Symposium on Algorithms and Problem Solving
Generic and Harder Problems: Teaching Problem Solving Catherine Middlecamp and Elizabeth Kean' University of Wisconsin-Madison. Madison, Wi 53706
Chemistry students encounter dozens of chemistry problems. They read examples in their chemistry texts, see others worked out by their instructors, and attempt assigned problems for themselves. In solvine these ~rohlems.students often act as if foremost in their minds were the question, "How do I solve this prohlem?" Their attention is focused on obtaining the answer. Much less frequently do we hear students asking questions such as, "What kind of a problem is this?" or "What strategy is useful for this kind of prohlem?" We believe these latter questions to be the mark of strone students can be taught to ask problem solvers. such questions, thereby improving their ability to solve problems. Consider the situation of a student attempting to solve 20 assigned thermochemistry problems. The magnitude of this task depends, in part, on how the student views these prohlems. Some students see these as 20 different problems. I t is unlikely, however, that a more proficient problem solver would see these prohlems in this manner. By asking the question, "What kind of a problem is this?", the student may come to see that there are onlv four or so different classes of problems. Rather than struggling to find 20 unique solutions, the student would master four basic solution Drocesses and some variations on each. In answering the question, "What kind of a problem is this?", we find it useful to have students divide chemistry prohlems into two general categories: generic ~ r o h l e m sand harder prd,lems. 'l'hese two ca&ories differ in whether the dtudrnt has an algorithm, that is, a well-defined solution process, to solve the prohlem:
ore over,
Ccnertr problernt "no irills", striyptd-down problems from which all uther pnhlems aw built They haw a standard procedure by nl~irhthey may hp solved, an algorithm. Thts algorithm consist*
of aseriesof steps that,performed in order, accomplish the goalof the problem. Harderproblems: more complex problems, made by combining several generic prohlems, by using more complex language, and/or by extending the problem into an unfamiliar situation. The solver has no slgorithm to solve these problems. These are not rigid categories. Rather, we see the classification of a problem as dependent on both the context in which the problem is presented and on the student who tries to solve it. For examule, one student mav see a articular chemistry problem a s a slight variation on; generi'c prohlem and solve it with ease, whereas another student may be totally baffled by it. By labeling some problems as "harder" prohlems, we also do not mean that generic problems are easy. While generic prohlems are easier than harder problems, they are not necessarily easy. In presenting these two categories, we intend that stndents see generic prohlems as a repertoire of basic prohlem types of which they have command. They can use this repertoire to assist in solving harder prohlems. As students pro-
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This paper is based on ideas found in Chapters 10 and 11 of Kean. E ; M#odlecamp.C A Success Manual for General Chemistry. Random House: New York, 1986. 516
Journal of Chemical Education
mess through their chemistrv courses. thev" e x .~ a n dtheir &rtoire oigeneric pro1)lrm;as each topic is studied. Armed with the auestion "\Vhnt kind oioroblem is this?". when confronted with a new prohlem, &dents begin the problem-solving process by seeking to relate the new problems to others that they have learned to solve. If the prohlem is similar to a generic prohlem in their repertoire, they already know an appropriate algorithm to use in its solution and can proceed. If the prohlem does not resemble one of their generic prohlems, the student's task is quite different. In this case, the student must either reason creatively and link concepts and rules to solve the prohlem, or locate a worked-out exam~ l ofe a similar ~ r o b l e mto use as a model. The latter method is especially useful. Realistically, when first encountering a new class of chemistry prohlems, students may need the help of a model to develop the required solution. Having found a solution to the ~ r o h l e mthe . student still must develop an algorithm for t h e problem, to assimilate i t into his or her repertoire of generic problems. Again, the student may need assistance in developing a generalized solution for a class of generic prohlems. Thus, the process of becoming a stronger problem solver in general chemistry involves two tasks: recognition of problems as belonging to classes of generic prohlems and generation of algorithms for new generic prohlems to add to the repertoire. Recognltlon of Generlc Problems Students can e x.~ e c eeneric t oroblems to he clearlv and simply worded, containing little or no extraneous information. Generic uroblems mav he mathematical in nature. requiring the use ot'mnthematicai formulas or conversion facturs. Kxamples of thrse types of prohlems follow.
