Submitted:
21 June 2026
Posted:
22 June 2026
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Abstract
This study addresses the protracted global crisis in mathematical education and investigates fundamentally new, high-technology pathways to overcome it, moving systematically from empirical diagnosis to technological renaissance. The first part of the work provides a rigorous cross-national analysis of cognitive decline and the degradation of mathematical pedagogy in traditionally dominant scientific nations, specifically the United States, France and Russia. Measures to stimulate youth engagement with mathematics through immersion into the historical context of mathematical concepts and operations, combined with the author’s empirical experiments in testing students using historical-mathematical problems, are described. The second half of the study (Parts II–IV) elevates the discourse to the contemporary technological landscape of 2026, demonstrating how Reasoning AI—an advanced class of artificial intelligence—can be strategically deployed as an autonomous digital mentor. We illustrate how these advanced models can automate the longitudinal diagnostic screening of underlying cognitive processes, pinpoint structural misconceptions, and act as a “digital Socrates” that forces students back to rigorous logical principles rather than providing ready-made answers. Furthermore, the paper shows how Reasoning AI decimates routine programming barriers, enabling students to channel their mathematical intuition directly into the empirical exploration of non-linear dynamic systems and fractal topologies. Finally, the study warns against the anthropocentric risks of “AI sycophancy” and the mechanical patterns of standardized testing, arguing for a radical paradigm shift that restores mathematics as a living language for investigating the real world.
Keywords:
mathematical education
; crisis in pedagogy
; reasoning AI
; cognitive resistance
; digital Socrates
; historical-mathematical problems
; visual-logic problems
; non-linear dynamics
; academic integrity
; Vladimir Arnold
Introduction
The present study offers a comprehensive, critical analysis of the protracted crisis in global mathematical education and investigates fundamentally new, high-technology pathways to overcome it. The structure of this work is deliberately divided into two logically interconnected parts, moving systematically from the empirical diagnosis of cognitive decline toward the technological renaissance of intellectual discipline.
The first part of this work is based on a rigorous translation and thematic expansion of the author’s original Russian-language study published in 2021 in the journal Original Research (Originalnye Issledovaniya, Belgorod) [1]. Drawing upon extensive empirical and historical data, it provides unique, well-documented examples illustrating the steady degradation of mathematical education in traditionally dominant scientific nations, specifically the United States and France. In this context, the critical ideas of leading mathematicians and educators who have raised concerns about falling educational standards are examined and explored.
This section articulates the author’s conceptual vision for restoring the quality of mathematical pedagogy to the gold standard observed in the USSR during its peak period. Furthermore, it details the author’s practical experience in testing students using a specialized methodology of historical-mathematical and visual-logic problems [2,3]. These tasks are designed not merely to evaluate basic computational proficiency, but to serve as a cognitive catalyst, stimulating students to engage deeply with both pure mathematics and the global history of science.
The second part of the paper elevates the discourse to the contemporary technological landscape of 2026. It explicitly demonstrates how Reasoning AI—a fundamentally distinct, advanced class of artificial intelligence—can be strategically deployed not as an instrument for academic dishonesty or sycophancy, but as an autonomous digital mentor. We illustrate how these advanced reasoning models can automate the longitudinal diagnostic screening of students’ underlying cognitive processes, pinpoint structural misconceptions (such as the inability to operate with elementary fractions and proportions highlighted in Part I), and assume the function of a “digital Socrates.”
Part three shows how Reasoning AI, by completely dismantling the routine technical barriers associated with complex computer programming, it enables students to channel their mathematical intuition directly into the empirical exploration of non-linear dynamic systems and fractal topologies, thereby resurrecting the authentic culture of rigorous, proof-based logical reasoning in the modern era.
The fourth part provides examples of how to solve various mathematical problems using Reasoning AI, including problems 5, 6 and 9 from the Appendix.
1. The Global Decline in Mathematical Education and Measures to Stimulate Interest in Mathematics
Due to the processes of globalization, which demand the unification and simplification of educational systems, the quality of mathematical education across all tiers has deteriorated sharply worldwide over the last 20–30 years, with only a few countries remaining exceptional. The crux of the matter is that globalized actors do not require sovereign states with advanced educational systems, nor do they need a critical mass of independently thinking individuals; instead, they seek mere cogs to maintain their business models and political frameworks.
In the United States, Americans themselves used to say: “No star wars – no mathematics”. Commenting on these words, Vladimir Igorevich Arnold (1937–2010) — an outstanding contemporary mathematician, Vice-President of the International Mathematical Union (1995–1998), and Academician of the USSR and Russian Academies of Sciences — noted [4]:
“The deplorable fact that with the (temporary?) cessation of military confrontation, mathematics, like all fundamental sciences, ceased to be funded, is a disgrace to modern civilization, which recognizes only "applied" sciences and behaves precisely like a pig under an oak tree [short-sightedly devouring the acorns while destroying the roots of the tree that feeds it].”
Arnold placed the word "applied" in quotation marks because he shared the stance of Louis Pasteur (1822–1895), who famously asserted that there are no applied sciences, but only applications of science. In our view, it is these very applications that are currently overfunded without any objective necessity, regardless of the damage they inflict on humanity and the environment. Today, the overriding priorities have shifted toward nanotechnology, biotechnology, and information and communication technologies (ICT), rather than the fundamental sciences that form their very bedrock.
Vladimir Arnold continues: “The planned suppression of fundamental science and, in particular, mathematics across all countries (according to American data, they will need about 10–15 years to achieve this) will inflict damage upon humanity (and individual nations) comparable to the harm that the fires of the Inquisition brought upon Western civilization (and Spain).” Arnold penned these words in 1997 in an article for the journal Mathematical Education [4], and six months later, he titled his piece in the Izvestia newspaper: “Mathematical Illiteracy is More Dangerous Than the Fires of the Inquisition” [5].
It is universally acknowledged that to engage in serious mathematics, a professional mathematician needs nothing more than a pen and a blank sheet of paper. Consequently, as V. I. Arnold notes, the budget of the Mathematics Division of the Russian Academy of Sciences was 40 times smaller than that of the physics divisions. Furthermore, according to the academic secretary of this division, President of the International Mathematical Union (1987–1990), and Academician of the USSR and Russian Academies of Sciences, Ludwig Dmitrievich Faddeev (1934–2017), the annual expenditures of the USSR on mathematics equaled the cost of a single military tank, while the expenditures of modern Russia amounted to approximately one-tenth of a tank per year [4].
All of this was written in the late 1990s, but today, there is a severe shortage of teachers in schools and universities capable of cultivating thinking mathematicians. This constitutes a genuine catastrophe. The wealthy West, while dismantling mathematical education in the rest of the world—and particularly in Russia, with the aim of minimizing the percentage of thinking individuals—simultaneously destroyed its own mathematical education under the pressure of transnational capital. However, the West could always import the finest mathematicians and other scientists from abroad.
When reading foreign papers across various fields of knowledge or watching scientific chronicles on television, one clearly observes that it is Asian researchers who are now driving science forward there.
V. I. Arnold noted that at the turn of the last century, only 1% to 2% of American schoolteachers, depending on the state, were capable of dividing one fraction by another, while American university students could no longer add two simple fractions, instead adding their numerators and denominators separately and dividing the former sum by the latter [6].
However, if one lacks the skill to operate with fractions, performing algebraic manipulations becomes impossible; consequently, one cannot calculate even the simplest derivatives and integrals, which in turn precludes the ability to solve differential equations, and so forth. For this reason, higher mathematics education in the United States and even in the most developed European nations, such as France, has been reduced to the rote memorization of tabular derivatives and integrals, or looking them up via smartphones. Even when an instructor attempts to explain the derivation of basic derivatives and integrals to students, the effort proves futile because the students completely lack a foundational knowledge of arithmetic.
About five years ago, the author of this paper was so astonished by V. I. Arnold’s description of the mathematics education crisis in the United States—specifically, the apparent absence of a ministerial standard for fractional computation—that he searched the Internet for this official standard. It turned out that the document did contain a fine-print entry, completely devoid of any commentary, illustrating how to find a common denominator: [(a/b)+(c/d)=(ad+bc)/bd]. It appears that V. I. Arnold’s persistence and his relentless crusade against global mathematical illiteracy during his tenure as Vice-President of the International Mathematical Union did eventually yield some results.
He cited the words of Henri Poincaré (1854–1912), who asserted that there are only two ways to teach fractions: to cut—at least mentally—either a pie or an apple. Poincaré argued that with any other method of instruction (whether axiomatic or algebraic), schoolchildren invariably preferred to add numerators to numerators and denominators to denominators [4].
Yet, that was the 19th century, whereas during the Soviet era, primary school teachers were so adept at explaining the addition of fractions that the vast majority of pupils faced no difficulties with it. Today, however, alongside the West, we have regressed back to the 19th century. Beyond the globalizing actors, mathematicians themselves have made a significant contribution to the dismantling of global mathematical education. This decline began with the introduction of the ideology championed by the famous French Bourbaki school. This approach, heavily anchored in set theory, was excessively formalized and systematically ignored standard branches of mathematics—the very disciplines without which real scientific and technological progress cannot occur.
Despite the universally acknowledged contribution of this school (group) to the development of abstract mathematics, its immense authority paradoxically facilitated the infiltration of its ideology into school and university curricula, and even into political discourse. As V. I. Arnold writes, the emasculated and formalized teaching of mathematics at all levels unfortunately became systemic, characterized by a proliferation of unmotivated definitions and incomprehensible (albeit logically flawless) proofs.
At the same time, the contemporary pedagogy of mathematics suffers from a total absence of concrete examples, diagrams, and illustrations, as well as a lack of extra-mathematical applications and conceptual motivations [4]. This trend has frequently bordered on the absurd. V. I. Arnold recalled a case involving a young fourth-year French university student who, after performing highly complex transformations, successfully arrived at the correct numerical criterion—equal to 4/7—to demonstrate the asymptotic convergence of an integral. However, the student proved entirely incapable of determining whether this number was greater or less than one! While he had diligently attended Arnold’s advanced courses on dynamical systems and differential equations, he had been taught basic arithmetic in school by inept didacticians and computer specialists [6].
