Ross, 1990). Enhancing instructional materials in terms of their potential for raising intrinsic motivation may lead to increased learning (Parker & Lepper, 1992).

Study Overview

This study was designed to examine the effects of particular aspects of word-problem context on students' ability to solve problems, measured by choice of appropriate solution plan, and their preferences for different problems in terms of, for example, story content. The sample included 273 students (127 fourth graders and 146 sixth graders) across twelve classrooms in a Midwestern small city and small town. Students were asked to solve four word problems constructed to have parallel mathematical structure (described below) but different problem contexts. A "distractor" problem that looked similar to the others on the surface but which had a different mathematical structure appeared in the middle of the four problems to minimize the likelihood that students would detect a pattern. Students also assigned preference ratings to problems and provided written comments to indicate their response (e.g., degree of interest in story content or perceived level of problem difficulty) to working with individual problems . Finally, 31 students participated in individual interviews to verbalize their thinking while working through problems and to answer questions related to word-problem context. Students with particularly low mathematics or reading ability either did not participate in the study or provided data that were excluded from analysis.

Problems in the study had different contexts ("stories") but the same essential mathematical structure. Mathematical structure refers to the nature of the mathematical relationships expressed in a problem, or situation, rather than the operation(s) the problem is likely to evoke in solving it. The problems in this study were part-part-whole problems in which subsets ccmbine to form supersets. By solution method, the problems are classified as two-step translation problems. Translation problems can be solved by converting verbal mathematical information to its symbolic equivalent and executing the arithmetic operation(s). The problems in this study also contained extraneous information, that is, quantitative information not needed to solve the problem. Although translation problems tend to be easier than process or open-ended problems, students experience difficulty with those that have more than one step and those that contain extraneous information.

In abstract form, all problems had the following general structure (highlighted terms are numerals, and the three subsets combine to form the superset):

There are a (superset name) and b (extraneous set name). C are (subset name 1), d are (subset name 2), and unknown quantity are (subset name 3). How many (superset name) are (subset name 3)?

The problems for each grade level included 6 each of fantasy and real-world contexts, with students solving two of each type. Numerous variables were controlled, within reasonable limits, to keep the problems fairly similar-for example, number of sentences, number of words, use of female and male names and their order of appearance, and number size. Two sample problems follow.

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

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

Each student solved a randomly assigned set of four problems. For each grade level, the problems came from a bank of 12 problems chosen from an initial 24. Students had assigned these 12 the highest preference ratings on an earlier occasion. These problems were used to minimize the effects of interest in individual problems. The four-problem sets distributed differed across students within each classroom so that the same, or nearly the same, number of students received each of the 12 problems.

The purpose of this article is to identify and interpret, qualitatively, aspects of word-problem context that influence problem-solving performance. Data for the analyses were obtained from students' written and oral problem-solving plans and, especially, their written and oral comments.

Quantitative Data

Although the main information examined for this article is qualitative, quantitative data from my study illustrates some important points about the influence of problem context. Problem preference ratings and problem-solving scores varied--sometimes markedly--across problems. It thus appears that problem context had influenced students' opinions of problems as well as students' success in solving problems.

Mean ratings of individual story ideas for problems ranged from 2.44 - 3.82 for grade four and 2.26 - 3.76 for grade six on a five-point scale in which 5 was the highest rating (see Table 1). Solved problems were scored 1 (appropriate solution plan) or 0 (inappropriate solution plan) to indicate whether or not students chose a plan that would lead to a correct answer if implemented properly. Mean scores for individual problems ranged from .41 - .83 for grade four and .57 - .89 for grade six (see Table 2). Females and males, as well as students from the small town versus the small city, performed quite differently on a few problems. For example, although girls and boys performed equally well overall, their mean score differences exceed .20 for three problems. These findings further point to the differential interaction of students with text.

Aspects of Word-Problem Context That Influence Problem Solving

Detailed analysis of students' problem-solving performance as well as their written and oral comments in this study--and consideration of the professional literature described earlier--provide insight into which aspects of word-problem context appear to impact students' problem-solving performance. Four major categories emerged from the data: readability, verbal structure, story concepts, and personal factors. The categories interact to some degree. For example, level of understanding of story concepts in a problem influences the problem's readability.

Readability

Readability refers to comprehensibility of problem context and relates to such elements as vocabulary and wording, rather than to standardized readability measures. Some words that were difficult for students to pronounce or to understand were weeded out during a pilot study. Others became evident in the main study. For example, reference to "write-in votes" was very problematic for students at both grade levels. Although students could pronounce the words, they did not know the meaning of the term. Even after discussing its meaning as a class, students used the term inappropriately in making problem-solving decisions. Similarly, at least two fourth graders seemed to think "unicorns" and "horses with wings" were the same thing. In these examples the interaction of readability and story concepts is evident. In still another problem, the concept was abstract (a fantasy problem about time skipping ahead one month) but may have been understandable had the wording not been so confusing.

