In his remarkable and beautiful little monograph entitled Liberal Education, Mark Van Doren [2] wrote '"Language and mathematics are the mother tongues of our rational selves"--that is of the human race...'. We, teachers and practitioners of mathematics, are well aware of the capacity of mathematics to empower the human mind; but I believe that we often forget or ignore the potential power of language, especially written language, as a tool for investigation. Van Doren reminds us that language is coequal with mathematics in its capacity to enable our minds. It is the purpose of this note to discuss an application of written language to the task of teaching (and learning) mathematics.
In Everybody Counts: A Report to the Nation on the Future of Mathematics Education [1], we read "Research on learning shows that most students cannot learn mathematics effectively by only listening and imitating...(and)...that students actually construct their own understanding based on new experiences that enlarge the intellectual framework in which ideas can be created." Let us first observe that writing about mathematics forces construction of understanding, because we cannot write coherently about something we have no understanding of. Professors of English tell us again and again of the deep interplay between language and thought.
It occurred to me that one way to harness that interplay to the task of "enlarging the intellectual framework" within which students of mathematics work is to require them to keep journals which they must periodically submit for critical evaluation. I have subsequently experimented with the idea, and I will describe the results of my experimentation in what follows.
In several classes over the last year and a half, I have required students to submit weekly journal entries. These courses have so far ranged from a survey course for liberal arts students, through beginning and intermediate algebra, and into first semester calculus. I have not yet used the technique in an upper division undergraduate course, but I foresee the possibility--indeed, the likelihood.
In the courses where I have used the technique, I have begun the semester by handing out a sheet, entitled "Your Journal", at the first class meeting. The handout describes in some detail what I expect them to do with their journals. As I refine my ideas, this handout continues to evolve. The appendix to this note is the version I used last; the version I use next will be somewhat different.
In the assignment, I have instructed students to prepare and submit weekly journal entries. I, and they, have found it most convenient to have the entries due on the first class meeting of each week. Each entry thus dealt with the work of the previous week. A weekly entry could be organized as the student wished, either as a single essay dealing with the week as a whole, or as a collection of smaller essays that each deal with shorter periods of time. Regardless of organization, the entry was to contain three kinds of writing: a short summary of the topics we covered during the week, a quick report on the student's own relevant activities for the week, and a lengthy analysis of the week's work. I have told students that their journal grades will depend on the quantity and quality of their reported activities; the content of the entry (especially the analysis); and grammar, spelling, and punctuation. I have not told them, although I plan to do so in the future, that their journal grades will not depend on the correctness of the mathematics they include.
In fact, I have been principally interested in their analyses, and I have based the bulk of their grades on them. I have not really graded the quantity or quality of their activities, nor have I graded grammar, spelling, or punctuation--except in cases of flagrant inattention. I have mainly used the threat of doing so to keep them honest. I think this especially important where grammar, etc., are concerned. They must pay attention to their grammar, to their spelling, and to their punctuation. These matters are part and parcel of the habit of precision--which is one of the things that I intend this writing assignment to help them develop.
I have recently required that each student analyze the solution to at least one problem that we did not solve in the classroom. In the absence of such a requirement, I have found that I see a good bit of stuff that is more than a little reminiscent of things I have put on the blackboard. That defeats another purpose of the assignment, which is to get them to attempt some mathematics on their own. As we all know, they find it much easier to regurgitate our thoughts than to have their own--and many of them will regurgitate, given the choice.
The first semester I used the journal technique--and every time thereafter--I included in their instructions a requirement that, in the entry that covers a week in which they sat for an exam, they devote a portion of their analyses to the exam and their performances on it. My intent was to get them to think about the exam after they had finished taking it--an activity most of them don't indulge in.
This requirement turned out to be an especially good one, owing to the fact that I habitually schedule exams for the last class meeting in the week. I do this routinely in order to get the entire weekend in which to do the grading, so this habit wasn't in the front of my mind when I included the requirement. But the happy result has been that I have required them to analyze their exams before they see my analyses of those exams. Mediocre students aren't very happy about this, but it is exactly what they need to do, and it is exactly what the better students have always done--though perhaps less formally than I now require.
An outgrowth of the requirement that they analyse their performance on an exam before I have returned their papers deserves discussion. On one occasion, one of my better students in a Calculus I course complained in her analysis that she couldn't recall what she had done on a particular problem. It turned out that the problem in question was one that she had botched--owing rather clearly to her failure to comprehend a central idea; she had, on the other hand, very satisfactorily recalled the correct work she had submitted for the other problems. In my comments on her entry, I suggested that perhaps the fact that she couldn't recall what she had submitted was a danger signal. I further suggested that she might be able to use her ability to recall her solutions to problems she had worked while studying as a measure of her understanding of underlying ideas. She agreed that the criterion had potential, and continued to experiment with it. In her final journal submission for the semester, she reported considerable success in using this criterion to identify matters that needed further attention.