In the same paper, Arnold provided another stellar example of a political nature. When the committee tasked with revising California’s school education standards, chaired by Glenn Seaborg (1912–1999)—a 1951 Nobel Laureate in Chemistry and a foreign member of the USSR Academy of Sciences (1971)—decided to introduce a university admission requirement stating that applicants must be able to divide 111 by 3 without using a computer, this level of difficulty proved insurmountable for the high school students. Such a modest requirement is deeply telling. Any Soviet schoolchild would have instantly calculated the answer—37—in their head. Yet, this initiative provoked immense outrage among politicians, who simply could not fathom how anyone could divide one number by another without a calculator.
The Washington federal authorities demanded the abolition of this "anti-constitutional, racist" standard. According to V. I. Arnold, one of the senators declared that he would never permit anyone, in any part of the United States, to teach anyone anything that he himself did not understand—such as dividing 111 by 3.
Another senator announced that the true purpose behind introducing these Californian standards (which, for instance, required familiarity with three phase states of water in a physics course instead of the two federally mandated states) was to create a racist obstacle to university admission for Black students, because "no Black person would ever comprehend your water vapor, which has neither color, nor odor, nor taste."
In this context, Arnold brings forward a conclusion published in 1781 by the third President of the United States, Thomas Jefferson (1743–1826), who asserted that no Negro would ever understand the geometry of Euclid or any of his modern commentators. Jefferson believed he spoke with authority, as he had raised many Negro children on his plantation [6].
Today, amid the ongoing momentum of the "Black Revolution" (Black Lives Matter movement) in the United States, a crusade against mathematics has been revitalized with a new force, as vividly reflected in the American mass media. For instance, the Department of Education in the State of Oregon identified "ideas of white supremacy" within the traditional mathematics curriculum. Objectivity and a singular focus on achieving the correct answer are now deemed manifestations of racism; instead, the authorities seek to introduce a new curriculum rooted in "ethnomathematics," as reported by Fox News on February 19, 2021. Remarkably, this is occurring in a state where African Americans constitute only 1.6% of the population, and Hispanics account for 8%. One can easily deduce what is transpiring in the Southern states under these conditions.
In another article, V. I. Arnold shared an anecdote about the French Minister of Education, who was deeply outraged by the inability of Paris’s top schoolchildren to add the numbers 2 and 3. The pupils routinely responded that "two plus three equals three plus two, because addition is commutative." The Minister then declared that the teachers propagating such methods should be expelled from schools, suggesting that anyone else—be it a chemist, an engineer, or a practical specialist—should teach children how to count instead. However, the influence of "Bourbakism" was so deeply entrenched in France that the Minister himself was dismissed from his post, and the Ministry was subsequently split into two independent departments [7].
Other egregious examples detailing the state of mathematics in France were documented by Dr. Viktor Stepanovich Dotsenko, a Professor of Physics and Mathematics who spent five years teaching mathematics and physics at one of Paris’s premier institutions, the University of Paris VI (Pierre and Marie Curie University) [8]. Echoing Arnold, Dotsenko noted that operations with fractions represented the most insurmountable barrier in students’ mathematical training, with up to 90% of undergraduates unable to handle them. Furthermore, many students struggled with the addition of natural numbers. For a significantly large percentage of students, the operation of raising any given number to the power of minus one yielded a result of zero, while three-sixths (3/6) was routinely calculated as one-third (1/3). This absurdity had been memorized dogmatically since their school days; the students exhibited a total lack of intellectual curiosity or surprise. When corrected and told that three-sixths actually equals one-half, they would merely respond with complete indifference: "Very well."Consequently, the author titled his study "The Fifth Rule of Arithmetic," implying that schoolchildren and university students now accept anything their instructors tell them purely on faith, without a shred of critical reflection. Dotsenko recalled a female student who used a calculator to compute the radius of the Earth and arrived at a result of 10 millimeters. She showed absolutely no surprise or skepticism. When explicitly asked to recalculate the figure, she pressed the correct sequence of buttons and obtained the right answer, yet she remained utterly indifferent to either outcome [8].
As far as can be inferred, university instructors—fully aware that secondary school graduates entering undergraduate programs via BAC (the French Baccalauréat, an equivalent to the Russian USE) arrive with immense foundational gaps and are practically untrainable—simply cater to their students’ deficiencies. Anticipating that these students will fail to solve any genuine university-level problem, professors conclude that exerting rigorous educational effort is futile. Instead, they assume that during university exams, much like during the BAC, students will successfully pass by utilizing smart calculators, a phenomenon previously highlighted by V. S. Dotsenko. Consequently, universities must continue to process this mass influx of students; after all, if the institution were to expel them all for academic ineptitude, the professors themselves would face unemployment. Furthermore, the faculty members are often heavily influenced by the dogmas of the Bourbaki school, whose core concepts frequently yield the most preposterous conclusions.
Proponents of this school posit that if two arbitrary propositions, A and B, are both true, then A logically implies B. To this, V. I. Arnold countered that from his supposedly "unenlightened" perspective, the statement "two times two equals four" would therefore imply that the Earth revolves around the Sun. A student trained under such a paradigm becomes utterly incapable of learning any natural science. As Arnold bluntly concludes, this brand of scholastic obscurantism completely decimates the very foundations of natural science [7].
When V. S. Dotsenko asked a student to solve the linear equation 5x+3=0, the student filled an entire page with dense theoretical arguments regarding the structure and cardinality of the solution set for this class of equations, yet proved completely incapable of computing the actual solution. In their mathematical analysis seminars, the concept of a derivative is introduced merely as a "prime" symbol placed above a function, while an integral is taught as a mere "squiggle" placed before it, accompanied by a set of formal, abstract manipulations performed upon them [8].
A well-known mathematical sophism asserts that any number is equal to any other number. Its premise unfolds as follows: suppose we have the equality a=b+c. Multiplying both sides by (a−b) and performing simple transformations yields a(a−b−c)=b(a−b−c), which seemingly implies that a=b. In standard mathematics, however, a−b−c=0, and division by zero is strictly prohibited. Yet, under the "Bourbakist" paradigm, zero is routinely classified as both a positive and a natural number; hence, for them, anything is permissible. This very mindset appears to be the root cause of the mathematical absurdity observed in countries heavily influenced by "Bourbakism."
Nevertheless, the "Bourbakists" themselves followed their own historical predecessors, such as Michel de Montaigne (1533–1592) and René Descartes (1596–1650), a historical lineage brilliantly outlined by V. I. Arnold in his essay “The Mathematical Duel Around Bourbaki” [7].
I vividly recall the immense popularity of the multi-volume translated works of Nicolas Bourbaki in the USSR, when every progressive mathematician or theoretical physicist considered it a matter of prestige to have these books on their shelves—though whether they actually read them is an entirely different matter. There was an undeniable fashion surrounding these writings at the time. Testing the keyword "Bourbaki" within the Google Books Ngram Viewer analytical tool reveals that the peak of this phenomenon’s popularity in the USSR spanned a twenty-year interval from the mid-1960s to the mid-1980s. Tragically, much like in other nations, this phenomenon exerted a deeply negative impact on mathematics education within our own country.
One can readily recall the Kolmogorov school mathematics reform of the 1970s, which had been preceded by a twenty-year period of preparation. Driven by benevolent intentions, it was conceived amidst a wave of euphoria following Soviet triumphs in science and technology, the apex of which was marked by the launch of the first human into space. This reform was spearheaded by our finest mathematicians. However, lacking insight into child psychology and the age-dependent cognitive capacity of pupils to assimilate mathematical concepts, they began authoring overly esoteric, abstract textbooks that proved incomprehensible not only to the schoolchildren but to their teachers as well.
The grim yields of this reform became fully apparent by the late 1970s, prompting severe grievances from university mathematics faculties who found themselves saddled with practically untrainable first-year students. This reform, its lengthy preparatory period, along with all its internal turbulence and key historical actors, has been thoroughly documented in literature [9,10].
To illuminate this outcome, we cite the definitive conclusion from the study by I. P. Kostenko [10]:
“The ‘Reform of 1970’ thoroughly purged pedagogy and methodology from textbooks, effectively banishing the Student from the educational process. It is directly responsible for the cognitive degradation of the students’ thinking, and consequently, for the erosion of their overall intellectual personality. It was this precise reform that provoked a mass aversion to learning among students. Furthermore, it gave rise to state-sponsored deception—the so-called ‘percentomania’ [the institutionalized manipulation of pass rates to fulfill bureaucratic quotas]—which effectively blocked any administrative path to correcting the crisis, thereby triggering progressive corruption throughout the entire educational sphere. To this very day, our school system groans under the heavy burden of that reform. One of the paramount lessons to be drawn from this historical analysis is that the quality of instruction is inextricably linked to the preservation of our national pedagogical tradition; breaking this continuity is utterly impermissible. In mathematics, this very tradition is synthesized within the textbooks of A. P. Kiselev. Consequently, a necessary—though perhaps not entirely sufficient—condition for the revival of our mathematical education is the definitive return of Kiselev’s textbooks to the school system.”
Incidentally, V. I. Arnold also frequently championed this perspective.
Andrey Petrovich Kiselev (1852–1940) was an outstanding Russian and Soviet mathematical educator whose textbooks across all branches of mathematics served as the gold standard for over sixty years. This enduring success spanned not only the Soviet era; even prior to the revolution, his Systematic Course of Mathematics for Secondary Educational Institutions (1884), Elementary Algebra (1888), and Elementary Geometry (1892) were widely recognized as the finest pedagogical works available. Despite the disruption caused by the Kolmogorov reform and the systemic shock that rippled through the educational sphere following the collapse of the USSR, the caliber of school-level mathematics education in Russia remained relatively high well into the mid-1990s due to historical inertia. This resilience was thoroughly documented by Dr. Murat Ashirovich Choshanov, a Professor of Mathematics and Education at the University of Texas at El Paso [11].
In 1994–1995, as part of his research under the Fulbright Program, Choshanov conducted a comparative analysis of the state of school mathematics education in Russia and the United States. He presented primary school students from both countries with the following word problem: “A shepherd, accompanied by 5 dogs, is guarding a herd of 125 sheep. How old is the shepherd?”
