Two-Year College Mathematics and Student Progression in STEM Programs of Study
Debra D. Bragg
Professor and Director,
Office of Community College Research and Leadership
University of Illinois at Urbana-Champaign
In spite of the strident pursuit of standards-based reform of twoyear college mathematics, implementation of reform has been slow and uneven. National studies show student enrollment in two-year college mathematics is growing in proportion to overall enrollment growth in higher education, but a substantial portion of these students are taking pre-college mathematics courses. Research suggests many of these students never reach college-level mathematics. Improvements need to be made to two-year college mathematics to prepare more students for science, technology, engineering, and mathematics (STEM)-related careers. Specific recommendations to support this goal are
Take a P-20 approach to reforming the entire mathematics curriculum. Without a strategic, collaborative endeavor, it will be difficult for two-year colleges that are caught between K-12 education and higher education to implement and sustain meaningful change.
Conduct more research on the teaching and learning of two-year college mathematics. Finding ways to support two-year college faculty to engage in professional development that reinforces innovative pedagogies is important. Included in this list are topics linked to quantitative literacy, accelerated and contextualized instruction, and college placement and related assessments that need to be better linked closely to student learning.
Investigate more fully the characteristics, experiences, and aspira-
tions of students who enroll in two-year college mathematics. More information is needed about how diverse learners, especially women and minorities, experience their two-year mathematics courses (pre-college and college level) and how these experiences influence their subsequent enrollment, completion, and career decisions related to STEM.
Engage practitioners in action research on mathematics education to facilitate the adoption and scale-up of innovation. Two-year faculty would benefit from opportunities to engage in action research that helps them to understand how various pedagogical and assessment strategies impact the learning of diverse students, and then employ these strategies in their classrooms.
There is wide consensus that mastery of mathematics is essential to progressing into and through STEM programs of study, yet many students are unsuccessful at navigating the normative mathematics course sequence (Cullinane and Treisman, 2010) that is fundamental to their advancement into STEM-related careers. Recent concerns about international competition and the struggling economy have focused attention on this important issue and renewed concerns about the challenges that many students, particularly women and minorities, face succeeding in mathematics coursework (National Academy of Sciences, National Academy of Engineering, and Institute of Medicine, 2010). Resolving this problem is an urgent priority if the nation is to see growth in student enrollment and success in STEM programs of study, placement of graduates in STEM-related careers, and rejuvenation of the nationâ€™s economy.
This paper examines the influence of the two-year mathematics curriculum on studentsâ€™ progression into and through STEM programs by drawing upon extant literature, materials on the Internet, and personal communication with two-year college mathematics experts and practitioners. It acknowledges the expansive developmental mathematics curriculum offered by two-year colleges, but even more importantly, provides insights into college-level mathematics that has been overshadowed by a preoccupation with developmental education. The paper begins with a brief historical perspective and then proceeds to address such questions as: what is the status of two-year mathematics courses, who teaches them, and how are they taught? What standards-based reforms are associated with two-year college mathematics, what curricular and pedagogical innovations are capturing the attention of mathematics reformers, and what do we know about the impact of these reforms on student success? This paper concludes with recommendations for future research, policy, and practice on two-year college mathematics that is intended to enhance
student progression through STEM programs of study and into STEMrelated careers.
PERSPECTIVES ON MATHEMATICS CURRICULUM IN THE TWO-YEAR COLLEGE
A useful framework for understanding two-year mathematics curriculum comes from Cullinane and Treisman (2010), who label the mathematics curriculum in the United States the “normative mathematics course sequence” (pp. 7-8), which they claim is ubiquitous to the P-20 (primary through grade 20) education system. The normative mathematics course sequence extends from basic arithmetic, to pre-algebra, algebra, and intermediate algebra on to trigonometry, pre-calculus, calculus, and other calculus-based courses, with a fuzzy demarcation between precollege and college-level mathematics that starts with college algebra. Geometry may be part of the sequential mathematics continuum, or it may be omitted, to the detriment of studentsâ€™ advancement into calculus and calculus-based sciences such as physics. Because this framework represents the dominant schema for which mathematics is taught and for which student competence is assessed at the secondary and postsecondary levels, I use this framework as the basis for discussing the literature. Later, in my discussion of reforms of the two-year college mathematics curriculum, I again cite Cullinane and Treisman (2010) who are studying alternatives to the normative mathematics course sequence. First, however, I provide a brief historical foundation and then move to contemporary developments in two-year college mathematics.
Liberal arts and sciences courses, including mathematics courses, have been part of the two-year college curriculum since creation of junior colleges in the early 1900s. Cohen and Brawer (1982) observed that, by the time two-year colleges arrived on the U.S. higher education scene, the academic disciplines were already “codified” (p. 284) by the rest of the educational system. Junior colleges that emerged to fill the void between high schools and universities adopted the prevailing curriculum structure advocated by the mathematics discipline and were therefore caught in between the K-12 sector and the four-year college sector from the start. To this end, Cohen and Brawer observed that, “the liberal arts [courses of two-year colleges] were captives of the disciplines; the disciplines dictated the structure of the courses; [and] the courses encompassed the collegiate function” (1982, p. 285). To facilitate the acceptance of college credits at the university level, two-year colleges reproduced the curriculum as well as the pedagogical methods used by universities to which their students sought entry.
