Thursday, November 22, 2007

Remedial Math: For Dummies?

These are some early thoughts on the growing debate about remedial math instruction in higher education. I'm still coming up to speed on the data and the issues in this area; in the meantime I welcome all comments, corrections, and criticisms of the ideas that follow (which are really more like instincts at this point...).


Background: A 2006 article by Bettinger and Long, "Institutional Responses to Reduce Inequalities in College Outcomes: Remedial and Developmental Courses in Higher Education," in Dickert-Conlin and Rubenstein (Eds.), Economic Inequality and Higher Education: Access, Persistence and Success, New York: Russell Sage Foundation Press; Achieve's 2006 report Ready or Not: Creating a High School Diploma that Counts; Sternberg's must-read 1997 book Beyond the Classroom; and Phipps's 1998 Institute for Higher Education Policy report, College Remediation: What It Is, What It Costs, and What's at Stake.


The article by Bettinger and Long gives a good overview of the research and debate about remediation in higher education. Some notable quotes (see the article for the references):

Greene and Foster (2003) found that only 32 percent of students leave high school at least minimally prepared for college. The proportion is much smaller for Black and Hispanic students (20 and 16 percent, respectively). Greene and Foster (2003) define being minimally "college ready" as: (i) graduating from high school, (ii) having taken four years of English, three years of math, and two years of science, social science, and foreign language, and (iii) demonstrating basic literacy skills by scoring at least 265 on the reading NAEP.



By 1995, 81 percent of public four-year colleges and 100 percent of two-year colleges offered remediation (NCES, 1996).


It appears that states and colleges know little about whether their remediation programs are successful along any dimension.


Moreover, a study of 116 two-year and four-year colleges found only a small percentage performed any systematic evaluation of their programs (Weissman, Rulakowski, and Jumisko, 1997).


On one hand, the courses may help under-prepared students gain the skills necessary to excel in college. On the other hand, by increasing the number of requirements, extending the time to degree, and effectively restricting the majors available to students (due to the inability to enroll in advanced coursework until remedial courses are completed,) remediation may negatively impact college outcomes such as persistence and long-term labor market returns.


Achieve's Ready or Not report has this to say:

Most high school graduates need remedial help in college. More than 70 percent of graduates quickly take the next step into two- and four-year colleges, but at least 28 percent of those students immediately take remedial English or math courses. Transcripts show that during their college careers, 53 percent of students take at least one remedial English or math class. The California State University system found that 59 percent of its entering students were placed into remedial English or math in 2002. The need for remedial help is undoubtedly surprising to many graduates and their parents — costly, too, as they pay for coursework that yields no college credit. (p.3)


This issue has been percolating for a while in both K-12 and higher ed. Sternberg in 1997 listed ten policy priorities, the eighth of which was to end remedial courses at four-year institutions. Sternberg writes (p. 192),

The current practice of providing remedial education in such basic academic skills as reading, writing, and mathematics to entering college students is disastrous. It has trivialized the significance of the high school diploma, diminished the meaning of college admission, eroded the value of a college degree, and drained precious resources away from bona fide college-level instruction.

I realize that despite whatever [reforms] are put into place at the elementary and secondary school levels, there will be students who pass through the system without some of the requisite academic skills. But these students should not be entering four-year colleges and universities. Rather, students who have managed to complete high school but who lack the necessary college entry skills should be required to pursue remedial coursework at local community and two-year colleges before they can apply for admission to more advanced institutions of higher education.


Weighing against Sternberg's proposal are the findings in Phipps's 1998 IHEP report. With respect to college remediation, Phipps argues that it is a core function of higher education; that it is not expanding in size or scope (Bettinger and Long disagree); that it is actually quite cost-effective; and that a number of undesirable consquences would ensue if it were to disappear.

The back-and-forth gets confusing, in part because some authors (like Sternberg) make a firm distinction between four-year vs. two-year institutions, while others (like Phipps) do not.

