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.
I freely admit that this is a difficult question for the students that are being asked to answer it. In order to find a solution, they need to be able to apply properties of real numbers (such as distributivity, commutativity, and associativity), correctly manipulate variable and constant terms, recognize a contradiction, and remember the correct notation for dealing with said contradiction.
What surprised me was how many of my students came very, very close to getting the correct answer, yet stumbled on the last little leap of logic. Nearly every student managed the algebraic manipulations, and ended up with the result \(-18 = 10\). It was the last little step from there that proved to be the most difficult part of the question.
In general, I saw two types of wrong answers: some students reported that the solution was \((-10,18)\), while others gave me solutions of the form \(x = 28\) or \(x=-9/5\). Both of these answers are understandable, and I think that they each give some insight into how the students were thinking.
The first incorrect result can be addressed by giving a little context. The lecture material that went over solving equations was from last week. My students did have the weekend to work on homework problems related to solving linear equations in a single variable, but the last time they saw the material in lecture was last Thursday. By contrast, the lecture material leading up to the quiz was on linear equations in two variables. Solutions to such equations are ordered pairs of numbers, with one number representing an \(x\)-value, and the other number representing a \(y\)-value.
When the students who gave ordered pairs as solutions were confronted by a contradiction, they knew that something special needed to happen, but they couldn’t dredge up last week’s lecture. Very understandably, they groped for the best they could come up with, and created an ordered pair. My hope is that his mistake will be a very easy one to correct—a few examples in class before moving onto the next section should clear up that confusion.
The other answer, I think, stems from an over-adherence to a procedure that the book emphasizes, and which I taught in lecture. The strategy is to first simplify each side of the equation independently, then collect terms, and finally to perform a division to get a solution.
This works well when there is a single solution, but, as I attempted to point out in class, there are pitfalls. Specifically, if the equation has either no solutions or infinite solutions, it breaks down somewhere. I think that many students simply applied the procedure without thought, and tried to either collect the constant terms, or divide the constant terms. In either event, they took the result to be their answer.
In order to correct these kinds of errors, I think I need to make two points to my students. The first is simply one of notation: generally, when I talk about solving equations, I talk about terms “canceling” each other. For instance, in the equation \(4x+1 = 3x-1\), I might subtract \(3x\) from each side. Instead of explicitly writing the that \(3x-3x = 0\) on the right side, I will just cross out the terms. I know that there is an “invisible zero” there, but it seems that my students need the explicit reminder.
Second, I need to emphasize the special cases of identity and contradiction a bit more. These cases disrupt the algorithm that the students are attempting to apply, and need to be recognized. If the variable terms cancel each other out, we’re done! What remains is either a contradiction or an identity.
Hopefully, it will be possible to correct those mistakes, and to emphasize that the goal of solving equations is not to just follow a procedure to an answer (both mistakes stem, I think, from blindly following a list of steps), but to find values for the variable which make the equation true. The procedure is just a means to an end, and we have to understand what the various kinds of results that we get mean.