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Thoughts on My Teaching At NDSU
Brian Rude April 04
My experience at NDSU has been different than at St. Cloud State, where I taught from 2001 to 2003, and from South Dakota State, where I was a graduate student from 1998 to 2001 and taught college algebra on a graduate assistantship. Most of these differences are imposed from above. Lecturers at NDSU are given comparative little freedom. There are a number of requirements that must be met. I understood this at the time of my initial interview.
There are basically two freshman math courses at NDSU, in addition to the usual remedial courses. Math 103 is college algebra and Math 104 is Finite Math. In the fall semester I taught four sections of 103 and one of 104. In the spring semester I taught three sections of 104 and one of 103.
College algebra at NDSU is supposed to prepare students to take calculus, and presumably a substantial number of students do go on to take either practical calculus or the regular calculus sequence. It covers the usual topics, a review of basic algebra, functions, combining functions, inverse functions, polynomial and rational functions, and logarithmic and exponential functions. This is similar to the corresponding courses at SDSU and SCSU, but we put much greater emphasis on the use of the graphing calculators, and our tests are quite different.
Finite math covers a bit of algebra, systems of linear equations and matrices, linear programming, probability, very elementary statistics, and consumer finance. It does not prepare students for any further math courses.
Each course has a course supervisor, responsible for coordinating the efforts of the staff teaching that course, and responsible for making many decisions that are not left to the discretion of the individual teacher.
We have group testing in the freshman math courses. Each course has three tests and the final. The three tests, which I will call “mid-course tests” take place in the evening, from 6:30 to 8:00 on the designated test dates. The date of the tests are given on the course syllabus and students are expected to make their plans accordingly. The testing locations are announced a week or so in advance of each test. We are to announce early and often in the course about the tests, and are expected to emphasize to students that they are students first and workers second. Work schedules should be arranged around the scheduled test times, not the other way around. In theory the tests are constructed by a group effort of the teachers teaching the course. As a practical matter the tests are mostly made out by the course supervisor. In one course the supervisor asked each teacher to submit two problems several weeks before the test. The supervisor then made out the test and gave it to the teachers for feedback before making the final version. In the other course the supervisor asked for input, if we had any. I did have some imput into this procedure, but I felt we fell short of the ideal of getting together and working out each test cooperatively.
Each mid-course test consists of two parts. The in-class part consists of eight regular questions and a bonus question. Then there is a two question, twenty point, take home part of the test, which the students may, or must, do together in groups. In one class there was a two point penalty for students who work alone on this part of the test. In the other class there is no penalty for working alone, and many students do. Students are told they can not ask for help from the teacher or anyone outside their group on this take home part of the test. However there is considerable evidence that students take liberties with this rule.
There is no formal rule that the problems on the tests will always be application problems, but it seems to be a well established pattern. I believe this is motivated by the ideal that students must be able to apply the mathematics to real world situations. For example we would not give a purely algebraic problem like “Find the equation of the line that goes through the points (7, 3.2) and (10, 2.8).” Rather the question would be something like, “A seven year old car has a value of $3,200. When it is 10 years old its value is $2,800. Find an equation that relates V, the value of the car to t, time.” I have felt that this ideal is carried too far. Many students have an aversion to “word problems”, because they are less successful at them. I think it makes sense to also have a number of smaller, purely algebraic, problems. I have noticed that as we progress through the book we depart more and more from this application ideal and ask more purely algebraic questions.
Students write their tests in bluebooks. Grading is assigned among the teachers of the course. If I am assigned to grade problems 9 and 10, then I grade those problems for every student taking the course, which can be over 600. Another teacher will grade everyone’s problem 1 and 2, and so on. Each sections’ tests are put in an large envelope and passed among the teachers until the grading is complete.
There is an undeniable and powerful argument for this system. Students in all sections of a course are held to the same standards, and, in theory at least, the results of one section can be compared to the results of the other section. But this situation for testing has proven frustrating to me. I have concluded that this system presents many problems that we have not adequately addressed. It takes some time to know what to expect. A new teacher is at a disadvantage for a while. One important requirement to make this system work, in my opinion, would be that all tests must be available to all teachers of a course before the semester begins, and each test must be available in a number of different, but corresponding, forms. It can be argued that this would cause teachers to “teach to the test”. Of course we have the ideal that we do not “teach to the test”, and this is an important consideration, but that is not all there is to be said about the matter. Students feel cheated if they are not given a clear indication of just what will and will not be on the test.
