=Paper=
{{Paper
|id=Vol-2770/paper1
|storemode=property
|title=Digital Transformation of School and the Role of Mathematics and Informatics within It Problems and Paradoxes of Mathematics Education and their Digital Solution
|pdfUrl=https://ceur-ws.org/Vol-2770/paper1.pdf
|volume=Vol-2770
|authors=Alexei Semenov,Sergei Polikarpov
}}
==Digital Transformation of School and the Role of Mathematics and Informatics within It Problems and Paradoxes of Mathematics Education and their Digital Solution==
https://ceur-ws.org/Vol-2770/paper1.pdf
Digital Transformation of School and the Role of
Mathematics and Informatics within It
Problems and Paradoxes of Mathematics Education
and their Digital Solution1
Alexei Semenov1[0000-0002-1785-2387] and Sergei Polikarpov2[0000-0002-3423-0950]
1
Lomonosov Moscow State University, Leninskie Gory 1, 119991 Moscow, Russia
Axel Berg Institute of Cybernetics and Educational Computing, FRC CSC RAS,
44 Bld. 2, Vavilova st., 119333 Moscow, Russia
Herzen State Pedagogical University, Moyka Embankment, 48,
191186 St. Petersburg, Russia
Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny,
141701 Moscow Region, Russia
alsemno@ya.ru
2
Steklov Mathematical Institute of RAS, Gubkina St., 8, 119991 Moscow, Russia
polik@mi-ras.ru
Abstract. Mathematics is the backbone of digital technology, critical to our entire
civilization. The demand for mathematical competence in professional activity as
well as in everyday life is raising. Paradoxically, interest for studying math decreas-
ing in general education. The paradox is caused by the huge digital gap dividing pre-
digital, even anti-digital school from digitally accelerating society. The paper revises
the goals of mathematical and computer science education in the 21 st century, and
the role of this education in the general school. Using digital instruments for the
main activities for students’ work in learning and applying mathematics, as well as
for communication with teachers and collaborative work on digital platforms for
learning, should be the critical factor of improvement and sustainable development.
At the same time mathematics as a field of discovery of novel and un-expected can
play an important role in the whole agenda of formation of an individual in the
VUCA world. The paper considers the case of general education as well as teachers’
preparation in Russia in the perspective of the framework described.
Keywords: digital transformation of education, mathematics education, math-
ematical literacy, digital (computational) competence.
1 Introduction
There is a “paradox” of mathematics education:
1
Supported by the Russian Science Foundation, grants No. 17-11-01377, 19-29-14152.
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0
International (CC BY 4.0).
Proceedings of the 4th International Conference on Informatization of Education and E-learning Methodology:
Digital Technologies in Education (IEELM-DTE 2020), Krasnoyarsk, Russia, October 6-9, 2020.
Mathematics is becoming an increasingly important element of modern civiliza-
tion: all digital technologies are built on mathematical methods and results.
The attitude of schoolchildren to basic mathematics in many countries is deteriorat-
ing: children lose interest in it and perceive it as meaningless. The level of mathe-
matics education of different categories of high-school graduates is falling.
To deal with this paradox, it is useful to understand how mathematics is actually used
in the work and daily life of a person in the 21st century.
The number of professionals working in the field of fundamental mathematics is
growing, but remains insignificant. This does not mean that children, who could be-
come future mathematicians, need not be found, motivated and supported, for exam-
ple, by organizing specialized schools for them. But this is not about them, but about
mainstream schools.
In Russia, as in many other countries, there is a shortage of IT professionals in the
broadest sense, ranging from chip developers to applied mathematicians, creators of
new algorithms and models of reality and behavior. This entails working with mathe-
matics at school on a larger scale than previously. Some successful attempts to intro-
duce “coding” in the kindergarten exemplify such work [1]. The variability of the
Technology school curriculum, which will be discussed below, is especially im-
portant.
It is widely (and probably reasonably) accepted that the mainstream school should
take into account the presence of these two categories of professionals (pure mathe-
maticians and the users of mathematics), give all children a certain minimum level
and interest in mathematics, involve most children in its study, etc. Then, there will be
more chances to find future researchers. And this is one of the reasons why it is im-
portant to stop the decline in interest in mathematics at school. The country that man-
ages to do this will receive a competitive advantage.
There is a significant number of professionals (although it is still a minority) who
use sophisticated software in their work. These are, for example, designers, engineers,
doctors, lawyers, and financial analysts. For many of them, it is not so mandatory to
understand “how it works inside”. It is also not mandatory to understand this for an
ordinary “user”, pulling a mobile phone out of his pocket to make a call or recalculate
the tax amount.
We said “not mandatory”, but still believe that it is “desirable”. For a car mechan-
ic, it is essential to understand how an internal combustion engine works, for a taxi
driver, this is advisable, for a car enthusiast, it is desirable, for a taxi passenger not at
all. But sometimes it is important for everyone to understand “what is going on” when
“the spark is failing”. It is also desirable to understand how graphs are arranged in the
Esquire or Bloomberg Businessweek magazine, or how video files are archived. This
issue becomes vital in the 21st century, and every year (month, day) it is becoming
more and more urgent, as the development of digital technologies is accelerating, and
people are constantly facing new situations. One has to build models from ready-
made mathematical bricks, for example, choosing the shortest route or calculating
their expenses, or planning a renovation, etc. At the same time, one usually holds a
calculator in their hand. But it is even more important that one is increasingly sur-
rounded by artificial intelligence, which will increasingly be entrusted with modeling
reality and decision-making. It becomes critical to understand how artificial intelli-
gence works, and understand what decisions it makes and how these decisions are
justified. And at the heart of this understanding (as well as at the heart of AI construc-
tion) is mathematics.
Thus, we sometimes need, albeit very rough, models of reality, and mathematics is
a part of these models.
But in addition to the external reasons for studying mathematics, there is also an
internal motivation, which is more significant for an average student. The math prob-
lem may be interesting in itself, and not because its content will be useful “in real
life”, but more especially if it comes in handy in ten years’ time. This means that the
task should look new and unexpected, and have the right individual level of difficulty
for the student (be in the zone of his proximal development).
Another goal of school mathematics, significant for everyone, is usually spoken
of. Ivan Yakovlevich Depman apocryphally ascribed the following motto to
Lomonosov: “And mathematics should be taught at least because it puts the mind in
order” [2]. This motto corresponds to our dream that our high-school graduates, hav-
ing learned the mathematical way to reason, give definitions, find a mistake in proof,
give a disproving example, etc., would be able to do it independently, and not only in
the field of mathematics and its application in real life, but also in a wider context,
e. g. a legal one. It is clear that in order to achieve the goal of transferring the methods
of reasoning from mathematics to new contexts, it is necessary to provide such new
contexts for students, at least in mathematics. Another reason for the importance of
the novelty factor in mathematics is the usefulness of the human quality of pre-
adaptability — the readiness to face something UNexpected, UNforeseen, and cope
effectively with it.
2 The situation in Russian schools
So, here are the goals that are desirable to achieve, and which, hopefully, will be ef-
fectively achieved and contribute to an increased interest in mathematics, and the
motivation to study it. It is necessary to form the ability of students:
to reason logically, even outside of mathematics.
to model reality using ready-made mathematical models and creating new ones.
The undoubted advantage of Russian, in particular Soviet school mathematics, is that
it is “problem-based” [3]. This means not learning facts, but applying them to solving
problems. Today it is commonly called the “competency-based approach”, minus the
significant fact that these problems are taken not from real life, but from a problem
book. In the school course of algebra, there are few “theorems” and many problems,
whereas in the geometry course there are many theorems, but there are also many
problems. However:
The problems of school mathematics are monotonous. Of course, when solving
trigonometric equations, there may be some “subtle, unexpected moves”, but the
average student does not get to the subtleties. It is significant that the introduction
of the problems on integer numbers, with some restrictions into the word problems
of the Russian Unified State Examination (EGE), such as “how many boxes will be
enough”, was perceived almost as revolutionary. Of course, these problems quickly
became “standard”, but still expanded the “scope of the standard”.
From an “applied” point of view, all school equations can be solved by computer
algebra systems and it is precisely these systems that a professional uses if neces-
sary. Word problems, as already mentioned, are monotonous and oversimplified.
The applied value of school geometry can be boiled down to some facts that occu-
py only a small portion of the curriculum.
There is almost no logical reasoning in school algebra. In fact, most students simp-
ly learn to follow a given pattern. This blind following causes the logic to disap-
pear. In school geometry, the proofs of theorems are also learned by heart, and not
found independently, there is not so much reasoning in geometric problems, and
the degree of novelty in these problems is insignificant.
School mathematics, the EGE, textbooks, and teachers alike, ignore digital tools
for mathematics. It is noteworthy that, for example, school “mathematical statis-
tics” and data analysis in mainstream Russian textbooks, do not involve any digital
means. Although this section already appeared in Russian textbooks in the digital
age, at the beginning of the “era of big data”, schools still remain in the pre-digital
world.
