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General education students can learn to read mathematics more thoughtfully and critically by writing questions over the reading, and short response papers—without enormous investment of faculty time grading.
Background and Purpose
Montclair State University was founded as a public two-year normal school in 1908 and has evolved into a teaching university with a Master's Degree program and plans for a doctorate. Almost 13,000 students are enrolled. Most have a mid-range of collegiate abilities, but there are a few super-stars. Most are first generation college students; many are working their way through college.
MSU requires all students to take mathematics, and specially designed mathematics courses support many majors. Students in other majors have a choice of courses to satisfy their mathematics general education requirement. One, "Development of Mathematics" has as its aim "to examine mathematics as a method of inquiry, a creative activity, and language, and a body of knowledge that has significantly influenced our culture and society through its impact on religion, philosophy, the arts, and the social and natural sciences." It attracts majors in religion, philosophy, psychology, the arts, foreign languages, English, journalism, elementary education, pre-law, political science, and history.
I was first assigned this course in 1993. How could one grade students in such a course? Computation doesn't have much of a role in achieving the stated aims, but downplaying computation essentially eliminates traditional methods of mathematics assessment. A friend who teaches collegiate English and gives workshops nationwide for teachers of introductory college writing courses urged me to consider giving short writing assignments and just checking them off, as is common in introductory writing courses. This flies against our custom in teaching mathematics. When I balked, she admitted she gives "check plus" to those who show special insight in their writing.
Wanting more variety, I decided to include weekly questions on the reading and a term paper as major contributors to the semester grade. A smaller weight is given to a final paper, attendance, and class participation, although (to avoid intimidating the students) I de-emphasize the oral component.
The two aspects that are most unusual in mathematics courses are discussed here: short papers and weekly questions on the reading.
Method
Short Papers. Typically, the weekly 250-word essay counts three points toward the final grade. The syllabus that I hand out on the first day of classes says, "You may explore ideas further (if you want an `A' in the course) or merely summarize the author's comments (if you are content with a `B-')." Many students choose to "explore ideas further," and when they do this creatively, they obtain a "check plus," worth four points instead of the usual three.
Some papers, but relatively few, are skimpy, and deserve only two points. The level of most papers is more than adequate. For lateness I subtract points, but I stay free of an algorithm, since some papers are more justified in being late than others.
Weekly Question. A more explicit device for discovering what students want and need to learn is to require students to hand in questions at the beginning of each week on that week's assigned reading. Each question counts one point toward their final grade, and ensures that most students have read the assignment before the week begins. These questions form the basis of much of our in-class discussion (see Use of Findings).
Findings
One of the surprises is the variety in the students' papers as they try to write something that is more than routine. I inevitably learn a great deal, not only about the students, but about new ways of viewing and teaching mathematics, and about other subjects related to mathematics that they know better than I do.
Another surprise has been how easy it is to decide for grading purposes which papers demonstrate originality. Excerpts from one of the first papers I received indicates how one student explored ideas further. (Rudy Rucker is the author of the text.)
....Suddenly, I'm drowning in base 27's and numbers I've never heard of before. I cry out for help as I notice a googol chasing me across the room.
My senses eventually come back and I realize that [the author] has made one major blunder... [He claims] that if we were to count two numbers a second for the better part of six days, one would reach a million.
....One could count two numbers a second very easily if it were up to one hundred. [but]...the numbers in the one hundred thousand to nine hundred ninety-nine thousand, nine hundred ninety-nine range are going to take more than a half a second. They are actually going to take the better part of a second. This does not even take into account that a person will probably get winded somewhere along the line and have to slow down. Fatigue may also factor into this because a person is bound to be tired after not sleeping for six days.
Nevertheless, I did want to make sure that my point was legitimate, so I took different sequences of numbers in a clump of ten and said them, checking the second hand of my watch before and after the set of numbers in the one hundred thousand range. My average was considerably greater than a half second per number. For example, try counting the sequence 789,884 through 789,894.
Mr. Rucker, your question is good. Its answer isn't. I'd tell whoever decided to attempt this to set forth a couple extra days. Six days is not enough.
Such a paper clearly merits an extra point beyond those that merely regurgitate the reading.
