Factors Associated With Completion: Pathways Through Developmental Mathematics

Abstract

This study examines two 5-year longitudinal data sets of community college students (n = 595 and n = 593) to explore factors associated with successful outcomes in developmental mathematics. Logistic regression models consider the role of demographic factors, course format, and student support structures on the likelihood of a student completing the developmental sequence and subsequently passing a credit-level mathematics course. Additional linear regression models examine the time required to complete developmental coursework. Tutoring has a strong association with positive student outcomes, as does full-time enrollment and developmental mathematics coursework grades. Alternative course formats are also associated with increased likelihood of success for students, but some alternative formats delay a student's time line. Implications for developmental mathematics programs in community college settings are discussed.

Keywords

developmental mathematics community college course format tutoring

Recent reports show that only 39% of high school seniors are prepared for entry-level college mathematics courses (Heitin, 2014), and more than half of community college students are required to enroll in at least one developmental mathematics course to demonstrate college readiness (Bailey, Jeong, & Cho, 2010). Developmental mathematics courses are typically noncredit courses that do not satisfy any mathematics credential but reinforce and strengthen basic mathematical skills. Entering college students may be mandated to enroll in a series of two to three of such courses to demonstrate readiness.

Although the intent of developmental programs is to provide alternative access to higher education, evidence suggests that these programs can instead serve as a barrier (Bonham & Boylan, 2011; Edgecombe, 2011). In a study that examined a typical cohort of developmental mathematics students, only 30% completed the prescribed sequence of developmental classes, and of those, only half were able to successfully complete a credit-bearing mathematics course within 3 years (Bailey et al., 2010). This situation has led some to refer to developmental mathematics courses as a “graveyard of dreams and aspirations” for college students (Merseth, 2011, p. 32). In 2012, the national nonprofit organization Complete College America released a report surveying the landscape of developmental education with a grim title: Remediation: Higher Education’s Bridge to Nowhere. The report described cohorts of beginning college students and the bleak outcomes realized with developmental coursework (Jones, 2012). A study of administrative records from Texas in the 1990s examined students at the margin for placement into remediation and found no evidence that taking developmental mathematics courses had a positive effect on degree completion, labor market earnings, or transfer to a 4-year college (Martorell & McFarlin, 2011). Boatman and Long (2010) looked at a wider range of students in Tennessee, found students taking developmental mathematics courses took fewer college credits by the end of their third year, and also experienced a negative effect on degree completion within 6 years. Scott-Clayton and Rodriguez (2015) in a large analysis of an urban community college system found no evidence that developmental mathematics courses help students and saw negative effects for developmental mathematics on students’ likelihood of passing college-level mathematics. They additionally found that negative effects were slightly greater for students they deemed to be lower risk.

The existing research on developmental coursework is not all negative, however. Chen (2016) examined a national sample of students required to enroll in developmental courses at both 2- and 4-year institutions and found that developmental courses were positively associated with persistence and degree completion. The developmental courses were especially crucial for students with the weakest academic preparation: Without developmental courses, these students completed less credit hours and dropped out of college at a higher rate. Similarly, Bettinger and Long (2009) examined a data set of 28,000 students and found that students enrolled in developmental mathematics courses were more likely to persist, controlling for selection issues. They found that positive effects were largest for students on the margin of needing remediation.

With large numbers of underprepared students seeking community college as a gateway to higher education, the problem of underpreparedness in mathematics has reached a critical juncture. While there has been research activity surrounding effective approaches to developmental education, the need persists to build an evidence base consisting of studies conducted in a variety of contexts that take into account many different student- and school-level variables. Indeed, the recent IES Practice Guide (Bailey et al., 2016) on best practices in developmental education suggests that there are too few studies for recommendations to be rated as having a strong evidence base supported by research.

This study adds to this research by longitudinally examining two cohorts of students enrolling in developmental mathematics in a diverse Southern college system. We use correlational analyses that account for a variety of student-level factors, college support structures, course formats, and course placement procedures. We explore what factors may be associated with success in developmental mathematics in a context where a high proportion of students are failing to become college ready. Rather than suggesting that developmental mathematics does not help students, this study suggests that students with weak mathematics preparation can be supported and reach readiness milestones in mathematics.

Literature Review

This section reviews research on key areas associated with success in developmental mathematics—course placement, course formats, tutoring, and full-time status.

Course Placement

Historically, the developmental mathematics sequence was established to reinforce algebra skills to prepare students for calculus (Cullinane & Treisman, 2010). The developmental sequence typically was composed of three courses, starting with arithmetic and advancing through algebraic concepts. Completion of the sequence renders a student eligible to enroll in an entry-level credit mathematics course. The most common first-year credit-bearing mathematics courses are College Algebra, Elementary Statistics, or College Mathematics (Quantitative Literacy). Many students still choose College Algebra based on long-standing tradition and familiarity (Adelman, 2006; Johnson, 2007). However, low completion rates coupled with a push for increased graduation rates have given rise to alternatives (American Mathematical Association of Two-Year Colleges, 2006; Cullinane & Treisman, 2010; Dana Center, Complete College America, Education Commission of the States, & Jobs for the Future, 2012; Zachry & Schneider, 2010). Reform efforts (Statways) suggest that an introductory statistics course can promote well-informed citizens and educated consumers (Dana Center et al., 2012). A recently conducted study found that students placing into developmental mathematics can succeed in a statistics course offered with support, even when students bypass remediation and enter statistics directly (Logue, Watanabe-Rose, & Douglas, 2016). Similarly, a course that embeds study skills in a credit quantitative literacy class (Mathways) has shown promising results with increased completion (Yamada, 2014).

Numerous placement tests exist for developmental mathematics—ACCUPLACER and COMPASS are common national tests (U.S. Department of Education [DOE], 2005)—and states often develop assessments as well (e.g., Texas Higher Education Assessment [THEA; n.d.] and Texas Success Initiative Assessment [TSIA]). In addition, institutions use results from other benchmark tests (e.g., SAT, ACT, or end of course examinations; Texas Higher Education Coordinating Board, 2012). Although studies provide evidence to the usefulness of the test results for placement (Mattern & Packman, 2009), wide variations exist between institutions, and even institutions offering a common assessment may have different cutoff scores (Bailey et al., 2010; Higbee, 1999; Parker, Bustillos, & Behringer, 2010; Rutschow & Schneider, 2011). Analysis of placement decisions from Accuplacer and Compass shows that both tests have weak predictive validity (Belfield & Crosta, 2012; Dana Center et al., 2012; Scott-Clayton, 2012). There are a variety of reasons why students may perform poorly on a placement examination, including an actual weakness based on difficulties, a weakness based on limited coursework in high school, a time delay since the last course was taken, test anxiety, restricted calculator use, or language barriers (Bailey et al., 2010; Higbee, 1999).

Multiple measures, such as high school grade point average (GPA) and high school coursework, may improve placement decisions (Belfield & Crosta, 2012; Boylan, 2009; Boylan & Saxon, 2006; Dana Center et al., 2012; Higbee, 1999; Rutschow & Schneider, 2011; Scott-Clayton, 2012). Some states are moving away from placement testing in favor of mainstreaming all incoming students into credit-bearing classes with appropriate supports (Fain, 2012; Jenkins, Jaggars, & Roksa, 2009). Further evidence suggests that allowing students to self-select into developmental courses may be more effective than placement testing (Kosiewicz, 2014), although results from recent reform efforts in Florida provide reason to pause: After legislation in Florida made developmental coursework optional, students enrolled in credit mathematics courses at higher rates, but the likelihood of passing dropped sharply (Hu et al., 2016). Evans and Henry (2015) found that allowing students to retake their placement test from home, and giving them access to review material before doing so, can increase students’ likelihood of enrolling in college-level mathematics or of passing college-level mathematics.

