How to respond to “solve this equation” in a basic algebra class

Question

I asked this question once on math.se, but don't follow the link unless you want to risk biasing your own response: https://math.stackexchange.com/questions/444696/how-to-respond-to-solve-this-equation-in-a-basic-algebra-class

It is really a question that belongs here though. Maybe I posted it there before I knew about this site. I'd like to see responses from this community. So here it is, restated, somewhat edited.

Imagine yourself teaching a basic algebra class: maybe to grade schoolers or to high schoolers, or in my case, to adults ages 18 and up in a community college. You will encounter "problems" like the following, where for now I am intentionally leaving out words:

$$2x+3=6−x$$

The "answer" to a question like this somehow communicates that $1$ is the only solution, that $x$ needs to equal $1$, that the solution set is $\{1\}$, or $\{x\mid x=1\}$, etc.

Some of my colleagues feel that if the task was to "solve this equation", that "$x=1$" is not an acceptable final response from a student. They say that "$x=1$" is an "equivalent equation" to the original equation, because it has the same solution set. They say that to "solve this equation", to the exclusion of other ways a student might respond, is to write a set as part of an English statement. They are happy with: "The solution is $1$", "The solution set is $\{1\}$", or "The solution set is $\{x\mid x=1\}$". But to them, "$x=1$" or "The solution to the equation is that $x=1$" cannot count as fully correct responses. Answers like these may not get full credit on their exams, and always get commentary on written homework. These colleagues would say the student has not learned the difference between an "equivalent equation" and the solution.

I support answers of the form "$x=1$", even better as "The solution to the equation is that $x=1$", because it is a whole statement that explicitly states what is equal to $1$, instead of relying on the implication that since $x$ is the only variable, that is what we must intend for the $1$ to take the place of. I counter the idea that "$x=1$" has to be interpreted as an "equivalent equation" by saying that sometimes "$x=1$" is an assertion/assignment rather than an equation; I'm asserting that $x$ has to equal $1$ for the equation to be true. I point to any number of programming languages, where you have one symbol for testing equality, and another for assigning a value (sometimes "==" and "=", sometimes "=" and ":=") so these are two different ideas but mathematics tends to overload them onto one symbol, "=".

So my question to this community is how do you feel about formatting answers for questions like this?

This has important (to me) implications. For one, I code problems for WeBWorK, and I strive to make the experience simulate the pencil-and-paper experience with feedback as close as possible to how a human teacher/grader would respond. I can do a lot with responding conditionally to this format or that, but I need to know what formatting people want most. Second, I am working on a massive OER for content at this level (draft, ever in progress), and its expository sections and WeBWorK homework should be consistent with each other and format answers the "right" way. It's most important that I get my own colleagues to be OK with how this OER treats this issue, but I also have an eye toward the broader community.

Answer 1

This is a really interesting question, because similar issues---the question of how demanding to be about formatting of answers---come up a lot, at all levels, and the answers aren't always straightforward.

In this case I think your colleagues are straightforwardly wrong, for the following reasons:

• They're wrong about what students who write "x=1" think. This is easily testable: if students actually can't tell the difference between an equivalent equation and a solution, they should think that, say, $x=3-2$ is an equally acceptable solution. I'm comfortable predicting that they don't. In fact, the notation "x=1" probably mostly means "1 is the solution" to them, and they'd have to stop to think about the fact that it's also an algebraic equation they could manipulate.
• They're wrong about what most students who write "the solution set is {1}" think. It's very unlikely that students who are writing that are actually expressing a meaningfully different thought than the ones who don't; it's much more likely that they've memorized the particular format that this teacher is demanding.
• They're wrong about convention. Writing "the solution is x=1" is standard throughout mathematics, and students will see it in later classes (including weirder things like "x=2,-3" as the solution to a quadratic, which, while notationally weird, is easily understood and widely used, even if "$x\in \{2,-3\}$" would theoretically be preferred).

Putting those last two together means that I think they're actively doing harm by being doctrinaire about this. They're not wrong that these are different concepts, but they're probably not succeeding at teaching students to distinguish them. Instead, what the students are probably learning is that math is mostly about doing arbitrary steps to please their teacher, and that you can get a problem totally right and lose points anyway because your teacher made up some rules. (It doesn't help that, in this case, the rules really are made up, in that no one else, including other mathematicians, actually follows them.)

Answer 2

Ask your pedantic colleagues to reconcile their formatting expectations with the context

"solve $F=m\cdot a$ for $a$".

Would they prefer to see just $\frac{F}{m}$, or maybe this expression jazzed up with set brackets somehow? This formatting of the answer looks jarring to me. I think most educators and scientists expect to see the answer presented as $a=\frac{F}{m}$.

Answer 3

I was never a professional mathematician, and I am no longer a professional educator. I hope this answer isn't too far off the mark.

I think the problem is the expression "solve the equation." I simply would never use that expression in a basic algebra class. I didn't come across it myself until using differential equations in college. (Before that I would have physics professors as us to "solve the equation" in class, and that expression left the class scratching their heads. Solve it in what way?)

At least from my experience, such a request was always to "solve the equation for x."

Having said that, if your colleagues or you wish to use the expression "solve the equation," then the trick is to clearly and unambiguously define what that means. That will almost certainly require a compare and contrast to the more explicit "solve for x" request. As long as you define your terms, and don't expect the students to just intuit the distinction, then the rest is a matter of how pedantic you want to be about "standard form."

For students that are still learning basic algebra of linear equations, I have a feeling that the distinction between "a solution set with one member" and "an equivalent equation" may be stretching their ability to take on too much abstraction at once. A better time might be after they have mastered quadratic equations. Then the distinction between a solution set, and a particular solution, might make more sense to them.

Answer 4

These "colleagues" carry pedantry to asinine levels. How many of your kids could/would "solve" the problem: $x = 1$?

There are a lot of problems. First is the expression of the problem isn't explicit. It doesn't state what domain x may come from. It's not necessarily true that x=1 even has a solution. But I see little value in teaching this subtle point to adult learners (unless they are going quite a bit further in their math education than algebra and differential calculus). I suspect that some of this "theory" is taught in K-12 education, but for adult learners? Why?!

When someone gives me a test which says "Solve the following equations" and then there are a list of equations, it does not require the explicit directions for me to reduce the equations to such a form where the values, if any, for which the equations are true are made explicit. (Technically, one doesn't "solve" an equation, one solves a problem (and the problem may be to determine the values for which the equation is true, but may also include other things, such as domain, range, etc.) So, would it hurt to explain this? I don't think so, but I'd bet it will be useless for 99% of your students.

Which gets me back to the question that you didn't address, but requires an answer prior to answering your question: what is the purpose of not only your class, but the educational tracks these students are on? Is learning silly pedantry concerning formal mathematics likely to be useful? It seem to me so blindingly obvious that this is a complete waste of time and energy, and demotivating as well, as to beg the question: do you really have to ask?