Teaching Students To Use Algorithms for Solving Generic and Harder Problems in General Chemistry Elizabeth Kean, Catherine Hurt Middlecamp, and D. L. Scott1 Chemistry Tutorial Program, University of Wisconsin, Madison, W1 53706
What makes one student a eood nroblem solver and another a poor problem solver? w h a t ;an a teacher do to help the . Door ~ r o b l e msolver im~rove?In our 14 vears of tutorial experience with general chemistry students; we have found that good problem solvers:
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(1) are able to recognize basic classes of generic problems (stan-
dard problems), (2) build strong, systematic procedures (algorithms) for solving
generic problems, and (3) can extend algorithmsto solve harder (nongeneric)problems.
In this paper, we shall describe teaching strategies that help students to improve these skills. Generlc Problems and Thelr Algorithms Generic ~ r o b l e m are s the "no frills". s t r i ~ ~ down. e d standard probiems that students are expected-to master.^ generic problem may he solved by use of an alaorithm, a series of steps that are performed, in sequence, t~accomplishthe goal of the problem ( I , 2 ) .Algorithms and their use in solving chemistry problems were the focus of a symposium a t the September 1985 ACS meeting. The reader is directed to papers from that symposium that recently appeared in this Journal (3). Here is an example of a generic problem: Problem I: How many kilometers is equivalent to 16 miles?
This is an example of a conversion factor problem. All members of this class of problem have the form: Given n mpnaurrmrnt of a quantity, transform the description
inu8an ait~rnntive,but erluivalrnr, description of the quantity.
When students confront a conversion factor problem, they need not create a unique process to solve it. Rather, they are expected touse a standard process, an algorithm, tosolve the problem. One algorithm for solving conversion factor problems, the factor label method or dimensionalanalysis, will be described later in this paper. A nongeneric (i.e., harder) problem can be illustrated by the following conversion factor problem:
fJroblem2. The hear of combusrion of 1 mol of acetylene is -1300 kd mol. HowmanygramsofCO~would br produced byo combustion reaction that evolved 67i k.l?
T o solve Problem 2. the student must write the correct equation for the combustion of acetylene, form a conversion factor linkine amount of heat and amount of Con, and then apply the factor label algorithm correctly. The Ho~utiouto Problem 2 thus requires more than a straightforward use of the appropriate algorithm. Typically, more difficult problems require that students combine algorithms, alter a step within the algorithm, or extend the algorithm to a new context. In learning to solve generic and harder chemistry problems, students begin by mastering the basic algorithm. Students work with the aleorithmuntil thevcanuse it with ease. Thereafter, use of t& algorithm reqiires little attention from workinz memorv (the Dart of one's mind that is activelv engaged during thinking). When solving harder problems, students are thus able to concentrate on how to modify the process to account for variations on the generic problem (4). Developing Useful Algorlthms
A useful algorithm is one that allows students to solve problems with meaning, rather than by rote. Such an algorithm will contain sufficiently complete and correctly sequenced steps to allow solution of problems with an expected range of difficulty. For example, the factor label or dimensional analysis algorithm is sometimes given in texts as a one-step process: m u l t i ~ l vthe known cluantitv and unit(s) bv one or more convemion factors to obtain ah answer in the Iequired units. For some students, this algorithm is insufficient, e.a., it does not contain directions fo; how they will discriminate the known quantity from other quantities in the problem. Below, we illustrate a more robust algorithm for conversion
' Formerly at the Australian National Unlversiiy, Canberra, Austra-
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factor Drobhns and comment briefly on each of the steps. The raiionale for each step is import& in suggesting teaching strategies to anticipate and overcome errors, as illuitrnted for the use of the factor label algorithm to solve Problem 3:
Step 6: Plot the solution path.
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METRIC VOLUME
llETRlC MASS
Ploblern 3: How many milligrams would 15 mL of methanol weigh?
(0.79 g methanol = 1.0 mL methanol) Algorithm: The Factor Label Algorithm for Conversion Factor Problems 1. Identify the problem situation. 2. Identify the stated and wanted descriptions. 3. Equate the stated and wanted descriptions. 4. Analyze the descriptions.
