Paper:
Main theme:
Secondary theme:
Oral Presentation
Learning Communities
Interdisciplinary Curriculum: Foreign Language and Fractal Geometry
Flying Tetrahedral Kites in French
Dr. Eliane Keane, Miami Dade College, U.S.A. ekeane@mdc.edu
and
Professor Joaquín G. Martínez, Carlos Albizu University, U.S.A. jomartinez@albizu.edu
Abstract
In this paper, we provide an overview of successful educational practices used in FrenchMathematical Learning Communities (LC) and discuss the rationale for future integration of
mathematics into ESL classrooms in community colleges. We framed our teaching techniques
on both Krashen’s theories and constructivist approaches to education, facilitated collaboration,
and found that a major learning outcome was retention of student knowledge. Students in the LC
explored complex geometrical concepts while these concepts were infused into a foreign
language course. Through this unique exploratory activity linking humanities, language
acquisition, and mathematics, students engaged in solving complex geometrical problems, linked
problem solving to a foreign language, and learned to perceive the study of mathematics in
association with another academic discipline. Although the French-mathematical LC provided an
active learning experience for students, we claim that ESL students may especially benefit from
similar learning communities.
Introduction
Research in mathematics education, language acquisition, cognitive learning, and general
pedagogy highlights the importance of collaborative learning and teaching to achieve student
success (Piaget, 1926; Vygotsky, 1978; Schoenfield, 1992; Pappas, 1993; Menezes, 1997;
Bransford, 2000). Further, learning processes in mathematics and foreign languages are alike, in
the sense that context, total immersion, and cognitive learning play a significant role in
comprehension, thus knowledge retention. Accordingly, the belief that students need to
understand mathematics in context and to apply complex concepts to meaningful real world
situations is based on that preliminary premise. Therefore, the idea of a learning community
linking French and Mathematics was born.
Mathematics educators have been aware that many processes associated with student
learning, the development of critical thinking, discovery learning, and student engagement in
“doing” mathematics are facilitated when mathematics is integrated across different areas of the
curriculum (Pappas, 1994; NCTM, 1989; Papert & Harell, 1991; Webb , 1991; Paas, &
Merrienboer, 1994; Rosenthal, 1995). In this paper, we discuss successful educational practices
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in a French-Mathematical learning community, where educators integrate Krashen’s (1985)
notions of the input hypothesis as a theoretical framework for student success in learning the
beginnings of fractal geometry, fractals, platonic solids, and French.
Argument
Our students experienced success comprehending complex geometrical concepts while
immersed in a foreign language environment. Krashen (1985) stresses the notion of
comprehensible input (i+1) as a pivotal aspect of language acquisition. Building on the notion
that languages should not be taught in a vacuum (Edelhoff & van Bommel, 1980), this learning
community established a relationship of mutual reinforcement between the second language (L2)
outcomes and the intended mathematical objectives. Further, our students faced an additional
level of cognition as they attempted to master new mathematical concepts in the L2. Aside from
learning the second language, the students also had to make additional connections to understand
geometrical concepts of self-similarity, fractals, and platonic solids by processing knowledge
through the additional filter (L2). We claim that the additional cognitive process contributed to
the effective retention of both skills; they relied on the assimilation of one as a prerequisite for
mastery of the other.
The teachers assumed the role of monitors in this learning community and defined both
extremes, from actual to potential development, allowing students the opportunity to define their
own zone of proximal development (Vygotsky, 1978). Krashen’s notion of +1 was used to
facilitate growth in both mathematics and in the target language (French), whereby the
“mathematical knowledge +1” and the “French knowledge +1” were generated by both the
teachers and by those students operating at the higher levels of their potential development zone
(Vygotzky, 1978). As monitors of a dynamic classroom environment, the teachers also adopted
a methodology advocated in the Van Hiele model which recognized a hierarchy in levels of
geometrical thinking (Teppo, 1991). The cohesive sense of community supporting the learning
environment, the meaningful interaction between students, and the pertinence of the content area
knowledge facilitated the learning experience; students successfully acquired knowledge of
French applied vocabulary, culture, and some concepts associated with fractal geometry while
processing other mathematical concepts at a higher level of cognition (Cummings, 1979; Bloom,
1984; Larsen-Freeman, 1997).
