Links and Nodes in Problem Solving H. G. Elliott Memorial University of Newfoundland. St. John's, Newfoundland, A 1 6 3x7 Interest in the study of prohlem solving has been renewed in recent years (1-6). Researchers throughout the world are making attempts to identify psychological processes which are occurring.when expert and novice problem solvers attempt to .. solve ..-- orohlems in t h e areas of science and mathematics. Students who do introductory chemistry are constantly presented numerical oroblems. One area in which oroblems ~ - with ~ ~ ~ ~ are given is that of emdirical formula determination. For the past three years, I have been conducting a study in which I am attempting to gain a deeper understanding of the ways British sixth-form and Canadian first-vear universitv students solve problems in chemistry. One eifort which has been made to obtain this understanding is to construct representations of problem solution routes. The following problem was given to approximately seventy-five sixth-form and first-year university students.
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A sample of a gas containing only nitrogen and oxygen has a mass of 66.0 g and occupies a volume of 3fi.7 cubic decimeters (liters) at a temperature of 298 K ( 2 5 T )and a pressure of 101.3 kPa (1.0atm). When this gas is decomposed, it yields 18.3 cuhic decimeters (liters) of oxygen at 298 K (25W and 101.3 kPa (1.0atm).Deduce the empirical (simplest) formula of this gas containing nitrogen and oxygen. T h e solutions were then analvzed with resoect to determining the processes used and thk entities obtiined and used in attemoting . .,to solve this nrohlem. Processes refer to ooerations performed while doing the problem. Some of the brocesses carried out include the use of mass, mole and volume conservation, determination of moles from volume, mass from moles, moles from mass. and volume chance " with temoerature variation. Entities represent information taken from the problem statement, information hrought from outside the problem statement such as from memory or the periodic table, derived expressions, and values calculated including the answer obtained. Some of the entities include mass of gas given, atomic mass of oxygen, volume per mole of gas a t STP, mass of nitrogen, and chemical formulas. When a listing was made of the processes used and entities
used or ohtained. attemots were made to reoresent the routes taken by the students through the prohlemsolutions. Figures 1and 2 renresent the results of these efforts. I'igt~re 1 depicts correct appruachrs taken to ihr 4 1 1 1 i ( m is one 111 u,hi(h all tht! SIPDS of t h t ~ixuhlem. A correct aonruach .. or processes used are valid or "legal" ones. Students using correct anoroaches mav or mav not arrive a t the correct answer. ~n;ncorrectanswer mairesult from mathematical errors. Figure 2 reoresents aooroaches taken in which a t least one stepor proce& used is G a l i d or "illegal." All the answers obtained using incorrect approaches were wrong. Both figures consist basically of an arrangement of nodes and links. A node is a graphical representation for an entity used or ohtained in the solution of the problem. A link is a graphical representation for the process used to transform one entity into another. Information from the problem statement is given inside circular shaped nodes and placed a t the top left-hand corner. The goal state is represented by an expression inside a double border node and placed a t the top righthand corner. Derived expressions and calculated values are represented inside rectangular shaped nodes. Information hrought from outside the problem statement and used is contained inside diamond-shaped nodes and directed toward the appropriate link. The links are solid and "wave-like" lines which represent the orocesses used in the orohlem solution. The numbered solid lines represent approved or "legal" processes. The numbered "wave-like" lines in Figure 2 reoresent invalid or "illegal" processes used. The dashed lines represent the needed association of entities to obtain another. T h e solution paths can he represented by listing numerically the appropriate processes used in solution to the prohlem. Rlr example, using Figure 1, the correct complete solution path taken by one student can be represented by: A comma is used to separate the processes used. A colon indicates the existence of a "dead end" or an abandoned route in the student's attempt. Information ohtained up to the
Volume 59
Number 9
September I982
719
PROBLEM
Figure 1. Correct solution routes. process indicated by a semicolon is used in the approach carried through to the end. A completed approach is indicated by the use of a square a t the end of the numerical list. If the student fails t o complete the solution and arrive a t an answer, this is represented by a dotted line at the end of a numerical list. A study of problem solving using networks has been reported by Ashmore, Frazer, and Casey ( I ) . They were particularly interested in presenting networks to show the interconnections of items of information and as a tool to he used in the instructional prmess. The objective of the nrtivity described was to enhance our understanding ot'the way st~ldenlc solved a specific problem in chemistry. The experience gained has enabled us to apply similar analyses to other chemistry prohlems. The information obtained allows a determination of the knowledge requirements for a specific problem domain. Areas in which misconceptions occur can be identified easily. One way in which this information will he of particular value to us is its use in a,njunctiw with other information ol~tilinrd from semisrructured interviews to determine the thought processes occurring while students are solving chemistry
720
Journal of Chemical Education
Figure 2.
Incorrect solution routes.
problems. Similar information is being ohtained from the same students solving different problems in an attempt to determine the similarities and differenceswhich exist in the ways students solve prohlems which differ in the degree of structure and numerical involvement. Literature Cited D,Frszer. M. J., and Casay, R J., "Problem Salvingand Problem-Solving Networks in Chemistn."J.CHEM. EDUC.,56 161.377-379 (19791.
(1) Ashmure. A.
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Bre~.D.S.."UnderstandineofSt~unuredProblemSoiutionr."ln~lructionolScienee. 3.3n-350l1975).
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(3) G ~ e n oJ., 6.."Pruee~~orundemfandingin Problem Solvinp"ln Cstellan, N. J., Pisoni, D. &and Pot&,C. R. IEdilom),"CognitlveThear)c-Volumc2:'London: John Wiley
and sona, 1877. (4) Mayer. R. E., "Thinking and Problem Salving: An lntraduction to Human Cognition and 1,eaming:' Illinois: Scatt. Poreman and Company, 1517. (51 Newail. A , and Simon,H. A,. "Human Prohiem soluinq:' New Jersey: Prentiee-Hall,
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(61 Wicholeren, W. A,, "How ta Solve Problems." Ssn Francisco: Cnmpany, 1974.
W. E. Freeman and
