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DOCUMENT RESUME
ED 242-550 SE 044.
AUTHOR TITLE Berger; Dale E.; Wilde; Jeffrey M.Solving Algebra Word Problems.
PUB DATE NOTE Mar 843013; Paper presented at_the Claremont Conference on
Applied March 3; Cognitive Psychology (3r-di Claremont; CA; 1984).
PUS TYPE Reports Speeches /Conference Papers (150)Research/Technical 0.43)
EDRS PRICE DESCRIPTORS MF01/PCO2 Plus Postage.Alggtra; Educational Research;_Error.Patterns;
Education; *SeCondary School Mathematics^ *Mathematics Instruction; *Problem Solving; Secondary
IDENTIFIERS *Mathematics Education Research; *Word Problems (Mathematics)
ABSTRACT Algebra word problems were ana1v7ed in terms of the
problems.information These tasks were classified into three levels: 4a1ue^ integration tasks that are required to solve the
assignment;first year algebra students) and experts (13 analytic geometryvalue derivation; and equation construction. Novices (
students) were comparedan the proportion of tasks completed at_each
levelnovices in'their attempts to solve six word problems. As predicted, showed greatest weakness on the tasks from the second and the
thirdproblems. levels, which required an appreciation of the structure of the Consistent with this finding, novices performed at chance
levelsthree wereon a most similar. Experts performed very well on this task.task that required them to identify which two problems of
Instruction focused on the structure of the problems was successful
in improving performance of a group of novices. (Author)
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Algebra Word Problems 1
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Solving Algebra Word Problems 1)ale E. Berger and Jeffrey ML Wilde Claremont Graduate School Paper presented at the Third Claremont Conference on Applied Cognitive Psychology, March 3, 1984 Claremont, CA 91711
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Alge...,ra Word Problems
Solving Algebra Word Problems If you haven't repressed the memory; you may recall the sense of frustration and feeling of incompetence that accompanied your first encounter with algebra word problems. AlthoUgh some. students quickly overcome these feelings as they gain a degree of mastery over word problems# many other students are left with the impression that mathematics is beyond them. As a high school algebra teacher some years agoi I^ had^ the^ opportunity^ to^ encounter^ the^ frustrations^ with^ word problems from another point of view. :I was struck by the difficulty many_ students had with translating word problems into equations. Even good students who were expert at solving algebraic equations were often baffled when the same problems were cloaked in a verbal cover story The extent of the problem has been well documented. In 1980# .a large national survey to as:2-ss educational progress (see Carpenteri Corbitt; Kepneri & Liniquist; 1980) posed the following problem to 17 year olds: 2 Present Figure 1 (^) here (Lembnade problem)
This problem is not especially difficult. One bottle will fill 7 cups. At 20 cent:. per cup# one bottle will yield $1.40 for a profit of 45 cents on each bottle. (^) Virtually all 17 year old students have had at least one course in algebra where tffey were taught how to splve problems such as this, and many of the students have taken several mathematics courses. Yet the solution rate for these students was (care to guess?)
Algebra Word ;Problem's
only 29%;
The persistent difficulty students have with word problems has attracted the intt rest of many educators and researchers. An early line of research focussed on the linguistic structure of the word problems' (cf. Loftus & Suppes, 1972); (^) While much has :been learned about how the phrasing of a question affects problem difficulty, it (^) can be argued that this approach is not very helpful to a teacher. (^) Although comprehenion may be facilitated by rewriting problans, teachers naturally are more concerned-with how they might help their students' develop their problem solving skills. An alternate approach which emphasizes the role of the 'student's prior Icnowled e assumes processing based on schemata. (^) The priemise is' that solvers are able to retrieve information about the formal structure of a problem upon recognizing the problem's relationship to a fami liar prototype; (^) Jill Larkin (1980; Larkin, McDermott, SiMon, & Simon, 1980) \ , has applied this approach to the study of problem iscilving in physics. She concluded that expert problem solvers make use of larger structural units than do novice problen solvers; Richard Mayer (1981a) made an important and rather heroic contribut ion to the study of algebra word problems when-he compiled and categorized a set of about 1200 problems from major'algebra texts used in California public schools. 01; the basis of underlying source formulas, (^) he' identified 25- fainiiiesi 'such as time-rate and unit-cost problems. (^) Families were diV ided into categories which she variables, derivations, and methods of formula constructions (^) For example, the time,rate family was divided into 13 categories,, such as motion,
Al,ehra Word Problems
The taxonomy consists of nine tasks organized into three levels of information integration with three types of tasks at each level. The levels are value assignment, value derivation, and equation
construction. For exlple, the first problem illustrAtes equivalence