Course Formats

The structure of developmental courses has undergone a transformation. Previously, the lecture format had become synonymous with college instruction. Knowledge of how students learn (e.g., Bonham & Boylan, 2011) and the presence of transformative technology have led to alternate course structures that may teach students more effectively and in less time; however, research is needed to investigate the impact of these alternate routes (Mesa, Wladis, & Watkins, 2014; Zachry & Schneider, 2010). Three broad types of promising designs are discussed: online/hybrid, self-paced or modular, and accelerated models.

Online/hybrid

Perhaps the most popular alternative to lecture-based mathematics courses are those delivered in an online or hybrid format. A course is considered hybrid if it retains elements of lecture as well as 30% to 79% online course delivery (Bendickson, 2004; Jaggars, 2011).

Online and hybrid courses are typically delivered via a computer software program, which includes student practice, videos, an e-text, and other interactive features.

Many experts have cited technology as an important innovation in developmental mathematics (American Mathematical Association of Two-Year Colleges, 2006; Le, Rogers, & Santos, 2011; Mesa et al., 2014; Rutschow & Schneider, 2011). However, the push for innovation “has clearly outpaced research and evaluation” (Epper & Baker, 2009, p. 4) in that there exists a scarcity of rigorous research (Means, Toyama, Murphy, Bakia, & Jones, 2010; Mesa et al., 2014; Sitomer et al., 2012; U.S. DOE, 2005). More than 40% of community colleges nationwide offer developmental mathematics courses in an online/hybrid format (U.S. DOE, 2005), yet the numerous descriptive reports evaluating the impact of course-related technology provide mixed outcomes and limited generalizability (Sitomer et al., 2012). On one hand, online/hybrid courses offer the greatest flexibility for students, which can contribute to fewer withdrawals and a higher incidence of successful course completion (Florence-Darlington Technical College [FDTC], 2011; Knewton, n.d.; Lancaster, 2001; National Center for Academic Transformation, 2009), as well as higher passing rates, improved test scores, and the development of better problem-solving skills (Atkinson, 2003; Means et al., 2010; Stylianou & Shapiro, 2002). When students persist in technology-aided courses, they perform at similar or higher levels than students enrolled in traditional lecture courses (Rutschow & Schneider, 2011).

However, not all studies found such promising results. Students in some technology-enhanced courses showed no gains in course completion or pass rates (Spradlin, 2009; Taylor, 2006; Weems, 2002) or may pass at a lower rate than their traditional lecture counterparts and have higher withdrawal rates (Bendickson, 2004; Jaggars, 2011; Keller, 2013; Rutschow & Schneider, 2011; Zavarella & Ignash, 2009). Studies citing positive results for online/hybrid courses may not generalize to academically at-risk students because these students require a high level of support (Jaggars & Bailey, 2010). Thus, research evidence is indeed mixed.

Self-paced

Another manner in which technology can assist learners is by creating individualized and flexible pacing, accelerating, or slowing courses to allow students to progress through concepts at their own pace. Self-paced designs provide flexible access to course materials, the instructor, and accompanying support services (Biswas, 2007; MDC, 2012; Rutschow & Schneider, 2011; Zachry & Schneider, 2010). Students can regulate and monitor their own progress (Kenner & Weinerman, 2011; Kinney, 2001) and gain advanced levels of knowledge efficiently through scaffolding and ownership (Caverly, 1999; Higbee, 2000), while the teacher’s role shifts to allow for more individualized and meaningful assistance (Fletcher, 2013).

Research suggests that self-paced formats are associated with increased retention rates and higher pass rates than lecture (Epper & Baker, 2009; Knewton, n.d.; Speckler, 2008). A self-paced format may also lead to a reduction in mathematics anxiety (Diaz, 2010; Epper & Baker, 2009), the ability to complete developmental courses in less time (Epper & Baker, 2009; FDTC, 2011; Knewton, n.d.), and most importantly, a higher success rate in credit-bearing mathematics courses (FDTC, 2011). Each of these cited studies is limited in generalizability, however, due to a lack of rigorous research methods and small sample sizes (Rutschow & Schneider, 2011).

Because a self-paced classroom has no lecture component, students and faculty both may feel uneasy initially, and students might perceive that the teacher is not actually teaching (Brothen & Wambach, 2000; Higbee, 2000). Teachers must provide structures and support to assist students in maintaining momentum to complete the course. Students may have anxiety about the heavy use of technology in a self-paced classroom, especially returning nontraditional students with less technology practice (Higbee, 2000). The U.S. DOE (2005) included a self-paced format in its review of promising alternatives and found no clear consensus. Some studies showed no effect, others a slight increase in course success rates, and still others a decreased course completion rate. Practically speaking, self-paced or modularized formats may be cost prohibitive for community colleges with limited resources and infrastructure, as these courses necessitate both a computer laboratory for students to meet in and increased personnel costs for tutors to assist in class (Epper & Baker, 2009; FDTC, 2011).

Accelerated

In an accelerated course, students complete a semester-long class in a compressed format, such as an 8-week course or a 5-week summer class. When appropriate, acceleration provides an alternative pathway for students that quickens remediation and shortens the developmental sequence (Bailey et al., 2010; Biswas, 2007; Dana Center et al., 2012). Indeed, Adelman (2006) touted summer enrollment as a critical factor in increasing the likelihood of eventual degree completion. Developmental students perform better in credit courses if taken immediately after the required developmental mathematics course (Boylan & Saxon, 2006), and so pairing a developmental course with a subsequent credit course in an accelerated format shows promise (Cullinane, 2012; Rutschow & Schneider, 2011).

The accelerated model has the potential to improve academic outcomes in two distinct ways: by minimizing exit points in the sequence and by more closely aligning the developmental course with credit-level learning objectives (Edgecombe, 2011; Jaggars, Edgecombe, & Stacey, 2014). However, if the premise of a developmental program is to allow learners to build prerequisite skills (Dana Center et al., 2012), then acceleration may not allow sufficient time, leading to an increased likelihood of a student failing subsequent credit course (Jaggars et al., 2014). As with other course formats examined, the limited research on accelerated models tells a conflicting story. While some studies (Cafarella, 2013; Zachry & Schneider, 2010) caution that acceleration is only appropriate for more academically prepared students with a strong work ethic, others (Hern & Snell, 2010; Jenkins et al., 2009) suggest that acceleration can assist developmental students at any level. Students enrolled in an accelerated class have shown higher retention, persistence, and comparable or improved pass rates in credit-level mathematics courses compared with students in a traditional developmental mathematics sequence (Cafarella, 2013; Edgecombe, 2011; Hern & Snell, 2010; Jaggars et al., 2014; U.S. DOE, 2005; Zachry & Schneider, 2010).

Tutoring

Access to higher education without the needed support structures is a “false opportunity” (Casazza, 1999, p.7). In the case of developmental mathematics, learning assistance centers (also known as tutoring centers or mathematics laboratories) are one common type of tutoring support, and students can visit the center for assistance on mathematics coursework without prior appointment. Warner, Duranczyk, and Richards (2000) found that the presence of a mathematics laboratory on campus was an important support structure for students. With the proliferation of computer-assisted or self-paced classes, tutors may also assist students alongside instructors within a classroom (Higbee, 1996). Research on the effect of tutoring on developmental students is limited and provides mixed results (Rutschow & Schneider, 2011). The effectiveness of tutoring is related to the amount and quality of tutor training (Boylan, Bliss, & Bonham, 1997) and depends on a tutor’s ability to apply individualized strategies (Boylan & Saxon, 1999). However, a small-scale examination of community college students attending 5 hours of tutoring over a semester found that attending tutoring led to a GPA boost of over one point and increased next semester retention by over 20 percentage points (Klohn, Martinson, & Gregory, 2016).

Full-Time Status

Research findings related to full-time/part-time enrollment status are surprisingly consistent: Part-time enrollment is bad for student success (Tinto, 2006). Adelman’s (2006) large-scale longitudinal study found that part-time status at any point along a student’s educational path was negatively associated with degree completion. Similarly, Penny and White (1998) found that part-time enrollment status at the time that students took their last developmental mathematics course was negatively correlated with their performance in developmental mathematics and had an even more negative impact on College Algebra outcomes.

