Mathematics in the Classroom and Mathematics in the Real World Reflections on
Diverse Views (Zalman Usiskin, University of Chicago, Chicago IL).
In most schools in the world, the mathematics taught in kindergarten through
grade six6 is considered necessary for everyday survival. The arithmetic of
whole numbers, fractions, decimals, and percents; measurement of length, area,
and volume; reading graphs of various kinds; and identifying geometric figures
are unquestioned as fundamental aspects of mathematical literacy. With the
exception of division of fractions, good teachers in
classrooms at these levels find it relatively easy to relate the mathematics
they are teaching to situations in the real world. Numbers, devices for
measurement, statistics, and common geometric shapes abound both in and outside
the classroom.
This situation changes when algebra and formal geometry enter the mathematics
classroom. Most parents do not consider mathematical proof, equations, variables,
and functions necessary for their children’s survival except in the sense that
getting good grades, scoring well on a college entrance test, and taking
advanced courses look better on college applications. The notion that the
mathematics might be important to learn because it is important for
understanding the world takes a back seat to a topic being important to learn
because it is needed for the next course or because it will help surmount
educational hurdles. The mathematics becomes abstract, and most teachers acclaim
that abstraction.
Top students buy into the beauty and power of the abstractions. They are amazed
by the quadratic formula; they are struck by the ways in which changes in
coefficients in a formula for a function are reflected in changes in the graph
of the function; they see how the strict rules of deduction make mathematical
truth an awesome concept.
By the time the interested student enters college, evidences of any relationship
between the mathematics being studied and the mathematics of the real world are
few and far between. Except for the calculation of compound interest, the use of
trigonometry to determine unknown distances and the use of combinatorics to
determine probabilities, the mathematics experience becomes devoid of
applications. Polygons and polynomials, rational numbers and rational functions,
complex numbers and complex fractions, and differential
and integral calculus are studied as theoretical concerns whose realworld
connections, if any, are dismissed in a line or two: “The study of (insert a
topic) has applications to (insert a field).”
The college mathematics student who wants to know how the mathematics in a
course is applied often has to go to another department’s courses for that
knowledge. The mathematics needed in engineering, business, and even statistics
in many places is not taught in the mathematics department, but outside it. And
some mathematics departments are proud of this. As a result, many if not most
mathematics teachers in the world learn to be proud of the purity of the subject
they teach. They view applications as messing up good mathematics and as
imprecise where real mathematics is precise. They realize that applications
might be appropriate for people who could not understand the abstractions of
mathematics, but not appropriate for people like them.
Other individuals view the applications of mathematics as an essential
ingredient in everyone’s mathematics education at all levels. They see applied
mathematics as enhancing pure mathematics just as pure mathematics is necessary
to do applications. As a result, they see no distinct separation between pure
and applied mathematics and view each as having its clean and messy aspects.
They express disdain for curricula that avoid applications just as others
discourage curricula that give strong attention to applications.
Most of these individuals had mathematics educations similar to those with
contrary views, so what happened to cause this reversal of attitudes? My
conversion went through a number of stages. First was the realization that many
of the standard textbook problems that were touted to be applications were more
aptly described as puzzle problems than applications. Few people in the real
world care about finding the ages of people, determining when one train will
pass another, or identifying a number by properties of its digits.
Second, nearly simultaneous with the first, was the realization that there were
some nice real problems that could be solved with standard mathematics. These
included the estimation of the number of fish in a lake, the calculation of the
number of miles one needed to drive in a rented car before unlimited mileage
became cheaper than paying per mile, and the determination of monthly mortgage
payments.
Third was the discovery that these nice real problems were not isolated but the
tip of an iceberg that encompasses virtually all of mathematics. Part of that
realization for me was that everything in the physical world – from the veins of
leaves to the structures of galaxies – was grist for geometry, a realization
that was assisted by my earlier work with geometric transformations. And just as
the study of transformations had itself completely transformed my views of
congruence, similarity, and symmetry in geometry, so this view of the
relationship between geometry and the real world transformed my view of
mathematics
itself.
The path I took to my current views towards applications and modellingmodeling
may not have been the same as those of the organizers and most of the
participants at ICME10 of TSG 20, Mathematical Applications and Modelling in
the Teaching and Learning of Mathematics, but I think we have arrived at the
same position. We believe very strongly that everyone’s mathematics education
needs continual exposure to applications and modelling. The presentations and
panel included fine examples of nice real problems and broader experiences that
demonstrate why mathematics is so important for consumers and educated adults,
and in so many different and varied fields of study. The plenary session of
Andreas Drees was in the same mold, painting a broad picture of how mathematics
has become useful in biology and DNA research.
