&what Can We Do About Sue: A Case Study of Competence J. Dudley Hemm Purdue Unlverslty, West Lafayette, IN 47907 Thomas J. G r e e n b e Southeastern Massachusetts University, North Dartmouth, MA 02747 Once again the public is aroused about the quality of American education, particularly in the areas of science and mathematics (1-3). A serious problem in education is that individuals clear the hurdles of the educational system without developing the competence required for success. Students pass exams and accumulate credits without developing the ability to solve prohlems that they will face in the real world. If we are to improve education, we must develop a clearer picture of the kind of competence needed, and we must develop better ways to detect incompetence where it exists. This oaper a case studv of "Sue", a college - oresents . freshman who was "successful" in secoidary school and wgo continues to "succeed" in college. However, the case study shows that Sue had serious weaknesses in areas that are critical t o success when unfamiliar prohlems are encountered. Specifically: 1) Successful problem solvers have a good command of basic fads
and principles. 2) Successful problem solvers construct appropriate representa-
tions of problems. 3) Sueeeasful pruhlem solvers have general reasoning strategies
that permit logical connections among elements of the prohlem. 4) Successfulproblem solvers apply a number of verification strate-
gies to insure that a) the representation of the problem is consistent with the facts
given b) the solution is logically sound, C) the computations are error free and, d) the problem solved is the problem presented. Context of the Study Sue was one of 31 individuals who participated in a study of problem solving in chemistry (4). Participants in the studv ranged from freshman chemistrv students to advanced graduate students in chemistriand included one professor of chemistry. All subjects were a t Purdue University, and there is no suggestion that they are representative of chemists in general. In fact, comparisons of SAT scores for the students who participated in the study with scores of other students a t Purdue suggest that the subjects were more able than chemistry students in general. Stoichiometry problems that varied in difficulty were given to the participants in the study. The participants were asked to talk out loud as they solved the prohlems. What tbev said was recorded on audio tape, and their written work was collected and analyzed along k i t h transcriptions of the, audio tapes. Follow-up interviews were conducted in which probing-questions w&e asked to clarify the participant's understanding of chemistry concepts or to clarify the thought patterns utilized as participants solved various prohlems. Prior to the first interviews, participants weregiven several written tests to assess competence in basic mathematics, understanding of chemical concepts related to the prohlems to follow, and general level of intellectual functioning as described by ~ i & e (5). t 528
Journal of Chemical Education
Details of the overall studv are resented elsewhere (4). several of the problems used in thk study are presented in the footnotes. I t is sueeested that the reader solve several of the prohlems (paniruiarly the second and fourth prohlems oresented). This will urovide a useful context for interpreting Sue's difficulties. ' Sue: A Case Study Sue, one of the participants in the study of problem solving, was a freshman enroiled in general chk&y. Her major was pre-veterinary science. Sue had two years of high school and four years of chemistry, one year of high school mathematics (algebra 1 and 11. geometry, and calculus). She wan enrolled in advanced calculus at the time of the study. Chemlcal Facts In spite of having completed two years of high school chemistrv. Sue's understanding of several chemistry facts was poor: o n the ~toichiometr;$uiz administered as a pretest in the studv. and during the study itself, Sue failed to write the correct formula f& a comp&nd when given the chemical name. For example, Sue wrote &SO4 for silver sulfate and MgFl for magnesium fluoride. During the solution of Problem # 3 in our study', Sue wrote ShIx for autimony(II1) iodide. This error in nomenclature (or writing formulas) resulted in failure to solve the prohlem. Using her incorrect formula, Sue wrote a balanced chemical equation to represent Prohlem #3, Sb + I,
= SbI,
calculated a correct mole mass (376 g) for her incorrect formula (Sh12), and used the incorrect ratio from her balanced eauation (1mol b:l mol ShL) t o calculate the mass of antimoni(111) iddide that could heproduced from the reactants. Althoueh her procedure for solving - the limiting- reagent prohlem was correct, she arrived a t the incorrect answer because her chemical facts were wrong.
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Representatlon On two other occasions Sue had difficulty solving the stoichiometry problems presented in our study because she represented the prohlems in a manner that is inconsistent with chemical facts. In her attempt to solve Prohlem #42,
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Problem #3 stated: "When 16.00 g of antimony (Sb)and 46.00 g of iodine (I2) are placed in a suitable solvent such as dichloroethane and refluxedat moderate temperatures, antimony(lll)iodide is formed. Assuming complete reaction, how many grams of antimony(lll)iodide can form in this reaction?" Problem #4 stated: "A l.OO-g mixture of cuprous oxide. CuQ, and cupric oxide, CuO, was quantitatively reduced to 0.839 g of metallic copper by passing hydrogen gas over the hot mixture. What was the mass of CuO in the original mixture?" (From Mahan, B. H. "University Chemistry," 2nd ed.: Addison-Wesley: Reading, MA, 1969).