3 What can we do?
The EGE is often blamed for the loss of the quality of mathematics education and the
loss of interest in it. Indeed, it is hard to both achieve our goals and prepare for the
mandatory exam in traditional mathematics, whether it be the EGE or the final exam
of a Soviet school. Perhaps we need to reconsider the content of both the school
course and the exam.
In short, the most important thing that needs to be done in school mathematics is
to give a lot of essentially new and unique problems, taking into account topics that
are more directly focused on the modern world, in particular, digital technologies.
This is the world of logic, language, combinatorial objects — finite symbolic se-
quences (chains) and finite multisets (bags).
Such problems can be found in the so-called “recreational mathematics”. Note that
the word “recreational”, interestingly, is not only an advertising gimmick in this case.
There is a real difference between these problems and the flow of monotonous school
problems.
In connection with the discussed issues, it is worth dwelling on the Russian course
of computer science, first of all, on its mathematical component. This course was
designed by mathematicians (including the author of these lines) who in one way or
another came into contact with digital technologies, primarily with programming and
teaching it. For us at that time, the introduction of computer science to schools was,
among other things, a way to update school mathematics “from the outside”. To some
extent, this attempt was successful:
1. The problems on algorithm design and the possibility of their visual execution on
the computer screen significantly expand the field of mathematical objects, the es-
sential novelty is achieved much easier than in school algebra.
2. Some “recreational” problems arise naturally, essentially consisting in algorithm
design. Problems on the development of “algorithmic thinking” often take on a
visual form, e. g. for a robot in a maze.
3. Many techniques in algorithm design, on the one hand, relate precisely to those
models that arise in mathematics and can be transferred to other areas (for exam-
ple, the “divide and conquer” strategy), and on the other hand are also used in seri-
ous, adult programming, and in real life.
Of course, the computer science course offers an abstract mathematical descrip-
tion of the functioning of a computer, along with a mathematical model of program
execution.
Does this mean that “continuous” mathematics should be completely abandoned?
Of course not. The abstraction of a real number is the most important achievement of
mathematics, as well as elementary functions — the sine, the exponential function.
But the center of gravity should be shifted from “simplifying expressions for taking
logarithms”, to independent discovery of the properties of functions, basic formulas
and identities, general formulas for solving an equation, as well as solving the sim-
plest equations in an amount sufficient to understand the general principle, but with-
out the obligatory achievement of a high level of faultlessness, and the technical so-
phistication in symbolic transformations. After that, the student can delegate the solu-
tion of the equations to a computer algebra system and, with the help of this system,
solve all school and non-school equations.
We can approach the goal of teaching and learning modeling in the Russian course
of physics, primarily in middle school. Ideally, there are real experiments: the ball is
rolling along the gutter, the rheostat engine is moving, the pendulum is swinging.
Note that this ideal can be achieved thanks to digital technologies: digital sensors of
position, temperature, pressure, current, etc., and computer algebra systems. On the
one hand, this reasonably reduces the time spent on conducting an experiment and a
visual presentation of its result. On the other hand, students actually use some real
objects — digital sensors equipped with a large number of physical effects and tech-
nological principles. Computer algebra systems also allow students and teachers to
overcome algebraic difficulties and errors more easily and focus on the physical es-
sence of phenomena and their mathematical models. Another role of a physical exper-
iment, both in the physics classroom and remotely via telecommunications, is to be a
source of data for analysis using statistical and machine learning methods. Thus, we
are moving towards an important goal of mathematics education — learning the ele-
ments of artificial intelligence. The topic of school physics as the right place for
mathematical modeling deserves separate consideration.
What about geometry? Its importance at school fell sharply during the couple of
years when it was not included in the EGE. Now it is being restored. As evident from
the above, we believe that the role of geometry is primarily the solving of various
problems that are appropriately difficult for every student. The number of geometry
lessons can even be slightly increased. At the same time, the course should be de-
signed for students of different levels to have a sufficient number of problems to
solve. It is important that logic is supported by visualization. This property of school
geometry is greatly enhanced in dynamic geometry systems, where one can accurately
and beautifully create a drawing, then transform it maintaining the configuration (in-
cidence of elements).
What should the “Mathematics, Computer Science, Physics, Technology” school
education sector look like? The following vector of development of school mathemat-
ics education emerges:
1. Expansion of the range of problems and a significant increase in their novelty,
which is even more significant than the inclusion of a particular area of mathemat-
ics into the curriculum. Inclusion of the principle of novelty in the EGE frame-
work.
2. The use of a computer as a tool for mathematical activity, in particular, for experi-
ments, visualization, data analysis (statistics), and algebraic calculations.