Astonishingly, 70% of the Russian schoolchildren immediately suspected a flaw in the formulation of the problem and stated that it had no logical solution. Conversely, 75% of the American pupils actively attempted to calculate a numerical answer, reasoning along these lines: 125+5=130 (far too old for a shepherd); 125−5=120 (still too old); 125:5=25 (now that looks okay! Answer: the shepherd is 25 years old).
For a person raised in our culture, reading such findings is almost incomprehensible—it suggests an extreme degree of cognitive decline among the younger generation in the world’s most developed nation. Salvador Dalí once wrote that the Catalan family is a paranoid community welded together by systemic delirium. This dictum from a true Catalan patriot characterizes, in the best possible way, the American school mathematical culture, which seamlessly allows for the addition of dogs to sheep.
M. A. Choshanov explains this phenomenon as follows:
“American schoolchildren are simply not taught how to solve problems properly. For them, the paramount goal is to guess the answer as quickly as possible or to arrive at just any solution. In U.S. schools, the system of standardized testing transforms mathematics education into a simple lottery: you either guess right or you don’t. Children are not conditioned to engage in prolonged analytical reflection over a problem’s solution or a theorem’s proof. Furthermore, mathematics teachers themselves contribute significantly to this issue.”
Conversely, Russian schoolchildren, unlike their American peers, are systematically trained to verify every analytical step. They inherently understand that summing the number of sheep and dogs will never yield the chronological age of the shepherd [11].
However, within the next generation of Russian schoolchildren, these analytical skills began to evaporate rapidly following the uncritical duplication of American mathematical instructional methods—specifically, through the introduction of the deeply flawed Unified State Exam (EGE). In this regard, M. A. Choshanov provides a telling example [11]:
“According to the results of a pilot testing initiative, 30% of Russian secondary school graduates proved unable to solve a rudimentary, two-step practical mathematics problem designed to calculate an electricity bill: ‘What will a person’s electricity payment be if on January 1st the meter read 88,742 kWh, and on February 1st it read 88,940 kWh, given that the cost of one kilowatt-hour is 3.5 rubles?’ Astoundingly, one examinee calculated that his monthly bill should amount to a staggering 260,000 rubles. Yet, this problem requires only primary school-level arithmetic.”
Judging by the publication date of that study, this distressing situation within Russian school mathematics had already manifested systematically by at least 2011–2013. Furthermore, Choshanov identifies systemic flaws within American mathematical education, famously characterizing them as being “mile-wide, inch-deep.” The U.S. school curriculum routinely introduced superficial, additional branches of mathematics instead of fostering a deeper comprehension of foundational topics. Similarly, in our own country—dating back to the Kolmogorov reform and persisting into the present day—elements of mathematical analysis have been continuously injected into high school curricula. Fortunately, we have at least been spared the overt introduction of the set-theoretic approach in the pure spirit of "Bourbakism."
However, since today virtually anyone is permitted to publish textbooks and educational manuals, we can observe how this very approach—albeit in a more elementary guise—has been propagated among primary school children since the mid-1990s through the textbooks authored by Professor L. G. Peterson, published under the auspices of the developmental education paradigm and the "I am Learning to Learn" project. Many parents find themselves utterly dismayed by this project, and for at least half of the student population in Russia, learning via these textbooks appears to be entirely counterproductive. Such materials are inherently tailored for "Olympiad-tier" children, participants of elite mathematics circles, and specialized science academies; indeed, a figure was once cited in the media indicating that 75% of the Russian National Mathematics Team members had been trained using Peterson’s curriculum.
One can empathize with professors of mathematical pedagogy: as time progresses, they naturally strive to innovate and introduce elements of higher mathematics into secondary schools. Yet, when instructing children, one must strictly account for child psychology and their capacity for the gradual, sequential assimilation of new knowledge—a domain of pedagogical mastery in which A. P. Kiselev remains entirely unrivaled.
But let us return to the observations of Professor M. A. Choshanov. He notes that a Russian sixth-grader could successfully teach mathematics to ninth-graders in an American high school, despite the fact that U.S. general education curricula are contextually broader—incorporating, for instance, topics like Probability Theory, Statistics, and Discrete Mathematics. Crucially, however, this comparison reflects the reality of the mid-1990s, prior to the systemic implementation of the EGE (Unified State Exam).
Choshanov’s description of American mathematics textbooks is particularly illuminating. They are bulky, vividly colored, and printed on high-gloss paper, replete with a myriad of illustrations. They contain an array of extraneous material—maps of the United States, portraits of American presidents, the rules of American football, and so forth; in short, they feature everything except substantive mathematics. As the author points out, such textbooks cause children to experience cognitive overload, rendering them unable to concentrate on the actual mathematical concepts. Conversely, Russian, Japanese, and Korean textbooks contain nothing superfluous, focusing strictly on "black-and-white mathematics" [11].
Following the implementation of the EGE (Unified State Exam)—which effectively reduced testing to a guessing game or a lottery—Russia rapidly deteriorated to the baseline level of mathematical education observed in the United States and France, as detailed above. For a stark comparison, one can readily find online the 1991 high school graduation exam papers covering algebra and the fundamentals of calculus, alongside the 11th-grade basic-level EGE mathematics tests administered in 2015 [12].
During the Soviet era, as many graduates of that time well remember, students were presented with rigorous, substantive problems. These required solving complex algebraic, trigonometric, and logarithmic equations and inequalities, calculating the areas of geometric figures bounded by two curves, and resolving intricate problems in solid geometry. Later, under the auspices of the Kolmogorov reform, relatively straightforward tasks involving the identification of domains and extrema of functions were introduced. The exam comprised a modest number of problems—typically around six—all maintaining a consistently demanding level of complexity.
What do we observe today? The exam has ballooned to 20 tasks, yet the first 6 or 7 of them could easily be solved by a fourth-grade primary school pupil. At the opposite end of the spectrum, the final problems are practically insurmountable, and there is no institutional expectation that a regular student will ever solve them. Nevertheless, successfully answering the rudimentary questions at the top of the list is entirely sufficient to pass the exam and graduate from school [12].
I had never previously taken any interest in this notorious EGE in mathematics, knowing intuitively that it was merely an absurdity that had migrated from the United States to Western Europe, and was subsequently exported across the entire post-Soviet space under the banner of the Bologna Process, with the sole objective of dismantling one of the finest educational systems in the world. However, upon reading an interview with Professor Aleksandr Ivanov, Head of the Department of Geometry and Topology at Petrozavodsk State University—whose work I have cited twice above—I saw for the very first time the exact types of problems presented to secondary school graduates under the generic title: “Establish a correspondence between the magnitudes and their possible values.”
In the first instance, the magnitudes are described as follows: the height of a child (A), the thickness of a sheet of paper (B), the length of a bus route (C), and the height of a residential building (D). These are paired with their possible values: 1) 32 km, 2) 30 m, 3) 0.22 mm, and 4) 110 cm. The student is then provided with a table consisting of eight cells; the top row contains the letters A, B, C, and D, while the bottom four cells are left blank, waiting for the student to allocate the suggested numerical values or their corresponding numbers.
The second task follows an identical format: the weight of an adult human (A), the weight of a cargo truck (B), the weight of a book (C), and the weight of a clothing button (D). The possible options provided are: 1) 8 tons, 2) 5 grams, 3) 65 kg, and 4) 300 grams, which must likewise be distributed among the empty cells [12].
This is an absolute farce; such material is fit only for an institution catering to intellectually disabled children. It is simultaneously laughable and tragic!
In another interview, Aleksandr Ivanov addresses the underlying causes of this total degradation in mathematics education. At the outset, he provides further examples of the problems featured on the EGE [13]:
“High school seniors are asked to solve tasks such as ‘determine the area of a room measuring 4 by 5 meters’ or ‘match a fly and an elephant to their respective sizes’ (given the measurements of 5 mm and 5 m, the student must indicate which size belongs to the elephant and which to the fly). This is precisely what schools occupy themselves with today—drilling children to answer such questions.”
The very inclusion of such tasks implies a recognized, high probability that a high school graduate might otherwise designate the elephant’s size as 5 mm, a residential building’s height as 110 cm, or a cargo truck’s weight as 65 kg. This perfectly mirrors the case in France, where a female student at a highly prestigious institution—the Pierre and Marie Curie University (Paris VI)—calculated the Earth’s radius to be 10 mm and saw no error in it.
Furthermore, Ivanov argues that the observed lack of motivation lies not with the children, but with the teachers. Because their performance is evaluated solely against a baseline graduation exam (the EGE) consisting of rudimentary problems on basic "number sense" and calculating the area of a room, instructors have no professional incentive to exert extra effort or to teach mathematics with genuine engagement. The title of this interview is highly telling: “You ask them to write ‘x squared,’ and the student draws an ‘x’ and encloses it inside a literal square!?” Coincidentally, back in 2009, I documented an identical, anecdotal case that had occurred at Kharkiv State University in my paper “Education and Science in the Post-Soviet Space (Critical Reflections)” published in the journal Alma Mater [14].
Consequently, it is imperative to urgently abandon the EGE and finalize the exit from the Bologna Process. Under the current reality of seemingly permanent sanctions, implementing such a transition is both highly pertinent and strategically viable. Although numerous petitions against the EGE and OGE (Basic State Exam) have circulated across the Internet, they garner weak public support primarily because they lack the endorsement of Russia’s prominent scientific and cultural figures. Following this grim diagnostic overview of mathematical education both globally and within Russia, we face the paramount task of identifying how to revitalize children’s interest in studying mathematics and how to foster their logical-mathematical reasoning.
Regarding instructional materials, as noted previously, many experts advocate for a definitive return to the classic Soviet textbooks compiled by A. P. Kiselev and A. S. Pchelko. We further posit that integrating the original chains of reasoning utilized by the historical pioneers of specific mathematical concepts directly into the pedagogy could profoundly stimulate the educational process. For instance, when introducing students to the core essence of the derivative, an instructor could concurrently elucidate how G. W. Leibniz (1646–1716) and Sir Isaac Newton (1642–1727) arrived at this fundamental concept from entirely different analytical standpoints. To heighten the emotional and cognitive resonance of the lesson, this narrative could be reinforced by displaying high-resolution digital facsimiles from their original mathematical treatises, many of which have been thoroughly digitized via projects like Google Books.
