The Solution Is Not the Problem Julien Oendell Oakland University. Rochester. MI 48036 We solve some problems because we have a practicalinterest in the answer. However, a much more important reason for working problems in an introductory chemistry class is that this activity is a most valuable way for students to increase their understanding of scientific concepts. A typical student obtains an initial vague understanding of a scientific concept from atext or a lecture. No matter how good the text or the lecture may be, the student's understanding remains vagueuntil he or she uses the concept in laboratory work and to solve written nroblems. For this uuruose. uroblems that demand for theirsolution that the &dent consider carefully the uhvsical situation to which the uroblem refers are a . necessity. Students should have to contend with a sufficient number of problems for which they do not immediately "know what to do". In addition we must help students to see that uroblem solving involves more than merely using algorithms and plugging numbers into equations. The five-stage problem-solving approach I use includes the following tasks for the student.' Stage 1: Use the information given in the problem to create a clear picture of the physical situation to which the problem refers and describe for vourself that situation in aualitative terms. Stape 2: Consider the physical principles or mathematical equations that relate the quantities involved in the problem. Stage 3: Devise a series of calculational steps that will enable you to determine what you want to know from the information that is given and the relationships among the quantities involved. (Algorithms are often of great value in obtaining answers to various parts of the problem.) Stage 4: Carry out the appropriate calculations. Stage 5: Verify that the answer or answers obtained in stage> are reasonable. In my experience, the hardest step for students is to give Stage 1 adequate consideration. It seems that they try t o avoid this activity a t all costs. T o help students increase their ability to restructure given information into a form that is meaningful and useful, I assign questions that delibPresented at the 190th National Meeting of the American Chemical Society. Symposium on the Use of Algorithms in Problem Solving, Chicago. IL. September 9, 1985. ' Genyea, J. J. Chem Educ. 1983, 60,478.
erately contain a great deal of information. Question 1below is one example. Question I : A, B, and Care labels for three different suhstanees. In anexperiment, 25.0 gof A went from the gas to the liquid state. All of the energy released in this process was used to convert a mixture of some B and 15.0 g of C from the liquid to the gas state. Determine the mass of B in this mixture. The heats of vaporization ofsubstancesA, B, and C are 0.175 kJIg, 0.130 kJ/g, and0.210 kJIg, respectively.
I have also found it extremelv useful to uresent students with a number of "qualitative problems'; dealing with a uarticular scientific conceut. A "aualitative urohlem" is a pnhlem that requires an adequnte explanation for a qualitat;veathwer and fa,r which there is insufficient information to obtain a quantitative answer. The absence of sufficient numerical data forces the student to think about the physical situation to which the problem refers. For examp& ~ u e s tion 2 below is a qualitative problem that is designed to help students increase their understanding of the concept of specific heat and the relationship between the heat flow into or out of a sample of a pure substance (Q), the specific heat of the substance (C), the mass of the sample (m), and the temperature change (AT), Q = C X m X AT. Question2 Consider twosubstanceslaheled Aand B. Thespecific heat of A is larger than the specific heat of B. A 25-g sample of A initially at 27 OC, and a separate 25-g sample of B initially at 27 O C , each absorb 48 J of heat. Which sample has the higher final temperature? Give an adequate explanation for your answer. It is hoped that a student whu is c~mfrontedwith Question 2 will realize that if a substance has a large specif'ir heat then it is relatively difficult to increase or decrease the temperature of a sample of the substance, and the student begins to appreciate specific heat as a measure of what might-be referred to as the "relative thermal inertia" of the substance. Similarly, I believe that the discussion of every significant equation should include a series of qualitative questions that are desiened to deeuen the student's understanding" of the connection between the mathematical equation and the specific ohvsical situation to which the eauation audies. other technique that I have found usefuiis to discuss
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several examples of incorrect approaches to problem solving, in particular, the dangers of merely plugging numbers into an equation without considering whether the equation is applicable. Quite often students get stuck in their attempts to solve a problem because they have overlooked a critical relationship that cannot be recalled from memory but applies to the specific problem under consideration. We should make students aware of how often this occurs, and encourage them to think more carefully about the physical situation to which a problem refers, with the specific intention of exposing a key relationship that they have previously overlooked, if they do reach an impass in trying to solve a particular problem.
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Often when I discuss the use of a particular algorithm, I also ask my students several questions to which the algorithm does not apply. Thus, they learn to be selective in their use of a particular algorithm. The factor-label method is, for examnle.. a useful aleorithm if one knows whv and how it u works. However, blindly applying proportionality factors in situations that do not involve directly proportional quantities will of course yield perplexing answers. Whether a particular question is an algorithmic or nonalgorithmic problem depends not only on the question itself, but also on the student's previous familiarity with the material, the training the student has had with similar questions, and on the form in which the question is asked.
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