Mathematics Self-Efficacy of College Freshman By J. Michael Hall and Michael K. Ponton ABSTRACT: The purpose of this study The number of students enrolling in col- is to determine differences in mathematics leges and universities, and consequently in de- self-efficacy between students enrolled in a velopmental courses such as developmental developmental mathematics course and mathematics, has continued to increase over those enrolled in a calculus course. Data the past 30 years to a level where 3 out of 10 from a sample of 185 freshmen students at first-time freshman students are enrolled in a single 4-year institution using the Math- such a course (Breneman & Haarlow, 1998; ematics Self-Efficacy Scale are analyzed. Smittle, 2003). With the number of students Results indicate that calculus students pos- requiring developmental courses growing sess not only better mathematical skills but yearly, many colleges and universities are con- also a more powerful sense of self-belief in tinuing to invest money in these courses by their ability to succeed in a college math- creating additional courses, hiring new fac- ematics course. The results of this study ulty, and sometimes creating new depart- Students lack the ability suggest that future teaching methodologies ments for developmental studies. As a result should be designed specifically for students of colleges spending valuable resources on to identify factors that developmental coursework, many in the fac- enrolled in developmental courses that not limit their success [in only develop mathematics capability but also ulty and surrounding community want assur- ance that the funding for developmental a self-awareness of increased capability. mathematics]. courses is not wasteful. In addition, Arendale Efficacy-enhancing instructional strategies (2003) notes that both legislatures and boards should be tested for effectiveness, thereby of higher education desire to make institutions improving the teaching and learning process more accountable for remediation of students. for all learners. Therefore, creating quality instruction in de- Students’ ability to learn and succeed in velopmental courses has become a priority at mathematics has been a concern of educators many institutions. for many years, especially since mathematics At its core, developmental education seems to be a determinate of not only choice should attempt to expand the academic skills of a college major but also serves as a deter- of students. Hence, relationships between the minant in the acquisition of a college degree. students enrolled in developmental courses Trusty and Niles (2003) assert that high school and cognitive factors such as self-efficacy need students earning high school credit in rigor- to be established. Research (Breneman & ous math courses have a much greater likeli- Haarlow, 1998; Higbee & Thomas, 1999; hood of success in acquiring a bachelor’s de- Stanley & Murphy, 1997; Wheland, Konet, & gree than students not completing such a Butler, 2003) indicates that, for developmen- course. Related research has been conducted tal mathematics students, academic self-con- in an attempt to establish a relationship be- cepts, attitudes toward success in mathemat- tween success in mathematics courses and ics, confidence in ability to learn mathemat- success in college. For example, several stud- ics, mathematics anxiety, self-efficacy, and ies (Campbell & Hackett, 1986; Hackett, Betz, locus of control are all variables that affect J. Michael Hall O’Halloran, & Romac, 1990) have determined Assistant Professor student goals, performances, and attainments that previous mathematics performance and Department of Mathematics and Statistics in mathematics. Higbee and Thomas opine Arkansas State University-Jonesboro perceived ability are both key elements for that for educators to be effective, they must P.O. Box 70 success in mathematics. Furthermore, re- have an understanding of how students State University AR 72467 search (Dorner & Hutton, 2002; Moreno & cognitively process information. [email protected] Muller, 1999; Hagedorn, Siadet, Fogel, Nora, Wheland, Konet, and Butler (2003) sug- & Pascarella, 1999) indicates that, although gest that many students attribute poor per- Michael K. Ponton many courses aide students in the completion formances in mathematics not to themselves, Professor of Education of a college degree, mathematics is the sub- but to factors out of their control. Such fac- School of Education ject most essential to students’ choices in de- tors include instructors, time and day of class, Regent University 1000 Regent University Drive termining college majors and ultimately to and instructional style. At the same time, stu- Virginia Beach, VA 23464 success in attaining a college degree. dents lack the ability to identify factors that 26 Journal of Developmental Education limit their success. Furthermore, Higbee and forming tasks and lack motivation to do so randomly selected clusters from the total num- Thomas (1999) assert that correlations be- have no incentive to exert effort (Schunk & ber of sections offered. tween mathematics anxiety, test anxiety, and Pajares, 2002). Therefore, self-efficacy is a Students enrolled in the class sections lack of confidence in one’s ability to complete mediating factor for academic outcomes, cog- chosen