Transfer was born from these early replication efforts. A landmark
study of junior colleges conducted at mid-20th century by Medsker (1960) confirmed the lengths to which two-year colleges mimicked university curriculum to enhance studentsâ€™ ability to transfer. He noted, “the junior college forfeits its identity and its opportunity to experiment in the development of a program most appropriate for it” (p. 53). Looking back to the start of the comprehensive curriculum of the two-year college, Cohen and Brawer (1982) cited findings from a very early study of 58 junior colleges conducted in 1921 and 1922 by Koos (1924, p. 29) that showed liberal arts, sciences, and humanities courses dominated the early junior college curriculum, with three-fourths of all courses representing these disciplines. Across a broad array of the liberal arts and sciences, mathematics represented about 8% of all course offerings. Whereas mathematics was not as dominant as English, communications, and the sciences, it was nearly universally offered in the two-year college. By the late-1950s, a national survey conducted by Medsker (1960) of 230 two-year colleges in 15 states confirmed mathematics courses were ubiquitous to the two-year college curriculum, but still, only a relatively modest proportion of students enrolled in them. In fact, only about one-quarter of two-year colleges required their students to take at least one mathematics course to meet general education requirements. Medskerâ€™s study was also important because it was one of the first to document the prevalence of pre-college courses in reading, writing, and mathematics, foreshadowing a phenomenon that would continue to grow to the present day.
Several decades subsequent to Medskerâ€™s study, Cohen and Brawer (1987) studied the two-year college curriculum and found remarkably similar findings about mathematics course-taking. Their analysis showed 9 percent of total course enrollments in the liberal arts, sciences, and humanities curriculum were in mathematics, and again reflective of Medskerâ€™s results, the survey revealed a high proportion of mathematics courses were at the pre-college1 level. Subsequent studies conducted by Cohen, Brawer, and colleagues included a curriculum mapping study conducted by Cohen and Ignash (1992) about two decades ago. This study examined courses offered by a national sample of two-year colleges by scouring the spring 1991 catalogs and course schedules of 164 community colleges, balanced by small (less than 1,500 students), medium, and large (over 6,000 students) institution size. Cohen and Ignash mapped the liberal arts and nonliberal arts curriculum into broad subject areas of which mathematics and computer science were combined into one area. Their study showed the preponderance of mathematics enrollments were
1Consistent with other literature on two-year college mathematics (see, for example, Blair, 2006), I use the term pre-college to refer to mathematics courses offered below the college level, including courses often referred to as developmental and remedial education courses.
in classes offered at the introductory and intermediate course levels, with enrollments at the introductory or intermediate level being nearly 9 times larger than enrollments at the advanced level (about 766,000 enrollments in the former and approximately 87,000 enrollments in the latter). Though enrollments were substantially lower in advanced mathematics courses, this study confirmed that two-year colleges offered a substantial array of mathematics courses, including courses extending from the developmental level to calculus, as well as statistics. The number of sections of mathematics accounted for 10.7 percent of the total liberal arts curriculum, which ranked mathematics just behind humanities at 13.4 percent and English at 12.8 percent.
Another important aspect of two-year college mathematics curriculum that is evident in the curriculum mapping study of Cohen and Ignash (1992), and that also has relevance to this discussion, pertains to the rise of nonliberal arts curriculum, a trend that began in the 1970s (Cohen and Brawer, 1987). Since much of mathematics course-taking in the two-year colleges relates to the majors that students choose in nonmathematics subjects, it is important to understand the ways mathematics is used to fulfill general education requirements. Cohen and Ignash (1994) identified the emergence of occupational-technical fields of study (many having a technical focus and having some STEM-related content) beginning in the 1970s, and they documented the growth of technical education, trades and industrial education, and other programs of study offered by two-year colleges that require various levels and forms of mathematics. Whereas the offering of liberal arts and sciences courses has been relatively robust over the years, by the 1990s nonliberal arts and sciences courses accounted for about 45 percent of the two-year college curriculum and occupational-technical education course were prominent among them. Technical mathematics and courses tailored for other majors such as elementary education were apparent in the curriculum as well. One implication of this trend is that the teaching of mathematics, which had been the purview of the mathematics discipline, spread to other instructional units and efforts to integrate mathematics with other subjects emerged as a strategy to increase learnersâ€™ abilities to apply mathematics in diverse occupational settings (Grubb, 1999).
Concerns about studentsâ€™ lack of preparation for college-level mathematics were also growing during the latter decades of the 20th century, as noted by an American Mathematical Association of Two-Year Colleges (AMATYC) report that showed developmental mathematics courses were offered by 91 percent of two-year colleges (Baldwin and the Developmental Mathematics Committee, 1975). Literature documenting the growth in remedial enrollments in two-year colleges observed pre-college mathematics courses were necessary for “marginally literate students
emanating from the secondary schools” (Cohen, 1984, p. 1). In a national survey conducted in the early 2000s, Greene and Forster (2003) found only 32 percent of all high school graduates demonstrated the level of competence needed to enter college mathematics coursework. Among all learners, Greene and Forster identified Hispanics and African Americans as “seriously underrepresented in the pool of minimally qualified college applicants” (p. 3), and they attributed their lack of preparation to inadequate K-12 education rather than “financial aid or affirmative action policies” (p. 3). Their research points to the uniquely important role that community colleges play in transferring underserved students to the university level in STEM-related careers (Arbona and Nora, 2009). However, transfer is not possible if students are unable to navigate the mathematics curriculum that begins at the pre-college level. Attewell et al. (2006) found 60 percent of community college entrants are required to take one or more developmental courses, usually mathematics, and numerous studies by Bailey and colleagues at the Community College Research Center (CCRC) (see, for example, Bailey, Jeong, and Cho, 2010) reveal the dismal success rates of students whose placement test scores prescribe multiple developmental mathematics courses. Beyond the developmental level, Adelman (2004) noted failure and withdrawal rates of 50 percent or higher in college algebra and pre-calculus courses, demonstrating problems with student success extend beyond to the gatekeeper course level. This trend and other critical aspects of the two-year mathematics curriculum, including documenting enrollment in mathematics course sequences extending from pre-college to college level, are examined in the next section.
