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 real-world 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

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 ICME-10 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 tertiary-level 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 well-served 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 co-organizer of DG 1: Iissues, movements, and processes in mathematics education reform.
DG 1 had one other co-organizer (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, consensus-building 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 last-minute 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 well-established 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 K-5 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 carefully-planned 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 not-very-large lecture hallthose in the back of the not-very-large 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 non-technical 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 ICME-5 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 5-10 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 ICME-3 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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