for participation were approached mathematical tasks do exist and may possibly nitive engagement, and performance (Patrick during the first month of the fall semester indicate that student achievement is not only & Hicks, 1997). and asked to participate in the study. Since related to external factors, such as the faculty student participation was completely volun- member and their instructional style, but also Purpose tary, it was explained that failure to partici- to student attitudes toward mathematics. Self-efficacy has been shown to be a me- pate in the study would have no effect on their Furthermore, because of the mandatory en- diating factor in human achievement across grade or standing in the class. rollment associated with developmental numerous domains, thereby necessitating re- classes, previous research (Bassarear, 1986; search that focuses attention on the difficul- Sample Higbee & Thomas, 1999) indicates that there ties of accomplishing tasks. One such area of The sample consisted of 185 freshman is a stigma associated with being labeled as a difficulty for numerous individuals is the abil- students from four sections of Calculus I and remedial math student. The embarrassment ity to complete a college-level mathematics four sections of Intermediate Algebra. Of the of enrollment in remedial mathematics is es- course. Being able to establish a relationship total 185 participants, 80 were enrolled in pecially damaging to females and minorities between perceived ability to successfully ac- Calculus I and 105 were enrolled in Interme- (Green, 1990) and their perception of math- complish mathematical tasks and class enroll- diate Algebra (see Table 1). Eighty-five of the ematics (Glennon & Callahan, 1968). Re- ment (i.e., developmental vs. calculus) could students were male, and one hundred were search by Betz and Hackett (1983) suggests serve college and university programming female. It should be noted that questionnaires that educators need to gain a complete un- efforts. It would allow for the creation of pro- completed by nonfreshman students enrolled derstanding of how these individuals are af- grams and instructional methods aimed at in either of the courses were not included in fected academically by such stigmas because creating self-sufficient learners capable of this sample. Within the courses, a total of 43 many career decisions are based on the per- making choices concerning their ability to males and 37 females were enrolled in Calcu- ception of ability to excel in a given field. complete tasks and accomplish goals. Unfor- lus I, whereas a total of 42 males and 63 fe- It is hypothesized in this study that per- tunately, there is a lack of research compar- males were enrolled in Intermediate Algebra. sonal beliefs in capability may be one of the ing the self-efficacy of developmental stu- inhibitors to success for students enrolled in dents to nondevelopmental students. Table 1 courses such as developmental mathematics. Therefore, the purpose of this research Descriptive Data of the Sample Personal belief in capability to organize and was to examine the differences in the Class Gender n % execute actions to produce outcomes is de- mathematics self-efficacy of freshman Calculus I All 80 43.2 fined as self-efficacy (Bandura, 1997). Per- students enrolled in Developmental Male 42 52.5 ceptions of self-efficacy are derived from four Mathematics (Intermediate Algebra) and Female 38 47.5 sources of information: (a) personal accom- Calculus I. Intermediate Algebra All 105 56.8 plishments, (b) verbal persuasion, (c) vicari- Male 42 40.0 ous learning experiences, and (d) physiologi- Study Description Female 63 60.0 cal and affective reactions (Bandura, 1986). Note. N = 185 The subjects in this study were ran- To be more specific, each of these sources of dom cluster samples of intact classes en- self-efficacy serves as a primary determinant rolled in one of two freshman-level mathemat- of how individuals make choices, expend ef- Instrumentation ics courses (Intermediate Algebra, N = 375, fort to achieve goals, and persevere through Consisting of two subscales and a total and Calculus I, N = 400) at a medium-sized, the completion of these goals (Bandura, 1997; of 34 items, the Mathematics Self-Efficacy rural, state, research university in the South- Schunk, 1996). Thus, individuals with low Scale (MSES) was originally developed in 1983 east for the Fall 2001 semester. Placement of levels of self-efficacy feel as if they do not by Betz and Hackett and contained 75 items. the students into each of the classes is based possess the requisite skills necessary to per- Revised in 1993 for the interest of parsimony, on American College Test (ACT) math sub form given tasks. the current version of the MSES contains a scores. The ACT is comprised of four sec- Human behavior is motivated by antici- Mathematics Tasks subscale and a Mathemat- tions (i.e., reading comprehension, mathemat- patory thought processes in which the capa- ics, verbal, and science reasoning) each hav- ics Courses subscale. The purpose of the bility of forethought is used to cognize desir- ing a range of scores from 0-36. Similar to Mathematics Tasks subscale is to measure stu- able future states and select courses of action other U.S. colleges and universities, the Uni- dent confidence