Research Purpose

We describe four contributions of this research study by discussing the current major research efforts in developmental mathematics education. First, the Florida reform efforts indicate that bypassing developmental coursework is not optimal; additional support is warranted. The authors recommend that further research is needed on alternate modalities for developmental mathematics—particularly accelerated formats (Hu et al., 2016). In this study, an examination of the accelerated format is a primary piece of our analysis.

Second, one aspect of the student experience that Chen (2016) was unable to include in his analysis was how students are supported at colleges while in developmental coursework. Tutoring research in developmental mathematics has been limited, and Chen concluded that more research was needed on student supports like tutoring. This study adds to the evidence presented by Chen by including the presence of tutoring as an important indicator of success. Third, recent proposals address alternate curricular pathways (Mathways, Statways; Logue et al., 2016; Yamada, 2014) but do not make recommendations for students needing an algebra pathway. This study adds to the existing literature by specifically examining success in an algebraic pathway, which is still critical and required for many majors.

Finally, the recent IES Practice Guide (Bailey et al., 2016) reviewed best practices for developmental education. The report not only gives a series of recommendations but also shows that the evidence base behind all of these recommendations is either minimal or moderate. Studies are needed that add to this evidence base, and this study speaks directly to several of the recommendations—including those regarding placement decisions, compressing developmental courses, providing support programs, and students’ full-time versus part-time status. Although this study is correlational in nature, the Practice Guide did include reviews of correlational studies. Such studies, when combined with the program of work on developmental education, can offer insight into how these programs work across different contexts.

This study explores several variables associated with success for developmental mathematics students. We examine the following research questions (RQs):

How are variables in developmental mathematics, including course format, full-time status, and student support services, associated with community college students’ likelihood of completing the sequence of developmental coursework?

How are these variables associated with community college students’ likelihood of passing a credit-bearing mathematics course?

What is the relationship of these structures on the amount of time (number of semesters) required to successfully complete the developmental mathematics sequence?

Methods

Our analysis consisted of two stages: First, we built regression models to predict outcomes for a 2009 cohort of developmental mathematics students. Second, we applied these same models to a 2010 cohort and examined how the models performed with this new data set.

Sample

This study used archival data for two First Time in College cohorts enrolled in developmental mathematics at a suburban community college system. The student population was approximately 10,000 students systemwide, with 20% enrolled in developmental courses. The students enrolled in developmental mathematics were predominantly Caucasian, with 13% to 25% Hispanic and 8% to 10% African American students (Table 1).

Table 1.

Fall 2009 (n = 595) and Fall 2010 (n = 593) First Time in College Developmental Mathematics Students.

Race/Ethnicity	2009		2010
Race/Ethnicity	n	Percentage	n	Percentage
Caucasian	257	43	356	60
Hispanic	78	13	146	25
African American	45	8	60	10
Asian	9	2	2	0.3
Hawaiian/Pacific Islander	3	0.5	0	0
Not specified	203	34	29	5
Gender
Male	232	39	242	41
Female	363	61	351	59
Title IV/Pell eligible	207	35	331	56
Full-time enrollment	156	26	262	44
Developmental reading/writing	185	31	202	34
First developmental class taken
Pre-Algebra	258	43	203	34
Beginning Algebra	154	26	130	22
Intermediate Algebra	183	31	260	44
Credit mathematics course taken
College Algebra	170	29	180	30
Other (statistics, quantitative literacy)	58	10	73	12

Any faculty teaching developmental mathematics were required to have a baccalaureate degree in mathematics or a related field, and most had prior secondary teaching experience. Students’ developmental course placement was determined by a test, such as THEA, Compass, or Accuplacer. Developmental courses included Pre-Algebra, Beginning Algebra, and Intermediate Algebra, and completion of developmental coursework led to entry-level credit mathematics courses including College Algebra, Statistics, Quantitative Literacy, Trigonometry, and Precalculus.

Measures

Outcome variables

A dichotomous variable describing whether students completed the developmental mathematics sequence during the 5-year period was created as a 0/1 dependent variable. Similarly, a 0/1 dichotomous variable was created to capture whether or not a student ever passed an entry-level credit-bearing mathematics course. Withdrawals and grades of F were considered failures, and grades of A, B, C, or D were considered passing. The third outcome variable was a continuous measure reflecting the number of semesters that a student took to complete the required developmental course sequence. The first two variables were examined through logistic regression models, while the third outcome variable (number of semesters) was examined by linear regression. Summary statistics are provided in Table 2. We next describe each predictor variable.

Table 2.

Overall Course Taking Outcomes for 2009 Cohort (n = 595) and 2010 Cohort (n = 593).

2009 Cohort (Top)2010 Cohort (Bottom)	Number of Students (% of Cohort)	Number Eligible to Progress to Next Course (Pass Rate %)	Number Eligible to Progress to Next Course (Pass Rate %)	Number Eligible to Progress to Next Course (Pass Rate %)	Number Completing Developmental Mathematics (% of Cohort)
Number (%) starting in Pre-Algebra	258 (43)	186 of 258 (72)	94 of 186 (51)	82 of 94 (76)	82 of 595 (14)
Number (%) starting in Pre-Algebra	203 (34)	140 of 203 (69)	79 of 140 (56)	60 of 79 (76)	60 of 593 (10)
Number (%) starting in Beginning Algebra	154 (26)	104 of 154 (68)	79 of 104 (76)		79 of 595 (13)
Number (%) starting in Beginning Algebra	130 (22)	97 of 130 (75)	63 of 97 (65)		63 of 593 (11)
Number (%) starting in Intermediate Algebra	183 (31)	124 of 183 (68)			124 of 595 (21)
Number (%) starting in Intermediate Algebra	260 (44)	178 of 260 (68)			178 of 593 (30)
Percentage of students who complete developmental mathematics sequence				2009: 285 of 595 (48%) 2010: 301 of 593 (51%)
Percentage of students who enroll in a credit-bearing mathematics course				2009: 228 of 595 (38%) 2010: 253 of 593 (43%)
Percentage of students who pass first credit-bearing mathematics course				2009: 157 of 595 (26%) 2010: 174 of 593 (29%)

Course format

Predictors included whether a student ever enrolled in a developmental course of a specified format, coded as a 0/1. First, accelerated courses were those taught in a condensed or shortened time frame, including 8-week fast track courses and 5-week summer classes. Approximately 10% of each of the cohorts enrolled in an accelerated course format (57 of 595 in 2009 and 62 of 593 in 2010). Second, self-paced classes were those in which students progress through a series of assigned computer-based content modules by demonstrating mastery in order to proceed. This was the least popular format for the 2009 cohort with only 40 of 595 enrolled in this format, but slightly more than 20% of the 2010 cohort participated in a self-paced class (120 of 593). Third, hybrid or online courses were combined to form a category of courses using technology for over 50% of the course delivery. Over 18% of the 2009 cohort (110 of 595) enrolled in this format, and 16% (93 of 593) of the 2010 cohort enrolled in a hybrid or online course. All developmental mathematics courses at the study college used the same textbook (Tobey, Slater, Blair, & Crawford, 2009), which encompassed content for all three levels (Pre-Algebra, Beginning Algebra, and Intermediate Algebra) and utilized MyMathLab software.