5. Find conversion factors in the problem. 6. Plot the solution. 7. Multiply the stated description by conversion factors. 8. Compute the numerical answer. Step 1: Identify the problem situation. This prohlem is about a quantity of methanol. Students need to form an image (on paper or in their minds) of the situation described in the problem. Step 2: Identify the stated and wanted descriptions. STATED
WANTED
Conversion factors are now laced in the solution ~ a t and h the partial solution evaluat2 to identify the transformations that must still be accomplished. In this example, the student must supply from memory the gram-to-milligram conversion factor to end with the mass unit required by the problem. Step 7: Multiply the stated description by the required conversion factors. Errors in writing conversion factors upside-down are selfcorrecting. Step 8: Compute the final answer. Students must compute accurately and check for correct units and number of significant figures. Finally, they must decide whether the maenitude of the answer makes sense. An elaborate algoritlhm such as this is appropriate for a mouD of naive students. As students become more accomplished problem solvers, shortcuts and omissions may he iustified. In all cases, the algorithm should be taught so as to maximize meaningful, rather than rote, use b y t h e target group of students. . T o develop complete algorithms, the instructor needs to get "inside thestudents'heads",identifying all thestepsand decisions thev must make in oroblem solvine. This mav be surprisingly difficult, because the instructor's long familiaritv with the Drocess tends to obscure manv of the comoonent steps. Two strategies are useful in helping identify hidden algorithm steps. First, ask students to solve some generic problems out loud. Note where they get stuck. Question students about their difficulties until you uncover the missing operation and decision steps. Alternatively, write a flowchart for a generic problem as if you were writing a computer program for teaching that problem. Computers are so dumb that you have to tell them evewthine. . -. and in the Droner . . order. Such an exercise may uncover assumptions you have made about the prohlem solving steps needed by students. ~
Students must read and interpret the problem statement (the auestion) for this int'ormatiun. Hoaaard's semantic al-. gorithm (5)emphasizes this skill. Step 3: Equate the wanted and stated descriptions. ?mg=15mL Thisstep is not trivial; it states the essence of the problem: The two descri~tionsmust be alternate but eauivalent descriptions of thk same quantity. The form is important because the student will later m u l t i ~ l vthe stated descri~tion by some conversion factor(s) to effek the required transformation. Without this step, students may start the problem solution by writing a conversion factor (with a 50-50 chance of writing it in the correct fractional form). Step 4: Analyze the stated and wanted descriptions for properties and systems of measurement. 15mL ?mg = metric mass metric volume This key step abstracts the nature of the conversion, i.e., what properties are to be transformed. This analysis forms the basis for planning the conversion path. A prerequisite skill is the ability of students to identify which property is described by each unit. For instance, milliliters YmL") must be recognized as aunit that measures the property of volume in the metric system. Step 5: Find and analyze conversion factors in the prohlem. 0.79 g methanol = 1.0 mL methanol metric mass metric volume Students shouldanalyze each unit in the conversion factor for both properties and system of measurment (e.g., the density conversion factor above can transform a metric volume to a metric mass and vice versa). 988
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Teachlng an Algorlthrn For a group of students with well-developed problemsolving skills, exposure to a single, worked-through generic prohlem during a lecture may be sufficient to communicate the essentials of the algorithm. The algorithm itself may be implied, rather than explicitly stated. The instructor may describe what is done, but may omit why each step is performed or how decisions are made during completion of steps. Students with strong problem-solving skills are able to extract and generalize the process from the one example. They then complete mastery of the process by working through additional ~roblems. unFortunately, &my general chemistry students are not good problem solvers. To develop and master algorithms, they will need explicit instruction. The instruction used to communicate an algorithm to this group must not present it as a mindless, arbitrary series of steps to be memorized. Rather, students need to be taught the rationale and basis for each s t e n as well as the context in which the aleorithm will be used: An algorithm is useless unless students know when it