CONTEMPORARY TWO-YEAR COLLEGE MATHEMATICS
Most of what we know about two-year college mathematics in the United States comes from a few large-scale national surveys. The fullest depiction of the curricular landscape about two-year college mathematics is the national inventory of the mathematics curriculum in U.S. higher education that has been conducted every five years starting in 1965 by the Conference Board of Mathematical Sciences (CBMS) with support from the National Science Foundation (NSF). The CBMS2005: Fall 2005 Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States is the last published installment of the national survey results on the two-year college mathematics curriculum (Lutzer et al., 2007). However, preliminary results of the 2010 CBMS national inventory on two-year college mathematics were shared with me by Ellen Kirkman and Rikki Blair (personal communication, December 4, 2011) to update
this paper.2 Taken together, the 2005 and 2010 reports (as well as selective use of earlier CBMS surveys) provide the most detailed description of two-year college mathematics curriculum offered by public community colleges in the United States, including trends in student enrollments, courses, faculty, and instructional practices.
The CBMS survey used a stratified, simple random sampling design, with strata based on the three variables of curriculum, highest degree level offered, and total institutional enrollments to address three distinct universes: two-year college mathematics programs, mathematics departments in four-year colleges and universities, and statistics departments in four-year colleges and universities. The stratum specifications used in the 2005 CBMS administration exactly replicated the ones used in the CBMS 2000, and closely emulates the specifications of previous CBMS surveys that were adjusted over time to improve national estimates. With respect to the 2005 CBMS administration, the most recent date for which a full report of methodology is available, a total of 600 public two-and four-year colleges and universities were surveyed during the period of September 2005 to May 2006. Minor adjustments are made to the CBMS at each administration, but the core of items included on the survey remains constant to address enrollments, instructional strategies, faculty demographics, and so forth.
Figure C-1 reveals fall enrollments in mathematics and statistics at a 5-year interval from 1975 to 2005 (Lutzer et al., 2007), compared to total enrollments in public four-year and two-year colleges obtained from the Community Colleges, Special Supplement to the Condition of Education, 2008 report (Provasnik and Planty, 2008). Results suggest the enrollments in mathematics and statistics are growing commensurate with the increase in enrollments in public four-year and two-year colleges over the last 25-year period. Total enrollment growth in public two-year college is highly correlated (r = .96) with mathematics and statistics enrollment over this time span. More recent enrollment figures for Fall 2010 for mathematics and statistics show enrollment climbed to an all-time high of 2,096,000 (E. Kirkman and R. Blair, personal communication), mirroring the enrollment growth in public two-year colleges to an unprecedented high of 7,101,000 for 2009, the latest year statistics are available from the U.S. Census Bureau (2012). With respect to the growth in mathematics enrollment, the CBMS 2010 survey showed a 26 percent increase from 2005, which is about the same percentage increase that was observed between 2000 and 2005.
2I want to express my sincere appreciation to Ellen Kirkman and Rikki Blair for their generosity in sharing preliminary tables from the forthcoming CBMS2010: Fall 2010 Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States.
FIGURE C-1 Total public enrollment in two-year and four-year colleges and twoyear mathematics and statistics enrollment (Fall 1975–Fall 2005).
Figure C-2 shows two-year college mathematics enrollments are critical to the overall U.S. postsecondary education system. Whereas two-year mathematics enrollment has seen some fluctuation over the 15-year period from 1990 to 2005, the 2005 mathematics enrollment figure confirms substantial growth from earlier years to the point where there were only modestly fewer enrollments in mathematics at the public two-year colleges (n = 1,580,000) than four-year public and private colleges (n = 1,607,000) by Fall 2005, based on 2005 CBMS results (Lutzer et al., 2007). These totals take into account dual enrollment, which has grown substantially over the last decade (Waits, Setzer, and Lewis, 2005); however, they do not take into account mathematics courses taught outside of mathematics disciplinary units, including centralized pre-college education units that are responsible for teaching pre-college mathematics classes. Therefore, these figures almost certainly underestimate enrollments in pre-college mathematics (arithmetic, pre-algebra, elementary algebra, intermediate algebra, and geometry) and possibly other mathematics-related courses taught on two-year college campuses, suggesting the actual enrollment in two-year college mathematics may be higher still.
Figure C-3 shows the percentage enrollment in two-year college mathematics courses by type of course and by the year the survey data were collected, based on the most recent CBMS 2010 data supplied by Kirkman and Blair (personal communication). Looking at the overall curriculum delivered by two-year college mathematics units, the preponderance
FIGURE C-2 Total enrollment in four-year college mathematics and two-year college mathematics (Fall 1990–Fall 2005).
FIGURE C-3 Total two-year college enrollment in mathematics and percentage of total enrollment at the pre-college level (Fall 1990–Fall 2010).
of enrollment is at the pre-college level, with a persistent percentage of about 57 percent of all enrollments in mathematics units over the last two decades. Other survey results reveal only modest changes in the distribution of enrollment across the mathematics curriculum since 1990 (not shown in tabular form), with a small but persistent decline in enrollment in pre-calculus (college algebra and trigonometry) courses since 1995, a slight drop but also fluctuation in calculus enrollment from 1990 to 2010, and a modest increase in enrollment in statistics courses since 1990 and in other mathematics courses since 1995, including classes for nonmathematics majors (e.g., mathematics for liberal arts and mathematics for elementary school teachers).