Crucially, this historical-scientific contextualization would require no additional hours carved out from the standard mathematical analysis curriculum. Similarly, when schoolchildren are taught the foundational mechanics of long division and column multiplication, teachers could display digitized pages from Newton’s “Universal Arithmetick; or, a Treatise of Arithmetical Composition and Resolution” published in 1720, allowing pupils to visually discover that the exact arithmetic and algebraic methods they are learning perfectly mirror the techniques once formulated by the greatest minds of scientific history (the first edition of this book was published in Latin in 1699).
When introducing a course in algebra, it is highly instructive to begin with Diophantus of Alexandria (3rd century AD), who pioneered the use of algebraic notation and introduced specific symbols for raising unknown quantities to powers up to the sixth degree (the "cubo-cube"). Utilizing Google Books, instructors could showcase “L’Algebra opera di Rafael Bombelli da Bologna Diuifa in tre Libri” (1579) by the Italian mathematician Rafael Bombelli (1526–1572). This allows for a poignant narrative: during Bombelli’s era, a standardized algebraic symbolism had not yet coalesced, save for the early foundational framework proposed by Diophantus, the "father of algebra." Furthermore, it was Bombelli who first arrived at the concept of complex (imaginary) numbers while solving cubic equations—a breakthrough that mathematical consensus refused to recognize until the 18th century. It is equally vital to mention that Bombelli was among the first in Europe to systematically open up and boldly operate with negative numbers. Intriguingly, these quantities were already well known to the Chinese as early as the 2nd century BC; they meticulously dyed negative values red to distinguish them from positive ones. The ancient Chinese understood negative numbers as representations of loss or deficit, and these concepts naturally emerged during their calculation of systems of linear algebraic equations, which they were uniquely adept at solving.
Students would undoubtedly find it fascinating to learn how the ancient Chinese formulated such equations. A prominent example is a classic riddle that reduces to a system of two linear algebraic equations with two variables:
“There are pheasants and hares in a cage. From above, one counts 35 heads; from below, 94 feet. How many pheasants and hares are there?”
This exact problem is preserved within the ancient text The Nine Chapters on the Mathematical Art, a monument comprising a compilation of scientific works written in China between the 10th and 2nd centuries BC, encompassing 246 practical problems. Under such a pedagogical paradigm, school and university students would naturally develop a profound interest not only in mathematics but in global history as well.
In his study, V. I. Arnold provided highly intriguing data revealing that Ancient Greek mathematicians borrowed almost their entire body of knowledge from the Ancient Egyptians, effectively operating as industrial spies (this applies to Archimedes, Plato, Eudoxus, and others). While this historical reality is well known to specialists who have directly analyzed Ancient Egyptian mathematical papyri, it remains largely unrecognized by contemporary mathematicians [7]. It is highly probable that this fact is deliberately obscured, given the centuries-old tradition of idolizing Pythagoras, Archimedes, and Euclid.
Regarding Pythagoras specifically, Arnold wrote the following: “Pythagoras was one of the world’s first industrial spies, as we would call them today. He spent about twenty years in Egypt. The Egyptian priests instructed him in their sciences but demanded a non-disclosure agreement (consequently, he never published anything during his lifetime). The Pythagorean theorem had been published in Babylon via cuneiform a couple of thousand years before him, complete with its proof and the formula for calculating Pythagorean triples (such as 32+42=52), which describe all right-angled triangles with integer side lengths.” Furthermore, Arnold noted that practically the entirety of Euclidean geometry, with the sole exception of the fifth postulate, was already known to the Ancient Egyptians [7].
Crucially, the Ancient Greeks themselves, unlike our contemporaries, explicitly acknowledged the Egyptians as their masters in mathematics. It is well established that the Pythagorean theorem possesses a vast array of proofs, with approximately 400 distinct derivations compiled to date [15].
We succeeded in rapidly developing four entirely new proofs of this theorem using both algebro-geometric and purely algebraic methods. Based on this outcome, we hypothesize that the Pythagorean theorem serves as an exemplary vehicle for the mathematical education of school and university students, precisely because the potential number of its proofs is virtually infinite. Deriving this theorem via diverse methodologies serves as an excellent algebro-geometric exercise. Schools and universities could systematically launch competitions challenging students to produce the greatest number of independent proofs for the Pythagorean theorem. Such contests would likely yield entirely novel derivations. Ultimately, this initiative could crystallize into an intellectual youth movement under the banner of "Pythagoreana," which would prove highly beneficial in elevating the prestige of mathematical education among the younger generation [15].
Pedagogical science recognizes various methodologies for constructing didactic problem-based situations aimed at enhancing students’ motivation to study mathematics [11,16]. A particularly compelling approach to creating such a problem-based scenario involves the historical and practical derivation of the number π. Ancient Egyptian, Babylonian, Indian, Chinese, and Greek geometers were well aware that the ratio of a circle’s circumference to its diameter remains constant for any given circle, and that this ratio is slightly greater than 3. For instance, in the Egyptian Ahmes Papyrus (also known as the Rhind Mathematical Papyrus) dating from the Middle Kingdom period, this relationship is recorded as 81256≈3.16.
As noted in [16], scholars since antiquity observed through the empirical monitoring of handicraft—specifically, the art of basket weaving—that to construct a basket of a desired diameter, artisans consistently needed to harvest reeds exactly three times longer than that diameter.
To ensure that children genuinely absorb and internalize this geometric invariant, L. I. Bozhovich suggested a hands-on exercise: pupils take three distinct circular objects, measure their diameters using a ruler, determine their circumferences using a piece of string, and subsequently record their empirical measurements and the calculated ratios of circumference to diameter in a structured table [16].
It is worth noting that Lidiya Ilinichna Bozhovich (1908–1981) was a highly distinguished Soviet psychologist and a direct student of the seminal Soviet psychologist Lev Semyonovich Vygotsky (1896–1934).Furthermore, didactic problem-based situations can be seamlessly engineered using mathematical sophisms and paradoxes, such as the one examined earlier. The previously analyzed example belongs to a broad class of mathematical fallacies whose resolution relies strictly on the prohibition of division by zero. One of the earliest documented sophisms of this specific type was formulated by Bernard Bolzano (1781–1848), the renowned Czech mathematician, pioneer of the rigorous theory of real numbers, and one of the early architects of set theory. His sophism unfolds as follows: start with the identities a−b=a−b and b−a=b−a. Adding these two equations together yields a−a=b−b, which can be factored as a(1−1)=b(1−1). Consequently, if an individual is unaware that division by zero is strictly forbidden, they will erroneously conclude that a=b.
It is worth noting that one of the finest publications dedicated to sophisms and paradoxes is the book by V. M. Bradis and his co-authors, entitled Errors in Mathematical Reasoning, which saw its second edition in 1959 [17], though its initial edition had been published jointly with V. L. Kharcheva as early as 1938.
Within this book, one discovers an excellent historical overview of the problem alongside a rigorous classification of sophisms and paradoxes. This work genuinely allows an individual to study elementary mathematics and the foundational elements of analysis (such as limits and series) with immense pleasure, serving as a stark contrast to the multitude of today’s dull, uninspiring textbooks. The previously discussed sophism by B. Bolzano was extracted precisely from this volume. As demonstrated in this text, Bolzano’s fallacy corresponds to the first category of false proofs (the covert allowance of division by zero) identified by the prominent German algebraic geometer Hermann Schubert (1848–1911). Schubert distinguished four distinct categories of such flawed proofs, though the authors of the book convincingly demonstrate that his taxonomy fails to encompass even the minimum scope of the phenomenon in question.
Another beautifully described paradox is the sophism asserting that “the length of the hypotenuse of a right-angled triangle is equal to the sum of the lengths of its legs.” Any rational person who is unversed in the strict technical subtleties of limits and infinitesimals will find themselves utterly bewildered by this puzzle. I once encountered an online article by an American author who had independently arrived at this very sophism and was actively pleading with readers to explain where the error lay.
This volume features a profound insight by the outstanding popularizer of science, Narodnik, and Honorary Academician Nikolai Aleksandrovich Morozov (1854–1946). He posited that a sophism of the type "the hypotenuse of a right-angled triangle is equal to the sum of its legs" “possesses genuine scientific value, for it draws our immediate attention to the critical peculiarities of mathematical methodologies, or to our very mathematical conceptions themselves, and the genesis of these conceptions within our minds.”
Regrettably, this specific phrasing contains an oversight, as the words "the length of" and "the lengths of" were omitted in reference to the hypotenuse and the legs. I highly encourage school and university students to independently wrestle with this particular paradox before seeking out its formal explanations. The aforementioned book is readily available via open access on the educational platform mathedu.ru.
Equally illuminating and highly instructive is the exposition of the sophism regarding the sum of the infinite series 1−1+1−1+… (the Grandi’s series). If an individual is entirely unacquainted with the rigorous definitions of convergent and divergent series, any reader can, by simply grouping these units in alternative ways, derive four entirely distinct answers: 0.5, 0, +1, and −1. Throughout the 18th century, the titan mathematicians of the era—including Leibniz, Wolff, Nicolaus Bernoulli I, and Euler, among others—engaged in fierce intellectual debates surrounding this paradox, occasionally even invoking metaphysical arguments to support their claims. These disputes were decisively resolved only after a precise, formal definition of convergent series was established. The series in question was ultimately demonstrated to be divergent.
A brief note is warranted regarding the principal author of the aforementioned book. Vladimir Modestovich Bradis (1890–1975) was a remarkable Soviet mathematician, educator, and master of computation, who was elected a Corresponding Member of the Academy of Pedagogical Sciences of the USSR in 1955. He was the creator of the legendary "Bradis Tables"—or, more precisely, the Four-Digit Mathematical Tables. For roughly seventy years, this compendium served as an indispensable reference book for every school pupil, university student, and practicing engineer. It represents an entire scientific era, undergoing more than twenty major re-editions since its inception in 1921.
In their day, the Bradis Tables performed the exact function that computers do now. When calculating the structural and technical specifications of their skyscrapers—among countless other projects—American and European engineers routinely relied on these Soviet Bradis Tables.
