\title{Trends in Mathematics:\\ How they could Change Education?}
\author{L\'aszl\'o Lov\'asz}
\date{March 2008}
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\begin{abstract}
Mathematical activity has changed a lot in the last 50 years. Some of
these changes, like the use of computers, are very visible and are
being implemented in mathematical education quite extensively. There
are other, more subtle trends that may not be so obvious. We discuss
some of these trends and how they could, or should, influence the
future of mathematical education.
\end{abstract}
\section{Introduction}
Mathematical activity (research, applications, education, exposition)
has changed a lot in the last 50 years. Some of these changes, like
the use of computers, are very visible and are being implemented in
mathematical education quite extensively. There are other, more
subtle trends that may not be so obvious. Should these influence the
way we teach mathematics? The answer may, of course, be different at
the primary, secondary, undergraduate and graduate level.
Here are some of the general trends in mathematics, which we should
take into account.
\smallskip
{\bf 1.} {\it The size of the community and of mathematical research
activity} is increasing exponentially; it doubles every 25 years or
so. This fact has a number of consequences: the impossibility of
keeping up with new results; the need of more efficient cooperation
between researchers; the difficulty of identifying ``core''
mathematics (to be mastered at various levels); the need for better
dissemination of new ideas. How can mathematical education prepare
future researchers and appliers of mathematics, future decision
makers and the informed public for these changes?
\smallskip
{\bf 2.} {\it New areas of application, and their increasing
significance.} Information technology, sciences, the economy, and
almost all areas of human activity make more and more use of
mathematics, and, perhaps more significantly, they use all branches
of mathematics, not just traditional applied mathematics. How can we
train our students to recognize problems where mathematics can help
in the solution?
\smallskip
{\bf 3.} {\it New tools: computers and information technology.} This
is perhaps the most visible new feature, and accordingly a lot has
been done to introduce computers in education. But the influence of
computers on our everyday life and research is also changing fast:
besides the design of algorithms, experimentation, and possibilities
in illustration and visualization, we use email, discussion groups,
on-line encyclopedias and other internet resources. Can education
utilize these possibilities, keep up with the changes, and also teach
students to use them in productive ways?
\smallskip
{\bf 4.} {\it New forms of mathematical activity.} In part as an
answer to the issues raised above, many new forms of mathematical
activity are gaining significance: algorithms and programming,
modeling, conjecturing, expository writing and lecturing. Which of
these non-traditional mathematical activities could and should be
taught to students?
\smallskip
I will say some more about these trends, and discuss the question of
their influence on mathematical education. I will make use of some
observations from my earlier articles \cite{LL,LL1}.
\section{The size of the community and of mathematical research
activity}
The number of mathematical publications (along with publications in
other sciences) has increased exponentially in the last 50 years.
Mathematics has outgrown the small and close-knit community of nerds
that it used to be; with increasing size, the profession is becoming
more diverse, more structured and more complex.
Mathematicians sometimes pretend that mathematical research is as it
used to be: that we find all the information that might be relevant
by browsing through the new periodicals in the library, and that if
we publish a paper in an established journal, then it will reach all
the people whose research might utilize our results. But of course
$3/4$ of the relevant periodicals are not on the library table, and
even if one had access to all these journals, and had the time to
read all of them, one would only be familiar with the results of a
small corner of mathematics.
A larger structure is never just a scaled-up version of the smaller.
In larger and more complex animals an increasingly large fraction of
the body is devoted to ``overhead'': the transportation of material
and the coordination of the function of various parts. In larger and
more complex societies an increasingly large fraction of the
resources is devoted to non-productive activities like transportation
information processing, education or recreation. We have to realize
and accept that a larger and larger part of our mathematical activity
will be devoted to communication.
This is easy to observe: the number of professional visits,
conferences, workshops, research institutes is increasing fast,
e-mail is used more and more. The percentage of papers with multiple
authors has jumped. But probably we will reach the point soon where
mutual personal contact does not provide sufficient information flow.
