Improving Students' Problem-Solving Skills A Methodical Approach for a Preparatory Chemistry Course Julien Genyea Oakland University, Rochester, MI 48063 Recently there has been increased interest in teaching problem-solving techniques in chemistry course^.',^ Many students beginning their study of chemistry have a great deal of difficulty with problems that involve quantitative or semi-quantitative considerations. The difficulties that these students experience are generally due not so much to a serious lack of specific mathematical skills, hut to a combination of: (1) .. andlor , , deficiencies in formal reasonine ahilitv. .. (2) . inahilitv. unwillingness to construct an appropriate physical picture given a problem in written form, (3) the mistaken belief that for any problem there is a formula, or set procedure (such as the factor-label method) that one can simolv wlue numbers ~~~
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tention to helping our students develbp problem-solving skills in our introductorv courses. then thev will he able to acquire a deeper understanding of the coursematerial. They wilialso have much less difficulty in subsequent courses that require quantitative considerations. In addition, much of what our students learn about abstract reasoning and urohlem solving in chemistry is also applicable to other-disci&nes and to life situations in generaL1 At Oakland University we have recently instituted a special preparatory chemistry course in which problem solving is a major aspect. This course was designed for those students who subsecluentlv intend to take our standard two-semester sequence of i&oductory chemistry courses (and usually two semesters of organic chemistry as well) hut are not yet sufficiently well prepared, as judged by their performance on a placement examination. For these students, we have found this preparatory course highly preferable to a survey-type elementary course. The latter course is more suitable for students who plan to complete a total of just one or two semesters of chemistry. Our reasons for designing a special oreoaratorv course with orohlem solving as one of its maior p r t ~ l h m.,,lvi!ig lc,pic>, d l as 1 he W I t~hi>vtnphxi, intwr,.t~du it11 the ~ ~.~ateriql. will 11, tilt, d ~ l ~ i e d of a k h s e q u e n t paper. Here, I would like to put forth some ideas and suggestions for helping students increase their problem-solving skills. These suggestions are directed mainly, hut not exclusively, toward a preparatory chemistry course.
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General Framework The key element determining whether a student significantly increases his or her problem-solving skills is, of course, the effort that he or she mikes. However, we as teachers can have a profound influence on both the magnitude and direction of that effort. We must holster the confidence of anxious students by presenting them with prohlems that they can handle successfullv with a reasonable effort. I t is uossible to motivate strongly Buch students to put forth a greater effort Presented at the American Chemical Soctety Metrochem '80 Meeting, South Falisburg, NY, October 3, 1980. 'Gilbert, G. L., J. CHEM. EDuC., 57, 79 (1980). Metter, C. T. C. W.. Pilot, A., Roossink. H. J.. and Kramers-Pals, H., J. CHEM.EDUC., 57, 882 (1980).
Callewaelt, Oenis M., and Genyea, Julien. "Fundamentals of College Chemistry," Worth Publishers, Inc., New York, 1980, p. 362. 478
Journal of Chemical Education
if we take a gradual approach to prohlem solving in a preparatory course. We should begin with prohlems that are simpler in terms of the type and number of physical principles that are involved, the number of calculational steps required, and the mathematical complexity of these steps. We can then oroceed to oroblems that are more complex. Many of the students in ;I pr I I I ~m.. \ \ ' t nwd t o t 111phasize repeatedly that one is not expected instantly to see a method for solvine-a orohlem from a first readine. - Rather, . wemust point out that i t is acceptable, desirable, and often necessary to try various approaches to a particular prohlem before a means of solving it becomes apparent. We must help students to understand that problem solving is not merely the application of a standard set of procedures. Too often sample urohlems that are worked in class or in texts are presented in H form something like: "we do this, then this, andnow you see we have the answer." Many students can follow a presentation of this type but cannot extract the underlying logic that was used to solve the prohlem. As a consequence they cannot solve even a similar problem on their own. I agree with Gilbert' that we must expose for students the reasoning process we use to solve particdar prohlems. Throughout a preparatory course we should devote a significant fraction of the class discussions to this activity. I t is my experience that it is quite useful to introduce early in a . nreoaratorv course a general oroblem-solvine . - avoroach -and some general problem-solving strategies with reference to a series of prohlems dealing with a particular topic. Then, throughout the course, one can refer repeatedly to this general approach and expand the set of general problem-solving strategies with reference to numerous other problems involving many different suhjects. Students (especially those who must discard inappropriate approaches to prohlem solving that they have unfortunately acquired) need such reinforcement of principles with reference to a variety of topics if they are to increase their problem-solvingskills. I have found the following general problem-solving approach to he quite useful for introductory chemistry problem^.^ General Problem Solving Approach Step One: Creation of a clear picture of the physical situation involved Students must come to realize that i t is essential for them to consider both the precise meaning of the terms and the specific physical situation descrihed in a given problem before thev trv to use one or more eauations to obtain an answer. It is quite important to emphasize, by using appropriately chosen samole oroblems. that crude sketches andlor eraohs can be extreme6 useful. Students readily agree that this K i a good idea, hut getting them actually to begin solving a problem hy drawing an appropriate sketch is quite another matter. Making an initial qualitative estimate of the answer to a problem (when this is possible) can also he very useful in creating a clear picture of the physical situation referred to in the prohlem as well as in designing a method for solving the problem, and in subsequently checking that the final nu-
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merical answer that one obtains is reasonable. We should train students to include this step in their initial consideration of the prohlem.
