How to help students understand/remember that $x^2 = a$ has two solutions?

Question

I teach math in university, in France. This semester I have first-year bachelor students. I am becoming increasingly annoyed that they cannot remember the simple fact that $x^2 = a$ has two solutions for positive $a$, $x = \sqrt{a}$ and $x = -\sqrt{a}$.

I don't know what to do. I've told them several time during lectures and exercise sessions. We had a whole class on determining square roots of complex numbers. And yet, I'm grading partial exams right now, and about half of them still wrote $x^2 = 3/2 \iff x = \sqrt{3/2}$ when trying to find out the domain of definition of a rational function (assuming they wrote the "iff" sign at all, but it's a different question). And I have the nagging suspicion that if the equation was written as $x^2 - 3/2 = 0$, they would realize it's a quadratic equation that has two solutions (some of them do solve the question this way and painstakingly find $x = \pm \sqrt{6}/2$).

What can I do?

Answer 1

There are possibly two different issues here.

Issue 1 is that some students are under the mistaken impression that the symbol $\sqrt{5}$ actually designates two different numbers, one positive and one negative. I have even heard high school teachers (in the United States) tell students this; it seems to be a very pervasive belief and you can find many examples of it on MSE. If this is the case, then the problem is not that they don't know that there are two solutions, it's that they don't know the correct way to denote it. Notation, of course, is just a convention, so this would be more of a failure to adapt to the common culture of mathematical writing than it would be a failure of comprehension.

One way to identify if this is, in fact, what's happening would be to give an example in which the square root is a simple whole number. If you give the equation $x^2=9$, would they say $x=3$ or $x=\pm 3$? If the issue is purely notation -- that is, if the problem is that they think the $\pm$ symbol is already "built in" to the radical sign -- then they will include it explicitly when the square root can be evaluated exactly, and drop it when the square root cannot be completely evaluated.

The second issue is this:

I've told them several time during lectures and exercise sessions.

Maybe the problem is that you're telling them during exercise sessions. I don't know what the format of your exercise sessions is -- in particular I don't know whether you're presenting the solutions, or having students present solutions -- but if the students are the ones showing their work, and such a mistake occurs, you can stop and ask "Are you sure that's right?" In my experience this is even more effective when the answer is extremely simple. I have had Calculus students go from $x^2=25$ to $x=5$; asking "Are you sure that's right?" typically catches them off-guard, precisely because of course they're sure that $5^2=25$. If that doesn't give them enough of a hint, prod them "Is $5$ the only number whose square is $25$?" However you get them there, the goal should be for them to tell you that there's a second, negative solution. If they figure it out themselves once, they will be much more likely to remember it than if they are told it fifty times.

Answer 2

Telling and explaining is really not enough. Working exclusively in square roots of numbers all but guarantees blocking vs interleaving problems and encourages shallow robotic thinking and generalizing involving roots.

A few ideas:

1. Error analysis. Have them look at a variety of equations that boil down to something of the form $x^n=m$, where $n$ is an even or odd whole number and $m$ is a real number. Show them various (fictitious) students' work of varying qualities. Have them check and assess the work, then discuss. Why do some have two solutions and others have only one? Why do some have more?
2. Multiple representations. Have them graph $y=x^2$ and have them use the graph to show the solution(s), if there are any, to $x^2=9$, $x^2=0$, $x^2=-9$, etc. Then use the graph of $y=x^3$ to solve $x^3=1$, $x^3=8$, $x^3=-8$, etc. Compare and contrast. Encourage them to generalize as to how many solutions the equations will have depending on the degree of the polynomial. Relate this to the error analysis above.
3. Positive incentivizes. Make a list of the 5 or 10 common mistakes that you see in class. Create a bonus quiz where students, on a blank piece of paper, recall all 10 mistakes from memory, explain and/or prove the wrongness in each, and demonstrate and check the right way to think about it. Another positive incentive: Tell them that regardless of the topic the class is on, at least 5% of every assessment will involve one of those 10 mistakes, so they have to keep studying them. Distributed practice makes a huge difference.

4. Negative incentives. Given the list of 5 or 10 most common mistakes has been released, tell them that any mistakes of that sort on the test will result in a zero (or a max score of 50% or something like that) for that portion of the test regardless of all other work.

5. A gentler version of those negative incentives. Only give feedback and grades on work that does not contain those common mistakes. e.g. Write something like this on their quiz. "On question 8, you made one of the common mistakes. Search for it and fix it, then I will grade it. If you choose not to fix it, you get zero for question 8."

Good luck with this!

Answer 3

We are all dealing with this, of course. In a theoretical/philosophical sense, you can't "make" students learn anything ("You can lead a horse to water..."). In a practical/punitive sense, this is what course grades are for.

So, consider one or more of the following options:

• Simply grade exams rigorously on this concept.
• Give a lead-in multiple-choice or short-answer question on the Fundamental Theorem of Algebra.
• Award only half the maximum score for the question if only one of two solutions is given.
• Require in applications where only a positive solution is practically useful (e.g., solving a right triangle) that the mathematical two solutions be expressed (for extra practice), and then a separate natural-language statement using only the positive result.
• For the most radical approach, consider mastery grading; if this particular item is absolutely critical, then do not pass any student from the course until they can successfully answer such a question.