assignment of values, transformation of these values to derive new values, and construction of an equation by applying a function rule.^ _ The second.problem illustrates assignment of an unknown, construction of a representation of an unknown using the values assigned^ \ at level 1, and then applicAtion of a source formula to generate an equation. The^ third problem shows a value assignment based on a relationship with another value, application of a source formula to derive new values, and combination c: these values into a final equation. There is a hierarchical relationship between the three levels of information integratiom The value assignments from the first level are often operated upon in the second level to derive new values, which are then used in the third level for the construction of the equations; However, problem solving can begin at any of the three levels. Activities at each' level place constraints on activities to be completed at each of the other levels. For example, if one can determine the form of the final equation, the range of possibly appropriate value assignments and derivations may be reduced.^ - The present research was designed to compare novice and expert problem solvers in term's of their facility with -information integration tasks at each level, and on their awareness and use of problem structure. We expected to find novice problem solvers to have more difficulty with aspects of problem solving thatArequire an appreciation
Algebr'a Word Problems 6
for the structure of the problem. This would be reflected in greater difficulty with the second and third levels of information integration than.with the first level, which is assignment of values and unknowns. We also examined the effects of instruction on problem structure for- novice problem solvers; We were hopeful that the training would improve r_ performance on the higher levels-of information integratiom
Method Three groups of High School studehts were recruited for thisstudy; Volunteers from first semester algebra classes were split-into two groups of novice problem solvers, an instruction group and a control grolip. A third group of experienggal;Obleth savers was comprised. of volunteers from analytic geometry classes; Each student was' paid $2. for participatin& Figure 3 shows the reilearch,design and tasks performedby each group.
Present Figure 3 hero (Research design)
.The initial task for all groups was a test set of six word problems, 1A6out a month later half of the novices Were given speciOl instruction on word problems while the other half served as a control group and were given a filler task; A posttest followed for both groups; A^ special^ test^ of^ ability^ to^ identify^ the structure of word problems was given to the control group of novices at the end of the second session and to the experts at the end of their first arid only sessiom The six word problem3 it the initial test were two problems each
Algebra Word Problems
specific integrations. Value^ assignments^ at^ Level^ 1 were easiest^ while formula constructions at Level 3 were the most difficult; This was dramatically true for the novices, who were moderately good at ,setting up givens but very poor at applying procedures which depend on the problem structure. Problem solving (^) and verbal abilities. One might expect verbal comprehension to be a good predictor of algebra word problem solving k success. To evaluate this notion, we gave X11 students the first part of Vocabulary Test II from the ETS Kit of Factor Referenced Cognitive
Tests (Ekstrom, French, Harman, & Derman, 1976). The correlation
between the proportion of information integration tasks completed and verbal comprehension for the entire sample was a highly significant .75. This high correlation was the result of large differences between the groups on both measures. The average score on the 18 item vocabulary test w;--.8 114;3 for the experts, and only .8;2 for, the novices; The correlation between problem solving and vocabulary for the experts alone was ;17 and for the novices it was .01; both nonsignificant. A high eves of verbal ability may be reqtkircd to become established in the high^ (= math performance group; but verbal ability does not account for the
variability of math performance found within a group.
fgmpargyIQIILAJuLAKQtron information intgatkmv-Uwi%
We next examined performance of the experts and novices on spe-cific
tasks. (^) The t-sks at Level 1 are assignment of unknowns, relational assignments, and equivalence assignments. Taole 2 3hows the mean proportion of success for each Level 1 integration.
Algebra Word Problems
Present Table 2 here (^) (Group by Le \fel (^) 1):_.
Both main effects and the interaction were-all highly significant. For the novices, performance on unknown assignment and relational assignment did not differ significantly, but both (^) were easier than equivalence assignment.- This pattern is somewhat different froim^ _^. (1981b) finding that equivalence propositions (^) were easier than relational propositions for college students to remember.. :0 (^) At Level 2 were the value derivation tasks using transformations, construction, and source formulas. Transformations were completely speAfied by the probleth, in that the initial value, a transfOrming value, and the ,transforming operation were all stated expli,citly. Constructions and source formulas, however, Involve combining information based on ideas about the problem structure that were not stated exricitly in'the problem. (^) This led to the prediction that transformations would be easier than contractionsactions and sourp.e formulas for novices. (^) Table 3 shows the proportion of Level 2 integrations completed for each group.
Present Table 3 here (^) (Group by Level 2)
For the novices, the construct//on task.s were significantly more difficult than either the aburce formula or construction tasks, which did not differ significantly from each other. No differences were reiiabl.e for the experts.
problem structure;
Al t,ebra Wc,rd Prot,: ems
test more direcUy the students' ability to
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detect and compare the structure of problems; we administered a special structural task. The structure task consisted of five triads of problems, where each triad was oianstructed of three problems from the same category, ith^ two
from one template and the third from a different template. Students
were asked to determine, for each triad, which two problems were most
alike. An example of a problem triad is shown in Figure 4. The first
two problems here are isomorphs which differ only in values of- varieties. (^) The third problem presents the second proposition is a form different from the first two problems.
