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

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.

• 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. --- Perhaps have a policy of NOT erasing mistakes, but rather crossing them out with a big 'X'. You could also incorporate the idea of learning from mistakes by including questions/problems (verbal in-class, as well as written homework) that consist of solutions designed with specific intended mistakes that students are to identify, explain, and correct. – Dave L Renfro Sep 26 '19 at 7:10
• Grothendieck was used $57$ as an example of a prime number, so $57$ is now the Grothendieck prime. The effectiveness of that anecdote will depend on the level of the students, but I think it shows that even the very best mathematicians make silly mistakes. (Comment because it doesn't directly answer the question.) – user12756 Sep 26 '19 at 7:33
• Two Books: A great idea my maths teacher had was to have 2 writing books. One is the normal one where you put all the examples you try yourself. Then the other book is the special book which the teacher writes the notes for the subject on the board which we copy. He would then then go through some worked examples he had made up to highlight many of the issue you would find in the subject. We would copy those into the special book as well. It was really great. This meant you had a perfect writing-book to revise from with absolutely perfect examples. – Rewind Sep 27 '19 at 16:41
• There is a problem when a student has a high error rate and also lacks the skills to check their work or detect the mistakes. The ethos I try to instill in my students is that everyone makes mistakes, and it's OK to make them, but it's not OK not to catch them. Many students are extremely reluctant to check their own work, even when there are easy ways to do so, such as checking an integral by differentiating, or checking a factorization by multiplying it out. For a student who will be a pharmacist, it is not OK to make a mistake in converting grams to milligrams -- without catching it. – Ben Crowell Sep 27 '19 at 21:41

• 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.
• I didn't know of Jo Boaler, looks like she has written very interesting stuff on mathematics education. – dietervdf Sep 26 '19 at 9:03
• The Jo Boaler quote - When I start with a student studying for the SAT, I ask them to take a sample test. Not interested in the score. Just which questions they got wrong. I then focus on the source of those mistakes to identify any potential weakness in their skills. – JTP - Apologise to Monica Sep 26 '19 at 9:15

"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

• It seems that the 10,000 is not exaggerated. The number of designs tried by Edison before a workable electric light was found. – Gerald Edgar Sep 27 '19 at 11:01
• Should come with an obligatory note that Edison had an army of assistants who did the dirty work for him, and from whose ideas he profited. Tesla is more sympathetic to me than Edison. – Rusty Core Oct 1 '19 at 0:24
• @Rusty Core, I don't think we have precise information about any idea in history to judge such issues, but myself I don't think that Edison would have been famous and well known for his inventions if he just stole ideas from others, he must have been pretty smart and ofcourse he did most of the work himself using his own ideas and not only the ideas of others. Also I just found this article edison.rutgers.edu/tesla.htm, and I see that it is objective. – Fareed Abi Farraj Oct 1 '19 at 6:15
• @FareedAF I don't agree with your logic. There are a number of named maths theorems that are not the work of the people they are named after. – Jessica B Oct 22 '19 at 6:07
• @FareedAbiFarraj, those who are culturally in power often get the credit for the work of the less powerful working with them. Check out Rosalind Franklin's work on DNA, for one example. Many women and POC have not been given the credit they are due. Many white men have gotten credit where it was not due. – Sue VanHattum May 26 at 19:14

“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.”

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.

• Tapestry, TNG S6e15. "There are many parts of my youth that I'm not proud of… there were loose threads… untidy parts of me that I would like to remove. But when I pulled on one of those threads… it unraveled the tapestry of my life." – J... Sep 27 '19 at 17:29
• Thank you for the quote, great life lesson from Proust! – dietervdf Oct 23 '19 at 19:38

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.

Mistakes Allow Thinking to Happen

"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

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

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.

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

Anonymous

• I can't find this one online – aloisdg moving to codidact.com Sep 26 '19 at 13:42
• @aloisdg, true, but now you can!--Anonymous – user52817 Sep 27 '19 at 3:46

"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.

"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.

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

• Please don't use codeblocks for anything that is not actual code. Quoteblock exists specifically for quotes. – Nij Sep 28 '19 at 4:24

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.”

• I like the "slightly bad memory" remark. I think we (humans in general) tend to underestimate and undervalue the brain's capacity to forget, which is very important in learning mathematics. Here forgetfulness is shown as a positive virtue. There are other positive aspects of forgetfulness, but generally we are taught that forgetfulness is a negative defect. Indeed, there are things to remember in order to be able to solve mathematics problems. Given the brain's tendency to forget, being able to remember them takes preparation and work. – user1027 Oct 13 '19 at 15:41

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.

• Thank you for the answer. (This question got a lot more useful/interesting comments then I anticipated, love the community!) The quotes I was looking for aren't going change the pupils mind. But they are a nice introduction into the theme. Also to clarify, I don't want students to make mistakes, but they shouldn't erase them from their notes when they do. I have the feeling that some (or most) don't realize the learning potential in leaving (and marking) the mistake in their notes. – dietervdf Oct 23 '19 at 19:36
• @dietervdf But you should want them to make mistakes! That's one of the themes of all these quotes. We need to dare to make mistakes, if we are to freely explore. – Sue VanHattum May 26 at 19:20
• @WeCanLearnAnything I am inspired by your first point. Perhaps I will change my grading scheme somehow to reflect this idea! – Sue VanHattum May 26 at 19:22
• @SueVanHattum - Awesome! I've failed to think clearly about how students perceive incentives many times. I would love to hear about how grading scheme evolves. :) – WeCanLearnAnything May 27 at 19:40

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

Unknown

Here are a few of my favorites:

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

• I don't see how this is related to making mistakes. – Dirk Sep 28 '19 at 15:18

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."

• Thank you for adding some details to your answer. It is much improved now. – Xander Henderson Sep 27 '19 at 16:16
• The quote about Einstein is absurdly misleading (or even false, although I can't check the claim about speaking and reading). Einstein was an excellent mathematician, even from a very young age. Although he was refused admittance to Zürich Polytechnic initially (at age 16), it was because of his scores on other subjects than physics and mathematics, and he did "eventually" get in at age 17. – Mees de Vries Sep 28 '19 at 7:21

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 ignorant most of the time and to often make the many mistakes I criticized in correcting my students’ homework.”

Article in Quanta about Yves Meyer, 2017 Abel Prize Winner

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.
• Thank you for the answer. I like your suggestions too. Just to clarify, I don't want pupils to mix wrong calculations with the correct ones. The problem I am/was facing, was that pupils are working on problems and when the correct answer was being worked out on the blackboard, a lot of pupils who made mistakes just erased their answer and wrote the correct, blackboard version down. Instead of tracing down the error in their own solution, marking it, and then writing the right solution next to it. They didn't see the learning potential of their wrong answers. – dietervdf Oct 23 '19 at 19:44

“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

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
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.)

• Seems more like a life lesson, but I love it. Thanks! – dietervdf Oct 23 '19 at 19:32

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: