Career Choice: Do Students\' Precollege Mathematics Programs Deny It?

Career Choice: Do Students5 Precollege Mathematics Programs Deny It? Bob Kansky and Melfried Olson Science and Mathematics Teaching Center University of Wyoming La ram ie, Wyoming 82071

It was the purpose of this study to gather information concerning the status of the mathematical preparation of high school seniors in Wyoming and to determine the mathematics necessary for entrance into University of Wyoming programs. At the time of the study,’ Wyoming had 73 public high schools; the number of seniors enrolled in these schools ranged from 597 to 5. These schools were divided into five types (Table 1) based upon senior class size. Forty-seven of these schools were selected for inclusion in the study using an intact-groups modification of the stratified random sampling technique employed by the various studies of the Wyoming Educational Needs Assessment Program.2 The objective of the technique was to identify a sample which included at least 60 percent of all Wyoming seniors and which was comprised of 60 percent of the seniors in each of the five types of schools. In fact, the sample selected included 70 percent of the senior population. Because "whole schools" were selected and a minimum representation of 60 percent was desired, the actual percents for the five School Types ranged from 62 (Type IV) to 76 (Type I). A school’s agreement to participate in the study meant that a contact person would complete a general information form about the school and would administer the survey instrument to all seniors in the school using written directions and definitions provided by the investigators. It should be noted that many schools administered the survey selectively rather than to all seniors as requested. The schools engaging in such "selective" administration reported using the instrument only with seniors currently enrolled in mathematics classes. It was hoped that the selection procedures described above would ensure returns which included at least 40 percent of the seniors in each of the five types of schools. Returns were received from 40 of the 47 schools; the 2010 individual respondents represented a response rate of 45 percent of the original sample of 4477 seniors and 32 percent of the total population of 6380 seniors. 1. Kansky, Bob & Olson, Melfried. Mathematical Preparation Versus Career Aspirations: A Study of U’yoming ’s 1978 High School Seniors. Laramie, Wyoming, ERIC: ED 186263 2. Wyoming Needs Assessment Project in mathematics. Science and Mathematics Teaching Center, 1977.

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TABLE l FIVE SCHOOL TYPES

School Type

I II III IV V

Enrollment Range of Senior Class

250-600 150-249 100-149 50- 99 5- 49

Instruments Two instruments were used to gather information; a Student Survey instrument and the College Program Interview Form. The Student Survey instrument asked the student to indicate sex, future educational plans, proposed field(s) of future study, the total mathematical content of the high school mathematics program pursued and the grade level at which the study of mathematics was terminated. This data was gathered in late April or early May (depending upon the school’s choice) so that it could accurately reflect the total program pursued by the student while in high school. The College Program Interview Form was developed for the purpose of assessing the minimal mathematical entry level skills expected of each of 99 academic programs at the University of Wyoming. Although some of this information appeared in University publications, it was recognized that such publications are usually two years behind current thinking of program faculty. Therefore, interviews were conducted with each program director to verify the mathematical expectations listed in the publications.

ANALYSIS

Sex Distribution in the Sample The first two items of the Student Survey served (1) to identify the student’s sex and (2) to determine whether or not the student planned to begin study at a University or community college within the following two years. Table 2 shows that the total sample had an even distribution between sexes. The college-bound group closely approximated the distribution of the total sample; moreover, the ratio of 52 percent female to 48 percent male remained relatively constant across the five types (sizes) of schools. The larger proportion of males in the noncollege-bound group also was maintained across the School Types.

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TABLE 2 SEX DISTRIBUTION OF SAMPLE

