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After years of experience in some area of expertise, you can easily forget how difficult it can be for the uninitiated to grasp some fundamental concepts, and, indeed, people often edit out of their own personal history memories of their initial confusion and fears.

History can re-teach us about the inherent difficulties in assimilating new concepts, and, I think, help in tempering the novices' fears by assuring them that initial confusion and making mistakes are quite natural and in encouraging them to persevere.

Using the evolution of the concept of negative numbers as an example, I'd like to present a historical note from Mathematics: The Loss of Certainty by Morris Kline (Oxford, pg. 115):

"An interesting argument against negative numbers was given by Antoine Arnauld (1612-1694), theologian, mathematician, and close friend of Pascal. Arnaud questioned that -1 : 1 = 1 : -1 because, he said, -1 is less than +1; hence, how could a smaller be to a greater as a greater is to a smaller?"

The history of mathematics is replete with similar examples. What are some of your favorites?

Edit in response to comments:

I'm asking for historical examples that illustrate the natural confusion present in the evolution of fundamental concepts in math and that ideally could serve both to mitigate the remedial or novice learner's reaction to his own confusion and allied fear and as a stepping stone for exploring the concept.

(I know of other examples involving Leibniz and the product rule (cf. Humanizing Calculus by M. Cirillo), and Euler and log(-1), but I'll leave them for others to describe.)

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    $\begingroup$ In the past, I have asked questions related to mathematics education and history at the History of Science and Mathematics StackExchange site. I'm not sure if your question would be more appropriate here or there. $\endgroup$ – Joel Reyes Noche Mar 6 '15 at 0:54
  • $\begingroup$ @JoelReyesNoche Certainly overlap. Why impose a dichotomy? History phobia of math educators? $\endgroup$ – Tom Copeland Mar 6 '15 at 0:59
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    $\begingroup$ I think this would work for both SE, but using it for a teaching tool is a nice fit here. I do like the idea of "unencapsulating" math from "here's another algorithm" to "here's how the algorithm fits into the scheme of math." $\endgroup$ – Chris C Mar 6 '15 at 1:37
  • $\begingroup$ I am not quite clear on the question ("What are some of your favorites?") and its connection to "math phobia." It seems to me that one question is about whether some sort of historical information could help reduce math anxiety; another, separate question might be one looking for a particular type of quotation or conception of the past (with the type more carefully specified in the question). $\endgroup$ – Benjamin Dickman Mar 6 '15 at 3:09
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    $\begingroup$ There is no question in my mind that introducing the math in a historical context would help to reduce a student's anxieties. That clearly is not part of my question. $\endgroup$ – Tom Copeland Mar 6 '15 at 3:45
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The biggest example to me of this is the confusion brought into math when calculus was first developed by Newton and Leibniz, with Bishop Berkeley saying, "And what are these fluxions? The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?"

It took mathematicians 150 years to develop a sound logical foundation (the limit) for the basic ideas used in calculus. I talk about this extensively in calc class, so my students can realize that even the people best at math can struggle with it. I also want them to realize both the importance and the difficulty of the topic of limits. I also think this history shows how important calculus was to science. People kept using it for that 150 years, even though it didn't have the logical rigor they knew was possible with geometry and algebra.

I follow the historical path in my course, presenting the book's material on limits and continuity after we spend a few weeks exploring the idea of the derivative (using the limit notation, but without any more backup than "this means infinitely close"). We do a few derivative properties, graph polynomials, and then look at the limit chapter, where the squeeze theorem is introduced, before we deal with the derivatives of trig functions.

I think calculus textbooks presenting a whole chapter on limits before students have any idea of their value shows that the texts were written by mathematicians who aren't thinking much about how most people learn. (Many of us who love math are willing to learn about the weirdest things, just because it's a big adventure, and a great puzzle. But that's not how it works for most of our students. And math lover that I am, I still would have benefited from a class like the one I teach.)

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    $\begingroup$ Good example! (The historical dialogue behind Robinson's nonstandard analysis.) $\endgroup$ – Tom Copeland Mar 6 '15 at 18:41
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    $\begingroup$ You know I've literally never thought of it this way. If I ever get a chance to teach AP Calc I will have to try this. $\endgroup$ – Jhecht Mar 7 '15 at 8:00
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The one I love and and I design my class around its history frame by frame is the notion of complex number. Just consider this question as a starter:

Find two numbers whose sum is equal to 10 and whose product is equal to 40.

Students' reactions are telling, in particular, those who calculate $b^2-4ac$ for $x^2-10x +40=0$ and put the pencil down as soon as they get $-60$ for the discriminant. But, exactly when they are happy with the end, comes this crucial and for most of them shocking new start: "what if we carry on with our beloved quadratic formula for the roots". At that very moment you can witness the "mental torture" attributed to Cardano: "No way" they say.And, here is me repeating the words of Cardano saying "Dismiss the torture" And a few moment later, they are adding and multiplying those "objects" being completely meaningless two minutes earlier: $5 + \sqrt{-15}$ and $5 - \sqrt{-15}$ without even remembering their initial reactions!

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    $\begingroup$ Another fine example. One contagious aspect of this interactive style of coaching is the enthusiasm the teacher has. $\endgroup$ – Tom Copeland Mar 7 '15 at 1:33
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In my experience some students have difficulty with where $m$ and $b$ come from in the slope-intercept form of a line. (In the US, we use $y=mx+b$ but this is not the same in other countries.) Apparently, the origin of $m$ is still disputed (see here for a more detailed explanation). The origin for $b$ is less ambiguous, and appears to follow from the case where $y=ax+b$ is used.

Some quick Googling appears to show that different countries use different variables, so I can't speak for those cases, but it seems to me that students who struggle with this concept are further put off by mathematics because it doesn't make sense to them. If I tell a story about where $m$ and $b$ come from, they seem to accept it a little bit more because there is a reason behind it (even if my story may not be 100% true).

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  • $\begingroup$ Historical context-Meaning-Memory-Basic Mnemonics. Nice. $\endgroup$ – Tom Copeland Mar 6 '15 at 18:50
  • $\begingroup$ Which concept are you talking about students struggling with — the concept of a linear equation, the concept of letters being used to represent numbers, or something else? $\endgroup$ – Daniel Hast Mar 7 '15 at 2:38

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