It is important to note that grades on journal entries do not depend on the correctness of the mathematics included. This is vital! What matters is the effort that a student puts into thinking about mathematics, into thinking about thinking about mathematics, and into organizing those thoughts for presentation in writing. Any bona fide effort to analyze a problem: any effort that shows organization, willingness to experiment, effort to trap error; any effort that displays any of the things that you or I do as we do our mathematics deserves credit--whether the mathematics is correct or not. It is primarily their approach to mathematics and their presentation of the result that I grade.
That is not to say that I let incorrect mathematics slip by uncorrected. I comment extensively on the mathematics they are writing about--as well as on the writing itself. At first, my policy of correcting their mathematics without penalizing puzzles them. Students expect to be punished for their mathematical mistakes. They are used to trying to obfuscate potential errors, or at least to trying to deny responsibility for them--often by using the passive voice. (Sample: "A mistake was made in calculating the derivative, so that the second critical point was not found." Note that no-one is responsible for this mistake--it just happened! We might ask ourselves who taught them to obfuscate, and after we recover from the ensuing discomfort, we might accept our own responsibility.)
This issue is the main problem that I must deal with during the first weeks of the semester. I must re-train students to attempt to present their thinking, potential errors and all. They must accept responsibility for their mistakes, when they would much rather conceal them and blame nobody in particular--least of all themselves. This re-training takes time, and they find that I get fussier and crankier about their evasions as the semester lengthens and they catch on. They must display their mistakes, because those mistakes are the high-grade ore I mine; they are what I use to teach.
At the end of the semester, I discard the lowest fifth of each student's scores and average the rest. I base one-third of the semester's grade on the resulting average. This is a substantial portion of their grade, and I expect a substantial effort from them for it. There are always a few who learn this the hard way. Every semester, the first entry includes several one- or two-paragraph submissions, hastily scrawled on a scrap of paper--even though the assignment clearly indicates that I expect several pages at least. Those submissions get what they deserve: A mark of zero and a note "This is not acceptable; please re-read the assignment." Second offenses are rare.
This is an effective approach to learning mathematics. The connection between language and thought is real, and students can use it to help themselves "construct their own understanding".
Students repeatedly remark that they have just succeeded in solving, as they worked on their journal entries, a problem they didn't think they would be able to do when they sat down to write. Of course, I anticipated that this would happen, or I would not have made the assignment. But I did not anticipate the frequency with which it happens, and I do not tell students that it will happen because I think that the lesson is more effective when they discover it for themselves."It never seems to fail that no matter how often I'll work a problem incorrectly, I'll invariably do it right when I go to prepare my journal. It really works to have just one problem in front of you, and to walk slowly through it and explain each step." --V.S., Calculus I
"In selecting problems for discussion in these entries I try to select ones that interest me and give me an opportunity to explain something to myself or puzzle out a seeming contradiction." --R.T., Calculus I
"At first I hated this journal idea and could not make it personal. After a while I began to see how it aided me in figuring out calculus and learned to use it as a means of putting my thoughts together in a useable form. It has helped me to test my level of understanding which has helped to enforce my skills and confidence. Although it is a lot of work...I feel that these journal entries are of great benefit." --H.E., Calculus I
A second advantage: The journal entries open up a new and useful channel between student and instructor. I see firsthand how students are thinking; I see what misunderstandings they are laboring under; and I see what they really do comprehend. And having seen, I can intervene. I can head off developing misconceptions; I can dispel them while they are no more than nebulosities. Of course this depends very much on students' coming to realize that their journals are an environment where they can explore and make mistakes without paying a penalty. Once they have this understanding, I find that they will discuss almost anything.
"I look at [studying calculus] as a journey, and these entries were like taking an experienced guide along." --R.T., Calculus I
Students cannot escape exploring mathematics in their journals. This is another advantage of the technique. Successful journal entries always involve personal exploration of some aspect of mathematics--however trivial. These explorations may fail in one of the narrow senses that the explorer gets lost or fails to find anything of real interest. But they are successful in the broader sense that the student becomes an explorer. For many students, exploring mathematics with any kind of success is a new experience. They are amazed and pleased with themselves and their newly discovered capabilities.
Still another advantage arises from the fact that students are writing about mathematics. Writing about mathematics is just a small step away from writing mathematics itself, and in fact, many students really do write mathematics for the first time. In working on their journals, all of them begin to learn how to write mathematics.