Apparently “cooperative assessment” is a popular idea in math education now. We have as many as three group projects in each course, and each midcourse test includes a 20 point “take home” part that is to be done in groups. This is something new to me, and not something that I would think of trying on my own. On theoretical grounds I would question if cooperative assessment can be considered “authentic assessment”, which is another term sometimes seen in current educational thought. It seems to me that authentic assessment would require false positives and false negatives to be held to a minimum. This, it seems to me, requires individual assessment. When I get a project from John, Jane, and Jo, I wonder who did what part of the work. When the semester is half over and I have an idea of the capabilities of individual students I may form the opinion that Jane is the brains behind the group effort and John and Jo are passively benefiting thereby. I have observed that in many cases students seem to enjoy their group efforts. But I also have observed plenty of frustrations associated with their group efforts. When group work is encouraged, but not required, as many as half of the students choose to work individually. I am not sure if this is primarily that they want to work individually, or just a consequence of the difficulty in getting together. On a number of occasions students have come to me with frustrations and problems arising from their group projects. One can conjecture that the problems brought to the teacher are only a small fraction of the problems and frustrations students encounter. But also one can also conjecture that when students do not come to the teacher, we may infer success of the process.
I think we overemphasize calculator use in these courses. We expect students to be able to solve equations by graphing. A simple way to do this is to let Y1 be the left side of an equation and Y2 be the right side of the equation, and find the intersection graphically. This method has some value, of course. It allows students to solve equations that they would find simply impossible to do algebraically. But there are problems. Successfully doing a few equations by this method may come quickly and easily. However students do not have a background that allows them to do any trouble shooting when things go wrong. Perhaps the biggest problem is finding a suitable window. Without a suitable window the calculator is worse than useless. And students at this level don’t know enough algebra to always know how to find an appropriate window.
We also teach students to do maxima and minima problems graphically. This can be done, but I question the appropriateness of doing so. We do such problems only as applications. This means students must take a written problem and derive a function to be maximized or minimized. This presents substantial difficulties for many students. By having a poor understanding of the problem they are again not well equiped to find an appropriate window.
We also introduce the idea of least squares regression equations and teach them how to find them with the calculator. But this seems mostly an exercise in rote learning, a bit of unrelated statistics thrown in for unknown reasons. And again students have little ability to deal with the many unexpected calculator problems that can arise.
I think our calculator emphasis comes from current thinking in math education to “utilize appropriate technology”. My thought right now is that we are missing the mark on “appropriate”. It is not at all clear that calculator use aids in the understanding of the important mathematical concepts that form a basis for continuing in math.
Teachers at NDSU are expected to use “Blackboard”, which is an internet course management system. This took a bit of learning at the beginning, but has been very helpful. It has eliminated the need to prepare and copy handouts for students. Documents can simply be posted on Blackboard and students can read them there, or they can print out their own copy. I posted all homework assignments on Blackboard. In fact I gave up trying to choose problems for homework before class. It was much more convenient to wait until after class and just put the assignment on blackboard. At that time I had the advantage of knowing just what was covered in class, which is not always exactly what I had planned beforehand. Also posting homework on Blackboard allowed me to give more detailed instructions than would be convenient to write on the board in class. I could give hints on problems, or change problems. Also, as I did many times the second semester, I could write out an entire worksheet and post it as an attachment under the assignments section of Blackboard, and then, if I wanted, follow up the next day by posting the same worksheet with answers under the course documents section.
Blackboard enables grades to be posted. Students, since they must log in with identifiing information, have access only to their own grades. However I made only tentative attempts to use this capability of Blackboard. I could not find an efficient way to enter the grades, compared to entering data on a spreadsheet, and, even more importantly, I could not find a way to have the grades weighted appropriately. I am not sure if these are really limitations of the system or if I just never had the time to fully learn to use the system.
A course managment system, such as Blackboard, has both advantages and disadvantages compared to simply having a web site, as I had in SCSU. I don’t know which I prefer, but I can’t imagine teaching without one or the other now.
In spite of the many peculiarities of what we do here, most days are routine. Class time is spent in going over homework and explaining new topics. I continue to think that the most important part of teaching math at the college level is to give careful and thorough explanations. I continue to think that the next most important part of teaching math is to assign and promptly grade well chosen problems for homework.