Sue wrote the following equations:
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Cu,02+2H,=3Cu+2H,0 Annarentlv Sue viewed the mixture of the two oxides as a new Eompouid formed by equimolar amounts of the two oxides combining in the form of CusOn. which then reacted with hydrogen gas to produce free cop& rather than as two separate compounds that react independently. In Problem #I3,Sue used the chemical formula, CrzMg3, to represent a solution containing chromium(II1) ions and magnesium(I1) ions. Sue's equation for the reaction described in Problem #I is: CrZMg,+ 12 NaF = 2 CrF,
General Problem-Solvlng Skills This is not a trivial task and suggests that Sue's general nroblem-solving skills were relativelv strone. Further evidence of Sue's general problem-solving prowess is found in her score of 39 out of a nossible score of 42 on the Loneeot " Test, a measure of f o r t h operational reasoning that was used as a pretest in the study (6-8).
+ 3 MgF, + 12 Na'
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Verlflcation Strateales Although Sue's score on the Longeot Test and her performance on Problem #5 (and elsewhere in the nroblem-solving interview) suggest that she has developedgeneral intellectual skills required for problem solving, she did not always use them effectively. In addition to the difficulties she had representing prohlems in a manner that is consistent with chemical facts. she occasionallv arrived a t an answer that was correct, but not the answe;called for in the question. Fur example, in Problem ir7 of the Aleebra and Trieonometry ~ u i z ~~ "; correctly e uied the general formula, ai+ bZ= cZ, and wrote: ~
Notice that Sue's balanced equation violates the normal requirement of balancing charge as well as utilizing the strange compound containing chromium and magnesium. Although there is no assurance that attention to rudimentarv chemical facts and nrincinles would have led Sue to solve this problem correctiy--ohy 26%of the participants in the study were successful with this problem-the fact remains that Sue proceeded to solve the problem incorrectly in the face of information that should have indicated to her that something was wrong with her procedure. Other portions of the interview suggest that Sue's difficulty stemsfrom ignorance of chemicaifacts (or possibly from overload of working memory); she had mastered the rules or algorithms that are followed in applying facts. For example, Sue appeared to know the rules for writing chemical formulas as indicated by her response to Problem #5 in the study.& In solving that problem, Sue wrote the following balanced equation: MCI,
+ 3 AgNO, = 3 AgCl + M(N0313
Manv narticinants in our studv had difficultv constructine the fokmla ;or the unknown product, M(NO&, from thz information given. Sue correctly inferred from the formula, MC13, that the unknown metal has an oxidation state of +3 and that it would combine with three nitrate groups to form the product. The ability to use information about the 1:3 mole ratio is critical to the solution of this problem.
Problem #7 stated: "A solution contains Cr3+and Mg2+ions. The addition of 1.00 L of 1.51 MNaF solution is just required to muse the complete precipitation of these ions as CrF,(s) and MgF,(s). The total mass of the orecioitate is 49.6 a. Find the mass of CPf in the oriainal solution." (From Boikess, R. S.: Edeison, E. "Chemical Principles": Harper & Row: New York. 1976). Problem #5 stated: "The chloride of an unknown metal is believed to have the formula, MCI,. A 2.3953 sample of the chloride is dissolved in water and treated with excess silver nitrate solution. The mass of the AgCl precipitate formed is found to be 5.168 g. Find the atomic weiaht of M. the unknown metal." (From Boikess. R. S.: Edeison, E. 'Chemical Principles"; Harper 8 Row: New York. 1978). Problem #7 on the Algebra and Trigonometry Quiz stated: "In a right triangle the length of the hypotenuse is 1 ft greater than the length of one of the legs. The length of the other leg is 7 ft. Find the length of the hypotenuse." Problem #I stated: "How many grams of an alloy containing 25% silver should be melted with 50 g of an alloy containing 60% silver in order to obtain an alloy containing 50% silver?" (From Nichols. E. D.; Heimer, R. T.: Garland, E. H. "Modern Intermediate Algebra"; Holt, Rinehart, 8 Winston: New York, 1965). Problem #2 stated: "A chemist wishes to obtain 80 mL of a 30% solution by mixing a 50% solution and a 25% solution. How many milliliters of each solution should be used?" (Note: Although not stated in the proolem, tne assumption that volumes are additive must oe made.) (From Sworowski. E. W. 'Fundamentals of Algeora and Tr!gonometry": Priole, Weoer 8 Schmidt: Boston. 1978.)
