Much has been written about the failure of “developmental education” in mathematics. Failure has not been our experience at Worcester State University. In response to concerns about both the placement rate into developmental math courses and the failure rate in those courses, we made substantial changes in our placement program and in our course delivery. We have decreased by 50% the number of students placed into developmental math courses. The success rate in these courses has increased from around 30% to around 80%.

Our program is based on several key principles:

• Students can be successful in mathematics with the correct entry point. Mathematics is a cumulative subject. Because students who take courses they are not prepared for are seldom successful, we rigorously maintain the prerequisite standards for our classes. However, we do not want to place students in developmental classes when all they need is a brief review. We work to ensure that students take placement seriously and are prepared to do as well as possible when they take the placement tests.
• We provide clear, consistent standards for all students. It doesn’t help students by pretending they have competencies that they don’t. In particular, different sections of developmental math must use equivalent grading techniques. All students must pass the same final exam to pass the class.
• We provide a nurturing and supportive environment for students who have often had negative experiences in mathematics. Students need to know that their instructors are there to help them when they struggle. However, the most supportive thing we do is placing students appropriately.
• We encourage all students to enroll in required math classes as soon as possible. Math proficiency atrophies over time. This means we have to offer sufficient seats for first-year students in both developmental and introductory credit-bearing classes. We work with our advising center to place students in these courses.

Reducing need for remediation

The Commonwealth of Massachusetts Department of Higher Education mandates that all incoming students in the state’s public higher education system attain a “passing” score on the College Board’s Elementary Algebra Accuplacer exam or pass an appropriate developmental math class before enrolling in a college credit-bearing math courses. In fall 2004, 54% of our first-year students received a “failing” score.

For the class entering in fall 2005, we required students to take a mock Accuplacer exam before they could register for orientation, where the actual exam was given. This mock exam was taken at home on the student’s own computer. It was not proctored. We saw this as a consciousness-raising activity—a way to give students a sense of what to expect as well as to let them know about the importance of the exam. With this change, our “failure” rate dropped from 54% to 36%.

The following year, we made additional changes. Before a student could register for orientation, he had to achieve a “passing” score on the mock Accuplacer exam. If he didn’t get a “passing score” after two opportunities, he had to come to campus for a two-hour math review session. With this additional change, the “failure” rate dropped to 24%. Since then it has been consistently around 25%

The placement process

The initial Department of Higher Education mandate for developmental math in 1998 set a single passing score of 82 on Elementary Algebra Accuplacer for determining whether a student was ready for college-level math classes. In 2001, the department added a second cut score of 72 for courses that used minimal amounts of algebra, such as a math for liberal arts courses.

At Worcester State’s Mathematics Department, we decided we needed more detail to appropriately place students. Many students needed developmental work in arithmetic as well as algebra. And while a score of 82 on the Elementary Algebra Accuplacer might indicate readiness for a college algebra class, it told us nothing about whether a student was prepared for calculus. We want each student to begin mathematics coursework at the best entry point. As a result, all first-year students begin by taking two Accuplacer exams: Arithmetic and Elementary Algebra.

• If they need developmental math work, we use a combination of the two scores to determine whether they need to take an arithmetic course before taking a developmental algebra course.
• If they score 82 or higher on the elementary algebra Accuplacer, they then take the college-level math Accuplacer. This score is used to determine the possible starting points for the student’s college-level math classes.

Logistically, each student is assigned a placement code of 1 through 7 based upon their scores on the two or three Accuplacer exams. Mathematicians call this a function of three variables where the range is: {1,2,3,4,5,6,7}. For example, a code of 1 means a student begins with our developmental arithmetic class. A code of 7 means a student may begin with calculus. During the registration process, placement codes are examined as part of the process of checking prerequisites. A student who wishes to take calculus needs either a code of 7 or successful completion of precalculus with a grade of at least C-. (Our experience has been that a student with a D seldom passes the subsequent course.)

The Developmental Math Program: philosophy

The WSU Developmental Math Program is designed to meet the academic needs of students who scored below 82 out of 120 on the Elementary Algebra Accuplacer exam. Many of these students have negative emotions and thought patterns around mathematics that needed to shift before they would be able to learn the subject matter. Some are so used to failing math that they don’t believe that they have the ability to succeed. They would rather walk away than face the challenge, despite the fact that this would severely limit their ability to earn a bachelor’s degree. For students who had already incurred significant student loans, failure to complete their degree would leave them with increased debt and decreased income potential. This heightens the anxiety associated with learning math.

Our program strives to create a classroom environment where students believe they can succeed and know they will have the support of the instructor. In each new class, the instructor’s initial goal is to build a relationship of mutual trust and respect. When these students enter the developmental math class, many things are different from how they were in their previous math classes. Since they are in a class with students at similar skill levels, most are no longer at the bottom of their class. Furthermore, the students are older and more mature than the last time they took a math class. With a positive environment, they are more likely to persevere and succeed. We find that as student anxiety begins to subside, they relax and start learning. All these benefits are only possible because the students are placed in a class that is being taught at their current proficiency level.