in the ability to perform ev- that are ideated as paths to these states. This versity of Mississippi uses the ACT as a eryday mathematics tasks; the purpose of the process is mediated by capability perceptions method to evaluate student ability in courses Mathematics Courses subscale is to assess stu- perhaps without the presence of actual capa- such as mathematics. Calculus I is designed dent confidence in their ability to earn a B or bility (Bandura, 1997; Zimmerman, 1990). for engineering, physical science, and math- better in a college course that requires math- Hence, although students may lack all the nec- ematics majors with ACT math sub scores ematical skills. essary skills to perform tasks, motivation with above 25, whereas Intermediate Algebra is Betz and Hackett (1993) note that the self-efficacy can serve as a mechanism through taken by students in nonmathematics majors content validity for the MSES has been dem- which students overcome finite ability and suc- with ACT math sub scores of 16 or less. At onstrated through research that validates each cessfully achieve the desired goals (Bandura, the University of Mississippi Intermediate area measured by the instrument. They note 1997; Zimmerman, 1990; Zimmerman, Algebra is the course designation for devel- that there were positive correlations between Bandura, Martinez-Pons, 1992). Conversely, opmental mathematics. The sections of each the MSES and other mathematics scales such students who do not feel efficacious in per- class chosen to participate in the study were as math anxiety (r = .56), confidence in doing Volume 28, Number 3, Spring 2005 27 mathematics (r = .66), perceived usefulness of self-efficacy to past experiences Table 3 mathematics (r = .47), and effectance motiva- and how those experiences relate t-Tests of MSES Score by Gender tion in math (r = .46), thus enhancing the con- to them personally. Self-reflection current validity of the instrument. Variable Mean MSES SD n t p of exposure to, or lack of exposure Calculus I to, mathematics classes is therefore Results Male 7.111 1.0452 43 .254 .800 the primary source of mathemat- Female 7.046 1.2571 37 ics self-efficacy. In essence, it is An independent t-test was conducted to Intermediate Algebra difficult for students to objectively determine if there was a significant difference Male 5.392 1.3011 42 .337 .737 between the mathematics self-efficacy of stu- Female 5.294 1.5448 63 evaluate themselves on topics for which they have little knowledge. dents enrolled in Intermediate Algebra and Calculus I as measured by the MSES. A two- Due to two independent variables on the de- Therefore, exposure to mathemat- sample Kolmorgorov-Smirnov Test was con- pendent variable MSES, a two-way analysis of ics with positive outcomes increases math- ducted to validate the assumption of normal- variance was performed to determine the ematics self-efficacy, whereas exposure to ity. The results indicated (p = .580) that the main effects and interactions (see Table 4). mathematics with negative outcomes de- data were indeed normal, thereby allowing for The results indicate no gender main effect (F creases self-efficacy, provided the positive the use of the two-sample t-test. The mean = .167, p = .683) and no significant interac- outcomes are attributed to increase in per- MSES score for the students in Intermediate tion between gender and course enrollment sonal capability and/or effort by the student. Algebra was 5.33 (SD = 1.4464); the mean on mean MSES score (F = .007, p = .936). MSES score for the students in Calculus I was Consistent with the t-test, the two-way ANOVA Implications for Practice 7.08 (SD = 1.1411). The results of the t-test (t shows (F = 75.753, p < .001) that the mean Most educators would agree that a given = 8.902, p < .001) suggested that the means MSES score of Calculus I students is unequal amount of mathematical knowledge is neces- are not equal (see Table 2). Thus, a signifi- to the mean MSES score of Intermediate Al- sary for all college graduates. Although not cant difference between the level of mathemat- gebra students, thereby suggesting a greater all students need to learn calculus, all students ics self-efficacy between freshman students en- self-efficacy for the Calculus students. do need a comfortable level of mathematical rolled in Calculus I and Intermediate Alge- ability that does not limit life-altering choices, bra was demonstrated: Calculus I students Discussion such as the choice of a major. Thus, the role exhibited a higher mathematics self-efficacy With 40% of all freshmen in 4-year col- of educators should be to do whatever is nec- than the Intermediate Algebra students. leges and universities requiring developmen- essary to aid students in increasing their per- tal education of some ception of actual ability. In classes such as Table 2 kind (Smittle, 2003), Intermediate Algebra, raising the mathemat- faculty at these institu- ics self-efficacy of all students to a level where t-Tests of MSES Score by Class Enrollment tions are charged with students’ choices are not limited should, there- Class Mean MSES SD n t p the task of not only fore, be a primary concern of educators. Calculus I 7.08 1.1411 80 8.902 <.001 being able to recog- Research conducted by Trusty and Niles Intermediate Algebra 5.33 1.4464 105 nize deficiencies