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Meyer began his career as a high school math teacher, an experience for which he felt ill-suited but which nonetheless, he said in the interview, “shaped [his] entire life.” He “felt guilty to be the only one who is always right while kids are wrong most of the time,” he explained, adding that he preferred the role of student: “To do research means to be ...

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“Success is not final, failure is not fatal: it is the courage to continue that counts.” ― Winston S. Churchill “There is only one thing that makes a dream impossible to achieve: the fear of failure.” ― Paulo Coelho, The Alchemist “Failure is the condiment that gives success its flavor.” ― Truman Capote

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I think that the most interesting application of (sum of numerators/sum of denominators) “addition” is to continued fractions. For example, if you want to calculate the continued fraction expansion of the square root of 2,start with 1/0 and 0/1 and “add” them in the following way: In the top row you put the results of the “adding” that are greater then the ...

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It is possible for an expression to be undefined. If you work in $\mathbb{N}$, you can consider $2 - 3$ to be undefined. We just have meaningless expressions. Similarly when working in $\mathbb{R}$, we can also consider the expression $\frac{1}{0}$ to be a meaningless expression. $\frac{1}{0}$ is considred undefined because $\frac{1}{0}$ means the number ...

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I appreciate Ben Orlin's "Math with Bad Drawings" post about Andrew Wiles, for which the theme is not mistakes but rather being stuck. I have the sense that it contains some relevant content for the question that is asked here; for example, consider the following excerpt (emphasis added by me): For example, take Wiles’ musings on the value of ...

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I'm not sure if I understand your point here: your students want to keep their course notes clean and don't want to mix correct calculations with wrong ones. You, however, want to add wrong calculations, in order for them to learn from those mistakes, and you are looking for quotes to support your point. I, however, see that your students have a valid point....

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The problem is with the question, not with the students' answers. The question is ambiguous and I think the students' answer is actually much better than yours. Suppose I drive a thousand miles at 25mpg and you drive one mile at 35mpg. What's the average fuel efficiency? Your answer is 30mpg but I honestly can't think of any situation in which that is a ...

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Think of an example with two ratios: 1/3 and 4/5. When you add the numerators, and divide this by the sum of the denominators, you get (1 + 4)/(3 + 5) = 5/8. Now, think about what is happening with the denominators - the denominator of the first ratio should only act on the first numerator. But instead, when you add the ratios in this way, the denominator ...

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Allow me to offer another example: Imagine you and your best friend both want to buy a new smart phone. The phone you have chosen will cost you 300€ but your friend chooses a phone that will cost as much as 600€! Luckily, you have two vouchers that will give you a discount: The first voucher will give you the cheaper phone for free, if you buy two phones. ...

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One observation is that (sum of numerators) divided by (sum of denominators) is not well defined. For example, let's work with the two ratios $a=\frac01$ and $b=\frac11$. The ratio of the sum of numerators to sum of denominators is $\frac12$. However, we can also write $a=\frac03$ and $b=\frac22$. Now the ratio is $\frac25$, which is not equal to $\... 12 I like guest's answer. To elaborate, here is a possible question to ask them. You take two trips in your car: Trip 1 is a 100 mile drive that takes you 2 hours. Trip 2 is a 200 mile drive that takes you 1 hour. (a) What is the average speed of your car? (b) What is the average speed on an average trip? The answer to (a) is$\frac{...

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Mistakes Allow Thinking to Happen

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It actually depends on exactly what you're asking. Or even what you SHOULD be asking. If you want the average profitability of all the 500+ operators in the Permian, you could just average all the profit margin percentages. This is taking the ratios (profit/revenue) for each company and averaging them. It corresponds to your expected (mean) profit margin ...

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Good decisions come from experience, and experience come from bad decisions. Unknown

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"In the remaining sections of this paper we briefly discuss various occurrences of the stability and pinching phenomena in differential geometry. The results we present are, for the most part, not new and we do not provide the detailed proofs. (These can be found in the papers cited in our list of references). What may be new and interesting for non-experts ...

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Sometimes life's going to hit you in the head with a brick. Don't lose faith. - Steve Jobs

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The one I'm using is: An expert is a person who has found out by his own painful experience all the mistakes that one can make in a very narrow field. Attributed to Niels Bohr, quoted by Edward Teller, in Dr. Edward Teller's Magnificent Obsession by Robert Coughlan, in LIFE magazine (6 September 1954), p. 62.

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Here are a few of my favorites:

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There was a post over at academia where somebody essentially said that after starting by investigating other people's mistakes: Eventually I got better — I started making my own mistakes. As I said over there, I liked that a lot and it stuck with me. Making your own mistakes is a sign of growing up.

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The But They Did Not Give Up site contains a large number of quotes and anecdotes regarding successful people from many different walks of life who achieved success only after many repeated failures. For example: Albert Einstein did not speak until he was 4-years-old and did not read until he was 7. His parents thought he was "sub-normal," and one of his ...

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The road to wisdom? — Well, it's plain and simple to express:         Err         and err         and err again         but less         and less         and less.” — Piet Hein

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Is a quote about failing in topic? I think so. Here we go. “Ever tried. Ever failed. No matter. Try again. Fail again. Fail better.” Samuel Beckett More about this quote on booksonthewall

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"Human is the only animal that trips twice over the same stone" - Anonymous "Next time you trip over a stone, instead of stepping over it, place a big flashing sign that will remind you where you fell to the ground. The next time you travel the same path you will remember your past mistake and be able to avoid the hazard" - Me This is trying to be funny to ...

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"I hope that in this year to come, you make mistakes. Because if you are making mistakes, then you are making new things, trying new things, learning, living, pushing yourself, changing yourself, changing your world. You're doing things you've never done before, and more importantly, you're doing something." Neil Gaiman "Failure is simply the opportunity ...

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"I have not failed. I've just found 10,000 ways that won't work." "Our greatest weakness lies in giving up. The most certain way to succeed is always to try just one more time." "Many of life's failures are people who did not realize how close they were to success when they gave up." Thomas A. Edison

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“There is no man,” he began, “however wise, who has not at some period of his youth said things, or lived in a way the consciousness of which is so unpleasant to him in later life that he would gladly, if he could, expunge it from his memory. And yet he ought not entirely to regret it, because he cannot be certain that he has indeed become a wise ...

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Johann Wolfgang von Goethe: "By seeking and blundering we learn." Original German, 1825. Albert Einstein: "Anyone who has never made a mistake has never tried anything new." (However, attribution to Einstein is weak. See quoteinvestigator.com.) Jo Boaler: "When I have tutored people in math, I've always started by saying, 'By the way, I just want you to know ...

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If you know what you are doing, then you are wasting your time. Anonymous

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