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If it's acceptable practice on math.se, I'd like to really only ask this question of math educators as opposed to students or mathematical researchers. Some researchers will undoubtedly think the whole question is not worthy of consideration, and some students will not understand the issue. But if you really have something to contribute, please do.

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 they are emphatic that "$x=1$" or "The solution to the equation is that $x=1$" cannot count as appropriate responses. Answers like these don't get full credit on their exams.

This matters because we are creating a library of higher quality online problems (for WeBWorK) and we need to set a standard for how the solutions should be entered. I am opposed to having answer blanks where the student merely enters a number. So for example I am opposed to having something like "The solution is __." appear on screen and the student only fills in the number. This trains the student that those words don't matter; they don't have to write them and they won't generally pay them any attention. We could program the question to understand a whole sentence, but there are too many issues with alternatively worded correct sentences, not to mention poor spelling. Remember, these responses are to be automatically evaluated.

I support making the students have to enter "$x=1$", because it is a whole statement. And if they enter something simpler, like "$1$", it is an easy matter to shoot back an automated message that their response is not the form we are looking for. And I counter the idea that "$x=1$" has to be interpreted as an "equivalent equation" by saying that sometimes "$x=1$" is an assertion (aka an assignment) rather than an equation to solve; I'm asserting that $x$ has to equal $1$ for the equation to be true.

So my question to this community, assuming you support my position, is: can you help me make better arguments for my case? I suppose to stay within math.se guidelines, I will say that I am open to being just plain wrong, and definitive mathematical vocabulary expertise can be used to prove it.

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A brief comment not on your question, but on your opening disclaimer: I don't think your declaration that you "only ask...of math educators as opposed to students or mathematical researchers" is helpful or constructive. A big part of Math.SE and MO is that mathematicians, despite all their differences, comprise one community, and it is that unity which makes the sites so powerful. And as a side note, I'd have a hard time finding a mathematical researcher (especially one active here) who is not also an educator. –  bgammage Jul 16 '13 at 3:13
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Also a comment on the question itself: what would your opponents in this dispute require as an answer to the question "Find the solution to 5x+3y=2,2x-y=1"? –  bgammage Jul 16 '13 at 3:15
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I think you can have it one way or the other but not both: either automated grading or grading based on ability to communicate mathematically. I also disagree with your colleagues: formalities like solution sets only make sense given a formal treatment of polynomials, which I don't think you're likely to be dealing with in that context. Note: I'm a mere student. –  dfeuer Jul 16 '13 at 3:16
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If I were asked to solve that equation, my answer would be "$x=1$". It is, indeed, as your colleagues claim, a statement equivalent to the given equation. And that's what I mean by "solving" an equation: Find an equivalent statement that explicitly tells what the values of the unknowns are. –  Andreas Blass Jul 16 '13 at 3:27
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@dfeuer Online homework should never completely replace other homework. But an intelligent system like WeBWorK can give pretty decent feedback immediately instead of after a week. And if wrong, the student is free to keep at it rather than have a permanent poor mark. Also, this is the only way to hold students accountable for the appropriate number of exercises when you don't have TA's or something to properly grade it all. –  alex.jordan Jul 16 '13 at 3:40

4 Answers 4

To put it bluntly, those of your colleagues who don’t accept ‘$x=1$’ as a solution don’t speak English. That sort of hyperpedantry accomplishes nothing beyond making students think that mathematics is all about invoking the right (incomprehensible) magic formula(s). I’ll go so far as to say that I think it excessive pedantry to object to $x=\frac12\left(1\pm\sqrt5\right)$ as an answer to a question asking for the solution(s) to $x^2-x-1=0$.

It would be another matter if the question were written The solution set is __: that question clearly calls for a set. In that case, however, the script should accept $\{1\}$, $\{x\in\Bbb R:x=1\}$, $\{x:x=1\}$, $[1,1]$, and any other reasonably straightforward variant, but not $x=1$, $1$, or the like.

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+1 for "[Pedantry] accomplishes nothing beyond making students think that mathematics is all about invoking the right (incomprehensible) magic formula(s)." –  Eric Stucky Jul 16 '13 at 3:24
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Agreed. It falls on deaf ears when I point out that this distinction between "equivalent equations" and solutions to equations is something that didn't even occur to them until years after they had earned higher level degrees. Why do some try to teach the highest degrees of subtlety to entry-level students? When do they forget that they became expert mathematicians and didn't learn all that subtlety until long after they themselves had started out? –  alex.jordan Jul 16 '13 at 3:32
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I agree. I think this kind of excessive nitpicking detracts and discourages far far more than it helps, and is part of the reason students struggle so much with mathematics early on. Instructors at my previous university did this kind of thing all the time and it irritated the hell out of me. The students don't even understand how to solve the equation let alone how the answer should be expressed to be "pedantically correct." Who comes up with this %&@#$?? –  icurays1 Jul 16 '13 at 3:47
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In my experience, the types of individuals who insist on this type of semantic minutiae are individuals who try to make themselves feel smarter by distancing themselves from the "mean" through pedantry, rather than through accomplishment. –  Arkamis Jul 16 '13 at 4:42
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@Arkamis: Sometimes it’s a misguided attempt to be sure of doing no harm or an exaggerated case of put their feet on the right track now, and maybe they’ll stay there. –  Brian M. Scott Jul 16 '13 at 4:44

If they are happy with "The solution set is $\{x\mid x=1\}$", they should also be happy with $\{x\mid 2x+3=6-x\}$" which is equivalent.