It is clear that there is a chasm between these views and those of some others
who presented at ICME 10. Perhaps the extreme example was provided by a plenary
speaker who, in addressing mathematical literacy, made the following as
essential to what students should learn: “Know why elementary mathematics has to
be abstract.” “Appreciate that mathematics is ‘the science of exact calculation’.”
(emphasis the speaker’s) “Use ideas of exact calculation to approximate
effectively.” For that speaker, applications would seem not to be a part of
elementary mathematics, and modelling would seem to have no place in mathematics.
One has merely to look around to see that elementary mathematics is everywhere,
and usually not abstract. A look at any tertiarylevel catalog should force
anyone to realize that more advanced mathematical study is both pure and applied.
In between, the secondary schools of various countries differ markedly in their
attention to these aspects of
mathematics. Yet our field obtains its richness and its high place in school
curricula because of its combination of theoretical power and practical utility.
There were ample opportunities at ICME 10 to hear about each aspect of this
combination, but seldom were they discussed together.
The relationship between pure and applied mathematics in the education of
students still seems to be an unsolved problem in many parts of the world. And
it seems related to the tension between the use of calculator/computer
technology and the use of paper and pencil. Our students are not wellserved by
policies of exclusion of any of these aspects. We need thoughtful discussions at
the highest levels to work out these relationships and tensions.
Reflections on the Congress.
I am grateful for the The International Congress on Mathematics Education (ICME)
meetings. I believe strongly that our field and the world in general benefits in
many ways as a result of interactions of individuals from different countries.
This was my ninth ICME congress; I missed only the first (in 1969, in Lyon)
because I was in the process of finishing my doctoral dissertation. I have had
the opportunity to have my reflections on earlier ICMEs published on several
occasions1. I also have had the opportunity to be involved in four University of
Chicago School Mathematics Project (UCSMP) international conferences and
at two of them I wrote about international mathematics education.2 As I wrote
these notes, I looked back at what I wrote for earlier ICMEs and it is clear
that my perspective has changed from that of an attendee to that of an organizer,
even though I had nothing to do with the overall organization of ICME 10.
The ICME conferences can be viewed as the world counterpart of NCTM annual
meetings in the U.S. and Canada. They are the largest international meetings;
they attempt to cover the entire panoply of activities in the field; they
include sessions of a wide variety of types; they are attended by experienced
and new people in the field and every amount of experience in between. These
features can be anticipated and come with the territory of the organizing body.
We come to a congress expecting to hear the best and latest work in the world,
and often our expectations are met but sometimes they are not met. The goal of
any organizing team is to maximize the ratio of good to bad.
DG 1: Issues, Movements, and Processes in Mathematics Education Reform.
ICME 10 was the first ICME to have only one official language. Thus, for the
first time, the plenary lectures were not simultaneously translated into French,
Japanese, Russian, and Spanish. As a result, the plenary sessions felt less
exotic than usual, and a little bit of the feel of internationalism was lost.
But the decision to have only one language of discourse made it possible to
initiate the idea of discussion groups (DGs) on a variety of areas of
mathematics education. I was the coorganizer of DG 1: Iissues, movements, and
processes in mathematics education reform.
DG 1 had one other coorganizer (from China) and three associate organizers
(from Chile, Japan, and Sweden). Of these five, only I and Bengt Johansson from
Sweden were at the Ccongress. The Chinese organizer was ill, and the associate
organizers from Chile and Japan both had to remain home because they were
leading figures in mathematics reforms that needed attention in their countries
even as the Ccongress was going on. Bengt also was in the position of having to
do work at home during the Ccongress, but being from Gothenburg he was able to
go back home and return during the Ccongress. Thus one could argue that, for the
most part, the unfortunate absence of these people was an outgrowth of the
territory they represented.
Theis expertise of the other organizers was matched by the expertise of many of
the people who attended one of the three meetings of DG 1. A number of attendees
at DG 1 were in charge of the testing programs, curriculum frameworks, or
development projects in their countries.
No formal presentations were allowed in the DGs. And although we asked for
papers to be sent to us before the conference, we received only one paper
written for the conference, from Margaret Kidd of the U.S. We did receive papers
from two others. The papers from one of the people were all old papers; the
paper from the other was more appropriate for another group and was sent to that
group. The absence of papers on the web may have been
a boon for our group, because unlike the Topic Study gGroups (TSGs) and posters,
everyone could participate without preparation.
About 50 people attended one or more of the sessions; 45 at the first; and about
28 at the second and third. And we did discuss. Participants from 20 countries
contributed views and, with only a few exceptions, everyone in attendance said
something. I heard later that the leading ministry person in one Asian country
remained silent, but he seems to have been an exception.
At the first two sessions, we discussed the following questions.
1. Who is mostly responsible for mathematics curriculum reform?
2. How do these individuals get together?
3. What are the goals of mathematics education reform?
4. What developments in mathematics curriculum reform are currently being
undertaken?
5. What forces inside the mathematics community have had significant effects on
curriculum reform?
6. What forces outside of mathematics have had significant effects on curriculum
reform
7. What is the role of various kinds of documents in instituting reform?
Thus the discussion centered more on process than on the substance of the
reforms (which operationally were defined as changes). Most of the contributions
were informational in direct response to the questions, but in the first session
a disturbing commonality appeared.
In a number of countries, consensusbuilding that has been carefully reached
over an extended period of time (often a number of years), in committees whose
individuals have been carefully selected to represent different viewpoints, is
sabotaged by lastminute changes by people whose competence is questionable and
whose identity may not even be known.
In recent years we have seen this phenomenon in the United States at both the
national and state levels. The phenomenon seems to occur most often when there
is a change at the top, and the new leaders in education want to place their own
stamp on the reforms, or when the education leaders disagree with the
consensuses that have been reached.
I should note that this phenomenon is not universal. In areas where education is
separated from politics and where wellestablished procedures are in place for
decision making (e.g.,, Japan, where reform in the system follows a schedule
planned years in advance), reform proceeds in a more orderly way.
In the third session, we discussed reforms that people felt were working in
their countries. A number of examples were offered and it was good to end the DG
on a positive note. One of mentioned reforms was the National Numeracy Project
in England, a project whose main goal is to get children in grades K5 to think
about mathematics (rather than to view mathematics as all memorization and rote)
by working with their teachers. This project has been adapted in Australia and
New Zealand. In general, the discussion about these reforms was positive about
these reforms. Thus it was interesting that at the poster
discussion immediately after, a presenter began by decrying the effects of the
very same project in England on the teaching of geometry and probability!
Mathematics and technology.
This ICME was the first in which all presenters were asked to use PowerPoint and
download their presentations in advance onto a main server. We were treated to a
host of very carefullyplanned carefully planned and effective presentations
combining text, photographs, and video. Furthermore, those of us who are still
editing our presentations up to the last minute were able to connect our own
computers to the technology without uploading to the server. I think that every
one of the approximately 40 lecture halls on the DTU campus used by the congress
was equipped to handle such technology.
However, it would be incorrect to conclude that the technology always worked. I
attended two sessions in which specialized computer software for mathematics was
to be demonstrated. In one of the sessions, the software had disappeared from
the server. In the other, the material on the screen could not be seen by those
in the back of the notverylarge lecture hallthose in the back of the
notverylarge lecture hall could not see the material on the screen. The
requirement that technology be reliable and effective is one of the main reasons
why mathematics classes in high schools use calculators more than computers. If
we, the experts, cannot get our act together with regard to technology, how can
we expect schools to do so?
In other sessions, we learned that Singapore is going heavily into the use of
technology in mathematics classrooms (with 1 computer for every two2 students
and one1 laptop for every two 2 teachers) and Japan is also increasing its
commitment to technology.
At the USACAS 2 conference two2 weeks before ICME (CAS = computer algebra
systems) I had learned that the curriculum in Austria makes heavy use of CAS.
The need for a workforce that can deal with the latest technology, competition
in international trade, and the continual advances in information technology
have been quite influential in these and other countries. In this environment,
it is particularly distressing to continue to hear a few loud voices within the
United States discouraging the use of technology in mathematics classrooms.
The plenary lectures.
In their plenary lectures, Hyman Bass and Andreas Drees demonstrated two of the
best ways in which mathematicians contribute to mathematics education. Hyman
Bass
pointed out that the mathematics needed by mathematics teachers is different
from that
needed by mathematicians, and that what may seem to be quite elementary
mathematics can involve deep mathematical ideas of some complexity. This is
something that many of us in mathematics education have felt for many years and
it was nice to hear it from a mathematician who relatively recently has come to
work in mathematics education. Andreas Drees gave a beautiful nontechnical
lecture in which he showed why the mathematics of networks and trees has become
important in biology and DNA research.
