Reflections on Applied Mathematics and Mathematics Education at ICME10
(Martha J. Siegel, Towson University, Towson, MD).
The International Congress on Mathematics Education (ICME)10 held in
Copenhagen, Denmark from July 411, 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 afterschedule
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 preservice 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
welldeveloped 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 counterintuitive 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 openended questions, etc.) Research shows that
openended questions work well with middle income students and not so well with
those of lower socioeconomic level. Since much of the process of modeling and
applications involves openended 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 AtlantoScandian 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, 1014 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 realworld 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 realworld, 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 openended 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 1317, 2004 ICMI14 in
Dortmund, Germany is available and has nearly 50 papers on the subject, covering
the topic at all levels (elementary through tertiary). Editor, HansWolfgang
Henn of the University of Dortmund may have some copies available (10 euros).
ICME10 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!