The Learning Community Model
Students enrolled in Geometry and French courses. They were engaged in a series of
collaborative activities between the two disciplines under the auspices of both the Mathematics
and French professors. In the geometry classroom environment as well as in the French
classroom, the professors exposed students to a variety of strategies commonly used in contentcentered second language instruction: cooperative learning and group strategies, multi-tasks
based on experiential learning, whole language strategies (Goodman, 1986; Crandall, 1992), and
the Natural Approach (Krashen & Terrell, 1983).
For example, in the Geometry class the students built tetrahedral kites while they learned
about fractals ( Mendelbrot, 2002). We promoted cooperative learning and students worked in
groups of four to build the kites. Each team member was assigned a specific responsibility
during the building phase of the kites. For instance, in one of the teams, one student was
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assigned to record new vocabulary and the geometrical concept associated with the word,
another student did research on additional uses and applications of these new words. All four
members were given the task to observe, describe, and record additional geometrical concepts
involved in the kite building process. All students were asked to research and identify real world
applications of kites. The idea was to allow each member of the group to feel that they were
making meaningful contributions to the project while learning together (Johnson, Johnson, &
Stanne, 2000). While the teams were engaged in building the kites, the geometry teacher
introduced concepts of self-similarity, recursion, and fractals. These concepts were formally
discussed in English in the geometry classroom and informally discussed in French in the foreign
language classroom. This way, students in this learning community subconsciously processed
geometrical concepts and later produced them as target language output.
While all students built similar tetrahedral kites in the geometry class, worked in the
same activity, were introduced to the same geometrical concepts of platonic solids, fractals, and
recursion, groups further investigated these various concepts with the support of the French
professor. In the French classroom, different groups focused on different aspects of kites and
each had different research projects related to kites: history of flying kites, kites in different
cultures, mechanics of building kites, physical aspects of flying kites, and effectiveness of
different geometrical designs other than tetrahedral kites.
In the French classroom, as students conducted their research, each different group
presented findings from their research questions to the other teams. The French professor then
conducted a whole group discussion on kites, allowing students to adopt didactic measures to
explain the geometrical concepts that they had studies and developed during the kite building
activities in their mathematics lessons. Throughout this activity, the French professor applied
methodological approaches following a communicative-based syllabus to moderate discussion of
student topics and promote classroom interaction, using mathematical applications of
geometrical concepts as a vehicle for effective communication in the target language, assessing
language acquisition and level-appropriate progress (ACTFL, 1985) in the L2.
The entire activity lasted three weeks and was developed both in and out of the classroom
environment. In the French classroom, the students worked with their team members to enhance
skills in each of the five areas of language: listening, speaking, reading, writing, and culture.
Aside from speaking in French to discuss concepts learned in the geometry class, they researched
and wrote short paragraphs in French about the history of kites, kite-building techniques,
aerodynamics, and design. Students also collaborated beyond the mathematics and French
classroom settings to complete the written kite project.
A Cultural Grand-Finale
The learning community celebrated the success of the Geometrical-French collaboration
by flying the tetrahedral kites in a field trip to the beach. In this field trip, students were
encouraged to communicate only in French with each other and with the French and Geometry
professors. Furthermore, while flying “French Tetrahedral Kites,” students and professors
discussed fractals, platonic solids, volume, surface area, and various other geometrical concepts
learned in the process of building the kites. Students and both professors brought French food to
this grand-finale celebration. Amidst baguettes, brie, and Orangina, all participants of this
learning community immersed in the French culture, flew the kites, and discussed the successes
and failures of the kite construction activity as well as the mathematical implications in the kite
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design and collaborative project. Some students also shared knowledge of further applications of
fractals with the teachers and their peers, which indicated further interest in fractal geometry
beyond the focus of the original activity of flying kites. This attitude in not commonly found in
regular or advanced geometry classes and we attribute it to the success of the integrated LC.
