Transforming and Accelerating Developmental Mathematics at
El Camino College
Success of the Traditional Developmental Mathematics Sequence – The Pipeline Problem
The diagram below illustrates, rather bluntly, why we were motivated to change the way we think about developmental mathematics, not simply as a sequence of courses, but as a program. After three years, among first-time arithmetic students in Fall 2009, only 11% of the original cohort had passed intermediate algebra (developmental completion rate) and only 3% had passed a transfer-level mathematics course (transfer-level completion rate). Other three-year tracking studies reveal similar results. Dismal progression rates are also an issue of equity; results are even worse for under-represented students. One aspect of this unacceptable situation has been deemed the pipeline problem. Long sequences of courses offer students too many premature points of exit. The members of the Developmental Mathematics Committee set themselves the following goal:
The Challenge: Redesign our curriculum so that ALL El Camino College developmental math students have a pathway to be ready for a transfer-level mathematics course after
at most two semesters.
Our Response to the Challenge:
Basic Accelerated Mathematics and General Education Algebra
(BAM) (GEA)
The members of the Developmental
Mathematics Committee at El Camino
College created two new courses, Basic
Accelerated Mathematics (BAM) and
General Education Algebra (GEA).
Integrating BAM and GEA into our
program offers students the opportunity
to complete developmental mathematics
by taking no more than two courses.
For students who place below the
elementary algebra level, BAM provides
a pathway to the three courses that are
one level below transfer: GEA, Inter-
mediate Algebra for STEM (IA-STEM),
or Intermediate Algebra for General
Education (IA-GE).
For students ready for elementary algebra, GEA offers a one-semester alternative to the two-semester elementary/intermediate algebra sequence and serves as a pre-
requisite for all entry-level transferable mathematics courses except college algebra and trigonometry. The elementary/intermediate algebra sequence remains the appropriate pathway for students considering a STEM (or Business) major or who have no plans to transfer and only needs a course to satisfy the associate degree mathematics requirement.
Students placing at the intermediate algebra level will take GEA, IA-STEM or IA-GE, depending on their education plans.
Guiding Design Principles for BAM and GEA:
The design of BAM and GEA reflects several beliefs we hold due to research results, our conversations in our faculty cohorts, and our personal experience:
· Students develop arithmetic and algebra skills more easily when these skills are contextualized and intertwined. Students need not be proficient at basic arithmetic before they can successfully set up and solve algebra problems.
· Students have and can create greater capacity than we give them credit for.
· Student learning is deeper and longer lasting when students engage in fewer topics at greater depth, within authentic and meaningful contexts, and using multiple mathematical representations.
· Basic skills students do better with a greater number of structured contact hours.
· Basic skills students benefit when college-readiness skills (affective domain skills) are embedded in the course.
BAM Nuts and Bolts
BAM is a pass/no pass, degree-applicable, five credit-unit course, in which students gain the arithmetic and algebra competencies needed for success in an intermediate algebra-level course after one semester of work. This course is recommended for all students who place below the elementary algebra level.
There are four hours a week in a classroom, where students engage themselves and each other with the activities that strengthen numeracy, problem solving skills, and conceptual understanding. There are four hours a week in a computer laboratory, in which students use a self-paced, mastery-learning online program designed to reinforce procedural knowledge.
The course has three content levels, which students encounter in a blended fashion. Level A tackles the arithmetic skills used in algebra courses. Level B covers basic algebraic concepts and graphing in the context of studying linear functions, first with integer and later with fractional and decimal coefficients. Level B culminates with a linear modeling project. Level C addresses the rest of the algebra topics from a typical introductory algebra course.
Students demonstrate mastery at an 80% level on the online mastery quizzes and traditional in-class exams. Students may retake new versions of the quizzes and exams.
Depending on each student’s educational goals and demon-strated competencies, BAM has multiple target courses. Completing all three levels qualifies a student for any course one level below transfer: General Education Algebra (GEA), Intermediate Algebra for General Education, or Intermediate Algebra for Science, Technology, Engineering and Mathematics. Completing Levels A and B successfully qualifies a student for GEA or Elementary Algebra. Students who do not successfully complete Levels A and B have a number of options, including repeating the course and continuing where they left off.