Underlying the program development, we have had a commitment to maintaining consistency of standards for all students and all course sections. Lowering standards for some students is not supportive and nurturing, but propagates student beliefs that they cannot succeed at mathematics. These beliefs reinforce societal perceptions of mathematical reasoning and skills as optional and only obtainable by a select few. Sadly, many higher education administrators and policymakers encourage these negative viewpoints

Implementation and design

Our current program was developed over the past 10 years and evolved through a series of iterations from a computer-based algebra review to one where students are placed according to their arithmetic and algebra skills into one of two developmental math courses that address topics required for success in WSU’s college-level math courses.

The developmental courses meet three hours per week, carry three institutional credits and are taught in a more traditional face-to-face format. (Institutional credit counts toward maintaining full-time status so students are eligible to receive financial aid and live in the residence halls, but not toward graduation.) We have used feedback from assessment data as we sought effective ways to teach and support our students. As we have developed these classes, the success rates of our courses have increased from 31% in 2003 to about 80%.

To maintain consistent standards across students and sections, we use the Arithmetic or Elementary Algebra Accuplacer as the final exam for each class. Students must pass this final exam to pass the course. Since the instructors no longer decide whether a student passes, they become more like coaches, working with the student to increase skills and achieve a common goal. Instructors meet with individual student, assign extra problem sets and arrange for tutoring. While success is ultimately the student’s responsibility, we want to provide as much support as we can.

We believe our students need structure and a series of smaller goals before the final exam. Therefore, we require that all students have a 70% average in the course in order to qualify to take the final exam, the Accuplacer. This requirement is made clear on the syllabus and the instructors discuss this throughout the semester. In the last three weeks of the semester, students with averages below 70% are invited to work with tutors to address topics on which they are struggling. They are given an additional quiz that provides the opportunity to raise their average and qualify for the final. Of course, our real goal is to get them to review the material so they pass the final. It’s a learning activity. The underlying principal is that we want to promote success without lowering standards and expectations.

If students qualify to take the final exam, but do not pass it, we discuss a re-test opportunity with them. The instructor offers these students a set of review problems and gives them a limited amount of time to complete it. This is another learning activity. Once the students complete the review material, they are given a pretest to determine if they have improved their skills. Students who perform favorable on the pretest may retake the final exam. This “review–pretest–retest” process helps most of the students pass and move forward with their mathematics program.

Changing minds

Most of our students who score into the developmental math program are in majors that require only one college-level math course. Many students enter the developmental math program intending to complete their developmental math and a single college level math class; but after experiencing success, they reevaluate their options. This was the situation for Jeremy Hart, a 30-year-old military veteran who entered the developmental arithmetic class as a history major. He had many doubts about his ability to succeed at mathematics and had chosen a major with a minimal math requirement. He planned on finishing his mathematics requirement as quickly as possible by taking our most basic college-level course, called “Survey of Math.” When the arithmetic class began with fractions, Hart found the class a safe place to openly express his confusion and frustration. He became so comfortable with his ability to learn mathematics that he changed his major from history to business administration. He successfully completed many courses that required mathematical and quantitative reasoning including statistics, college algebra, mathematical economics, microeconomics and three accounting classes. He is currently employed in a managerial job that brings together the skills he developed at Worcester State and in the military. He manages a multimillion-dollar budget and performs cost and statistical analyses as he contributes to his organization’s success.

Our program works, but we are constantly looking for ways to minimize the need for remediation. We work with Massachusetts high schools through the state GEAR UP (Gaining Early Awareness and Readiness for Undergraduate Programs) so students can take our placement tests while still in high school. And we are currently studying how students who successfully complete our developmental courses perform in the first college-level math class.

Developmental math education does not have to be a failure, as long as we are all willing to meet the challenge.

Richard Bisk is a professor of mathematics at Worcester State University and was math department chair from 2004-2012. Mary Fowler is an associate professor and current chair of the math department at Worcester State. Eileen B. Perez is Developmental Math Program coordinator and lead instructor at Worcester State.

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Many students begin higher education unprepared for college-level work in mathematics and must take non-credit developmental courses. Furthermore, many are math-phobic and avoid courses, majors and careers that involve quantitative work. Yet science, technology, engineering and mathematics (STEM) fields are among the few job-growth areas in the U.S. Many companies are lobbying the federal government to expand the number of H-1B visa positions in order to bring overseas hires to the U.S to fill STEM positions.

At Worcester State University, we have learned to do developmental math better, but that’s not enough. We shouldn’t need developmental math programs at all. Our country needs more students prepared for STEM majors and careers. This article will address issues related to fostering student interest and success in mathematics.

What is college-level work?

At some selective institutions, the first college-level math course is calculus. Many other institutions offer for credit two or three algebra-based courses consisting of topics that are prerequisite to calculus and normally taught in high schools. Given the push to teach all students algebra in grade 8, this is perplexing: it doesn’t matter how early students take algebra, if they arrive in college needing to repeat it.

Many of my colleagues report that their students not only have weak algebra skills, but also struggle with arithmetic that should have been mastered in elementary school. Some of my calculus students would prefer that I avoid any mathematics that involves fractions.

