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High school calculus andachievement in universityengineering and applied sciencecoursesDiane M. Spresser aa Department of Mathematics and Computer Science , JamesMadison University , Harrisonburg, Virginia, U.S.A.Published online: 09 Jul 2006.

To cite this article: Diane M. Spresser (1981) High school calculus and achievement inuniversity engineering and applied science courses, International Journal of MathematicalEducation in Science and Technology, 12:4, 453-459, DOI: 10.1080/0020739810120415

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INT. J. MATH. EDUC. SCI. TECHNOL., 1981, VOL. 12, NO. 4, 453-459

High school calculus and achievementin university engineering and applied science courses

by DIANE M. SPRESSERDepartment of Mathematics and Computer Science,

James Madison University, Harrisonburg, Virginia, U.S.A.

(Received 16 September 1980)

The purpose of this study was to determine what relationships, if any, existbetween high school calculus and achievement in first semester engineering/applied science courses. Subjects were 145 first year students in the School ofEngineering and Applied Science at the University of Virginia. These subjectswere partitioned into three groups, representing different levels of calculusbackgrounds for entering first year students. Three first semester courses in theuniversity curriculum were considered: 'Introductory Calculus,' 'Chemistry forEngineers,' and 'Computers in Engineering'. Achievement in each of these threecourses was measured by the letter grade received. The findings of the studyfavoured those students with a year of high school calculus over those withoutthat preparation in all three engineering/applied science courses under consider-ation. When certain other variables were taken into account, however, alldifferences in achievement disappeared.

1. Statement of the problem and review of related literatureThe teaching of calculus in the high school has been a subject of controversy in

the United States for many years. While some topics in the calculus were consideredappropriate material for certain high school students at least as early as 1923 [1], itwas not until the post-Sputnik era that the national demand for acceleration inscience and mathematics created a real 'push' for calculus in the high school. The1959 report [2] of the Commission on Mathematics, appointed by the CollegeEntrance Examination Board (CEEB), was a very thorough and influential report,but quite cautious on the subject of calculus in the secondary school. TheCommission recommended that secondary schools with well-qualified staffs offer, aspart of the CEEB Advanced Placement Program, a full year of college-level calculusto their most talented twelfth grade students. They advised against teaching a shortunit of formal calculus to college-bound students at the end of grade twelve sincesuch a practice "tends to breed over-confidence and blunt the exciting impact of athorough presentation" [2, p. 15].

In the 20 years which have passed since the CEEB report appeared, high schoolcalculus has continued to be a source of controversy. Critics of high school calculushave sometimes questioned the worth or the value of the high school calculus courses[3] and have observed that, in the hurry to get to calculus, basic pre-calculus skillsmay not be receiving full and proper emphasis [4, 5]. They argue that there are othergood mathematical alternatives to calculus in the high school [4].

Advocates of high school calculus, on the other hand, argue (1) that success in andmastery of the calculus while still in high school can lead to advanced placement incalculus at the university level, thus creating flexibility in the student's university

()(>20-739x 81 1204 0453 S0200 c 1981 Taylor & Francis Ltd

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454 D. M. Spresser

schedule and the opportunity to study more advanced mathematics than wouldotherwise be the case; (2) that, even if the student does not receive advancedplacement, and essentially repeats his calculus course, he is still likely to achievemore in his university calculus than would otherwise be the case; and (3) that earlystudy of the calculus increases the student's understanding of and appreciation forthe sciences. They say that "calculus is a necessary tool in many college physicscourses taught in high school" [6, p. 452] and that, even beyond physics, "hardly abranch of science . . . fails to benefit from the application of the concepts of thecalculus . . . " [7, p.538]. The second argument above has motivated a sizeablenumber of empirical studies (see [8-17]) of the effects of high school calculus onstudent achievement in beginning university calculusf. There has, however, beenvery little research into the effects of high school calculus on achievement inintroductory university science courses. In fact, with the exception of the studyreported in the pages which follow, this researcher is aware of only one study [18],completed in 1967 by Puzzuoli, which specifically investigates any aspect of thisquestion. The Puzzuoli study found that a high school calculus experience wassignificantly related to success in university chemistry.

While the 11 empirical studies cited above [8-18] are not unanimous in theirfindings, they do provide substantial evidence on behalf of the calculus. It appears, ingeneral, that students who study calculus in high school achieve more in theirbeginning university calculus and science courses than do their peers with lesserpreparations%. When the studies cited above are considered as a group, there is anapparent gap in the literature regarding the effects of high school calculus—specifically, a full year of college-level calculus—on student achievement inintroductory university science/engineering courses. The experiment which followswas undertaken to provide needed data on that question §.

2. Description of the experimentThis investigation was undertaken to determine what relationships, if any, exist

between high school calculus and achievement in first semester engineering/appliedscience courses. For purposes of this study, three courses were of prime consider-ation: 'Introductory Calculus' (Applied Mathematics 101), 'Chemistry forEngineers' (Chemistry 111), and 'Computers in Engineering' (Engineering 109).These three courses are part of the general curriculum for first year students in theSchool of Engineering and Applied Science at the University of Virginia.