should and should not be used. As the instructor teaches the aleorithm. s h e must identifv clearlv the initial and eoal &es of the problem and t h i p r e r e q k i t e knowledge i;ase 161 Srudents must then be taueht the ~ u e hthat identify the class of problem and signal the-algorithm's use. For example. in a conversion factor problem, a auantitv is desckbed h i it's name, magnitude, and units. students m i s t transform the aiven description to obtain an alternative description, without changing the quantity itself. The transformation may be partial, involving a change of unit only, as in Problem 1 above. Or, the modification may be complete, as in Problem 3, involving changes in name, magnitude, and units. To be a conversion factor problem, the solution must reauire use of a t least one nairwise eauivalent representations df a specific quantity, i.d., a conveision f a c t o r . conversion factor is based on directly proportional representations of .. . the same quantity. In Problem 3 above, t h e equivalency needed is 1.0 mL methanol = 0.79 e methanol In other words, the amount of methanol that has a volume of 1.0 ml. is the same amount of methanol that weighs 0.79 g. Studentsrarely seem confused by the use of the equal sign in the equivalency statement, hut are readily taught ro read it as "describes the same amount". Students need also to recognize when the conversion factor algorithm is not appropriate. For example, a problem might identify three properties of a gas sample (e.g., P, n,T ) and ask for the value of a fourth . DroDertv . - (V). . . Such a nroblem would not use the conversion factor algorithm, hut a mathematical formula alaorithm, since the formula P V = nRT relates these four va;iables. Prerequisite concepts may be necessary for both correct use of an algorithm and for recognizing when the algorithm should be used. For example, the concept of "conversion factor" is essential to the students' understanding of the factor label algorithm. Teaching students to recognize conversion factors in alternate forms (e.g., concentrations, densities) is also a critical step in the recognition of when the factor label algorithm should be used. Given the above framework, a possible instructional sequence for teaching an algorithm follows: (1) Confront students with multiple examples af the generic problem; discuss the similarities and differences of these problems. (2) Present the algorithm in written outline form to give students an overview of the solution process. (3) Work through the algorithm using a simple example; discuss how to ~erformeach steo and its rationale. (1) Revrew, as nrreswry, the prerequisite information and skills needed fur each step. (5) Girt students practice in using the algorithm to solve two or three problems without assistance. (6) Discuss contexts in which the algorithm can and cannot he used, usina examples when appropriate. This sequence directly attacks the question of when to use the algorithm. I t encourages students to see the similarities among apparently dissimilar problems, facilitating the application of the algorithm to novel situations. By attending to prerequisite concepts and processes, it links the algorithm to knowledge the student already has. By providing a rationale for indkidual sreps, the entire process becomes more meaningful, reducing the number of examples students must work in-order to master the aleorithm. Providina a written ~-~~~~ algorithm allows the student toreview the steps .neriodically. Finallv. bv reauirine students to work throuah several problems &&t assistance, i t acknowledges the difference between understandine a nrocess when someone else (the instructor) does it andinternalizing the process so as to be able to do i t oneself. Students become proficient by practicing the process, not by attending to thelist of steps: ~
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Teachlng Students To Form Algorlthms The instructional sequence ahove leads to mastery of an algorithm when the algorithm is presented in complete form hv the instructor. An additional instructionaleoal is to teach students to form useful algorithms indep&dently, from shorter and less complete presentations. Creative ~ r o b l e m solving often depends on the recognition of patterns from limited numbers of ~ r o b l e m sor data. Forming alaorithms solutions gives students practice in from modeled searching for patterns. Students tend to make up algorithms implicitly, whenever confronted by related problems. Good problem solvers appear to make up functional algorithms spontaneously. Moreover, they need little help in mastering and extending the aleorithm. Weaker students likewise make uv aleorithms. but theirs often are "buggy", that is, contain flaws ;hat limit their usefulness (7). Thus, instruction in formina alaorithms focuses on making the piocess explicit and also in trouble shooting the resultant algorithm. T o shift responsibility to students for forming their own algorithms, gradually give them less information about algorithms for generic prohlems. Solve a standard problem and then select some of the following types of activities: Ask students to write a set of directions for solving the problem that could be used for any problem of that type; let students share their algorithms with each other. Given a set of problems of various sorts, have students select the ones that use the current algorithm for the solution. Ask students to identify the starting information that is needed for the algorithm and the information that is obtained or calculated at the conclusion of the problem. Have students make up an algorithm to teach another student. Use the algorithm in working assigned problems. Have students critique each other's algorithms. Note especially when algorithms fail, indicating that there is amissing or incorrect step. As you teach students to form algorithms independently, tell them that this is the goal of the instruction. How rapidly students develop this skill depends on the maturity and background of the students and the instructor's ability to select clear examples and challenging exercises.
