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This recent question reminded me of a question: this year several students expressed concern about the expression $\sqrt{-x}$, on the grounds that it must be undefined because $-x$ is a negative number, and negative numbers don't have square roots. This is in the context of a calculus class for first-year college students, so these are students who ostensibly know algebra (and most of whom have taken a year of calculus already). (In the context of this class, characterizing negative numbers as not having square roots is correct; they know about complex numbers, and also know they're off-topic.)

My experience so far has been that these students can handle it after being reminded that it's possible that $x=-2$, so I can address it in an ad hoc way when it comes up, but I suspect it indicates a bigger conceptual gap. So my question is:

  1. Why do students think this?
  2. What algebra should I be reviewing to head this off before the specific case comes up?
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    $\begingroup$ I think they always assume $x$ to be nonnegative inside a square root, despite the fact that what's correct is that what is inside the square root should be nonnegative. $\endgroup$ – Mark Fantini May 5 '15 at 20:16
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    $\begingroup$ If students are coming in a bit rusty, then you could start off the semester with a simple Always True / Never True / Sometimes True list of statements (where "Sometimes True" requires specifying when it is true of, e.g., real numbers -- a non-rigorous development of $\mathbb{R}$ ought not to be a big deal). Examples might be statements such as $x = -x$ or $|x| = |-x|$ or $\sqrt{x}$ is well-defined or $\sqrt{-x}$ is well-defined or $x^2 + 1 = 0$ or the latter set of examples I left at MESE 1403. $\endgroup$ – Benjamin Dickman May 6 '15 at 2:28
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    $\begingroup$ For me, this connects to another recent question - language gets in the way. Most students will read -x as "negative x", which is hard not to think of as negative. These days, I often read it as "the opposite of x". $\endgroup$ – Sue VanHattum May 6 '15 at 3:49
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    $\begingroup$ Interestingly @SueVanHattum, many students here in Australia are taught to say "negative 2" for "-2" with dire injunctions to never say "minus 2". So it is indeed natural for them to think of -x as negative. $\endgroup$ – DavidButlerUofA May 6 '15 at 6:53
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    $\begingroup$ @JpMcCarthy: Yes, in $\mathbb{R}$, $\text{negative}(x) = \text{minus}(0,x)$ and $\text{minus}(x,y) = x + \text{negative}(y)$, so each can be recovered from the other in $\mathbb{R}$. As above, this can fail in other algebraic structures, such as $\mathbb{N}$. And Serge Lang would not let you forget it. :) $\endgroup$ – Michael Joyce Nov 17 '15 at 13:51
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Why do they think this?

I can think of a few intertwined reasons why the students think that $-x$ is negative.

Firstly, psychologically the $-$ sign has a very strong pull towards thinking of subtraction and negative numbers. When they were small, your students had more than five years to get used to the idea that $-$ meant subtraction and that subtraction always made things smaller. Then on top of that they probably had a few years to connect $-$ to negative numbers before they were ever introduced to letters representing numbers. Those instincts were forged very strongly and are still stronger than anything that came after, so when they see $-x$ it is natural for them to think of negative numbers.

Even more than that, their first and most strongly-forged experience of numbers is the counting numbers, and that will always be the first thing they think of unless specifically directed to think otherwise. If you ask any person -- even a mathematician -- to name three numbers I daresay they will pick three positive whole numbers. They won't even pick 0. In fact, in my experience, 0 takes up a very small amount of mental space and is often forgotten as an option in all sorts of situations.

Secondly, over in other courses like economics and chemistry, variables usually do represent positive amounts and graphs only exist in the first quadrant, so it is not unnatural for that thinking to transfer over to maths.

Finally and most importantly, I don't think that your students think of the $x$ as standing for a range of possible numbers at all most of the time. I'm pretty sure you don't either. If you really analyse your own thinking, I'm pretty sure you find that when doing algebra, you mostly think of $x$ as an object of its own kind that moves around the page, rather than a number at all. You can substitute numbers in its place, sure, and make a statement that it is equal to a number, sure, but they are not the same as actually conceiving of it as a number at all times.

