The educator must be an anthropologist. The educator
as anthropologist must work to understand which
cultural materials are relevant to intellectual development.
Then, he or she needs to understand which trends are
taking place in the culture.
Papert, 1980
How feasible are studying groups outside MDC campus?
In this paper, I intend to describe a few strategies to address a generic question:
What would make me a better teacher for my students? Specifically, I would like to
investigate how I can motivate students to create and maintain study groups outside the
college campus and how curriculum can be formatted to reflect interdisciplinary aspects
of mathematics teaching.
Introduction
Over the last fifteen years it became widely accepted among mathematics
educators that it is beneficial to students to incorporate a community sense in the
mathematics classrooms and across other disciplines too. Research in mathematics
education, cognitive learning, and general pedagogy highlights the importance of
collaborative learning and teaching to achieve student success ( Webb. 1991; Schoenfield,
1992; Menezes, 1997; Bransford, 2000; Roueche, 2003). In theory, and across many
educational institutions, researchers promote weaving together extracurricular learning in
and out of the classroom. New techniques involving active learning, doing mathematics,
engagement of students are spreading and representing new adopted paradigms. My
experience as an educator left me no doubts that a community concept is beneficial for
mathematics students, to their learning, and to the retention of knowledge (Keane, 2006).
Yet, in community colleges like MDC the wide adoption of these practices is still far from
becoming a reality. Poor performance on mathematics courses, attrition rates and low selfesteem are evidence that the road that mathematics educators need to build to prevent
further negative results is long.
Studying Groups and Cooperative Learning
Some of the good habits in studying mathematics or any other subject requires
students to be actively involved in managing their own learning process. For example,
many professors expect students to ask questions in class, to go to their office hours with
questions about homework problems that students could not solve. Even more empirical
is the instructor who dreams of students who try to work some of the problems before
they are actually covered in class. Instructors believe and advertise to students that good
study habits make it easier to prepare for the major tests and that asking questions in class
reflects positive attitudes because there are usually several other students who also want
to know answers to the same if not to similar problems to the ones students have.
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Nevertheless, the reality of many of our students is quite different from the ideal
learning environment. Students do not find time to attend office hours as much as the
instructor would like; they are usually too busy with their full-time jobs and cannot find
time to attend or call during these hours. Findings on attitudes and experiences of first
year students confirm that their engagement with university life has diminished
tremendously over the years (Varsavsky, 2006). Students’ priority, if not a necessity, has
shifted and they are spending more time on paid employment at their work and less time
on campus.
Furthermore, many times students do not ask questions in class because they do
not feel comfortable with mathematics vocabulary and cannot phrase the question to their
own satisfaction. They are usually so critical of themselves because of their own math
anxiety feelings of the past that they prefer not to ask anything. After all, many students
admit in privately that if they ask questions in front of the class, then others might
consider their question stupid and will probably think of them as not intelligent. Low
self-esteem and mathematics phobia may indeed result in confused thinking and passivity
(Mandler, 1989). Yet, as students work in small groups their familiarity and levels of
comfort with these groups usually increases. Anecdotal reports tell that professors hear
students communicating with each other by using informal language and not fearing to
express confusion or admitting that they do not know how to solve problems. Besides, as
students attempt to solve problems and collaborate toward a common solution, they
usually take risks and several times succeed. They may succeed through a succession of
failures but they take risks with the support of the group. What may be happening in
these circumstances is the development of a learning community among the students and
their groups.
Among the various definitions of a learning community, I especially relate and
adopt the following: In a learning community, each individual works with others in a
spirit of experimentation and risk taking to improve the educational experience (Mitchell
& Sackney, 2001).
Yet, learning environments which promote cooperative learning do not solve the
problem of student doing homework, studying for exams or preparing cooperative
projects. A review of the literature in learning communities (Tinto, 1998; Leigh, 2002)
and peer-tutoring (Webb, 1989; Kalkowski, 1995) highlights the success of these
practices. Students take advantage of the support of a peer and develop stronger ties with
their peers and instructors running the learning communities. Nevertheless, our students
at Miami Dade College are not meeting outside the classroom very often, are not
studying together for exams, are not frequently meeting face-to-face to elaborate group
projects or gathering socially. For these and many other similar reasons that I would like
to investigate more deeply how the concept of student communities can crystallize in
student success in a self-contained mathematics class.
