Learning Mathematics

A suitcase full of rags

1.In the summer of 1999 I enrolled in computer engineering at the University of Rome La Sapienza. I had spent the five years of secondary school at a modest provincial school specialising in the classics, and the collision with that new world came as a shock.

2.Mathematics has always been one of my favourite subjects, not because I had any particular gift for it, simply because it drew me more than Greek and Latin, more than biology and art history. Still, knowing that what I had learned up to that point fell short of the threshold required for a degree that was anything but easy left me disheartened.

3.I had no idea that sequences and numerical series existed.

4.I had never solved an integral.

5.I had only heard of them in my last year of school, from the cousin of a classmate of mine who was studying computer engineering at the same university I would later enrol in, and the way he described them made them sound like hostile things.

6.I knew complex numbers by reputation alone, essentially through the relation $i^2=-1$ and the fact that a second-degree expression with discriminant $\Delta \lt 0$ admits no real solutions.

7.I thought geometry came down to the study of right triangles and plane figures. I had no idea what a vector or a matrix was, nor how one solves systems of linear equations in several variables.

8.I realised that the baggage I was carrying looked nothing like a suitcase, but rather like a crumpled handkerchief, tied any old way around a stick, entirely unfit for that journey.

9.Classes began in September. The first six months were brutal. The gaps in my preparation kept me from following the lectures. I gave it everything I had, and the results were poor. At some point around April, near the end of the first year, I realised that if I wanted to give myself a chance, I had to start again from the beginning.

10.I had no idea how or where to begin, and I feared it would not be easy at all.

Learning by heart

11.At the time I lived with my parents in a small house a few kilometres outside Rome, two floors, a garden, several cats and a Maremma sheepdog. To get to the university I took the six o'clock train every day and moved around the city by public transport. Classes started at eight. The journey was tiring, but I had reading as a hobby, so I buried myself in books and the trip felt shorter than it was.

12.Not far from our house lived a couple, she a doctor, he an engineer and an executive at the largest Italian oil company. They were my parents' age, family friends of long standing, and the two couples saw each other regularly. Whenever the occasion arose, he made a point of asking how I was getting on. He was a kind man, always smiling, unlike me, who in that period had my morale on the floor, was set off by the smallest thing, and found that even curling my lips into the imitation of a smile felt forced. In any case, one day I told him about my difficulties and he listened to me for the whole afternoon.

13.Wise people do not give advice, and he gave me none. He only told me that if I wanted to understand mathematics, I would have to learn it. There is a substantial difference between understanding something and learning to do it. The distinction is not merely semantic, it concerns above all the neurological processes involved, which differ in part between the two cases. To understand is to grasp the meaning of things. To learn is a complex ability acquired through experience, practice, repetition and memory. It is a mechanism that engages the motor cortex, the cerebellum, the basal ganglia and the sensory systems, and it is built to stabilise a given piece of knowledge or a given skill over time.

14.At the base of all of it lies memory.

15.In primary school children learn to count, they do not understand how it is done. In time one moves from learning to add whole numbers to multiplying them together. That $2+2+2$ gives $6$ becomes as natural as learning that the same sum can be written in the compact form $2 \times 3$. Level by level one takes in that the product $2 \times 2 \times 2$ equals $8$, that the same result can be written in the abbreviated form $2^3$, that the exponent $3$ is the logarithm of $8$ in base $2$, or that the square root of $12$ can also be written as $\sqrt{12} = \sqrt{3 \times 4} = 2 \times \sqrt{3}.$

16.There is no way to know your times tables other than learning them by heart, and that means repeating them, repeating them and repeating them again to the point of exhaustion.

