proportional reasoning and achievement in high school chemistry

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Proportional Reasoning andAchievementin High School ChemistryJoseph S. KrajcikRichard E. Haney

What thinking processes are necessaryfor success in high school chemistry?What types of chemistry subject matterare most demanding? The answers tothese questions should provide guide-lines for helping teachers match instruc-tional activities to individual studentcapabilities.Many of the concepts traditionally

covered in high school chemistry are ofhighly abstract entities and thus requirethe student to function at the level offormal operations to attain comprehen-sion (Kavanaugh and Moomaw, 1981;

Herron, 1975 and 1977; Smith, 1975; Shayer and Adey, 1982). However, evi-dence indicates that students who are enrolled in high school chemistry are at avariety of stages of cognitive development (Chiapetta, 1976; Lawson and Ren-ner, 1974; McKinnon and Renner, 1971; and Shayer and Adey, 1982). Some stu-dents have been found to be concrete operational, some formal, and others in atransition between these two stages. Thus, it would be expected that concrete op-erational students, whose understanding is contingent upon reference to physicalexperiences and observable properties, would find much of chemistry^s subjectmatter difficult if not incomprehensible. As a result, it could be expected thattheir achievement would be below that of formal operational students who aremore able to apply hypothetical, proportional, probabilistic, prepositional andcombinatorial reasoning, and identify and control relevant variables. Which ofthese reasoning patterns are essential for success in high school chemistry?A reasoning pattern has been defined as "an identifiable and reproducible

thought process directed at a type of task" (Karplus, 1979) and can be classifiedas concrete or formal processes. Examples of reasoning patterns applied at theconcrete and formal levels are presented in Tables I and II respectively (Karplus,1979). A transitional student would be likely to apply these inconsistently in dif-fering situations.

School Science and MathematicsVolume 87 (1) January 1987

26 Proportional Reasoning

Table IExamples of Reasoning Patterns Applied

at the Concrete Level

Classification�separating a group of objects into several groups according to an ob-servable property (e.g., distinguishing consistency between acids and bases according tothe color of the litmus paper).

Conservation�realizing that an observable quantity remains the same if nothing isadded or taken away, even though it may appear different (e.g., when all the water in aglass is poured into an empty bottle, the amount originally in the glass equals theamount finally in the bottle�unless some was spilled).

Proportional Reasoning�making inferences from data under conditions of a constantratio equal to a small whole number (e.g., if two pieces of candy cost 25$, four piecescost 50$).

Interactional Reasoning�attributing an easily observable change to interaction amonga set of objects (e.g., magnet picking up a nail, rubber band stretching when it ispulled).

Additive Reasoning�making inferences from data under conditions of constant differ-ence or sum (e.g., when Mary is five years old, Joan is ten, so when Mary is ten yearsold, Joan is fifteen).

Table IIExamples of Reasoning Patterns Applied

at the Formal Level

Classification�arranging a group of items (objects or abstractions) into a multi-levelhierarchy according to observable or intangible properties (e.g., classifying the memberstates of the United Nations according to their form of government, economic system,and standard of living).

Conservation�realizing that certain properties of a system remain the same if nothingis added or taken away, but that this reasoning cannot be applied to all properties (e.g.,the total angular momentum of an isolated system is constant, but some students cangain knowledge without depriving others of knowledge).

Proportional Reasoning�making inferences from the data under conditions of a con-stant ratio not equal to a small whole number (e.g., if twelve pieces of candy cost 16$,then fifteen pieces cost 20$).

Correlational Reasoning�recognizing relationships among variables in spite of unpre-dictable fluctuations that mask them (e.g., drunk driving is associated with increasedaccidents even though sober drivers also have accidents, and many intoxicated peopledo not have accidents).

Prepositional Reasoning�Using postulates or axioms of a theory to deriveconsequences without regard to the factual basis of the postulates (e.g., making infer-ences from the theory according to which the earth’s crust consists of rigid plates mov-ing in relation to one another).


Proportional Reasoning 27

There have been several studies of the relationship between cognitive develop-ment and achievement in chemistry at the secondary school and college level(Martin, 1979; Wiseman, 1981; Herron, 1975; Bender and Milokofsky, 1982).Typically, positive correlations have been reported. However, none of thesestudies examined which reasoning patterns were necessary for success in highschool chemistry.