What can you do to prevent it?

It would probably be a good idea to review the classic functions and their behaviour early in your calculus course -- you probably do this anyway. What you can do to help yourself later on is to focus explicitly on reminding students that $x$ (or y or t or whatever) represent various numbers and that one way to understand a function is to ask yourself about what happens when $x$ is various possibilities. Basically this is considering the domain and range of the function, which you probably also discuss in your calculus course anyway. You can tell them that domain and range are not just information about the function that they need to be able to find for tests, they are a fundamental way of understanding functions.

As the course progresses, reinforce this idea: every time you write down a function, take a moment to consider the various x-values that could go in and the various function values that come out. (I'm not saying calculate the domain and range formally, just take a moment to think about it aloud.) I recommend asking "Can I put in 0? Can I put in positive numbers? Can I put in negative numbers? What about very small and very big numbers?" If they are used to this happening every time, they are more likely to do it themselves and come up with the idea that $\sqrt{-x}$ is ok by subbing in $-2$ on their own.

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    $\begingroup$ This is exactly the sort of idea I was looking for. Tying the topic back to domain and range connects it with other things in the course, and thinking about the domain of x regularly is the sort of useful skill I'd like students to make progress towards learning. $\endgroup$ – Henry Towsner May 7 '15 at 11:59
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The following "definition" given in the Glossary of the Common Core State Standards for Mathematics (2010) give you an idea why students think this:

Integer. A number expressible in the form a or – a for some whole number a.

The problem is not about algebra alone, rather it is about disconnection of teaching of algebra and negative numbers in schools. It has also a long history in which some great mathematicians of the past knowingly or unknowingly avoided negative values (particularly, as input) for an algebraic letter. The following is an interesting passage from Euler (taken from an under review paper of mine that is about the exact same issue you asked).

The question which we have hitherto considered lead all to an equation of the form $ax+ by=c$, in $a$,$b$, and $c$, represent integer and positive numbers, and in which the values of $x$ and $y$ must likewise be integer and positive. Now, if $b$ is negative, and the equation has the form $ax-by=c$, we have questions of quite a different kind, admitting an infinite number of answers.

At the first glance, it seems Euler is being careless when he is saying "if $b$ is negative, and the equation has the form $ax-by=c$..."; if $b$ is negative in the sense that we understand it today, the equation $ax-by=c$ would be like the equation $ax+dy=c$, in which $a$,$d$, and $c$, represent integer and positive numbers, and Euler has already considered such an equation. But, Euler is not neglectful: when he is saying that "if $b$ is negative", he uses the letter $b$ as a label denoting the coefficient of $y$, and then the letter $b$ in the equation $ax-by=c$ is a parameter that represents integer and positive numbers.

If there is a way to bypass a negative number as an input of a letter, people find it and use it! Thus, a short answer to your second question would be to review the algebraic nature of negative numbers.

Added. The following Figure taken from a paper of Warwick Sawyer, The Importance of the Unbelievable, beautifully shows the importance (though often taken for granted) role of negative numbers in our most basic algebraic conceptions.I often use it in my classes to bring the unifying nature of accepting negative numbers to the fore.

enter image description here

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  • $\begingroup$ Sorry to come back to this 3.5 years later, but is this the paper you are referring to in your answer? Making Associativity Operational, International Journal of Science and Mathematics Education, 15 :1559-1577 $\endgroup$ – Chris Cunningham Nov 29 '18 at 17:46
  • $\begingroup$ @ChrisCunningham No. I stopped working on that paper for some reason. However, I have just recently come back to it and I would be more than happy to send its first draft to you in one week or two. $\endgroup$ – Amir Asghari Mar 17 at 17:01
  • $\begingroup$ I am still interested! But you do not have to spend any special effort on me. Whenever there is a way for me to access it, drop it here. Thanks! $\endgroup$ – Chris Cunningham Mar 18 at 13:22
  • $\begingroup$ @ChrisCunningham It is on its way :) $\endgroup$ – Amir Asghari Mar 18 at 15:27
  • $\begingroup$ @ChrisCunningham dropped you an email with the paper. $\endgroup$ – Amir Asghari Mar 22 at 20:21
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I think it's appropriate to lead the student(s) through answering their own objections with some leading questions - perhaps questions like

"When is $-x$ negative?"