The CBMS also examined faculty and instruction, which is an important issue for two-year colleges where part-time faculty are well documented and an important part of the teaching workforce (Townsend and Twombly, 2008). The Fall 2005 survey reveals the extent to which part-time faculty are engaged in mathematics instruction (Lutzer et al., 2007), with the percentage of two-year college mathematics sections taught by parttime faculty being 44 percent. (Although a percentage was not apparent in preliminary results from the Fall 2010 survey, it is clear that part-time faculty numbers remain high in the 2010 CBMS survey.) Part-time faculty members are most evident at the pre-college level, with 56 percent of these sections being taught by part-time faculty, and part-time faculty are least involved in teaching of mainstream calculus and advanced mathematics compared to other courses in the two-year mathematics curriculum, with only 12 percent and 9 percent, respectively.
These results suggest students taking the advanced college-level mathematics curriculum are most likely to be taught by a full-time faculty, a finding that seems to recognize the importance of advanced mathematics curriculum being taught by professionally trained mathematics specialists as well as the need to align standards with disciplinary requirements that support student progression (transfer) to the university. Noting this advantage, there is little evidence to suggest students who take even the most advanced two-year college mathematics courses intend to continue their study of mathematics at the university level as mathematics or STEM majors. Looking at all the CBMS data for 2005, Lutzer et al. (2007) suggested relatively few two-year college students intend to transfer to the university and major in mathematics at the four-year college level, although community college science and engineering (S&E) students do transfer to the baccalaureate level to pursue S&E and engineering technology baccalaureate degrees. Indeed, associatesâ€™ degrees in S&E and engineering technology constitute about 11 percent of all associateâ€™s degrees awarded in 2007 (National Science Board, 2010), and presumably most of these students are interested in transferring. Looking
retrospectively, about 44 percent of S&E graduates attended community colleges (Tsapogas, 2004). The proportion of associateâ€™s degree holders who are racial-ethnic minorities is higher among associateâ€™s degree than bachelorâ€™s degree holders in S&E fields, making these programs a rich ground for recruiting of STEM majors at the baccalaureate level (Handel, 2011). From 1995 to 2007, the number of S&E associateâ€™s degrees earned by racial-ethnic minority students more than doubled from 7,836 to 19,435. The Fall 2005 CBMS survey data report on instructional approaches that provide insights into how two-year mathematics curriculum courses are taught (Lutzer et al., 2007). These data show over three-quarters of on-campus sections of college algebra and trigonometry, two courses core to the mathematics curriculum for many transfer students, are taught using the standard lecture method. The standard lecture method was less evident in pre-college course sections such as arithmetic (64%) and more evident in mainstream and nonmainstream calculus, elementary statistics, differential equations and technical mathematics (calculus), ranging from 81 percent to 93 percent. Given that calls for reform of mathematical pedagogy have been made for many years (see, for example, Wubbels and Girgus, 1997, and the authors of two-year college standards-based reform mentioned in the next section), it is perplexing that so little change has occurred in the teaching of such important two-year college mathematics courses.
Looking at both instructional and outreach methods in the Fall 2005 and Fall 2010 CBMS surveys (E. Kirkman and R. Blair, personal communication, December 4, 2011; Lutzer et al., 2007), results show college placement testing in mathematics is nearly universal in two-year colleges, although the preliminary finding from CBMS 2010 shows a 7 percent drop from 2005 to 2010 that deserves further study. Blair indicated she and her colleagues are still exploring the reason for this drop, but point out that all of the 90 percent of two-year colleges that report diagnostic testing report requiring mathematics placement testing for all incoming students. These results also show an increase in K-12 outreach opportunities and undergraduate research, but a decline in honors sections and special programs to encourage women and minorities to enroll in two-year college mathematics. In terms of the use of distance and online learning, the CBMS 2010 surveys show relatively modest use of online instruction, with most courses showing less than 10 percent of the sections using online learning systems. These results are consistent with previous results from CBMS 2000 and CBMS 2005.
Given the dearth of information about student enrollments and outcomes in two-year college mathematics, results of a national study by Horn and Li (2009) on postsecondary awards (credentials) below the baccalaureate level provide some insights into the scope and status of
degree attainment. Most importantly, this research shows only .5 percent of all subbaccalaureate awards conferred in 2007 by Title IV postsecondary institutions (public community colleges as well as private for-profit institutions that award credentials below the baccalaureate level and also participate in federal financial aid) are in mathematics and science fields. Whereas this statistic is alarmingly low, it represents an 11.5 percent increase from 1997 to 2007. Also, in spite of the fact that females earn a majority of all subbaccalaureate credentials, enrollments in mathematics and science by males and females shifted toward males, from 46.9 percent in 1997 to 50.3 percent in 2007. Results on subbaccalaureate awards were not disaggregated by discipline and race; however, Horn and Liâ€™s study confirms that most subbaccalaureate awards are conferred to whites. Among minority groups, Hispanics showed the largest increase (75%) in overall subbaccalaureate-conferred awards from 1997 to 2007; followed by African Americans, with a 54 percent increase; and whites with an 11 percent increase only. Results of national studies by Tsapogas (2004) and Dowd, Malcom, and Bensimon (2009) also report the tendency of Hispanics to use the two-year college to pursue science and engineering degrees, though again, these results were not specific to mathematics.