This study identified test items on a nationally standardized chemistry test thatdiscriminated between formal and nonformal operational students and exploredthe reasoning patterns required by these items. The Classroom Test of FormalReasoning (the Lawson test) was used to classify students into formal and non-formal (Lawson, 1978). Student scores on the 1981 version of the AmericanChemical Society-National Science Teachers Association High School ChemistryAchievement Examination (the ACS exam) was used as a measure of achieve-ment in high school chemistry (Examination Committee, 1981). A more com-plete description of this study can be found in Krajcik (1982).

Procedures

The subjects used in this study were students from a private, all-male, collegepreparatory school in Wisconsin. Approximately 98% of the graduates go on toattend a four year college program. Complete data were obtained from 170 stu-dents. This sample was composed of 15 twelfth grade students and 155 eleventhgrade students. The ages of the students ranged from 15.9 to 18.7 years with amean of 16.9 years.One hundred forty-nine students were enrolled in six sections of first year

chemistry and 21 were enrolled in accelerated chemistry. The contents of the twochemistry courses were similar. The distinguishing features of the acceleratedcourse were greater student independence, more laboratory work, and individualstudent research projects. The textbook used in all sections was Chemistry (Mas-terson, Slowinsky and Walford, 1981).The Lawson test was administered by the first author in each of the chemistry

classes. It is composed of 15 items which tests for the conservation of weight andvolume, proportional and combinatorial reasoning, probability and the controlof variables. Each test item was demonstrated in front of a group using simpleequipment such as clay balls, pendulums and ramps. The test administrator pre-sented a problem illustrated by a demonstration. Students could ask questions toclarify the problem. These questions were answered as long as the correct solu-tion to the problem was not revealed. Each student selected one of the multiple-choice answers and then wrote a brief explanation for his selection. In order foran item to be correct, both the answer and the explanation had to be correct.Testing time ranged from 70 to 80 minutes in the different classes.The ACS Exam was administered to all the chemistry classes during the final

examination period. Students were informed prior to the examination date that astandardized examination consisting of 80 multiple choice items designed to



measure achievement in first year chemistry would be used as the final exam.They were given exactly 80 minutes to complete the exam and were allowed touse calculators.

Results

The scores on the Lawson test ranged from 5 to 15 with a mean of 12.37 and astandard deviation of 2.07. On the basis of directions supplied with the test, 2students were classified as concrete operational, 42 as transitional, and 126 asformal operational. Lawson test scores for the entire group are presented in Ta-ble III. The mean ACS score was 56.31 with a standard deviation of 11.39.

Table IIIFrequency and Percentage of Students Receiving

A Given Score on the Lawson Test andPiagetian Classification

Score on theLawson Test

151413121110987654321

Fre(N

�quency= 170)

213638311415551220000

Percentage

12.321.022.218.78.28.82.92.90.61.21.20000

Classification Basedof the Lawson Test

FormalFormalFormalFormal

TransitionalTransitionalTransitionalTransitionalTransitionalTransitionalConcreteConcreteConcreteConcreteConcrete

Since there were very few persons who scored at low levels on the Lawson test,all scores of 5 through 9 were pooled. Thus, seven groups of students were identi-fied on the basis of Lawson test scores. Next, the means of the ACS test scoresfor these groups were examined. One-way ANOVA was used to test the nullhypothesis that there were no significant differences between these seven means;this hypothesis was rejected. The F statistic for the test was 7.33, with 6 and 167degrees of freedom, which is significant below the .01 level.The Duncan and Student-Newman-Keuls (SNK) tests were used to determine

which ACS means were significantly different from each other (Edwards, 1972).The results are shown in Table IV. The SNK test is more conservative and foundonly three significantly different groups, but the Duncan test found four. How-



TABLE IVDuncan and SNK Tests

Four Duncan Three SNKLawson Mean ACS Groups* Groups*

N Score Score (P=.05) (P=.05)

211564.3A A361459.3AB AB381356.9BC AB311257.3BC AB141154.5BC BC151041.5CD BC159-547.0D C

* Means having the same letter or with different letters that overlap are not significantlydifferent from each other.