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What is the pH of a 0.1 M solution of potassium hydroxide? A pencil has a mass of 3.0 oz. Express this mass in milligrams Generic problems can also he nonmathematical, involving the manipulation of chemical formulas and symbols, as in the following examples: Draw the Lewis dot structure for PCls. Write the net ionic equation for the reaction of nitric acid and sodium hydroxide, While these latter prohlems involve no computation, a series of sequential steps is required to solve them. Furthermore. thereis a standaid s o l u t i k process, an algorithm, which the student can be expected to learn. Harder Chemistry Problems Harder problems are ones for which the solver does not have an algorithm. Harder prohlems can be made by:
combining two or mare concepts, rules, or algorithms in one problem, extending an algorithm or rule into an unfamiliar context,
making one or more steps in an algorithm more difficultthan usual, wording the problem with complex language, or all of the above.
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Harder prohlems in introductory chemistry courses often involve familiar processes and knowledge put together in unfamiliar ways. The manner in which the material is taught may determine whether the solver finds the problem to he a harder or a generic prohlem. Harder nroblems can he cateeorized hv the tvoe of harrier of which blocks solution of the p~ohlem.The the information eiven bv the orohlem statement may he a harrier. If information is missing, hidden, duplicated; complex, or simply overwhelming, a student may find the prohlem to he a harder prohlem. For example, in the following prohlem, one set of vapor pressure-temperature data is "missing":
the ideal gas law. Some forms are useful in prohlems with one set of gas conditions (PV = nRTj; other variations are more efficient when two sets of gas conditions are given (e.g., PIVl = P2V2).The student must he ahle to select the appropriate form for a specific prohlem. Similarly, students may learn several algorithms for calculating the pH of a solution, each algorithm appropriate for a specific type of solute (strone - acid.. weak acid. neutral salt.. etc.).. When students are unable to classify solutes correctly, or are unaware that such classification is necessarv. thev will not he ahle to solve the prohlem correctly. A generic prohlem may become a harder prohlem by varying the form of the final answer. For example, in nuclear chemistry, the student may he taught formulas that calculate the amount of radioactive material remaining after a time period. However, a harder prohlem might ask the student to find the amount of radioactive material that has decayed. Reasonine chains.. seouences of " I f . . . . then . . . " statements, mayYbe harder prohlems. For example, a student may he asked t o correlate the nature of aligand with the color of a coordination compound. T o solve the prohlem correctly, the student must keen track of the following reasonina: "If the ligand is strong fieid, then delta islarge. If deltais large, then a relatively high energy of light is absorbed. If a higher energy of light is absorbed, then . . . . " Students make mistakes when their record-keeping strategies are inadequate. Such record-keeping strategies can he taught. ~~~~
The vapor pressure of water at 25 'C is 23.8 mmHg. What is its heat of vaporization? In order to use the Clausius-Clapeyron equation t o solve this prohlem, the solver must supply a second set of temperathe vapor pressure of ture-vapor pressure data, water, 760 mmHg a t 100 "C. Having duplicate sets of information can also make a prohlem harder. For example: A 31-g block of metal requires 85 J to raise its temperature by 1.5 C. How manyjoules would be required to raise the temperature of a 54-g piece of that same metal by 10 "C?
This type of prohlem is frequently encountered in the lahoratory, where one set of data enables the solver to calculate a constant that can then he used with a second set of data. In other cases, information can he "hidden". For example, in a kinetics prohlem, the units of the rate constant, k, may he the only indication of the order of the reaction. Information can also he complex, introducing new concepts, vocabularv. or svmholism. or i t can simnlv he too lenathv for the soger t o b e ahle td keep track of it. In each of'thise cases, this presentation of information may present a barrier to the solver, making the prohlem a harder prohlem rather than a generic problem. ~ r o h f e malso s become harder when the solver must choose the correct alaorithm from among a number of possible alternatives. FO; example, studentsencounter many forms of
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lmpllcatlons for Teachlng Problem Solving Traditional instruction in college chemistry rarely offers direct teaching of prohlem solving skills. Few of the insights of cognitive psychology about problem solving have been integrated into course^.^ We believe, however, that students can become better problem solvers if teachers will help them become aware of the t ". w e s of eeneric orohlems and the demands of harder chemical problems. The learning process will he more efficient when such learning strategies are explicitly t a ~ g h tEventually, .~ students wili gain the powerful tools needed to cateaorize the problems thev encounter, to build a useful repertoire of basic algorithms, and to recognize when thevmust modifv those algorithms to solve harder prohlems.
Reif, F. J. Chem. Educ. 1983, 60.948. McKeachie, W. J.: Pintrich, P. R.: Lin. Yi-Guang. Educ. Psychol. 1985, 20, 153.
Volume 64
Number 6
June 1987
517