GEA Nuts and Bolts
For many of our students, a statistics course is a more appropriate transfer-level general education mathematics course. We believe that students will use their numeracy, algebra and mathematical reasoning skills most often in the context of making or understanding the decisions based on data and statistics. We set out to design a course that tailors the elementary and intermediate algebra skills to fit the needs of the student learning how to pose questions about data and to interpret data in a way that is meaningful.
GEA is a graded, degree-applicable, four credit-unit course in which students, using descriptive statistics as the primary application, develop the algebraic and mathematical reasoning skills that are necessary to succeed in most transfer-level mathematics courses and which are important for a generally educated populace. This course is open to anyone who is eligible for elementary algebra and is appropriate for any student not planning to take college algebra or trigonometry. (GEA is not currently approved for the associate degree mathematics requirement, so students who only need a course to satisfy this graduation requirement are encouraged to take IA-GE or IA-STEM.)
In GEA, students spend three hours a week in a classroom working on group activities that explore algebraic concepts in the context of real-world situations. For the reading and writing assignments, we used a monograph written by Edward Tufte, in which he discusses useful and detrimental displays of data, comparing Dr. John Snow’s illuminating diagram tracking the 1854 cholera epidemic in London with the confusing communications between NASA engineers and decision-makers in the hours before the Challenger disaster. Students also spend three hours a week in a computer laboratory, practicing algebra skills through contextual problems.
Evaluation Results
BAM and GEA do not form an isolated accelerated sequence, but rather these courses are integrated into our existing program. Therefore, the heart of our long-term evaluation plan is to compare two-year and three-year developmental completion rates and transfer-level completion rates in our program before and after BAM and GEA were introduced.
Currently, we have results comparing the progression of students in the Fall 2011 BAM and Arithmetic cohorts over two years, as well as comparing the progression of students in the Spring 2012 Elementary Algebra and GEA cohorts over one-and-a-half years. These cohorts include students enrolled in BAM, GEA and their comparison groups on census day in the indicated semester. The students in BAM and GEA were enrolled in these courses for the first time, but students in both of these cohorts may have taken mathematics at El Camino in earlier semesters. In particular, many of the students in GEA in this spring 2012 had successfully completed BAM the previous fall. It is too soon to conclude much about the long-term success of BAM and GEA, but we have some promising initial data that also suggest some unexpected challenges and lessons.
Fall 2011 BAM and Arithmetic Cohorts:
It is already well-known that shortening the pipeline leads to higher progression rates, but comparing the percent of students in these cohorts who complete developmental mathematics after two years (46% of BAM students versus 8% of Arithmetic students) drives this fact home in a dramatic way.
In contrast, some caution is required when we look at student progression from developmental mathematics to transfer-level mathematics. At first glance, it is tempting to conclude that students starting in BAM persist to and succeed in a transfer-level mathematics course at a much higher rate. But what the numbers tell us may be colored by some confounding variables.
The persistence rate of 35% for student in the Arithmetic cohort may be artificially low, since this cohort includes students pursuing an associate degree who may never attempt a transfer-level course. Currently, we do not have the ability to control well for this variable.
Returning the BAM cohort, it may be surprising that the persistence rate for students in the BAM cohort drops from 93% moving from BAM into the courses one level below transfer to just 52% moving from developmental mathematics into transfer-level mathematics. What may account for this is one of our unexpected challenges. Most of the students in the BAM cohort were new students at El Camino College. Instead of spending four semesters or more accumulating units, the successful BAM students were transfer-level ready after just two semesters. As a result, most of these successful students have much lower registration priority than peers competing for seats in a limited number of transfer-level courses.
Spring 2012 GEA and Elementary Algebra Cohorts:
The data for the Spring 2012 GEA and Elementary Algebra Cohorts tell an interesting story. In terms of the developmental mathematics completion rate and transfer-level mathematics completion rate, the GEA cohort outperformed its comparison group. Again, we have not controlled for students in the Elementary Algebra cohort who have no intention of taking a transfer-level mathematics course, so the persistence rates moving from one level below transfer to transfer may introduce some noise into the picture. However, the comparatively low pass rate for GEA students in the transfer-level course is a concern we must investigate. Anecdotally, we have heard from GEA students that have attempted statistics more than once that GEA prepares them better for statistics courses where the context of statistical analysis and communication are emphasized more than formulas and rules.
