I appeal to all my readers for their personal stories of difficulties they experienced in (early) learning of mathematics. I have discovered that such stories provide a fascinating insight into psychology of mathematical thinking and are frequently linked to astonishingly deep issues of mathematics. I’ll give you a few examples of what I am looking for.
Story 1. A girl aged 6 who could easily solve “put a number in the box” problems of the type
by counting how many 1’s she had to add to 7 in order to get 12 but struggled with
because she did not know where to start. Worse, she felt that she could not communicate her difficulty to adults. Her teacher forgot to explain to her that addition was commutative.
Story 2. A girl aged 6 who was also afraid of subtraction. She could easily visualise subtraction of 4 from 100, say, as a stack of 100 objects; after removing 4 objects from the top (by reverse counting: 100, 99, 98, 97), 96 are left. But what will happen if you remove 4 objects from the bottom of the stack?
Story 3. A boy aged 9 who could easily factorise two-digit numbers and knew that, say 42 is 6 times 7, but had difficulties with the times tables and could not answer what was 6 times 7. In his learning of mathematics, division preceded multiplication.
Please send me your stories. You can write at
or leave your comments at this post. Please also give me the following details:
1. Your age when a particular episode had happened.
2. Your gender.
3. What was the language of mathematical instruction? Was it different from your mother tongue?
4. What is the level of mathematical education that you have eventually got? If your occupation is mathematically related, what is it? Are you a teacher?
Do not be afraid that your story perhaps repeats an already told one. In the examples above, Stories 1 and 2 are interesting exactly because they deal with the same underlying mathematical difficulty.
My warmest thanks! Alexandre Borovik
Since Physics is the natural operative math of the universe
should we not solve the physics riddles first that will allow better survival by giving those riddles mathematical clarity?
(As a kid I always had trouble with 5+7.)
18, Male, English (same), About to start postgraduate mathematics.
The first conceptual hurdle I can recall having was that of treating functions as objects to manipulate. During my real analysis course I was OK with discussing series of real numbers but found it required a quantum/discontinuous leap to become comfortable with things like the Weierstrass M test. In addition, I initially found it strange to apply the Banach contraction mapping theorem to obtain a function as the fixed point.
“Pick any function you like and apply this transform over and over; regardless of your initial choice the result is unique.”
“What if you start with something weird? There are a lot of functions to choose from.”
“It doesn’t matter.”
“I now control functions!”
“Unlimited power is yours….”
[...] I need your stories [...]
1. About 10-11
2. Male
3. Russian, the same.
4. Graduated from Meh-Mat Department of MSU (MGU), PhD in (theoretical) computer science. Lecturer in CS.
I read about the Fundamental Theorem of Arithmetic when I was about 10 or 11. I felt very uncomfortable about the proof of the Theorem. The statement seemed so obvious that the proof appeared to be absolutely redundant, not adding anything to understanding and making me worrying that I am missing something. I would be then much happier rather to accept the Theorem as a self-evident axiom.
1. about 9-10 / grade 4
2. Male
3. Indonesian
4. graduated from Mathematics Department of Bandung Institute of Technology (BSc). was a teacher in one of IB school in Indonesia. right know in a math-course.
one of my reason i really, really love maths whenever i buy a math book from Brian Bolt, about maths games and amusements. it’s really cool. at that time of course for me! “)
it’s all about numbers. i don’t know why i really like numbers. from ordinary numbers, prime numbers, fibonacci, pascal’s triangle until about decoding / cryptography. it’s awesome!
well. i think that’s all from me.
hello from Indonesia.
and feel free to visit my blog “)
[c] Master.Mister.Matematik! das Wunder!
[...] I need your stories [...]
[...] I need your stories [...]
Age: 10 or so. I think. Gender: Male. Language: Finnish.
I am an undergraduate mathematics student right now and will continue studying further.
I had trouble remembering if a/b means a divided by b or b divided by a. There’s no way to tell. It is arbitrary.
Likewise, some years earlier, remembering which side is left and which is right.
[...] I need your stories [...]
[...] I need your stories [...]
1. 9 years
2. male
3. American English, same
4. currently a second year student in a PhD program in pure math
When I was in the fourth grade (about 9 years old), we learned long division. I had enormous difficulty learning the method, though I could divide 3 and 4 digit numbers by 1-2 digit numbers in my head. I don’t recall exactly how I did the divisions in my head, though I suspect that the method was similar to long division. I recall that it was broadly based on “seeing how much I needed to add to the result to move on.” However, I couldn’t seem to remember long division, despite being able to follow a list of instructions on homework. My homework on long division took hours to finish. I think the issue was that my teacher and parents never explained *why* long division actually worked, so it seemed like a disconnected list of steps that had little relation to one another. On a quiz, my teacher thought I had cheated, since I had written no work on any of the problems.
