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I'm not entirely how best to pose this question, so that it fits within the guidelines (so edits / suggestions for modification are warmly welcome).

I'm interested in exploring effective strategies for helping students solidify their basic ability to recognize nonsensical expressions and was curious to know what has worked for others or what others have tried. I'll illustrate the kind of "nonsense" I mean with two examples.

e.g. 1: nonsense commonly recognized by students.

In my experience, most HS and university students working only within the set of real numbers have no trouble recognizing something like $\sqrt{-2}$ as being "meaningless" or "nonsensical" or simply undefined.

e.g. 2: nonsense not always recognized by students.

Again, drawing from experience, I find a surprisingly large number of students who would recognize the above are less likely to notice that "$y = \sqrt{3-x}$ for $x > 1$" doesn't make sense (i.e. assuming one is not using a domain convention in which this would be understood to actually mean "for $1 < x \leq 3$").

N.B.: I'm not interested in focusing on the particular issues with this specific example, I'm interested in any effective strategies others have used for helping students develop the instinct to reflect & question what they are reading, to make sure it makes some sort of sense; i.e. to get them in the habit of thinking about the math they are working with and not just "going through the motions."

Again, I know this is quite fuzzy and subjective; however, it seems like it falls under the acceptable type of subjective questions; happy to edit or remove, if I'm mistaken.

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    $\begingroup$ FYI, here's an example similar to what I saw a couple of years ago on a certain standardized test (not USA) that I was involved with as a consultant, a test in which all numbers on all problems were assumed to be real numbers. Given that $x>0$ and $x^2 + \frac{1}{x^2} = \frac{1}{4},$ what is the value of $x+\frac{1}{x}$? Intended solution: $(x+\frac{1}{x})^2=(x^2+\frac{1}{x^2})+2=\frac{1}{4}+2=\frac{9}{4},$ so $x+\frac{1}{x} = \frac{3}{2}.$ However, note that from $(x-1)^2 \geq 0$ we get $x^2+1\geq 2x,$ or $x+\frac{1}{x}\geq 2$ (when $x>0),$ so how can we have $x+\frac{1}{x}=\frac{3}{2}$? $\endgroup$ Nov 9 '21 at 20:40
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    $\begingroup$ There's (A) syntactic nonsense and incoherence, and there's (B) "exists only in $\mathbb C{\setminus}\mathbb R$", undefined, etc. I refer only to A as nonsense, as B can typically be "plugged". I completely agree with the goal of cultivating the habit of making sense. $\endgroup$
    – ryang
    Nov 10 '21 at 5:41
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    $\begingroup$ @DaveLRenfro I'm curious to know how you noticed the problem with the example you shared? I'm wondering if there is something in the process that you went through that could be extracted into an exercise for students. $\endgroup$
    – Rax Adaam
    Nov 11 '21 at 18:47
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    $\begingroup$ I was curious how difficult it was to explicitly solve for $x$ and use the value(s) to determine the value of $x + \frac{1}{x},$ or to sufficiently estimate $x + \frac{1}{x}$ to identify the correct multiple choice option, each of which would be an unintended solution that could muddy the statistical results for the item. (This is a no-calculator test.) After seeing the implicit use of non-real numbers, I determined for these types of problems the item writers need to ensure that $x + \frac{1}{x}$ is $\leq -2$ or $\geq 2,$ which also allowed us to quickly identify the problematic items. $\endgroup$ Nov 11 '21 at 21:18
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    $\begingroup$ Misusing notation may not be ungrammatical (and abusing notation isn't being incoherent); by (A), I rather meant ungrammatical unintelligible mathematics. On the other hand, nonsense maths isn't all worthless, hehe. $\endgroup$
    – ryang
    Nov 12 '21 at 2:51
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I don't have experience teaching this skill directly, but in my calculus teaching I always tried to develop in students the ability to visualize the graph of a function, given its symbolic expression. With your example I think you want to get students to the point where they immediately picture shape of $\sqrt{3-x}$, which instantly tells them where it doesn't make sense. Exercises in which they are required to sketch a graph without explicit plotting or use of software should develop this skill. This would happen after teaching the shapes of a variety of standard functions and how these shapes are affected by basic algebraic transformations.

