The Mainstreaming College Mathematics Project^{1, 2, 3}

Harry P. Allen
Department of Mathematics
The Ohio State University

What is MCM? The Mainstreaming College Mathematics Project was established in 1994 to determine if a single ten-week pro-active intervention, to mathematically activate entering students who test into our remedial Mathematics program, was possible. By way of background, Ohio State gives two Mathematics placement exams based on ACT MATH scores. Students with ACT MATH at most 24 take the B-test, and those who score at most 37% (Math code T) are required to take a remedial Mathematics course^{4, 5}. Almost all of our freshman who test at this level have had at least three years of college Mathematics]preparatory Mathematics⁶ in high school. These students tend to do poorly in subsequent Mathematics courses, and about half leave the university after two years.

Students who enroll⁷ in MCM are placed into Math 104⁸, and are required to have concurrent enrollment in Math 103⁹. The format of each class session of Math 103 is one of informal student centered active learning in small groups of three or four students each. Each class meeting is guided by a worksheet containing relatively routine problems (from Math 104) that are framed by detailed instructions that attempt to evoke oral-aural-cognitive interactions with a focus on strategy and process^10. The primary role of the teaching staff is to oversee the activity of the groups, and to facilitate, redirect, and intervene as needed. Homework is collected regularly and is corrected and returned. Each section is limited to 32 students meeting twice a week, and is staffed by a graduate teaching associate together with an undergraduate student classroom assistant.

Students who take Math 050 may take either Math 075^11, or Math 104 as their next Math course. Math 075, although similar to Math 104, is not as demanding.

Has MCM made a difference? The outcomes^12,13 thus far have been very interesting. Not unexpectedly MCM students do better in Math 104 than students who start out in Math 050. Table 2 lists the percentage of each cohort (Math 050, MCM and 104) that received a grade of at least C- in Math 104 on their first attempt. Table 3 lists the percentage of each cohort that eventually received a grade of at least C- in Math 104 by the end of Spring Quarter 1998. We note that the separation between Math 050 and MCM is significant.

Table 4 lists the average number of Mathematics courses taken beyond Math 104 that each student attempted in each cohort-year, while Table 4 lists the average number of such courses in which a grade of at least C- was received. This data is inclusive through Spring Quarter 1998. In both instances MCM students, on average, took more math courses than 050 students and, on average, consistently received grades of at least C- in such courses as compared with 050.

One could easily be tempted to conclude from this data that MCM:

• Has had a residual positive impact on the willingness of remedial Mathematics students to take additional Mathematics courses beyond Math 104.
  
• Has had a residual positive impact on their ability to earn a grade of at least C- in such courses, and,
  
• Has successfully advanced remedial Math students at least one Math placement level¹⁴ in OSU's Mathematics placement scheme.

As tempting as this is, we will defer judgement on these issues until we will have at least four years of data on each of four cohort-years.

Tables 6-9 detail four-year retention data for the Math 050, MCM, and Math 104 cohorts, and for the corresponding Autumn Quarter entering classes at OSU. In this data (provided by the Office of the University Registrar) a student is considered retained if they are enrolled during Autumn Quarter of the year in question. This data is current through Autumn Quarter 1998. The retention data for MCM is markedly higher than that of the Math 050 cohorts.

At this stage, the underlying data set concerning graduation rates is somewhat sparse, encompassing only the 1994 cohort. We will wait several years to gauge the effect of MCM on graduation rates and at this time offer in the following table listing graduation data as of Autumn Quarter 1998.

Cohort			050 94		MCM 94		104 94		OSU 94
4 year Graduation Rate			7.9%		14.0%		13.3%		19.3%

Table 1:Graduation Rates for the 1994 Math 050, MCM, Math 104, and OSU Cohorts as of Autumn Quarter 1998.

MCM continues to evolve
During Autumn Quarter 1998, MCM expanded in two directions. Math 104 has traditionally been populated by the two Math placement levels (Math code S and Math code R) above Math 050 (Math code T). Historical data indicated that the mathematics performance of the code S students has been comparable to that of code T students. In an attempt to better serve these higher ranked students, the Mathematics department made MCM mandatory for code S students who enroll in Math 104 as of Autumn Quarter 1998. Departmental data indicates that the Math 104 GPA for code R students has been at least 0.3 higher than that of code S students. In Autumn Quarter 1998, for the first time ever, these students were out performed by the code S students by 0.6. This represents about an 18% gain in GPA for code S students.