Group college noncollege

total

Number of Seniors in Sample

Percent Females in Sample

Percent Males in Sample

1415 595

52 46

48 54

2010

50

50

Terminal Year of Secondary School Mathematics Another item of the Student Survey asked the student to indicate the last year of school in which he took a mathematics course. Again, the data was examined across two categories: college-bound and noncollegebound. Each of these categories was further divided according to sex. The data showed that college-bound women terminated their study of mathematics much earlier than the college-bound men. By the end of Grade 10 an average of 15 percent more women than men had ended the study of precollege mathematics, while 17 percent more men than women took mathematics during their senior year. Moreover, whereas the female/male enrollment ratio in Grade 12 mathematics classes was about 2 to 3 for the total college-bound group, the ratio was 1 to 2 in the largest (Type I) schools. The greater number of elective courses available to women in Type I schools could be a major factor supporting this difference in ratios. The comparisons above are with respect to college-bound students who enrolled in any mathematics course during the senior year. When one examines the level of advancement of the content of the material studied in the senior course, the difference in termination rates is amplified. That is, whereas the enrollments of senior men may be characterized as being in advanced mathematics courses, many senior women may be enrolled in career-, consumer- or business-oriented courses in mathematics. Moreover, the terminal-grade pattern ofnoncollege-bound students was very close to that of the college-bound with regard to sex. This suggests that the decision of when to terminate the study of mathematics may be independent of intent to pursue postsecondary education.

Levels of Mathematical Preparation The Student Survey also asked students to describe their pre-college mathematical education. Specifically, they were asked to indicate whether they had completed certain familiar courses: General Mathematics, First-Year Algebra, Second-Year Algebra, and Geometry.3 In addi3. A definition of the last three of these courses was provided to the school person administering the survey, along with the written and oral (telephone) suggestions that these be used to guide student responses.

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tion to such "whole course" descriptions, the survey specified other mathematical topics which often are part of a mathematics program but which may find themselves titled differently at different schools. Students were asked to indicate which of these topics they had completed. They also recorded the year each course or topic was completed. The courses and topics of the Survey were used to define the six levels of mathematical preparation found in Table 3. These six levels were then used to classify, from highest (Level 1) to lowest (Level 6), the summative TABLE 3 DESCRIPTION OF SIX LEVELS OF MATHEMATICAL PREPARATION AMONG WYOMING SENIORS LevelDescription

Algebra I and II; synthetic and analytic geometry; trigonometry; logarithmic functions (common and natural) and their graphs; mathematical induction; algebra of functions; basic operations on matrices; limits, continuity and differentiation of polynomial functions. Algebra I and II; synthetic and analytic geometry; trigonometry; logarithmic functions (including common and natural) and their graphs, mathematical induction; algebra of functions; basic operations on matrices. Algebra I and II; synthetic geometry. Algebra I; synthetic geometry. Algebra I. General mathematics.

TABLE 4 PERCENT OF COLLEGE-BOUND SENIORS ATTAINING EACH OF Six LEVELS OF MATHEMATICAL PREPAREDNESS Level of Mathematical Preparation 1

2 3 4

5 6

School Type Subgroup (

female male total female male total female male total female male total female male total female male total

12 34 24 19 44 31 55 71 62 75 85 80 93 98 95 100 100 100

19 29 24 20 32 26 65 71 68 72 79 76 91 94 93 100 100 100

III

IV

8 24 17 18 33 26 49 63 56 58 67 63 89 83 86 100 100 100

22 26 23 23 28 25 65 48 57 78 71 75 97 91 94 100 100 100

Combined (I-V)

14 31 23 24 33 29 38 52 46 63 76 70 93 93 93 100 100 100

15 30 22 21 36 28 54 64 59 71 78 74 93 94 93 100 100 100

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precollege mathematical experience of each college-bound student. The classification of student responses into these six levels is summarized in Table 4. TABLE 5 Six OCCUPATIONAL CLASSES:

DEFINITION IN TERMS OF MATHEMATICAL ENTRY SKILLS Class

Definition

A

College study should begin with calculus; MATH 303D 5 is viewed as a deficiency or remedial. Students taking these courses may be unable to complete program degree requirements in four years. MATH 303D 5 is prerequisite to the study of the additional mathematics, statistics or computer science courses required by the programs in this class. MATH 303D 5 is a specific program requirement. No additional mathematics, statistics or computer science courses are listed as degree requirements. While MATH 302F is not specified as a degree requirement, knowledge of its content is prerequisite to program degree requirements in mathematics, statistics or computer science. MATH 302F is a degree requirement. No additional work in mathematics, statistics or computer science is specified. No college mathematics is listed as a degree requirement. The program descriptions do not indicate that the use of any high school mathematics of the level of Algebra I(or above) will be assumed in the courses of the program.