Placement procedures

An additional predictor included which of the three developmental courses the student began with—Pre-Algebra, Beginning Algebra, or Intermediate Algebra. The weakest students tended to place into the first class (Pre-Algebra) based on entry test scores, while the stronger students placed into the third class (Intermediate Algebra). A student’s placement test score on the THEA or Compass was also considered as a predictor in the model. In the 2009 cohort, 272 of the 595 had THEA scores, and 306 had Compass scores. Seventeen students had no score (three students), or an alternate test score exemption, such as SAT (one), Texas Assessment of Knowledge and Skills (TAKS; nine), or an Accuplacer score (four), and these students were excluded from the analysis. For the 2010 cohort, 313 of the 593 had THEA scores and 261 had Compass scores. Nineteen students had no score (one student), an alternate exemption (nine), or an Accuplacer score (nine); again, these students were excluded from the analysis. To compare student mathematics scores for the THEA and Compass tests, the mean and standard deviation (SD) for each test were utilized to create a normalized measure that reflected how many SD units away from the mean a particular student’s score was. The mean and SD for the Compass test were publicly available. The mean and SD for the THEA test were not publicly available, but we were granted access to state-level data in order to calculate these measures (THEA, n.d.). The weighted average for all THEA and Compass scores combined for the 2009 cohort had a mean of −0.23, with an SD of 0.76. The scores for the 2010 group had a mean of −0.20 and an SD of 0.79. As we consider the placement variables to be control variables, we present them in the regression tables but do not discuss main effects for them in the narrative describing our results.

Support services

Continuous variables were created representing the amount of time a student attended tutoring on campus and the total time spent specifically in mathematics-related tutoring while enrolled in developmental or credit-level mathematics coursework. A log transformation was applied in order to minimize the effect of extreme values for a handful of students with over 100 hours of tutoring. Dichotomous variables were created describing whether a student attended on campus tutoring in any content area or in mathematics specifically. This category of variables is limited in that tutoring was voluntary, and the students may have attended tutoring off campus, which was not reported. For the 2009 cohort (n = 595), 395 students never attended tutoring, and the 200 students who did attend tutoring on campus spent approximately 8 hours on average (SD = 52 hours) over the 5-year study period. Of the 200 students who attended tutoring, only 142 students attended mathematics-specific tutoring, for an average of about 3 hours (SD = 13 hours) over the study period. In the 2010 cohort, a similar number of students attended on-campus tutoring (n = 238), but the average was only about 3 hours (SD = 11 hours). The number of 2010 cohort students who attended mathematics-specific tutoring was identical (n = 142), but the average amount of time spent in mathematics tutoring was only approximately 2 hours (SD = 9 hours).

Other variables

Additional student demographic data (see Table 1) were added to the model as predictor variables to control for background characteristics and to examine interactions. The demographic variables previously described in Table 1 include gender (0/1, 1 = male), full-time/part-time status (0/1, 1 = full time), enrollment in a student success course (0/1, 1 = enrolled), and years since high school (continuous). A proxy for financial need was based on eligibility for federal Pell grant funds, an indicator of limited socioeconomic status (0/1, 1 = Pell eligible), and a proxy for literacy skills was created based on required enrollment in developmental reading or writing coursework (0/1, 1 = developmental reading/writing required). Initial results indicated that approximately 26% and 44% of the 2009 and 2010 cohorts, respectively, were full-time students, approximately 25% were required to take developmental reading/writing courses, and the number of years since high school ranged from 0 to 39 years, with an average time of 2.6 years since high school (SD = 6.33 years). In each cohort, approximately 38% enrolled in a student success course, a one semester course designed to bolster a student’s overall academic behavior by addressing study skills, time management, and learning styles. A student’s GPA (on a 4.0 scale) in developmental mathematics coursework was calculated and included in the analysis as a proxy for student effort. Due to a large amount of missingness in the data, race/ethnicity was not included in any analysis. Finally, a predictor was created for whether the credit-level mathematics course that students enrolled in was the traditional College Algebra, as opposed to Statistics or Quantitative Literacy.

Although we included all of these variables in our model selection, we limited the discussion in the “Results” section to only main effects that are malleable or partially under the control of the college system—part time/full time (as colleges can offer incentives for students being full time), which credit-level course students enrolled in (as colleges can determine which course should be required for which students) and enrollment in a student success course.

Analysis Techniques

Logistic Regression (RQs 1 and 2)

To model the binary outcomes related to the first two RQs, logistic regression was employed. Logistic regression models the probability of a dichotomous outcome when predicted by other variables (Gelman & Hill, 2007). To conduct logistic regression using R, the glm(family = binomial) command was used (Crawley, 2007). Descriptive statistics as well as zero-order correlations were examined first in order to understand the variables being measured and the relationships (if any) existing between variables. We examined the model assumptions, checking for multicollinearity of predictor variables, independence of error terms, and sparseness when considering interaction terms in a logistic model. To correctly specify a logistic model, necessary assumptions were checked: The dependent variable is binary, a forward method was utilized in order to include only important predictors and exclude trivial ones, and a sufficiently large sample exists (Meyers, Gamst, & Guarino, 2013; Pituch & Stevens, 2016). Due to the archival nature of the data, ensuring that observations were independent of one another and that explanatory variables were measured reliably and without error was not possible. However, variables were checked for correlations and multicollinearity to minimize the risk associated with violating this assumption. In addition, residuals for the outcome variables were plotted post hoc to visually verify randomness. To verify little to no multicollinearity among the independent variables in the models, variance inflation factors were checked and confirmed to be in the desired range (Meyers et al., 2013).

Allison (2014) cites two ways to measure fit for logistic regression models: predictive power as measured by R² measures, and goodness-of-fit tests considering outcomes such as deviance. For goodness of fit, the tests indicate a high p value (p > .05) as showing the model is correctly specified (Allison, 2014). To compare models, the Akaike Information Criterion offered a relative measure of the quality of the model relative to the parsimony of the model, and a model with lower Akaike Information Criterion values was considered a better fit. Once a model was specified, the Huberty index I was computed as a measure of correct classification (Huberty, 2002). A desirable Huberty index value is I > 0.70. The area under the curve measures the probability that a randomly sampled positive observation has a predicted probability greater than a randomly sampled negative observation. An acceptable area under the curve score exceeds 0.70, while a score greater than 0.80 is considered very good (Edman & Runge, 2014; Meyers et al., 2013).

For ease of interpretation, coefficients were transformed to odds ratios, which describe the change in the likelihood of the dependent variable relative to the predictor variable. Odds ratios calculated in the logistic regression model were also transformed to d-type effect sizes (standardized mean differences; Chinn, 2000). In addition, 95% confidence intervals were computed for each odds ratio.

Linear Regression (RQ3)

To examine the number of semesters required for students to complete developmental coursework, linear regression was used. Residual plots were utilized to diagnose violations of the assumptions for linear regression as well as to determine model fit. Linear regression models were fit in R using the lm () function. Results for the linear regression include R² values.

Analysis Techniques

Potential predictor variables were added sequentially and retained only for p values less than .05 and when model fit was improved as measured by a χ² test that tested for reduction of model deviance (the anova () command in R). This forward stepwise approach gives a conservative way to guard against overfitting or underfitting the model. When a variable displayed a statistically significant effect, it was added to the model, and all previously deleted variables were reconsidered for statistical significance. Models considered both main effects and two-way interactions for the predictor variables on the outcomes. To reduce the false discovery rate for multiple comparisons, the Benjamini and Hochberg p value correction method was utilized (Benjamini & Hochberg, 1995; Horton, 2012). The Benjamini and Hochberg correction method is a post hoc procedure that adjusts p values in order to minimize the expected proportion of falsely rejected hypothesis yet retain power when testing multiple significance.

Replication Analysis

A replication analysis with the second cohort (2010) was conducted using the regression models derived from the first cohort (2009), and the models were examined for any changes in statistical significance for key predictor variables. The model was then used to predict whether each member of the second cohort will successfully pass a credit-level mathematics course.