3. Physics as a natural field for mathematical modeling and data analysis using digital
technologies.
4. Algebra — achievement of all the results required by the Federal State Educational
Standard and curriculum guidelines, and many other results achieved by students
using computer algebra systems.
5. Geometry — formation of a system of goals and a system of tasks (as well as re-
search tasks, projects) of various difficulties. These systems, placed on the digital
learning platform, allow to build individual educational routes designed for the
mandatory achievement of all set goals. Such a personalized approach to teaching,
if compliant with the Federal State Educational Standard, is also applicable to other
areas and subjects, but it is especially important for geometry. In addition, dynamic
geometry should be used for experiments.
6. Computer science — the use of algorithms as a source of a wide range of new tasks
and puzzles, and a computer as a tool for experiments, debugging, finding errors in
one’s own work. A common system of basic objects should be used for both ele-
mentary mathematics and mathematical computer science.
4 What is already being done?
How does the content of modern Russian mainstream school mathematics relate to all
this? This content is mostly determined by the EGE, mainstream textbooks, an aver-
age teacher, and the way of training new teachers in pedagogical (and other) universi-
ties. Unfortunately, the above-mentioned development trends are implemented to a
very small extent in all these elements, even if we consider (as formulated in the Fed-
eral State Educational Standard) mathematics and computer science combined. More-
over, the above-mentioned decline in interest is accompanied by a “decline in mor-
als”. Students don’t see any point in solving problems independently and use answer
books from the Internet, at best complementing them with an effective Russian com-
puter algebra system UMS [4], and more recently, the Croatian product Photomath
[5].
Nevertheless, some changes are taking place. For example, the international Kan-
garoo contest has been very popular in Russian primary schools for decades, in many
ways it is the creation of our compatriot, Professor Mark Bashmakov from St. Peters-
burg [6]. The problems of this competition represent a wide range of mathematical
questions of varying difficulty. “Kangaroo” sets an example for motivating students.
The problems of this competition are solved every year by millions of children in
Russian primary schools. At the same time, the competition is not supported by the
state, and even meets certain resistance from the educational authorities.
The systematic development of the modern content of mathematics + computer
science has been going on for three decades. The textbooks are published by
Prosveshchenie publishing house [7] and are used in hundreds of Russian schools.
They attempt a balanced introduction of modern, combinatorial, logical, and algo-
rithmic content, along with the traditional numerical one.
Since the 1960s there is a system of mathematical schools in Russia that has
gained worldwide fame. One of the powerful branches of this system implements the
methodology of N. N. Konstantinov. Without trying to describe it as a whole, let us
pay attention to just one of its features. Within this framework, students “create”
mathematics (not so important what kind of mathematics) themselves. The problems
they receive have a high degree of novelty — these are individual lemmas, steps in
proving important, understandable, motivating theorems, and not monotonous solu-
tions of equations that do not make sense to the student.
As already mentioned, we largely associate the progress in mathematics education,
in particular, the resolution of our initial paradox, with the practice of using digital
technologies in school. One of the approaches to this practice is the Framework of the
Subject Area Technology at School [8]. This framework implies significant variability
in the content of school technology education and the application of technology in
various school subjects. Based on these provisions, a team of authors is preparing for
publication, a textbook on computer science and ICT. In its digital form there are
modules on the use of digital technologies in the process of learning various school
subjects. In particular, in the module on mathematics, there will be an introduction to
the dynamic geometry system GeoGebra, which is becoming the most popular digital
mathematical tool in school [9], and there will also be a tutorial on the Mathematical
Constructor [10, 11]. In the physics module, digital sensors and computer algebra will
be taught.
Of course, the EGE will apparently remain the main regulator and, to a certain ex-
tent, a drag on the modernization of school mathematics education. However, there
are also modest victories here. In 2021, for the first time, the computer science exam
will be run on computers, and real programming will be found in the corresponding
tasks. Hopefully, it will be allowed to use an ordinary fraction as an answer in the
EGE in mathematics, and the computer will understand this.
The number of participants in mathematical modeling Olympiads is growing [12,
13].
Of course, training new mathematics teachers is key. For several years, the princi-
ples of the above approach have been implemented in the training of primary school
teachers and mathematics teachers at Moscow State Pedagogical University [14]:
the problem component was strengthened, primarily due to the “school” problems,
computers were used for mathematical activity.
The work was supported by: Russian Science Foundation, grant number 17-11-01377
(A. L. Semenov, parts 1 and 4), Russian Foundation for Basic Research, grant number
19-29-14152 (S. A. Polikarpov, parts 2 and 3).
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