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One of my main research areas is algebraic thinking at different levels. Yet, from time to time, I observe something that I cannot relate to anything else that I know.

This is the story of one of these things that happened today in my calculus class. As part of using the ratio test, students needed to work with factorials. One of them interpreted $(2k-1)!$ as $1\cdot3\cdot5\cdots(2k-1)$. It took quite a while for me to help him to see the correct interpretation.

Thus, it does not look like it was just a mistake. If not a mistake, what was it? Can you see it as an example of a more general algebraic (or something else) misconception? Or alternatively, can you point to a "similar" phenomenon (mistake, error, misconception, or whatever you want to call it)?

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    $\begingroup$ Were they ever exposed to the double factorial? $\endgroup$ – Joel Reyes Noche Nov 20 '15 at 12:23
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    $\begingroup$ to me, it looks like the student saw $2k -1$ which is a pretty standard mathematical way to say "odd numbers" and then saw the factorial and thought to themselves "looks like i need to factorial only the odd numbers" $\endgroup$ – celeriko Nov 20 '15 at 12:34
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    $\begingroup$ @celeriko This describes the situation and explains well why he had no difficulty with say 5!. If you could also include one "similar" situation, I was getting closer to an answer :) $\endgroup$ – Amir Asghari Nov 20 '15 at 12:52
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    $\begingroup$ The student seems to have imprinted on 2k-1 being the way to describe odd numbers, as celeriko suggested. I wonder if this student would think (2k)! = 2*4*...*(2k). And would (3k+1)! for them be 1*4*7*...*(3k+1)? For them an algebraic ending is determining not just endpoint, but also pattern. I might get excited and tell them they invented a new operation. ;^) $\endgroup$ – Sue VanHattum Nov 20 '15 at 16:13
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    $\begingroup$ This response is essentially equivalent to the others, but my guess is the misconception is around what the notation of $!$ means; specifically, believing that it means multiplying together elements in that sequence. So, e.g., $(2k-1)! = \prod 2k-1$, though even this product notation is confusing (rather, one might use $\prod_{n=1}^{k} (2n-1)$ instead). I do think that the idea of the product they wrote is important, which is why it does have its own notation! But a different notation. [I see now S. Gubkin has remarked equivalently around the $\prod$ notation.] $\endgroup$ – Benjamin Dickman Nov 20 '15 at 20:43
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It seems clear that there is a certain conceptual gap in the student's understanding. My suspicion is that the student is essentially running the following program in his mind:

Initialize factorial = 1

For $j = k$ to $1$

  factorial = factorial * (2j - 1)

Output factorial

(The correct program should read: for $j = 2k - 1$ to $1$, factorial = factorial * $j$.)

That is, the student is decreasing $k$ by 1 in $(2k-1)!$ to form each successive factor, rather than decreasing $n = 2k - 1$ by $1$ at each step of the recursive algorithm. (This is not quite the right way to phrase it, but I'm at a loss for better words at the moment.) Perhaps explaining the difference between the two algorithms will solve the issue.

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    $\begingroup$ In fact, the part that was not "the right way to phrase it", was the right way to phrase it! Your answer suggests that we can think of, say Sigma i from 1 to (2k-1) as 1+3+5+...+(2k-1). Thus, your answer shows that the difference is not between the notations of factorial and sigma, it is about the way we interpret these notations. $\endgroup$ – Amir Asghari Nov 20 '15 at 16:37
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    $\begingroup$ In other words, the student is thinking of $\prod_{j=1}^k (2j-1)$. They may have thought all along that $k! = \prod_{j=1}^k j$, and just used the obvious substitution. $\endgroup$ – Steven Gubkin Nov 20 '15 at 18:58
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    $\begingroup$ going with @StevenGubkin's definition of the factorial, they would be substituting $2j-1$ for $j$, when in fact they should be substituting $2k-1$ for $k$. $\endgroup$ – alexw Nov 20 '15 at 23:24
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As per my comment above, it looks to me like the student saw $2k−1$ which is a pretty standard mathematical way to say "odd numbers" and then saw the factorial and thought to themselves "looks like i need to factorial only the odd numbers". While I have never encountered this particular misconception, I categorize it into a general category of an "illogical leap" (not an official term, just one that I use when thinking about these classroom issues).

Basically, the illogical leap occurs when a student recognizes individual pieces of an equation/expression/relation but they don't understand the whole or how to correctly connect the pieces. Because of this, they take what they know about the individual pieces and try to connect them in an incorrect and illogical fashion. In this case, they recognized the factorial sign and the common expression $2k-1$ and made the illogical connection that this should mean to factorial the odd numbers. I don't think that this is an incredibly unreasonable jump, but it should obviously be corrected. I have found the best way to correct a student who makes an illogical leap is to first recognize the information they understood. This is big because it encourages them that they did get something right and makes them more responsive to your corrections. Make sure that you have a legitimate explanation for exactly why they were wrong and how to correct their thinking. Simply saying "This is not correct, here is the correct way" is not very helpful and will probably not actually help their thinking.

Some other examples I have seen of this:

  • Thinking that there is no real number solution to $ - \sqrt{4} $. They recognize that the square root of a negative number is imaginary but they fail to recognize the negative needs to be under the radical for this to be true. There are plenty of others involving radicals (ex. $\sqrt[3]{8} = \pm 2$)
  • Take the equation: $$8x - 9 = 3x$$ I have seen many students rewrite this as $$-9 = 3x + 8x$$ They recognize that the variables should be consolidated but they think "I'll just move it over here" without recognizing the actual operation that needs to occur to make this "move" possible, i.e. subtracting $8x$
  • Thinking that $2^{-3} = -8$. They recognize that there is a cubic in there but they mistake the negative sign to mean the sign of the answer not that the cube should be performed in the denominator.

There are probably hundreds of other examples out there but these were the first three that came to mind. Unfortunately, there is no one-size-fits-all answer that will correct students thinking every time. The best that I can suggest is to:

Look at their mistake and try to approach the problem from their point of view. Ask yourself what they got correct and how that manifests in the answer provided. Use that to determine what "leaps" they would have to take to get from their correct knowledge to the incorrect answer. Then make sure to correct those leaps. Lastly, if you have no idea what they were thinking, then ask them!

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    $\begingroup$ I am not sure I can agree with you that the examples you have mentioned are similar to each other. The equation example is quite conceptual, but the power example (third example) is conventional. The same is true for the difference between seeing the cube root of 8 as plus or minus 2, and seeing the square root of 4 as plus or minus two. The former is conceptually wrong, the latter is conventionally wrong. $\endgroup$ – Amir Asghari Nov 20 '15 at 17:12
  • $\begingroup$ im not sure what you mean by conventional vs. conceptual? for the third example, it is not a mechanical mistake (i.e. forgeting a sign or transposing two numbers) but it is a mistake in the understanding of what a negative exponent is $\endgroup$ – celeriko Nov 20 '15 at 17:15
  • $\begingroup$ You can see the difference between conventional and conceptual in cube root and square root example. It is just a convention that we take positive 2 as the answer to the square root of 4. Consider that if we interpret square root of 4 as "a" number whose square is 4, the answer would be plus and minus 2. Thus, giving minus 2 as an answer to the cube root of 8 is conceptually wrong, but as an answer to square root of 4 is conventionally wrong. $\endgroup$ – Amir Asghari Nov 20 '15 at 17:28
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    $\begingroup$ got it, i edited my answer $\endgroup$ – celeriko Nov 20 '15 at 17:29
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    $\begingroup$ Just to expand one one theme here... when a student verbally answers a question wrong, I personally try to think of what related question it would have been the right answer to. This allows me to give encouragement, i.e., "That would have been the right answer if the question had been X, but instead it's Y, do you see the difference?", and also serves as kind of a mental game for myself in-class. $\endgroup$ – Daniel R. Collins Nov 21 '15 at 3:28

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