Common misconceptions held by students when they enter college
(Items are in the form Erroneous assertion --- Explanation of misconception.)
Misconceptions about high school and college
1. Only a jerk could get less than a B-grade in a course!
. . . If that's true, then 40 or 50% of freshman in math courses in previous years are jerks!
2. I had a good math teacher in high school who taught at my level.
. . . But most of the freshmen admitted here say they were in the top quarter (say) of their math class, yet agree that a teacher [in high school] is supposed to make sure that even the weakest students in the class learn. Now, whose level was the course run at?
3. I did well in math, even calculus, in a good high school. I'll have no trouble with math at Hopkins.
. . . There is a different standard at the college level. The student will have to put in more effort in order to get a good grade (or equivalently, to learn the material sufficiently well by college standards).
4. My AP calculus course in high school was like a calculus course in college.
. . . The student here is expected to do much more learning outside of the classroom; see also #3 above. (On the other hand, the Advanced Placement Tests are college-level exams.)
5. College will be like high school, just a little harder.
. . . (See all of the above)
Misconceptions about learning mathematics
6. In a calculus course, theory is irrelevant, for what is really at stake is doing the problems. The lectures should be aimed just at showing you how to do the problems.
. . . We want you to be able to do all problems--not just particular kinds of problems--to which the methods of the course apply. For that level of command, the student must attain some conceptual understanding and develop judgment. Thus, a certain amount of theory is very relevant, indeed essential. A student who has been trained only to do certain kinds of problems has acquired very limited expertise.
7. The purpose of the classes and assignments is to prepare the student for the exams.
. . . The real purpose of the classes and homework is to guide you in achieving the aspiration of the course: command of the material. If you have command of the material, you should do well on the exams. On the other hand, some students act as though the exam problems have been decided in advance, and expect the lectures and assignments to be leading up to performance on those problems, or ones just like them. The latter would constitute the avoidance of our goal.
8. The best way to study mathematics is to just memorize everything very carefully.
. . . As a colleague in the Physics Department once put it, "You can't memorize problem-solving!" Here, problem-solving refers to the ability to take a problem and attempt to carry out whatever methods might be relevant to solve it. This is a skill that grows with experience. (You might keep in mind as an analogy that memorizing the dictionary of a foreign language is not enough to achieve fluency in that language.)
9. Students learn best when everything they have to know is presented slowly in the classroom.
. . . If everything the student has to know is presented slowly in the classroom, the total amount of material in the course will be rather little. Thus, students actually learn least that way.
10. It is the teacher's job to cover the material.
. . . As covering the material is the role of the textbook, and the textbook is to be read by the student, the instructor should be doing something else, something that helps the student grasp the material. The instructor's role is to guide the students in their learning: to reinforce the essential conceptual points of the subject, and to show the relation between them and the solving of problems (cf. #6).
11. Since you are supposed to be learning from the book, there's no need to go to the lectures.
. . . The lectures, the reading, and the homework should combine to produce true comprehension of the material. For most students, reading a math text won't be easy. The lectures should serve to orient the student in learning the material.
12. A good teacher is one who can eliminate most of the struggle for the student, making the material easy to learn.
. . . Of course, it is possible to direct the students toward correct ways of thinking, but a certain amount of struggle is inevitable. Experience cannot be taught. Moreover, many topics are inherently difficult enough that they cannot be understood either passively or quickly. Eliminating the struggle can only be achieved by excising substance from the course (e.g., constricting the scope of the course, or reducing the means for recognizing where the methods of the course apply). Then, the fraction of the material that remains could well be easier to learn, but the student will be acquiring diluted skills.
13. When the students are happy with the instructor's lectures, they learn the material better.
. . . This statement is wishful thinking. According to the evidence I've seen, once threshold requirements are met the perceived quality of the instructor makes little, if any, difference in learning. What makes a real difference in learning is appropriate effort by the student. The best thing that a decent instructor can do, in order to get the students to learn better, is to hold high yet reasonable expectations of them.