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Brian Rude, 2010
I have argued that the study of education needs more description. There have always been plenty of theories and ideas about how teaching should be done, but not much communication about how teaching actually is done, by normal everyday teachers in normal everyday situations. The following is a short description of one day of teaching in my own experience. The day I will describe is Wednesday, April 14, 2010. There's nothing special about that day. It's a few days ago as I write this. I will try to reconstruct the outline of what I did that day, and add details as best I can. There are no immediate lessons to be drawn from this, simply a bit of description, field notes one might say.
I teach math in a community college. The courses I teach are basic algebra, but with the exception of geometry, are not remedial. My method is lecture. The essence of my job, as I see it, is to use class time to give a careful and thorough explanation of one topic at a time. I try to stick closely to the textbook. However all the text books for my courses are a disappointment. I make a lot of handouts for homework, simply because that is usually the best way to make a good assignment, an assignment in which the students apply and practice the mathematical ideas that I explained that day. "Worksheet" or the term "worksheet teacher" is often used as a put down, as if the teacher doesn't have anything better to offer than to hand out busywork to keep the students occupied and out of the way. To me the situation is quite the opposite. It's a lot of trouble to make out a worksheet, or handout as I usually call it. There is only one reason to go to that trouble, and that reason is that the textbook does not have the problems I want. A good textbook, to me, is one in which I can find the problems I need in order to feel that I have made an assignment in which the students' time will be well spent. Often I don't find those problems in the text. There may be too few of some problems I need. There may be problems that are basically what I want but with something I don't like. The problem may be worded in the wrong way, or the numbers used are not what I want. Sometimes a problem is just what I want, but I want to use it for an example in class, so then I need another, similar, problem for homework. So, for one reason or another, most of the homework I assign is in the form of a handout, not problems out of the book.
In its essence math is a system of mathematical ideas. Math is definitely not, in its essence, a collection of procedures for solving different types of problems. But that perspective is a little misleading. We want to teach mathematical ideas, but normally that is done by teaching students to solve problems. Problems are central to learning mathematics. Problems are the stuff of everyday teaching and learning of mathematics. Real mathematics is the logical structure of mathematical ideas and their connections. But the road to that goal is through problems. Teachers know this, though perhaps only intuitively. Students know this, to some extent at least, but again mostly intuitively. Teachers can see math as more than a collection of problem solving strategies. Students can do this only to a much lesser degree. It is understandable that students' first question in many cases is a "how" question. "How do I do this problem?" But the evidence that students know that math is something more than that is in the inevitable follow-up question, asked again and again, "How come?" "Why do I divide at this step?" "Why do we throw away that seven?" "Why can't I just multiply this by that?", and on and on and on.
So problems are very important for learning math. Practice is very important for learning math. So every homework assignment is very important in learning math. A poor assignment wastes time and effort. A good assignment is productive. It does not waste the student's time and effort. It produces results, which brings satisfaction of accomplishment, which, I have always argued, is the engine that drives learning year after year for the vast majority of students.
I use direct instruction, in the generic sense, not in Engelmann's specific sense. That means I explain a topic, I work problems as examples, I explain why a problem is solved in a particular way, I try to elicit feedback as I explain so that I can guide and shape the students' thinking, and then I make an assignment in which students can apply what I explained, and then I grade that assignment when it is turned in and return it the next class period, and then I follow up on all that with quizzes and tests. I do not claim that everything can be taught by direct instruction. Some learning has to come from indirect instruction. Music appreciation, I presume cannot be taught directly. Music appreciation is a by-product of learning music. Learning to play a clarinet comes best by direct instruction, but when that is done well, music appreciation will naturally follow. I hope the same is true in math. I hope I teach math appreciation. But I don't think too much about that. I've first got to simply teach the math, as directly and effectively as I can.
On Wednesdays I get to school about 8:00 am. Actually my goal is 8:00 am, but I am usually a few minutes late. There is nothing scheduled for that time but office hours. Office hours are important. I designate two hours a day as office hours and post my weekly schedule on the window of my office. But I also tell all my classes that I'm pretty much in my office from 8:00 to 5:00 everyday except when I'm teaching, and that Tuesdays I don't try to come until 10:00 because I have an evening class, and Fridays I may leave as early as 3:00 if I am not expecting anyone. Students don't come for help nearly as often as they should, for many of them at least. So my office hours are not much distinguishable from other hours in the day. But, I do consider it important to be there when I say I am going to be there.