There is another consequence of the increase in mass: the inevitable
formation of smaller communities, one might say subcultures. One
response to this problem is the creation of an activity that deals
with the secondary processing of research results. For lack of a
better word, I'll call this expository writing, although I'd like to
consider it more as a form of mathematical research than as a form of
writing: finding the ramifications of a result, its connections with
results in other fields, explaining, perhaps translating it for
people coming from a different subculture.
Are there corresponding changes in mathematical curricula and, more
generally, in the way we teach mathematics? The first, and most
pressing, problem is the sheer size of material that would be nice
(or absolutely necessary) to teach. In addition, as we will see, we
should put more emphasis on (which also means giving more teaching
time to) some non-traditional mathematical activities like algorithm
design, modeling, experimentation and exposition. I also have to
emphasize the necessity of preserving problem solving as a major
feature of teaching mathematics.
How to find time to learning concepts, theorems, proofs, especially
with the rapid expansion of material, and at a time when class time
devoted to mathematics is being reduced in many countries? Which of
the new areas should make its way to education (on the secondary or
college level), and which of the traditional material should be left
out? This is not a one-time crisis: mathematical research is not
showing any signs of slowing down.
One possible answer to this question is to leave the teaching of any
recently developed area of mathematics to later in the education, to
Masters and PhD programs. The trouble with this approach is that many
educated people will never meet the mathematics of the last 200
years, which will contribute to the unfortunate but persistent
misconception that mathematics is a closed subject. Many of the new
areas of mathematics are important for understanding developments in
technology and science, and by not teaching them we give up
illuminating the increasing role of mathematics in modern life.
The other possible answer is to remove from the curriculum
traditional material that is deemed less important. This approach has
the negative effect of eroding well-established methods for teaching
mathematical thinking. For example, elementary geometry has been
purged from the curriculum in many countries. While this kind of
geometry is indeed peripheral in modern mathematical {\it research},
it is of course still important in {\it applications}, and, perhaps
even more important, its study is very instrumental in the
development of spatial conception, and, perhaps even more
significantly, in understanding the real nature of mathematical
proofs, the ``Aha'' event when an incomprehensible connection becomes
clear through looking at it the right way.
I have no easy answer to this question. Probably one must concentrate
on mathematical competencies like problem solving, abstraction,
generalization and specialization, logical reasoning and use of
mathematical formalism, along with the non-traditional skills
mentioned above (see e.g. \cite{NISS}). One could select a mixture of
classical and more modern mathematical topics that are best suited to
develop these competencies and (of course) basic skills, and at the
same time give some sort of picture of the historical roots as well
as contemporary applications.
Another question raised by the increasing complexity of the world of
mathematics is whether exposition style mathematics has any place in
education. One aspect of this is teaching students to explain
mathematics to ``outsiders'', teaching them how to summarize results
without getting lost in the details. This is not easy to do, but to
teach such skills would be very useful indeed.
A more heretical thought is to do some expository style teaching. In
most sciences like chemistry or astronomy, it is natural to teach in
high school or even college the facts without explaining all the
technical details of their discovery (or even of their exact
meaning). Some of this is done in mathematics too: many students
learn that the regular pentagon can be constructed with ruler and
compass but the regular heptagon cannot, or that equations of degree
5 or more cannot in general be solved by radicals. But these examples
are almost 200 years old! Can we solve the problem of exposing
students to modern mathematics by working out appropriate non-exact
but still mathematical blocks of material? I hesitate to answer
``YES'', but the question is valid.
\section{New areas of application, and their increasing
significance}
The traditional areas of application of mathematics are physics and
engineering. The branch of mathematics used in these applications is
analysis, primarily differential equations. But in the boom of
scientific research in the last 50 years, many other sciences have
come to the point where they need serious mathematical tools, and
quite often the traditional tools of analysis are not adequate.
