A. D. Ashmore, and M. J. Frazer The University of East Anglia Norwich. England R. J. Casey Footscrav Institute of Technology Melbourne, Australia
Problem Solving and Problem Solving Networks in Chemistry 1
The term problem is defined in the Oxford Dictionary as "a doubtful or difficult question; a thing hard to understand; a. nrooosition in which somethine" has he constructed: an enquiry starting from given conditions to investigate a fact, result, or law . . . . " Problem solving, then, is the result of application of knowledge and procedures to a prohlem situation. Obviously there are personal variations on what is "doubtful," "difficult." or "hard to understand," hut teachers of chemistry invariably agree that ability to solve chemistry problems is an important but poorly achieved aim and that there is a need to improve problem solving skills of chemistry students at every level of the educational system. This paper outlines an ongoing progpam a t the University of East Anglia. Problem solvine has been investieated bv manv authors who attach various meanings to the te&. ~ e r l i n (1j e has outlined the variations in definition ~ r o d u c e dhv different authors. Definitions range from those of ~ackson($,who takes a broad view that prohlem solving is bridging a gap between a prohlem state and the solution state, to those of Gagn6 (3) and Ausubel (4). Gagn6 considers problem solving to occur as a result of assembling rules, already known, to create a new (tothe solver) superior rule which is learned and which allows solution of the nrohlem. Once this rule has been learned. subseouent "problems," requiring reapplication of the rule to yield a solution. are no loneer "orohlems" within his definition. and the activity of generating the solution is not "prohlem solving." Ausuhel's definition is similar in that he insists that no frequently practiced procedure or strategy can be called prohlem solving. Problem solving as defined by Ausubel (4) is a form of discovery learning, bridging the gap between the learner's existing knowledge and the solution to the problem. The latter form of definition may be too restricted to he of
.
great use to chemical educators. The range of prohlem situations encountered by chemists, indicated in Figure 1,is from "chemical puzzles," where there is a unique answer which can he generated from information given in the prohlem statement, through to the hiahest levels of research work, where the answer may not he unique and the information needed to solve the problem must be generated by the solver's ohservation/experimentation. Ideally, a definition of problem solving ought to encompass this range. In solving any prohlem, both the information needed for arriving a t a solution (content) and the reasoning whichgoes toward the solution (nrocess) are i m ~ o r t a n tfactors. The majority of studies oiprohlek solving have attempted to overcome variahilitv in the hackeround knowledee - of the solver by using coutent-free problems. Examples of this approach are those of Maier (5) (Pendulum Problem), Duncker ( 6 ) (Cancer Problem) and de Bono (7)(Pebbles Problem), while other authors (8-11) have used the well known "Tower of Hanoi" puezlt: to study p r d ) l m solving processes. 'l'he addcd dimension of attempting to study processes of problem solving in a chemical context has been attempted at thr ilniversityuf East Anglia hy .\shmore (121,hlcCa111( J J ) , Powell (141, Morris (J;ij, and Ikhcrtson (16). Numerous approaches to the proress of pwhlem solving have been pmpnsed since Wallas (171in l Y ' 6 proposed a lour
- .
- - - - I- - - I
I
I
- -
- -
fundamental research
Stages of Problem Solving Staoe
Comments i. Many people failat problem solving because they are not clear about the objective.
[problem.
I
I
----- -----SELECT
appropriate
ii.
lli.
Most frequently a problem Is put in the form of a statement a question. k oflen helps to rephm me problem (into one or more questions). Oflen a problem needs shucturing (i.e.subdividing into a number of smaller problems). This stage has to be alternated with stage 3. As progress is made, pieces of relevant chemical informationmay need to be Incorporatedandlor become meaningful to the salver.
i. The simplest problem requires combining two pieces of information(A and 0 ) to reach solution (S). I
I I I
I
I
i
I
I
I
separate pieces 01
I
I i
!
SOIRES OF INFORMATION
Figure 1. A classificationOf chemical problems.
p q EVALUATE
db
ii. Most problems require piecing together several triangles of this type to farma network or "tree" (Figs.2.3).me way the ~rablemis structured (see Stage 1. comment (/ii)) will onen give intormation on the way the tree is formed. iii. Stages 2 and 3 are alternated until a solution is reached. it IS essential always to check that Me solution is: (i) a solution lo lhs problem defhad in Stage 1: and (ii) consistent with all the information stated in the omblsm.
