Many students believe that writing has nothing to do with mathematics. This is false. In any field, it is important to be able to communicate your ideas and results to others. Exactly how you communicate them varies from field to field, but in nearly all fields it is through writing. Mathematical writing, however, has its own particular style. The emphasis is on clarity and precision, and not on the clever turn of phrase. The “Good Problems” method is designed to teach you how to write about mathematics coherently. It provides you a set of specific writing skills and gives you enough practice through regular assignments so that writing well will become a habit. It is also designed to be as painless as possible.
The main goal of this method is to teach mathematical writing. It is motivated by our experience grading technology-based laboratory reports in sophomore-level courses. It seemed that most of them had never before been asked to write a coherent report on a mathematical subject. They had not mastered even a basic set of writing skills. Furthermore, they resented being asked to write, because they believe that it is not necessary for engineers and scientists to know how to write. For most college courses this is true, but we know that after college, it is crucial to be able to write about what you do.
The method itself is very simple. Each week, in conjunction with the regular homework assignment, one homework problem will be designated a “good problem.” The student must write the solution to this problem carefully, demonstrating specific skills at presenting mathematics. A set of six skills will be taught during the course of the semester or quarter. At any given point in the term, the student will be graded only on those skills that have already been taught. The main mechanism for teaching these skills is a set of handouts. Each handout identifies and explains a specific skill, and provides examples of good use of this skill and examples where this skill is lacking. A secondary mechanism for teaching these skills can be discussions or demonstrations during the class or recitation. After a skill is officially learned, the regular feedback from these weekly assignments will coax the students toward mastery of that skill.
One beneficial side effect of this program is to encourage organization. College presents new found pressures on the students’ time and energy. It is expected that to survive the student will have to learn time-management and organizational skills. This topic is beyond the scope of our endeavor, but we require the student to present one well-organized sheet of paper each week. A second beneficial side effect is to encourage logical thinking. This is another skill that is vital, but not taught directly. By teaching the students the proper uses of logical connectives, we can teach them to recognize when their argument has gaps or contradictions. It is again beyond our scope to teach logical thinking properly, but we can require one logical sheet of paper each week, and provide some basic skills.
One very important aspect of this method is that it is designed to be as painless as possible. One drawback of many teaching reforms is that they require significantly more effort on the part of both the instructors and the students. When performance gains are noted, it is not clear how much is simply due to spending more time thinking about mathematics. We aim here not to add effort to the system, but to re-align a small amount of effort to a more productive area. There is some initial added labor involved of course, but then instructors need merely assemble the ingredients provided and set the program in motion. The instructors or teaching assistants will need to spend time brushing up on their own writing skills, but this will benefit them, and so should not be considered a burden. In the short term, we are requiring the students to learn additional skills and they will therefore have to expend additional effort. We expect, however, the returns to be rather quick. Many of these ‘presentation’ skills transfer directly to problem-solving skills, and also help with reading mathematics. We expect these skills to carry over to future mathematics courses.
If the course is taught without recitations, then these tasks fall on the instructor. The first time you use this method, there are extra tasks, which we list first.
1. Read and understand the material in this packet. This should be performed in the week before the term starts.
2. Attend training. The instructor should provide some training during the week before the term starts.
3. Deal with unanticipated problems.
Each time the method is used there are the following tasks.
1. Prepare for the new skill to be taught that week. It may also be useful to review the old skills. This is part of the weekly preparation time.
2. Present difficult skills in the recitation. Sometimes students will have particular difficulty on some specific skill. The teaching assistant will need to re-explain this skill and perhaps demonstrate it on a relevant problem. It is important to note that the teaching assistant is not expected to be the main teacher of these skills. The handouts should provide all the information the students need.
3. Grade the good problems. It seems fair to grade two less regular homework problems to compensate for the good problems. After some initial trauma, it is expected that the good problems will be more pleasant to grade than ordinary problems. There should be no net gain in grading time as a result of this.
4. Provide feedback on the handouts. Any deficiencies in the method or the handouts should be noted. At the end of the term these comments should be collected by the instructor and the materials corrected.