Given the need to reach more women, minorities, and other underserved student populations, it is not surprising that professional mathematics groups have made recommendations to change mathematics education. Among various groups to respond to calls for reform, the American Mathematical Association of Two-Year Colleges (AMATYC) played an especially important leadership role, as this next section describes.
REFORM OF THE TWO-YEAR MATHEMATICS CURRICULUM
Following a wave of reform agendas at the K-12 level, mathematics professionals associated with two-year colleges eagerly jumped into the discussions. For many years, AMATYC has provided guidance to professionals who teach in two-year colleges throughout the United States. Consistent with efforts to improve mathematics education at the K-12 level, most notably the National Council on Teachers of Mathematics (2010) reform agenda on Principles and Standards for School Mathematics, AMAYTCâ€™s Crossroads in Mathematics (Cohen, 1995), and the subsequent Beyond Crossroads (Blair, 2006) initiatives conceived the two-year college position on reforming mathematics curriculum and instruction in community and technical colleges. The premise of “crossroads” identified in the Crossroads in Mathematics (Cohen, 1995) is that more citizens need to be prepared for STEM-oriented careers, including the “mathematics, science, engineering, and technology” workforce, but many students who seek a postsecondary education are not adequately prepared to perform
at the college level. With respect to college preparation in mathematics, the Crossroads in Mathematics report anchored college readiness in studentsâ€™ needing the mathematics fundamentals as well as their needing to advance to studying calculus. A statement that captured the major focus of the Crossroads in Mathematics report is that:
More students are entering the mathematics “pipeline” at a point below the level of calculus, but there has been no significant gain in the percentages of college students studying calculus (Albers et al., 1992). The purpose of Crossroads in Mathematics is to address the special circumstances of, establish standards for, and make recommendations about introductory college mathematics. The ultimate goals of this document are to improve mathematics education and encourage more students to study mathematics.
The AMATYC taskforce associated with Crossroads in Mathematics called for a “flexible framework for the complete rebuilding of introductory college mathematics” (p. 5), emphasizing student growth in knowledge of mathematics by enhancing the meaning and relevance of mathematics, the importance of laboratory teaching of mathematics, the use of technology as essential to up-to-date curriculum, “balanced” content and instructional strategies that include “viable components of traditional instruction,” the contribution of mathematics to studentsâ€™ educational and career options, and the inclusion of diverse students (pp. 4-5).
The report also noted three related yet distinct sets of standards: standards for intellectual development, standards for content, and standards for pedagogy. Standards for intellectual development pertain to problem solving, modeling, reasoning, etc.; standards for content address such topics as real numbers and basic properties, solving linear equations, whole-number exponents, quadratic equations, etc.; and standards for pedagogy include teaching with technology, interactive and collaborative learning, connecting with other experiences, and experiencing mathematics (Cohen, 1995). The approach taken in the Crossroads in Mathematics report was that all students should grow in their fundamental knowledge of the normative mathematics content, supplemented with probability and statistics.
AMATYCâ€™s second standards document, entitled Beyond Crossroads (Blair, 2006), extended the goals, principles, and standards set forth in the earlier Crossroads in Mathematics report by calling for fuller and more strategic implementation of standards-based reform. Beyond Crossroads placed more emphasis on assessment of studentsâ€™ learning and promoting quantitative literacy (a topic discussed more fully later in this paper), meeting the needs of diverse learners, promoting active learning and online learning, promoting professionalism among full-and parttime instructors, and recognizing and involving more stakeholders in
implementation of mathematics reforms. The Beyond Crossroads report acknowledged the complexity of reforming two-year college mathematics in ways not addressed in the first Crossroads in Mathematics text to facilitate “student learning and the learning environment” and encourage faculty, departments, and institutions to improve all facets of two-year college mathematics education” (p. 7). Whereas the normative mathematics course sequence was still dominant in the second Beyond Crossroads report, the content standards seemed to place more emphasis on the application of mathematics to solve problems and to collect, analyze, and use data to help faculty make informed decisions and grow as professionals. Blair (2006) also observed that the Beyond Crossroads standards were intended to enhance access to college for underserved students, noting that two-year college mathematics “holds the promise of opening paths to mathematical power and adventure for a segment of the student population whose opportunities might otherwise be limited” (p. 7).
College Renewal Across the First Two Years (CRAFTY) (Ganter and Barker, 2004) is a third initiative that has focused on improving two-year college mathematics, in this case by focusing on college algebra. CRAFTY recognizes that most college students enroll in college algebra to fulfill a general education requirement and never see the relevance of the subject to the rest of their college education. Very few of these students ever move beyond college algebra to enroll in calculus, a point consistent with the Crossroads in Mathematics report. CRAFTY is a subcommittee of the Committee on the Undergraduate Program in Mathematics (CUPM) at the Mathematical Association of America. The project looks at the introductory mathematics courses for the broad range of students who enroll in postsecondary education, most of whom will not be mathematics majors, and it solicits input from disciplinary groups (e.g., biology, engineering, computer science, etc.) on what mathematics departments can do to best prepare students for those disciplines. Ganter and Barker (2004) advanced the notion of a series of disciplinary-based workshops known as the Curriculum Foundations Project that were conducted across the country between 1999 and 2001 to discipline partners to state their views of the mathematics curriculum and engage with mathematics practitioners in a dialogue about ways to reform the curriculum. Through these and other interactive strategies, CRAFTY encouraged faculty to engage colleagues, college administrators, employers, and other local business leaders “to improve the role of College Algebra in our educational system and in the effectiveness of the present programs” (Small, 2002). Consistent with the CRAFTY approach, faculty are encouraged to use small group projects and technology applications that engage students in active use of mathematics to solve real-world problems.
