Student Conceptions and Competence Concerning QuantitativeRelationships between Variables Mailoo Selvaratnam and Sudath Kurnarasinghe University of Bophuthatswana, Southern Africa A large part of chemistry, and also of many other sciences, consists of the study of the relationships between variables, that is how one variable changes when another is changed. These relationships may be expressed in three ways: statements, equations, and graphs. For the effective learning of many areas of chemistry, one must be competent in the concepts and the skills required for dealing with relationships between variables. The main aim of this study was to find out how competent chemistry students in some of the Southern African universities are in these concepts and skills. One hundred nine students were tested. Of them. 65 were second-year chemistry students from two of the newer universities in the "homelands" (Group A-almost exclusively black students), 19 were second and third years from two well-established universities (Group B-almost exclusively white students), and 25 %,erepostgraduate teacher diploma students at a well-established university (Group C-a mixtureof black and whitestudents,. The diploma students had offered chemistry as a subject for their first degrees. Previous studies in this field have lareelv- been restricted to just one aspect: proportional reasoning and how proportional reasoning ability affects problem solving (1-5). The present study is broader. We investigated student conceptions of direct and inverse proportionality and proportionality constants andalso their competence in the skills required for interconverting the information provided by equations, statements, and graphs. Furthermore, we investigated their ahilitv to recognize the necessitv for derivina certain equawhen certain types of tions;and for drawing linear calculations have to be done. The study method adopted was the analysis of student answers to carefully designed test irems. Since the objective was mainly to test skills, any knowledge of subject content required for solving a test item was provided in the item itself. This ensures that anv failure was not due to lack of knowledge of subject conteht. Furthermore, the test items were desiened from areas unfamiliar to the student to exelude the possibility of a correct answer being due to mere recall of existing knowledge.
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The test items used and the results obtained are indicated below. In the actual test the items were placed randomly in order to prevent their comparison and correlation with each other. In the discussion below, however. related items have been grouped together to aid the analysis. Test Items and Results I t is our experience that many students have only a partial and qualitative understandingbf the concepts of direct and inverse proportion. Misconception exists that two variables x and y-are directly proportional if y increases when x increases and that they are inversely proportional if y decreases when x increases. Questions I and 2 were designed to find out the extent of this misconception. Question 1 Some experimental results of the variation of a physical quantity Y with another physical quantity X are given below:
Is Y directly proportional to X? Indicate briefly your reasoning Question 2 Some experimental results of the variation of a physical quantity Ywith another physical quantity X are given below:
Is Y inversely proportional to X? Indicate briefly your reasoning. Over 80% of the students in Groups A and C and about 50%in Group B thought, incorrectly, that the data in questions 1 and 2 indicate direct and inverse proportionality, respectively. Of these students, about a third indicated that Y was directly proportional to X (Question 1) because when X increased by 1, Y increased by the same amount. Another third thought that since Y increased with increasing X, they were directlv orooortional. Analvsis of the answers to auestion 2 gave similar results. earl; 40% of the erring students indicated that since Y decreased with increasing X, they u
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were inversely proportional. About 25% reasoned that the inverse proportionality was indicated by the fact tbat Y decreased by two when X was increased by 2. I t is also our experience that manv students tend to assume intuitively, dithout there beingany justification for it, that two auantities have to be either directlv or inverselv porpotional to one another. Questions 3,4, a i d 5, which ail deal with intensive pro~ertiesand which hence do not depend on amount of sample, were designed to test this aspect.
(c) for a fined mass of gas at constant temperature, pV = canstant. Rewrite all the incorrect statements in the correct form.
Over 50% of the students in Groups A and C had difficulty with all three parts of this question. Group B students fared better, but they too had difficulty with part (c). The understanding of the information provided by equations is vitally important for the meaningful study of the quantitative aspects of science and the inability of such a large proportion of university students to extract the information contained Question 3 this relatively simple equation is indeed very disturbing. in Consider 10 g of a sample of a solid that has a density 8.8 g ~ m - ~ . Question 7 also probes another aspect of the phrase directWhat would be the density of 20 g of the solid under the same ly proportional: it tests one's ability to convert a statement conditions of temperature and pressure? (Note: density is the mass involving this phrase into a n equation. Often the relationper unit volume.) s h i between ~ variables is expressed as a statement and its Question 4 conversion into an equation is necessary before a calculation A closed vessel contains a mixture of two gases A and B at a total can be done. This abilitv is hence an important skill. This pressure of 2 atm. The mole fraction rA of A in this mixture is then question also examines bne's understanding of the signifi0.20. Whatwonld be themole fraction of A if the pressureafthis gas cance of a proportionality constant. ~ ~ ~nA~is, the mixture is increased to 4 atm? (Note: X A = n ~ l n ,where moles of A and ntotsiis the total moles present in the mixture.) Question 7 The absorbance of a solute (in a solution) is found to be directlv Question 5 proportional to the mass of solhe, and inversely proportional to the The conductivity of a solution of a weak acid HA of concentration square of the volume of the solution. 0.100 mol dm-, is 1.20 S m-1. The conductinty of a 0.200 mol dm-' (A) Express this information as an equation solution of this acid would then be (B) Suppose that 100 cm3 of a solution contains 10 g of solute. ( c ) 0.60 S m-' (a) 1.20 S m-' (b) 2.40 S m-I The absorbance of the solute in this solution is (d) cannot he calculated since sufficient data has not heen given. (b) 0.10 g ~ m - ~ (a) 1.0 X 10V g Select the correct answer. State also whether you made any assump(c) 1.0 X g (d) 10.0 tion in arriving at your answer. (e) cannot be calculated since sufficient data have not heen given. Nearlv 75%of the students in Group A. 25%in Group B. and Briefly explain your answer. 70% i; Group C assumed, erroneously, tbat density [s di'rectly proportional t o mass of sample (Question 3), tbat mole About 50% of Group A students were unable to convert fraction is directly proportional to pressure (Question 4), correctly the statement given in question 7 into an equaand that conductivity is directly proportional to electrolyte tion. However, in Groups B and C, 85%were successful. Most concentration (Question 5). The format of questions 3 and 4, students who answered incorrectly gave their answer as A = which resemble auestions that are generally used to test m l V (where A = absorbance, m = mass of solute, and V = proportional relationships, may have contributed to stuvolume of solution) instead of the correct equation A = kml dents hastily jumping to the conclusion, without much V ,where k is a constant. Furthermore, some who wrote the thought, that there is direct proportionality. T o check this, correct eauation did not a m e a r to understand the sienifithe multiple choice question (Question 5) was given with the canceof t i e constant k. In aiitbree groups, a highpercekage correct answer as one of the responses (response d). I t was of students did not recoenize that the correct resDonse for thought that the correct response should alert some of the part (B)is alternative ( e r ~ o soft them assumed, implicitly, students to recognize their error. This happened only for a tbat k = 1 and hence calculated the absorbance using the very small percentage (about 10%); these students did not incorrect equation A = mICR. They did not explain why k answer questions 3 and 4 correctly but solved question 5. was assumed to be 1. The majority of students (those who answered question 5 Questions 8 and 9 are essentially the same; the only differincorrectlv). however, seem to have a tendencv to assume, ence being a "hint" in 9. The main objective of this pair of without tl&e being ahy reason for it, direct proportionalit$ questions is to determine what percentage of students do not between two quantities. Almost all the students who went recognize the important fact that to deduce, or to calculate, wrong here gave response (b) as the correct answer. Some of how one variable changes when another variable is the reasons given for selecting this answer (i.e., for assuming changed, it is necessary to first derive the equation that direct proportionality) were: the two solutions bad the same shows the relationship between these two variables. solute: the higher the concentration the higher would be the Question 8 condukivity;-when concentration increases twofold, conductivity should increase twofold. I t appears that many stuAt constanttemperature, a nonideal gas obeys the equation p l W dents do not recognize the important principle tbat to calcu= km, where p, V, and m represent, respectively, the pressure, volume, and mass, and k is a constant. Consider a fixed mass of this late how one quantity changes with another, i t is necessary gas at 1 atm pressure; let ita density be represented by d. If the to know the equation that relates the two quantities. pressure of this gas is increased (at the same temperature) from 1 Question 6 also tests ones undemtanding of the concepts atm to 4 atm, the density of the gas would (note: d = mlV) of direct and inverse proportionality. The aspect tested is (a) not change (i.e., remain at the initial value d) the recognition, and the ability to state in words, of the (b) increase to 2d proportional relationships huilt into an equation. (c) increase to 4d (d) increase to 8d Questlon 6 (e) decrease to 0.25d Suooose that a nonideal eas obevs the eouation D'W = k m P . . Selectthe correct answer from the above alternatives. Show also the where p, u, m , and T represent the pressure. volume, mass, and steps in your deduction. temperature of the gas, respectively, and k is a constant. \Vhich of the following statements in connection with this gas are true? Question 9 ra) the volume is directly proportional to the temperature Anonidealgasobeys the equationpl@J = km, wherep, V, m, and (hl the volume is inversclg pnqortional to the pressure ~
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371
T,are, respectively,the pressure, volume, mass, and temperature of the gas, and k is a constant. (A) Derive an equation to show how the density d of this gas would depend on the pressure (note:d = m/V). (B)Make useof thisequation toanswer the followingquestion: A sample of this gas has a density dl at 1 atm pressure. Which one of the following would be the correct expression for the density d2of this gas if the pressure is increased from 1 atm to 4 atm? (the temperarture is kept constant). (h) d~ = 2dl ( c ) dz = 4d, (a) dz =dl (e) d2 = 0.5dl (d) dp = 3dl Show the steps in your deduction.
relating the relevant variables, either by deriving an equation or by drawing a graph, before certain types of calculations could be done.
