Attitudes toward mathematics of precalculus and calculus students.

Moldavan, Carla C.

Introductory college mathematics courses comprise a large percentage of course offerings in postsecondary institutions, serving over half of all students who ever study mathematics in college (Cohen, 1995). In a report of mathematics classes offered in fall, 2000, 14% of the sections were remedial and another 38% were introductory level, including precalculus (Lutzer & Maxwell, 2000). Many students are ill-equipped for introductory college math courses. Many degree programs in non-technical fields require math prerequisites, which are often stumbling blocks for students.

A matter of scientific interest is the nature of students' attitudes toward mathematics and the relationship between attitudes and achievement in mathematics, especially as it relates to the achievement gap in mathematics between males and females, and the lack of interest by females in science, technology, engineering, and mathematics majors (STEM). In the past decade the American Association of University Women (AAUW) and the National Science Foundation (NSF) have invested nearly $90 million to fund hundreds of projects aimed at increasing the participation of girls and women in STEM (AAUW, 2004). During the past few years, SAT math scores indicate that the gender gap is narrowing because females on average gained 19 points while males gained 13 (Hoover, 2001).

Explanations of the math gender gap have focused on social and cognitive differences. Males do better on multiple choice tests in mathematics, while girls are better on open-ended or essay questions that involve verbal skills (Beller & Gafni, 2000). Boys have better spatial ability (Collins & Kimura, 1997; Nordvik & Amponsah, 1998). Differential treatment of males and females in math classes has also been used to explain the difference, because females are not supported in math aspirations by their instructors and their parents (Hammrich, 2002). Efforts to create equal educational opportunities for females are primarily based on changing the attitudes of females about the study of math and pursuit of technical careers, because there are only social impediments to women entering technical fields and professions. Some researchers maintain that it is important to foster safe and nurturing environments in order to encourage female students' success in science and mathematics (Allen, 1995; Hammrich, 2002; Mann, 1994).

Research has cast doubt on explanations that account for cognitive differences, because achievement in mathematics courses in middle school and high school is virtually the same for males and females (Davis-Kean, Eccles, & Linver, 2003). Data from the National Assessment for Educational Progress (NAEP) also confirm that at all grade levels there is little difference in the overall performance of males and females (Campbell, Reese, O'Sullivan, & Dossey, 1996; Kenney & Silver, 1997). Performance in specific content area also reflects little difference between males and females; the only statistically significant gender difference appeared at grade 12 for items in the areas of measurement and geometry, with males having statistically significantly better performance. NAEP (Kenney & Silver, 1997) reported little overall difference between males and females for those who enrolled in core college preparatory courses, with the exception of calculus, which was taken more frequently by males. These data reflect a national trend toward increased course taking by high school students in response to increased graduation requirements, and they attest to a change in the achievement of females. NAEP data regarding affect toward mathematics showed that males in grades 8 and 12 were significantly more likely than females to agree that they liked mathematics, but there was little or no difference between males and females in their perception of being good at mathematics. Students at all grade levels appeared to view mathematics as having considerable social and economic utility. In responding to a belief statement regarding mathematics being more for boys than for girls, the vast majority of females did not see mathematics as a male domain, but considerably more of their male counterparts did view it that way. Thus, NAEP points to some attitude differences between males and females but almost no performance differences. The Third International Math and Science Study (TIMSS) also reports no performance differences between males and females in the participating countries (U.S. National Research Center, 1996).

Less than 1% of undergraduates major in mathematics (Haycock & Steen, 2002). The number of bachelor's degrees awarded in mathematics fell 19% between 1990 and 2000, although undergraduate enrollment rose 9% (Lutzer & Maxwell, 2000). While girls have virtually identical abilities in mathematics, by age 13 they already have quite difference career aspirations. Boys are intent on careers in science or engineering, but girls express preference for business, professional, and managerial occupations (U.S. Department of Education, 1990). Women earned 57% of the bachelor's degrees awarded in 2003, but only 20% of the degrees in technical fields in 1999 went to women (Hacker, 2003). Attitudinal research among college students has not been thorough. In this study, attitudes of male and female students enrolled in college introductory general education courses (precalculus and calculus) were compared.