Answer 4

Not a perfect solution, but I would insist on the meaning of $\sqrt{\cdot}$: I have observed that this symbol quickly becomes a meaningless mantra to students. I regularly ask (notably in 1st year of master for future high school teacher, also in France) what $\sqrt{2}$ means. First time students stare at me, and I go with them through the definition: $\sqrt{2}$ is the unique positive real number whose square is $2$. Then I question the definition: does it make sense? Why is there one such number? Why only one? Is there a non-positive one? (intermediate value theorem / monotony / look at the graph of the square function and draw a horizontal line).

It will not replace drill, but it complements it. Often, this kind of mistake come from the loss of meaning. Bring meaning back any way you can.

Answer 5

Use repetition and carrot/stick (e.g. weekly period-long exams, daily one question pop quizzes, in class games).

Question shows an unconscious assumption that clear explanation is the key criteria. ("I told them several times.") It's not. We are not computers that get fixed forever with a line of code. We are physical, imperfect beings. Our brains adapted to make sense of our universe but are subject to various fallacies (e.g. optical illusions). We learn to avoid various errors by "imitation and practice" (cf. Aristotle).

Instead of taking the attitude "how can these students make this mistake" have a position that is more sympathetic and tough at the same time. "I get why you made the error but I will keep hammering you until you get it right." Drill sergeant. Or even the "yeah, this stuff is tough...let me tell you a sneaky trick" (like you are a fellow criminal, fellow imperfect human).

Make little silly names for the errors. A computer would not need them. It would grok the iff symbol. But humans are humans. If you call it "offsides" or "penalty kick" or whatever, it will make a weird association that helps them.

Answer 6

The square root symbol $\sqrt {\cdot}$ means "the principal square root". For positive numbers, it's understood that it means the positive square root. But your students clearly haven't advanced enough for this to be implicit. Insist that if they want to represent the positive square root of $x$, they have to write $\sqrt[+] x$. Note that I didn't write $+\sqrt x$. I'm treating positive sign as part of the square root symbol. When you take the square root, write it as $\sqrt[\pm] x$ rather than $\pm \sqrt x$ to emphasize that you're not taking "the" square root and then taking plus and minus that number; the square root operation itself gives two outputs. You should eliminate as much as possible the concept of "the" square root from your students' minds.

Answer 7

I have similar issues, and handle it in the following way:

1. On the first day of class I do a review of basic algebra, things like: $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} $$ $$\sqrt{a + b} \neq \sqrt{a} + \sqrt{b} $$ $$\sqrt{ab} = \sqrt{a}\sqrt{b}$$ $$a(b + c) = ab + ac$$
2. Give a list of what I call 'fatal errors', things like what you mention, give demonstrations of why these things are the way they are, e.g.: if $n$ even: $$x^n = b \hspace{.5in} (b > 0)$$ has 2 solutions, namely $$\pm\sqrt[n]{b} $$ if $n$ odd: $$x^n = b \hspace{.5in} (b >0)$$ has 1 solution, namely $$\sqrt[n]{b}$$ and use some numbers that work out 'nicely' as examples, e.g. $$(-3)^2 = (-3)(-3) = 9 = (3)(3) = 3^2 $$
3. Tell them I will stop grading their solution the moment I see one of these errors.
4. Give them lots of examples to work, circulate to help individual students
5. Work out solutions while being sure to explain my reasoning
6. Make sure to write a specific comment on their exams about having stopped grading because they made one of these fatal errors

Answer 8

When you just know the basics of complex number theory, then you know that $x^n = a$ has $n$ solutions, so for $n=2$, $x^n = a$ has two solutions.

What's the big deal?
For $n=2$, all roots (in case $a>0$) are real, which means you have 2 real roots.
For $n=3$, there is one real root and two purely complex ones.
For $n=4$, (in case $a>0$) two roots are real, and two are purely complex.
...

You can easily show this in a nice drawing:
For $n=5$, the roots are on a regular pentagon.
For $n=4$, the roots are on a regular quadrilateral.
For $n=3$, the roots are on a regular triangle.
For $n=2$, the roots are on a line segment.

I truely believe, once you've shown this in a graphical manner, they won't forget it anymore.

Answer 9

Maybe the first question to ask is : WHY are they doing that mistake.

As says a famous 17th century philosopher, falsehood is nothing in itself, but just incomplete truth. So to understand the mistake, it can be usefull to look for the incomplete truth that lies in that mistake.

I think the " incomplete truth" here is the "rule":

       IF a²=b then  sqrt(a²) = sqrt (b)   and then    a = sqrt(b). 

Yes, this rule is true, but with that complement : IF AND ONLY IF it is first given that the number a is positive.

The complete truth is that, when one does not know whether the number a is positive or not, one has to write :

                              sqrt (a²) = |a|. 

Applying what precedes to the specific question, one has :

(1) a²= b

(2) sqrt (a²) = sqrt (b)

(3) |a| = sqrt (b)

(4) a = sqrt(b) OR -a = sqrt (b)

(5) a = sqrt (b) OR a= - sqrt (b)