Other Unexpected Challenges:
The success rate in BAM may seem quite high, but BAM has multiple target courses, so we expect most students to take BAM once and move on to a course that is appropriate to the student’s need and achievement. So, who are the students who do not complete BAM? BAM is the first mathematics course for roughly two-thirds of all students who enroll. We have informally investigated the remaining students and we suspect that many of them have exhausted their attempts at arithmetic or pre-algebra. Unfortunately, almost all of these students have not found the structure of BAM helpful.
Professional Development for Accelerated Courses
Will this acceleration model scale up? Yes, but it will require a great deal of professional development support. Both of these courses are dissimilar to courses in a traditional developmental mathematics program. In order to scale up, it is imperative that some professional development accompany the implementation of these courses. We have used three different strategies for introducing new instructors to the courses and for ongoing assessment of the courses.
1. The instructors in both courses work together as teams, meeting weekly to discuss the effectiveness of instruction and further refine classroom activities and common assessments.
2. Adjunct instructors interested in teaching BAM or GEA have been funded from BSI funds to “shadow” a class to see how it is conducted. Shadows are invited and paid to attend multiple classes per week and to participate in the weekly team meetings.
3. In BAM, a single section can be a shared assignment by two instructors. New instructors can be paired with a seasoned instructor to help ease the transition to teaching a course that some might characterize as a “one-room schoolhouse”.
Cost-Benefit Considerations
Cost comparisons are difficult – viewing costs from different perspectives might lead to a variety of conclusions. Below we have the results of an analysis that compares different measures of the cost of both the traditional pathway in four terms and accelerated pathways in two terms, based on the BAM Fall 2011 cohort and its comparison group.
Traditional Developmental Sequence / Accelerated Developmental Sequence / ApproximateChange from Traditional to Accelerated
Minimum Time to Transfer-Level-Readiness / 4 terms / 2 terms / half the time
Units to Transfer-Level-Readiness / 15 units / 9 units / 6 fewer units
Average Weekly Instructor-Student Hours / 4.25 hours / 7 hours / 65% increase
Sequence Faculty Load / 1.13 / 0.83 / 25% decrease
FTEF per 100 Beginning Students (counts repeat rates) / 1.5 / 2.1 / 20% increase
FTEF per 100 Transfer-Level-Ready Students / 34 / 7 / 80% decrease
Transfer-Level-Ready Students / 8% of beginning cohort / 46% of beginning cohort / 6 times greater
Transfer-Level-Ready Students per Faculty Load / 6 / 22 / 3.5 times greater
(Instructors made the calculations in this section. The El Camino College Office of Institutional Research and Planning has not
verified these calculations.)
When looking at costs, it is helpful if we choose a desired outcome and estimate the cost of students achieving that outcome. We will focus on the developmental completion rate for the Fall 2012 BAM and Arithmetic cohorts. We found that in order to achieve the 8% developmental completion rate, the 250 students in the Arithmetic cohort required 3.9 FTEF; in order to achieve the 46% developmental completion rate, the 109 BAM students required 2.3 FTEF. While the faculty load per 100 beginning students was 37% higher for the BAM students (2.1 FTEF/100 students compared with 1.5 FTEF/100 students), the faculty load per student completing developmental mathematics (scdm) was 76% lower among the BAM students (0.05 FTEF/scdm compared with 0.19 FTEF/scdm). We are currently working to verify these calculations.
If we assume that all students who succeed at one level persist to the next level, then the advancement rates and success rates at each level are equal. It is then possible to predict the relative costs of replacing a four-course sequence with a two-course sequence. Again, we focused on target developmental completion rates. In the first row of the table, we see that if we desire a 10% developmental completion rate, the average advancement rates in the four-course and two-course sequences must be the fourth and square roots of .10, respectively. We can then approximate the sequence faculty load (SFL) for both the 4-course and 2-course sequences. It is clear that increasing the target developmental completion rate increases the SFL in both cases, but the percentage increase in the SFL for the two-course sequence is far less than for the four-course sequence. In other words, not only does a two-course sequence result in better completion rates, but it also costs less in a two-course sequence to increase the target completion rate than in a four-course sequence.