Why are students unprepared for college-level courses that require a math background?

The problem begins in the early grades in both curriculum and instruction.

The Massachusetts state standards were considered among the best in the nation. Yet they still had problems. For example, the third-grade framework listed 33 standards—far too many.

One of the more important ones was:

... Know multiplication facts through 10 x 10 and related division facts, e.g., 9 x 8 = 72 and 72 ÷ 9 = 8. Use these facts to solve related problems, e.g., 3 x 5 is related to 3 x 50.

How were third-grade teachers supposed to teach 33 math topics in addition to all their other responsibilities? How were they to recognize the importance of mastering single-digit multiplication when a topic so important is not even on state tests?

New standards

The new Common Core State Standards (CCSS) address this  issue. They begin by stating:

For over a decade, research studies of mathematics education in high-performing countries have pointed to the conclusion that the mathematics curriculum in the United States must become substantially more focused and coherent in order to improve mathematics achievement in this country. To deliver on the promise of common standards, the standards must address the problem of a curriculum that is “a mile wide and an inch deep.”

Forty-five states, including all six in New England, have adopted the CCSS, but it will be years before the effort’s promise is fully implemented and we see a real improvement in college readiness.

Why are teachers unprepared for K-8 mathematics?

The CCSS is part of a potential solution, but far more important is the mathematical preparation of our teachers. The CCSS emphasize mathematical reasoning and understanding. However, teacher-preparation programs have not provided most elementary teachers—nor many middle-school teachers—with the depth of mathematical background needed to effectively teach for understanding. We want students to readily divide 12 by ½. An effective elementary teacher must understand why the procedure works and be able to create word problems to illustrate the calculation.

Massachusetts began to address this issue in 2007 with a new certification test in mathematics for elementary and special education teachers (the failure rate on the first test administration was 73%). In the past, we have taken the view that the math of elementary school is simple, so we really don’t need to provide our teachers with much coursework. People will say: “How hard it is to teach how to add 27 + 18 or multiply 7 x 8?” Yet, the same people would not ask, “How hard is it to teach the Cat in the Hat?” In the debate leading up to this new requirement, I frequently heard the question “Why does an elementary teacher need math beyond the level she teaches?” My answer was simple: “Suppose that your child’s 3^rd-grade teacher reads at the 4^th-grade level—is that acceptable?”

It’s clearly not acceptable: Effective teachers must understand the mathematics they teach to a much greater depth than their students, understand subsequent levels that their students will soon encounter, and be able to engage students in real mathematical discourse.

For example, suppose a 2nd-grade student writes: 27+18=315

This is a common error that indicates confusion about place value. The teacher must provide experiences that help students develop an understanding of place value, not simply tell the child to memorize a procedure.

We send our elementary teachers into the workforce with strong backgrounds in English language arts, but minimal backgrounds in mathematics. Educators at all levels from kindergarten through higher education must take ownership of this problem. Finding fault or being defensive is not helpful.

The CCSS, the Massachusetts certification changes and the state universities’ new entrance requirement of four years of high-school math have the potential—over time—to minimize the need for remedial math programs in higher education. Meanwhile, we still have to work with the students who arrive at our doors with substantial mathematical deficits and phobias.

Despite much noise to the contrary, it is possible to have a successful remedial math program. At Worcester State University, we have cut our remediation rates in half primarily through awareness activities. Now substantially fewer students need to take remedial courses. Success rates in these classes have increased from around 30% to approximately 80%. We did this by providing clear and consistent standards as well as a nurturing, supportive environment. Students, including many nontraditional learners, gain tremendous self-confidence when they discover that they really can understand math and be successful.

We face several other challenges

The excessive use of calculators is a problem at all levels of math education. Yes, use a calculator to accurately divide 12.567 by 2.154, but not to divide 12 by 2. Otherwise how will you be able to tell if your calculator’s answer to the first question is even roughly correct? Every math professor can tell stories of students using calculators to multiply by 10. A child learns to dribble a basketball, by dribbling a basketball—a lot. Our students will not be comfortable working with numbers unless they spend lots of time working with numbers.

There is pressure at all levels on educators to pass students who have not mastered concepts. My concern is that the focus on higher education graduation rates is leading to a lowering of standards so that more students can be “successful.” We want our students to graduate, but they must have the quantitative skills and knowledge to be successful in a highly competitive world.

Steven Pinker, a Harvard cognitive scientist, wrote: “Mathematics is ruthlessly cumulative, all the way back to counting to ten.” When our students have significant gaps in their mathematical knowledge, it becomes impossible for them to move forward. If I know nothing about World War I, I can still take a class about World War II and be successful. However, a student who doesn’t understand arithmetic will struggle with algebra and statistics. A student who doesn’t understand algebra will struggle with calculus and many courses in the sciences. We can substantially improve our system of mathematics education, but there are no quick fixes.

Richard Bisk is a professor of mathematics at Worcester State University. Thanks to Tom Fortmann for his many suggestions to improve this article.

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