And what do we observe today? Modern engineers no longer possess a foundational command of the strength of materials (sopromat); they are inherently incapable of calculating the structural stability of a rudimentary beam by hand. Instead, they offload every calculation onto computers, yet no one truly understands how the software arrives at its conclusions, because the underlying algorithms function as an impenetrable black box. Consequently, we witness the collapse of bridges and residential buildings, explosions at power plants, failures of dams, and instances where landslides and mudflows sweep away everything in their path, or where marine waves demolish coastlines and hydraulic engineering structures. Regrettably, with each passing year, the frequency of such catastrophic events across the globe will only continue to escalate.
However, let us return directly to our core theme: how can we stimulate secondary and higher education students to engage with mathematics while actively developing their logical-mathematical reasoning?
In 2016, we conducted an empirical testing initiative among students at Belgorod State National Research University and V. N. Karazin Kharkiv National University. The objective was to evaluate the caliber of their logical-mathematical thinking through the lens of solving historical mathematical problems (the comprehensive inventory of these tasks is provided in the Appendix). Given that resolving the majority of these problems fundamentally necessitates a fluent ability to operate with simple fractions, a rigorous analysis of the test metrics led to a striking general conclusion: only approximately 50% of the Kharkiv and Belgorod students demonstrated the capacity to perform arithmetic operations with simple fractions, specifically when finding a common denominator for addition or subtraction. These empirical insights were gleaned from a cohort comprising senior undergraduate ecology students at V. N. Karazin Kharkiv National University and senior undergraduate economics students at Belgorod State National Research University. Remarkably, when juxtaposed against the performance metrics of the American and French university students detailed previously, this outcome can still be interpreted as exceptionally high.
Our subsequent testing of 223 students specializing in technical, socio–economic, and humanitarian sciences across four faculties and institutes of Belgorod State National Research University (2016–2018) demonstrated that the easiest problems to solve were those involving the formulation and solution of a basic linear algebraic equation with one unknown (Problem 1, scoring 247 out of 446 points), the formulation and solution of a system of two linear algebraic equations with two unknowns (Problem 5, 182 points), and Gauss’s logical problem on summing the first 100 natural numbers (Problem 7, 138 points) [2,3].
Problems of moderate difficulty included those requiring the formulation and solution of a linear algebraic equation with one unknown, further complicated by fractional calculations: Problem 2 (95 points), Problem 8 (63 points), and Problem 10 (104 points). Notably, setting up the equations in these tasks demands a reasonably high level of logical thinking from the student. This category also encompasses the fourth, geometric problem (Archimedes’ problem), which involves calculating and comparing the areas of a square and its circumscribed circle (54 points).
The most challenging tasks were those requiring proofs of propositions in elementary number theory (Pythagoras’ problem, numbered as Problem 3 – 41 points; Leibniz’s problem – 13 points; and Sophie Germain’s problem – 15 points).
Thus, for the simplest task (Problem 1), which was solved either fully or partially, the students managed to score approximately 60% of the maximum possible points. Only a quarter of the students successfully handled the problems involving arithmetic operations with common fractions. The majority of respondents do not know how to approach proof-based tasks, lack the ability to formalize problem conditions and logically structure their solutions, and possess no skills in performing algebraic manipulations. Nearly a quarter of the students (60 individuals) failed to solve a single problem.
The calculated problem-solving performance by age leads to the conclusion that younger undergraduate students perform best. However, it should be taken into account that international students generally enter the university at an older age.
Given the unequal sample sizes among the tested students based on their countries of origin (152 students from Russia, 19 from the near abroad, and 23 from the far abroad), a direct comparison of the overall problem-solving performance is challenging. Furthermore, it should be noted that international students who come to study are often among the top performers in their home countries.
Nevertheless, the data obtained from this study indicate a fairly weak mathematical background among students across various fields of study, ranging from the humanities to technical disciplines.
Scoring was determined as follows: 2 points for a complete solution, 1 point for a partial solution, and 0 points if the problem was not solved. The time allocation for completing the tasks was 1.5 hours.
To systematically cultivate the logical-mathematical faculties of both school pupils and university students, instructors would be well-advised to complement historical mathematical tasks by collaboratively solving problems from the celebrated collection compiled by V. I. Arnold, entitled Tasks for Children from 5 to 15 Years Old. Intriguingly, certain riddles within this compendium routinely prove insurmountable even to university professors and members of academies, whereas sharp-witted children resolve them with intuitive ease [18].
As an exemplary illustration, V. I. Arnold presents problem number 13:
“Two volumes of Pushkin’s works, Volume I and Volume II, are standing next to each other on a bookshelf. The pages of each volume collectively measure 2 cm in thickness, while each cover is 2 mm thick. A bookworm chewed a hole (perpendicularly to the pages) from the first page of Volume I to the last page of Volume II. What distance did the worm chew? [This topological problem with an unbelievable answer—4 mm—is utterly insurmountable for university professors and academicians, yet some preschool children handle it with intuitive ease].”
Two other tasks, numbered 2 and 3, are equally revealing, and their specific formulations appear to trace their lineage back to deep antiquity:
“2. A bottle with a cork costs 10 kopecks, with the bottle being 9 kopecks more expensive than the cork. How much does the bottle cost without the cork?” and “3. A brick weighs one pound plus half a brick. How many pounds does the brick weigh?”
As Arnold notes in the preface to his slim volume, he compiled these and other tasks in Paris during the spring of 2004, responding to requests from expatriate Russian Parisians who sought his assistance in helping their young children acquire a rigorous culture of thinking—a tradition native to Russia that profoundly surpasses all Western educational customs. The eighth edition of his problem book, which we have cited, encompasses 79 such tasks. It is precisely these types of problems that school and university students should be actively resolving, rather than being subjected to stupefying and meaningless standardized tests that cannot even legitimately be labeled as mathematics.
It must be emphasized that under the current educational paradigm, the more time a child spends within the walls of a conventional school, the more their cognitive faculties decline. Consequently, affluent and forward-thinking parents are entirely preoccupied with hunting for specialized academies, employing top-tier private tutors, or transitioning their children to homeschooling frameworks.
What else could ignite a profound interest in the study of mathematics? It is the revelation of its intrinsic beauty. Consider the exquisite elegance of Euler’s identity, exp(iπ)+1=0 which simultaneously encapsulates five fundamental mathematical constants. Every professional mathematician knows that a well-formulated problem, even after highly cumbersome and intricate transformations, ultimately yields a compact and beautiful result.
A prime example is the simplest non-linear first-order ordinary differential equation, known as the Verhulst equation. We have previously detailed the history of its creation based on original papers in [19]. When expressed in a discrete form (the logistic map), this equation reveals behaviors far beyond the standard solutions inherent to its continuous differential analogue (such as population extinction or stabilization at a steady-state carrying capacity). Specifically, it exhibits regimes of self-oscillations and deterministic chaos, which manifest sequentially as the growth rate parameter of the equation is incrementally increased. This map is mathematically formulated as: Xn+1=rXn(1−Xn).
In a historical context, these breakthroughs trace back to the foundational work of Feigenbaum [17], while in Russia, this specific phenomenon appears to have been first illuminated in the study by A. P. Kuznetsov and S. P. Kuznetsov [21,22].
The Internet is replete with simple Python scripts that allow students to visually observe the astonishingly beautiful transitions in the behavioral regimes of this discrete equation’s solutions. However, an even more breathtaking and dynamic visualization is unlocked when the logistic map is extended into the domain of complex numbers. In one such formulation, it can be elegantly reduced to the following form: Zn+1=λ−Zn2.
The most accessible description of the resulting non-linear dynamics portraits within the complex plane can be found in the textbook compiled by O. B. Isaeva, which is openly available on the Internet [23].
If λ=0 and the initial point resides within the unit disk, it remains confined within this boundary throughout the iteration process; otherwise, it escapes to infinity. Consequently, the boundary separating these two sets of points (solutions) is a circle of radius one. However, when λ is not equal to zero, the mapping operations yield highly complex fractal boundaries of extraordinary aesthetic beauty. By varying the value of λ, one can generate an infinite array of diverse fractal structures. In a historical context, these phenomena trace back to the Mandelbrot, Fatou, and Julia sets. As early as 1905, the French mathematician Pierre Joseph Louis Fatou (1878–1929) investigated recurrent series described by the aforementioned quadratic map in the complex plane.
Closely aligned research was conducted by another French mathematician, Gaston Maurice Julia (1893–1978). In the 1970s, Benoit Mandelbrot (1924–2010) vigorously popularized Julia’s findings, which, alongside Fatou’s pioneering work, formed the bedrock of his fractal geometry. In his seminal paper on fractal geometry, which focused on measuring the coastline of Great Britain, Mandelbrot demonstrated that the calculated length inherently depends on the scale of the measuring segments used; as the length of these segments approaches zero, the total length of the coastline diverges toward infinity. This groundbreaking study was published in 1967 in the journal Science under the title “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension” [24].
In 1975, Mandelbrot provided a mathematically rigorous description of the sets generated via quadratic mapping, which had been originally explored by Fatou, and coined the term “fractals” to define them. Later, the French mathematician Adrien Douady (1935–2005) named this specific class of topologies the Mandelbrot set. Following the advent of high-performance computing systems (IBM), the first high-resolution visualization of the Mandelbrot set was successfully rendered in 1986 by Dick Pountain [25].
In principle, this entire historical trajectory is thoroughly documented in the Wikipedia articles dedicated to Mandelbrot and his celebrated set, where one can also examine a multitude of computer-generated, multi-colored fractal images. Furthermore, an individual interested in this field can easily discover various online interactive simulators that dynamically generate fractal structures. Nevertheless, this phenomenon is so profoundly breathtaking—blurring the line between rigorous mathematics and visual art—that we intentionally decided to dedicate considerable attention to this subject.
Due to the fundamental simplicity of the underlying quadratic map and the recursive relationships constructed upon it, engaging with these concepts is entirely within the reach of secondary and higher education students. For instance, students can independently experiment with any other basic non-linear discrete mappings within the complex plane. Through computer modeling, such experimentation will consistently yield a highly diverse array of novel fractal structures.