ever, the results of the two procedures are qualitatively the same. Persons whoscored high on the Lawson test scored significantly higher on the ACS test thandid those who scored in the middle and lower ranges of the Lawson scores. And,those in the mid-range of the Lawson scores achieved a higher mean ACS scorethan did those in the lowest range of the Lawson scores.The next step in this study was to identify and explore the characteristics of in-

dividual items on the ACS exam which discriminated between high and lowachievers on the Lawson test. For this purpose two groups of students were iden-tified: a high or formal operational group consisting of those whose scoresranged from 13-15 and a low or nonformal operational group whose scoresranged from 5-10. Those whose scores were 11 or 12 were excluded to minimizethe possibility of overlap between the two groups. The mean scores on the Law-son test for the high and low groups were 13.82 and 8.80 respectively. The meanACS score for the high group was 59.47 and the mean for the low group was42.00.For each ACS test item, a four by four contingency table was generated show-

ing correct or incorrect answers versus high and low Lawson scores. If a chi-square test of homogeneity was significant, the item was classified as a "dis-criminator," that is, students with high Lawson scores performed differentlythan students with low Lawson scores. For all of the questions that were foundto be discriminators, the difference was the same: the students with high Law-son scores did better than those with low Lawson scores. On the basis of the chi-squares tests, 31 items were found to discriminate high from low achievers on theLawson test (p > .05).The question then arose as to whether the discriminators required the use of

different reasoning patterns than did the nondiscriminators. The answer was aqualitative one but the analysis provided insights into the different thinking



processes of students in the two groups. Each of the 80 items on the ACS examwas inspected by the two investigators to identify the possible reasoning patternsrequired for their solution.As a result of this analysis, it was found that 18 (58%) of the 31 discriminators

required the use of proportional reasoning; however, only three (6%) of the 49nondiscriminators required this form of reasoning. Of the 13 remaining dis-criminators, six required the recall and application of chemical principles, threedealt with information not covered in the course, two required simple recall, onerequired combinatorial reasoning and one required the application of a func-tional relationship. Of the remaining nondiscriminating items, a majority re-quired the recall of factual information and often its application to a given situa-tion. Because of the sharp contrast between the percentage of discriminators re-quiring proportional reasoning and that of the nondiscriminators requiring thisreasoning pattern, it appears likely that the ability to apply proportional reason-ing patterns is a major factor that differentiates the ACS exam performance ofstudents with high Lawson scores from that of students with low Lawson scores.Discriminators and nondiscriminators are listed in Table V.

TABLE VDiscriminators and Nondiscriminators in the

ACS Exam

Discriminating Test Discriminating Test NondiscriminatingItems not involving Items involving the Test Items involvingthe use of proper- use of proportional the use of propor-tional reasoning reasoning tional reasoning

6,8,12,22,24,26,39,53, 4,9,10,13,14,15,17,31, 46,49 and 7356,63,64,70,77 32, 33, 40,43,44,45,50,

51,54, and 58

Discussion

Since high scores on the Lawson test indicate that a student has the ability to useformal operational structures, this study furnishes evidence that students whocan use formal operational reasoning patterns are capable of a greater degree ofachievement in high school chemistry than students who cannot use these reason-ing patterns. Greater facility with the use of proportional reasoning is associatedwith high ACS scores. However, there are limitations to this study which must beconsidered.

First, there were few students who achieved low scores on the Lawson test. Avery useful and insightful extension to this study would be using a more hetero-geneous sample. The use of interviews to assess the reasoning patterns studentsuse to solve chemistry problems would also be a significant enhancement.



Second, the nominal level of significance used for the chi-square tests causessome difficulty with interpretation. Since there were 80 separate hypothesestested, the overall probability of making an error in classification is much largerthan the alpha-level for each separate test. Since the overall alpha was picked tobe .05, there is a fairly large, but unknown, probability of making at least onemistake in classifying the items. But since our goal was to find qualitativesimilarities among the discriminators, this was not a major problem.

Lastly, it is possible that students could have answered the discriminatory testitems with the aid of an algorithm. However, the fact that so many discrimina-tory items (18 of 31) involved proportional reasoning while so few of the nondis-criminatory items did (3 out of 49), reinforces the notion that proportional rea-soning plays an important role in the study of chemistry.As chemistry teachers, we must consider these results in our teaching. As Her-

ron (1975) stated, good study habits and hard work may allow students who arenot formal operational to obtain moderate success in chemistry; however, theyare not likely to be as successful as students who are formal operational becausethese groups differ in their cognitive development.