In addition, MCM was introduced at Ohio State's regional campus in Mansfield for code T students only. The Mansfield results were similar to those at the Columbus campus. In Autumn Quarter 1999, our regional campus in Newark offered MCM as well.

In 1999, two new features were added to the MCM program. The first was a requirement that students have at least one consultation with one of their teachers during the first three weeks of the term, and at least two consultations within the first six weeks. We wanted to see if a positive office-hour interaction would effect the willingness of these students to seek help in the future. The second new feature was a course project. Students were required to maintain a file of problems (from homework, quizzes and exams, in either 103 or 104) that they had difficulty with. They had to identify the issues or skills that were lacking from the following list: solving equations, factoring, simplifying, working with exponents, graphing, or computing (i.e., arithmetic). Frequency counts were used to determine the three areas (the BigThree) where they had the most difficulty. During the first eight weeks of the quarter, students collected examples, where available, (same sources as above) that demonstrated that they were able to overcome an issue or skill that were having trouble with earlier. Finally they were asked to review the examples that they had been collecting to determine whether or not progress had been made in overcoming their BigThree. For each of these, students were asked to write a brief paragraph indicating why progress was either made or not made

How does MCM differ from traditional Mathematics instruction? A small digression is needed in order to fully answer this question. Since 1987 or so, I have, out of sheer frustration, been interviewing new first quarter freshman who fail the initial hour exam in my sections of mainline calculus. I was willing to try almost anything in order to improve the quality of learning in my classes, and decided to start by attempting to get a better understanding of how/why these students were not functioning. The following summary reflects the most commonly shared characteristics of students who had failed the first calculus exam:

• A belief that their course obligation consists of getting answers (any way that they can, even copying answers from a solution manual) to the assigned homework problems and that this will be sufficient to do well on exams.
  
• They admit¹⁵ to spending about three hours/week doing homework and getting help in the course.
  
• When they sit down to try to solve a problem they immediately begin to write something down.
  
• When they sit down to do homework, they begin with the homework problems. Encountering difficulty, they either look for an identical example that can be used as a template, or they skip the problem altogether hoping that someone else will ask the recitation instructor to provide a solution in the next recitation.
  
• An expectation that the problems on exams will be exactly like the homework problems.

These characteristics indicate that these entering freshman who had tested at the top of our Mathematics placement exam, had not developed a constructive scheme in high school for learning Mathematics. The initial design of MCM assumed that students who had passed (at least) Algebra 1, Algebra 2 and Geometry in high school, and who tested at the remedial level, shared many of the above characteristics. These interviews also convinced me that MCM had to be very different from the high school experience, that it needed to focus on strategy and process, and that this required a new pattern for interacting with Mathematics. We began by structuring a non-traditional environment characterized by informality, small group student centered active learning, with tolerance for a modest amount of socializing. As indicated earlier, each class had an instructional team consisting of a graduate student and an undergraduate classroom assistant. From the beginning, each class meeting was structured by a daily worksheet that assumed that students did not know how to ask themselves questions that could lead towards a solution to a problem. So the worksheets asked the questions for them.

As an example, consider the problem of solving a linear equation in one variable. The worksheet version of the problem, solve 2x + 3 = 11 would begin with the following instructions. Group Discussion: Put down your pencil and do not write anything! Is there a first computation that will simplify this problem? Is a second computation needed? Can you determine the answer to this problem without writing anything down? Several similar problems would follow. Students would be asked to solve each of these individually (or in pairs), check their answers, and discuss discrepancies until they are resolved.

This pattern of directing behavior continues throughout the worksheet as the problems become more difficult, e.g. solve 2x - 4 = 8 -x for x, 1/2 y = y - 3/2 for y, and 3ac - 4bz = 5a +2 for a.

