SEAMEO MEETING
UNIVERSITY OF TSUKUBA, TOKYO 2020, FEB 13

MATH EDUCATION IN CHANGING WORLD
Math vs MthEd. Conflict or collaboration?

Ivan Vysotskiy, Moscow, MCCME
http://ptlab.mccme.ru

AN INTRODUCTION AND A HISTORICAL REVIEWIn the modern world of school education, one can see
a significant gap between the scientific and educational
communities, which has developed over the past 150 —200 years. 200 years ago the situation was different: in anera when classical, in particular mathematical, education
was the destiny of a few, school mathematics was basedon the enthusiasm and enlightening activity of leading
contemporary scientists.

HISTORICAL REVIEW

It is enough to recall Legendre, who wrote «Elémentsde géométrie», which served as the basic geometry
textbook in all educational institutions not only in France
but also in other European countries, in particular in
Russia, until the beginning of the 20th century. Legendclaims that it was Legendre who introduced the division
sign «: » or « ÷ ». Another striking example is Leonard
Euler. His textbook, «A Guide to Arithmetic», translated
into Russian by Vasily Adadurov (1740), was written for
the gymnasium opened at the St. Petersburg Academy ofSciences and for many decades was one of the main
textbooks of Russian gymnasiums.

HISTORICAL REVIEWHISTORICAL REVIEW

Speaking about the Russian school in the 19th – 20th
centuries, one must not forget the contributions of P.L.
Chebyshev, A.N. Kolmogorov, A.Ya. Khinchin, I.M.
Yaglom, V.I. Arnold. Recently, much attention is paid toschool mathematics by the Fields prize winner S.K.
Smirnov. He is one of the authors of the modern Conceptof the development of mathematical education in Russia(2013).

CURRICULA AND TEXTBOOKS

Figuratively speaking, if curricula and textbooks
were written only by educators and professional
educational specialists, then school mathematics would
degenerate into scholastic knowledge, full of errors,
incorrect interpretations of mathematical facts, andartificial problems.

SCHOOL MATH AS AN ALGORITHM

Mass school education is based on using andcombining algorithms: long multiplication and division,
multiplication, opening brackets, factorization etc.
A teacher who does not see the global goals ofmathematical education naturally seeks to fill the content
with algorithms that are easy to teach, not fully
understanding or taking into account their real value.
It is easy to imagine that there are teachers who
dream of teaching schoolchildren solely how to solve
quadratic equations because this is an easily achieved andeasily controlled goal.

SELF-CLOSED DIDACTICS

We observe a lot of biases towards the technologicaleffectiveness of training both in Russia and in othercountries.
Just an example of a "self-closed didactics". In amass school, logarithms appeared for practicalcomputational needs.
Since the 1950s, the practical importance oflogarithms began to shrink rapidly. At the same time, awhole genre of mathematical didactics “logarithmicequations and inequalities” appears and develops, where
the properties of the logarithm are used as a way to create
sophisticated puzzles that have nothing to do with actual
applications of logarithms.

NEW MATH WAVE GIVE NEW SCDs

Here is an example from a modern collection of tasks
for preparing for a university entrance exam:
2log1-3 x ( 42 x 2 - x + 1)£125öæ
log1-3 x ç x - ÷6øèRETROSPECTIVE

We will not comment on this, we only say that in theclassic textbook by A. P. Kiselev “Algebra 1” you canfind many tasks “reduce the expression to a form
convenient for taking a logarithm”, natural for that time,and that exactly one page of the textbook is devoted to thelogarithmic equations and just mentions that the equationlog ( x + a ) + log ( x + b ) = log ( x + c )can be easily reduced to quadratic.

SIMPLIFICATION AND SLOWDOWN

This is not the worst. Better that than simplification which
grows from the same root of “professional approach” tomath education instead of approach of mathematicians toeducation.
Excluding professional mathematicians out the education
accelerates another harmful tendency we observe in theworld: there is excessive slowdown of the learning
content, bias towards the basic component, an attempt to“remove stress” from the learning process, dangerous
simplification and gamification.
Teachers and curricula developers give up to populist
trends.

ROLE OF PROFESSIONALSFor these reasons, the role of professional mathematiciansin the formation and ongoing support of mathematicaleducation is extremely important.
Not only curricula and textbooks but also the preparationof teachers should take place in close cooperation withprofessional mathematicians. They must lead futureteachers toward they profession.
People can argue endlessly about the competition between
classical universities and specialized pedagogicaleducational institutions in the field of teacher preparation.

HOW TO GET PROFESSIONALS TO SCHOOL?

Our experience shows that both trajectories areneeded and find their rightful place. The most qualifiedteachers that work in specialized schools and are able tointellectually load motivated students often come from theacademic environment.
An analogy seems appropriate here: a good farmer
does not sell the best fruits but replenishes his seed fundwith them. So, in the field of teacher training, some of thebest specialists should return to school.

DON’T YOU KNOW LETTERS?

Let us give just one example: in elementary school,
equations traditionally appear in the first grade in theform of the task
Find an unknown number x if x + 1 = 3 .
In the modern textbooks, letters are for some reason
replaced by boxes:
Insert a number in the box: + 1 = 3 .
The task did not become either simpler or more
complicated but ceased to bear an important propaedeuticload, taking the time and doing harm.

DISGUISED SABOTAGE

Simplification processes are usually presented as acommon good that ensures “equal rights andopportunities”, “universal access to quality education”,
etc. In fact, the availability of high-quality education andthe very quality of education suffer, because both thegeneral educational background and the level of rational
thinking are reduced in society.

VICIOUS CIRCLE

In addition, the process of simplification is
unstoppable: the lower the final requirements inmathematics, the lower the prestige of mathematicalknowledge for the next cohort, and the lower the final
requirements are.
This is a vicious circle that is difficult to break. Thiscan be broken out only by joint efforts of all the forces
acting in education: professional mathematicians,
educational organizers, methodologists, teachers, children
and parents.

DID PCK WIN THE FINAL VICTORY?

In recent decades the idea that the knowledge ofmathematics is not the main thing for a mathematics
teacher has been rooted in many societies. The balance ofCK (content knowledge) and PCK (pedagogical contentknowledge) is clearly leaning towards the latter.
Moreover, it was inclined to such an extent that many
young teachers who come to school have a vague idea ofclassical calculus, cannot form an image of amathematical object corresponding to a definition, andhave no idea of how different sections of schoolmathematics are connected with each other.

CONCLUSION THESES

The gap between mathematics and mathematicaleducation is extremely dangerous at any stage of society'sdevelopment — whether it is society #3 or #8.
Now we are harvesting the bitter fruits of thisapproach.
The mathematical Outlook of a math teacher should
be broader and the knowledge deeper than that of anengineer.
For the teacher brings mathematical culture to people,
while the engineer brings only a wrench with him.

MUST TEACHERS KNOW MATH?

What is from the fact I don’t know
mathematics? But I know how toteach!
An old non-jokeYES!

When you teach mathematics to people, please don't
forget about teachers – they are also people.

Maxim Gorky said this about writers
and the Russian language, but this is also
relevant for us.

CONCLUSION THESES

Education policymakers should be clearly aware of
the dangers of simplification of the math teacher training
course for countries that are planning to develop theirtechnology and industry.
I keep and nurse a faint hopethat artificial intelligence can helpus
here.
Becausenaturalintelligence is probably powerless.

THANK YOU