Three groups, representing different levels of calculus backgrounds for enteringfirst year students, were defined as follows:

(1) Group I—those with less than a full year of high school calculus and analyticgeometry.

(2) Group II—those who had a full year of high school calculus and analyticgeometry yet did not receive advanced standing in university mathematics.

fThe first eight studies cited here appeared in the literature prior to the experimentreported in this paper and formed its theoretical base.

% This was the conclusion of a review of research on high school calculus which recentlyappeared in these pages [19]. One additional study [17], which has since appeared in theliterature, also supports that conclusion.

§ This paper reports the results of the author's Ph.D. dissertation (University of Virginia,Charlottesville, Virginia, 1977).

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High school calculus and achievement in university 455

(3) Group III—those who had a full year of high school calculus and analyticgeometry and who received advanced standing in university mathematics.

Groups I and II took the introductory calculus course in their first semester at theUniversity. Group III took a more advanced calculus course during the firstsemester.

The following variables were considered:

(1) Mathematical aptitude (SAT—M scores).

(2) Previous mathematical proficiency (average grade received in all mathe-matics courses taken in grades 9-12, inclusive).

(3) Previous mathematical experience (number of semester-courses in collegepreparatory mathematics or college level mathematics completed prior toentering the University).

A fourth variable, percentile rank in high school graduating class, was includedprimarily as a control for general academic proficiency.

Achievement in each of the three engineering/applied science courses in thestudy was measured by a numerical equivalent of the letter grade received (A = 4,B = 3, C = 2, D = l and F = 0).

Subjects for this study were 145 first year students in the School of Engineeringand Applied Science at the University of Virginia. Of these 145, 68 had less than afull year of high school calculus (group I); 47 had a year or more of high schoolcalculus but did not receive advanced standing in mathematics at the University(group II); 30 had a year or more of high school calculus and received advancedstanding in mathematics at the University (group III).

The subjects generally had strong high school backgrounds in the physicalsciences. 139 (approximately 96% of the 145 subjects had taken a year or more ofhigh school chemistry, and 129 (approximately 89%) had taken a year or more ofhigh school physics.

Multiple linear regression techniques were used to test the effects of a year of highschool calculus on achievement in each of the three university courses underconsideration. Criterion (dependent) and independent vectors, together withappropriate means and standard deviations, are shown in the Appendix. All nullhypotheses were stated at the 005 level of significance.

3. Findings

The findings of this study can be summarized as follows:

(1) Those students with a year of high school calculus achieved significantlymore than those without that preparation in all three engineering/appliedscience courses under consideration.

(2) With respect to university chemistry achievement, those students who had ayear of high school calculus and who received advanced standing inuniversity mathematics achieved significantly more than their counterpartswho did not receive advanced standing.

(3) With respect to engineering achievement, there were no significant dif-ferences between those students who had a year of high school calculus and

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456 D. M. Spresser

who received advanced standing in university mathematics and those whodid not receive advanced standing.

(4) When mathematical aptitude was taken into account, all differences inuniversity calculus achievement and in university chemistry achievementdisappeared, although some differences remained for engineering achieve-ment (see table 8, Appendix).

(5) When any one of the variables (a) previous mathematical proficiency, (b)previous mathematical experience, or (c) percentile rank in high schoolgraduating class, was taken into account, all differences in achievementdisappeared.

4. Conclusions

As a result of the findings of this study, the following conclusions can be drawn:

(1)' Students who take high school calculus for at least a year achieve more intheir first semester university engineering/applied science courses than thosewithout a year of high school calculus.

(2) Any advantage which the high school calculus students have in theiruniversity engineering/applied science courses appears to be more closelyrelated to academic backgrounds of the students than to the actual content ofthe calculus. When certain academic background variables are taken intoaccount, there are no differences in achievement between those students witha year of high school calculus and those without that preparation.

(3) Of the students who take at least a year of high school calculus, those whoreceive advanced standing in university calculus achieve to an equal orgreater extent in their first semester university engineering/applied sciencecourses than their counterparts who do not receive advanced standing.

5. Summary and discussion

The findings of this study, in general, favour those students with a year of highschool calculus over those without that preparation in all three engineering/appliedscience courses under consideration. At the same time, the findings yield no evidenceto suggest that high school calculus makes any greater contribution to later studentachievement than that made by other mathematical subjects studied at a comparablelevel. Study of the calculus does, however, present one additional advantage overstudy of these other mathematical subjects: the possibility of advanced placement. Ingeneral, a talented and well-prepared student (one who has participated in anaccelerated mathematics programme and who has successfully completed a full pre-calculus programme of studies) who has the opportunity to study a full year ofcollege-level calculus while still in high school should be encouraged to do so.