Limitations

As with any correlational study, the relationships uncovered in the statistical analysis do not imply causality or directionality (Springer, 2010). Caution should be taken when attempting to generalize the model to other populations in dissimilar schools or to students with differing academic backgrounds, as there was sampling bias toward the population of one community college system. Furthermore, there may be a variety of background variables that are influencing the relationships (Gelman & Hill, 2007). The possibility of confounding variables not captured in the data set may contribute to the likelihood of completing developmental mathematics and passing of a credit-level mathematics course. Omitted variables may cause endogeneity concerns where model variables are correlated with the error term. This may cause predictors to have significant relations to the outcomes when other unmeasured variables are in fact driving the effect. Measurement error in the independent variables (e.g., placement test score) used in this study may also cause endogeneity concerns.

Based on the 5-year nature of the data set, there may be an undetected maturation effect that led to bias in the model for students over multiple semesters enrolled. Community college students choose courses and so the threat of selection bias is present in terms of the faculty teaching the course, the format, and the time of day. Furthermore, the data set is archival and there may problems with data accuracy or missing data that cannot be addressed. Longitudinal data often lead to problems with attrition, especially for college students who may have relocated by the end of the 5-year period (Shadish, Cook, & Campbell, 2002).

Results

The outcomes for the 2009 and the 2010 cohorts are shown in Table 2. Around half (48% and 51%) of the 2009 and 2010 cohort students successfully completed the developmental mathematics sequence, and approximately a quarter (26% and 29%) successfully passed a credit mathematics course in the 5-year study.

Completion of Developmental Mathematics Sequence

To address the first RQ about successful completion of the developmental mathematics sequence, logistic regression models were fit. Fifteen predictor variables were considered, including variables related to course format, student demographics, and tutoring. Main effects and two-way interactions were considered. Results are shown in Table 3.

Table 3.

Logistic Regression Model Predicting Completion of Developmental Mathematics Sequence.

	2009 Cohort (n = 578)						2010 Cohort (n = 583)
	Β (SE)	Odds	95% CI (Odds)	d	z Value	p	Β (SE)	Odds	95% CI (Odds)	d	z Value	p
(Intercept)	−1.63 (0.28)	0.20	[0.11, 0.33]	−0.90	−5.91	<.001***	−1.51 (0.28)	0.22	[0.13, 0.38]	−0.84	−5.37	<.001***
Test score (normalized)	0.67 (0.21)	1.96	[1.30, 2.96]	0.37	3.22	.002**	0.30 (0.18)	1.35	[0.97, 1.94]	0.17	1.70	.089^†
First class
Pre-Algebra	(Ref)
Beginning Algebra	0.67 (0.32)	1.96	[1.05, 3.70]	0.37	2.10	.036*	0.96 (0.29)	2.61	[1.47, 4.66]	0.53	3.25	.002**
Intermediate Algebra	1.71 (0.34)	5.50	[2.84, 10.90]	0.94	4.99	<.001***	1.68 (0.30)	5.39	[3.03, 9.68]	0.93	5.68	<.001***
Course format
Lecture	(Ref)
Accelerated	2.45 (0.42)	11.53	[5.25, 27.62]	1.35	5.81	<.001***	1.33 (0.32)	3.77	[2.02, 7.26]	0.73	4.09	<.001***
Gender
Female	(Ref)
Male	−0.56 (0.22)	0.57	[0.37, 0.88]	−0.31	−2.55	.012*	−0.35 (0.19)	0.70	[0.48, 1.02]	−0.19	−1.83	.076^†
Enrollment status
Part time	(Ref)
Full time	1.90 (0.26)	6.72	[4.06, 11.42]	1.05	7.24	<.001***	0.76 (0.19)	2.14	[1.47, 3.14]	0.42	3.95	<.001***
Tutoring	1.48 (0.24)	4.39	[2.79, 7.05]	0.82	6.26	<.001***	0.85 (0.20)	2.34	[1.60, 3.44]	0.47	4.35	<.001***
AIC: 570.17AUC: 0.8493437Huberty index = 0.753							AIC: 684.6AUC: 0.7681031Huberty index = 0.679

Note. SE = standard error; CI = confidence interval; AUC = area under the curve; AIC = Akaike Information Criterion.

*p < .05. **p < .01. ***p < .001. ^†p < .1.

For both the 2009 and 2010 cohorts, enrolling in an accelerated developmental mathematics course was associated with an increased likelihood of completing the developmental sequence (2009 cohort odds = 11.53, d = 1.35, p < .001; 2010 cohort odds = 3.77, d = 0.73, p < .001). To better understand the impact of accelerated course format, additional logistic models were created using a subset of students restricted to those who initially placed into Beginning or Intermediate Algebra or those who progressed from Pre-Algebra to Beginning Algebra (see Table 4). This secondary analysis intended to minimize the confounding effect of the accelerated format not usually being offered for Pre-Algebra. Because taking an accelerated course essentially meant that the student had advanced to at least Beginning Algebra, we conducted the secondary analysis removing all students who had only taken Pre-Algebra and had not advanced from this course. When limited to this subset of students, the association between accelerated format and developmental mathematics completion was still observed for both cohorts (2009 cohort subset odds = 7.55, d = 1.12, p < .001; 2010 cohort subset odds = 2.07, d = 0.40, p = .051).

Table 4.

Logistic Regression Model Predicting Completion of Developmental Mathematics Sequence for Students Enrolled in Beginning/Intermediate Algebra.

	2009 Cohort (n = 316)						2010 Cohort (n = 362)
	Β (SE)	Odds	95% CI (Odds)	d	z Value	p	Β (SE)	Odds	95% CI (Odds)	d	z Value	p
(Intercept)	−0.67 (0.23)	0.51	[0.32, 0.80]	−0.37	−2.88	.007**	−0.39 (0.20)	0.67	[0.45, 1.00]	−0.22	−1.95	.051^†
Test score (normalized)	1.12 (0.30)	3.08	[1.75, 5.67]	0.62	3.77	<.001***	−0.05 (0.16)	0.95	[0.69, 1.31]	−0.03	−0.33	.741
Course format
Lecture	(Ref)
Accelerated	2.02 (0.44)	7.55	[3.34, 18.96]	1.12	4.60	<.001***	0.73 (0.35)	2.07	[1.07, 4.18]	0.40	2.11	.051^†
Gender
Female	(Ref)
Male	−0.59 (0.28)	0.55	[0.32, 0.95]	−0.33	−2.12	.034*	−0.32 (0.23)	0.72	[0.46, 1.13]	−0.18	−1.42	.154
Enrollment status
Part time	(ref)
Full time	1.45 (0.30)	4.26	[2.40, 7.74]	0.80	4.87	<.001***	0.06 (0.22)	1.06	[0.68, 1.64]	0.03	0.25	.805
Tutoring	0.88 (0.31)	2.40	[1.32, 4.48]	0.48	2.82	.007**	1.02 (0.27)	2.77	[1.65, 4.74]	0.56	3.81	<.001***
Test Score × Tutoring	−0.94 (0.41)	0.39	[0.17, 0.86]	−0.52	−2.31	.024*	0.41 (0.33)	1.51	[0.79, 2.93]	0.23	1.25	.212
AIC: 348.59AUC: 0.8024089Huberty index = 0.728							AIC: 481.66AUC: 0.6406498Huberty index = 0.608

Note. SE = standard error; CI = confidence interval; AUC = area under the curve; AIC = Akaike Information Criterion.

*p < .05. **p < .01. ***p < .001. ^†p < .1.

Full-time enrollment status was also associated with an increased likelihood of completion (2009 cohort odds = 6.72, d = 1.05, p < .001; 2010 cohort odds = 2.14, d = 0.42, p < .001), as was whether or not a student attended on-campus tutoring (2009 cohort odds = 4.39, d = 0.82, p < .001; 2010 cohort odds = 2.34, d = 0.47, p < .001). Other support services considered (student success course, developmental reading/writing coursework, and time spent in tutoring) were not statistically significant.

