We often come up with solutions to problems that lend themselves to various interpretations. Exploring the different interpretations can be useful, as such exploration has the potential to provide deeper insight into the problem at hand, or lead to an extension of the result to an entirely different class of problems. A particularly gruesome example came up this week in a course that I am taking.
One of the great joys of mathematics—and one of the things that I think is often lost in lower level math classes—is that problems, especially the interesting ones, rarely have a single solution. When a given question is asked, there may be many routes to a single correct answer, or there may be a multitude of correct formulations and answers.
As a brief example, let’s consider the homework assignment for an upper division applied math course that I just finished grading. One of the problems on this assignment presents a production situation: a dairy buys two kinds of milk, which it processes and uses to make two kinds of cheese. These two cheeses have differing production costs and sell for different amounts, and there are a plethora of other details that constrain the precise mixture of cheeses that the dairy should produce and sell. The homework problem asks how the dairy can maximize its profits given all of the constraints presented.
For whatever reason, I have been unable to get into the student database on campus. What this means is that I have no access to up-to-date rosters for my algebra classes. Because I do not know who is still taking the class and who has dropped it, I have tried to be careful about keeping track of grades and whatnot, even for students who (for instance) never show up to class or missed the first exam.
In general, however, I have been marking those students who have missed major assignments as having withdrawn. I had assumed that if they were going to miss exams and such, they probably were not planning on finishing the course. This seems like a good system to me.
Of course, the deadline to drop classes is looming large—any class which is not dropped by the end of next week cannot be dropped without appealing to a higher authority. This means that I have been filling out a fair number of progress reports this week—these are mostly for student athletes and “at risk” students. Much to my surprise, I am seeing the names of many students that I have not seen since the first week of class.
These ghost students were once enrolled in the class, but have disappeared. This is understandable. Many students are not prepared for algebra, and the course has a very low pass rate. What I don’t understand is why these students have not dropped the class. It requires two minutes of mouse clicking to withdraw. I simply do not understand why they are still enrolled…
I am always fascinated to see how students manage to incorrectly answer exam and quiz questions. Not only does this provide a great insight into my own deficiencies as an instructor, but it also gives me some idea about how my students are thinking about and analyzing problems.
Take the quiz question that I gave yesterday in my remedial algebra classes:
Solve the following equation. Give your answer using set notation, and check your solution.
\(6(x-3) = 2x + 2(5+2x)\)
The correct answer is that there is no solution. The equation ultimately reduces to something like \(-18 = 10\), which is a contradiction. The correct interpretation of this result is that no matter what value is substituted for \(x\), the equation will never yield a true statement, which means that there is no solution.
