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It's very common in learning mathematical induction to prove statements like

$$ 0^2+1^2+2^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}{6}.$$

I've found that very frequently, on this sort of problem, when students get to the induction step they will write the induction hypothesis as "$n^2= \frac{n(n+1)(2n+1)}{6}$" and the goal to prove as "$(n+1)^2= \frac{(n+1)(n+2)(2n+3)}{6}$", totally ignoring the ellispis "$\cdots$" and everything before it. I have never been able to figure out where this mistake comes from. More importantly, what can I say to help them overcome this mistake?

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I suspect that the issue is not so much the ellipsis per se but a problem with notation in general, and in particular with the correct use of the equals sign. At the risk of repeating what I wrote in this answer, students often regard the equals sign not as a symbol meaning "these two expressions are the same" but rather as a symbol separating a question (on the left) from an answer (on the right).

To illustrate what I mean, let me change contexts for a moment to something that may seem at first to be unrelated. Think of how students often write down their work when solving a story problem: If the problem is "Pens cost 75 cents each and pencils cost 20 cents each. If Andy spends a total of $5.15 on a combination of 12 pens and pencils, how many of each did he buy?", it is not at all uncommon to see student work like:

Pens = x

Pencils = 12-x

Pens = 75x

Pencils = 20(12-x)

Now all of us cringe at the fact that the student does not distinguish between "number of pencils" and "cost of pencils." We say things like "'Pencils' is not a number!" and "How can Pens be equal to $x$ and also equal to $75x$?" But students do it anyway. What are they thinking? What are they doing?

What they are doing is using "Pens = " as a label for a formula. When they write "Pens = " they internally verbalize it as "The formula for pens is..." They are essentially using the equals sign like a colon -- what they mean is something like

Quantities

Pens: x

Pencils: 12-x

Cost

Pens: 75x

Pencils: 20(12-x)

but they omit the headers (more precisely, the headers are "mentally present" but not written down, because you don't 'do' anything with them) and use an equals sign instead of a colon.

Notice that if a student wrote it down this way (with the headers and the colons) we would probably be inclined to regard their work as correct. So the issue is really not that they do not understand the problem, but that they do not know an appropriate way to write down their understanding.

Okay, let me bring this back to the actual OP's question. I think when students write $$n^2 = \frac{n(n+1)(2n+1)}{6}$$ what they are thinking is something like this: There is a list of formulas. Each formula on the list is named or labeled by the last term in a sum. So, there is the $1^2$ formula, the $2^2$ formula, the $3^2$ formula, and so on. If that list of formulas were written out, it might look something like this:

$1^2: \frac{1 \cdot 2 \cdot 3}{6}$

$2^2: \frac{2 \cdot 3 \cdot 5}{6}$

$3^2: \frac{3 \cdot 4 \cdot 7}{6}$

and the $n^{th}$ formula in the list would be:

$n^2: \frac{n(n+1)(2n+1)}{6}$

But instead of using a colon (:), they use an equals sign (=). The fact that $2^2$ does not equal $\frac{2 \cdot 3 \cdot 5}{6}$ does not bother them at all, because (a) when they write down $2^2$ they do not literally mean $2^2$ but just "the $2^2$ formula", i.e. the formula that ends with / is labelled by $2^2$; and (b) they are not thinking of "equals" as meaning "they are the same thing", but simply as indicating a correspondence of some sort.

I think a solution to this problem might lie not in fighting the tendency but in recognizing that students are looking for some way to label the $n^{th}$ formula in a list, and that impulse is not a bad one. It's just that they don't know a correct way to do it, so they are grasping at whatever notation is available and misusing it. So let's teach them a proper way to write what they want to say:

$P(1): 1^2 = \frac{1 \cdot 2 \cdot 3}{6}$

$P(2): 1^2 + 2^2 = \frac{2 \cdot 3 \cdot 5}{6}$

$P(3): 1^2 + 2^2 + 3^2 = \frac{3 \cdot 4 \cdot 7}{6}$

Of course students will not know how to read such statements without being taught to do so. Notation does not speak for itself! Some of the key things to point out about this notation are:

  • $P(n)$ means "the $n^{th}$ property in the list" -- it is the label for each formula
  • The label for each formula is separated from the formula itself by a colon
  • The formula is an equation, so it contains an equal sign separating two expressions
  • The expression on the left side of each equation is not a single term but a sum.
  • The expressions on the opposite sides of the equals sign are equal to each other; they should be encouraged to actually compute them and verify that this is so (e.g. to compute both $1^2 + 2^2 + 3^2$ and $\frac{3 \cdot 4 \cdot 7}{6}$ and confirm that they are both equal to $14$)

Give them plenty of practice writing out statements like this by giving them formulas like

$P(n): n! - (n-1)! = (n-1)(n-1)!$

or

$P(n): 1 + 2 + 4 + \dots + 2^n = 2^{n+1}-1$

and asking them to write out the first several instances of the formula, replacing $n$ with natural numbers, and verify that they are true. (You might also want to mix in some formulas that are false, or that are true only for the first few values of $n$, and have them determine whether for a given value of $n$, $P(n)$ is true or false.) This will help them build their comfort with this kind of notation, and with the distinction between an expression, an equation, and a labeled formula.