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Extending Algorlthms Algorithms, as conceptualized in this paper, are directly applicable only to standard problems. Meaningful knowledge of an algorithm prepares the student to adapt it to solve nonstandard problems. Again we find that the abilities of students fall on a continuum, from those who can automatically adapt the algorithm tomore difficult problems to those who can use an algorithm only as presented. Proficient prohlem solvers mav reach the desired comnetencv level bv completing on thei; own practice problems'of incieased difficultv. Weaker students will reauire more direct teachine. ". such as in the following possible sequence of activities: i presentation and mastery of the basic algorithm confrontation with oroblems that have a sinele differencefrom the standard oroblk inrtruruu explication of the adaptations needed student-generated adaptationr on a new ret uf problems confrontation with prlrhlems that have multiple differenre8 from the standard problem instructorlstudent-generatedadaptations These activities can be continued until the class reaches the desired level of proficiency. To illustrate how extensions of algorithms could be taught, we present three common extensions for the factor label algorithm.
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Finding "Hidden" Conversion Factors Conversion factors may be hidden within prose statements, as in the following problem: Volume 65 Number 11 November 1988
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Problem4: The maximum amount of NaCl that can he dissolved in 100 mL of water at 0.00 'C is 35.7 g. What volume of saturated NaCl solution at this temperature would be obtained from the complete dissolution of 72.3 g of this salt? The equality, 100 mL water = 35.7 g NaC1, is contained in the first sentence. Many students need practice in writing verbal equivalencies first in equation form and then in conversion factor form. In practicing this skill, students need not solve the entire problem. Also give students practice in translating conversion factors into words.
ing students how to cope when algorithms are not enough. Such instruction might take the form of generating a set of "coping questions" that students can use whenever they become stuck.z Heuristics are hest taught not in isolation, but as part of the natural extension of algorithms to nonstandard problems.
Conclusions In developing teaching recommendations for algorithms, we have drawn upon the educational literature and upon our own 14 years of experience with nontraditional and educationally disadvantaged students enrolled in general cbemisidentification of Situation/Propertles/Units try courses. Evidence for effectiveness of our methods is shown in the statistics for student success in general chemisProblem 5: What is the mass, in grams, of one chlorine atom? try courses. When the Chemistry Tutorial Program at the The context for this problem, a single atom of chlorine, University of Wisconsin-Madison was started in 1974, educannot be experienced directly by students. You may need to cationally disadvantaged students were failing chemistry assist students to visualize the problem situation by asking courses at a rate of 85%. Since that time, students in our questions such as "What do you imagine a chlorine atom Program have had a D, F, and drop rate that averages 15% would look like?" Students may also have difficulty in idenless than that of the general student population. Our Protifying the properties and units for quantities (as, for examgram does more than teach problem solving; much of our ple, the property of number or the property measured by the work involves teachina conceotual knowledee and processes mole). of learning. Still, since the most advancedbentai skill required in our courses is problem solving, our students would identifying the Given Description not be succeeding if we did not have an effective program of Problem 6: The maximum allowable concentration of carbon monteaching problem solving, oxide in urban air is 10mg/m3over an 8-h period. At this In our work, we acknowledge that students are active Level, how many grams of carhon monoxide are present constructors of their own knowledge (12). The instruction