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Sue then wrote 24 as her answer to the problem, failing t o attend to the fact that the problem asked for the length of the hypotenuse, which Sue correctlv renresented as x 1. ~ e i i h e r snormal ' response t o such student errors is that the student has a reading nroblem or is not attendine carefully to reading the probi&, but this is not the c a s c Tape recordings of students solving problems virtually always indicate that the student reads the problem correctly, understands what the problem is asking, begins t o solve the problem presented, hut at some point in the solution process is diverted from the initial goal of the task and reports an intermediate result for the solution rather than the result called for in the problem. Successful problem solvers frequently have the same difficulty, but they have developed habits of verifying their work so that the error is detected and corrected before the solution is reported. Sue failed to do this. Sue made the same kind of error in solving Problem lt5. In that prohlem, Sue found the molecular weight of M C I ~ and reported that as her answer; the nroblem asked for the atomic weight of the metal, M. Sue's difficulty in representing word problems in terms that are consistent with physical reality was not limited to her performance on chemistry problems such as Problems # 3 and #I. She had the same difficulty with Problems #1 and #2, which require an understanding of percent but do not require an understanding of any chemical facts.6 Sue wrote the following mathematical equation t o represent Problem # 1:
+
A correct representation would he Sue's representation for Problem
# 2 was:
A correct representation would he
+
0.25% (80 - x)0.50 = 0.30 X 80 Sue's representations of these problems are inconsistent with the physical reality described, but she evidently did not realize that. Volume 63 Number 6 June 1986
529
Reviewers of an earlier draft of this manuscript commented that Sue had a diiiicult time applying what she learns. This is oln~ioudvtrue. hut that anal\,sis < . orovides no hint concerning the &ture of her difficulty or what we must do as teachers to heln her overcome it. We believe that there are two variables [hat operate here. First, we believe that Sue, like many other students, has a poor understanding of the meaning of mathematical representations. She does not make strong connections between t h e mathematical representation and the physical reality that she is attempting to describe. Second, we believe that, once problems have been represented, students like Sue seldom check the representation against other knowledge that they have to be certain that i t is an accurate description of the problem situation. This description of Sue's difficulty points to two areas of instructional emphasis that could lead to improved prohlem-solving performance. Discussion
After studying Sue's performance in our study, we are inclined to characterize Sue as a "rule learner". By this label, we mean to imply that Sue views her primary task in the educational system as memorizing rules and algorithms. She must then practice those rules until she can apply them flawlessly. When she is presented with problems such as those used in this study, she looks for clues in the problem statement to identify the rule that she must apply to solve the problem. She then recalls the rule, applies it, and reports an answer. Successful problem solvers in our study often did something similar to Sue's ~erformance:i.e.. thev often recalled rulesand applied them in the rontixt of the-problem given. The difference between Sue and successful ~ n h l e msolvers was that surcessful problem solvers did somethml! in addition. They frequenrlvcherked thevaliditvoi~roceduresand the applicability of rules by comparing the cbnditions given in the problem with their understanding of physical reality. They frequently verified their answers (interim as well as final) against information given in the problem statement. Sue did not. Whether our characterization of Sue as a "rule learner" is the most useful description is a matter of interpretation that may change in the face of further research. It is based primarily on the kind of performance that we have described in which Sue appears to be applying various rules incorrectly. Sue was quite successful on tasks that rewired nothing more than the application of a rule, even rules that many students have learned and forgotten. For example, Sue recalled and correctly applied the trigonometric rule, sin theta = opposite side divided by the hypotenuse, t o solve Problem #14 on the Algebra and Trigonometry Quiz.' Sue's work for Prohlem # 5 of our study also provided evidence that she was c a ~ a h l eof solvine tasks throueh the use of a familiar rule. Sue wrote a correct equation to represent the chemical reactions for this nroblem. She then used the factor-label method to calculate>orrectly the number of moles of MC13 present: 1ma1 MC4 1 mol AgCl 5.168 g AgCl X 143.32 g AgCl 3 mol AgCl
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Sue's procedural knowledge of how to convert grams of one substance to moles of another suhstance is adeouate. Sue can solve stoichiometry tasks that require simple conversions. She correctlv solved several such tasks on the Stoichiometry Quiz.
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Ouiz stated: "An 'Problem f114 on the Alaebra and Triaonornebv " isosceles triangle has two sides of length 10 in. and two angles of 77'. Find its altitude." (Sin 77- = 0.974; cos 77' = 0.225: tan 77' = 4.331.)