As a further bonus, I find that I get more out of my best students than I have in the past. I mean the ones who are capable of sailing through a lower division course unfazed. I have had the feeling many times in the past that I do not really give full measure to these people, and that they do not really give me full measure. The journal is a tool for changing that. Because the communication in the journal is one-on-one, I can teach these students more. And demand more of them in return for their A's.
Finally, students like the technique. They gripe, but generally in the good-natured fashion of one who is working hard, knows it, is accomplishing something worthwhile, and knows that, too. I must admit that this advantage of using journals isn't of great importance to me, personally. I suspect that if I believed that I could instill understanding or appreciation of the Mean Value Theorem by smacking someone up alongside the head with a two-by-four, I would try that technique, too.
The major disadvantage of using weekly journal entries has to do, as the reader has probably guessed, with grading them. I have perceived a two-fold disadvantage. First, grading weekly journal entries is a time-intensive chore. One must read them attentively--even more attentively than one reads exams. One must provide extensive commentary, especially during the first few weeks of the semester while one is trying to break their old habits and to instill new ones.
During the first semester I used journals, I went somewhat overboard. I used them in three courses simultaneously, and I think that I was reading somebody's journal at just about every odd minute of every day. (And during some of the even minutes, too!) I felt like I wasn't doing anything else! I learned from my mistake, and I now use the technique in just one course each semester. And I choose that course carefully as the one where I think the technique has the most to offer.
The second of the disadvantages associated with grading journal entries is perhaps more troubling to many of us. As mathematicians, we are likely to be uncomfortable with the idea of grading writing. We worry about our qualifications where writing is concerned, and to us the grading of writing seems more subjective than the grading of mathematics. These concerns are probably less justified than we think. We are, after all, highly educated folk, and we can reliably determine whether or not undergraduates are having reasonable ideas and whether or not they are expressing them reasonably well.
A final disadvantage: Students hate the technique. Not all of them, of course. By way of example: I announced on the first day of a calculus course that I would require them to hand in weekly journal entries during that semester, and I handed out the journal assignment sheet to 30 students. The following day, 18 students returned--I never saw the other 12 again. But then, I would probably have lost those 12 anyway, later on in the semester when both they and I had more substantial investments in failing enterprises. Come to think of it, maybe this one isn't a disadvantage after all...
I have no doubt that I will continue to use this technique. I find that the advantages clearly outweigh the disadvantages, and except for the drain on my time, the method fits my style. There are, however, some things that I will do differently.
Be explicit about what doesn't count toward the grade. In particular, I think that it will help me re-train them to present potential errors honestly and forthrightly if they know from the beginning that I will not punish them for their mathematical mistakes.
Provide more explicit instructions on what to do. There are several issues here. I need to encourage students to write about the problems they couldn't solve (that they are reluctant to do so is probably very closely tied to their fear of displaying their errors). I need to provide them with some instruction in dealing with seemingly intractable problems. Some of the material in [3] may be helpful here, either directly or indirectly. In general, I need to give them some concrete strategies for conducting what I have in the past called "analysis". It may even be helpful to give them some examples of what other students have accomplished before they have to submit their own writing.
Involve the best students in self-evaluation, with their evaluations of their own work being the principle determinants of their grades. This is an entirely new idea, which occurred to me during the preparation of this note. The idea, which I shall have to think about some more, is to return the student's entry, ungraded, after several days, with instructions to the student to evaluate his own work for his grade.
You are to keep a journal for this course. You must make at least one entry in your journal each week. At the beginning of the first class period of every week, you are to submit the entries you have made during the preceding week; late submissions are not acceptable. Journal entries should be submitted on ordinary 8 1/2 by 11 paper which is (or can be) punched for placement in a notebook.
A complete entry for a week will contain:
If you took an exam during the week preceding the entry, you should devote the bulk of this section to a thorough analysis of the problems on the exam, their solutions, and an evaluation of the solutions you submitted. In other weeks, you may choose to focus on a single problem or topic, or you may analyze the entire week's progress. In any event, you must analyze the solution to at least one problem that we did not solve in class.
You will find that a careful written analysis of the difficulties you encounter on problems you don't know how to solve is one of the most profitable things you can do in your journal.
Your analysis of your week's work is the heart of your journal; it should be at least several pages long.
I will grade your work on the quality and extent of your activities, the content of your remarks--especially in your analyses, and on your grammar, spelling, and organization. Within a reasonable time of your submission, I will return your work to you. You are then to undertake any rewriting I have asked you to do and place both the original entry and the rewritten entry into your journal notebook--which you should maintain separately from your classroom notes. I may ask you to submit these notebooks for inspection from time to time, so it is important that you keep them up to date and that you do the rewriting I ask you to do.
I may use copies of your work for in-class discussion; when I do so, I will not identify the author publicly.
The grade you earn on your journal will count for one third of your semester grade.