A companion initiative to CRAFTY led by AMATYC, called “The Right
Stuff” (funded by the National Science Foundation), opened the dialogue within the two-year college mathematics community to re-envisioning college algebra and redesigning curriculum to meet the needs of students enrolled in college algebra who might not be calculus-bound. Through traveling workshops, AMATYC assists faculty to use materials that were directed at encouraging students to engage in “meaningful activities that promote the effective use of technology to support mathematics, further provide students with stronger problem-solving and critical thinking skills, and enhance numeracy” below the calculus level (American Mathematics Association for Two-Year Colleges, n.d.). AMATYC also administered an initiative called Mathematics Across the Community College Curriculum (MAC3) (also funded by the National Science Foundation) that designed and shared materials that infused the mathematics curriculum with real-world problems and scenarios in collaboration with disciplines outside of mathematics (like science and economics) (R. Blair, personal communication, December, 8, 2011). Hillyard et al. (2010) summarized a number of studies from MAC3 projects for a special theme of the AMATYC journal to demonstrate that mathematics can be connected to other disciplines, that students increase their quantitative literacy, that faculty members are positively impacted, and that “academic ’turfâ€™ conflicts that emerge when we move towards interdisciplinarity” (p. 7) are overcome.
Together, these several initiatives—the two AMATYC standards-based reforms and the CRAFTY, The Right Stuff, and the MAC3 projects—represent important developments in improving two-year college mathematics curriculum in the United States. All of these initiatives have contributed to a national conversation to reform the curriculum. At a time when mathematics course enrollments have grown at an impressive rate in the twoyear college, particularly during the decade of the 2000s, strong consensus has emerged about the need to improve two-year college mathematics as a means of enhancing the STEM pipeline in the United States. Given the importance of two-year college mathematics to the overall P-20 education agenda of the United States, it is important to examine two-year college mathematics innovations to lay a foundation for recommending the next steps for research, policy, and practice.
WHAT WE KNOW ABOUT INNOVATIVE APPROACHES TO TWO-YEAR MATHEMATICS
Due to the considerable enrollment in pre-college mathematics in two-year colleges, many innovations and reforms are focused on the pre-college (or developmental education) level. Studies of the struggles that students face enrolling in and navigating the sequence of pre-college
mathematics courses have been immensely important to understanding student success (or lack thereof) in mathematics courses and the larger STEM pipeline. Researchers such as Bailey, Jeong, and Cho (2009); Perry et al. (2010); and many others have laid a foundation for understanding complex issues associated with pre-college mathematics. Despite a growing body of research, more rigorous research is needed on pre-college as well as college-level mathematics curriculum at the two-year college level. Research on mathematics teaching and learning at the classroom level is needed to provide a fuller and more nuanced understanding of content-based and pedagogically oriented reforms that may promote mathematics competency and positive student outcomes at the two-year level, including mastery of pre-college competencies, matriculating to and mastering advanced competencies, and advancing to and through the STEM pipeline.
An important recommendation that emerged from the AMATYC standards-based reform reports, particularly the 2006 Beyond Crossroads report, is consistent with the wider national and international conversations to emphasize quantitative literacy and quantitative reasoning as an element of or, in some cases, alternative to the normative mathematics course sequence. A leader in the dialogue about quantitative literacy, Steen (2001) argues that enabling students to use mathematics to solve real-world problems that are complex, ambiguous, and incomplete is the most important thing that college mathematics courses can do. She notes that “rarely will high school graduates be faced with problems that present themselves in the language of algebra” (Steen, 1992, n.p.), but just because students donâ€™t appreciate algebra in its traditional forms does not mean that it is not applicable or useful to them. Steen notes that quantitative literacy is rooted in real data that are part of lifeâ€™s diverse contexts and situations. She believes pedagogy should change to encourage quantitative thought that can help learners “to understand the meaning of numbers, to see the benefits (and risks) of thinking quantitatively about commonplace issues, and to approach complex problems with confidence in the value of careful reasoning” (Steen, 2001, p. 58). Students who experience quantitative literacy are empowered to think independently, to ask smart questions, and to confront complexities and challenges with confidence, and, as Steen concludes, “these are the skills required to thrive in the modern world” (p. 58).
Given the importance of this topic, it is unfortunate that the literature on quantitative literacy and quantitative reasoning is disconnected from literature on contextualized teaching and learning, integrated academic and technical curriculum, and problem-based learning. Referring to this collection of curricular and instructional approaches, Perin (2011) described contextualization as the “practice of systematically connect
ing basic skills instruction [in fields of study such as mathematics] to a specific content that is meaningful and useful to students” (p. 3). Her recent review of the literature includes findings of various types of contextualization employed in postsecondary settings, especially pre-college mathematics courses. Baker, Hope, and Karandjeff (2009) have explored the wide range of definitions that are used for contextualized instruction, and, to their credit, they link practices associated with contextualization to theories of learning and pedagogical strategies. Among the recommendations made by Baker et al. is the importance of exploring alternative formats for delivering the normative mathematics curriculum. Whereas rigorous research has not been performed on contextualized math at the two-year college level, an experimental study that examined the effect of training of high school math and career and technical education (CTE) teachers to work cooperatively to make math explicit in CTE classrooms produced statistically significant outcomes, including higher scores on standardized and college placement tests without negatively impacting technical skill attainment (Stone, Alfeld, and Pearson, 2008). This study has not been replicated in the two-year college context, but it would be very helpful to do so.