For example, biology studies the genetic code, which is discrete:
simple basic questions like finding matching patterns, or tracing
consequences of flipping over substrings, sound more familiar to the
combinatorialist than to the researcher of differential equations. A
question about the information content, redundancy, or stability of
the code may sound too vague to a classical mathematician but a
theoretical computer scientist will immediately see at least some
tools to formalize it (even if to find the answer may be too
difficult at the moment).
Even physics has its encounters with unusual discrete mathematical
structures: elementary particles, quarks and the like are very
combinatorial; understanding basic models in statistical mechanics
requires graph theory and probability.
Economics is a heavy user of mathematics---and much of its need is
not part of the traditional applied mathematics toolbox. The success
of linear programming in economics and operations research depends on
conditions of convexity and unlimited divisibility; taking
indivisibilities into account (for example, logical decisions, or
individuals) leads to integer programming and other combinatorial
optimization models, which are much more difficult to handle.
Finally, there is a completely new area of applied mathematics:
computer science. The development of electronic computation provides
a vast array of well-formulated, difficult, and important
mathematical problems, raised by the study of algorithms, data bases,
formal languages, cryptography and computer security, VLSI layout,
and much more. Most of these have to do with discrete mathematics,
formal logic, and probability.
One must add that which branches of mathematics will be applicable in
the near future is utterly unpredictable. Just 30 years ago questions
in number theory seemed to belong to the purest, most classical and
completely inapplicable mathematics; now many areas in number theory
belong to the core of mathematical cryptography and computer
security.
A very positive development in recent decades is the decreasing
separation between pure and applied mathematics. I feel that the
mutual respect of pure and applied mathematicians is increasing,
along with the number of people contributing to both sides. The
diversity of applications should also strengthen the flow of
information across all of mathematics. No field can retreat into its
ivory tower and close its doors to applications; nor can any field
claim to be ``the'' applied mathematics any more.
How to give a glimpse of the power of these new applications to our
students? Perhaps some nonstandard mathematical activities like
programming and modeling (to be discussed later) can be used here.
\section{New tools: computers and information technology}
Computers, of course, are not only sources of interesting and novel
mathematical problems. They also provide new tools for doing and
organizing our research. We use them for e-mail and word processing,
for experimentation, and for getting information through the web,
from the MathSciNet database, Wikipedia, the Arxives, electronic
journals and from home pages of fellow mathematicians.
Are these uses of computers just toys or at best matters of
convenience? I think not, and that each of these is going to have a
profound impact on our science.
It is easiest to see this about experimentation with Maple,
Mathematica, Matlab, or your own programs. These programs open for us
a range of observations and experiments which had been inaccessible
before the computer age, and which provide new data and reveal new
phenomena.
Electronic journals and databases, home pages of people, companies
and institutions, Wikipedia, and e-mail provide new ways of
dissemination of results and ideas. In a sense, they reinforce the
increase in the volume of research: not only are there increasingly
more people doing research, but an increasingly large fraction of
this information is available at our fingertips (and often
increasingly loudly and aggressively: the etiquette of e-mail is far
from solid). But we can also use them as ways of coping with the
information explosion.
Electronic publication is gradually transforming the way we write
papers. At first sight, word processing looks like just a convenient
way of writing; but slowly many features of electronic versions
become available that are superior to the usual printed papers:
hyperlinks, colored figures and illustrations, animations and the
like.
The use of computers is an area where often we learn from our
students, not the other way around. The question here is: how to use
the interest and knowledge in computing, present in most students
today, for the purposes of mathematical education? Most suitable for
this seem to be some nonstandard mathematical activities, which I
discuss next.
\section{New forms of mathematical activity}
\subsection{Algorithms and programming}
The traditional 2500 year old paradigm of mathematical research is
defining notions, stating theorems and proving them. Perhaps less
recognized, but almost this old, is algorithm design (think of the
Euclidean Algorithm or Newton's Method). While different, these two
ways of doing mathematics are strongly interconnected (see
\cite{LL}). It is also obvious that computers have increased the
visibility and respectability of algorithm design substantially.
