Matematica? È difficile
Ma non diamo i numeri

Perché la matematica ci fa tanto paura? Gli studenti delle Superiori, in Italia, si piazzano al di sotto della media europea, eppure l’Italia ha lasciato un segno profondo in questa disciplina, in ogni epoca. Ne parliamo domani, sul quotidiano, con il professor Gabor Toth, 66 anni, uno dei più brillanti matematici del mondo (ed egittologo, specialista di Medio Egizio), di cui sta per uscire un libro - intitolato “A Prelude to Advanced Calculus with a View to Mathematical Contests” - che smonta numerosi “miti” sulla matematica e offre un approccio al calcolo avanzato sulla base dell’esperienza maturata all’AoPS di Princeton (Academy Art of Problem Solving) come coach dei giovanissimi “geni” dei numeri, tra i quali si annoverano molti vincitori delle Olimpiadi Americane di Matematica. Per ragioni di spazio non tutta la conversazione con il professor Toth è pubblicabile. Di seguito il testo integrale della conversazione con La Provincia.

Gabor Toth (born in 1954) is a world leading mathematician. Distinguished Professor since 2017 at the Rutgers University in US, he is also a coach at AoPS, Princeton’s Academy Art of Problem Solving, where the most gifted and talented young students are trained to attend the USA Mathematical Olympiad. Worth noting that this beautiful mind, native Hungarian, who speaks currently English, Chinese (Mandarin), Russian, Franch, Hungarian is also the author of a Middle Egyptian Grammar and is member of Egypt Exploration Society. Like the scribe Irtysen, Gabor could really say: “I know hieroglyphs”.

A recent survey made by OCSE revealed that Italian students (15-years-old) scored 487, coming just under the European average, which is 489. Why is mathematics a so though subject?

The difficulty of studying mathematics lies in its high level of abstraction, and its symbolic expressions. For most students it is very hard to grasp abstract mathematical concepts and reasoning. This natural resistance to learning mathematics is not new, however. For example, the Roman numeral system is very cumbersome to do basic arithmetic, but it dominated Europe till the late 14th century, until it was replaced by the far more superior Arabic positional numeral system, largely by the efforts of the great Italian mathematician Leonardo Pisano Bigollo commonly called as Fibonacci. As another example, although the concept of irrational number can be traced back to the Pythagoreans, and it is implicitly discussed in Plato’s Theaetetus in a dialog with Socrates; it took mankind over two millennia to arrive at a clear understanding of these numbers and a precise definition of the real numbers in general. As yet another example, European mathematicians resisted using negative numbers until the advent of calculus (mostly because negative numbers represented debt, something to fear of); and this reluctance is clear in the Ars Magna of the Italian mathematician Gerolamo Cardano (as he refused to move a term with positive coefficient to the other side of the equation). So, when I see a child struggling with these abstract concepts, I always think of history.

Professor Toth, how to ease up the complexity of mathematics, in general?

There are many well-established methods to bring down the abstraction and symbolism mentioned above to a level that an average student would be comfortable with. One is to introduce “real-life problems” where the abstract concepts can be interpreted in easy and clear forms. Nowadays this method is often abused in the sense that authors of mathematical textbooks often invent artificial situations to achieve this goal. For example, to introduce the concept of “percent” in a pre-calculus textbook I have seen pictures of ladies shopping for clothes and busily calculating sales taxes. In this particular case a much better approach would be to explain the real origins of percent by going back to ancient Egypt, where a scribe would go out to the farmer’s land to assess the damage caused by the inundation of the Nile river, and to determine the amount of tax to be paid to the crown accordingly. Another, related method is based or real and imaginary associations of abstract concepts with some easier understood notions. I have seen, for example, explaining complex algebraic expressions involving multiple parentheses, such as 2(x+3(x+1)), by associating the first (outside) parenthesis to the “federal government” and the second to the “local government.” Although some students apparently like these, I have my own doubts in these artificial means.

Since most people are visual, in classroom I use the most basic association; visualization of algebraic concepts. Once again, this is not new; in fact, it takes us back to the ancient Greeks who created Geometric Algebra. For example, the binomial formula (x+y)2 =x2+2xy+y2 is best explained by taking a square of side length x+y and cutting it up into two smaller squares and two rectangles, and calculating areas. When a student understands this, the geometric picture will be imprinted in his or her mind, and will never make the typical mistake (x+y)2 =x2+y2 again.The ancient Greeks proved many beautiful results using this so-called cutting-and-pasting method, but almost none ended up in the present day school curricula. As a final example, it is not very well-known but easily understood that irrationality of the square root of 2 can be shown by playing origami; folding corners of a square paper.

How to encourage children to approach numbers and problems in a different way?

I would first generalize this question to “How to encourage children to take interest in mathematics?” First, most parents would like to see their child to show great talent in music and/or mathematics at a young age, and they spare no time, effort or money to this purpose. They compel the children to take music or mathematics lessons even though they may show no apparent inclination to these. They envision the child to become a great scientist, a famous concert pianist, or a noted mathematician. It is very important to recognize that there are many (at least nine) different types of intelligences other than what are called musical and logical-mathematical intelligences, including, for example interpersonal, naturalist, bodily-kinesthetic, etc. intelligences. An intelligent child “misdiagnosed” by parents would suffer through childhood by peer pressure continually exerted by the parents. On the other hand, if a child does exhibit early signs of number/reasoning smarts; for example, he or she spends a lot of time on quantitative games (for example, games promoting arithmetic), and/or shows ability of doing basic logical reasoning at an early age, then my best advice is to spend as much time with the child as possible, and steer and promote his or her interest in complex thinking. Italy has a long tradition in doing this, for example Fibonacci’s famous book, the Liber Abaci contains many interesting and elementary examples in number theory, and in studying geometry, Fra Luca Pacioli’s Da Divina Proportione is an excellent early example. I do not know the Italian system, but in the US there are many standard tests (beyond the various forms of the generic IQ tests) that specifically measure a child’s quantitative intelligence, and it is done at the age 3-5. If a child is classified as having above average intelligence1 (in the US it is given by a percentile compared to the entire population; for example, a child with 98% or above is considered to be a prodigy) then there are special programs and Summer schools that they can attend. It is also important to note that unless the parents have the time and ability to do it themselves then further training should be done by professional educators.

You teach at AoPS (Art of Problem Solving), an academy in Princeton for US talented high schools students among whom there are most of the winners of the USA Mathematical Olympiad. Are they “geniuses”? In any case, how they skills can be further enhanced?

I use the word “genius” very sparingly as geniuses are extremely rare, and we know most of them. The Academy in Princeton where I teach is a hub of many talented children who love mathematics. Most of them exhibit great analytical abilities, but only a few have extraordinary talents. These few I call “child prodigies.’ They compete fiercely in a class room in problem solving sessions and in mock contests, where I play the role of a moderator. They have very fast minds but need a lot of directing and steering. As for your second question; without previous training even a true genius would probably not do well in the USAMO since tackling Olympiad level problems require a wide ranging and well developed set of problem solving strategies. These strategies I teach at the Academy. Coming back to your previous question; all my students have been recognized as having mathematical talents at early ages, all had enormous influence from the parents, and some have been trained by parents who are either scientists or engineers. For example, in my class it became apparent in a discussion that one of my students, a 12-year old girl knew how to write computer programs in C language. I asked her how did she know and she said “my father taught me.” Finally, even though highly trained and mathematically inclined parents at home obviously have an advantage, there are many examples of sheer parental influence without the know-how. The best is probably one of the greatest mathematicians of the 20th century, the Indian mathematician Ramanujan, who rose from extremely humble beginnings to Cambridge through an early influence of her mother.

I am curious about how you became a mathematician. Were you an inquisitive student? Very intuitive?

My path to become a mathematician was somewhat circuitous. I was born, raised, and educated in Hungary. Although my father, who was an architect and would have liked me to become one, encouraged me to learn mathematics, in middle school I actually wanted to become a soccer player, so much so, that after school I usually headed for the nearby soccer field and played soccer until the evening. My grades in mathematics were usually C’s (which made my father very worried). It all changed when, as a freshman in high school, I was accidentally seated next to the Hungarian national champion in mathematics. We soon became friends, started competing, and in senior year I ended up doing mathematics round the clock. This strong love for mathematics is still with me up the present day. As to your second question, to become a tolerable mathematician one needs to have intuition and computational skills. Although there is a debate about this in the circles of early childhood psychology, I still believe that intuition is something that one is born with (or at least it is acquired at a very early age), and it may be recognized in a child at various ages. In my example, some fellow mathematicians told me that I have great intuition (which I was completely unaware of) only at around senior year at the university, when I started doing research. As for the second, computational skills can be learned in school, mostly in colleges and universities. Acquiring these skills one needs to do a lot of exercises and computations.

One of the special features of your book is that is a myth breaker. I was surprised that nowadays scholars have doubts whether Pythagoras of Samos ever did any mathematics… would you please quote briefly some of these historical inaccuracies?

It is a very long list. In fact, it is closer to the truth to say that up until the end of the 19th century, almost no new mathematical result was named after its own inventor. Coming back to your question, we simply do not know who proved first the “Pythagorean theorem.” Scholars guess that the first proof was given by one of the Pythagoreans, Hippasus of Metapontum, but almost nothing is known about his life apart from the fact that he was drowned at sea (as punishment of the gods for revealing an unrelated discovery). This result, actually the converse (if the side lengths a, b, c of a triangle satisfy a2+b2=c2 then it is a right-triangle), has been used in architecture by the ancient Egyptians, and, somewhat earlier, by the Babylonians. This was over 1000 years before the Pythagorean school was established in Southern Italy. It is fairly clear, however, that it was the ancient Greeks who recognized that this fact has to be proved.