one should first write the name or symbol for the quantity whose value is to he determined, (5) when the letter "x" is used
Step Two: Determination of a method for solving the problem Step Four: Verification that the answer is reasonable
and (i)making, as a guide, an expiicit list of those calculational stem that one intends to use before actually carryingout any calculations or algebraic manipulations is extremely useful. It is valuable to illustrate that general strategies exist that can be used to arrive a t a method for solving a problem, and that many of these general strategies can he phrased in terms of certain key questions. Some of these key questions are
Students should be encouraged to he satisfied with their work on a uarticular wrohlem onlv after thev are convinced
( 1 ) W h a t physical principles or mathematical equations relate the quantities inuolved i n a particular problem? A crucial step in a student's progress in developing prohlem-solving skills is the realization that any mathematical relationship involving physical quantities (such as Density = MassIVolume) is a relationship among the quantities, rather than a "set-up" that can he used only to find the value of one particular quantity. (2) A m I sure that the physical principles and mathematical equations that I intend to use are appropriate for t h e particular conditions of t h e problem? A very common student error is merely to memorize equations, such as P I V l = P2V2, without associating the equation with a specific physical situation. Many students fail to recognize that in order t o use a particular mathematical relationship involving physical quantities, a specific set of physical conditions must apply. (3) How many unknown quantities do I h a w , and how many
I use the term "aualitative urohlem" to refer to a orohlem that requires an adequate explanation for aqnalitativ; answer, and for which there is insufficient information to obtain a
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relations do I know connecting the unknown quantities and t h e quantities uhose ualues are known? Students often try to salve a problem using insufficient information. This error can be avoided if one first analyzes the problem and ascertains if one has enough numerical values and equations to make the prohlem determinant. In addition, making this type of analysis quite often suggests a method for solving the problem. (4) Can I break t h e problem down into a series of simpler steps? As students are confronted with increasingly mare difficult problems, i t is essential that they acquire the ability to break
sometimes referred to as working a problem in "reverse." In this approach, instead of aiking oneself "here is theinformation I have, how can I use it to determine what I want", one turns this question around. Students should realize that far some problems one approach is most suitable, while a different approach is more appropriate fur other problems, and for some problems a combination of approaches is the easiest way t o arrive a t a method for solving the prohlem. (6) Are there any physically reasonable approrimations t h a t I can make? Students should become familiar with the fact that most meaningful physical problems cannot be solved with mathematical exactitude. They should also recognize that making reasonable approximations and checking that these approximations are valid, is quite an appropriate problemsolving approach.
We can bring out the utility of these and other general strategies by confronting students with appropriately chosen prohlems. Step Three: Algebraic manipulation and arithmetical operations
Some of the specific things that we can point out to students urohlems are that: (1) with reference to annronriate .. . . . it is usually easier to consider a mathematical relationship in general form first, rearrange it algebraically if necessary, and then substitute the appropriate numerical quantities into the general form, (2) it is essential to use units correctly, and when this is done, it can he a valuable aid in solving prohlems, (3) i t is necessary to label quantities precisely and sufficiently (e.g., using subscripts to distinguish two or more quantities of the same kind), (4) when doing a specific calculational step,
their final numerical answer are con&tent with each other. Specific Examples Qualitative problems
B is a more abstract form of brohlem A: P ~ o b l e mA
Problem B
At 2 5 T the density of diethyl ether is less than the density of ethyl alcohol. If 25.0 g of ethyl alcohol a t 25'C can completely fill a certain flask, can this flask contain 25.0 g of diethyl ether a t 2 5 T ? Give an adequate explanation for your answer. Consider two substances A and B. The density of substance A is larger than the densitv of substance B. A " sample of A and a sample of B have the same mass. Which sample has the larger volume? Give an adequate explanation for your answer.
the physical content of the problem, (2) encourages students to a~nreciatethe utilitv of an aonrooriate .. . sketch. and (3) can help students to recognize (and use) the important principle that any mathematical equation involving physical quantities expresses a relationship among those quantities. Many stndents, when confronted with the necessity of giving an adequate explanation for their answer to prohlem B, for example, realize that they can provide a concise and precise explanation by referring to the equation Density = MassIVolume. These students have thereby made a vital connection between a mathematical relationship and its physical content. One can reasonably expect these students also to apply what they have learned from considering such qualitative prohlems to problems reauirine a numerical answer. Another type of problem with many of the same pedagogical advantages as a qualitative prohlem: (1)illustrates a physical principle with reference to a concrete physical situation, and (2) reauires a minimum of auantitative work. Consider, for example, the following probiem that illustrates the prin&le of conservation of energy.