Algorithmic mathematics (put into focus by computers, but existent
and important way before their development!) is not the antithesis of
the ``theorem--proof'' type classical mathematics, which we call here
{\it structural}. Rather, it enriches several classical branches of
mathematics with new insight, new kinds of problems, and new
approaches to solve these. So: not algorithmic {\it or} structural
mathematics, but algorithmic {\it and} structural mathematics!
What does this imply in math education? As we discussed above,
mathematical education must follow, at least to some degree, what
happens in mathematical research; this is especially so in those
(rare) cases when research results fundamentally change the whole
framework of the subject. So set theory had to enter mathematical
education (one would wish with more moderation and less controversy
than happened with ``new math''). Algorithmic mathematics is another
one of these.
However, the range of the penetration of an algorithmic perspective
in classical mathematics is not yet clear at all, and varies very
much from subject to subject (as well as from lecturer to lecturer).
Graph theory and optimization, for example, have been thoroughly
re-worked from a computational complexity point of view; number
theory and parts of algebra are studied from such an aspect, but many
basic questions are unresolved; in analysis and differential
equations, such an approach may or may not be a great success; set
theory does not appear to have much to do with algorithms at all.
Our experience with ``New Math'' warns us that drastic changes may be
disastrous even if the new framework is well established in research
and college mathematics. Some algorithms and their analysis could be
taught about the same time when theorems and their proofs first
occur, perhaps around the age of 14. Of course, certain algorithms
(for multiplication and division etc.) occur quite early in the
curriculum. But these are more recipes than algorithms; no
correctness proofs are given (naturally), and the efficiency is not
analyzed.
The beginning of learning ``algorithmics'' is to learn to {\it
design}, rather than {\it execute}, algorithms \cite{MAUR}. The
euclidean algorithm, for example, is one that can be ``discovered''
by students in class. In time, a collection of ``algorithm design
problems'' will arise (just as there are large collections of
problems and exercises in algebraic identities, geometric
constructions or elementary proofs in geometry). Along with these
concrete algorithms, the students should get familiar with basic
notions of the theory of algorithms: input-output, correctness and
its proof, analysis of running time and space, etc.
In college, the shift to a more algorithmic presentation of the
material should, and will, be easier and faster. Already now, some
subjects like graph theory are taught in many colleges quite
algorithmically: shortest spanning tree, maximum flow and maximum
matching algorithms are standard topics in most graph theory courses.
This is quite natural since, as I have remarked, computational
complexity theory provides a unifying framework for many of the basic
graph-theoretic results. In other fields this is not quite so at the
moment; but some topics like primality testing or cryptographic
protocols provide nice applications for a large part of classical
number theory.
One should distinguish between an algorithm and its implementation as
a computer program. The algorithm itself is a mathematical object;
the program depends on the machine and/or on the programming
language. It is of course necessary that the students see how an
algorithm leads to a program that runs on a computer; but it is not
necessary that every algorithm they learn about or they design be
implemented. The situation is analogous to that of geometric
constructions with ruler and compass: some constructions have to be
carried out on paper, but for some more, it may be enough to give the
mathematical solution (since the point is not to learn to draw but to
provide a field of applications for a variety of geometric notions
and results).
Let me insert a warning about the shortcomings of algorithmic
language. There is no generally accepted form of presenting an
algorithm, even in the research literature (and as far as I see,
computer science text books for secondary schools are even less
standardized and often even more extravagant in handling this
problem.) The practice ranges from an entirely informal description
to programs in specific programming languages. There are good
arguments in favor of both solutions; I am leaning towards
informality, since I feel that implementation details often cover up
the mathematical essence. For example, an algorithm may contain a
step ``Select any element of set $S$''. In an implementation, we have
to specify which element to choose, so this step necessarily becomes
something like ``Select the first element of set $S$''. But there may
be another algorithm, where it is important the we select the first
element; turning both algorithms into programs hides this important
detail. Or it may turn out that there is some advantage in selecting
the {\it last} element of $S$. Giving an informal description leaves
this option open, while turning the algorithm into a program forbids
it.