The highlight of the plenary lectures for me – and to some extent my highlight
of the congress  came from another person who was trained as a mathematician –
Ubi D’Ambrosio. In response to a question asking what he would like to see in
mathematics education in the near future, Ubi D’Ambrosio responded with great
emotion and eloquence: “Mathematics is the dorsal spine of civilization, but the
body supported by that spine is ugly.” He went on to say that we need to
transform mathematics (by which I assume he meant both the field itself and
mathematics education in schools) to use it to further society. As at ICME5 in
1984, where Ubi D’Ambrosio gave a plenary lecture, he rose to the occasion – but
here he rose above it to present us with a meaning of life.
Ubi D’Ambrosio’s comments came in a panel of mathematics educators capably
interviewed by Michele Artigue. It was quite a panel, with Gérard Vergnaud (a
student of Piaget), Jeremy Kilpatrick (a student of Polya), Ubi D’Ambrosio, and
Gila Hanna. Jerem Kilpatricky and Michelle Artigue both made the point that
mathematics education differs from mathematics in that in mathematics, when we
solve a problem, it is solved for all time unless someone finds a flaw in the
solution. But in mathematics education, the same problem we have solved in one
place and time will come up 510 years later at another place or another time
and need to be revisited. For Ubi D’Ambrosio’s remarks, Hanna Gila’s humor (she
was determined to say what she had prepared regardless of the question), the
insights of Jeremy Kilpatrick and Michele Artigue, and the history presented by
Gerard Vergnaud, the panel received a standing ovation.
Mathematics and its applications.
The theme and study group devoted to mathematical modeling and its applications
(TSG 20) exhibited three stages of the process in which a person with a
traditional mathematics education (typically devoid of applications) comes to
appreciate mathematical modeling. The first stage is the discovery of a very
nice problem or two, to teach that application, and to find out that students
can do the application and learn mathematics in so doing (surprise) and that
they like it (no surprise). The second stage is the discovery that there are
many such problems exist and to teach several such problems , achieve sing the
same results as the first. The third stage is the realization that when students
undergo an experience during which they are continually exposed to and/or
immersed in applications, and where the mathematics is intertwined, then the
conceptions of the students (and the teacher!) about that mathematics is
changed.
Sitting in TSG 20 was quite frustrating, because I and many others in the group
had been at the third stage for many years, if not decades, but we heard from
some individuals who were at the first stage. I had first attended this
particular group at ICME3 in 1976, when Henry Pollak was one of the main
organizers. Others in the group have been very active in the International
Conferences on the Teaching of Mathematical Modelling and Applications (ICTMA)
that have been conducted biennially since 1983, and many of us had attended some
of those conferences.
The frustration was not with the individuals who were less experienced. It was
with the slowness of movement in mathematics education to realize the importance
of applications and modeling to our subject. It still is the case that,
throughout the world, most mathematics teachers have had a very narrow education
in mathematics, one that essentially ignores engineering, operations research,
and many of the other largest fields of applications of mathematics, and gives
only a slight and sneering nod to statistics. Thus the changes that have
occurred in the last forty years in our field are being ignored. No wonder fewer
students in the United States are interested in majoring in mathematics now than
a few decades ago. Why major in a subject that ignores its most recent
applications?
General remarks about the program.
The organizers had hoped for 3000 to 4000 attendees; instead, there were about
2300.
I did not attend any session that was overcrowded and was amazed at how evenly
attended most of the sessions were. Despite the large number of participants and
a venue that was quite spread out, the smaller numbers at the sessions made this
a more intimate ICME for me than I had expected. The local organizing committee
was terrific and handled all requests politely and efficiently.
I would have liked to see more attention given in the lectures to the
engineering aspects of mathematics education: reports from curricular projects
(I think none from the
U.S. was represented), from testing programs (nothing from the Programme for
International Student Assessment or PISA?), and from entities that have recently
changed their mathematics guidelines. I would have liked to have received more
information regarding what is happening in the mathematics classrooms of the
world. AT here was a great deal of attention was given to mathematics education
research, but that research is fueled by problems in engineering and by what is
happening.
Yet, overall, the program was outstanding. The program committee did a fine job
at covering the field in the TSGs and the DGs. The several national
presentations (I saw all of them except the one from Mexico) were quite
interesting. The UNESCO presentation "Why Math?", was particularly engaging. I
enjoyed the one poster discussion session I attended.
There was ample time to renew friendships and to make new acquaintances. By the
end of the week most of the people I talked to were exhausted yet very pleased
with the conference. The ratio of good to bad was good!
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