Less than 30% of the students answered this question correctly. Subsequent discussion with some of the students revealed, however, that about one-third of them were able to draw the graph and obtain AD once they were told that a graphical method must be used and that AD must he obtained from the intercept. Their difficulty was hence due to their not recognizing that the solution required the initial drawing of a graph.
T h e skills and strategies indicated above are repeatedly required for the quantitative study of science a t university level. and a lack of comoetence in them would seriousfv handicap effective learning. The teaching of important skilis and strategies (e.g., visualization skills, mathematical skills, thinking, and problem-solving strategies) is often somewhat neglected in our courses. This is undesirable since the development of generalized skills and strategies is generally recognized to he an important objective of most courses: this is often the more permanent aspect of education. I t has been established ( 4 , 6 , 7) that about 50% of hiah school children are unable to use formal thought p r o ~ e s ~ e11 s . may he that indufficient r ~ m p e t e n r ein basic skills and rtrateaies atnone chemistry students a t universities is a consequeice of th; deficiency. Competence in skills and strategies is essential if one is to avoid unnecessary memorization, if one is to organize knowledee around a minimum number of conceots and hence re&e the load on our memory. Let us illus&ate this statement bv considerine. as an examde. the teachine and learning of ~ a o u l t ' slaw.;i'he usual way bf teaching t k s law is to give a verbal statement first, then to represent it as an equation, and finally to illustrate i t as a graph. Most students unnecesarily memorize all these representations, as if they were independent pieces of knowledge. These three types of representation (statement, equation, and graph) are, however, merely alternative ways of conveying the same information. I t is sufficient, and also better, for one to remember a law in just one of its representations, say either statement or equation. Conversion of a statement into an equation, or into a graph, involves general skills and does not depend on subject content. Students should he guided to do this on their own with carefullv desiened exercises. There is evidence that rule learners (tiose Gho memorize rules and algorithms) generally fail problems that require integration of algebra, subject matter, and reasoning (8).I t has also been established that leadine students to discover aleorithms and their extensions by th&elves make them recognize their knowledge in a more powerful wav and then thev become active cokributors to their own knowledge (9,10): T o adopt the approach suggested in the last paragraph, it is neces&ry for one to he cor&etent in certain geneial skills and strategies. The mastering of these skills and strategies would, of course, require time and effort. The skills required are, however, few in number; the same few skills being sufficient for dealing with large areas of subject content. The time spent on teaching the required skills and strategies to students would, in the long term, he more than compensated by the progressively increasing saving in time when we subsequently teach content.
Dlscusslon The results indicate that many university studenta have serious difficulties with the concepts, skills, and strategies reauired for dealine with relationshios between variables. ~rom the sample tested, 80% of the &dents from the newer universities. 40% from well-established ones. and about 70% postgraduate teacher diploma students:
Acknowledgment We thank J. Bradley, University of Witwatersrand, J. Rutherford. Universitv of Transkei. and J. Bull. Universitv of Cape l'own for assistance in conducting chis project. Thanks are also due to H. Selvaratham and K. R. A. de Alwis, both from the University of Bophuthatswana, for helpful discussions.
All students in Group H answered question 8 correctly, but onlvabout 50% in GrouosA and C could do so. About 2 6 q of those unable to answer question 8, however, solved question 9. This percentage of students therefore had all the mathematical skills required to solve question 8 (i.e., the ability to combine equations and to use the combined result for a calculation) but could not solve i t because they did nut know how to set about the solution. They did not recognize that to findout how density changes with pressure, i t is necessary to first derive the equation that relates density and pressure. T h e answers to question 9(A) also showed that only about 50% of Group A students were able to derive the equation relatine densitv and oressure. Some of them assumed direct proportionality between pressure and density in this case after derivine the correct eauation. Students in Grouos B and C answered the questions well. About 50% of the total students, however, had difficulties in one or more of the operations concerning combining, rearranging, and using equations. T h e skills required for the conversion of a linear equation into a graph, and the ability to recognize that for certain types of calculations one must draw a graph, were tested by question 10.
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Questlon 10 The variation of molar conductivity (A) of a solute in s solution with concentration (c) is given by the equation where A, B, and A0 are constants. Indicate how you would determine Ao (which is the molar conductivity of the solute when c tends to zero) if values of A at various values of e have been determined experimentally.
(a) thought erroneously that two quantities are directly proportional if both increase and that they are inversely proportional if one increases when the other decreases; (b) assumed, without justification, that two unrelated quantities were directlv,.oro~ortional to one another:. . (cl had difficulty in ronvrrring rhr i n f o r r ~ ~ a t i ogiven u in a stntemrnt inlo an equation and vicr versa; (d) had diflicully concerning r l w comhinaliun and rearrangement of equations; (e) had difficulty in recognizing the necessity for quantitatively 372
Joumal of Chemical Education
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