Method

Participants

The participants were 89 undergraduate students enrolled in precalculus and calculus at a small liberal arts college. The sample was predominantly Caucasian. Forty-six students were enrolled in precalculus and 43 were in calculus. Forty-nine students were male, 39 were females, and one student did not report the gender. There were 58 freshmen, 18 sophomores, 10 juniors, and two seniors. All were volunteers, and all students in the classes agreed to participate.

Instrumentation

The Attitudes Toward Mathematics Inventory (ATMI: Tapia & Marsh, 2004) is a 40-item scale. The items were constructed using a Likert-format scale of five alternatives for the responses with anchors of 1: strongly disagree, 2: disagree, 3: neutral, 4: agree, and 5: strongly agree. Eleven items were reversed, and these were given the appropriate value for data analysis. Total score is the sum of item ratings.

Exploratory factor analysis of the ATMI (Tapia & Marsh, 2004) resulted in four factors identified as Self-confidence. Value of mathematics, Enjoyment of mathematics, and Motivation. The Self-confidence factor consists of 15 items. The Value factor and the Enjoyment factor each consist of 10 items. The Motivation factor consists of five items. Table 1 shows sample items from each one of the factors. The complete inventory is available from the first author upon request. Alpha coefficients for the scores on these scales were found to be .95, .89, .89, and .88 respectively (Tapia & Marsh, 2004).

A Student's Demographic Questionnaire was also used. This questionnaire consisted of four questions. The purpose of these questions was for identifying gender, course, ethnic background, and undergraduate classifications (freshman, sophomore, junior, or senior).

Procedure

The ATMI and the Student's Demographic Questionnaire were administered at the beginning of the semester to students in three precalculus classes and two calculus classes. Students in these classes were informed that it was completely voluntary to participate and that there was no penalty for not participating. No incentive was provided to students for their participation. The instruments were administered in class by the class instructor. Directions were provided in written form and students recorded their responses on answer sheets that could be scanned by a computer.

Results

Using the four-factor solution found by Tapia and Marsh (2004), the 40 items were classified into four categories (Self-confidence, Value, Enjoyment, and Motivation), each of which was represented by a factor. A composite score for each category was calculated by adding all the numbers of the scaled responses to the items belonging to that category. Cronbach alpha coefficients were calculated for the scores on the factors and were found to be .97 for Self-confidence, .91 for Value, .91 for Enjoyment, and .89 for Motivation.

In the one-way design with gender as the independent variable, data were analyzed using four separate analyses of variance (ANOVA). For each one of the ANOVAs, the dependent variable was one of the four factors: (1) Self-confidence, (2) Value, (3) Enjoyment, and (4) Motivation, respectively. The assumption of homogeneity of variance was supported for Self-confidence scores, Levene (1, 86) = 1.619, p = .21, for Value, Leven (1, 86) = .753, p = .39, for Enjoyment, Levene (1, 86) = 3.288, p = .073, and for Motivation, Levene (1, 86) = 1.036, p = .31 according to the results of Levene's F tests of homogeneity.

The four one-way analyses of variance indicated that there were no statistically significant differences when the data were grouped by gender. Partial eta squared values indicated very small effect size. Table 2 shows the results of the analyses of variance and partial eta squared values. Table 3 shows means and standard deviations of the scores on the four dependent variables by gender. All statistical tests were performed using an alpha of .05.

In the one-way design with math course as the independent variable, data were analyzed using four separate analyses of variance (ANOVA). For each one of the ANOVAs, the dependent variable was one of the four factors: (1) Self-confidence, (2) Value, (3) Enjoyment, and (4) Motivation, respectively. The assumption of homogeneity of variance was supported for Self-confidence, Levene (1, 86) = .238, p = .63, for Value, Levene (1, 86) - 1.709, p = 20, for Enjoyment, Levene (1, 86) = .043, p = .84, and for Motivation, Levene (1, 86) = .52, p = .82 according to the results of Levene's F tests of homogeneity.

Data analysis indicated statistically significant differences among the scores on self-confidence, enjoyment and motivation when grouped by math course.

Partial eta squared values indicated small effect size for self-confidence and medium effect size for enjoyment and motivation. Students in precalculus scored significantly lower than students in calculus on self-confidence, enjoyment and motivation. No significant differences were found on the scores for value. Table 2 shows the results of the analysis of variance and partial eta square values. Table 4 shows means and standard deviations of the scores on the four dependent variables by math course. All statistical tests were performed using an alpha of .05.