Crucially, we noted that within the Saratov research group of theoretical non-linear dynamics—which originally operated under the prominent leadership of Doctor of Physical and Mathematical Sciences Sergey Petrovich Kuznetsov (1951–2020) and Alexander Petrovich Kuznetsov—a specialized youth scientific laboratory is successfully functioning. Within this framework, Saratov school pupils and university students are actively engaged in modeling various processes of non-linear dynamics, including projects at the intersection of non-linear discrete mappings and fractal geometry (http://www.sgtnd.narod.ru/studlab/rus/studlab.htm).
Everything described above, within the framework of secondary school education, could be readily adopted by advanced lyceums, gymnasiums, and enthusiastic instructors.
Yet, contemporary consumerist society and the liberal public persistently pose the question: why does a modern individual need mathematics at all? Who genuinely requires these derivatives, integrals, and differential equations? The answer, however, is fundamental. The crux of the matter lies not in the derivatives and integrals themselves—which, admittedly, the vast majority of people will never utilize in their daily lives—but rather in the fact that mathematics systematically builds an analytical mind, cultivates logical reasoning, and disciplines an individual’s intellectual faculties. Such cognitive traits are absolutely indispensable in any field of professional activity. Indeed, mathematicians themselves routinely achieve far greater success in the humanities and socio-economic sciences than traditional representatives of those respective disciplines.
However, for a consumerist society, an analytical mind and logical reasoning are, by definition, counterproductive. The defining attributes of mathematics—namely logic, rigorous proof, and intellectual discipline—are sought after and valued exclusively within a highly cultured, advanced society.
As the very same V. I. Arnold underscores, mathematical education must constitute an indispensable component of the cultural baseline for every school pupil, and the primary objective of mathematical education must be to cultivate the ability to mathematically investigate the phenomena of the real world [4]. It must be precisely the real world, rather than the abstract domain dogmatically imposed by the followers of the Bourbaki group. Furthermore, mathematics stands as the ultimate methodology capable of providing an adequate description of numerous natural and socio-economic processes and phenomena.
Government and ministerial bureaucrats across all nations must urgently comprehend a critical reality, regardless of their desire to manage systems they fundamentally fail to understand: if they do not immediately initiate the reintroduction of rigorous, healthy standards in mathematics, physics, and all other natural sciences, they will inevitably confront a rapid, aggressive degradation of their countries’ entire technosphere. Bridges and buildings will continue to collapse, power plants will suffer catastrophic failures, and so forth. Crucially, this destruction will devastate not only the technosphere but, first and foremost, the ecosphere itself. This is a reality that we are already acutely witnessing today.
2. Instrumental Decomposition of “Dead Algorithms” and Overriding the Conceptual Crisis
As demonstrated by our testing of students across various majors at Belgorod National Research University between 2016 and 2018, the core problem of the modern decline in mathematics lies in the fact that students memorize formulas as "dead algorithms," entirely lacking an understanding of their physical and logical meaning. Slightly less than 50% of the 223 surveyed students failed to solve Problem No. 1 (from the Moscow Mathematical Papyrus), which required performing basic operations with fractions (1 and 1/2), because they were conditioned to operate exclusively within standard, template-based tests. When confronted with a live textual problem, their cognitive processing stalled.
Even worse, as we noted in the first part of this work, according to observations by V.I. Arnold, the situation regarding operations with fractions in the United States is such that only a few percent of students, along with their teachers, manage to perform them successfully.
At this critical juncture, new-generation Generative AI fundamentally revolutionizes pedagogy by acting as a "Socratic interlocutor." Unlike a conventional digital solver that simply outputs a ready-made answer—thereby permanently stripping the student of any incentive to think—a reasoning AI operates via the method of guided decomposition:
- Diagnostic of Mental Gridlock: Upon identifying that a student is committing a fundamental error regarding fractions or proportions, the AI immediately halts the calculation process. This aligns with the critical shift identified in contemporary higher education by J.Y. Auh [26], who argues that as information and ready-made answers become infinitely abundant, the true scarcity in modern academia is no longer knowledge, but judgement—the capacity to navigate uncertainty and evaluate competing algorithmic outputs. It recognizes that this is not a mechanical typo, but rather a systemic failure in grasping foundational concepts. Consequently, instead of contributing to what Auh terms “epistemic dependence” by merely generating a textbook answer, the reasoning model focuses on cultivating human discernment.
- Return to Kiselev’s Visual Geometry: Instead of indulging in abstract manipulations of numerators and denominators, the AI restructures the learning interface. It prompts the student to visualize the problem geometrically—through the division of segments or areas—thereby returning to the gold standards of the pre-reform school system established by A. P. Kiselev. The AI forces the student to visually comprehend what "a number taken one and a half times" actually signifies, effectively resuscitating spatial reasoning.
A parallel breakthrough is achieved by AI in the domain of overcoming mathematical illusions. In the first part of this paper, we examined in close detail the famous geometric sophism regarding the stepped jagged line. As it “presses” tightly against the hypotenuse, it continuously preserves the sum of the lengths of its legs, seemingly proving that the length of a hypotenuse equals the sum of the two legs. For a modern student accustomed to flat, two-dimensional test sheets, this paradox becomes an insurmountable cognitive trap: the limit of the sequence of jagged lines visually merges with the hypotenuse, and intellectual discipline surrenders to an optical illusion.
In the pre-AI era, an enthusiastic instructor required hours of whiteboard explanations to convey the essence of this topological paradox to a classroom. Today, AI enables a student to instantly conduct an interactive experiment upon this very sophism. Based on a single natural language request, the AI generates a dynamic simulator. The student can independently scale the number of steps to thousands and millions, while simultaneously observing how the tangent vector behaves on the graph. The AI visually demonstrates the fundamental divergence between the geometric proximity of two lines and the proximity of their derivatives. What previously appeared as a dry, incomprehensible theorem of mathematical analysis is transformed by AI into a living, tangible physical experience.
3. Overcoming the Technical Barrier: AI as a Digital Telescope for Non-Linear Dynamics and Fractal Geometry
This exact pedagogical principle of cognitive autonomy unfolds when studying non-linear dynamics and fractal geometry. In the first part of our study, we highlighted the youth scientific laboratory of the Saratov research group of theoretical non-linear dynamics, operating under the guidance of the Kuznetsov professors [21,22,23]. In that environment, school pupils and university students achieve remarkable insights at the intersection of mathematics and visual art by modeling the discrete mappings of Verhulst, Fatou, and Julia.
However, within the traditional educational framework, a massive barrier has historically blocked the path to this fractal beauty: the necessity of complex programming. In order for an ordinary school pupil to actually visualize the Mandelbrot set or the fractal boundaries of complex mappings, they first had to spend months mastering the syntax of programming languages (such as Python or C++), visualization libraries, and numerical methods. In 90% of instances, this routine technical barrier completely extinguished the spark of genuine mathematical curiosity in a child.
The era of reasoning AI completely obliterates this routine by cleanly separating creative intent from technical implementation. This phenomenon perfectly exemplifies what J.Y. Auh [26] terms the rise of “cognitive partnership” and the paradigm shift from isolated learning to “networked cognition.” As Auh perceptively observes, educational technologies rarely eliminate thinking; instead, they shift where it occurs—much like calculators once reduced the need for mental arithmetic to free up cognitive space for higher-level reasoning. In this new collaborative matrix, the AI actively participates in the intellectual workflow, allowing the human student to move past syntax barriers and focus entirely on the uniquely human capacities of evaluation and strategic direction:
- Pure Mathematical Intuition: A school pupil no longer needs to be an expert programmer to conduct sophisticated synergetic research. They can interact with the AI as a high-level assistant engineer. The child formulates a hypothesis in plain language: “I want to utilize a cubic mapping instead of a quadratic one, and observe how the fractal boundaries shift when the lambda parameter is not equal to zero.”
- Instant Materialization of Ideas: Within seconds, the AI generates clean, fully functional code for computer modeling. The coding barrier vanishes. The student immediately transitions to the core substance: to the experiment, to the analysis of non-linear dynamics portraits in the complex plane, and to the contemplation of fractal topologies.
Consequently, Generative AI scales the elite experience of advanced laboratories across the entire globe. It mutates into a “Global Digital Mentor,” making a profound physical-mathematical culture accessible to any thinking child, regardless of how remote their province may be, completely insulating them from the degrading standards of mass bureaucratic education.
4. Practical Case Studies: AI-Driven Solutions for Diverse Mathematical Problems
AI and an Algebraic Approach to Understanding the Addition of Fractions
As we already know, the majority of students do not know how to find a common denominator for fractions: ba+dc. In this scenario, AI can offer them the following algorithm. Multiply this expression by b, which eliminates the denominator in the first fraction: a+(bc)/d.
Now, multiply this resulting expression by d to eliminate the denominator of the second fraction, yielding: ad+bc.
As a result, you have multiplied the original expression by bd. To ensure that the value of the expression remains unchanged, it must now be divided by bd. Ultimately, we arrive at the standard formula: (a/b)+(c/d)=(ad+bc)/bd.
As we can see, there is nothing complicated here.
AI Pedagogical Capabilities: Demystifying Long Division and Non-Integer Remainders
AI can transform long division from a mechanically memorized routine into a conceptually transparent process by leveraging the core concept of number place values (orders of magnitude) combined with the distributive law.
Instead of teaching students to blindly "bring down the next digit," an AI tutor can model the division of a dividend (e.g., 972÷4) by breaking it down into its constituent place values:
972=900+70+2
The division then proceeds sequentially from the highest order to the lowest (hundreds, tens, and then units), where any remainder from a higher order is logically converted into the next lower order (e.g., 1 hundred=10 tens). This approach eliminates the common student error of "missing zeros" in the quotient and grounds the algorithm in strict algebraic logic:
972/4=(800+160+12)/4=(800/4)+(160/4)+(12/4)=200+40+3=243
Furthermore, AI can elegantly clarify the transition to non-integer division when a number cannot be divided evenly (e.g., 975÷4). Instead of introducing a decimal point as an arbitrary structural rule, the AI explains the remainder as a fractional mandate: the remaining 3 units must still be divided by 4, represented as 243+(3/4).