On Wednesdays I have two classes, geometry at 9:00 am and algebra 105 at 12:00 noon until 1:15. Geometry is a fifty minute class that meets Monday, Wednesday, and Friday. Algebra is a seventy-five minute class that means only on Monday and Wednesday. There are only five students in the geometry class. Geometry is supposed to be a high school subject completed before being admitted to college, but we offer it each spring semester for those who for some reason or another did not get it in high school. I enjoy teaching it. Math 105 is our lower level algebra course. "College algebra" is the name given by most institutions for precalculus math. It is a multi-purpose course. For some it is indeed precalculus, for some students will go on to take calculus and other math courses. But for the majority of students it is a terminal course, required for most majors. Like English composition it is a freshman course to get out of the way as soon and as painlessly as possible. At my community college Math 110 is required for most majors, and Math 105 is prerequisite for 110 except for students who do well enough on the math placement test to start at a higher level. For those who are not ready for 105, as shown by the placement test, we also offer remedial math courses.
The topic for today in geometry is similarity of triangles. Homework 25 is due. Homework 25 is a handout I made. The essential problem on this handout is making a proportion from a pair of given similar triangles, usually given by a diagram. Homework 22, 23, and 24 were also about similar triangles and proportions, developing the needed postulates and theorems which we now apply to solve problems. One might ask how this can take up four homework assignments. Isn't that overkill? Can't you explain the math in ten or fifteen minutes, spend another five or ten minutes doing a few examples, assign homework for more practice, and be done with it? My conclusion is that you can't. At least I can't.
Again and again I underestimate what it takes to teach what seems to be a rather simple topic of math. Again and again I'll plan on taking fifteen or twenty minutes explaining a topic and doing examples, but then find those fifteen or twenty minutes have flown by and I need a lot more time. Did I use the time ineffectively? Maybe, but again and again I reflect back on what I did and said, and not find anything obvious to change. I felt I had to explain that point. I felt I had to give an example of this. A student asked a question and I felt it was important to clarify something. On and on it goes. My conclusion is that it takes a lot of time to explain a new math topic thorough and carefully, more time that it would seem.
I think this is an important cause of the frustration I feel in trying to pack in all the topics that we are supposed to teach in each course. Curriculum is made by those higher up the ladder than I. When the content of a course is reviewed it probably seems like we should have plenty of time to thoroughly cover the given list of topics. This topic can be explained in fifteen minutes. That topic can be explained in twenty minutes. A third topic should be pretty easy. Students were supposed to learn that in high school. Maybe we can squeeze in matrices. That would be good for students to know and wouldn't take too much time. Wrong, dead wrong, in my view, but that is another story.
So on this day I started the geometry class by answering questions and going over problems on homework 25. That seemed to go well so then I introduced the topic of the day, which is a theorem that says that if a line is drawn parallel to one side of a triangle, cutting the other two sides, the result is similar triangles and that proportions can be made from those similar triangles. Some of those proportions are not at all obvious. They require drawing another line, parallel to another side, and showing that the resulting little triangle down in the corner is similar to both the big triangle and the medium sized triangle produced by the first parallel line. All this seemed understandable to the students, so by the end of the hour I felt they were prepared to tackle homework 26.
At 10:00 I am scheduled to be a tutor in our "learning center" for an hour. The learning center is a room adjacent to my office. My office is my desk in a room shared by myself, another full time math teacher like myself, a part time math teacher, and a part time English teacher. The learning center is a place for students to come and ask for help in math or in English. The tutors who staff this learning center are advanced students who are paid, or teachers like myself. At this particular time, 10:00 on Monday, Wednesdays, and Fridays, I don't get too many students coming for help and I often have the entire hour for grading papers. On this day, as I recall, I helped one of my geometry students for about fifteen minutes, but otherwise graded homework.
Between 11:00 and 12:00 I was in my office preparing for algebra 105 at 12. My usual preparation is to grade the papers handed in the previous day, look in the text book to see what I should cover next, look in my notebook from last semester to see what I did next last semester in 105, and then decide what to do in class today. Last semester the 105 class was a Monday-Wednesday-Friday class with 50 minute periods, so I usually cannot directly translate last semester to this semester.
