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 ICME-10. 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 15-minute train ride away. From the train station we then had to take a 10-minute 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 pre-service 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 high-level mathematics performance. Speakers from T South Africa, Jjill Adler and Remuka Vithal, ICME-1wo 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.