Reflections on the Congress (Veronica Meeks, Western Hills High School, Fort
Worth, TX).
I had the privilege this summer to be a part of the group from the United States
sponsored by the NSF and NCTM to attend ICME10. I am a high school mathematics
teacher in Fort Worth, TexasX. I was a novice in two ways. This was my first
ICME and my first venture overseas. I did not know quite what to expect. The
structure of the ICME lends itself to the dissemination of research from
different parts of the world. Key speakers were chosen to share their thoughts
or findings with the whole assembly. Others presented their ideas or research in
smaller groups divided by different themes or interests. The first morning I
felt sort of out of place because the majority of the people I met were either
mathematicians or mathematics educators from colleges and universities. That
afternoon I had a discussion group on issues and topics in upper secondary
education and there was where I met other high school teachers from other
countries. My intent in this report is to share some of the insights that I
gained along with what I learned about applications and modeling. When I
initially signed up for the different sessions I did not know to which group I
was assigned, so my comments and observations will not deal exclusively with
modeling and applications.
The conference did not occur in Copenhagen, but in Lyngby, a small city a
15minute train ride away. From the train station we then had to take a
10minute bus ride to the Danish Technical University (DTU). We were serenaded
by the Danish Royal Brass and welcomed by the Minister of Education of Denmark.
The Minister set out two major questions, which she felt needed to be addressed
by the mathematics and mathematics education communities:
1) What mathematics do we need to learn and how to learn it?
2) Why learn mathematics?
The Mayor of Lyngby, who has a doctorate in mathematics, also greeted us. He
shared a wonderful application of modeling by describing the growth of his city.
He discussed what type of model could best model represent the data. It was very
important in city planning to have a realistic model to make good growth
projections. He demonstrated the way that mathematics can be applied outside of
a mathematics classroom setting and suggested that this maybe part of the “why”
we learn mathematics.
The plenary sessions raised many issues and issued challenges to those of us in
the audience. From a classroom teacher’s viewpoint, there were several important
points, many of them posed as questions.
1. What do students and teachers need to know, in what ways, and for what
purposes? How can they learn what they need to know them? What math problems do
that teachers and students solve in their daily work? Are they worthwhile? These
questions also echo the questions asked by the Minister of Education.
2. What is the connection between the different communities – mathematicians,
college mathematics education researchers and classroom teachers? It was argued
that rather than research just shaping practice through research was limited
and, that trends in preservice and inservice teacher education can influence
both research and practice.
3. Anna Sfard from Michigan State University cited an initial report of a survey
conducted on mathematics education research.
4. No matter where you teach in the world, very few curriculum projects have
been replicated successfully beyond the few classrooms that were initially
involved in their development. Several factors seem to stand in the way. There
is too much national, cultural, and economic diversity. Some parts of the world
are rarely researched in mathematics education. Typical classroom settings are
rarely researched. Note that 16% of the children of the world do not attend any
type of school. Where there is war, there is no interest in mathematics
education. One reason expressed for the problems of implementing curriculum
change on a large scale is that classroom practices refuse to change.
5. In my discussion group with other secondary mathematics teachers there were a
few differences. In many other countries when students reach high school there
are tests that determine the type of education they receive at the secondary
level. Very few of the countries even have special education students even in
the same building let alone mainstreamed into regular classrooms. After these
few points there were a lot of similarities. The group I choose to be with
wanted to deal with issues facing secondary mathematics teachers. We divided
into three groups to brainstorm, and we eerily we had about the same list of
concerns. The issues that seemed important were a lack
of time, too much in the curriculum, lack of motivation of students, decreasing
skills, high stakes tests, and coverage versus depth in teaching. We found this
surprising. Even the couple of representatives from Japan voiced the same
concerns.
6. There were several eloquent presentations and comments on the issue of equity
that should be mentioned. There was a heated plenary panel discussing the
balance between mathematics education “for all” and for highlevel mathematics
performance. Speakers from T South Africa, Jjill Adler and Remuka Vithal,
ICME1wo made two of the best comments were made by two mathematicians from
South Africa, Jill Adler and Remuka Vithal. They pleaded for equity and access
to a quality education as an issue of social justice. One powerful point was
made when pictures of students in classrooms from around the world were shown.
Some students were seated in rows; others using computers and others seemed to
be working in cooperative groups. It was pointed out that the students pictured
in South Africa were in a group because there was only one piece of paper and
pencil between among 5 five or 6 six students.
Even though many of the sessions I attended were not directly tied to
application and modeling, there were some ideas were raised that I think might
have an impact. I listened to a presentation from Doug Clarke from Australia on
“Understanding, Assessing, and Developing Young Mathematical Thinkers. The
comments that were shared that seem to apply at the secondary level. When it
comes to applications and modeling, teachers need to carefully choose the tasks
requested of students. How do we use questions? He suggested that questions that
are asked should be useful. Questions should be asked that ask about process and
justification of the answers. Questions should be constructed that prompt
students' thinking without giving the answer. When given a task do we make the
most of the opportunities for learning? What message do we give students when
assessing with using these tasks? He suggested that teachers need to encourage
persistence – staying on task and completing the task. Teachers need to take
into account the interests of students when choosing a problem. He also felt
that introducing algorithms before students develop their own strategies causes
a loss in flexibility and creativity in solving problems.
A big deterrent to application and modeling is the excessive testing that seems
to be occurring almost worldwide. High stakes tests seem to take away time in
from instruction, and teachers are less likely to give students rich problems to
solve. There are fewer opportunities to build conceptual links. Ubiratan
D’Ambrosio of San Paulo, Brazil, when sharing his vision of the direction of
mathematics education, lamented that “We need to stop using testing to
domesticate our children.” He is suggesting the testing is forcing students to
lose some of their individuality and creativity of thought so as they are
encouraged to learn to perform well on high stakes tests.
Another obstacle to modeling and application is the perception by teachers that
curve fitting is mathematical modeling. In a session with Peter Galbraith of
Australia, he stated that the use of curve fitting does not truly reflect
modeling.
Applications modeling problems should be derived from real situations. When
teaching modeling he suggested that we begin with modeling that uses simple
mathematics. As the experience of the students grows, they students gain the
ability to access more substantial mathematics. Students should be taught a
structural template to use as a tool. Some of the changes he suggested was were
to estimate the tasks' complexity by looking at what cues are given, what
contextual assumptions are made, what mathematical assumptions are made, and
finally identifying the mathematics procedure needed. Some of the problems that
have been used to demonstrate and are used to teach modeling and applications
require have more variety. He also suggested creating and writing problems that
are real world examples. He said that another problems concern with application
and modeling problems was that it is hard to create user friendly questions for
the student and still maintain the integrity of the real world setting. The
power of modeling is that it can shape the belief system of what mathematics can
do.