Passing Credit Mathematics Course

To examine the second outcome variable, passing a credit-bearing mathematics course, only the subset of students who completed the developmental sequence and subsequently chose to enroll in credit mathematics was considered (2009: n = 228; 2010: n = 253). At the study college, policy requires developmental students to enroll in the appropriate developmental course every semester in order to make continuous progress toward becoming college ready. However, once the sequence of developmental courses is completed, students may choose if and when to enroll in credit-level mathematics coursework. Thus, by only including those who actually enrolled in a credit-level course, we best capture the effects of our variables and not the effects of student choice of whether to enroll during the 5-year period. In addition, by separating our analyses into factors that allowed students to succeed in the developmental sequence, and factors that allowed students to pass a credit-bearing class, we allow for the possibility that different factors may be important at different stages of the students’ progression.

Results of the logistic regression analysis for the second outcome variable (passing a credit-level mathematics course) are shown in Table 5. A statistically significant effect occurred for developmental mathematics GPA with both cohorts (2009 cohort odds = 2.46, d = 0.50, p = .002; 2010 cohort odds = 2.23, d = 0.44, p = .009), and both cohorts showed a statistically significant interaction between GPA in developmental coursework and the amount of time spent in mathematics-specific tutoring (2009 cohort odds = 0.60, d = −0.29, p = .004; 2010 cohort odds = 0.66, d = −0.23, p = .048). In both cases, the amount of time spent in mathematics tutoring seems to matter more for students with lower GPAs and less for students with higher GPAs. For example, for 2010 students with a GPA one SD below the mean, the association of GPA with completion of developmental mathematics is increased from odds of 0.44, exp(−.80), for students who spend no time in mathematics-specific tutoring to odds of 1.28, exp(−.80) × 1.87 × exp(0.42), for a one-unit increase in the log of hours spent in mathematics-specific tutoring. In practical terms, a student with lower grades in developmental coursework has decreased odds of successfully completing a credit mathematics course, but time spent in mathematics-specific tutoring counteracts the detrimental impact of low grades and becomes an important factor associated with success in credit mathematics coursework. Note that in this analysis GPA is not simply acting as a proxy for whether students made it through developmental mathematics—all students included in this analysis had already made it through the developmental sequence, but their grades in the developmental courses still mattered.

Table 5.

Logistic Regression Model Predicting Completion of Credit Mathematics Course for Students Completing Developmental Coursework.

	2009 Cohort (n = 219)						2010 Cohort (n = 253)
	Β (SE)	Odds	95% CI (Odds)	d	z Value	p	Β (SE)	Odds	95% CI (Odds)	d	z Value	p
(Intercept)	0.58 (0.24)	1.79	[1.12, 2.94]	0.32	2.40	.025*	0.25 (0.25)	1.29	[0.80, 2.10]	0.14	1.03	.302
Test score (normalized)	0.13 (0.23)	1.13	[0.72, 1.79]	0.07	0.55	.585	−0.39 (0.23)	0.68	[0.42, 1.03]	−0.21	−1.67	.094^†
GPA (recentered)	0.90 (0.25)	2.46	[1.54, 4.10]	0.50	3.64	.002**	0.80 (0.27)	2.23	[1.33, 3.92]	0.44	2.91	.009**
Enrollment status
Part time	(Ref)
Full time	0.16 (0.35)	1.17	[0.59, 2.36]	0.09	0.45	.652	0.42 (0.32)	1.52	[0.82, 2.87]	0.23	1.32	.186
Mathematics tutoring hours (Ln)	0.16 (0.15)	1.17	[0.88, 1.59]	0.09	1.06	.291	0.63 (0.20)	1.87	[1.30, 2.86]	0.35	3.15	.008**
Credit mathematics course
College Algebra	(ref)
Other	0.55 (0.40)	1.74	[0.81, 3.97]	0.30	1.37	.171	0.96 (0.40)	2.60	[1.24, 6.07]	0.53	2.39	.028*
GPA × Test Score	−0.71 (0.25)	0.49	[0.30, 0.79]	−0.39	−2.85	.009**	−0.04 (0.28)	0.96	[0.56, 1.67]	−0.02	−0.16	.874
GPA × Full Time	0.90 (0.45)	2.47	[1.07, 6.25]	0.50	2.02	.044*	0.42 (0.35)	1.52	[0.77, 3.05]	0.23	1.20	.230
GPA × Mathematics Tutoring Hours (Ln)	−0.52 (0.16)	0.60	[0.43, 0.81]	−0.29	−3.19	.004**	−0.42 (0.20)	0.66	[0.44, 0.97]	−0.23	−2.07	.048*
GPA × Credit Mathematics Course (Other)	−0.85 (0.40)	0.43	[0.20, 0.94]	−0.47	−2.14	.039*	−0.27 (0.39)	0.76	[0.36, 1.71]	−0.15	−0.69	.488
AIC: 245.56AUC: 0.7706477Huberty index = 0.749							AIC: 274.62AUC: 0.7713627Huberty index = 0.752

Note. SE = standard error; CI = confidence interval; AUC = area under the curve; AIC = Akaike Information Criterion; GPA = grade point average.

*p < .05. **p < .01. ***p < .001. ^†p < .1.

The 2009 cohort shows an interesting interaction between GPA in developmental coursework and full-time enrollment status: An increase of one point in developmental GPA for a part-time student is associated with increasing the odds of passing credit mathematics by 2.46 (d = 0.50, p = .002), but a student with a one point increase in GPA who also attends college full time has odds of passing a credit mathematics course greatly increase to 7.11 (2.46 × 1.17 × 2.47). Similarly, a part-time student with a one point decrease in GPA is associated with lowered odds of passing a credit mathematics course, odds = 0.41, exp(−.90), while a full-time student with the same one point decrease in GPA again has a greater effect of GPA with low odds of 0.19, exp(−.90) ×1.17 × exp(−.90). Thus, being enrolled full time is associated with a magnified effect for GPA, but only in the 2009 cohort.

A second statistically significant interaction only in the 2009 model is between GPA in developmental coursework and placement test score (odds = 0.49, d = −0.39, p = .009). As a student’s test score increases, the importance of the grades earned in developmental coursework is lessened. Conversely, the higher a student’s placement score, the less important the grades earned in developmental coursework are. In practical terms, a student hoping to pass a credit mathematics course does not always require both high placement scores and high developmental coursework grades—one will often suffice. For students beginning the sequence with low test scores, the grades earned in developmental coursework are important. For example, for a student with a placement test score one SD below the mean, a one SD change in GPA is associated with odds of 4.39, 2.46 × exp(−.13) × exp(.71). For a student with an average test score, grades in developmental coursework seem to matter, but not as much, as reflected by the main effect for GPA (odds = 2.46, d = 0.50, p = .002). But for a student with high placement scores, grades in developmental mathematics coursework seem to have a smaller association with the likelihood of eventually passing a credit mathematics course. For example, for a student with a placement test score one SD above the mean, a one SD increase in GPA is associated with odds of 1.36 (2.46 × 1.13 × 0.49).

Finally, in the 2009 model, a statistically significant interaction occurs between GPA in developmental coursework and which credit mathematics course a student enrolls in (odds = 0.43, d = −0.47, p = .039). Due to the small number of students opting to enroll in Statistics or Quantitative Literacy (Table 1), these courses were combined into one category of courses (OTHER), and compared with enrollment in College Algebra. For students with a lower GPA (below the mean), there is a positive association between enrolling in Statistics or Quantitative Literacy (a nonalgebra pathway) and passing a credit mathematics class. For example, a student with a GPA one SD below the mean has odds that are reduced by 0.41, exp(−0.90), when considering passing College Algebra, but has a significantly greater odds of passing Statistics or Quantitative Literacy, odds = 1.66; exp(−.90) × 1.74 × exp(0.85). Conversely, for stronger mathematics students (indicated by a higher developmental coursework GPA), a high GPA is negatively associated with passing a non-College Algebra credit-level class. For example, a student with a GPA one SD higher than the average is associated with a reduction in odds of passing their credit-level mathematics course from 2.46 for College Algebra to odds of 1.84 (2.46 × 1.74 × 0.43) for the likelihood of passing the other credit mathematics courses. This is not surprising given that the developmental sequence is targeted to preparing students for College Algebra. For students who are at the mean level of GPA, the decision of whether or not to take College Algebra does not seem to be significantly associated with success (p = .171).