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High school calculus and achievement in university 457

APPENDIXIn formulating the regression model, four continuous independent variables

were considered:

(1) mathematical aptitude,(2) previous mathematical proficiency,(3) previous mathematical experience, and(4) percentile rank in high school graduating class.

The three groups, representing the different levels of calculus backgrounds, wereincluded as categorical vectors. Criterion (dependent) and independent vectors,together with appropriate means and standard deviations, are shown in tables 1—3.Table 4 lists the interaction vectors.

Multiple linear regression techniques were used to analyse the data. Table 5shows group means for each of the three criterion variables. Significant differenceswere found between groups I and II with respect to achievement in AppliedMathematics 101 (F-ratio of 28-13, with associated probability of approximately0-00). When significant differences were found among the three groups, as withachievement in Chemistry 111 and Engineering 109, further tests of hypotheses wererequired to determine differences between specific pairs of treatments. The results ofthose tests are reported in tables 6-8.

Table 1. Continuous criterion vectors.

Vector

123

Vector

GradeGradeGrade

receivedreceivedreceived

Table 2.

Definition Mean

in Applied Mathematics 101 2-49in Chemistry 111 2-69in Engineering 109 299

Continuous

Definition

independent vectors.

Standarddeviation

1-230-96111

StandardMean deviation

66-20f3-459-50

11-92

6-660-481-53

10-89

4 SAT—M score5 High school mathematics average6 Number of college preparatory mathematics courses7 Percentile rank in high school graduating class

f SAT—M scores were reported in tens; for example, a score of 570 was reported as 57.

Table 3. Categorical independent vectors.

Vector Definition Mean

8 1 if subject comes from group I and 0 otherwise 0-479 1 if subject comes from group II and 0 otherwise 0-32

10 1 if subject comes from group III and 0 otherwise 02111 unit vector 100

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458 D. M. Spresser

Table 4. Interaction vectors generated.

Vectors Product

12-14 Groups x SAT—M15-17 Groups x high school mathematics average18-20 Groups x number of college preparatory mathematics courses21-23 Groups x percentile rank in high school graduating class

Table 5. Group means for three criterion variables.

Criterion Group I Group II Group III

Applied Mathematics 101Chemistry 111Engineering 109

Table 6. Differences between pairs of groups for Chemistry 111.

2032-372-62

3-152-743-30

—3-333-33

Group

I, III, IIIII, III

F-ratio

4-71 f21-55f10-44f

+ Significant

Probability

003000000

Table 7. Differences between pairs of groups for Engineering 109.

Groups

I, III, IIIII, III

F- ratio

ll-02f7-95f003

f Significant

Probability

0000010-87

Table 8. Differences between pairs of groups for Engineering 109 when mathematicalaptitude is considered.

Groups .F-ratio Probability

I, II 1-56 0-21I, III 6-48f 001II, III 3-27 007

f Significant.

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High school calculus and achievement in university 459

References[1] THE MATHEMATICAL ASSOCIATION OF AMERICA, INC., 1923, The Reorganization of

Mathematics in Secondary Education, A Report by The National Committee onMathematical Requirements (The Mathematical Association of America, Inc.).

[2] COLLEGE ENTRANCE EXAMINATION BOARD, 1959, Program for College PreparatoryMathematics, Report of the Commission on Mathematics (New York: College EntranceExamination Board).

[3] NEELLEY, J. H., 1961, Am. math. Mon., 68, 1004.[4] RASH, A. M., 1977, High Sch. J., 60, 277.[5] SORGE, D. H., and WHEATLEY, G. H., 1977, Am. math. Mon., 84, 644.[6] FERGUSON, W. E., 1960, Maths Teacher, 53, 451.[7] BLANK, A. A., 1960, Maths Teacher, 53, 537.[8] TILLOTSON, D. B., 1963, Diss. Abstr., 24, 577.[9] MCKILLIP , W. D., 1966, Diss. Abstr., 26, 5920.

[10] THELEN, L. J., and PIPPERT, R., 1966, Clearing Ho. Bull. Res. hum. Org., 40, 478.[11] ROBINSON, W. B., 1969, Diss. Abstr. B, 29, 2990-B.[12] SHIMIZU, M. T., 1969, Maths Teacher, 62, 311.[13] SCANNICCHIO, T . H., 1969, Diss. Abstr. Int. A, 30, 1344-A.[14] PAUL, H. W., 1971, Diss. Abstr. Int. A, 31, 3396-A.[15] GRIGSBY, C. E., 1974, Diss. Abstr. Int. A, 35, 280-A.[16] AUSTIN, H. W., 1975, Ph.D. Dissertation, University of Virginia, Charlottesville,

Virginia.[17] AUSTIN, J. D., 1979, J. Res. Math. Educ, 10, 69.[18] PUZZUOLI, D. A., JR., 1968, Diss. Abstr. A, 28, 3380-A.[19] SPRESSER, D. M., 1979, Int. J. Math. Educ. Sci. Technol., 10, 593.

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