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?
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 third-grade teacher reads at the fourth-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 second-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.
Developing Story: A Forum on Improving Remedial Education