B C

D

E F

Table 4 reveals that the upper three levels of mathematical preparation are attained by a decidedly larger percentage of males than females.4 In general, the male and female subgroups are equal on Levels 5 and 6 only. The figures of Table 4 are of particular importance in that they show the populations reaching the highest level of precollege mathematical preparation. For all school types, the ratio of women to men who reach the highest level is 1 to 2 (i.e., 15 percent to 30 percent). This ratio, however, is not uniform across school types. For School Types I and III, it increases to about 1 to 3. University Program Requirements Versus Precollege Preparation

One objective of this study was to evaluate students’ precollege mathematical studies in terms of the mathematical entry skills set forth by 99 academic programs at the University of Wyoming. An estimate of these entry skills first was obtained by careful reading of University catalogs. These approximations were then revised and brought up to date by means of interviews with each program head. These entry-level expectations, subsequently expressed in terms of specific University of Wyoming precalculus mathematics courses, led to the definition of the six occupational classes found in Table 5. Each class is defined in terms of the 4. There is an exception in School Type IV in regard to preparation Level 3. 5. Or MATH 302F and 3020.

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mathematical skills expected of students entering the program; these skills are, in turn, expressed in terms of courses called MATH 302F (PreCalculus Algebra), MATH 302G (Pre-Calculus Trigonometry) and MATH 303D (Pre-Calculus Algebra and Trigonometry). MATH 303D compacts the content of the other two courses into a single semester. The fact that program entry-level skills have been defined in terms of college courses and that students’ preparation is expressed in terms of precollege courses needs some explanation. The general points to be made are: 1. The content of the three precalculus courses mentioned is precollege content. 2. The primary purpose of the precalculus offerings is to review and refine the concepts listed and not to introduce them.

At the time of the study, the content of MATH 302F included all of Algebra I and several topics from Algebra II; MATH 302G was comprised of selected topics of Algebra II in addition to an intensive study of circular, exponential and logarithmic functions.6 However, the rapid pace of the courses was a reflection of the fact that they were not intended for students who had not studied these topics already. The six occupational classes defined in Table 5 were used to classify 99 University of Wyoming academic programs. Thirty-five of the 99 programs require Class A or B entry skills in mathematics. Elementary school teaching (Level D) is a program in which over 90 percent of the students are female. A question might be raised as to whether females elect this program with forethought; or does their limited preparation in mathematics confine them to this level (or below) while many males are free to select programs which demand greater entry level skills in mathematics. On another note, the large number of secondary school teaching programs (ten) that fall into Level F may be viewed as indicative of probable attitude toward the importance attached to mathematics by many secondary school teachers. Transmitted to their students, this attitude surely can influence students’ decisions regarding the selection of advanced mathematics courses. To determine whether or not students’ precollege mathematics programs matched their occupational aspirations, the survey instrument asked the student to indicate, in order, his top three choices from among the 99 fields listed. These correspond, of course, to college programs at the University of Wyoming. While some students may enter these occupations without a college degree, many indicated that they planned to take the college preparation route. The analysis which follows is restricted to the first choices of these college-bound students. The interaction between the precollege mathematical preparation and 6. In September 1980, many Algebra I topics were deleted from MATH 302F to make room for the Algebra 2 topics which were in MATH 302G. MATH 302G was altered to include polar coordinates, complex numbers and more problemsolving.

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occupational aspirations of college-bound students is presented in Table 6. As a whole, the results indicate that many more males than females aspire to enter occupations in Class A (29 percent male versus 12 percent female) and Class B (33 percent male versus 20 percent female). When viewed across School Types, the male/female ratio for selection of Class A occupations ranges from a high of nearly 11 to 3 (School Type I) to 1 to 1 (School Type IV). There is greater homogeneity in the ratio of Class B choices which stand at about 2 to 1 for three School Types (III, IV, V) and, roughly, 3 to 2 for School Types I and II. These percentages are singled out because occupations in Classes A and B are those which have high entry-level expectations in mathematics. At the other end of the mathematical expectation spectrum are the occupations of Classes E and F. For these classes, the male/female selection ratios are, generally speaking, the reverse of those observed for Classes A and B. For the total population the composite male/female ratio is about 1 to 2. Moreover, this is roughly the ratio found for each School Type. About 25 percent of the seniors indicated an occupational choice which, at the time of the study, required at most a knowledge of enough algebra to complete MATH 302F. If one takes the liberal view that persons having Level 3 precollege mathematics are prepared to enter Class B college programs, we found that 72 percent of the students in the sample had at some time in their secondary school career been exposed to the minimal mathematics requird by the programs they wished to pursue. This rate is relatively uniform across the five School Types. While this figure accounts for the majority of the college-bound students, the 28 percent who are not prepared constitute a large minority.