On the other hand, the main problem with the informal presentation of
algorithms is that the ``running time'' or ``number of steps'' are
difficult to define; this depends on the details of implementation,
down to a level below the programming language; it depends on the
data representation and data structures used.
The route from the mathematical idea of an algorithm to a computer
program is long. It takes the careful design of the algorithm;
analysis and improvements of running time and space requirements;
selection of (sometimes mathematically very involved) data
structures; and programming. In college, to follow this route is very
instructive for the students. But even in secondary school
mathematics, at least the mathematics and implementation of an
algorithm should be distinguished.
An important task for mathematics educators of the near future (both
in college and high school) is to develop a smooth and unified style
of describing and analyzing algorithms. A style that shows the
mathematical ideas behind the design; that facilitates analysis; that
is concise and elegant would also be of great help in overcoming the
contempt against algorithms that is still often felt both on the side
of the teacher and of the student.
\subsection{Problems and conjectures}
In a small community, everybody knows what the main problems are. But
in a community of 100,000 people, problems have to be identified and
stated in a precise way. Poorly stated problems lead to boring,
irrelevant results. This elevates the formulation of {\it
conjectures} to the rank of research results. Conjecturing became an
art in the hands of the late Paul Erd\H{o}s, who formulated more
conjectures than perhaps all mathematicians before him put together.
He considered his conjectures as part of his mathematical {\it
{\oe}uvre} as much as his theorems.
Of course, it is difficult to formulate what makes a good conjecture.
(There is even a lot of controversy around Erd\H{o}s's conjectures.)
It is easy to agree that if a conjecture is good, one expects that
its resolution should advance our knowledge substantially. Many
mathematicians feel that this is the case when we can clearly see the
place of the conjecture, and its probable solution, in the building
of mathematics; but there are conjectures so surprising, so utterly
inaccessible by current methods, that their resolution {\it must}
bring something new---we just don't know where.
In the teaching style of mathematics which emphasizes discovery
(which I personally find the best), good teachers always challenged
their students to formulate conjectures leading up to a theorem or to
the steps of a proof. This is time-consuming, and there is a danger
that this activity too is eroding under the time pressure discussed
above. I feel that it must be preserved and encouraged.
\subsection{Mathematical experiments}
In some respects, computers allow us to turn mathematics into an
experimental subject. Ideally, mathematics is a deductive science,
but in quite a few situations, experimentation is warranted:
\smallskip
(a) Testing an algorithm for efficiency, when the resource
requirements (time, space) depend on the input in a too complicated
way to make good predictions\footnote{I do not include here
verification of the correctness of a program, which is not a
mathematical issue, but rather software engineering.}.
\smallskip
(b) Cryptographic and other computer security issues often depend on
classical questions about the distribution of primes and similar
problems in number theory, and the answers to these questions often
depend on notoriously difficult problems in number theory, like the
Riemann Hypothesis and its extensions. Needless to say that in such
practically crucial questions, experiments must be made even if
deductive answers would be ideal.
\smallskip
(c) Experimental mathematics is a good source of conjectures; a
classical example is Gauss' discovery (not proof) of the Prime Number
Theorem. Among the contemporary examples of this, let me mention the
most systematic one: the graph-theoretic conjecture-generating
program GRAFFITI by Fajtlowicz \cite{FS1,FS2}.
There are several excellent books about experimental mathematics (see
e.g. \cite{BBG}). Programs like Derive, Maple or Mathematica offer
us, and the students, many ways of experimentation with mathematics.
A simple example: a student can develop a real feeling for the notion
of convergence and convergence rate by comparing the computation of
the convergent sums $\sum 1/k^2$ and $\sum 1/2^k$.
