Reflections on Applied Mathematics and Mathematics Education at ICME-10 (Martha J. Siegel, Towson University, Towson, MD).

The International Congress on Mathematics Education (ICME)-10 held in Copenhagen, Denmark from July 4-11, 2004 was my third ICME. As before, the highlights of my experience at this conference were the personal relationships that developed with people from around the world. As ideas flowed between and after scheduled sessions, I felt that it would have been a better conference if all of us could have stayed in the same hotel so that the after-schedule activities might be a more prominent part of the week's interchanges. Nevertheless, there was much to be learned about Mathematical Modeling and Applications in Mathematics Education.

I attended all the plenary sessions, and was a member of Discussion Group 22, Topic Study Group 20 on Applications and Modeling, and Thematic Afternoon C. I found that I wanted to contribute more of my experiences but I had not taken the time to prepare written materials before the Congress. I think my experience with mathematical modeling over many years of teaching at the collegiate level could have informed the Topic Study Group. I was disappointed in myself for not having realized how rich my
own experience has been compared with that of many of those who contributed papers to the group. I probably underestimated how much personal knowledge I had gained over my career that might have added to the discussion. But I learned many new things nonetheless. And the best part of the entire meeting for me was meeting up with the Affiliated Study Group (ASG), The International Community of Teachers of Mathematical Modeling and Application (ICTMA) and the related International Commission on Mathematics Instruction (ICMI)-14 Study Group 14.

Of the morning plenary sessions, I thought that Renuka Vithal of Durban, South Africa did a very good job on Monday’s panel debate in addressing “Mathematics education for whom and why? The balance between mathematics education for all and for high level mathematics performance.” on Monday’s panel debate. Anna Sfard of Haifa, Israel and Michigan State University made many excellent points in her lecture on Tuesday as she remarked, “There is nothing more practical than good research.” The discussion of the Survey Team I illuminated many of the difficulties with the quality of current mathematics education research. I noted with regret that there were disappointing responses to questions about the extent of the effect that mathematics education research had in the classrooms in schools, even though there was agreement that the research results can do reach the pre-service teacher.

I found the material in Wednesday’s lecture by Erno Lehtinen (Turku, Finland) on “Mathematics education and learning sciences” relevant to the discussion of how students learn modeling and applications. Lehtinen pointed out that well-developed subject knowledge in mathematics frequently results in inaccuracy in predicting student’s
problem solving ability. Teachers with more mathematical background were somewhat less able to predict the source of a student’s difficulty. Studies show that many concepts are counter-intuitive for the learner. How do we effect conceptual change? We can enrich the prior knowledge base, but we frequently have to work with the knowledge base with which students come to us. This may be insufficient and may need radical restructuring, which is threatening to the learner and makes him or /her resistant. When we teach we need to maintain a meta cognitive awareness about the concept and help students make the cognitive changes they need in a visible and concrete way. Cognition takes place in an environment involving institutional space, interests, social and cultural traditions, and discipline. The ways to develop mathematical understanding must include (1) physical representation of the concept, though this only works when the student really sees the connection, (2) building conceptual links with stress on connections to the real world, and (3) verbalization (small groups, questioning, writing, use of open-ended questions, etc.) Research shows that open-ended questions work well with middle- income students and not so well with those of lower socio-economic level. Since much of the process of modeling and applications involves open-ended questions, we may need to make special effort to draw out responses from many of our students.

I enjoyed the Plenary Interviews as well as Jill Adler and the Survey Team 3 speaking on “The professional development of mathematics teachers,” but I will leave that for others’ comments. There was little in these sessions pertaining to the topic of this report on the teaching of applications.

Saturday morning’s plenary lecture by Andreas Dress of Bielefeld, Germany was a rare treat. He spoke on “Structure formation in nature as a topic of mathematics.” He described applications related to the galaxies, the continents, crystals, the general order of species, etc. He skillfully defined the role of mathematics in developing the tools for rigorously deducing the consequences of model hypotheses, and in developing tools for encoding and comprehending the characteristic features of resulting structures by using such tools as Fourier and wavelet analysis, linear algebra and analytic geometry, group theory, and combinatorics. He used examples in tiling, fullerenes, crystallography, and phylogenetic trees, and displayed the great beauty and utility of mathematics in the process.