530
Journal of Chemical Education
Sue's difficulty comes when she is confronted with unfamiliar problems that require analysis of the ~ r o b l e mto Droduce a sensible representation and subsequent use of f a h iar rules in this new context. This difficultv was observed repeatedly in Sue's work. For example, in her attempt to solve Problem # 4 in our study, Sue applied the same procedure that she used successfully in Problem #5. Sue's factorlabel solution for Problem #4 is: 0.839 g Cu X
1mol Cu
1mol CuO
19.53 g CuO
63.54 g Cu
1 mol Cu
1mol CuO
Sue then reported 1.05 g of CuO as her answer. However, I'roblem # 4 staws that the mixtureof theoxides has a total mass of 1.00 g. Sue was oblivious to the contradiction between her answer and the informationstated in the problem. She a ~ o a r e n t l v"reroenized" the form of the oroblem. recalledLihefactbr-label"algorithm, and applied with cdnfidence. Sue's work suggests that she does not associate the symbols and numerical answers that she generates with real objects and events. She applied rules correctly when the context of the application was clear from the problem statement. However, when problems required the integration of algebra. . chemistrv. and reasonine. she was unsuccessful. If Sue's perfor&kce were unusual, i t would be little cause for concern, hdt i t is not. It is twical of manv students in our study, and it is typical of many students k h o are passing science courses a t all levels. The problem is not that the amount of science studied i t too little or too weak, but that the courses she took did not result in the ahility to apply knowledee and skills in other than the one used durine instruction. How manv Sues do vou and I have in our current classes? What are we doing on" a routine basis to detect the kind of weaknesses revealed in Sue's interview? What could we be doing? What changes must be made in the kind of science that we teach and the way that we teach it so that the fundamental ideas of our discipline can be used outside the classroom? We are unable to provide proven answers for these questions, hut viewina our data in the light of standard presentations in high schiol and college texts leads us to these suggestions:
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1) Virtualb all problem-solving activities in standard courses focus on problems for which an algorithmic solution hes been taueht. We believe that students should encounter novel (thoueh not"necessarilv comolex) oroblems for which no set orocedke has been taught. The encounters should be part of homework, hour exams, and laboratory experiments. 2) Textbook solutions to problems and solutiona presented by teachers in class are almost always efficient,well-organized paths to correct answers. They represent algorithms developed after repeated solutions of similar problems. Our research reveals that expert problem solvers never follow such direct paths when confronted with novel tasks, and most problems in chemistry are navel to the students in the course. We believe that we must give far more attention to how experienced problem solvers go about making sense out of problems encountered for the first time. 3) Expert problem solvers make use of a number of general strategies (heuristics) as they interpret, represent, and solve problems. ohvsical Trial and error. thinkine of the orohlem in terms of the . . system d~rcusaed,n~lvmga special rase. solving a simple problem that seems related to s difficult pruldem, and then analyzing the procedure, breaking the problem into parts, substituting numbers for variables, drawing diagrams to represent molecules and atoms, and checking interim or final results against other information held in memory are common strategies used by successful problem solvers. Although teachers frequently use these strategies,little attention is given toteaching these strategies to students. We believe that we must, even if it means a reduction in the number of algorithms students are taught. 4) Much problem solving in chemistry is done without reference to real chemical systems, even to the point that conditions stated in
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the problem are totally unrealistic. This denies students the opportunity to develop proficiencyin using a potent strategy of successful problem solvers-asking whether a result "makes sense"in terms of physical reality. We believe that teachers should make deliberate attempts to relate chemical problems to the physical systems they describe and to encourage students to examine their answers in terms of physical hi^ implies,of course, that students must understand facts about physical reality hefore they are expected to solve problems related to it. The reader is cautioned that these suggestions are largely
untested. The changes we suggest may not lead to the improvement we need, but we are convinced that, until the science education community addresses these issues square-
ly and finds answers, the crisis in science education will remain. Literature Clted dof~cience,-~dueation ~ in~th.sciences: ~A evel loping crisis": washington. 1982. (2) American Chemical Soeiety. '"Tomarroar: Report of the Task Force for the Study of Chemistry Education in the United States"; Washington. 1984. :i ~ (3) N ~~ ~ ~~E ~ ~in~ ~ I ~I ~ I~~~i~~ ~~ s~t ~ i a~k ~h~ ~~ for ~ d ~ ~ R t & i ~~' :~U.S. a oepartment l of ~ d u r a t i a nwaahington, : 1983. ( I ) Gmonbow, T. "An Investigation of Variables Involved in Chemistry Problem Sol"ing': unpublished dodoral dissertation, purdue univcraity, ~ ~198s. ~ (5) hheider. B.;Piaget, J. '"The Growth of Logical Thinking from Childhood to Adoleaeenca": Basic Bwk8. Near York. 1958. (6) hngeot,F.Buii. lnstitvt Not. Etude 1962, 18, 153.
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Volume 63
Number 6
June 1986
531
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