Volume 56, Number 6,June 1979 1 377
stage sequence of "preparations," "incuhations," "illumination" with a final stageof "verification" to explain the invmtiveness of famous creators. These approarhes include the work of P d y a (18, 19)Guilford ( ~ O J Newell , and Simon 121) and .lacksun (2) ("logical methods", as well as rhr more radical approaches of de Bono (7,22,%31("lateral thinking"), Meadow (241 and Gordon (25) ("svnthetics"). , . ~"brainslorminr"). . We suggest that a?o& stage (logical) modei, as depicted in the table, is a useful aid to solving chemistry prohlems. Our thesis is that the best chances for success in chemical prohlem solvine rest on a combination of ularly those used in the sciences), 3) confidence.
We have devised a course (12) which, by using small group peer teaching and computer assisted learning, has attemped to give university students confidence in prohlem solving. Particularly in order to show the interconnection of items of information (see table), the concept of prohlem solving networks is useful. The conventional wav . of mesentine .a solution to a problem shows the reasoning in a sequence which is the order of the sentences and eauatious appearing on the .. written page. However, different silvers may "assemble" the solution to a problem in different ways dependent on such factors as their knowledge, cognitive styles, and confidence. Two examples of prohlems, together with the corresponding networks, are shown in Figures 2 and 3. These networks are derived by hreakipg down problems into unitary pieces of information and reirssembling them to show how the various pieces of information have to be connected to arrive a t a solution to the prohlem. A network can he written for any chemical prohlem. The orohlems used in our course (12) are not only numerical hut HlS0 include preparative and k r k d chemistry. "Closed" problems have networks the shape of a triangle with the
unique solution a t the apex: "Open" prohlems have a trapezium-shaped network. Advantages of the network approach include 1) Anetwork allows the presentation of the information showing
the interconnections in one diagram. 2) It reinforces the notion that there are (usually) a number of olternatiue "routes" to the solution. This, in our experience, encourages those students failing to make progresson a particular "route", to actively seek alternative "pathways" to the solution. This aspect has been stressed by Polya (18, 19) in his works on mathematical problem solving. 3) The network is useful for analyzingproblems. It shows how a mohlem can be broken down into a series of suh-oroblems.Case 1271 hna certninlv sneeested that a learnine- seauence ~ - ,. ~ . ~ -~ -. . like that in Figure 4(a) could be simplified by rearrangement to4(h).This has implications far both teaching strategies employed in prohlem-oriented courses and for assessing the progress and capabilities of students in these course< 4) Key items of information show up readily as centers of many connecting lines. (A deeper analysis of problem solving networks using network theory may be profitable). 5) Facility in perceiving student difficulties is enhanced when the teacher has analyzed the prohlem into a network. d
-~--~~ ~~
~
Further details of particular uses of prohlem solving networks in the courses a t the University of East Anglia are available from one of the authors (M. J. F.).
i Figure 2. Example problem and network 1.
378 1 Journal of Chemical Education
Figure 3. Example problem and network 2
(I)Ausuhel, D. P.. "Educational Psychology-a Cognitive View: Holt, Rinehart and Winston,New Y0.k. 1970. ( 5 ) Meier, N. F., "Reasoning in Humans?in "Selected Readingson the Learning P"(Mitors: Harris, T. L.,and Schwahn, W. E..) Oxford University Presr, 1961. lfi) Duncker, D.. "On Problem Solving" in "Thinking and Ressaning? (Editors: Wason, P. C.,and Johnson-Laird, P. N.,) Penguin, 1968. (7) de Rono,E.."The Use of Lateral Thinking? Pelican. 1971. (8) Gagn&,R.M.,andSrnith,E.C. JI., J. Expar Psyehol., 63,12(1962). 10,88(19651. I91 Harmsnn,A. M. R.,R~hauionralSeience, r l o l Haws. .I. R.. and Sirnun. H. A . in "Coenition and Instruction," Lawrence Edbaurn.
0
"'
",,ormation
""-"
0
Partial Solution to Problem
0
Solution to Problem
Figure 4. Two alternative networks (stylized)
Literature Cited 11) Redyne, D. E., "SLructure and Direction in Thinking," John Wiley, Now York. 1965. (2) Jackson, K. F.."Tha Art of Solving Problems? Heinemann, London, 1976. (31 Gasn6. R. M., "TheCondilionsoflaarning? Holt, Rinehart and Winston, New York, 1970.
Volume 56, Number 6, June 1979 1 379