Conclusions

Students at this "fairly selective" liberal arts school have for the most part been very successful in their study of mathematics (The mean SAT Math score for entering freshmen in fall, 2004 was 588, n = 514.) Thus, the females who attend would be expected to have more positive attitudes toward mathematics than their less successful counterparts from high school. It is not surprising that differences between genders in affect toward mathematics are not found in this population.

Students enrolled in precalculus scored significantly lower than students enrolled in calculus on items related to self-confidence, enjoyment, and motivation. No difference was found between precalculus and calculus students in the scores related to the value of mathematics.

Students who have below 620 on their mathematics SAT are recommended to take precalculus. Calculus I students have generally either had precalculus or had a mathematics SAT score greater than or equal to 620. Since the instrument was administered in the fall semester and 24 of the 43 students were freshmen, most of the students enrolled in calculus would not have taken precalculus in scollege.

The results support the notion that students who are more successful (as measured by initial placement in a college mathematics course) in mathematics are more self-confident, enjoy mathematics more, and are more motivated. Those affective characteristics are concerned with how an individual personally responds to the study of mathematics--whether there is anxiety, pleasure, willingness to study, etc. Differences of this nature are relevant in interpreting course evaluations, planning instruction, advising, etc.

The other affective characteristic--value--is a more detached view of the nature of mathematics. Whether mathematics is viewed as worthwhile, relevant, necessary, helpful, important, etc., is not dependent on one's success in mathematics. As confirmed by the NAEP data, students at all grade levels see mathematics as utilitarian. This is certainly encouraging to mathematics educators who have so often heard the question, "When am I ever going to use this?"

Recent literature on gender differences and the results of this study indicate that there is less concern for the gender gap than there was a few years ago. However, there is still a need to encourage the participation of females in the mathematical sciences. Faculty and administration in higher education should be aware of the differences in attitude toward mathematics for students at different levels of study.

Questions remain as to the relation between attitudes and achievement. Do students do well because they have good attitudes? Do students have good attitudes because they will do well? Instructors need to approach the teaching of mathematics by viewing the introductory courses as pipelines, not filters, in the study of mathematics. Can students' attitudes be improved by focusing on seeing mathematics as making sense and promoting conceptual understanding? If there is less focus on manipulation and algorithms and more value placed on mathematical thinking, will students' self-confidence increase or decrease? If persistence in problem-solving is made a goal, will motivation and enjoyment be more apparent? The standards of the National Council of Teachers of Mathematics (NCTM, 2000) make such questions extremely relevant. Much remains to be learned about the attitudes toward mathematics of students in introductory mathematics courses.

References

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American Association of University Women. (2004). Under the microscope of gender equity projects in the sciences. Washington, DC: American Association of University Women Educational Foundation. Retrieved April 1, 2004, from http://www.aauw.org/research/micoscope.cfm

Beller, M., & Gafni, N. (2000). Can item format (multiple-choice vs. open-ended) account for gender differences in mathematics achievement? Sex Roles, 42, 1-21.

Campbell, J. R., Reese, C. M., O'Sullivan, C. Y., & Dossey, J. A. (1996). NAEP 1994 trends in academic progress: Achievement of U.S. students in science, 1969 to 1994, mathematics, 1973 to 1994, reading 1971 to 1994, and writing, 1984 to 1994. Washington, DC: National Center for Education Statistics.

Cohen, D. (ed.). (1995). Crossroads in mathematics: Standards for introductory college mathematics before calculus. Memphis, TN: American Mathematical Association of Two-Year Colleges. Retrieved April 1,2004 from http://www.imacc.org/standards

Collins, D. W., & Kimura, D. (1997). A large sex difference on a two-dimensional mental rotation task. Behaviorial Neuroscience, 111(4), 845-849.

Davis-Kean, P., Eccles, J., & Linver, M. (2003, April). Influences of gender on academic achievement. Paper presented at the biennial meeting of the Society for Research on Adolescence.

Hacker, A. (2003, June 20). How the B.A. gap widens the chasm between men and women. The Chronicle of Higher Education. Retrieved April 1, 2004, from http: chronicle.com/prm/weekly/v49/i41/41b01001.htm

Hammrick, P. L. (2002). Gender equity in science and mathematics education: Barriers of the mind. In J. Koch, & B. Irby (Eds.), Defining and redefining gender equity in education. Greenwich: Information Age Publishing, 81-98.