Two variables showed associations with success in credit-level mathematics in the 2010 cohort where the 2009 model showed no effect. For 2010 students, the likelihood of passing a credit mathematics course was increased by odds of 1.87 (d = 0.35, p = .008) for a one-unit increase in the log of mathematics tutoring hours. Furthermore, an additional main effect was detected for which credit mathematics course a student enrolled in. When 2010 students opted to enroll in Statistics or Quantitative Literacy after completing developmental coursework, the likelihood of successfully completing the credit course was increased by odds of 2.60 (d = 0.60, p = .028).

As including developmental mathematics GPA in the models might dampen the effects of the different course formats (online/hybrid, accelerated, and self-paced), models were also fit without the GPA variable. Results were similar and did not show any effects for the course formats. In both cohorts, mathematics tutoring hours were significantly positively associated with passing a credit-level mathematics course. In the 2010 cohort, enrolling in a non-College Algebra credit course was also associated with higher odds of passing a credit-level mathematics course. No other effects were significant. Thus, the inclusion of the GPA variable did not seem to be functioning to mask other significant effects.

Time Required to Complete Developmental Mathematics

To answer the third RQ regarding the association between the variables and the amount of time (number of semesters) required to successfully complete the developmental mathematics sequence, linear regression was used. A subset of students who successfully completed the developmental sequence was created (2009: n = 285; 2010: n = 301).

Main effects for both cohorts occurred for enrollment status, first class enrolled, self-paced course format, and online/hybrid course format. In addition, a statistically significant interaction occurred for both cohorts between years since high school and gender. Results of the regression analysis are shown in Table 6. The mean number of semesters to complete the required developmental sequence is approximately three (3.2 semesters for the 2009 cohort; 2.6 semesters for the 2010 cohort), where one complete academic year includes fall, spring, and summer semesters, or 3 semesters. Based on the regression model, a student in the reference group for all variables in the table (recent high school graduate, female, average test scores, no self-paced or online format, starting in Pre-Algebra, attending part time) completes the developmental sequence in a much longer time line: approximately 6.1 semesters, or 2 academic years for the 2009 cohort, or approximately 4.7 semesters for the 2010 cohort.

Table 6.

Summary of Multiple Regression Model Describing Time to Complete Developmental Mathematics Coursework.

	2009 Cohort (n = 285)				2010 Cohort (n = 301)
	Β	SE	t Value	Corrected p	Β	SE	t Value	Corrected p
(Intercept)	6.06	0.35	17.52	<.001***	4.69	0.36	13.02	<.001***
Test score (normalized)	0.32	0.22	1.48	.141	0.00	0.20	0.01	.99
First class
Pre-Algebra	(Ref)
Beginning Algebra	−2.07	0.39	−5.34	<.001***	−1.56	0.39	−4.04	<.001***
Intermediate Algebra	−3.52	0.39	−9.02	<.001***	−2.83	0.38	−7.52	<.001***
Course format
Lecture	(Ref)
Self-paced	2.61	0.49	5.30	<.001***	1.26	0.33	3.79	<.001***
Online/hybrid	0.66	0.30	2.16	.039*	1.04	0.34	3.07	.003**
Gender
Female	(ref)
Male	−0.84	0.28	−3.05	.004**	0.31	0.27	1.16	.245
Enrollment status
Part time	(ref)
Full time	−1.25	0.24	−5.14	<.001***	−0.78	0.24	−3.27	.002**
Years since high school	−0.09	0.02	−4.11	<.001***	−0.01	0.03	−0.30	.77
Test Score × Self-Paced Format	1.30	0.64	2.02	.049*	0.52	0.51	1.03	.305
Years Since High School × Male	0.12	0.05	2.37	.026*	−0.150	0.07	−2.28	.024*
	R² = .4504; adjusted R² = .4297F(10, 265) = 21.72, p < .001					R² = .3555; adjusted R² = .3328F(10, 284) = 15.66, p < .001

Note. SE = standard error.

*p < .05. **p < .01. ***p < .001.

Several factors appear to be associated with a significant delay or increase in the time line for both cohorts, including enrolling in a self-paced format (associated with approximately 2.6 more semesters, p < .001 for the 2009 cohort; associated with approximately 1.3 more semesters, p < .001 for the 2010 cohort) or enrolling in an online/hybrid format (associated with 0.7 more semesters, p = .039 for the 2009 cohort; associated with 1 more semester, p = .003 for the 2010 cohort).

The 2009 cohort also had an interesting interaction between placement test score and self-paced format: for students with higher test scores, the effect of enrolling in a self-paced is associated with an additional increase of approximately 1.3 semesters to the already delayed time line (p = .049) for each one SD increase in test score. One interpretation is that the self-paced format could be detrimental for students with higher placement scores, implying that those students would be more efficient by enrolling in a lecture or online section. Alternately, for students with lower placement test scores, a self-paced format could provide a comfortable environment in which students may progress at a slower pace, resulting in a slowed time line but not as slow as if a student fell behind and had to repeat a course. For example, a 2009 student with a placement score of one SD below the mean would be associated with a 1 semester (−0.32 + 2.61 − 1.30) addition to the time line for completing developmental mathematics if the student took at least one self-paced course, compared with a student with an average placement score in a self-paced class (2.61 semesters added).

Several variables are associated with a shortened time line for completion with both cohorts. Not surprisingly, being enrolled full time is also associated with a shorter time line to complete developmental mathematics (associated with 1.3 fewer semesters, p < .001 for the 2009 cohort; 0.8 fewer semesters, p = .002 for the 2010 cohort).

Summary

The regression models specified for the 2009 cohort and replicated for the 2010 cohort sought to provide answers to the RQs of the study. The findings summarized here are those that replicated for both cohorts. First, regarding which structures in developmental mathematics are associated with students’ likelihood of completing the sequence, the models suggested that tutoring may be a significant predictor of success. In addition to tutoring, enrolling in an accelerated format was also associated with an increase in the likelihood of a student completing the developmental mathematics sequence. Even after removing the confound that accelerated format often implied progress to the second developmental course, enrollment in an accelerated format was associated with an increased likelihood of completion. Finally, full-time enrollment showed a statistically significant positive association with increased likelihood for completion compared with part-time enrollment.

When considering the second RQ, related to a students’ likelihood of passing a credit-bearing mathematics course, the models revealed that the amount of time that a student spent in mathematics-specific tutoring interacted with GPA to lessen the association with grades. For both cohorts, mathematics tutoring had stronger positive associations for weaker students and weaker positive associations for stronger students.

To answer the third RQ, the impact of developmental mathematics structures on the amount of time (number of semesters) required to successfully complete the developmental mathematics sequence, the predictor variables associated with an increased time line for students were course format and enrollment status. Students who opted for self-paced or online/hybrid course formats in both cohorts were associated with an increase in the amount of time required to complete the sequence, with online being associated with approximately 1 additional semester, and self-paced associated with an additional 1.3 (2010) to 3 (2009) semesters. Finally, being enrolled full time has a strong association with completing the sequence in less time, saving approximately one semester. Results are summarized in Table 7.

Table 7.

Summary of Results for Factors Associated With College Algebra Readiness, 2009 and 2010 Cohorts.