The topic of the day is fractional exponents, Chapter 7 in the book. In this case I was able to take a homework handout from last semester and revise it. I improved it by adding a few problems of one type and adding a type of problems that I should have had last semester. Many math textbooks are set up so that each chapter is divided into sections and each section is a good amount of material to cover in one lecture. But I have to average more than one section a day to cover everything I'm supposed to cover in the semester. And of course I have to cover more material in a seventy-five minute class this semester than I covered in the corresponding fifty minute class last semester. But revising the handout doesn't take to long. I'm soon ready to send it to the printer, but realize I'd like to get a quiz in today, the last ten minutes of the hour. But making out a quiz can be time consuming, and I've only got about a half hour until class. But I have quizzes on the computer going back five previous semesters. If I am lucky one of last semester's quizzes will be just what I need. This time I am lucky. I send the homework handout and the quiz to the printer from my computer, walk over to the faculty office where the printer is and pick up my copies. I'm ready for class with a few minutes to spare. As I recall during that 11:00 hour in which I prepared for the 12:00 class there was at least one student who dropped by for help on a problem or two. I was able to give the needed explanation quickly and return to my preparation.
So I start class with a general plan in mind. I will answer questions on today's homework for fifteen or twenty minutes, take the next half hour to carefully explain radical notation, just what we mean by a square root, cube root, and so on, do a few examples, then ask rhetorically what we might mean by a number to the power of one half, which leads to what I hope is a good explanation of fractional exponents. All the time I'm doing this I must keep an eye on the clock. I don't want to run out of time and realize that I left out something important, some idea necessary for the students to be able to make sense out of the homework. I do that more often than I like. This time it all goes well, and at five minutes after one I pass out the quiz. The quiz is meant to be a ten minute quiz, though I don't mind if a few students go over that. The room is not needed for the next class until 1:30, so if a few students need more time I just wait for them. Today, as I recall, all the quizzes were handed in before 1:15.
Today' quiz covered addition of algebraic fractions, not one of the easier topics in this course. This topic was covered a few days or a week ago. When I first started teaching college algebra about a dozen years ago I would explain algebraic fractions, and then wonder why so few students could do such problems on tests. Now I think I do a better job of teaching that topic, but it remains difficult for students. One answer, of course, and a very important answer, is that students don't remember what they learned about fractions from elementary school, or, worse, they never learned fractions in the first place. I have discussed this thoroughly elsewhere.
It's about 1:30 and I have the rest of the afternoon to prepare for tomorrow. Since I am interested in the quiz I just gave I grade that first. Quizzes often are very quick and easy to grade, at least in comparison to grading homework or tests. Other times quizzes are not at all quick and easy to grade, especially when I want to spend time analyzing the results and reflecting on how to teach it better. Today I probably spent about an hour altogether on this quiz.
The rest of the afternoon, as I recall, I prepared for my Thursday class, helped a few students who came by, did some routine things, and went home about 5:00.
If we count fifty minutes as an hour and seventy-five minutes as an hour and a half then I am in front of a class for ten and a half hours a week. The standard load for a college teacher at the places I have taught is four courses, twelve hours a week in front of a class. How does that come out to a forty-hour work week? I don't know. I put in more than forty hours a week. I am at school more or less from 8:00 to 5:00 everyday (10:00 to 7:00 on Tuesday), each lunch at my desk while working, and still come in on weekends to try to catch up. I assign daily homework in each of my classes and pride myself on regularly having it graded and back to the students the next class period. But teachers in high school or junior high do all that and are in front of a class at least twenty-five hours a week. How do they do it? I don't know.
For several months now I have made a video recording of my teaching in every class (other than when I forgot to put batteries in the camera, forgot the camera, and so on). I just set my camera down facing the front of the room so that I am on camera and students are not, click the start button and let it record. I later transfer it to a hard drive for permanent storage. If I have time I'll watch part of the video, but I usually don't. So I think I have a video record of both classes I talked about in this article. However I didn't review them for writing this article. Perhaps when I have time this summer I will watch them and revise this article. I wonder if things I described might then seem different. And I wonder which might be more accurate, things as I remember them now, several days removed, or things as I remember and review them several months removed.