Present Figure 4 here (Structure Task)
Chance performance on the structure task was 33% correct. The novices performed right at chance, 33% correct, while the experts were
correct on 88% of the triads; This^ is^ convincing^ evidence^ that^ the
novices had little appreciation for the structure of the problems, in contrast to the experts who were able to identify---treltructure quite consistently;
Ana-lvs-is- lot -problem- A third source of .data was the
problem protocols. All studenis w e asked to write down each step of
their solution attempt; and to "think out loud" as they Proceeded. These protocols showed striking differences between the novices and the
experts in terms of their use of the problem structure.
a
Algebra. Word 12
Present Table 5 here (Protocol Analysis)
Strategies that led to solution are listed in Table 5 as Type 1. Just over half of the problems solved by experts showed a "Workingdawn" strategy that started with the Level 3 integration, allowing the solver a relatively clear idea of what the goal path wot. d be. The novices
never started with the Level 3 integration. On the 18 problVms solved
by noVices, 17 showed^ It a workingup strategy where\all Level 1 integrations were listed first, then the Level 2 integrat ons, and fi nally the Level 3 equation emerged; The second strategy type was an incorrect application of a formula or procedure from a problem thought to be similar. Novices were likely to show formulas, while experts tended to show diagrams. The most
common strategy for the novices was Type 3. On 51% of the problems, the
novices produced only a simple listing of some or all of the Level 1 value assignments, with little else.
An examination of the 18 sUccessful protocols from novices showed
ithat 13 protocols included a complete labeled diagram. Of the 122
unsuccessful pcalkocols from novices on the same problems, only 3 included such a diagram. It seems likelit to us that the diagrams played a role in structuring the problem; CetnsIstent with: other information, the protocol. data suggest that the novices generally made little use lof the- structure of -the problems in determining their approach to the problems._ Experts, on the hand, made extensive use of their knowledge of the prdblem structure to
Algebra Word Problems 114
2 Present Figure 6 here (Training Aid)
The same procedure was repeated for the second motion problem, Cyclists. (^) The final step was to compare the two protlems using the
figure. The similarity of variables, value derivation, and equation
construction was pointed out. The control group of novices were given the two motion problems and asked to set them up. (^) A posttest for both groups consisted of four- problems, three of which were isomorphs of the training prciblems sharing category and template features. (^) The fourth was a generalization problem which was a motion problem from a different category. (^) The isomorphs involved combining two subdistances to equal a known total, while the generalization problem involved comparing two subdistances to find an unknown total. (^) The mean proportion of problexas solved for each group is shown in Table 6.
Present Table 6 here (Instruction vs. Control) The instruction^ ( group outperformed the control group on both the isomorphs and on the generalization problem. An examinaticsn Of performance at each of the three levels of information integration* showed that the Instruction group was better at all three levels on the isomorphs, and better on the first level on the generalization problem. The differences on the second and third levels of information integration were not significant for the generalization problem,
Algebra Word Problrms \
although the they were in the expected direction. Summary Overall, the clearest lesson to be drawn from our study is that an appreciation of problem structure is a crucial part of expertise in problem solving. (^) Experts are able quickly to identify the for of the equation to be solved and they use thiskriformation-to guide them on the path to solution; Novices are much more likely-to stop after they have generated a list of value assignments, unab e to see relationships inherent in the stru&t,Ure of the problem
tt
Some implications for instruction can also be drawn from the study; S Instruction on word problems should give attention to helping students^ v^ , c.f build schemata for the general structure of word problems and the
specific structures found within problem categories. Our small training
study suggests that detailed side-by-side comparisons of the structure of problems from the same category may (^) be a useful approach. Our datA also indicate that diagrams can play an important role in helping students. to organize information about a problem and to generate a structural representation of the problem. Perhaps students should 13:: trained to draw figures, at least for some categories of problems. We are encduraged by the results of our study, and are hopeful that teachers and designers of instructional materials will be able to put information like this to good Use.
Algebra Word Problems
Mayer; R. E. (1981a); Frequency norms and, structural analysis of
algebra story problems into families; categories, and templates.
Instructional ScienaQ, la., 135-175;
Mayer, R. E. (1981b). Recall of algebra story problena :Tech. Rep.
80-5). Santa Barbara, CA: University of California, Department of
Psychology.
Wilde, j. (1984).* Settirbz up algebra word problems: A task analytic
aPProach to Problem difficulty. Unpublished doctoral
dissertation, Claremont,Graduate School, Claremont, CA.
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. Algebra Word Problems
Author Notes This paper is based on a dissertation completed by J. Wilde under the supervisiOn of D. Berger., (^) We would like to thank Rich Ede and
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Ronnie Hardie; mathematics teachers at Claremont High School; for their help in this project;