17.A study published in The Times showed that out of 317,000 primary school pupils, 27% of those surveyed got the product $9 \times 6$ wrong. I cannot offer statistics to support this, but I have the feeling that if you stopped thirty-year-olds at random in the street and asked them point-blank what $7 \times 8$ or $8 \times 9$ comes to, the results would not be much more encouraging. Let us try, for example, to solve this simple indefinite integral:

$$\int \frac{1}{1+x^2} \ dx$$

18.If one does not know that this is an integral whose antiderivative is $\arctan x + c,$ with $c$ an arbitrary constant, there is no point trying to obtain the result through simple manipulation. It is not an integral that yields to routine steps. Knowing the list of the most common integrals by heart, on the other hand, lets you recognise it and solve it in an instant. Or let us try to solve this limit:

$$\lim_{x\to 0}\frac{\sin x}{x}$$

19.Using the bare definition and substituting $x=0$ into the expression, one obtains an indeterminate form that says nothing about how the expression behaves as the limit approaches $0.$ The one just proposed is a remarkable limit and the result is $1$. The proof is not particularly complicated, but when an exercise has to be solved it is far more useful, if not necessary, to remember that this limit is exactly $1,$ rather than losing yourself each time in pointless calculation.

20.This point about the role of memory may seem trivial, but in fact it is anything but. To learn mathematics it is essential to memorise a great quantity of information, rules, simplifications and problem-solving tricks. There is no other way. Theory has its use, but in the right dose.

21.It is like learning a language.

22.If you concentrate too much on the grammar, you will never speak it fluently.

It's not about talent

23.In my first year of primary school the teacher suggested to my parents that they enrol me in a music school. I remember that in class we did simple rhythm exercises with percussion instruments, and they came to me with a certain ease. I had learned to plunk out the melody of Vangelis's Hymne on a melodica, a tune that in those years had reached a peak of popularity in Italy thanks in part to a television commercial for a well-known pasta brand. I did not just get the notes right, I gave it a pinch of personality by lingering over the longer slurs and the rests, like a truly seasoned musician.

24.A few weeks later I began taking private piano lessons and discovered that, besides delighting me, it came very naturally. At eight I played Bach and Mozart better than many children my age. Hearing a piece on the radio a couple of times was enough for me to reproduce it on the keys by ear. In my high school years I had a Korg keyboard connected to the computer's sound card, and with it I composed, arranged and produced my own songs. Time is the foundation of music, which is quantised into quarters, eighths, sixteenths, triplets and so on. Fractions and music go arm in arm, and I was decidedly good at both.

25.Around the age of thirteen, an acquaintance of my father's heard me play and recognised talent in me. So he proposed an informal audition with a professor at the Rome conservatory. I was proud and thrilled at the idea of measuring myself against professionals from such a prestigious institution, and I spent an entire weekend practising at the piano.

26.The meeting took place a few days later in a bare little room at Santa Cecilia looking onto the cloister of the Ursuline convent. The professor had me play a piece and listened to the end without interrupting. Then, in the kindest way he could, he told me: playing is not striking the keys with your fingers. It means putting intention into it, reading the score and knowing how to interpret it as the composer would have wanted, judging the use of the pedals, and a heap of other things. Heart is not enough, technique is essential.

27.And I did not have it.

28.I went home disappointed and angry, because it seemed to me that I had wasted years merely getting by at something that, when it came down to it, I had not mastered nearly as well as I had imagined. So I decided to work as hard as I could to improve. I changed music schools and turned to material as boring as it was gruelling, made of mechanical exercises for the joints, scales and chords to train the mobility of the fingers and the muscle memory of the forearm. My repertoire swerved sharply towards Hanon, Clementi and Pozzoli. Anyone who has been through it will know what that means.

29.I wanted to master the technique and, in a short time, I saw that I would have to practise to the point of collapse. Within a few weeks I realised I had made incredible progress: the way I played the classical composers had improved noticeably, and a firmer command of technique let me concentrate more on expression and give the pieces a musical depth I had not until then realised I could reach.

30.The same holds for mathematics and, in general, from what people say, for every area of life. Each of us may have our own inclinations and preferences, but I am convinced that results worthy of respect can be reached even in the disciplines we love least, if we are willing to face them with the effort and the constancy they require.

Practice, practice, practice

31.Being able to solve a linear system with Cramer's rule or an integral by substitution is not merely a matter of mathematical intuition. Or rather, intuition has its use, but it is not an innate gift possessed by a few. It can be acquired and developed through practice. Take a very simple example:

$$\log_a(x \cdot \cos x )$$

32.If one does not know the property of the logarithm of a product, it is hard to guess that the expression can be rewritten in the equivalent form:

$$\log_a(x \cdot \cos x)=\log_a x+\log_a(\cos x)$$

33.It has to be taken in and memorised first. Only then can intuition establish the necessary connections and suggest to us that, in order to work through a given expression, we may fall back on one of the two equivalent formulations. Which of the two to use depends on the context, and the more one practises, the more one becomes able to pick the appropriate one automatically.