A second feature of MCM arose fortuitously. MCM had two class meetings to every three lectures in Math 104, creating a content flow issue. The solution that arose during the first years was based on the interviews with calculus students. I believe that mathematical learning is enhanced when ideas, skills and techniques interact with each other over time learning how to successfully blend and cooperate in problem solving. Consider the following analogy:

Frame a student's obligation in a Math course as that of putting together a giant jigsaw puzzle by the end of the term. Each lecture provides the student with a bag of puzzle pieces. If a student returns to the dorm after each lecture and simply attempts to either fit the new pieces together or to the current state of the puzzle (returning unused pieces to the bag which is then stored with the other bags), will the puzzle ever be completed? Even allowing for the possibility that the student will open all the available bags several days before each exam in an attempt to fit in more of the pieces?

The solution to the two versus three meetings was to have the Tuesday worksheet cover the previous Friday and Monday lectures, and attempt to point toward the Wednesday lecture. Similarly, the Thursday worksheet would cover material from the preceding two lectures and attempt to point toward the next lecture. We took this pattern a step further by establishing a rolling window of homework, i.e., one where problems from the previous two or three lectures of Math 104 were assigned for each meeting of MCM as well as a few simple problems from the next 104 lecture. This scheme mandates that students work on problems of a given type over a 1-2 week time period, with each iteration involving a different mix of nearby mathematics (topics, ideas, skills, etc.)

Where We Stand
Without making overly bold claims for MCM, it seems clear that its hands-on intensive highly directed coaching approach works better at helping remedial and near remedial students over the hump of their math deficiencies and into a more successful college experience. We haven't proven it, but we feel close.

Addendum
A correlation was observed in 1994 between the performance of MCM students on the first exam in Math 104 and final grades. Students who scored at most 55 on the first exam invariably failed the course. We soon learned that this correlation applied to the entire Math 104 audience to a slightly lesser degree. Since these MCM students were mathematically at risk by definition, and had enrolled in Math 104 despite their placement scores, we felt an obligation to advise the 1995 cohort of this correlation . One student took advantage of our offer to facilitate an immediate transfer to Math 050 and the remaining students who had scored at most 55 on the first exam, invariably failed Math 104. This was also the case throughout Math 104. Starting in 1996 we required that MCM students score at least 56 on the first exam in Math 104 in order to remain in the program. This was known as SafetyNet. Students who didn't pass this hurdle had to drop Math 103 and 104 in the fourth week of the term. We facilitated a change in registration, replacing Math 103 and 104 by Math 050 for those students who wanted to continue in a Math course. SafetyNet students on average performed at a level comparable to the average in Math 050, although their grades tended to cluster closer to the middle. This worked reasonably well in 1996 and 1997 when the number of students was still relatively small. However this process proved totally unmanageable in 1998 when the numbers were much larger, and the performance requirement on the first 104 exam was dropped as of1999

This of course presents a problem in reporting data and assessing significance. The only credible way to accomplish this is to add these students to the MCM data. All of the MCM data reported in this paper (unless there is an indication to the contrary) includes all students (1996-97) who had been required to drop their registration in MCM. As a consequence, it is very likely that the reported 1996-97 data is understated.

In retrospect, we suspect that these students would have been better served by remaining in MCM. Even if they eventually failed Math 104, the MCM experience might have had a beneficial impact on their performance in Math 050 and subsequent Math courses. It will be interesting to monitor these populations (with scores at most 55 on the first exam) to determine whether, and to what extent, MCM effects their subsequent Math performance. The department now requires (with few exceptions) MCM students who fail Math 104 to take Math 050 as their next Math course. The effected 1996-98 populations will serve as a comparison group for this study.

Cohort		1994		1995		1996		1997
Math 050		27.9%		27.6%		29.0%		39.6%
Math 050		38.2%		41.3%		49.1%		50.2%
MCM		50.0%		56.9%		57.1%		57.5%
Math 104		59.6%		65.0%		61.2%		68.9%

Table 2: Initial Grade of at least C- in Math 104

Cohort		1994		1995		1996		1997
Math 050		38.3%		40.6%		37.5%		45.6%
MCM		73.2%		75.9%		71.4%		68.8%
Math 104		75.7%		77.3%		78.8%		79.5%

Table 3:Eventual Grade of at Least C- in Math 104

Cohort		1994		1995		1996		1997
Math 050		0.7530		0.8078		0.5540		0.6263
MCM		1.3392		1.3448		0.8357		0.7708
Math 104		1.6365		1.6955		1.4826		1.0203

Table 4: Average Number of Math Courses Taken
Beyond 104 by each Student.