TL;DR version of the above:

Students are using the expression "$n^2 =$" as a shorthand for labelling "the $n^{th}$ formula in the list is: ". If you teach them an appropriate alternative way to write that label, they may stop misusing notation the way that they are.

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    $\begingroup$ More examples of "can't distinguish between colon and equals" -- In my statistics course, we need to properly distinguish between hypotheses $H_0: \mu = \mu_0$ and $H_a: \mu \neq \mu_o$. But instead frequently students write $H_0 = \mu = \mu_0$, etc., even after explicit warnings about it. Also, where a formula card says "Mean of $\bar x$: $\mu_{\bar x}$ = $\mu$", they will "copy the formula" as: "$\bar x = \mu_{\bar x}$ = $\mu$". $\endgroup$ – Daniel R. Collins Dec 16 '15 at 19:12
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    $\begingroup$ @DanielR.Collins I like those examples very much. One of the things they make me realize is that the colon is often read aloud as "is" (e.g., "The null hypothesis is that mu equals mu-zero"). Combine that with more than a decade's worth of experience being told that "is" (in a story problem) is denoted with an equals sign, and this kind of notational error becomes unsurprising. $\endgroup$ – mweiss Dec 16 '15 at 20:28
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    $\begingroup$ Interesting. I'm familiar with the fact that students don't understand the equals sign, but do you have any evidence to support your guess that this is what's going on here? Note that what we're doing is writing the inductive hypothesis and inductive goal, so if $n^2=$ is only a label for the formula $\frac{n(n+1)(2n+1)}{6}$, then they are "assuming $\frac{n(n+1)(2n+1)}{6}$" which doesn't make any sense. Of course, that doesn't always stop them, but I can't think of anything I've heard them say or write that supports this guess about their thought process. $\endgroup$ – Mike Shulman Dec 17 '15 at 19:08
  • $\begingroup$ (Also, with my type theorist's hat on, I don't like writing "$H:x=y$" to mean that $H$ is the statement $x=y$, since in type theory $H:x=y$ means that $H$ is a proof (or assumption) that $x=y$ is true. From the same point of view, I wouldn't object to $H=(x=y)$ to mean that $H$ is the statement $x=y$; it's just the absence of parentheses that's a problem.) $\endgroup$ – Mike Shulman Dec 17 '15 at 19:17
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    $\begingroup$ @MikeShulman No evidence whatsoever; just a strong hunch! Regarding your observation that "they are assuming $\frac{n(n+1)(2n+1)}{6}, which doesn't make sense" -- I think that students often (mis-)understand induction proofs as "Start with an expression, and do something with it until it becomes another expression". A correct understanding would require them to have a fully-formed understanding of the equals sign, which I think we both agree many students lack. $\endgroup$ – mweiss Dec 17 '15 at 19:37
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Encourage your students to actually read the problem to themselves, similar to how they would if they were reading a book. Starting from left to right, read the problem: "Zero squared plus one squared plus two squared plus dot dot dot plus n squared equals..." etc. I find that this not only reinforces that there is that "..." there and it needs to be addressed in further procedures, but it also helps students from making "stupid mistakes" like dropped signs etc. You might be surprised how many students cannot actually read mathematical expressions/equations out loud, at least at a high school level where I teach. Adding an auditory element to math can also help students who learn better by hearing (gardner's theory of multiple intelligences and such).

I would not only advise them to do this themselves but also model it for them by doing it every time a problem is discussed in class. If you use chalk/whiteboards it works very efficiently to say it out loud as you write it on the board. You can even have students say it out loud and write exactly what they say, which can lead to some interesting things on the board. Sometimes all I need to do is ask a confused student to read a problem out loud to themselves and they will instantly see the answer and go "Ahhh :)"

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    $\begingroup$ I like this suggestion. My existing habit is to ask, "What will I write?" for each advancing line of work; but even just reading the initial expression or equation seems like a good sanity-check, so I'll probably try more of that in the future. $\endgroup$ – Daniel R. Collins Dec 16 '15 at 3:04
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Perhaps they are misinterpreting the ellipses in $$ 0^2+1^2+2^2+\color{red}{\cdots}+n^2 = \frac{n(n+1)(2n+1)}{6}.$$ as meaning \begin{eqnarray} 0^2 & = & \frac{n(n+1)(2n+1)}{6} = \frac{0(1)(1)}{6} = 0\\ 1^2 & = & \frac{1(2)(3)}{6} = 1\\ & \color{red}{\cdots} &\\ n^2 & = & \frac{n(n+1)(2n+1)}{6} \end{eqnarray} So I hypothesize this is more a syntactic issue than it is a semantic issue.

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The other answers may indeed be right, but another thing just occurred to me, namely that when they prove the base case of the induction, the sum on the left-hand side does generally reduce to a single term (in my example, $0^2$). Perhaps they then approach the inductive step by looking at the base case and saying "okay, wherever there was a $0$, now I put an $n$" rather than correctly looking all the way back to the original statement to be proven. (Of course, different students might be making the same mistake for different reasons.)

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