In particular it helps reinforce to students that this is the sort of question they should be asking themselves.

Why do students think this?

I think it's because they're applying a "rule" that says something vaguely like "a minus in front of a number makes it negative" ... but it's an internalized idea they haven't articulated so they don't really notice the flaws in it.

Running into these sorts of errors is in some sense part of learning to think mathematically; it's okay to make these errors, as long as they're treated as something to recognize, and try to address.

It's no good drilling someone for weeks on how not to fall off a bike -- they're still going to fall off when they're learning the skill. You can try to anticipate some things, but I don't think you should try too hard to make sure they never make an error. Errors are useful!

I've been doing mathematics for a long time; I make new mistakes almost every time I learn something new. Being able to identify and correct misapprehensions is part of getting there. I make fewer now than I did 30 or 40 years ago, but that's because I made enough errors to finally internalize things like "definitions matter".

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Instead of thinking about $-x$, consider the expression $(0-x)$.

Having (hopefully) being exposed to expressions of the form $(a-b)$ for a long time, where $a$ and $b$ can be arbitrary reals, there's no reason to assume that the value of the expression is positive (or negative) without further information.

Therefore, when they see $\sqrt{-x}$, they can now think of $\sqrt{-x}$ $\equiv$ $\sqrt{(0-x)}$, and realise that there is a real root where $x$ has a negative value.

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I'd conjecture that classes have not given sufficient time to the specific translation skill between the English and Algebra languages, especially the meaning of relation symbols like equals (=). Two days ago I had to spend a few minutes clarifying this in my College Algebra class.

  • Q: "Algebraically, how would one say that $c$ is a negative number?"
  • A: "$-c$" (this from one of my best students)

This brings up a few problems. Primarily, this response isn't any kind of assertive statement; they fail to intuit relational symbols (=, <, >, etc.) as the necessary verbs in a sentence no matter how much I say, drill, or grade on it (obviously, "$-c$" is merely an expression, equivalent to a sentence fragment). Generally when I first meet students they place or omit equals signs pretty randomly on the page, as if it's mostly a decorative element, and are frequently bewildered at my grading for this on tests.

When I write that the English "c is a positive number" is equivalent to $c > 0$, then most of my students can correctly answer that "c is a negative number" is equivalent to $c < 0$. For at least one weaker student I had to go over this again after class, but she did seem to have an "aha" moment after I wrote down an explicit table (English vs. Algebra columns).

In summary: Practically none of my students can intuit that "c is a negative number" would be written $c < 0$ without some direct instruction on the issue. Without that, their first guess is indeed that $-c$ has that meaning.

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    $\begingroup$ Your answer is the only one that hit the problem on the head. It is that students are often not taught to think and express themselves logically, and has nothing to do with how they learnt about negative numbers or "$-x$", since the same kind of logical error repeats itself in all other areas of mathematics. This is why type checking should be enforced, not just emphasized. "$c$" without any context is ill-formed. In the context that $c$ is a real number then "$-c$" has type "real number", not "boolean", so it is ill-formed as an answer to a yes/no question. $\endgroup$ – user21820 Oct 31 '15 at 2:27
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I tutor many students whose teachers have been telling them to pronounce -x not as "negative x" but instead as "the opposite of x." I assume that this semantic change is aimed at instilling in students the idea that -x need not be a negative number.

You can ask questions such as "what is the opposite of negative 3" and "what is the opposite of 4" to get students used to this kind of thinking before you begin teaching about opposites of variables.

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