in (2003) indicates that high school students Note. N = 185 students but also ad- completing rigorous mathematics courses justing instructional have much higher levels of success in college In addition, a Pearson product-moment methods in a way that enables each student than students who do not earn credit in such correlation was also performed to determine the same opportunity to succeed in a college a course. Developmental education is a if MSES scores correlated with individual ACT classroom. If students have not achieved suc- method to offer students who have not had test scores. The results for all participants (r cess in high school mathematics, additional the opportunity to be successful in mathemat- = .580, p < .001) and for the students enrolled ics such an opportunity, espe- in Calculus I (r = .454, p < .001) showed mod- Table 4 cially important in a content area erate correlations, whereas the results for stu- Two-Way Analysis of Variance of MSES Score by recognized as a gatekeeper to dents enrolled in Intermediate Algebra (r = Class and Gender college retention and success. .052, p = .598) indicated no significant corre- By fusing the concepts of devel- Source SS df MS F p lation. opmental education and rigor of Since mathematics self-efficacy has been Class 133.977 1 133.977 75.753 <.001 mathematics, teachers of devel- shown to be a mediating factor influencing Gender .295 1 .295 .167 .683 opmental mathematics courses choices of academic major by females (Betz & Class*Gender .0115 1 .0115 .007 .936 could hold the key for college Hackett, 1983; O’Brian, Martinez-Pons, & Error 320.117 181 1.769 success. For this reason, it is sug- Kopola, 1999), additional t-tests were per- Total 459.194 184 gested that teachers of develop- formed (see Table 3) on data within each mental mathematics courses cre- course concerning gender differences. In pressure is placed on developmental math- ate a learning environment conducive to fos- Intermediate Algebra, results of an indepen- ematics instructors. This pressure is height- tering self-efficacy in developmental students dent t-test indicated that there was no signifi- ened by the identified link between success while keeping the rigor of the course compa- cant difference in the MSES score based on in mathematics and college success (Trusty & rable to other college courses. This may be the gender of the students (t = .254, p = .800). Niles, 2003). Also, self-efficacy is one area accomplished by several means. For example, In Calculus I, the results of an independent t- specifically shown to aid individuals in the giving students chances for success in small test indicated that there was no significant determination of how much effort they should increments can serve to improve their mas- difference in MSES score based on gender of expend in order to complete a task. Bandura the students (t = .337, p = .737). (1997) suggests that individuals attribute their continued on page 30 28 Journal of Developmental Education Volume 28, Number 3, Spring 2005 29 continued from page 28 et al.). Pointing out increases in capabilities of mathematical ability, thereby necessitating may be even more important for students in additional research to determine not only the tery experiences. Additionally, creating an developmental courses because, in order to sources of information each gender group environment that exhibits the magnificence increase choices that influence life trajecto- uses to determine self-efficacy but also to fur- of mathematics and its implications to other ries, changes in actual capability must be ac- ther establish the relationships between gen- fields can allow students to see practical ap- companied by adjustments to self-efficacy. der and mathematical ability. plications that may be of interest to them. Lastly, encouraging students to ask questions Future Research Conclusion regarding mathematical operations and ap- With the results of this research indicat- Unsuccessful attempts have been made plications and then serving as a guide through ing differences in levels of mathematics self- to enhance mathematics self-efficacy through discussion can augment student understand- efficacy for students enrolled in Intermedi- training (Rushing, 1996). Students in ing that mathematics holds the key to many ate Algebra versus students enrolled in Cal- Rushing’s treatment group were given special- fields of study and ultimately to choice of culus, prospects for future research concern- ized materials such as teacher-created vocabu- major. ing the mathematics self-efficacy and students lary lists, individualized study guides, and Past experiences, often times failures, in enrolled in developmental mathematics journaling materials aimed at improving their mathematics usually dictate student opinions courses are promising. First, specific instruc- understanding of the mathematical material concerning their perception of personal abil- tional strategies need to be evaluated longi- and their self-efficacy. Though such attempts ity in mathematics as well as their optimism tudinally to determine their influence on en- may seem futile, it must be noted that learn- about career choices for which mathematics hancing self-efficacy beliefs. Second, Cassazza ing mathematics has been a lifelong struggle is the basis of the