Fernando Arzarello (Torino, Italy) spoke on “Mathematical landscapes and their inhabitants: Perceptions, languages, theories.” Here again, there were lessons for the teaching of modeling and applications.

The meeting schedule was so packed that I did not get to the formal meeting of the ASG, the International Community of Teachers of Mathematical Modeling and Application (ICTMA) until their third and final session. Since many of the same people were in the Topic Study Group, I had the opportunity to talk with them informally earlier in the meeting.

So what did I learn? Not much has been discovered about how students at any level do mathematical modeling. The most problematic stages for students in learning modeling are that the student must
• translate the situation into mathematical language,
• must then choose appropriate variables and mathematical tools,
• must translate the mathematical results back into the language and context of the given situation,
• compare the reality with the model, adjust the model, and renew the process.
Though I remember discussions of the very same issues when I last attended an ICME (Quebec), I have not seen much progress in developing any new strategies that might better inform the process and aid students in learning the appropriate techniques.

Thorir Siggurdsson of Iceland asked the following question, “Could a mathematics student have prevented the collapse of the Atlanto-Scandian herring?” He explored the mathematics and the level at which a student might be able to handle some of the attendant problems associated with data and mathematical analysis. Some of the papers presented a theoretical framework for the teaching of modeling. Many of the relevant papers are available through ICTMA conference proceedings. The next ICTMA Conference will be held in London, 10-14 July 2005.

One of the distinctions that was discussed was the difference between a mathematical modeling course and a mathematics course in which mathematical models and applications are inserted. The former requires a different type of developmental activity, with the teacher helping to guide the process in the context of the student’s mathematical sophistication and skill level.

The European Society for Research in Mathematics Education (CERME) will hold its fourth 4th international congress 17 - 21 February 2005 in Sant Feliu de Guíxols, Spain. There is a Working Group on Applications and Modeling at that conference. Gabriela Kaiser of the University of Hamburg is the group leader (gabriela.kaiser@uni- hamburg.de). Her paper at TSG 20 mentioned the typical problem – a low level of reflection on the model and its relationship to reality and a low level of meta cognitive activity on the part of students. She concluded that small innovative projects that are related to the real - world related achieved limited impact but seemed to be headed in the right direction.

I chose my Lectures carefully and I enjoyed them. Among these, the highlights for me were Gil Strang’s lecture on “Linear algebra: A happy chance to apply mathematics” and Peter Galbraith’s “Applications and modeling (sic) in mathematics education: Progress to celebrate – so much more to do.” Galbraith’s lecture showed some progress in this
field as he covered some of the successes at various levels. It also addressed the problem of moving from a single successful practice to a theory and implementation of proven techniques for teaching and learning mathematical applications. One persistent question is “How have technology, assessment formats, and so on impeded progress in the field?”

Galbraith mentioned that we should recognize the need for research paradigms that might better reveal the proven efficacy of innovative programs in this arena. Though word problems and modeling problems may share the same verbal clothing by sharing links to real-world contexts, Galbraith pointed out the fundamental differences in teaching these. The challenge is to estimate the task complexity of a modeling problem, to ascertain the relevance of assumptions about the real-world, while making appropriate assumptions about the mathematical context. He pointed out that Gloria Stillman (2003) had described the modeling task complexity along several axes: mathematical, conceptual, linguistic, intellectual, representational, and contextual. To be able to measure each of these as we present students with open-ended modeling exercises will be a monumental task!

My visit to the ASG on Applications and Modeling was quite rewarding. At the last session of the ASG, the discussion focused on the ICMI Study 14. The volume of those papers that were presented at the February 13-17, 2004 ICMI-14 in Dortmund, Germany is available and has nearly 50 papers on the subject, covering the topic at all levels (elementary through tertiary). Editor, Hans-Wolfgang Henn of the University of Dortmund may have some copies available (10 euros).

ICME-10 provided me with an excellent chance to meet “birds of a feather” from around the world, to think seriously about my teaching of mathematical modeling in a new way, and to help me frame research questions about the teaching and learning of applications and modeling. It opened doors and also confirmed that much of what I have been doing over the years – writing textbooks with applications and teaching applied mathematics – is important and valuable. Now, if only my study of mathematical models had enabled me to control the weather – especially summer in Copenhagen!