I only became able to do long division in high school when I learned how to do long division with polynomials. At that time, the teacher went to great pains to explain why it worked. While doing a homework, I had an epiphany: long division for polynomials was a close cousin of long division for numbers. Suddenly I could do long division of numbers.
I now take pains when teaching calculus to try and explain why formulas are true whenever possible. This has lead to mixed reactions on my evaluations, with some students saying that they enjoy knowing why the formulas are true and others saying that they have enough trouble learning formulas in the first place without having to *also* learn why they’re true. For me, the two are deeply connected. If I don’t know why something is true (at least in broad outline), I find it significantly harder to remember.
I would like to point out that the stories are provided by the people who happen to stumble on this blog. I am sure that the readership of this blog is atypical in terms of mathematical thinking and learning. Most people who have responded have not only gone to deal with unusually abstract concepts in their career, but actually do mathematics. So, the examples here might represent not so much the major difficulties that need to be overcome (as in finding the correct way of thinking of division apples by apples) before an understanding can be reached, but the signs that the understanding has already been reached, and difficulty is purely semantic, i.e. how to express it.
My own story (11 or 12 years at the time, male, Russian, currently PhD student in pure math): In our math circle we covered induction (domino analogy, proofs of summation formulae such as 1+…+n=binom(n+1)(2), and varied other examples). I did passably well on the problems, but still I did not understand what the induction is really for, until the end-of-year competition. I failed to solve a single problem: arrange all binary strings of length 10 around the circle so that two adjacent differ in precisely one position (it is known as cyclic Gray code of size 10). It was when I was told the solution that I felt that I finally understood the induction. The missing element was probably the fact that I did not realize that the statement proved by induction is an honest mathematical statement that pertains to concrete numbers like 10, and not only to x,y,n,m and 1996, among which only the latter is a number, but so big and arbitrary that it could as well be denoted by n.
A slightly later example: From the time I learned matrices (age 16 or so) I cannot remember which are the columns and which are the row. Given that arrangement of coefficients in a linear transformations can be written equally well in a matrix in two ways, it is something that always takes me 10-15 seconds to recall even now.
Finally, a very recent tidbit (age 21): I read a very nice paper, where the author solves a genuine long-standing mathematical problem by reversing the binary digits of logarithms of primes. Psychologically it is revolting — primes are not meant to have their logarithms to be taken, and certainly reversal of digits of a number is such an unnatural operation that it cannot ever be used for anything! Well, that paper shows that, on the contrary, both operations can be very natural when facing the right problem. The paper in question is I. Z. Ruzsa: “An infinite Sidon sequence”, Journal of Number Theory, 68 (1998), 63–71. This relates to unnaturality of carries, of course.
1. 9-10
2. Female.
3. English (my native language).
4. Eventually did a PhD in maths, currently confused about career plans. Not a teacher.
When I learnt about square roots for the first time, I assumed that there must be a way to calculate the square roots of numbers (because we had a symbol for it I guess).
So I spent a lot of time trying to figure out what the rule was but never got anywhere. Mostly I just squared numbers and then adjusted. e.g., for 61 I’d try 7.7, 7.8, 7.9 and then see that it had to be between 7.8 and 7.9, and then guess the next number according how far away the square of each of the numbers was, all the while trying to figure out what the rule was for what adjustment I should make. I spent at least two weeks working on it each night, after which I knew the square roots of various numbers to several decimal places (which is why I remember this) but still no clue how to calculate them. All the potential “methods” I tried were ridiculously simplistic and unimaginative.
[...] I need your stories [...]
[...] I need your stories [...]
1. 9
2. female
3.UK
4. Studied mathematics at university but not very happily.
I was a competent at mental arithmetic but did it as a learned trick without real understanding. I understood multiplication/ division eventually by eavesdropping on teacher explaining the concepts using (not real) squares of chocolate to a child who was struggling.
This experience was useful to my when I taught secondary mathematics for a while.
When I read your request, I thought I’d never had problems with math when I was young. But, like Tommi, I couldn’t remember which way division went. Of course, it’s made harder by the fact that it goes either way, depending on the symbol: 6 divided by 3 is the same as 3 gazinta 6.
But my inability to remember that didn’t get in my way. If I saw a division problem with harder numbers, I just put 3 and 6 in their place for a moment, to orient me, and then I knew what was being asked.
[You asked for problems from when we were young, but in college, doing proofs was a huge hurdle.]
(Female, age: from when division was introduced on into high school or even college, English-English, Masters degree in math, prof at community college)