If the above $x=1$ answer is not acceptable for some of your collegues, but many students are doing it as I guess it's happening, they should first think about what they did wrong: in this case write a question stating clearly that they want the student performing two tasks. Putting the equation in some canonical form with the variable isolated on the left side, secondly states that they know that the result is a set. Outside pedantic mathematic, the second part is irrelevant, and it is disputable that may even be seen as a secondary goal of the exercise.

Which student could believe a corrector is so dumb the corrector may ignore that the result is a set ? If The result is say $x = 11$, should the student also state that he is using base 10 arithmetic ? If the base could possibly be binary it could be ambiguous.

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Your example here does indeed help strengthen my case, which is what I was hoping for. Thanks for that! –  alex.jordan Jul 16 '13 at 22:42

A lot of the previous answers seem to have overlooked the importance of making it clear to the user what format the answer needs to be in. If you just have a blank which will accept only "x=1", a lot of students will correctly solve the equation, enter "1" or {1} as their answer, and become very frustrated when the system doesn't accept it. By attempting to force students to enter an answer in a specific (unspecified) format, you are not only engaging in the same sort of "hyperpedantry" you want to avoid, but also causing a lot of unnecessary frustration.

After having tutored a lot of students who had to use this sort of online homework system, it is my belief that the best system is one in which the amount of information that the student has to enter is minimal, to ensure that they get credit for correctly solving the equation. Hence I would recommend the text "x=" followed by a blank, which accepts the answer "1". Perhaps even put reminder text which says something like "enter a single number".

Automated problem checkers are not the best venue for testing pedantic sort of things, e.g. the format of answers (as you describe), whether the student remembers a "+C" in integral calculus, whether the student remembers to include units, etc. They should be used only to assess whether the student can actually solve the problem.

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so true. And if you want to remind students solutions are a set just put brackets in the hardcoded answer form and write the numbers inside it. –  kriss Jul 16 '13 at 14:21
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In the past I would agree with your first point. But you may not be familiar with WeBWorK and how fancy it is. Properly written problems are coded so that all these these things are genuinely understood by the software as mathematical objects. You have Davide Cervone to thank for that, the same programmer responsible for MathJax on sites like this. –  alex.jordan Jul 16 '13 at 14:34
    
For example, in the current version of these problems, "x=1" would be the official answer. If the student types "1" or "{1}", they are told that "they have the right value, but please answer in the form x=__". If they type something incorrect like "2", they are told "please answer in the form x=__". If they type something that is in the wrong ballpark like "x^2" they are told "you are answering with a formula but this problem asks for a solution to an equation." Of course, we could make it so "x=1", "1", "{1}", "{x|x=1}" are all truly understood and accepted, and that is the debate here. –  alex.jordan Jul 16 '13 at 14:37
    
My philosophy in writing these is to make the experience more truly simulate pencil-and-paper. Extra special instructions about formatting detract from that, and are unnecessary given WeBWorK's feedback system. Also, providing "x = " before an answer blank is not true to the pencil-and-paper experience to my mind either. –  alex.jordan Jul 16 '13 at 14:40
    
Last comment: WeBWorK handles the "+C" thing with integrals beautifully. If you omit it, or replace it with a specific number, you are told you do not have the most general solution. Even more neat: there's no requirement to use "C". You could add "k", "c", "K", or any symbol for a constant that doesn't already have meaning in the context. And units are handled beautifully too. If you are asked how long is a meter, you can respond "1 m", "100 cm", "39.37 in", etc. If you leave out units, you are told that you have left off units. –  alex.jordan Jul 16 '13 at 14:42

This question reminds me of how I could be given the exact same word problem in Grade 2 and again in Grade 7. The difference in getting the same problem is that what is required to justify my answer is wildly different between these grades. In Grade 2, it is enough to just work out the number that is the right answer that a teacher accepts without challenge while in Grade 7 the same problem requires stating a variable, formulating an equation and solving it, not unlike what you have as a given problem here. There is something to be said for understanding how to justify an answer which could be a proof, counter-example, or something else to demonstrate an understanding of a concept and its use on some level.

Thus, I'd be tempted to consider what kind of formality are you expecting and would you want to be feeding someone the ideas of how to go through these kinds of problems. Thus, I'd suggest bringing in Mathematical constructs and setting a bar for what kinds of answers are reasonable. Otherwise you may run into people that want to claim what may otherwise seem ridiculous.

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