in a room that is 2.5 X 15 x 40 m in dimensions (a)? that we have sueeested activelv builds on students' nrior Students may have difficulty discriminating the stated or when &dents are led to discover knowledge. wanted descriptions from irrelevant information or from parts of alaoritbms (and their extensions) themselves.. thev conversion factors. Discuss with students oroblems containieorganizetheir knowledge of the material in a more powering extraneous information. ~ s them k to&d some contextuful way (13). When selecting - ~.r o b l e m sthat will be used in al information to a standard conversion factor problem. problem-solving instruction, instructors must consider careStudents may also' find i t difficult to disciiminate the fully the ultimate level of problem solving that is to be conversion factor, 10 mgIm3, from the stated description attained. The goal is to challenge students with problems that is to be transformed by the problem. Conversion factors that stretch their abilities to solve problems, without overare characteristics that remain constant. no matter what whelming them. magnitude of quantity is described in thd problem. In this We believe that instructors have the responsibility to problem, the allowable pollution level is the same. no matter teach problem solving whenever students enter their classes what size room might b e considered. 1 n t e n s i ~ e ' ~ r o ~ e r t i e s without the required proficiency. The teaching techniques (e.g., density and concentrations) may need to be defined as in this paper were derived from the questions students comconversion factors. If students have difficulty with this dismonly ask and the errors they often make as they struggle crimination, give students more practice in identifying conwith general chemistry problems. There are many ways stuversion factors, when embedded within complex problems. dents can go wrong in solving typical chemistry problems. Although it is useful for students to discover some of these Teachlng General Problem-Solvlng Strategles (Heuri4lcs) for themselves, i t is also possible t o teach students directly We teach problems in general chemistry because we want how to avoid some common pitfalls. T o do any less is to students to learn some basic processes useful to chemists. penalize students whose initial problem-solving skills are We may also choose t o use chemistry problems to teach weak, by not providing them with adequate opportunities to general problem-solving strategies, i.e., heuristics. For eximprove those skills. ample, the first steps in the factor label algorithm lead to a redescription of the problem in abstract terms. Such a redescription is an essential part of most chemical problem solv1. Kean. E.; Middlecamp, C. The Success Monuol /or Geneml Chemiafv; Random ing (9). Heuristics will be especially needed when the conHouse: New York. 1986. nection between known algo&thmi and a specific problem 2. Frederiksen, N. ~ s uEdue. : Rea. 1981.64.363. are obscure. T o solve more difficult problems, students may 3. Sympmivm on Algorithms and Problem Solving.J. Chem. Educ. 1967,64,513. 4. Schneider,W.:Shiffrin,R.M.Psyrhoi.Rou. 1917,84,1. need to reoreanize the ~ r o b l e minto a series of "submob5. Hoggsrd, F.J. Chem. Educ. 1987.64. 49. which contributes partially to the soiution lems", each 6. Rubenstein. M. Toola /or Thinking and h b k m Soiaing; Prentice-Hall: Englewiwd Cliffs, NJ, 1986:p 7. (10, 11). A "working backwards" strategy, focusing first on 7. Bmwn,J. 8.;Burton, R. R.Cognitive Sei. 1978.2.71. the final form of the answer, may also be effective. 8. Brown, T. L.: LeMay, H., Jr. Chamisfry The CenfmlSeienee. 2nded.;Prentice-Hall: Englewwd Cliffs, NJ, 1981; p 21. Effective problem-solving instruction must include teach9. Reif, F. J. Chem. Educ. 1983,60,948. 10. Newell, A : Simon, H. A. Humon Information Ploreaaing; Prenfiee-Hall: En&wood Cliffs.NJ. 1972. 11. Lsrkin, J. In Problem Solving and Educotian: Issues in Teaching and Remarch; Tuma, D.T.:Reif, F.. Eds.: Erlbaum: Hilldale, NJ, 1980:Chapter 8. 12. Bodner, G. J. Chsm. Educ 1986.63.873, See p 253 in ref 1for a possible list of such questions. 13. Wan, D. E.; Greeno, J. G. J. due. ~ s y e h o l1. 9 8 3 . ~ .85.
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