Not being able to pronounce a word certainly raises the difficulty level of text. Several students could not pronounce "elves," and others had trouble with the pronunciation and meaning of "eerie" and "residents." Several problems used people's names that were unfamiliar to students. Many students seemed bothered by the unfamiliarity of the names, their inability to pronounce them, or their uncertainty about the gender assignment of the name. For example, the unfamiliarity of the surnames Chen and Kline made them stumbling blocks in terms of pronunciation, despite their seeming simplicity in that regard. Many students could not pronounce the name Jamal and were unsure whether it referred to a female or male. A number of students commented unfavorably on the use of unfamiliar names, some of which I had included to introduce more ethnically diverse names to predominantly white student bodies. In general, students seemed put off by unfamiliar and stilted language. A fourth-grade girl said one problem used "words we really don't use when we talk." Presenting words in atypical written form also evoked negative reaction. Using a verbal rather than symbolic label to mirror other problems in the study, one problem referred to "148 dollars." The usage stood out to students, who wondered why the amount did not appear in its more familiar form, $148.

Finally, although it was not possible in this study to detect the influence of smaller words embedded in the problems' verbal contexts, it is important to note that small words--prepositions, for example--can be particularly problematic for readers (Allen, 1992; Bernhardt, 1984; Rastall, 1994). Moreover, use of prepositions in mathematical text may differ from the way we use them in everyday language (MacGregor, 1990). This is one reason to doubt use of standard readability formulas to assess text comprehensibility, probably to a greater degree for mathematical than literary text. These measures generally consider word and sentence length to determine the difficulty level of text.

Verbal Structure

Verbal structure includes visual and physical formatting of text. One example is whether or not subset names that are the same (e.g., "children") are repeated for each subset or are merely implied (as in "56 children on the sidewalk and 36 in the parking lot stayed dry, but those on the grass...."). Another variable related to set names or actions is whether the number quantifying a set precedes or follows it.

The physical proximity of numbers to each other and of set names or actions to each other and to related information might affect how well a problem solver associates those items. This may be advantageous (e.g., conceptually relating two physically close subsets that are to be combined) or disadvantageous (e.g., copying a wrong number from a remembered location in a sentence, as in choosing the nearby extraneous instead of the superset number in these research problems). Students apparently notice some aspects of format, as evidenced in a sixth-grade boy's favorable comment about a problem as compared with some of the other problems: "The numbers were not jumbled up and [were] more spread out." Size of numbers (e.g., those with the same numeral in both the hundreds and tens place) or visual similarity of numbers (e.g., 7 and 9) might also affect problem solving. For example, error in appropriate number choice seemed more likely in this study for nearby numbers of 139 and 137 than for 159 and 133.

Story Concepts

Story concepts involve meaning embedded in the story in a larger sense (e.g., the story line) and a smaller sense (individual terms and phrases) and the way that meaning is presented. Students reported that the story lines of a few problems in this study were hard to understand or were confusing, such as a fantasy problem about rain coming up from the ground, an unfamiliar concept. Some students said certain stories needed more information. They requested noun specifics (e.g., what kinds for the general term "toys"), background information about characters, reasons for story events, various types of story enhancements and expansions, and information about future events in the story. In general they wanted richer, more developed stories. Students reported liking problems that had "a background," "details," or "enough information," and those that allowed them to "get a whole picture" of what was happening. Students' general desire for rounded-out stories may highlight a need for an adequate context in which to conceptualize a problem or merely a substantial enough story to draw their interest. (Indeed, most word problems as we know them are not truly story problems.) Recall research mentioned earlier indicating that students perform better on word problems centered about a common theme, or unified "bigger picture."

Story concepts influence problem-solving performance by way of comprehensibility of key terminology and larger ideas. Not only the type but also the number and complexity of different concepts contained in a problem's context may affect level of understanding. For example, the first sentence of one problem in this study states, "139 children rushed to their favorite ride at the yearly carnival held by the 137 business owners in their town." A student may conceptualize the sentence as consisting of the two concepts "children rushing to a favorite carnival ride" and "carnival held by business owners in the children's town." Another problem, intended to be conceptually comparable, begins "149 children left their home in the sky and used their magic arm-wings to fly 135 miles to visit an Earth school." Students might interpret this sentence to have the three concepts of "children leaving their home in the sky," "children using magic armwings," and "children visiting an Earth school."