Another discovery is the extent to which students reread each week's assignment for their papers. Most read it reasonably carefully before submitting their question at the beginning of the week. Many refer to a second, third, or even fourth reading in their paper written at the end of the week. They are obviously seeking a deeper understanding based on the class discussions. Some students actually chronicle for me how much they comprehended at each reading, a useful account for my future teaching.
Use of Findings
Grading such papers is far more interesting than checking computations; it is worth the time it takes. I feel more in touch with the students than in any other course. Since I try to write at least one comment about the content of each paper, the papers become, in effect, a personal correspondence with each student.
I often read excerpts from the best papers aloud, and it is clear that the students also are impressed. Having one's paper read aloud (but anonymously) is an intrinsic reward that students work hard to achieve. Reading interesting passages to the class also sometimes generates discussions that probe the ideas further.
The papers that merely summarize the reading are less exciting to read, but they too inform my teaching. Sometimes I catch misunderstandings. More commonly, students mention confusions that need clarification. The written papers give me an opportunity to see how and to what extent they are absorbing the material.
Students' written questions are the backbone of my lesson planning. Some of their questions lend themselves to traditional teaching:
Explain what a logarithm is. I'm still a little confused.
or
Please explain the significance of 7=111two=1+2+4. I've never seen this before.
I usually begin the week's exposition by answering such questions. One advantage of basing lessons on the students' questions is that at least one student is curious about the answer.
After I have taught the strictly mathematical topics, I turn to other questions where I believe I know the answer better than the student. For example:
When I think of a computer, I imagine an overpriced typewriter, and not a math-machine. I especially don't understand the system that a computer uses to translate the math information into word information. In other words, what is a bite (sic) and how does it work?
This is not a traditional mathematics question, but it does guide me in helping the students understand how mathematics "has significantly influenced our culture and society" (the course's aim). I answer such questions in the lecture format of a liberal arts or social science course, keeping the lecture short, of course.
For the final group of questions, I have the students arrange their chairs in a circle. I read a question and wait. My first class let hardly a moment pass before the ideas were flying. The second class taxed my ability to extend my "wait time." Sitting silent for long gaps was not one of my fortés before teaching this course, but developing it paid off. Student discussion would eventually happen, and it was almost always worth waiting for. Subsequent classes have been between these two extremes; none have been disappointing.
Sample discussion questions students contributed from the same lesson as above included:
If we agree that "computerized polling procedures" give citizens a "new power over the government," is the reverse not also true?
Page 30-bottom: "I construct in my own brain a pattern that has the same feel as your original thought." I disagree with this. Information obtained doesn't necessarily have the same "feel" to it as the transmitter's thought intended, for we decipher it according to our perception and, therefore, can and often do distort the intended meaning or end product of information.
This last question is related to a fascinating student story. A clerical worker from Newark, she wrote at the beginning of the semester in her introductory comments to me that she was scheduled to graduate; this was her last course. Then she added, "I have enormous math anxiety. I hope to be able to ease this somewhat and sweat in class only from the high temperature!" The student spoke not a word in class during its first month, but she was always there on time and made constant eye contact. Class reaction to her questions (such as the one above) was strong, however, and I remember her response when another student exclaimed to one of her questions, "What a genius must have asked that one!" I saw her sit a bit taller. After a few weeks, she began to participate, very cautiously at first. By early December, her term paper, "From Black Holes to the Big Bang," about Stephen Hawking and his theories, was so fascinating that I asked her if she would be willing to present it to the class. To my amazement, she hardly blinked, and a few minutes later gave a fine oral summary that mesmerized the class and stimulated an exciting discussion. She received a well-earned "A" at the end of the semester.
The best part of using essays and questions as the major assessment devices is that students take control over their learning. The essays reveal what they do know, while traditional tests find holes in what they don't know. Students rise to the challenge and put real effort into genuine learning.
Success Factors
The sad fact is that the general public perceives traditional math tests as not objective because people can so easily "draw a blank" even though they knew the subject matter last evening — or even an hour ago. My student ratings on "grades objectively" have risen when using this method.
Students like to demonstrate what they know, instead
of being caught at what they don't know. As the
semester progresses, their efforts accelerate. Taking pressure
off competitive grades results in remarkable leaps
of understanding.
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