curriculum. Therefore, with- (1999) has shown that the fastest growing seg- for many students. Thus, development and out confidence in mathematical ability, stu- ment of higher education is the number of classroom implementation of new tools for dents’ choices of majors, and ultimately their nontraditional-aged learners, and research learning subjects that have continually futures, may be limited to nonmathematical regarding self-efficacy should be conducted plagued students is quite difficult and chal- areas. Though some students would naturally lenging for faculty and students. choose to major in such an area, the point is Past experiences...usually There are no easy solutions to the com- to broaden students’ choices rather than limi- plex problems that confront students with low dictate student opinions tations. Hence, institutions of higher learn- levels of mathematics self-efficacy. However, ing and educators themselves should imple- concerning their personal continual attempts should be made at enhanc- ment modes of instruction that develop and ing the learning experience for students who ability in mathematics. enhance self-efficacy in groups of students have been shown to have low levels of self-ef- with lagging self-concepts of mathematical ficacy thereby enabling individuals to master ability. Doing so would allow students to more the important concepts of mathematics while adequately gauge their actual ability, thereby with this population. Lastly, many students enabling them to become lifelong, self-regu- helping students to make better choices con- entering college are simply not prepared in lated learners. To maximize their impact on cerning their future. high school to be successful in college math- students’ lives, developmental courses must Because self-efficacy has been shown to ematics classrooms. Factors contributing to not only develop actual skills but also a posi- be a mediating influence on motivation and this unpreparedness may include, but are not tive self-image of capability. performance (Bandura, 1997; Ponton, Horine- limited to, the physical size and location of Edmister, Ukeiley, & Seiner, 2001), enhanc- the high school, mathematics courses offered References ing mathematics self-efficacy should be an at the high school, and the highest mathemat- Arendale, D. (2003, October). Developmental important part of any effort to aid in the aca- ics class completed in high school. Therefore, education: Recognizing the past, preparing for demic growth of students enrolled in lower- additional research should include the devel- the future. Paper presented at the Minne- level mathematics classes. Too often educa- opment of instructional methodologies that sota Association for Developmental Edu- tors attempt to teach students with low math- focus on increasing the mathematics self-effi- cation 10th Annual Conference, Grand ematics self-efficacy in the same manner by cacy and ability of each subgroup identified Rapids, MN. which students that have higher levels of math- as having substandard levels of each. Bandura, A. (1986). Social foundations of ematics self-efficacy are taught. By altering Contrary to previously established re- thought & action: A social cognitive theory. instructional methodologies educators can search (Betz & Hackett, 1983; Hackett, Betz, Englewood Cliffs, NJ: Prentice-Hall. assist students to create more meaningful O’Halloran, & Romac, 1990; Lent, Lopez, Bandura, A. (1997). Self-efficacy: The exercise of mastery experiences in mathematics. Ponton Brown, & Gore, 1996; O’Brian, Martinez- self-control. New York: W.H. Freeman and et al. propose that faculty consider two sug- Pons, & Kopala, 1999) suggesting that females Company. gestions when attempting to design mastery possess lower levels of mathematics self-effi- Bassarear, T. (1986, April). Attitudes and be- experiences for students: (a) What exactly do cacy than males, the findings of this study liefs about learning, about mathematics, and we want the students to master? and (b) How suggest that there is little difference in the about self which most seriously undermine per- are we going to let them know? It should be mathematics self-efficacy between males and formance in mathematics courses. Paper pre- the role of a faculty member to clearly define females. Various factors that might contrib- sented at the annual conference of the New for the students the importance of the ex- ute to differences in the findings include the England Educational Research Organiza- plicit/implicit material, create mastery expe- possibility that the females participating in the tion, Rockport, ME. (ERIC Document riences to enhance knowledge and skill, de- study do indeed possess higher levels of math- Reproduction Service No. ED 299147) velop modes of assessment that accurately ematics self-efficacy than their male counter- assess desired attainments, and highlight to parts. It is also possible that the participat- students actual increases in capability (Ponton, ing females inaccurately assessed their level continued on page 32 30 Journal of Developmental Education Volume 28, Number 3, Spring 2005 31 continued from page 32 continued from page 10 Betz, N. E., & Hackett, G. (1983). The rela- pursuit. Journal of Early Adolescence, 17, 109- Casazza, M. 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