Conceptualizing quantified sets as more or less similar--as in whether they bear the same label, belong to an easily associated superset, or manifest discrete actions--might help a problem solver make important distinctions for setting up and answering problems. The fact that students in this study chose the extraneous number for use in solution plans more frequently in some problems than in others might mean that the degree of distinctness between the two set types in the first sentence influenced their choice. It is understandable that some children in this research accidentally interchanged "residents" and "strangers," the extraneous and relevant sets in the first sentence of one problem. Distinctness among the three sets in the second sentence of the problems also may have influenced solution plans used. Such "distinctness" is difficult to determine for children or for any individual. For example, perhaps children envision music types (rock, country, rap) or school grade levels (first, second, third) as being more discrete than an adult would because of the greater salience of these concepts to children. In problems similar to those used in this study, Hembree's (1992) meta-analysis indicated that very young students (the specific age level is not defined) attained higher problem-solving scores on problems within which common set names were used.

Subsumed under "distinctness of set names" is whether or not subset names used in the second sentence of these research problems are the same within a problem (e.g., "days") or are different (e.g., "quarters," "dimes," and "nickels"). One potentially problematic area is that the third or missing set (subset) in the second sentence of the problems is sometimes named and at other times is either implied or loosely associated with the other sets. This places different demands on the problem solver in connecting it to the two given subsets, the superset, and the set solicited in the final sentence for finding an answer. Moreover, in some cases the subset names are the same as the superset name, whereas in others they are not. Marshall (1995) addresses this concern for problems of the type, or mathematical structure, used in this research (see also Briars & Larkin, 1984):

Given the emphasis on combinations and partitions and the requisite dual labels of things (a label for the combination [superset] as well as a label for each partition [subset]), it is clear that a large part of the identification knowledge for this situation needs to be semantic. The problem solver who does not have a large store of knowledge about class inclusion and hierarchical categories will be at a disadvantage. (pp. 93-94)

Another story aspect that might influence student problem-solving response is the activity level of the story line. Students addressed this factor, unsolicited, by stating that some stories either had a lot of action or needed more action. In his meta-analysis of problem-context studies, Hembree (1992) states that very young students performed better on problems that included some measure of action. Alternatively, one might consider whether action could serve as a distractor.

A few students in this study commented that they could "Imagine" or "picture" certain problems, suggesting response to the imagery aspect, or perhaps level of abstraction, of a context. Some concepts - such as the term "strangers," the idea of "playing tricks," or the concept of time skipping ahead - do not lend themselves as well to visualization as those that are more concrete, specific, or familiar. Improved ability to visualize information or invoke imagery has been associated with increased learning (Kaufmann & Helstrup, 1985; Reynolds, 1993; Shepard, 1988; Silver, 1987).

Finally, number size might also contribute to how students conceptualize a problem. As noted earlier, some numbers and story topics interact, as in conceptualizing dividing 100 by 4 in a money problem as a dollar partitioned into four quarters. Magnitude of number seems to affect a person's ability to conceptualize quantitative information, and particular numbers and ranges of numbers have different meaning to different individuals. Students also have real-world--often relational--ideas about number size that influences self-assessment of answers to problems. One sixth-grade girl in this study verbally justified her answer to an election problem thus: "They [number of votes] all look like they'd be reasonable for how many votes each person would get for three people.... Because usually when there's three people in a small town, it'd seem like they'd be almost equally divided between all." In justifying his answer to a problem, a sixth-grade boy noted that the answer made sense because there should be more tre es than people in a forest. He also noted that 33 seemed too small a number of animals to be living there.

Personal Factors

Personal factors in word problems are those that are personally relevant and, therefore, vary with the individual. They include interest, personalization (e.g., using a child's name), and familiarity. Students in this study expressed strong feelings and specific ideas about what did and did not interest them. Many said degree of interest in a problem's context affects how much they like the problem and how they solve it by affecting them in these ways: perceived difficulty level of the problem, interest in working the problem, attentiveness, degree of effort exerted, success in solving the problem, and feelings toward the problem and solving it.

Dominant group interests surfaced as well as highly individual ones. Students expressed interest in both fantasy and real-world contexts. In general, however, story specifics - such as story lines and particular story elements - outranked general story type, or genre, in importance (see also Haynes & Richgels, 1992). For example, a student who strongly disliked fantasy liked one that was about animals. A student who liked country music reacted favorably to a context in which the music type only received minor mention, whereas a student who detested it reacted negatively to the context. More significant aspects of stories, of course, also evoked student responses, such as issues related to demonstrated fairness or kindness, as in distaste for the idea of forest creatures playing tricks on passers-by.