Student Awareness of Mathematical Preparedness To ascertain how well informed the students were of their mathematical preparedness (or unpreparedness, as the case may be), the survey instrument included the question: Do you feel that the mathematics you have taken in high school has prepared you adequately to begin training in your chosen occupational field(s)?

Of those students identified as underprepared (28 percent of the total), 21 percent were unaware that they have chosen college programs for which they do not have entry level mathematical skills. This figure was relatively constant across School Types and sex.

SUMMARY This study was made in response to a concern often expressed to the authors by students at the University of Wyoming. Students from a broad range of fields of study reported that the mathematics courses re-

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TABLE 6 STUDENT CHOICE OF OCCUPATIONAL CLASS: COLLEGE-BOUND SENIORS (As a Percent of College-Bound Sample) Occupati onal Class

School

Type

I II III IV V

TOTALS

Subgroup

A

B

C

D

E

F

female *male *total female *male total *female male total *female *male total female male total

9 32 20 10 27 19 4 22 14 19 19 19 12 33 22

25 33 29 20 33 27 15 34 25 20 39 30 14 27 20

8 9 9 6 8 7 7 17 12 4 6 5 9 15 12

25 10 18 31 12 21 29 10 19 25 17 21 32 9 21

1 0

2 1

32 15 24 32 18 25 43 17 29 30 17 24 32 14 24

*female male total

12 29 19

20 33 26

7 11 9

28 11 20

1 0 1

33 16 25

0

quired for completion of their college programs were posing serious obstacles. These students had terminated their study of precollege mathematics as early as possible on the assumptions that (1) no mathematics was required for the fields they wished to enter or (2) college mathematics courses were available wherein they could correct any mathematical shortcomings. For many students, both assumptions had proved erroneous. In conclusion, they asserted that "someone should have told us we’d need so much mathematics." Was the problem reported by these students widespread or were these the complaints of a vocal few? Was the University remiss in providing adequate career information relative to the mathematical expectations of its many (99) programs? Were high school students uninformed or misinformed? This study sought paired information regarding both the precollege mathematical preparation and the occupational aspirations of students entering the University of Wyoming. (Since over 70 percent of the University’s students are graduates of Wyoming high schools, the study was confined to that population.) Using that paired information, we have been able to provide: * a general description of the mathematical programs pursued by students in Wyoming’s public schools;

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* an evaluation of the adequacy of the high school’mathematical preparation ot college-bound students relative to the occupational aspirations (99 fields) of those stu-

dents; and, * a measure of the extent to which students are aware of the adequacy of that preparation.

Each set of data was broken out along the dimension of sex. The results reported in this article raise serious concerns regarding the academic planning and mathematical preparation of Wyoming’s collegebound students. They indicate that both mathematical preparation and career aspiration are sex-related. Significantly more men than women achieve high levels of precollege work in mathematics and aspire to enter occupations requiring several college courses in mathematics, statistics and computer science. Conversely, freedom of occupational choice by women is severely limited by their earlier termination of the study of precollege mathematics. The study also found that about 1 out of 5 of Wyoming’s collegebound students is mistaken in his belief that he is mathematically prepared for the field of study he wishes to enter. Anyone in this group of students who know-not-and-know-not-that-they-know-not is faced with a high probability of disappointment in the college pursuit of a particular career. Moreover, the figure is a minimum in view of what the study accepted as adequate preparation for a Class B occupation. Finally, the investigators suspect that the observations of this study are unique neither to the course descriptions and program requirements of the University of Wyoming nor to Wyoming’s secondary school students. The courses described are standard precalculus offerings, the entry-level mathematical requirements of the 99 programs do not constitute an upper bound for universities in general, and there is no reason to believe that Wyoming’s secondary school students have below average instruction, motivation, or counseling with respect to mathematics. Anyone wishing to put these assertions to the test is invited to share the study’s instruments and procedures for replication in other locales.

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