34.Let us take another example, with an integrand containing the following expression:

$$\sqrt{a^2 - x^2}$$

35.Integrals of this kind are typically solved by substitution, and the most immediate one consists in setting $x=a \sin t$ and, differentiating with respect to $t$, setting $dx = a \cos t \ dt.$ The reason for this choice lies in the Pythagorean identity $\sin^{2}x + \cos^{2}x = 1$, which in this case allows the expression under the root to be simplified. Here too intuition is not much of an ally if it has not first been trained, through practice, to recognise this kind of recurring pattern in integrals.

36.Contrary to common belief, most of the mathematical problems one faces from secondary school through the first years of university involve little more than mechanical procedures. But, as with learning to play the piano, it takes a great deal of practice to apply those procedures with ease and acquire the necessary confidence.

37.In any case, by the end of the first year I had managed to pass only two of the five exams on the syllabus. I did not lose heart and decided that I would spend the three summer months starting again literally from zero. I dug out my old school books, stocked up on photocopies of the new topics and started over from the basics: powers, radicals, exponentials, logarithms, trigonometry, equations, inequalities, systems, complex numbers, series, sequences, derivatives, integrals.

38.I began early in the morning and finished in the late afternoon. I copied out obsessively on paper the formulas and the properties of the main mathematical objects, and filled pages with exercises until the results came out right. After the first few weeks I felt I had become steadier and more at ease. The more I learned, the more I wanted to take on increasingly complex problems, and before long my sessions stretched several hours past dinner.

39.When September came and classes started again I was at another level. I caught up on my exams and carried on, repeating the same approach of memory and practice I had adopted over the summer. In the operations research exam I got 28 out of 30. On 11 January, four of us turned up for the exam in discrete mathematics for electronic computers. I had spent all of December studying. I got 25, the highest mark of anyone.

40.I did nothing exceptional, mind you.

41.But I had realised that, by carrying on that way, I would reach the end of a journey that, only a few months earlier, had seemed impossible to complete.

It is not only a question of method

42.Each of us has at least once in life found a study method that works. There is no universal recipe for everyone, so steer clear of anyone who promises otherwise. Each person learns through a combination of experience and associative processes that depends a great deal on the way the individual brain organises and consolidates information. One thing, though, is beyond doubt: one cannot learn without training.

43.Modern large language models such as ChatGPT, Claude and Gemini are trained with billions of parameters before they are able to identify the correlations needed to produce an output consistent with the input they receive. This kind of training, generally very expensive in terms of both time and resources, involves numerous cycles during which, put very simply, such systems gradually learn to reduce statistical error and produce increasingly accurate results. The correlations learned during this process form a kind of distributed memory used to process new inputs with increasing accuracy.

44.Memory, repetition, training, error reduction.

45.The way large language models "learn" is, in certain respects, analogous to the way mathematics is "learned", at least in the sense described so far. I am not saying it is a universal model, but I believe it comes close, at least in its underlying principles. After all, no one would ever let themselves be operated on by a surgeon who had only studied books, without also training in an operating room.

46.What I mean is that the difficulties many people encounter when studying mathematics do not necessarily arise from the intrinsic difficulty of the subject matter—although that certainly plays a part—but above all from a lack of awareness of the fundamental role that memory and practice play in the process of understanding and learning.

47.When I started writing the first entries of Algebrica, in May 2023, I tried to structure the content so that it would be easy to use, easy to set out in schematic form and easy to memorise, cutting away the superfluous and keeping what I judged truly necessary. I hope this approach can help the many people who use the site regularly to understand mathematics better and to get past the barriers that often make it appear harder, more distant and more inaccessible than it really is.

48.

Author's note: this piece was translated from Italian and may contain some linguistic nuances that do not fully reflect the original version, as English is not my native language. Please don't be too harsh. Copyright © Antonio Lupetti. This material may not be reproduced without prior permission.