Typically, teaching strategies do not reflect that teachers are aware of thesedifferences. Since this study indicates that a likely factor contributing to the dif-ference in chemistry achievement is the ability to apply proportional reasoning,high school chemistry teachers must realize that nonformal operational studentswill probably have difficulty comprehending material requiring the use of thisreasoning pattern. This realization will require teachers to adjust their teachingstrategies to help these students. Typically, nonformal operational students willtry to memorize this material, often becoming confused and learning to dislikescience. For these students, teachers could provide numerous activities directedat developing this reasoning skill. These activities could include both a variety ofhands-on activities as well as computer-assisted instruction. Many of the labora-tory activities in Chem Study are very appropriate for developing this reasoningpattern. Currently, software is being developed that will help students learn thisreasoning pattern. Formal operational students can profit from these activitiesas well. However, they should also be challenged by more abstract problems andactivities.

References

Bender, D. and L. Milokofsky. College chemistry and Piaget: The relationship of aptitudeand achievement measures. Journal of Research in Science Teaching, 1982, 19, 205-216.

Chiapetta, E. Review of Piagetian studies relevant to science instruction at the secondaryand college level. Science Education, April 1976, 60, 219-237.

Edwards, Alien L. Experimental Design in Psychological Research (Fourth Edition). Holt,Rinehart and Winston, 1972.

Examinations Committee. American Chemical Society�National Science Teacher Associ-ation Examination�High School Chemistry Form 1981. Tampa, Florida: AmericanChemical Society, University of South Florida, 1981.



Good, R., E. K. Mellon and R. A. Kromhout. The work of Jean Piaget. Journal of Chem-ical Education, November 1978, 55, 688-693.

Herron, D. J. Piaget for chemists�Explaining what "good" students cannot understand.Journal of Chemical Education, March 1975,52,146-150.

Herron, D. J. Piaget in the classroom: Guidelines for application. Journal of ChemicalEducation, March 1978, 55, 165-170.

Karplus, R. Science teaching and the development of reasoning. Journal of Research in Sci-

ence Teaching, March 1977, 14,169-175.Karplus, R. Teaching for the development of reasoning. In A. Lawson (Ed.), The Psychol-

ogy of Teaching/or Thinking and Creativity. 1980 AETS Yearbook, 1979. (ERIC: ED184-894).

Kavanaugh, R. D., and W. R. Moomaw. Inducing formal thought in introductory chemis-try students. Journal of Chemical Education, March 1981, 58, 263-265.

Krajcik, J. S. The relationship between cognitive development of high school students andstudent achievement in chemistry. Unpublished Master’s thesis, University of Wisconsin-Milwaukee, 1982.

Lawson, A. The development and validation of a classroom test of formal reasoning. Jour-nal of Research in Science Teaching, 1978, 75 11-24.

Lawson, A. and J. Renner. A quantitative analysis of responses to Piagetian tasks and itsimplications for curriculum. Science Education, 1974, 58, 545-559.

Lawson, A. and J. Renner. Relationship of science subject matter and developmental levelsof learners. Journal of Research in Science Teaching, October 1975,12, 347-358.

Martin, D. A group administered reasoning test for classroom use. Journal of ChemicalEducation, March 1979, 56, 179-180.

Masterton, W.L., E. Slowinski, and E. T. Waifour. Chemistry. New York: Holt, RinehartandWinston, 1980.

McKinnon, J. W. and J. Renner. Are colleges concerned with intellectual development?American Journal of Physics, September 1971, 39,1047-1052.

Milakofsky, L. and H. Patterson. Chemical education and Piaget: A new paper-pencil in-ventory to assess cognitive functioning. Journal of Chemical Education, February 1979,56, 87-90.

Roberts, R. S. Concurrent validity in tests of Piagetian developmental levels. Journal ofResearch in Science Teaching, 1980, 77, 343-350.

Shayer, M. and P. Adey. Towards a science ofscience teaching. London: Heinemann Edu-cational Books, 1981.

Smith, Patricia. Piaget in high school instruction. Journal of Chemical Education, Febru-ary 1978,55,115-118.

Tobin, K. and W. Capie. The development and validation of a group test of logical think-ing. Education and Psychological Measurement, 1981, 41, 413-423.

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Joseph 5. KrajcikRichardE, HaneyUniversity of MarylandUniversity of Wisconsin-MilwaukeeCollege Park, Maryland20742Milwaukee, Wisconsin 53223


proportional reasoning and achievement in high school chemistry

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