What seems to be missing in any of these debates, however, is any serious discussion of what is actually being taught in remedial classes - especially in math. I would like to see someone take a careful, critical look at the typical content covered in college remedial math courses. I suspect that such a study would show us that most of the material in these courses is useless for most of the college students forced to take these courses.

Achieve's Ready or Not report contains the following lofty-sounding quote from a Purdue math professor:

The ability to understand and apply the mathematical content typically taught in an Algebra II course is vital to a student’s success in science and social sciences courses required by our university.


After reading this, just for the heck of it I visited Purdue University's website. I wondered what is actually required in order to graduate from Purdue with a degree in, say, nursing.

Nursing at Purdue is not exactly science or social science, but it is science-based, and at Purdue it turns out to be a very rigorous program. Here is the four-year plan of study.

BIOL 203, Human Anatomy and Physiology
CHM 111, General Chemistry
ENGL 106, English Composition
NUR 102, Dynamics of Nursing
BIOL 204, Human Anatomy and Physiology
CHM 112, General Chemistry
PSY 120, Elementary Psychology
SOC 100, Introductory Sociology
Humanities elective
NUR 104, Foundations for Nursing Practice
NUR 105, Foundations for Nursing Practice-Clinic
NUR 206, Health Assessment
NUR 207, Health Assessment-Clinic
PHPR 202, Introductory Pharmacology
Guided philosophy elective
BIOL 221, Introduction to Microbiology
F&N 303, Essentials of Nutrition
NUR 208, Lifespan Human Development
NUR 214, Introduction to Pathophysiology
PSY 350, Abnormal Psychology
NUR 302, Adult Nursing I
NUR 303, Adult Nursing I-Clinic
NUR 312, Nursing of Childbearing Families
NUR 313, Nursing of Childbearing Families-Clinic
Guided statistics elective
NUR 304, Psychosocial Nursing
NUR 305, Psychosocial Nursing-Clinic
NUR 306, Adult Nursing II
NUR 307, Adult Nursing II-Clinic
NUR 310, Public Health Science
Guided sociology elective
NUR 402, Public Health Nursing
NUR 403, Public Health Nursing-Clinic
NUR 412, Pediatric Nursing
NUR 413, Pediatric Nursing-Clinic
Free elective
NUR 404, Leadership in Nursing
NUR 408, Research in Nursing
NUR 409, Senior Capstone Clinic
NUR 410, Issues in Professional Nursing
Humanities elective
Free elective

If I managed to get the HTML right, then you'll see that I color-coded the courses. Courses with a low math demand are blue. Courses with a medium math demand are green. Courses with a high math demand are black. (The plural in this case turns out to be superfluous.) This was just a very rough take; I didn't look at any syllabi. But I claim there is a strong message here - and it's a message that may strike many as counterintuitive: Very little math is required to succeed in nursing at Purdue. What little math there is could probably be handled perfectly well through tutoring arrangements for those students lacking even the basics.

As far as that one statistics elective is concerned, it turns out that the stated prerequisite for the eligible statistics courses is a two-semester sequence called Math 153-154 - or else the equivalent in high school preparation. Now, like many universities, Purdue is wary of the word "remedial." Having a lot of "remedial" students tends to compromise a university's image. In Purdue's case, the only courses deemed remedial are courses at the satellite campuses; the preferred word at the main (West Lafayette) campus is "preparatory." This kind of parsing notwithstanding, I'm going to classify Math 153 and Math 154 as remedial, because (1) they are prerequisites for the plan of study, but not listed in the plan of study; and (2) they cover material traditionally taught in high school Algebra II/Trig courses.

Here is the first midterm exam for Math 154.