Presentation is very important in mathematics, just as it is in other fields. The main reason for this is clarity. Good presentation allows you to communicate more clearly the content of your work. A secondary reason is the appearance of competence. Careful presentation makes the reader think you were also careful with the content of your work. Although good presentation is rarely successful at bluffing past poor content, poor presentation can easily ruin good content. The first thing the reader evaluates is the overall visual layout of the problem. Exactly how polished this should be will depend on the purpose of your writing. For our purposes, we have the following rules:
• Put the good problem on its own sheet of paper, separate from any other problems. Regular quality paper is fine, but there should be no ragged edges or tears.
• If you need more than one page, staple (no folded corners!) them together and put your name and the page number on each page.
• Leave margins on all sides.
• Print neatly or type. Do not switch colors or from pen to pencil in the middle of the problem. (You can use different colors to highlight if you wish.)
• If you had to cross out material or erased a lot and left smudges, rewrite the problem. (It is a good habit to solve the problem first on scrap paper and then copy it neatly.)
The reader next needs to know what it is they are reading. Next is the format for the problem itself:
• Label the problem with the chapter, section, and problem number.
• Write out the entire question, including any instructions. If the question refers to another problem, include the relevant information from that problem. The goal is to make your work as self-contained as possible, so the reader does not need to look anything up.
• Do the problem in some logical order. Do not do the problem in several disjoint pieces connected by arrows.
All these rules may seem picky. Once you have learned this way to lay out your work, you will understand the principles behind these rules. Then you will know when to change the rules.
Written mathematics must be readable. This may seem trivial, but it is an important point. You should be able to read your work aloud to a classmate and have them understand your solution. If you need to add any explanations, these should be included in your written work. The most common mistake is to write mathematics without using enough words. All writing, even mathematics, should consist of complete sentences. These should explain the problem by providing both the method and justification for each step of the solution.
Why are sentences important in mathematics?
Although sometimes it seems hard to read textbooks, it would be much harder to understand if they only had equations and no sentences. The situation is similar in lecture: if the professor just listed formulas on the chalkboard without talking about them, leaving you to figure out what was being done in each step, how much could you understand from the lecture? Neither of these would be a good way for most students to learn, since sentences are necessary to explain the mathematics.
Why should students use sentences in a Mathematics class?
In a Mathematics class, you should explain homework solutions using complete sentences. That means linking together thoughts with words and embedding equations into sentences. Going through the extra work to do this will benefit you in several ways:
• Writing down your thoughts and organizing them into complete sentences will help you to understand the method of solution better.
• When you look back on homework to study for a test, or later on in another class, you will understand what you were doing on each problem and the mathematics behind it.
• Other people (teacher, classmates, grader,...) will understand what you are doing at each step, and why you are doing it. This way, you won’t lose points for skipping steps or solving the problem in an unusual way.
• Communicating your work will be essential in whatever field you choose. Even though the fields are stereo typically weak on writing, engineers and scientists spend a surprising amount of time writing reports and giving oral presentations.
Mathematics has its own language. As with any language, effective communication depends on logically connecting components. Even the simplest “real” mathematical problems require at least a small amount of reasoning, so it is very important that you develop a feeling for formal (mathematical) logic.
Consider, for example, the two sentences “There are 10 people waiting for the bus” and “The bus is late.” What, if anything, is the logical connection between these two sentences? Does one logically imply the other? Similarly, the two mathematical statements “r2 + r − 2 = 0” and “r = 1 or r = −2” need to be connected, otherwise they are merely two random statements that convey no useful information. Warning: when mathematicians talk about implication, it means that one thing must be true as a consequence of another; not that it can be true, or might be true sometimes. Words and symbols that tie statements together logically are called logical connectives. They allow you to communicate the reasoning that has led you to your conclusion. Possibly the most important of these is implication — the idea that the next statement is a logical consequence of the previous one. This concept can be conveyed by the use of words such as: therefore, hence, and so, thus, since, if . . . then . . . , this implies, etc.
Many students, especially those in science and engineering, are not able to effectively communicate mathematical ideas in writing. Furthermore, they generally believe that this ability is not important. Employers, however, consider this to be a crucial skill. Instructors would have to side with the employers, but usually do not attempt to teach the students how to write, or that writing is important. We present here a plan to make writing coherent mathematics a regular part of the students’ experience. This will both improve their writing and teach them that writing is valued in mathematics. We provide materials to teach a set of important writing skills, so the students will not have to guess how to write. This method is designed to be as painless as possible, for the instructors as well as for the students.
No comments:
Post a Comment