Success Findings
In both classes, students studied the history of flying kites and acquired new terminology
related to kites and their construction. Throughout the process of building kites, students
collaborated, communicated, and broadened their French and mathematics skills; students
learned French in a mathematical context and vice-versa. They linked the French language to a
mathematics activity done in a geometry class
Perhaps the greatest success of this collaborative practice was the retention of student
knowledge. Gass (1997) highlights the need for an interactive model of second language
acquisition that pays special attention to the affective aspects that influence student retention,
creating an atmosphere that is more conducive to learning, reactivating dormant knowledge, and
lowering the affective filter (Krashen & Terrell, 1983). In this exploratory interdisciplinary
activity, students were engaged in critical thinking processes which contributed to their learning
of Mathematics and French. By solving meaningful problems connecting two content-disciplines
the students were exposed to language and to geometry simultaneously. Like Gass (1997) and
Anderson, Reder, & Simon ( 2000), we believe that due to interactive practices in this learning
community and its affective domain, knowledge may have been carried from one environment to
the other. Successfully, students perceived mathematics as an extension of foreign languages and
not an isolated discipline of the curriculum. We claim that some of the positive outcomes of our
LC were the increased retention and knowledge on both fields: French and Mathematics.
This French-mathematical learning community provided an active learning experience
for students who explored geometrical concepts while immersed in foreign language training.
Like several other mathematics and language educators we also argue that most students learn
better and retain more if they engage in learning activities that require them to think and process
information rather than passively listen. Furthermore, students evaluated their own experiences
in this LC and most believed that being active, “doing” mathematics and working collaboratively
helped them understand fractals and French vocabulary associated with kites. Students’
feedback indicated that the LC improved students' attitudes towards geometry and mathematics
in general.
Students who participated in this learning community continued their studies of
mathematics and foreign languages, which is an indicator of student retention: All students
enrolled in the next mathematics course and continued their studies in French 3.
Students in this learning community were active in both mathematics and foreign
language learning. Their engagement backed the claim that it is feasible to attain greater
retention as a result of linking interdisciplinary content and language teaching by infusing
constructivist approaches to pedagogy (Kaufman, 2004). The learning community also engaged
faculty in a creative new dimension for instruction. The success of this learning community
emphasized the positive impact that teacher collaboration from different disciplines had in
student learning. The student engagement was reinforced beyond the geometry course because
the students used the French classroom as a background context to continue to develop and apply
knowledge of fractals, self similarity, and platonic solids in their discussions and research.
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Conclusions
Integrated activities like the ones described in this paper put the learning of mathematics
into a broader context because it facilitates and extends students’ experiences with mathematics
beyond the classroom. It has a positive impact in student learning because it integrates real-world
applications, such as construction of kites, flying kites, using the French language, and
integrating cross-cultural competencies across different areas of the curriculum.
Although our learning community focused on geometrical-French concepts acquired at
the secondary level, the findings suggest that undergraduate students in community colleges can
greatly benefit from similar experiences as it became evident that the collaboration and
interdisciplinary approaches were extraordinary factors contributing to student success. Also
crucial in this study was the underlying framework of Krashen’s model which may help explain
why these students, who were faced with the additional challenge of learning mathematics in a
second language environment, acquired geometrical concepts so well and further applied it to
mathematics and the real world. Furthermore, as the number of second language learners in U.S.
classroom increases, so do the possibilities for content-area teachers to integrate collaborative
efforts with various foreign language programs other than French (Kaufman, 1997). Further
research linking mathematics and other foreign languages is also needed.
Additionally, further research is needed to help maximize the effectiveness of other
learning communities linking geometry and ESL classrooms; Spanish and English as a second
language in particular. The experiences of this learning community are equally pertinent to the
field of English as a second language. It is essential to move away from the political forces that
continue to isolate ESL from the field of foreign language education in the U.S. As the number
of students whose second language is English abounds in our classrooms, students in ESL
classes both in Florida and in other states in the Unites States, in community colleges and in
secondary education, may also retain English and geometrical knowledge when placed in
supportive LC integrating mathematics and ESL.
4
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