1. Find the angle that is complementary to 26 degrees 9' 40''.
2. Express theta = 4.6 in degrees, minutes, and seconds, to the nearest second.
3. Find the reference angle for theta =122 radians, to the nearest hundredth of a radian.
4. Jupiter is the fifth planet from the sun and by far the largest. Jupiter is twice as massive as all the other planets combined (the mass of Jupiter is 318 times that of earth). Its orbit is 483,633,704 miles and it diameter is 88,846 miles. A nautical mile on a planet is the distance on the surface subtended by a central angle of 1' from its center. Approximate the number of land miles in a nautical mile on Jupiter to the nearest tenth of a mile.
5. Find the exact values of [the short leg and the hypotenuse] of the given right triangle [long leg 8 units, angle 30 degrees between long leg and hypotenuse]
6. Stonehenge in Salisbury Plains, England, was constructed using solid stone blocks weighing over 97000 pounds each. Lifting a single stone required 550 people, who pulled the stone up a ramp inclined at an angle of 8 degrees. To the nearest tenth of a foot, approximate the distance that a stone was moved along the ramp in order to raise it to a height of 34 feet above the level ground.
7. Approximate sec(78 degrees 18') to four decimal places.
8. csc(x)/cot^2(x) is equivalent to which of the following? csc(x)cot(x), sec(x)cot(x), csc(x)tan(x), cos(x)sin(x), sec(x)tan(x)
9. Find the exact value of tan(theta) if theta is in standard position and the terminal side of theta is in quadrant II and is parallel to the line 3x+5y = 9.
10. If sec theta = 8 and cot theta < 0, find the exact value of sin theta.
11. Let P(t) = (-7/25, -24/25) be the point on the unit circle that corresponds to t. Find the exact value of P(-t+pi).
12. Complete the statement: As x -> pi/2+, tan(x) -> ?.
13. Approximate, to the nearest 0.01 radians, all angles theta in the interval [0, 2pi) that satisfy the equation cot(theta) = -2.3412.
14. Which of the following statements are true about the graph of y = -2 + cos(x)? The graph intercepts the y-axis at -2; The graph intercepts the x-axis at the origin; (pi/2, -2) is a point on the graph; The graph is always below the x-axis.
15. Find the equation of the graph shown below, in the form y = a sin(bx+c), with a>0, b>0, and least positive real c.

I think there is a serious content-validity problem here. How many students at Purdue actually need to know any of this? In particular, is Math 154 any sort of reasonable preparation for future nurses - or future lawyers, psychologists, social scientists, doctors, or even cell biologists? To me, Math 154 looks like welfare for college math instructors. I think we should stop taking university faculty at their word when they say that large fractions of incoming students "need" remediation. And we should start questioning whether these faculty members' syllabi accurately define the skills necessary for college.

Achieve (and many others) seem to want to use the college remediation crisis as a hammer to swing at our high schools. I would certainly agree that high school math education could benefit from a couple of solid whacks; but I am not inclined to accept the universities' terms in this debate without question. My intuition is that the hammer has to swing both ways.

Personally, I suspect Sternberg is right (about math at least) when he says that four-year institutions should abolish remedial math courses. As Sternberg notes, such a policy would result in many students attending two-year colleges in order to build up their math skills. However, given the way college math requirements tend to get exaggerated, I would want to take extra care to ensure that the two-year/four-year bar is set in the appropriate place, based on realistic studies of what it actually takes to get a C or better in university gateway courses.

Instead of using SATs, ACTs, or other tests to place students in non-credit-bearing courses, universities could redirect their remediation expenditures towards studying how these test scores relate (or don't relate) to the grades students ultimately receive in the university's gateway science and math courses. Universities could publish these statistics, as well as statistics on how these scores relate to attrition rates in various majors. Knowing the proportion of students with ACT scores of X earning grades of Y in courses Z, students can make informed decisions about whether to attempt a given major, whether to get a tutor for a required course, and how to design a schedule that allows enough time to devote to the class in question. Advising could help ensure that students interpret the data properly.

My intuition is that many students who "lack algebra II/Trig" could just as well skip the two-year college altogether and go directly to State U. Let them sign up for those introductory engineering courses if they want. Let them get tutoring, use existing academic support mechanisms, learn the crucial content in real time, and work as hard as they can to catch up - and let them earn college credit when they pass. If, instead, they fail or withdraw, then they'll find that there are lots of interesting things to do in a university besides engineering and physics. And they won't have spent a year treading water in remedial classes before coming to that decision point.