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Problem C
It takes 79 cal tomelt 1.0 g of ice a t O°C. This e n e r a can he supplied in a variety of forms in addition to a direct input of heat energy. An experiment was performed in which 1.0000 g of ice a t O.OO°C was held over a pail of water a t 0.00"C and dropped into the water. After the ice hit the water and the ice and water stopped moving, the mass of ice in the water was determined to he 0.9990 g. The temperature of the ice and water in the pail was still found to be O.0OoC. The temperature of the air was also O.OO0C,so no ice melted as it fell. (a) How much ice melted? How much energy was required to melt this amount of ice? (h) Where did the energy to melt the ice come from? (c) Using the principle of conservation of energy, calculate the kinetic energy of the ice a t the instant before it hit the surface of the water and determine the change in the potential energy of the ice falling from its initial position above the pail to a position just a t the surface of the water.
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Number 5
May 1983
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The detail given in problem C is deliberate. The detail makes the problem concrete and unambiguous and i t implicitly encourages the student to make a sketch of the physical situation referred to. The quantitative calculations that are required to obtain the answers to problem C are minimal, but a student must give the physicalsituation considerable thought. The format of the problem (i.e., the division into three oartsi also leads a student to consider the . nhvsical sit" uation carefully. As presented here, problem C is a useful exercise for students early in a preparatory course. If presented later in the course, when students have increased their abilities to think carefnllv about the nhvs~calcontent of a problem, parts (a) and (hjneed not bdexilicitly asked. In a similar manner. the format of other uroblems can be adiusted according to the progress that students have made in acquiring problem-solving skills. Sequence of Problems The set of prohlems discussed below illustrates how one can use a carefully designed sequence of problems involving a single relationship among physical quantities to help students increase their problem-solving skills and at the same time increase their understanding of that relationship. In this, or any comparable sequence of problems, there should he a gradual development along two lines, (1) from the physically simpler and concrete to the physically more complex and abstract, and (2) from the mathematically simpler to the mathematically more complex. The particular sequence of prohlems discussed helow utilizes the relationship among the heat flow into or out of a sample of a single pure substance, Q, the specific heat of the substance, C (assumed to he temperature independent), the mass of the sample, m , and the change in temperature, AT. The relationship among these quantities is Q = C X m X AT (1) The physical situation described by eqn. (1) is relatively simple and concrete, so that it is easy to discuss problems involving this eauation auite earlv in a nrenaratorv course. . . eqn. and the physiEal situation it refers to, are sufficientlv comnlex so that nrohlems involvine them can be used to il1;strati many problem-solviig strategies.
ow ever,
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Problem I
Two samples ofgold,labeled A and B, are both at 25°C. The mass of sample A is larger than the mass of sample B. Both samples absorb some heat energy and the temperature of both samples increases to 30°C. Which sample absorbed the larger amount of heat energy? G~ve an adequate explanation for your answer. ~ r o b l e m2
The specific heat of gold is 3.1 X 10F eal g-'OC-'. A sample of gold lost 18 cal of heat energy and the temperature of this gold sample changed from 37'C to 25'C. (a) Was the mass of the gold sample less than, equal to, or greater than 1.0 g? (b) Calculate the mass of the gold sample. Problems 1and 2 involve the relationship between heat flow and mass of the sample. I t is often quite useful to present students with such a pair of prohlems, dealing with the same asnect of a eeneral relationshin., that are related in the following manner: A qualitative problem (problem 11,followed by a problem with numerical data that is broken down into two parts-an initial qualitative part (problem 2, part (a)), and a part that requires anumerical answer (problem 2, part (b)).Many students when presented with a problem that requires a numerical answer merely look for an equation that they can plug numbers into without thinking about the uhvsical situation. Paired ~rohlems.such as vrohlems 1and can help students replace this inappropriate approach in favor of a stepwise approach in which they first carefully consider the problem in qualitative terms, and make a qualitative estimate of the answer, before doing anv numerical calculations. As students adopt aproper approach to problem
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solving and gain more confidence during a preparatory cours it should become unnecessary to explicitly include an initial qualitative part (like part (a) of problem 2) as part of the problem. Problem 3
The specific heat of aluminum is 0.21 cal g-lSC-l, and the specific heat of calcium is 0.15 cal c l Y - I . A 10.0 e samole of aluminum and
answer. This qualitative problem is quite a bit more sophisticated than Problem 1.It is sufficiently complex so that students can readily appreciate the value of an appropriate sketch. Problem 3 also requires students to think more deeply about equ. (11, and, in particular, to realize how the relationship between Q and A T depends on the product of the sample's mass and snecific heat. Problem 3 has another advantaee. This oroblem contains numerical values for the specific heat and mass for each sample, but there is not sufficient information to obtain a numerical value for the final temperature for either sample. A student must recognize this important aspect of the problem. Many students try to obtain a numerical answer to a uroblem usine an insufficient amount of numerical information, and/or relationships among the quantities involved in the problem. Prohlem 3, and others like it, can be useful in helping students stop using this undesirable problem-solving approach, and start by asking the questions: "Before I start plugging numbers into equations, do I have enough numerical values and eqtl?liw- s.3 lhni ihe pnlld