Examples of personalization include using a student's birthday month, a friend's name, students' grade level, or the name of a nearby town in a story. These contexts appeared to be quite salient to students. For example, students who currently had or had plans to get dental braces reacted favorably to a problem about a student with braces.

Familiarity of content, as noted earlier, has been associated with enhanced problem-solving performance. Some students associated concepts with personally familiar ones. A sixth-grade boy who liked a problem about an art show said, "I like seeing art. It reminds me of my Grandma's art that she made." Two students liked a problem about a family who had started their own business because their families owned or were starting a business.

Individual response to problem context may vary according to a problem solver's gender, age, community type and geographical location, family background (including values, preferred activities, race/ethnicity, socioeconomic status, and religion), individual personality (which includes interests), and academic ability, in addition to other personal--as well as random and coincidental--factors. Although the problems in this study were constructed with much attention to creating "gender-fair" contexts, females and males reacted differently to several problems in terms of both interest and their success in solving them. For example, females expressed greater interest than males in a problem context that was about helping younger children, whereas males showed greater interest than females in a problem about making money and one about starting a business.

Closing Comments

In my study, analysis of salient problem-context aspects in problems students liked best and least and those on which they attained the highest and lowest problem-solving scores did not reveal consistent factors that in themselves account for student response to the problems. It appears that problem-context factors are many and that they interact in highly complex ways to influence students in their problem-solving attempts, such that no individual factor alone causes differential response to various problems.

Although many variables were controlled across the word problems used in my study, many others were detected in conducting the study. These variables, in conjunction with those discussed by other researchers, are categorized under four umbrella headings in this article: readability, verbal structure, story concepts, and personal factors. It seems impossible to control all variables, particularly in light of the intricate interactions of the many problem-context variables. It is important to continue to analyze student problem-solving performance on individual problems both quantitatively and qualitatively to increase understanding of these variables' impact and interrelationships and to screen for the validity of particular problems. Further, it is important to attend to the affective and cognitive response of various student subgroups, such as those formed by gender, race/ethnicity, or socioeconomic status. The needs of some groups may be overlooked with attention only to group data.

Table 1

Preference Ratings for Story Ideas

                              Fantasy Story Ideas


Grade
Four
M         3.68      3.12      3.22      3.41      3.14      3.36
SD        1.98      1.42      1.33      1.31      1.17      1.24
M         3.39      3.64      3.82      3.32      3.66      3.63
SD        1.20      1.34      1.12      1.17      1.45      1.39
Six
M         3.09      3.46      3.14      3.45      2.97      3.09
SD        1.31      1.36      1.27      1.29      1.15      1.03
M         2.72      3.33      3.76      3.09      2.96      3.59
SD        1.24      1.33      1.03      1.26      1.37      1.27

                             Real-World Story Ideas


Grade
Four
M         3.73      3.34      3.68      3.39      3.63      3.24
SD        1.24      1.04      1.14      1.34      1.13      1.10
M         2.69      2.75      2.93      2.94      2.44      2.81
SD        1.09      1.27      1.23      1.23      1.10      1.12
Six
M         3.25      3.29      3.23      3.26      3.50      3.35
SD        1.01      1.02      1.13      1.13      0.94      0.98
M         2.26      2.72      2.49      2.56      2.36      2.26
SD        1.16      1.02      0.96      1.12      0.92      1.14

Note: Means (M) and standard deviations (SD) are listed for the 24
problems rated. The number of raw scores per story idea is 58-59 for
grade four and 69-70 for grade six (each student rated three story ideas
per category). Raw scores range from 1 ("really dislike") to 5 ("really
like").
Table 2

Problem-Solving Scores by Individual Problem

                                Fantasy Contexts

Grade
Four
M         .83       .45       .57       .44       .72       .47
SD        .38       .51       .50       .50       .45       .51
Six
M         .64       .87       .73       .80       .84       .72
SD        .49       .34       .45       .41       .37       .46

                              Real-World Contexts

Grade
Four
M         .53       .41       .75       .69       .54       .47
SD        .51       .50       .44       .47       .51       .51
Six
M         .77       .82       .89       .57       .82       .64
SD        .42       .39       .31       .50       .39       .48

Note: Means (M) and standard deviations (SD) are listed for the twelve
problems solved. The number of raw scores per problem ranges from 31-36
for grade four and 44-46 for grade six (each student solved two problems
per category). Raw scores are 0 (inappropriate solution plan) or 1
(appropriate solution plan).

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This article is based on a doctoral dissertation completed under the direction of Frank K. Lester, Jr.

AUTHOR NOTE