Haycock, K., & Steen, L. A. (2002). Add it up: Mathematics education in the U.S. does not compute. Thinking K-16, 6, 1.

Hoover, E. (2001, September 7). Average scores on the SAT and the ACT hold steady. Chronicle of Higher Education. Retrieved April 1, 2004 from http://chronicle.com/prm/weekly/v48/902/02a05201.htm

Kenney, P. A., & Silver, E. A (1997). Results from the Sixth Mathematics Assessment of the National Assessment of Educational Progress. Reston, VA: National Council of Teachers of Mathematics.

Lutzer, D. J., & Maxwell, J. W. (2000). Statistical abstract of undergraduate programs in the mathematical sciences in the United States. Washington, DC: Conference Board of Mathematical Sciences.

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National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: Author.

Nordvik, H., & Amponsah, B. (1998). Gender differences in spatial abilities and spatial ability among university students in an egalitarian educational system. Sex Roles: A Journal of Research, June, 1998. Retrieve April 1, 2004 from http://www.findarticles.com/cf_dls/m2294/n1112_v38/21109782/p1/article.jhtml

Tapia, M., & Marsh, G. E., II (2004). An instrument to measure mathematics attitudes. Academic Exchange Quarterly, 8(2), 16-21.

U.S. Department of Education, Office of Educational Research and Improvement, National Center for Education Statistics. (1990). A profile of the American eighth-grader. NELS:88 Student descriptive summary, Washington, D.C.: Government Printing Office.

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Dr. Martha Tapia & Dr. Carla C. Moldavan

Berry College

Table 1. Sample items by factors

Items by Factor

Self-confidence
 Mathematics does not scare me at all.
 Studying mathematics makes me feel nervous.
 My mind goes blank and I am unable to think clearly when working
 mathematics.
Value
 Mathematics is a very worthwhile and necessary subject.
 Mathematics courses will be very helpful no matter what I decide to
 study.
 Mathematics is important in everyday life.
Enjoyment
 I really like mathematics.
 I have usually enjoyed studying mathematics in school.
 I am happier in a math class than in any other class.
Motivation
 I am willing to take more than the required amount of mathematics.
 I plan to take as much mathematics as I can during my education.
 The challenge of mathematics appeals to me.

Table 2. Analysis of Variance Summaries of Self-confidence, Value,
Enjoyment, and Motivation by Sex and Math Course

Source df SS MS F P Partial [[eta].sup.2]

Self-confidence
Sex
 Between 1 231.47 231.47 1.36 .25 .02
 Within 86 14686.98 170.78
Math Course
 Between 1 754.51 754.51 4.47 .04 .05
 Within 87 14683.38 168.77

Value
Sex
 Between 1 2.76 2.76 .07 .80 .00
 Within 86 3673.32 42.71
Math Course
 Between 1 110.42 110.42 2.69 .10 .03
 Within 87 3566.07 40.99

Enjoyment
Sex
 Between 1 47.46 47.46 .70 .40 .01
 Within 86 5810.99 65.57
Math Course
 Between 1 469.06 469.06 7.57 .01 .08
 Within 87 5390.54 61.96

Motivation
Sex
 Between 1 .65 .65 .03 .87 .00
 Within 86 1937.85 22.53
Math Course
 Between 1 203.61 203.61 9.99 .00 .10
 Within 87 1773.51 20.39

Table 3. Means and Standard Deviations by Gender

 Male Female
Construct Mean SD N Mean SD N

Self-confidence 55.37 12.36 49 52.10 13.91 39
Value 39.51 7.31 49 39.51 5.39 39
Enjoyment 34.73 7.42 49 33.25 9.14 39
Motivation 16.33 4.47 49 16.15 5.08 39

Table 4. Means and Standard Deviations by Math Course

 Precalculus Calculus
Construct Mean SD N Mean SD N

Self-confidence 50.85 12.88 46 56.67 13.11 43
Value 38.28 4.88 46 40.51 7.71 43
Enjoyment 31.85 7.97 46 36.44 7.76 43
Motivation 14.72 4.46 46 17.74 4.58 43