Mathematical experimentation has indeed been used quite extensively
in the teaching of analysis, number theory, geometry, and many other
topics. The success seems to be controversial; my feeling is that,
similarly as in the teaching of algorithms, the development of large
well-tested sets of experimental tasks takes time, and is the most
crucial element of the success of these teaching methods.
\subsection{Modeling}\label{MODEL}
To construct good models is the most important first step in almost
every successful application of mathematics. The role of modeling in
education is well recognized \cite{APMOD}, but its weight relative to
other material, and the ways of teaching it, are quite controversial.
Modeling is a typical interactive process, where the mathematician
must work together with engineers, biologist, economists, and many
other professionals seeking help from mathematics. A possible
approach here is to combine teaching of mathematical modeling with
education in team work and professional interaction.
A good example is the course ``Discrete Mathematical Modeling'' at
the University of Washington \cite{UWMOD} (similar courses are taught
at several other universities, e.g. at the E\"otv\"os University in
Budapest). The main feature of this course is that the students, in
groups of 2 or 3, must find a real-life problem in their environment.
They have to develop a model, gather data, find and code the
algorithms that answer the original question, and give a presentation
of the results. The real-life problems raised are quite broad in
scope, from problems on favorite games to attempts to help family or
friends in their business, and some of the answers obtained turn out
quite useful.
\subsection{Exposition and popularization}
The role of this activity is growing very fast in the mathematical
research community. Besides the traditional way of writing a good
monograph (which is of course still highly regarded), there is more
and more demand for expositions, surveys, minicourses, handbooks and
encyclopedias. Many conferences (and often the most successful ones)
are mostly or exclusively devoted to expository and survey-type
talks; publishers much prefer volumes of survey articles to volumes
of research papers. While full recognition of expository work is
still lacking, the importance of it is more and more accepted.
On the other hand, mathematics education does little to prepare
students for this. Mathematics is a notoriously difficult subject to
talk about to outsiders (including even scientists). I feel that much
more effort is needed to teach students at all levels how to give
presentations, or write about mathematics they learned. (One
difficulty may be that we know little about the criteria for a good
mathematical survey.)
\begin{thebibliography}{99}
\bibitem{BBG}
J.M.~Borwein, D.H.~Bailey, R.~Girgensohn: {\it Experimentation in
Mathematics: Computational Paths to Discovery}, A.K.~Peters (2004).
\bibitem{FS1}
S.~Fajtlowicz: On conjectures of Graffiti, {\it Discrete Math.} {\bf
72} (1988), 113--118.
\bibitem{FS2}
S.~Fajtlowicz: Postscript to Fully Automated Fragments of Graph
Theory \url{http://math.uh.edu/~siemion/postscript.pdf}
\bibitem{UWMOD}
{\it Discrete Mathematical Modeling}, undergraduate course at the
University of Washongton, http://www.math.washington.edu/~goebel/381/
\bibitem{HAL}
P. R. Halmos (1981), Applied mathematics is bad mathematics, in {\it
Mathematics Tomorrow} (ed. L. A. Steen), Springer, 9-20.
\bibitem{LL}
L.~Lov\'asz: Algorithmic mathematics: an old aspect with a
new emphasis, in: {\it Proc.\ 6th ICME, Budapest, J.\ Bolyai Math.\
Soc.} (1988), 67--78.
\bibitem{LL1}
L.~Lov\'asz: One mathematics, The Berlin Intelligencer,
Mitteilungen der Deutschen Math.-Verein, Berlin (1998), 10-15.
\bibitem{MAUR}
S. Maurer (1984), Two meanings of algorithmic mathematics, {\it
Mathematics Teacher} 430-435.
\bibitem{APMOD}
{\it Modelling and Applications in Mathematics Education} (eds:
W.~Blum, P.L.~Galbraith, H.-W.~Henn and M.~Niss), ICMI Study No. 14,
Springer, 2007.
\bibitem{NISS}
M.~Niss: Quantitative Literacy and Mathematical Competencies,
http://www.maa.org/Ql
\end{thebibliography}
\end{document}