To achieve a decimal representation, the AI simply extends the place-value logic to sub-orders (tenths and hundredths). The remaining 3 units are conceptually broken down into 30 tenths (yielding 7 tenths with a remainder of 2 tenths), and the remaining 2 tenths are converted into 20 hundredths (yielding exactly 5 hundredths). This mental model ensures that students comprehend decimals not as detached numbers following a dot, but as a continuous extension of the base-10 place-value system:
243+0.7+0.05=243.75
AI and the Concept of Equilibrium: Demystifying Basic Algebraic Equations
The conventional pedagogy surrounding basic algebraic equations often relies on rote rules, such as “move the number to the other side and change its sign.” This mechanical approach obscures the underlying mathematical logic. An AI-driven instructional model can replace this "dead rule" with the dynamic equilibrium concept (the balance scale analogy), grounded in the axiom of maintainable equality.
An equation is modeled not as a static sequence of symbols, but as a perfectly balanced scale where the equals sign (=) represents the fulcrum. The objective is to isolate the unknown variable (x) while strictly maintaining this balance.
For instance, when solving an addition-based equation:
x+5=12
Instead of teaching the arbitrary “transfer” of the number 5, the AI explains that to isolate x, one must remove 5 from the left side of the scale. To maintain equilibrium, the exact same operation must simultaneously be performed on the right side. Thus, the operation is executed symmetrically on both sides:
(x+5)−5=12−5 or x=7
The shift in sign is therefore demonstrated to be a logical consequence of preserving equality, rather than a magical property of moving across the equals sign.
Similarly, for a multiplication-based equation: 3x=15.
The AI conceptualizes this as three identical unknown masses balancing a 15-unit weight. To find the value of a single x, the content of the left side must be reduced threefold. Consequently, to preserve the balance, the right side must also be divided by 3:
3x/3=15/3 or x=5.
By embedding the concept of symmetrical operations into the learning process, AI transitions the student from blind algorithmic execution to a deep structural understanding of algebraic logic.
AI Pedagogy in Systems of Linear Equations: The Classical Chinese Problem Framework
To demonstrate how AI can bridge the gap between intuitive logical thinking and formal algebraic structures, we can utilize Problem No. 5 from our appendix—the classical Chinese puzzle from the Sunzi Suanjing (The Mathematical Classic of Sun Zi): "There are pheasants and rabbits in a cage. From above, there are 35 heads; from below, there are 94 legs. How many pheasants and rabbits are there?"
Conventional pedagogy often forces students to blindly apply the substitution or elimination method without mapping the symbols to physical reality, causing cognitive blockages. An AI tutor, conversely, introduces a two-tiered cognitive transition.
First, the AI activates a logical mental model based on physical constraints: if all 35 animals were pheasants (or if rabbits lifted two of their paws), there would be exactly 35×2=70 legs on the ground. The excess legs (94−70=24) must belong exclusively to the rabbits, meaning there are 24÷2=12 rabbits and, consequently, 35−12=23 pheasants.
Second, the AI formalizes this vivid reasoning into a system of two linear equations with two variables, where symbols maintain strict physical meaning: let x represent the number of pheasants and y represent the number of rabbits.
The student is guided to construct two independent equilibrium equations—one for heads and one for legs:
x+y=35
2x+4y=94.
Rather than treating the system as an abstract computational puzzle, the AI models the substitution method as a logical flow. Expressing x from the first equation (x=35−y) is explained as "viewing the number of pheasants as the total population minus the rabbits." Substituting this into the leg-balance equation yields:
2(35−y)+4y=94.
Applying the distributive property and the balance-scale principle discussed earlier, the equation unfolds transparently:
70−2y+4y=94 or 70+2y=94.
Subtracting 70 symmetrically from both sides isolates the variable:
2y=24⟹y=12.
Thus, the AI successfully transitions the student from a localized arithmetic trick to a universal algebraic tool, preventing the formation of "dead algorithms" and fostering structural mathematical literacy.
AI-Driven Deductive Reasoning in Number Theory: Leibniz’s Divisibility Problem
To further illustrate how AI transitions from mechanical computation to semantic mathematical synthesis, we examine Problem No. 6 from our appendix—a classical proposition by G.W. Leibniz: Prove that if n is an integer, then n5−n is fundamentally divisible by 5.
Traditional university-level instruction frequently addresses this problem through mathematical induction or modular arithmetic (mod5), which often leaves students executing steps without grasping the structural harmony of numbers. An AI pedagogical model, however, guides the student through a structural decomposition of the polynomial, using basic algebraic transformations to reveal its underlying logic.
First, the AI demonstrates how to factor the expression n5−n using the difference of squares identity:
n5−n=n(n4−1)=n(n2−1)(n2+1)=(n−1)n(n+1)(n2+1)
Second, the AI shifts the student’s focus from symbol manipulation to cognitive interpretation. It highlights that the first three terms—(n−1), n, and (n+1)—represent three consecutive integers. Within any sequence of three consecutive integers, at least one number must be a multiple of 2, and exactly one must be a multiple of 3, meaning the entire expression is inherently divisible by 6.
To establish strict divisibility by 5, the AI introduces a comprehensive yet intuitive case analysis based on the base-10 number system and terminal digits (remainders upon division by 5):
- Case 1: If the integer n itself is a multiple of 5 (ends in 0 or 5), the term n guarantees the divisibility of the whole product.
- Case 2: If n yields a remainder of 1 or 4 (ends in 1 or 4), then either (n−1) will terminate in 0, or (n+1) will terminate in 5, rendering the product divisible by 5.
- Case 3: If n yields a remainder of 2 or 3 (ends in 2 or 3), the AI directs attention to the final quadratic component, (n2+1). If n ends in 2, n2 ends in 4, making n2+1 end in 5. If n ends in 3, n2 ends in 9, making n2+1 end in 0 (a multiple of 10).
Consequently, the AI models a elegant deductive framework: regardless of the choice of n, at least one of the structural components within the factored polynomial is guaranteed to be divisible by 5. By decoupling the proof from rigid, abstract schemas and anchoring it in structural logic, the AI fosters genuine mathematical intuition and high-order cognitive processing.
AI and Creative Problem Solving: Deconstructing Sophie Germain’s Identity
To demonstrate how AI can demystify "magical" mathematical tricks and replace them with purposeful, structural logic, we analyze Problem No. 9 from our appendix—the famous Sophie Germain’s identity: Prove that every number of the form a4+4 (where a is an integer and a>1) is a composite number.
In standard educational frameworks, this proof is typically introduced abruptly: students are told to "add and subtract 4a2," which leaves them executing the algorithm without understanding the cognitive origin of the strategy. An AI pedagogical model, however, frames this not as an arbitrary trick, but as a deliberate completion of a geometric and algebraic square.
The AI guides the student’s reasoning through three distinct cognitive phases:
First, the AI highlights that the expression a4+4 consists of two perfect squares: (a2)2 and 22. It then asks the student what component is missing to form a perfect trinomial square. By defining the missing middle term as the doubled product of the bases (2×a2×2=4a2), the AI justifies the structural intervention based on the law of equilibrium: to add 4a2, one must simultaneously subtract 4a2 to preserve the value of the expression:
a4+4=(a4+4a2+4)−4a2
Second, the AI demonstrates the transformation of the newly balanced expression by factoring the trinomial and recognizing the resulting difference of squares pattern (A2−B2):
a4+4=(a2+2)2−(2a)2
Using the fundamental difference of squares identity, the AI transparently factors the polynomial into two distinct quadratic components:
a4+4=(a2−2a+2)(a2+2a+2)
Third, the AI leads the student to the final deductive conclusion regarding primality. For any integer a>1, both factors are strictly greater than 1 (e.g., for the minimum bound a=2, the factors evaluate to 2 and 10, respectively).
Consequently, the AI effectively models the cognitive leap from seeing a rigid binomial to discovering a hidden composite structure. By transforming a rote algebraic manipulation into an intuitive, goal-oriented search for symmetry, the AI prevents cognitive stall and fosters true mathematical fluency.
AI Pedagogy in Higher Mathematics: Solving the Verhulst Equation via Separation of Variables
To demonstrate the capacity of AI to resolve cognitive blockages in higher-level undergraduate mathematics, we examine the classical analytical derivation of the Verhulst differential equation. Rather than forcing students to execute integration sequences through rote memorization, an AI-driven instructional framework models the solution by anchoring the separation of variables method in strict algebraic symmetry.
The historical Verhulst model is formulated using linear and quadratic growth parameters as follows:
dx/dt = ax - bx2
The AI guides the student to isolate all variables associated with x on the left side of the equilibrium and time (t) on the right side. Factoring the denominator yields:
dx/[x(a−bx)]=dt.
Instead of introducing the integration step as an arbitrary procedural rule, the AI explains how the left-hand side can be meaningfully split into a sum of partial fractions to decouple the algebraic components.
The decomposition is initiated by introducing two undetermined constants, A and B:
1/[x(a−bx)]=(A/x)+B/(a-bx).
To determine the values of A and B, the AI guides the student to recombine the right-hand side over a common denominator:
1/[x(a−bx)]=[A(a−bx)+Bx]/ x(a−bx).
Since the denominators are identical, their numerators must be mathematically equivalent. Equating the numerators yields:
1=A(a−bx)+Bx⟹1=Aa−Abx+Bx.
The AI then demonstrates the principle of polynomial identity by grouping terms relative to the powers of the variable x (x1 and x0):
1=(B−Ab)x+Aa.
For this equation to hold universally for all values of x, the coefficients of corresponding powers on both sides must be equal. Since the left side contains no x term (coefficient is 0) and a constant term of 1, this directly generates a system of two basic algebraic equations:
B−Ab=0; Aa=1
The system is solved sequentially through basic substitution. From the second equation, we isolate A: A=1/a.
Substituting A into the first equation yields the value for B:
B−(1/a)b=0⟹B=b/a.
Substituting these calculated coefficients back into the partial fraction template transforms the original expression:
1/[x(a−bx)]= (1/a)/x+(b/a)/(a-bx)=(1/a)⋅[(1/x)+ b/(a−bx)].