In solving equations, getting students in the habit of checking their solutions, and giving them carefully selected examples where standard manipulations (such as squaring both side) lead to spurious solutions, should alert them to the fact that not all manipulations preserve truth value for all values of the variables. In such examples you can then ask them to pinpoint the step where the spurious solution entered, and have them identify why that happened. These are habits, and can only be developed by practice.

As for Dave Renfro's example in the comments, you want students to reach the point where they immediately become uneasy on seeing such a problem. The graph is more complicated than in your example, but students should be able to notice that $x^2+\frac{1}{x^2}$ is large both when $x$ is close to $0$ and when $x$ is far from $0$. So if a solution to $x^2+\frac{1}{x^2}=\frac{1}{4}$ exists, it can only lie in some delicately chosen sweet spot. But they should quickly come to suspect that no such sweet spot exists. You could try guiding their thinking as follows: ask if they can use $x$ real, $x^2+\frac{1}{x^2}=\frac{1}{4}$, and basic knowledge of algebra to put bounds on the two terms separately, with the goal being to get them to see that both terms have to lie between $0$ and $\frac{1}{4}$. Also try to get them to notice that the two terms are reciprocals of each other. So if one term is less than $\frac{1}{4}$, what can they say about the other?

I think that the common element in these two examples is that potential problems are recognized when students see an algebraic expression as representing a computational process. In $\sqrt{3-x}$, a square root is going to be taken, and students know that there will be a problem if the radicand is negative. So what is the radicand? Well, $3-x$. So when is that negative? The equation $x^2+\frac{1}{x^2}=\frac{1}{4}$ tells me that two (clearly positive) numbers add to $\frac{1}{4}$. So both numbers are $\le\frac{1}{4}$. But they are reciprocals of each other. Can two reciprocals both be less than $\frac{1}{4}$?

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  • $\begingroup$ Re: intended statement: no, I meant what is written, there. The (real-valued) root wouldn't be defined for $x > 3$ -- I was meaning that some students won't even question whether the stated domain makes sense (in my experience, students generally ignore domains altogether and are not really aware of domain conventions). Thank you for your thoughtful reply! $\endgroup$
    – Rax Adaam
    Nov 13 '21 at 23:11
  • $\begingroup$ Thanks for explaining that. I should have paid more attention to the quotation marks, which, now that I reread that statement, do make things clear. $\endgroup$ Nov 14 '21 at 14:23
  • $\begingroup$ examples where standard manipulations (such as squaring both side) lead to spurious solutions, should alert them to the fact that not all manipulations preserve sense. It's not that squaring both sides creates nonsense (i.e., is invalid), but that solving an equation (saying that $x=\alpha$ satisfies the equation) ultimately is about asserting that when $x=\alpha,$ the given equation is true; so, just a chain of forward implications is an incomplete argument that one has solved the equation. @RaxAdaam $\endgroup$
    – ryang
    Nov 16 '21 at 16:53
  • $\begingroup$ Good point. I've modified the wording, but the result is a bit awkward. $\endgroup$ Nov 16 '21 at 17:33
  • $\begingroup$ not all manipulations preserve truth value for all values of the variables Haha yes awkward, and still not quite accurate: the point is that the squaring step, being a forward- rather than a bi-implication, is rigorously understood as creating merely candidate solutions; as such, there is no invalidity or introduction of falsity, since only after the extraneous solutions (if any) are subsequently filtered out will the remaining solutions be claimed as actual solutions. $\endgroup$
    – ryang
    Nov 16 '21 at 18:14
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Students at the high school level expect to see examples somewhat similar to test questions. This is also true to some extent at the first/second year undergrad level as well. Therefore, you should clearly give examples that highlight domain restrictions or extraneous roots. These should be accompanied with a graph of the function.

It's also a good idea to always do a "reality-check" at the end of questions you do as teaching examples. For example if the question asks for number of people, ask the class if "23.6 persons" or $17.82343 is the final answer. When solving an equation quickly plug the answer back into the equation to show that it works.

Then when you get to one with an extraneous solution, plug answer back as usual but pause right before attempting to divide by 0 or evaluate square root of negative numbers and ask class "I think something's wrong. Can someone help me out?". A little bit of drama and overreaction may help keep students engaged thus more likely for them to remember to spot "nonsense". As usual homework questions should be curated to reinforce this habit of verifying the solutions.

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