A few other historical inaccuracies in this long list are as follows. (1) Gaussian elimination appears first in the ancient Chinese text “The Nine Chapters on the Mathematical Art” (10th-2nd century BCE) predating Gauss by about two millennia. (2) The Archimedean Property of natural numbers can be traced back to Eudoxus of Cnidus. (3) The famous Bernoulli inequality is not due to Jacob Bernoulli, but it is probably due to Sluse communicated to him by Newton’s great teacher Isaac Barrow. (4) The Fibonacci sequence was known to the Indian mathematicians around the 6th century CE. (5) The Pascal triangle was known to the Indian mathematician Pingala around 200 BCE from the Vedic period, and it also appears explicitly in Persian and Chinese text about 600 years before Pascal.

How to emphasize analytical thought? Have you some basic recommendations for Italian teachers? (You, for instance, encourages student to ask questions.)

First and foremost, for the Italian teachers I recommend reading the popular writings of the Nobel laureate Richard P. Feynman such as “The Meaning of It All,” “The Pleasure of Finding Things Out,” the “Six easy Pieces”, etc. He discusses physics but most the material applies to sciences and mathematics as well. More specifically, I not only encourage students to ask questions, but make it clear to them that if they have no questions then they will get nowhere. But asking questions is just the beginning of the process. Mathematics and most of the theoretical sciences require a great deal of individual thinking, not just attending the lectures and reading the textbook. A good mathematics student “lives” with the problems, continually thinks about complicated structures, plays with complex ideas, and tries to find different ways to approach them. The role of the teacher is absolutely vital as a guide and as a mentor. In this role they should never tell the final solution but lead the students toward it. I would also recommend to make many side tours, avoid standard clichés, and be as original and inventive as possible. Encourage the students to take unknown or untrodden paths, let them discover things on their own, and shock them with interesting mathematical ideas. Finally, two very important principles: (1) The idea to solve a problem is more important than the actual (often numerical) result; (2) The process to get through a problem should be the real enjoyment, not the final result.

Your text takes a fresh look at many mathematical concepts. One of the main challenges is the concept of infinity and limit: What is new about it?

Yes, one of the greatest challenges in mathematics a student can face is the concept of limit and, related to this, the concept of infinity. The challenge is twofold. First, as the ancient Greek philosopher Zeno of Elea was already aware of, the concept of infinity can quickly lead to contradictions. Second, on the college level, the concept of limit, for an average student, is very complex both theoretically and symbolically. What I propose in my book is the exact reverse of what is usually taught in colleges; first introduce the concept of limit superior and limit inferior, which always exist and in many cases computable, and then, as an application, derive the concept of limit from these. This path is longer and more tedious, but at the end, it is more rewarding. Finally, another pedagogical novelty is that I treat limits is multiple levels, both naïve and axiomatic, and in multiple aspects, through infinite decimals, infinite sequences and sums, etc.

In chapter 4 of your book you give an interpretation of the Babylonian method of extracting square roots. Would you please give us some details in brief?

The Baylonian method to approximate the square root of 2 by rational numbers is important in at least two aspects. One is that it is a very fast algorithm in the sense that in a few steps one gets a very close approximation; for example, the 6th step approximates square root of 2 by the fraction 886731088897/627013566048 which is about 10-24 precision! The Babylonians did not get this far, they did not need it, but they were able to obtain the correct value of square root of 2 up to 5 decimal digits. The second is that it is a special case of the Newton’s Method, which is widely used today to obtain good approximations of roots of polynomials and other functions.

You quote some Italian mathematicians of the past, as benchmarks for this subject (Peano, the inventor of letter Q, quoziente). Have you ever worked or studied in Italy?

I am a great admirer of the achievements of Italian mathematics, and have great respect for the Italian mathematicians. Italy played a crucial role in my early career since, while a young budding mathematician of the Hungarian Academy of Sciences, I visited Italy multiple times. Between 1980-1990 I had several visiting professorships, invited lectures, and conferences in Italy, for example, I visited the ICTP in Trieste, the Universita di Cagliari, and the Istituto Nazionale in Rome. I worked with several Italian mathematicians and published a few papers in Italy. In my early days I went to Italy so many times that my mentor, James Eells, a renowned American mathematician and the director of the mathematics section of ICTP then, desired me to settle down in Northern Italy and become a professor of mathematics. At that time I was already in the US, and so this idea did not realize. Many years later, on a tour in Europe, I did go back to Italy and visited the Scuola Normale di Pisa.