Quote to show students don't have to fear making mistakes

Question

I have some high school students which seem to be afraid of making mistakes. They are hesitant to make exercises in class because they want their course notes to be super clean, without any mistakes. The following has often happened in my class. A student writes (when $a$ is a positive real) $\sqrt{16a^2+9a^2} = 4a+3a = 7a$. When the correct solution is shown and the mistake the student made is discussed, the student erases his mistake and writes down the correct solution. However, when studying the contents, he is not reminded of the mistake. (Which would be very valuable)

Does anyone know of a nice quote which shows that there is great learning potential in making mistakes (and figuring out why!). (Especially in math)

I know of the following quote by prof. Francis Su. A nice quote, but it's more about the value of persistence.

Struggling is a good thing… it’s where learning happens, it’s what we professors are always doing in our research… the struggle is the most interesting place to be.

Answer 1

• 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 that I love mistakes the most. They are the time that your brain grows, when you really learn, so it's really great to make mistakes.' [...] people immediately relax and breathe a sigh of relief and are much more willing to jump in to problems and persist longer." Cited here.
• Keith Devlin: "Learning occurs when we get something wrong and have to correct it." From Devlin's Angle, Oct 1st 2019.

Answer 2

"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

Answer 3

“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 man — so far as it is possible for any of us to be wise — unless he has passed through all the fatuous or unwholesome incarnations by which that ultimate stage must be preceded. I know that there are young fellows, the sons and grandsons of famous men, whose masters have instilled into them nobility of mind and moral refinement in their schooldays. They have, perhaps, when they look back upon their past lives, nothing to retract; they can, if they choose, publish a signed account of everything they have ever said or done; but they are poor creatures, feeble descendants of doctrinaires, and their wisdom is negative and sterile. We are not provided with wisdom, we must discover it for ourselves, after a journey through the wilderness which no one else can take for us, an effort which no one can spare us, for our wisdom is the point of view from which we come at last to regard the world. The lives that you admire, the attitudes that seem noble to you are not the result of training at home, by a father, or by masters at school, they have sprung from beginnings of a very different order, by reaction from the influence of everything evil or commonplace that prevailed round about them. They represent a struggle and a victory. I can see that the picture of what we once were, in early youth, may not be recognisable and cannot, certainly, be pleasing to contemplate in later life. But we must not deny the truth of it, for it is evidence that we have really lived, that it is in accordance with the laws of life and of the mind that we have, from the common elements of life, of the life of studios, of artistic groups — assuming that one is a painter — extracted something that goes beyond them.”

Marcel Proust, here: https://ebooks.adelaide.edu.au/p/proust/marcel/p96w/complete.html

It is about a wise man who has committed foolish acts in his youth. Perhaps a little bit wordish (hey, it's Proust), but you might extract something like the part I have highlighted.

Answer 4

Mistakes Allow Thinking to Happen

Answer 5

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.

Answer 6

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.

Answer 7

There are plenty of quotes here...

But if you're trying to increase willingness to make and learn mistakes, I doubt that quotes will change a lot of minds, because a quote will be a tiny trickle of words pushing in the direction you want versus a raging river pushing the other way. Even Michael Jordan's quote about being cut from his high school basketball team or a Jo Boaler video just isn't going to make a lick of difference against the forces pushing against you.

A few considerations:

• How much of their letter grades for the course are explicitly about making mistakes and learning from them? E.g. "Exams are 50% of your mark, quizzes are 25% of your mark, and the other 25% of your mark will be how you recorded and learned from your mistakes." Most teachers don't do this. Every student sees the gap between what teachers say and what they incentivize. The normal reaction to this is to ignore the words and internalize the "obvious" message from the incentives: "You'll get high marks for correct answers and low marks for mistakes so don't make mistakes!" I am aware that many teachers are not allowed to have grading schemes that incorporate work habits. And I am also aware that the message you want to get across is to make the mistakes before the stakes are high, but...
• A huge share of math students, in the short-term, can't really learn from their mistakes because they lack the background knowledge to do so. If you're highly confused by the difference between $x=2$ and $x+2$, then how do you learn from your mistakes in a 9-step algebra problem? You can't. But you've probably been told to learn from such mistakes a thousand times, all to no avail. The normal reaction to this is to believe that most attempts to "learn from mistakes" in math class are a total waste of time and to be annoyed by your teacher who tells you to waste a huge amount of your own time for the thousandth time. Frustrated students will just say "I don't care what this means. Just show me the one right way to do it." [This problem is obviously solvable by backfilling knowledge gaps, but that's an enormous effort in and of itself.]
• What happens to the social status of students who make mistakes publicly?
• Do you exemplify learning from mistakes as a teacher? For example, do you video record what happens in your classroom, then share that with students, parents, colleagues, administrators, etc. all in order to find out how to improve? Do you try to make sure lots of people find all of your mistakes as a teacher? Would you focus your recording efforts on your least favorite lessons and topics to teach? Most teachers don't do this.

Overall, if it's a quote vs incentives, weak prior knowledge, countless bad experiences, lessened social status, and "do as I say not as I do", it's pretty obvious which side will win out.

Answer 8

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

Anonymous

Answer 9

"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 to begin again, this time more intelligently."

Henry Ford

Answer 10

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

Answer 11

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

Unknown

Answer 12

"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 convince them to mark their own mistakes so they can learn from them, instead of just getting over them. Please adjust to proper English where needed.

Answer 13

"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 is an exposition of the stability/pinching philosophy which lies behind the basic results and methods in the field and which is very rarely (if ever) presented in print. (This common and unfortunate fact of the lack of an adequate presentation of basic ideas and motivations of almost any mathematical theory is, probably, due to the binary nature of mathematical perception: either you have no inkling of an idea or, once you have understood it, this very idea appears so embarrassingly obvious that you feel reluctant to say it aloud; moreover, once your mind switches from the state of darkness to the light, all memory of the dark state is erased and it becomes impossible to conceive the existence of another mind for which [the] idea appears non-obvious.)" --Mikhail Gromov, "Stability and Pinching" pp. 64-65. Bold emphasis added.

This quote (from an absolutely towering figure of modern mathematics) pretty much guides my entire pedagogical philosophy. In particular, we must make a conscientious effort as both instructors and students of mathematics to combat this binary nature of mathematical perception. As instructors, we need the empathy and humility to remember that a topic was not always blindingly obvious and that we too once struggled with it. As students, we need the perseverance to push through until the current struggle becomes blindingly obvious in hindsight.

I do what I can to encourage this metacognition. Much of mathematics education can feel like an endless treadmill, of being perpetually stuck in the dark of new ideas and concepts. Seek out time to reinforce and reflect upon topics that once seemed impossible: "Remember when multiplication tables / equations of parabolas and lines / the product rule / linear differential equations / ... seemed hard? This too shall pass...". As grating as we might find it when students write $\sqrt{a+b} = \sqrt{a} + \sqrt{b}$, who among us didn't make this mistake at least once upon our own journeys? While we are tasked with correcting this mistake, don't fall into the trap of believing it to be trivial or the sign of an inferior intelligence. Embrace and encourage acknowledgement of mistakes.

That even an Abel prize winning mathematician can feel this way about mathematical understanding only further hammers this point home. If even Gromov can admit being embarassed about not grasping a subject at first pass, then what shame should the rest of us possibly bear? The only crimes would be to either lie to ourselves about having ever struggled or to write off those who struggle with it.

Edit: Another quote

"Understanding what and why did not work may be more instructive than celebrating our successes." ~ Mikhail Gromov

I am having trouble finding a rigorous citation for this quote, but I will edit if I can track it down.

Answer 14

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 forgetfulness. “I think it’s bad to have too good a memory if you want to be a mathematician,” Wiles said. “You need to forget the way you approached [the problem] the previous time.”

It goes like this. You try one strategy on a problem. It fails. You retreat, dispirited. Later, having forgotten your bitter defeat, you try the same strategy again. Perhaps the process repeats. But eventually—again, thanks to your forgetfulness—you commit a slight error, a tiny deviation from the path you’ve tried several times. And suddenly, you succeed.