Comparison	2009 (n = 578)	2010 (n = 583)
Completion of developmental mathematics sequence	Positive associations between accelerated course format and completion of the developmental sequence (d = 1.35) Positive associations between full-time enrollment and completion of the developmental sequence (d = 1.05) Positive associations between completion of the developmental sequence and attending on-campus tutoring (d = 0.82)	Positive associations between accelerated course format and completion of the developmental sequence (d = 0.73) Positive associations between full-time enrollment and completion of the developmental sequence (d = 0.42) Positive associations between completion of the developmental sequence and attending on-campus tutoring (d = 0.47)
Success in credit mathematics course	Negative associations for the interaction between GPA and test score: GPA has a stronger association for passing credit mathematics course for students with lower standardized test scores (d = −0.39) Positive associations for the interaction between GPA in developmental coursework and passing credit mathematics class is higher for full-time students compared with part-time students (d = 0.50) Negative associations for the interaction between GPA and time spent in mathematics-specific tutoring: Time spent in mathematics-specific tutoring matters more for students with lower GPAs in developmental coursework and less for students with high GPAs in its association with an increased likelihood of passing a credit mathematics course (d = −0.29) Negative associations for the interaction between GPA and which credit mathematics course students enroll in: For students with lower grades in developmental coursework, enrolling in Statistics or College Mathematics is associated with an increased likelihood of passing a credit mathematics course (d = −0.47)	Positive associations between time spent in mathematics-specific tutoring and likelihood of passing credit mathematics class (d = 0.35) Positive associations between which credit mathematics class enrolled in and likelihood of passing increases when students opt to NOT take College Algebra (d = 0.53) Negative associations for the interaction between GPA and time spent in mathematics-specific tutoring: Time spent in mathematics-specific tutoring matters more for students with lower GPAs in developmental coursework and less for students with high GPAs in its association with an increased likelihood of passing a credit mathematics course (d = −0.23)
Time required to complete developmental mathematics remediation	Enrolling in a course format other than lecture is associated with additional time for a student to coursework complete developmental (self-paced B = 2.61; online/hybrid B = 0.66) Full-time students are associated with increased completion times compared with part-time students (B = −1.25) The interaction between self-paced format and standardized test scores is associated with increased time lines for completion when students have higher test scores (B = 1.30)	Enrolling in a course format other than lecture is associated with additional time for a student to complete developmental coursework (self-paced B = 1.26; online/hybrid B = 1.04) Full-time students are associated with increased completion times compared with part-time students (B = −0.78)

Note. Results that replicate in both data sets are bolded. GPA = grade point average.

Implications

The models specified in the various analyses suggest several critical variables with the potential to improve outcomes for developmental mathematics students. The structures examined in this study can be loosely categorized into three broad areas, as described later.

Course Format

Analyses revealed that accelerated format was associated with an increased likelihood of completing the developmental sequence and passing a credit mathematics course, supporting the positive findings of the other researchers examining accelerated format (Cafarella, 2013; Edgecombe, 2011; Hern & Snell, 2010; Jaggars et al., 2014; U.S. DOE, 2005; Zachry & Schneider, 2010). These findings for accelerated course format echo those of prior research, which suggests that accelerated format can shorten the developmental coursework sequence and quicken remediation (Bailey et al., 2010; Biswas, 2007; Dana Center et al., 2012).

Colleges should offer accelerated courses as an option for students, and train advisors with the needed information to be able to address any student questions related to enrollment in an accelerated course. The role of advising in offering accelerated courses cannot be emphasized enough. While some students may be overwhelmed by the idea of a condensed 8-week course during a typical 16-week semester, institutions could instead encourage students to attend school year-round, thus enrolling in summer classes, which are also accelerated by design (5 or 10 weeks). Summer coursework assists students in maintaining momentum to completion of the sequence and provides a means for continuous coursework to minimize the loss of memory that comes after a hiatus in coursework. Due to the replicated negative associations between self-paced and online/hybrid formats and the time to complete developmental mathematics, these options should be investigated further and could potentially be limited.

Support Services

Prior research indicates that institutional supports such as tutoring can play a critical role in assisting underprepared students (Adelman, 2006; Boylan, 2009; Boylan & Saxon, 2006; MDC, 2012; Tinto, 2006). Tutoring had a strong positive association in a student’s likelihood of completing developmental mathematics and passing credit mathematics for both cohorts. Attending any on-campus tutoring was associated with the completion of developmental coursework, supporting the findings of Tinto (2006) that the connectedness gained from participating in college life beyond the classroom walls is a huge indicator of persistence. A direct positive association also occurred between the likelihood of passing a credit mathematics course and the log of hours spent in mathematics-specific tutoring for the 2010 cohort. Furthermore, an interaction between GPA and log of mathematics tutoring hours replicated for both cohorts, supporting the notion that students can counteract the detriment of lower grades by attending mathematics-specific tutoring. However, in both cohorts examined in this study, relatively few students (less than half) participated in any tutoring, even though tutoring was available at no additional cost on every campus. This finding supports prior research that many students do not take advantage of available resources, even when it is clearly in their best interest to do so. As one researcher laments, “Students don’t do optional” (Couturier, 2010, p. 4). Institutions could transform the known benefit of tutoring from optional to routine or mandatory by working through academic departments to incorporate this practice into coursework (Klohn et al., 2016).

Other Factors Related to Success

Outside course format and support services, two other predictor variables showed statistical significance across analyses. For both cohorts, full-time enrollment was associated with increased likelihood of students completing developmental mathematics coursework and in less time. This supports the findings of Adelman (2006) regarding the improvement in student completion rates and graduation rates for full-time students. Institutions could create tuition incentives to encourage students to enroll full time, such as offering a discount for each 12 hours competed successfully or offering a flat-rate tuition for a student who enrolls in either 9 or 12 hours, making the switch to full-time enrollment more attractive.

In addition, the decision to enroll in a credit mathematics course other than College Algebra was associated with an increased likelihood of passing a credit mathematics course for students in the 2010 cohort. For the 2009 cohort, this finding was combined as an interaction with GPA, meaning that the option to enroll in Statistics or College Mathematics had an associated effect that was especially significant for students with lower GPAs. While the findings for which credit mathematics class did not replicate exactly across cohorts, in both cases, there were situations where these alternate credit offerings were associated with student success. This suggests that weaker students could have a greater likelihood of passing a credit-level class if they avoid the College Algebra path and seek a nonalgebraic path through mathematics, echoing recent reform efforts to provide nonalgebraic options as a means to improve success in mathematics (Charles A. Dana Center, 2012; Dana Center et al., 2012; Logue et al., 2016; Yamada, 2014).

Conclusion

More and more underprepared students are arriving at community colleges, looking for access to high education and an advanced credential. Developmental mathematics programs provide a means for underprepared students to attain credit-bearing coursework in mathematics, yet the potential is all too often unrealized when students fail to complete required remediation. Rather than allowing access, developmental programs can serve as barriers that separate students from the promise of higher education and improved job opportunities. Multiple factors, such as assessment routines, placement protocols, course offerings, and academic support programs, all come together to determine the efficacy of developmental education programs. When these myriad processes work together, the outcome can be increased success for college students in mathematics. Alternately, when the multiple processes do not culminate in a collaborative and networked effort focused on student success, frustration and potential failure ensue. In order for the promise of developmental education to be realized, the system must be repaired.

This study identified several factors associated with increased likelihood for success, including tutoring, accelerated format, and full-time enrollment. GPA and initial course placement were also revealed as important considerations, along with standardized test scores. Other potential factors almost certainly exist, and future research may reveal alternate variables that are highly associated with success for developmental mathematics students. It is our responsibility to continue the search for the multitude of future students who are relying on community colleges and developmental mathematics as a path to a successful future.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Elizabeth Howell

Author Biographies

Elizabeth Howell is a mathematics faculty member and Division Chair at North Central Texas College. She teaches both developmental and credit mathematics, and has worked in the area of developmental mathematics for 19 years. Her research interests focus on smoothing the transition from secondary to post-secondary mathematics classes for all students.

Candace Walkington is an associate professor in Mathematics Education and Learning Sciences at Southern Methodist University. Her research focuses on ways to make high school mathematics courses more relevant to students’ lives, interests, and experiences. She teaches courses for pre-service and in-service mathematics teachers.

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