Examples of other innovative mathematics curriculum formats that are being studied include modularization, which involves delivering instruction in manageable segments or “chunks” (Rutschow and Schneider, 2011, p. 25), rather than traditional, semester-long courses. Mostly applied to pre-college mathematics, this strategy of chunking the curriculum could be extended to college-level mathematics. When implemented properly, students can achieve success in shorter time periods than traditional courses, which also motivates them to persist to the next shortened segment. Bailey et al. (2003) evaluated modularization in six NSF Advanced Technological Education (ATE) projects and four centers, and they noted that instructors praised the method for its flexibility and adaptability. The National Center for Academic Transformation Mathematics Emporium model, which Twigg (2011) described as a “silver bullet,” combines modularization with technology-supported instruction (G. Reese and C. Kirby, personal communication, October 18, 2011).
Another innovation that is being attempted in mathematics, particularly pre-college mathematics, involves compression of the curriculum, meaning compressing the amount of time it takes for students to complete mathematics course sequences, and accelerating them toward their next course or completion. Compression often requires scheduling courses more hours a day for shorter amounts of time, and pairing courses that complement one another, including pairing mathematics and science courses or pairing multiple mathematics courses (including pre-college and college level) to create an intensive learning experience. Though
most of the research on compression and acceleration is focused on the pre-college level curriculum, this strategy may be useful to attempt with college-level mathematics courses (e.g., college algebra and statistics). For example, two forms of acceleration were used by the FastStart Program at the Community College of Denver, wherein FastStart accelerated students through the mathematics course sequence by allowing students to enroll in a developmental course concurrently with a college-level course. Results from 11 student cohorts who began developmental mathematics at various levels revealed encouraging outcomes on retention and credit accumulation (Bragg, Baker, and Puryear, 2010). Synthesizing the literature on acceleration, Edgecombe (2011) noted evidence of the impact of acceleration on developmental education is limited but promising based on evaluations of FastStart and other similar programs.
Change of not only how mathematics is taught but also what is taught is also important for mathematics reformers. One of the most notable efforts in this regard are the Carnegie Foundation for Learning and the Dana Center's QuantwayTM and StatwayTM projects (Carnegie Foundation for the Advancement of Teaching, 2011a, 2001b) that are attempting to replace the normative pre-college mathematics courses with mathematics courses focused on quantitative literacy and statistics. Using an accelerated timeframe, the Quantway and Stat way projects seek to prepare students for college level mathematics instruction. Quantway and Statway “enable developmental mathematics students in community colleges to complete a[n accelerated] credit-bearing, transferable mathematics course in one academic year while simultaneously building skills for long-term college success” (Cullinane and Treisman, 2010, p. 4). The Statway course sequence assists students to develop statistical literacy and engages them in mathematical reasoning using data, and it provides them with college credit in statistics. Cullinane and Treisman hypothesize that the adoption of a statistics sequence such as Statway will support many more students to engage in mathematical reasoning, especially when the curriculum is institutionalized from K-12 education and extended to the postsecond-ary level. Quantway is similarly focused on increasing the mathematical literacy of students who need to take pre-college mathematics by replacing traditional textbook-based, procedural instruction with numerical reasoning that is necessary to solve real-world problems. The Quantway pathway promotes an accelerated format, allowing students who place into elementary algebra to gain access to and move through a college-level quantitative reasoning course in one year.
Reform of instructional materials such as those associated with Statway and Quantwav address a disconcerting problem that Kays (2004), Mesa (2010), and others have noted in the literature: the reliance on textbooks to structure and guide classroom instruction of mathematics.
These studies demonstrate the ways classroom teaching that relies on procedural-based textbooks reinforce the memorization of procedural knowledge at the expense of quantitative reasoning. Many mathematics texts are also tied to the normative math course sequence, and traditional pedagogies associated with teaching the mainstay courses in that sequence (e.g., algebra, geometry, trigonometry, and calculus) provide a valuable backdrop for a discussion of the critical needs that lay ahead as two-year college mathematics educators delve more deeply into reform. These studies suggest that systemic reforms, including curriculum, instruction, and instructional materials, are need to be pursued if mathematics education is to be responsive to the diversity of learners who seek the opportunity to pursue STEM-related programs of study in the two-year college.
RECOMMENDATIONS FOR FURTHER RESEARCH, POLICY, AND PRACTICE
In spite of strident pursuit of standards-based reform of two-year college mathematics, implementation of reform of the mathematics curriculum has been slow and uneven. National studies show more students are enrolling in two-year college mathematics, but a substantial portion of these enrollments are at the pre-college level, and many of these students never reach college-level mathematics. Thus, the STEM pipeline appears to be widening at the start, which is encouraging, but it also seems to narrow rapidly as students attempt to advance to college-level mathematics, a prerequisite to pursuing STEM programs of study and STEM-related careers.
To facilitate the role that two-year mathematics can play in providing access to the STEM pipeline and preparing larger numbers of postsecondary students, mathematics instruction needs to be sufficiently engaging and useful to support their interests and to assist them to make the commitment necessary to pursue a STEM program of study. A whole host of issues need to be addressed with respect to two-year college mathematics and the preparation of students who seek subbaccalaureate credentials and who desire to transfer to universities in STEM fields. Specific recommendations for research, policy, and practice to support this goal are described below.
