The Role of Algorithms in Teaching Problem Solving

The Role ol Algorithms in Teaching ProЫem Solving Recently, students in my course were assigned the following question (J). А sample of indium bromide weighing 0.100 g reacts with silver nitrate, AgNO3, giving indium nitrate and 0.159 g of AgBr. Deter­ mine the empirical formula of indium bromide. In virtually every recitation section, the students asked the ТА to do this proЫem. Time and time again, the TA's told the students that the proЫem could Ье worked more or less like this Status as а proЬ!em is not an innate characteristic of а question, it is а suЬtle interaction between the question and the individual trying to answer the question. It is а reflection of experience with that type of question more than intellec­ tual ability.For example, I would expect that many chemists for whom the indium bromide question was an exercise would find the following question to Ье а proЫem, and vice versa. Design а synthesis for the following compound. Start Ьу converting grams of AgBr into moles of AgBr. Convert moles of AgBr into moles of Br, and then convert moles of Br into grams of Br. SuЬtract grams of Br from grams of indium bromide to give grams of In. Convert grams of In into moles of In. Then divide moles of Br Ьу moles of In to get the empirical formula of the compound. During the next staff meeting, I asked my TA's to stop lying to the students. I suggested that (1) the technique for solving this proЬlem that they had presented to their stu­ dents had little to do with the process that they had used to solve the proЬlem for the first time, (2) they had confused the process used to solve exercises with the process used to solve proЫems, and (3) the description given above was an algorithm for solving similar questions that they had con­ structed after they had solved this proЫem. Deflnltlon of an Algorlthm Algorithms have been defined as " ... rules for calculating something, especially Ьу machine" (2). They are rules that can Ье followed more or less automatically Ьу reasonaЬly intelligent systems, such as а computer. А common algo­ rithm among chemists is the factor-label approach to unit conversions, such as converting feet into inches or inches into centimeters. Other examples might include the fast Fourier transform, the road-map approach to stoichiometry questions, the rules some chemists use for automating the writing of Lewis structures, and the step-by-step process often used to predict the shape of а molecule from its Lewis structure. The Dlfference between Exerclses and ProЫems Hayes's (3) definition of а proЬlem reads, Whenever there is а gap between where you are now and where you want to Ье, and you don't know how to find а way to cross that gap, you have а proЫem. His definition provides а basis for distinguishing between two closely related concepts, exercises and proЫems. Ac­ cording to this definition, if you know what to do when you read а question, it is an exercise not а proЫem.The indium bromide question was а proЫem for every student in my general chemistry class, and for most of the teaching assis­ tants as well. It was an exercise, however, for several of my colleagues who knew exactly what to do wheq they read the question. Based in part on а paper presented at the 190th National Meeting of the American Chemical Society, Chicago, IL, September 1985. The Role of Algorithms in Working Exercises Algorithms are useful for solving routine questions or ex­ ercises. In fact, the existence of an algorithm constructed from prior experience (4) may Ье what turns а question into an exercise. Ву listening to us in lecture, Ьу reading exam­ ples in the text, and most importantly Ьу working similar questions on their own, many if not most of our students construct algorithms that turn the following question into an exercise when it subsequently appears on an exam (5). What is the empirical formula of а compound of xenon and oxy­ gen which is 67.2% Хе and 32.8% О? Unfortunately, algorithms are not sufficient for answering exam questions that are more likely to Ье proЬlems for our students, such as (5): 9.33 grams of copper metal was allowed to react with an excess of chlorine, and it was found that 14.6 grams of а compound of copper and chlorine \vere formed. What is the empirical formula of this compound? It's not surprising, therefore, that most students in а survey of beliefs about chemistry agreed with the statement: Good teachers shouldn't ask students to figure out proЬlems on an exam that they have not seen in class (б). The Role of Algorlthms in Worklng ProЫems Johnstone (7) has suggested that а common source of difficulty in science is the overload that occurs when the demand on working memory exceeds its capacity. One solu­ tion to this overload is to help the student build strategies that decrease а task's demand on working memory. John­ stone described these strategies as tricks or techniques for simplifying proЬlems and schemas for organizing prior knowledge. То some extent, these strategies are algorithms that automate individual steps in а proЫem. Students who have not built algorithms for at least some of the steps in а proЬlem, such as converting between grams and moles, will never solve the proЫem. There is more to working proЫems, however, than applying algorithms in the correct order. Before turning to the next section, try the following question (J). А sample of а compound of xenon and Пuorine was confined in а bulb with а pressure of 24 torr. Hydrogen was added to the bulb until the pressure was 96 torr. Passage of an electric spark through the mixture produced Хе and HF. After the HF was removed Ьу reaction with solid КОН, the final pressure of xenon and unreact­ ed hydrogeп iп the bulb was 48 torr. What is the empirical formu­ la of the хепоn Пuoride in the original sample? Now write down а summary of what you would tel1 your students if you decided to work this question in class. Stages or Steps ln ProЫem Solvlng Forty years ago, Polya (8) proposed four stages in proЫem solving: (1) understanding the proЫem, (2) devising а plan, (З) carrying out the plan, and (4) looking back. While work­ ing the question given above, you undoubtedly spent а con­ sideraЫe amount of time on the stage Polya described as "understanding the proЫem". What is less certain is wheth­ er you went through а separate stage in which you devised а plan to solve this proЫem before carrying out the plan. The steps that many people go through while working а "proЫem" such as this might Ье represented more or less as follows. You began Ьу reading the proЬlem, perhaps more than once. У ou then wrote down what you hoped was the key information, reread the question or а part of the question, drew а picture to help represent the question, and then tried something. Then you tried something else, and looked at where this led you. Ву gradually exploring or playing with the question you got closer and closer to the answer. lt's possiЫe that you never fully "understood" the question un­ til you had an answer. If you compare this reconstruction of the steps many of us take while solving а proЫem with the description we all too often give our students of how we solved the proЫem, you may understand why I asked my TA's to stop lying to the students, and you may also understand the role of algo­ rithms in solving exercises versus proЫems. Llterature Clted 1. Holtzclaw, Н. F.; RoЬinson, W. R.; NeЬergall, W. Н. General Chemistry, 7th ed.; Heath: Lexi11 gton, 1984. 2. Ehrlich, Е.; Flexner, S. В.; Carruth, G.; Hawkins, J. М. Oxford American Dictionaгy; 3. 4. 5. 6. Oxford University: Oxford, 1980. Hayes, J. The Comp/ete РrоЬ/ет Soluer; Franklin Lnstitute: Philadelphia, 1980. Bodner, G. М. J. Chem. Educ. 1986, 63,873. Carter, С. S.; LaRussa, М. А.; Bodner, G. М. J. Res. Sci. Teach. 1987, 24. Carter, С. S.; Bodner, G. М. Abstracts of Papers, 192nd National Meeting of the American Chemical Society, Anaheim, СА; American Chemical Society: Washin gton, DC, September 1986; CHED 5. 7. Johnstone, А. Н.; EI-Banna, Н. Educ. Chem. 1986, Мау, 80. 8. Pol ya, G. How to Solve lt: А NeuJ Aspect о{ Mathematical Method; Princeton Universi­ ty: Princeton, NJ, 1945.