In general, in a college level math course, the instructor/professor has no idea how prepared the students are. He/she presumes that they are ready for the course because they are enrolled in it. However, students come to any course with a variety of levels of preparedness. The courses are not designed to meet the needs of students (although colleges will claim that they are). They are instead designed to fulfill the curriculum requirements for various majors, STEM (science, technology, engineering, and mathematics) and non-stem. In other words, the college says, “These are the courses you need to graduate with degree X.” And presumably, if you do well in, or at least pass these courses, you will have the minimum knowledge that was transmitted in these courses. As we shall see, however, particularly in math courses, such claims by higher education institutions are often false.
All students take math courses that are determined by their major. For students who are not math majors, the math requirements can be rather arbitrary. For example, a student may be required to take pre-calculus not because they intend to take calculus at all, but because the student’s major, such as nursing, architecture, or pre-dental, requires all their students to do so. The faculty of such majors give little or no thought to the mathematical knowledge their students need to succeed in their programs, but instead want their students to have some knowledge of math (not a bad idea) beyond high school. Precalculus probably does a very poor job of serving that need. As a college professor who regularly teaches precalculus (two semesters of it, the latter being essentially trigonometry), there are very few topics that I teach that would directly benefit such students in their non-stem major.
For students who are stem majors and therefore must take calculus for their desired degree, it would seem self-evident that such students should take precalculus before taking calculus. But do such students actually perform better in calculus in college for having done so? Sonnert and Sadler found the answer to that question, based on a sample of over 5000 students at over 100 different institutions, to be “no.” They write, “We find that students who take college pre-calculus do not earn higher calculus grades.” https://www.tandfonline.com/doi/abs/10.1080/0020739X.2014.920532?tab=permissions&scroll=top
So, should STEM majors be taking pre-calculus at the high school level? How can they be prepared for calculus, which they need to complete their desired degree, if taking a college pre-calculus doesn’t do the job?
In my opinion, STEM majors clearly need to know the content of almost everything we teach in the two semesters of pre-calculus. But taking the courses and learning the content are not the same! So what are the issues that plaque many such students?
(1) The course goes too fast. We “cover” the material, meaning it was presented in class, but we don’t assess well whether the students are learning the material. We give quizzes and tests, but they contain only a sample – sometimes a very small sample -- of the topics covered. Students can do well on such assessments and still have “holes” in their understanding. Even students who earn an A in both pre-calculus courses may not be ready for calculus, even though the grade would suggest otherwise.
(2) Students learn how to do specific kinds of problems in these courses, but lack conceptual understanding. For examples, students may know how to “complete the square” on a quadratic expression, but not understand why the process works or why one would perform it. They memorize formulas, techniques, and procedures without ever understanding why they work. If they work a problem incorrectly, they are often unable to find their mistake. Mathematics must be seen by many of them as a collection of unrelated facts, techniques, or algorithms that strain the memory, produce anxiety, and crush any curiosity. This is a poor way to learn mathematics and it isn’t enjoyable for most. Learning mathematics doesn’t have to be this way, but colleges and universities seem hopelessly tied to the idea that “if you want to be X, take these courses leading to degree Y.”
(3) The students taking a math course, e.g. pre-calculus, are doing so because it is required for their major, whether in STEM or not. In my many years of teaching, I have observed that many such students are grossly unprepared. They fail or do poorly on the first few quizzes and the first test. If I suggest that perhaps they should master more elementary mathematics, such as adding fractions, before taking pre-calculus, they usually tell me that they must have the course for their major, they must pass it in the current semester, and even sometimes, that they must receive a grade of A in the course! Such a student is invariably being set up for an academic disaster. Why can’t we build courses around the needs of students, rather than simply herd students into courses?
(4) Colleges and universities attract students because they offer a credential that supposedly signals to the world they have some knowledge in a particular field or that they are generally “educated.” Bear in mind, however, that even if a college is excellent at educating its students, there are almost certainly other ways to acquire the desired knowledge, although they may come without the certification, the degree. Nowadays, almost any subject taught at the undergraduate level can be learned online by watching videos and doing practice problems. (Whether or not a student has the self-discipline to do so, is another question.) Money not spent on tuition can be used to hire a private tutor and/or take courses that require payment. And, although prestigious colleges and universities compete for students at admission, there are only very subjective ways of comparing the graduates of different colleges. If a college is very selective in admissions, short of destroying their brains, such an institution is likely to have brighter, more capable students graduate simply because they were brighter, or perhaps more motivated at the beginning of college.
(5) It is easy to teach math poorly. It is easy, as a professor, to present topics in a logical manner if one understands the material well. It is much, much harder to look out a sea of faces and tell whether anyone is understanding. “Does anyone have any questions?” is often met with blank stares. It is quite understandable when few, if any, students respond to such a question. Students are notoriously shy, perhaps even embarrassed, about displaying their lack of understanding in front of other students. Students also rarely ask for help outside class. Perhaps they feel so overwhelmed with what has been presented in the last hour and fifteen minutes, that they just want to escape the classroom as soon as class concludes.
(6) Students often have a myriad of responsibilities and distractions outside of learning my course of precalculus. They have other classes. Many work an excessive number of hours. Some have children or family responsibilities. Some are sleep deprived. Some eat a diet that probably interferes with their ability to concentrate (too many carbohydrates?). Some cannot focus on the class for an hour and fifteen minutes without checking their phones multiple times. Many need help in learning how to take notes in a math class. (Professors could and should do a better job of providing outlines of notes, to help students move their attention quickly from looking at the presentation in class, to watching what they are writing at their seats.)
Thank you for reading this, the first of my blogs here on www.calculussucks.com. Please give your comments below.
Peter H. McCandless