Cohort		1994		1995		1996		1997
Math 050		0.5033		0.4760		0.6563		0.1965
MCM		1.0357		0.9138		0.5643		0.5313
Math 104		1.2649		1.2784		1.1316		0.7842

Table 5: Average Number of Math Courses Taken
Beyond 104 with Grade at Least C- by each Student.

Cohort Year		1 year		2 years		3 years		4 years
050 94		67.1%		48.4%		43.6%		43.7%
050 95		68.1%		52.1%		44.4%
050 96		61.3%		49.9%
050 97		67.0%
Total		66.2%		50.1%		43.6%		43.7%

Table 6 : Four Year Retention Data Math 050

Cohort Year		1 year		2 years		3 years		4 years
MCM 94		80.7%		68.4%		71.9%		66.7%
MCM 95		81.0%		60.3%		53.4%
MCM 96		78.2%		66.0%
MCM 97		70.3%
Total		76.7%		65.3%		62.6%		66.7%

Table 7: Four Year Retention Data MCM

Cohort Year		1 year		2 years		3 years		4 years
104 94		74.0%		59.9%		54.2%		52.0%
104 95		75.3%		61.5%		57.0%
104 96		75.5%		63.8%
104 97		78.3%
Total		75.6%		61.6%		55.5%		52.0%

Table 8: Four Year Retention Data Math 104

Cohort Year		1 year		2 years		3 years		4 years
OSU 94		77.7%		66.1%		61.2%		60.8%
OSU 95		79.0%		68.4%		63.4%
OSU 96		79.1%		70.0%
OSU 97		81.8%
Total		79.4%		68.2%		62.6%		60.8%

Table 9: Four Year Retention Data OSU

Footnotes

1. Professor Gale Watson (OSU Ph.D. 1996 The Use Of Small-Group Instruction To Support Remedial Students In College Mathematics) was very instrumental in helping to organize MCM during 1994-96. The program benefited significantly from her many insights.
2. Data for this project has been provided by the Office of Enrollment Management and the Office of Admissions .
3. Mathematics counselors Judith Berenstein and Judith Monson, and Mark Garner Mathematics program associate, made valuable contributions to the development and operation of MCM.
4. Math 050 is taught in individual sections of 25 students meets daily and covers chapters 1-5 of Beginning Algebra by K.Elayn Martin-Gay.
5. In addition, the university requires that these students pass a Mathematics course beyond Math 050 in order to graduate..
6. This usually means Algebra 1, Geometry ,and Algebra 2. Some have even had Calculus!
7. Up until 1998, enrollment in MCM was voluntary. In 1998, it was made mandatory for the cohort who tested just above Math code T.
8. Math 104: has a 3 lectures and 2 recitations per week covering chapters 1-3, 5-6 of Essentials of Intermediate Algebra (2nd edition -OSU version) by Larson, Hostetler & Neptune.
9. Math 104 and Math 103 are coordinated by different faculty and are run independently of each other. The MCM program and staff are not involved in any aspect of Math 104 course policy, lectures, recitations, exams, the grading of exams, or the assigning of final grades.
10. The author's experience with Autumn Quarter NFQF's in the main line calculus sequence over the past four years suggests that interactions of this type can provide significant opportunities for learning for Math code T students .
11. Math 075 is taught in a three lectures and two recitations per week covering chapters 9-10 of Beginning Algebra, 2nd edition by Martin-Gay. Math 075 does not count toward graduation and is the minimal level of mathematics competency required by the General Educational Curriculum of the Colleges of the Arts and Sciences. Students who pass Math 075 are, in general, eligible only for the BA degree in areas which do not use mathematics. The department is currently phasing Math 075 out.
12. On going multi-year data collection for each cohort of each population, Math 050, MCM and Math 104, has been provided by the Office of University Registrar. We have just started analyzing an extensive data set ,that was recently received, containing the full academic records of the 10,500 students in these three populations.
13. The Statistical Consulting Service at OSU and its director, Dr.Panikos Paletas ,have provided invaluable support in assembling the data that appears in this report.
14. Math 104 is populated by two placement levels on the B Mathematics placement exam. The performance of the lower of these two in Math 104 and Math courses beyond 104 has been comparable to that of Math 050 students. This will be discussed later in this article.
15. I believe that this is the largest number of hours that they think will be credible.