How to cure students from the idea that root and squaring are identity operators?

Question

I tutor high school algebra and I’ve noticed that a lot of my students don’t seem to understand what they’re doing when they “convert” between different ways of writing numbers involving perfect squares. For instance, I have seen students simplify $\sqrt9$ to $\sqrt3$ time after time, even after I’ve just corrected them on $\sqrt4\neq\sqrt2$.

Similarly, if I show them the difference of squares formula $(a^2-b^2)=(a-b)(a+b)$ and then write $(x^2-9)=?$ I will inevitably get $(x^2-9)=(x-9)(x+9)$ (sometimes with the preliminary step of $(x^2-9)=(x^2-9^2)=(x-9)(x+9)$, which is truly baffling). Showing similar examples with other numbers doesn’t seem to help.

Does anybody have any tricks on how to teach this subject and snap students out of the idea that $\sqrt3 = \sqrt9 = 9 = 9^2=\ldots$?

Answer 1

I suspect that there are several different and interacting things going on here.

1. It's likely that these students don't understand the "equals" sign in the same way that you do. An extensive body of research over several decades has documented that for many students the symbol $=$ does not indicate a relationship (that the thing on the left is identical in value to the thing on the right) but rather an operation: They interpret $=$ as a signal to do something, produce a result. (I don't have access to a bibliography right now but Carolyn Kieran's 1981 article on elementary schoolchildren's conceptions of the equals sign was the first in this field; it has been extensively followed up on by researchers at the secondary and college level who have found that the operator conception can significantly impede students' success in algebra.) So your students may not be understanding the expression $(x^2-9)=(x-9)(x+9)$ as meaning that the two quantities are equal to each other; rather they see the LHS as a question, and the RHS as an answer.

2. Second, it's almost certain that your students' understanding of what they are doing is limited to a set of rules about what you are supposed to write down where, and is disconnected from any understanding of what the symbols mean. A phenomenon related to the one you describe: Students are asked to simplify $\sqrt{12}$. They write $\sqrt{12} = \sqrt{4 \times 3} = 3 \sqrt{4}$. When asked what they mean, they say "Wait, which number is supposed to go on the outside and which one is on the inside?"

3. A third issue (closely related to both of the above) may have to do with decoding the written symbols into verbal form and vice versa. I have often heard students read $\sin(x)$ aloud as "sin times x"; it may be that your students are doing something similar, using the notation $\sqrt{3}$ to say "The square root is 3". ("Is", not "Of"). In more detail, they may be reading the initial expression $\sqrt{9}$ as an instruction ("Find the square root of 9"), know (remember) that the answer is 3, saying to themselves "the square root is 3", and write that down as $\sqrt{3}$.

The connection among these is (in part) that the square root symbol designates completely different things on the different sides of the equals sign: On the left, it means a question, and on the right, it labels the answer. In neither case is it connected with what square roots or squaring actually mean.

One strategy that can help with all three of these issues is to emphasize that these operations ought to be reversible. If the student thinks that $(x^2-9)=(x-9)(x+9)$, ask them to multiply the RHS out and see if they get the LHS. (They are less likely, I think, to repeat the error in the other direction, because they will be thinking $9 \times 9$ rather than $9^2$.) If a student thinks that $\sqrt{12} = 3\sqrt{4}$, ask them what they should get if they multiply the RHS times itself, and then ask them what they do get if they multiply the RHS times itself.

(Aside: I once asked a group of 8 high school teachers how they would respond to a student who made the mistake of writing $\sqrt{12} = 3\sqrt{4}$. Almost all of them said that they would go over the correct process again. One creative thinker suggested putting both the LHS and the RHS into a calculator and showing that they are not equal. When I proposed multiplying the RHS times itself and getting not 12 but 36, several of them expressed surprise, not at the method but at the fact that it had never occurred to them before.)

Another strategy that you may find helpful (over the long term) is to try to get into the habit of giving them problems that are like those they are accustomed to, but with the LHS and RHS swapped (i.e., of the form ____ = $\sqrt 9$). This may help break the associations "left side = question" and "right side = answer"; at the very least, it may help to bring it to the surface, which can help confirm the diagnosis.

Answer 2

First: I do not think this is really an issue with a lack of understanding of the square root function. When someone writes $\sqrt9 = \sqrt3$ it means they are not thinking about what the equals sign even means.

I have had a small amount of success with the following method, which relies on the students "believing in their calculator" as a source of truth.

Suppose a student has the following work, with the square root error that you outlined:

$\frac{-2 + \sqrt{20}}{2} = \frac{-2 + \sqrt4 \sqrt 5}{2} = \frac{-2 + 4 \sqrt 5}{2} = -1 + 2\sqrt5$.

They look up the answer and find that the correct answer is $-1 + \sqrt5$, and ask for assistance.

You can tell them to check each step with their calculator. They know they were supposed to get $-1 + \sqrt5$ as the answer, and their calculator happily translates that correct answer into 1.236.

• When they type the first fraction into their calculator, it also comes out 1.236. This means they were correct at that step.

• When they plug the second fraction into their calculator, it also comes out 1.236. This means they were correct at that step too.

• But when they plug the third fraction into their calculator, it comes out 3.472. This shows that the simplification error was made between the second and third steps.

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What is a student supposed to take away from this?

I'd argue that when using this method, you are teaching the student to begin to grasp that each mathematical expression they write is actually a value, and that as they simplify, they should be doing things that do not change that value.

Expert-level mastery of this concept would allow them to look at the 4 and the $\sqrt4$ next to each other and realize "wait... $4$ and $\sqrt4$ are different numbers -- but first students need the insight that $4$ and $\sqrt4$ are numbers that could be different at all!

Answer 3

Late answer, but since it comes up every semester I have a stock response that I'd like to share. The most common mistake in this vein is for a student to write $\sqrt{16} = \sqrt{4} = 2$ or $\sqrt{81} = \sqrt{9} = 3$. Frequently if I try to correct this they'll ask, "Where did the radical sign go?".

When this happens, I go to the board and write something like $5 - 3 = 2$, and get them to agree that's true. Then I ask "Explain to me, where did the minus sign go? Why didn't we need to rewrite the minus sign?" And thinking about explaining that does seem to help most students. When they're done I try to frame it as, "When we perform a particular operation, then the symbol goes away because we're done with it."