Symposium on Algorithms and Problem Solving
Using Algorithms To Teach Problem Solving C. L. Schrader Dover High School. Dover, OH 44622 An algorithm is a mechanical computational procedure that provides a simplified set of directions for solving a complex problem. The purpose of using an algorithm is to increase the reliability, accuracy, and efficiency of obtaining answers. If one can follow directions carefully, one can use complex algorithms susccessfully. Problem solving has been described as "what one does when one doesn't know what to do". Problem solving must make use of the highest cognitive processes of analysis, synthesis, and evaluation. Often the problem posed can be recognized as a certain type, and the method for solving that type of problem can he recalled and used to produce a solution. This procedure is not problem solving; it is doing an exercise. I)oiny vxrrrises requirts analysis, recall, and applicatim ofthr rechniquesor skills required. Problem solving requires all of these plus synthesis, evaluation, and creativity. Technlclans vs. Problem Solvers Algorithms are used to simplify the process of obtaining answers. T o use an algorithm, one needs only to follow directions. For example, an algorithm may instruct a technician to titrate 5.00 mL of a sample to blue-black endpoint and then multiply the number of milliliters of the standard titrant used bv 7.25. The answer is then reuorted as the number of k i ~ ~ , ~ r i m a o f t ~ i c ksulfate r l ~ l I ,per l;trr.'l'he technician rnrrly understnnds the origins nnd hasis of the algorithm. If the operation is a rirrntinn, the trchnician would pn~bablv nor knl,a, how to calculnte the number that is mulriplied by the exeprimental answer to get the data in the desired units. The algorithm is usually devised by the head chemist and just used by the technician. One of the most powerful, versatile, and widely taught algorithms in chemistry is the factor-label method. This method is taught in most introductory chemistry courses. Unfortunately, students often learn to "get the units to cancel" without understanding the logic behind the algorithm.' If the students are then faced with problem-solving tasks, they often apply the algorithm inappropriately and arrive a t incorrect answers that thev are unable to check.2 The use of algorithms may he appropriate for technicians, but problem solvers can do more. They can construct and utilize their 0a.n algorithms. T o do 50, they must understnnd what algw rithmi are and how ru use them. Algorithmi hare their place in science teaching. (1)
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Algorithms are efficient procedures that provide a simplified set of directions for solving complex problems reliably and efficiently. We need algorithms to avoid confusion when we are faced with complex tasks. If algorithms are used sensibly, they must be connected to the chemical phenomenon to which they apply. In order to give correct answers, any algorithm mu& he based on sound science andlor math principles. In teaching algorithms, care must he taken so that students focus on the principle from which the algorithm is derived rather than the algorithm itself. Whenalgorithms areapplied in anewcontext, thevalidity of the procedure must he checked against our knowledge of general principles to he sure that the algorithm is valid and that logical or computational errors have not entered. Journal of Chemical Education
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Algorithms are taught as an important part of problem solving, hut application of an algorithm alone seldom solves prohlems.
Students must he taught general procedures that are important in solving problems. In addition to being taught to check results against principles that are assumed to hold, they are shown that careful organization of work is important when we tackle complex problems such as those encountered in chemistry. Without such organization, we are often unable to recall interim results that we have calculated, and we find it difficult to check one value against another. lnventlng an Algorlthm Limiting-reactant problems are usually quite difficult for most students.3 Determining which reactant is in excess and which is the limitine reactant reouires the annlication of .. chemical principles and ability to use assumptions (i.e., the chemical system under consideration is assumed to be one in which the equilibrium constant is large, so the reaction goes to completion). The laboratory provides a valuable key to this process. Students are required to develop a method for calculating which reactant is in excess, compare their calculated values with the experimental results, and explain the implication of these results. (For example, zinc is reacted with 1.50 M HCI, and if the student predicts that zinc is in excess, some zinc must remain the following day.) Several chemical systems are investigated to provide ample reinforcement of this concepL4 Students are then assigned an extensive work sheet containing a dozen reactions with the amount of each reactant provi$ed. From this they are asked to calculate which is the limitine reactant. the amount of each uroduct formed. and the amount of excess for two of the equations each day: The first few davs. the results are checked. nut on t h e chalkboard, and discussed, and subsequent eq;ations are graded. The calculations can he time consumina and comulicated, hut the goal is to provide ample practice so that the skills and processes involved become routine. In all this, however, the majority of the students carry out individual calculations for each reactant, determinina the amount of product that could be formed and comparing results. Although correct, this lengthy procedure is not often used by experts. Students are subsequently informed t h a t a much simpler computational method can he used to obtain these results, and they are asked to try to devise an algorithm that can he used in this way. While the concent of aleorithms has been ureviouslv described, discussed, anrl used in a mathematical and a chemical contcxt, onlv about lor( of mv studrnts are successfi~lin , an algorithm is preinventing an algorithm. ~ o w e v e ionce
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' Snider, Richard C. Sci. Teach. 1985, 5.34), 24-28.
Grenbowe. Thomas. PhD Dissertation, Purdue University, Dec.
1983.
Kalantar. A. H. J. Chem. Educ. 1985, 62,106. Herron, J. D.;Greenbow. T. J. J. Chem. Educ. 1986,63,528-531.
sented or invented by another student, most of the rest can understand the logic of the algorithm and many can see how i t was, in fact, devised. Thus, successful algorithms are shared among the students. The use of algorithms is not in itself significant. We should try to teach students so that they not only know how an algorithm is used but also why the algorithm works. I t is of
greater importance to provide the students with opportunities and challenges to create algorithms, for this will enhance their problem-solving skills. The author would like to acknowledge the many valuable suggestions and comments of J. Dudley Herron.
Symposium on Algorithms and Problem Solving
Volume 64
Number 6
June 1967
519