A systemic, P-20 approach is needed to reform mathematics curriculum. Recommendations offered by a plethora of professional groups, including AMATYC and the Mathematics Association of American (MAA), and at different levels of the educational system are logical, reasonable, and substantive, and equally importantly, they consistently argue
for a multi-level yet coordinated P-20 approach. Without such a strategic, collaborative endeavor, it will be difficult for two-year colleges that are caught between K-12 education and higher education to engage in reform, except in isolated ways. Given the national imperative to enhance the STEM pipeline, and the critical role that mathematics needs to play in that work, this recommendation may be the most important of all to emerge from the Summit on Realizing the Potential of Community Colleges as Pathways to STEM Education and Careers.
More research is needed to improve two-year college mathematics instruction. Although numerous pedagogical strategies are emerging that offer promise to change the way mathematics is taught at the twoyear college level, CBMS survey data confirm the prevalence of lectureled, teacher-centered instruction rather than the sorts of contextualized, problem-and project-based approaches that support quantitative literacy. Finding ways to support two-year college faculty to engage in professional development that reinforces innovative instructional reforms is important. Included in this list is the importance of helping faculty to adopt curriculum and instruction that draws upon studentsâ€™ everyday life experiences in the workforce, their communities, and other aspects of their lives. Mathematics instructors also need to understand how to integrate technologies to deliver instruction in the classroom or from a distance. Moreover, mathematics instructors need to understand how college placement tests can either impede or advance students through the mathematics curriculum. Involving faculty in decisions about assessment may help them to understand how college placement testing impacts student learning and ultimately, improves student outcomes.
More research is needed on the students who enroll in two-year college mathematics, especially college-level mathematics (college algebra and beyond), and how their experiences and performance in college-level mathematics courses influences subsequent enrollment, completion, and career decisions. Because two-year colleges are the gateway to postsecondary education for diverse learners, these schools have an important role to creating pathways that prepare students to advance to higher levels of postsecondary education. More research is needed to support the study of mathematics pathways, other than the normative mathematics sequence, and to understand how students “develop the ’habits of the mathematical mindâ€™ that are required to be successful in mathematics and science and engineering and technology courses” (R. Blair, personal communication, December 8, 2011). Students need to know what these new mathematical pathways look like and how they lead to STEM careers, and they cannot be expected to understand or navigate them on their own, without encouragement and support. Systemic change is needed to ensure that all students who have aspirations for STEM careers get the chance to learn
mathematics in ways that fully and respectfully support their goals. If the nation expects more women and minorities to participate in STEM programs of study, fulfilling this recommendation is essential.
More and better data are needed to support practitioner engagement in active research on mathematics education at the local level, where two-year college mathematics faculty and other stakeholders engage in the teaching and learning process. Beyond participating in training, many two-year faculty would appreciate and benefit from opportunities to engage in active research that encourages them to try out new pedagogical strategies in the classroom and determine how they impact student learning. The Equity ScorecardTM and Benchmarking projects of the Center for Urban Education at the University of Southern California provide valuable examples of ways that professional development of two-year college faculty can be integrated with action research to address equity issues for minority students who seek to participate in STEM programs (Baldwin et al., 2011). The Achieving the Dream initiative has established a strong track record of engaging practitioners in using data to improve pre-college mathematics (Rutschow et al., 2011). Lessons learned from this initiative and other newer ones, such as Pathways to Results in Illinois (Bragg and Bennett, 2011), offer the potential to improve two-year college mathematics and support student success in STEM programs of study.
This paper relies on existing literature available from a number of sources. Most importantly, academic databases were queried to identify peer-refereed articles as well as books, monographs, reports, papers, and conference presentations on two-year college mathematics. Databases included in this review were ERIC, EBSCO, Education Full-Text, JSTOR, Dissertation Abstracts, and Sociological Abstracts. In addition, Google and Google Scholar were queried to identify relevant documents and materials that appear outside of the traditional scholarly databases. Searches of websites maintained by organizations known to research and publish on the topic of two-year college mathematics were conducted, including the National Center for Education Statistics, the National Science Foundation, the Community College Research Center at Teachers College, and Charles A. Dana Center at the University of Texas at Austin, the AMATYC website, and others. Keywords used in these searches included the following words singularly and in combination with one another: math, mathematics, mathematics education, developmental, remedial, pre-college, algebra, calculus, advanced mathematics, statistics, etc.
Keywords used to understand how the scholarly literature situates two-year mathematics curriculum in the broader liberal arts and sciences context included the following: liberal arts and sciences, liberal arts, science, STEM, STEM education, technology, technology education, engineering, engineering education, technician education, etc. Also, to ensure that the full spectrum of literature on two-year colleges was included in this literature review, I used an extensive set of keywords to capture the institutional context, including the following: two-year college, community college, technical college, and junior college. I also entered keywords related to four-year college and university, transfer, and articulation to determine whether literature was available to compare the two-year context to the four-year context, including transfer.
In addition to the above methods, I reached out to several two-year college mathematics experts, including David Lutzer, Ellen Kirkman, and Rikki Blair, all authors of the Fall 2005 and/or Fall 2010 CBMS surveys. Rikki Blair also served as editor of the 2006 Beyond Crossroads report of AMATYC and was an especially thoughtful and gracious contributor. I also sought guidance from several two-year college mathematics practitioners and colleagues at the University of Illinois, including George Reese, director of the Office of Mathematics, Science and Technology Education, and Catherine Kirby, assistant director of the Office of Community College Research and Leadership, who collaborated recently on a literature review on this same topic and brought numerous sources on two-year college mathematics to my attention. Finally, I offer my gratitude to Dr. Julia Makala, research specialist at the Office of Community College Research and Leadership, who offered a critical review that was invaluable to the final draft of this paper.
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