Wiles has a nifty analogy for this: it’s like a chance mutation in a strand of DNA that yields surprising evolutionary success.

“If you remember all the false, failed attempts before,” said Wiles, “you wouldn’t try. But because I have a slightly bad memory, I’ll try essentially the same thing again, and then I’ll realize I was just missing this one little thing.”

Answer 15

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

Answer 16

Here are a few of my favorites:

Answer 17

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

Answer 18

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 teachers described him as "mentally slow, unsociable, and adrift forever in foolish dreams." He was expelled from school and was refused admittance to the Zurich Polytechnic School. He did eventually learn to speak and read. Even to do a little math.

and:

Thomas Edison's teachers said he was "too stupid to learn anything." He was fired from his first two jobs for being "non-productive." As an inventor, Edison made 1,000 unsuccessful attempts at inventing the light bulb. When a reporter asked, "How did it feel to fail 1,000 times?" Edison replied, "I didn’t fail 1,000 times. The light bulb was an invention with 1,000 steps."

Answer 19

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. You do too. So, instead of forcing one point of view, I'll try to mix both:

• You can tell your students to write wrong calculations in their course, but they need to do this in another colour (green, for instance). That way, their course (written in blue or black) will still be completely correct, taking into account that the green text needs to be understood in another way (as the way NOT to calculate something).
• You can tell your students to flip their course notes, and write the wrong calculations in the backside of their notes. Like this, the wrong calculations will still be in their notes but in such a way that it does not disturb the rest of their notes.

Answer 20

“Success is not final, failure is not fatal: it is the courage to continue that counts.” ― Winston S. Churchill

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“There is only one thing that makes a dream impossible to achieve: the fear of failure.” ― Paulo Coelho, The Alchemist

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“Failure is the condiment that gives success its flavor.” ― Truman Capote

Answer 21

Although (as with several other answers) this one is more about persistence, I think it's relevant enough ("depressingly slowly" sure implies intermittent failure to me) to post.

This is a poem by Piet Hein.

T.T.T.

Put up in a place
where it's easy to see
the cryptic admonishment
T.T.T.

When you feel how depressingly
slowly you climb,
it's well to remember that
Things Take Time.

(Amazingly, you can even buy plaques with a shorter version of the Danish original (?) of this on Etsy, with the famous picture that goes with it.)

Answer 22

I had a poster of Jake the Dog (from Adventure Time) in my high school geometry classroom, with this phrase:

I also gave frequent closed-notes quizzes, and then made my students correct their quizzes on an error analysis sheet like this one (source):

I had a poster with a description of different types of errors, in order to encourage them to analyze why they made the mistake they made and how to avoid it in the future. I didn't use the same types of errors as the above blog, instead I used the following:

Answer 23

I would challenge the premise of your question, that nice quotes are particularly useful.

At best, they are neutral, that is, they are on the wall somewhere and no one ever notices them. At worst, they are downright misleading and misrepresenting reality.

Take your quote as an example:

Struggling is a good thing… it’s where learning happens, it’s what we professors are always doing in our research… the struggle is the most interesting place to be.

Sounds nice, but it's dubious at best. Struggling isn't always conductive to learning. Students often struggle but don't learn anything. To the contrary, much learning happens when students are given tasks they can be successful with.

If you want to get your students stop worrying about making mistakes, then the most important thing you have to do is make sure, they are not punished for making mistakes.

Avoid even the most subtle things, that make mistakes look like a personality defect. Try to prevent situations where students are likely to make mistakes publicly, but when it exceptionally happens, make sure, that students do are not mean to each other, and enforce this rule consistently.

The second thing is to teach them well and set them up for success, so even if they make mistakes occasionally, it reduces over time, so they start having a reason to believe in themselves.

This will be difficult, as most students have prior negative experiences in math class, so you have to undo previous pedagogical mistakes.

Decoration is superficial, behavior is what matters. And I argue, that quotes and posters have much less of an impact on behavior than people think. I would even go so far as to remove most unnecessary decoration from the classroom, to reduce possible distractions.