This transformation directly prepares the dynamic equation for symmetrical integration:
∫(dx/x) + ∫bdx/(a-bx) = ∫adt
Integrating both components yields the logarithmic balance:
ln(x)−ln(a−bx)=at+C⟹ln[x/(a−bx)]=at+C
To transition from logarithms to the continuous time domain, the AI eliminates the natural logarithm by applying the exponential function, utilizing the clean exp(…) notation:
x/(a−bx)=exp(at+C)=C1exp(at)
Where C1=exp(C) is an integration constant. In the final phase, the AI algebraically isolates x(t) through systematic distribution and factorization:
x=aC1exp(at)−bxC1exp(at)
x [1+bC1exp(at)]=aC1exp(at)
Dividing both the numerator and denominator of the right-hand side by C1exp(at) yields the classic analytical solution:
x(t)=a/[b+C2exp(−at)].
By anchoring each calculus phase in algebraic notation transparency and structural symmetry, the AI prevents the formation of "dead operational algorithms," enabling the student to perceive the exact mathematical mechanism underlying non-linear growth constraints.
To determine the constant C2, we write down the initial condition under which the population size at the initial moment of time (t=0) is equal to the known value x0:
x(0)=x0.
Substituting these parameters into the general solution yields:
x0=a/[b+C2⋅exp(−a⋅0)]
Since exp(0)=1, the expression simplifies immediately, minimizing cognitive load for the student:
x0= a/( b+C2).
Using fundamental algebraic transposition, we isolate the denominator component:
b+C2=a/x0⟹C2= a/x0−b = (a-bx0)/x0.
Substituting this precise value back into the general template outlines the specific trajectory of the Cauchy problem:
x(t)=a/[b+((a−bx0)/x0)⋅exp(−at)]
To eliminate the compound fraction in the denominator, both the numerator and the denominator are symmetrically multiplied by x0, resulting in the final, clean analytical solution:
x(t)= ax0 / [bx0+(a−bx0)exp(−at)].
Once the specific analytical solution is fully constructed, the AI tutor transitions the student’s focus toward long-term qualitative system dynamics by evaluating its behavior as time approaches infinity (t→∞).
Given that the intrinsic growth parameters is positive (a>0), the temporal expansion causes the negative exponent to decay toward zero:
lim (t → ∞) exp(-at) = 0.
Applying this mathematical limit directly to our specific analytical solution isolates the behavior of the denominator:
x∞ = lim (t → ∞) (ax₀) / [bx₀ + (a - bx₀)exp(-at)] = (ax₀) / [bx₀ + (a - bx₀)0] = a/b.
This zero-decay property completely neutralizes the transient historical modifier (a−bx₀), leaving a simplified rational baseline:
x∞= ax₀ / bx₀ = a/b.
Canceling out the initial state variable x0 from both the numerator and the denominator yields the clean, fundamental limiting solution:
x∞=a/b.
Through this analytical finalization, the AI successfully delivers a crucial epistemological takeaway: the ultimate carrying capacity of the system (x∞=a/b) is determined strictly by its internal structural parameters (a and b). The initial historical condition x0 (provided x0>0) merely dictates the early trajectory, while the long-term asymptotic equilibrium remains a robust, invariant property of the Verhulst mathematical architecture.
Similarly, a reasoning AI, such as Gemini, can explain to a student any more complex mathematical problem, for instance, one involving the qualitative analysis of autonomous nonlinear dynamical systems.
All of the above shows that with the advent of reasoning AI, the situation is evolving in such a way that students no longer need large math lecture halls, electives, or seminars alongside. university instructors—particularly those who received their education after the end of the Cold War and the collapse of the socialist system.
Conclusion: From Cognitive Extinction to Intellectual Renaissance
The systemic crisis of mathematical education, documented in the first part of this study, takes on the character of an anthropological catastrophe in the era of artificial intelligence. As T. Cook [27] in LSE Impact profoundly observes, contemporary Generative AI has caused a devastating "severance of competence from thinking": an impeccably structured text or a competent academic paper is no longer evidence of an underlying active mind. Encountering "AI sycophancy," modern students and researchers—whose cognitive foundations have already been eroded by the mechanical patterns of standardized testing—are thoroughly losing the very instinct for when something needs to be thought through at all. The cycle of simulation is complete: the educational establishment, as represented by mainstream commentators R. Mason, R. Jalkebro, and Z. Hani (2026) in University World News [28], attempts to combat this crisis either through futile technological policing (such as Turnitin) or through utopian appeals for manual human mentorship (KYS), both of which collapse against the realities of a commercialized, assembly-line mass education.
We argue that the salvation of academia lies neither in the prohibition of technology nor in passive appeasement, but in a radical paradigm shift in how AI is utilized. The tragedy described by T. Cook applies strictly to AI used as a mere "text generator." However, the advent of advanced Reasoning AI allows us to transform this technology from an instrument of cognitive degradation into an autonomous trainer of logic and intellectual discipline.
It is precisely this type of AI that can shatter the educational facade by acting not as a sycophantic flatterer, but as an uncompromising “digital Socrates.” It assumes the crucial function of a deep, longitudinal diagnosis of the student’s cognitive deficits:
- Instead of providing ready-made answers, the AI isolates conceptual gridlocks (such as the inability to operate with basic fractions demonstrated in our empirical tests) and forces the student back to the rigorous, visual geometric principles of the school of A. P. Kiselev.
- It completely decimates the routine technical barrier of complex programming, enabling students to directly channel their mathematical intuition into investigating the living beauty of non-linear dynamics and fractal topologies, as pioneered by the Saratov research school.
Our warning remains absolute: without a return to a fundamental physical-mathematical culture, the global technosphere and ecosphere are mathematically condemned to aggressive degradation. Yet, through the strategic adoption of Reasoning AI, the thinking segment of humanity—independent lyceums, enthusiastic educators, and rigorous scholars—has acquired a formidable weapon of cognitive resistance. AI restores to mathematics its true destination as envisioned by V. I. Arnold: to serve not as a repository of dead algorithms, but as the living language for investigating the real world.
Appendix
Historical and Mathematical Problems for Testing the Logical-Mathematical Reasoning of Students
1. Problem 19 from the Moscow Mathematical Papyrus of Golenishchev (comprising 25 problems), compiled in Egypt around 1850 BC. The papyrus was purchased in 1892 or 1893 at a market in Thebes by the prominent Egyptologist Vladimir Semenovich Golenishchev (1856–1947) and is currently preserved in the Pushkin State Museum of Fine Arts in Moscow.
Task: To a given number, taken 1 and 1/2 times, add 4 to obtain 10. What is the initial number?
2. The problem regarding the lifespan of Diophantus from the Palatine Anthology. Diophantus of Alexandria (circa 2nd–3rd centuries AD) was the last of the great mathematicians of Ancient Greece. The Palatine Anthology is a celebrated collection of ancient and medieval Greek epigrams, compiled by the 10th-century Byzantine grammarian Constantine Cephalas.
Task: Diophantus’s boyhood lasted 1/6 of his life; his beard began to grow after another 1/12 of his life; he married after another 1/7 of his life, and 5 years later a son was born to him. This son lived exactly 1/2 of his father’s total lifespan, and Diophantus died 4 years after his son’s passing. What was the lifespan of Diophantus?
3. The Problem of Pythagoras (570–490 BC).
Task: Prove that every odd number, except for 1, can be expressed as the difference between the squares of two integers.
4. The Problem of Archimedes (circa 278–212 BC).
Task: Prove that the area of a circle circumscribed about a square is exactly twice the area of a square inscribed within that same circle.
5. Problem 31 from the final book of the Chinese treatise of Sun Tzu (3rd–4th centuries AD). This identical problem is also found in the classic treatise The Nine Chapters on the Mathematical Art, which represents a compilation of scholarly works written in China between the 10th and 2nd centuries BC (encompassing 246 problems).
Task: There are pheasants and rabbits inside a cage. From above, one can count 35 heads; from below, there are 94 feet. How many pheasants and how many rabbits are in the cage?
6. The Problem of G. W. Leibniz (1646–1716). Gottfried Wilhelm Leibniz was a monumental German philosopher, mathematician, physicist, inventor, jurist, historian, and linguist. Alongside Isaac Newton, he independently developed the foundational frameworks of integral and differential calculus.
Task: Show that if n is an integer, then n5 - n is always divisible by 5.
7. The Problem of Carl Friedrich Gauss (1777–1855). This exact problem is also documented in the ancient Chinese treatise (Problem 36 of the final book) by Zhang Qiujian (5th century AD), composed in connection with the Chinese invention of the concept and formulas of an arithmetic progression. Resolve this task via a purely logical pathway, without utilizing the standard formula for the sum of the terms of an arithmetic progression.
Task: Compute the total sum of the first one hundred terms of the natural sequence: 1 + 2 + … + 99 + 100 = ?
8. A problem from the "Compendium of Arithmetic" (Ganitasara) by the Indian mathematician and philosopher Sridhara (circa 950–1000, Bengal). Sridhara authored a series of innovative mathematical problems that were subsequently widely utilized by later generations of Indian mathematicians.
Task: A fifth part of a swarm of bees settled on a kadamba flower, and a third part settled on a silindhra flower. Three times the difference between these two groups directed its flight toward the kutaja flowers, leaving a single small bee hovering back and forth, drawn by the sweet aroma of jasmine. How many bees were in the swarm in total?
9. The Problem of Sophie Germain (1776–1831). Sophie Germain was one of the pioneering French women mathematicians who maintained an active scientific correspondence with all the preeminent mathematicians of her era. This particular task is closely linked to her lifelong fascination with number theory.
Task: Prove that every number expressed in the form of a4 + 4 is composite—meaning it can invariably be factored into distinct multipliers.
10. A problem from the "Arithmetic" of Leonty Filippovich Magnitsky (1669–1739). By decree of Emperor Peter the Great, L. F. Magnitsky (Telyashin) compiled the first Russian textbook-encyclopedia on mathematics, entitled Arithmetic, That Is, the Science of Counting (1703).
Task: A master hired a laborer for a full year, promising to pay him 12 rubles and a caftan. However, having worked for only 7 months, the laborer decided to leave. Upon settling the account, he received the caftan and 5 rubles. How much does the caftan cost?
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