Student Community in and out of Campus
There are approaches that teachers can take, within the constraints, which may
increase student engagement with their peers, their studies and their learning. Cultural,
historical and practical connections to mathematics, particularly as these relate to the
students’ cultural backgrounds may be incorporated as part of the curriculum of any
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mathematics classes. The premise is that students learn mathematics best by doing
mathematics in a meaningful context. For example, in the very first week of the course,
instructors might assign an individual essay paper in which the student has to research
and write about a place in the world where a famous mathematician lived and in which
the student would like to visit or have visited. This assignment would require research but
also a 5-minute oral presentation so that other students begin at an early stage in the
course to share some common interests with the other students of the learning
environment. After the presentations, the instructor may facilitate allocation of groups of
students who shared interest in the same countries. These students may end up working
on additional projects involving mathematics and these countries or continents. There is
room for expanding the student learning culture since many students would like to share
their personal views about similar or different countries. The mathematics research as one
perceives it served mainly as the initial catalyst to help develop the student community.
The goal of a curriculum is to serve students. Curricular activities like those
mentioned, which deliberately tie mathematical topics of the course to some of the
students’ interests, need to be developed so that groups of students learn about each
other’s culture and personal interests. Possibly, students may decide to socialize after
class. Ultimately, the more often they meet the more likelihood there is that they will
find time to develop studying groups and form student learning communities. Integrated
mathematics education programs incorporating activities like those ought to be designed
as part of the mathematics curriculum because they aim at student success and our MDC
students need all the support they can get. Attrition in mathematics classes needs to
decrease. Indeed, researchers across the globe are incorporating peer tutoring activities
and collaborative tutorial activities with successful increase in student retention (Edwards
2006; Keane, 2006; Phillips, 2006; Reilly, 2006).
When instructors involve students in campus life, students tend to find a reason to
stay longer and keep developing a student mathematics community. For example why not
invite our students to attend departmental seminars on topics in mathematics? Sometimes,
these topics are difficult for the student to fully understand but the purpose is not to
receive instruction on another mathematics topic. The aim is to instill reflection, to offer
vision of more sophisticated thinking processes that faculty expresses during these
gatherings, and ultimately to involve students in the campus life. After all, who does not
remember a friend or a family member who attended a presentation but had no idea what
the presentation was all about but who came in the name of moral support? Why not
encourage students to attend more conferences and attempt to create a mathematics
learning identity where higher number of students participate and then share with their
peers positive experiences? Students’ testimony will generate reliability to our efforts
convincing them to stay longer at the campus and study together. As students hear from
their own peers positive feedback about “belonging to the college community” the more
they will find excuses to stick together as a student community. Encourage students to
meet for studying sessions at coffee shops and attend too. This way, they perceive the
studying aspect as a fun activity occurring outside the campus and may meet more often.
Attend the first couple of sessions so the students believe that we find these important
too. Let students see that we are attending to have fun with them too.
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Interdisciplinary Activities
The discrepancy between conceptual mathematical learning in many classes and
the use, its modeling, and applications of real world situations, suggests that students will
continue to perceive mathematics detached from reality unless instructors help change
this approach. One strategy to bridging this gap is to help students see meaning in their
learning and to integrate the subject content across different disciplines. Integration of
mathematics into the different academic curriculum depends on mutually supportive
environments such as successful learning communities. In these, both faculties interact as
peer teachers and students continuously interact with the instructors and with each other.
Mathematics by itself will not develop into knowledge. Students will not
automatically use knowledge constructed in one domain and transfer it into the other
domain. Vygotsky’s theory of proximal development (1978; 1985) supports the need for
teacher intervention at a critical moment when cognitive learning is about to happen. Yet,
in a learning community, many times students define their own zone of proximal
development. Using this analogy with the establishment of student communities, I claim
that students may need help from faculty, at critical moments, to form their studying
groups and student communities. For example, the instructor will be responsible for
facilitating the immediate exchange of emails, of telephone numbers, and for creating
simple activities which involve student interaction outside the campus. Assign a
homework assignment that demands classroom postings. Comment on these postings and
encourage students to do the same. Monitor the thread of discussion. Intervene while ediscussions are happening, Raise critical thinking questions and steer the discussion if
necessary. For instance, ask students to write about a mathematical concept learned in
class that was clear, one that was exciting and one that was not clear.
As interactive activities involving students become the norm, a student
community often develops. Students feel free to take risks and sometimes taking risks
becomes a priority. Students become familiar with adding to the whole and sharing a part
into the classroom community success. By communicating, students are developing
reasoning skills, learning and getting to know others in the class. They share success
stories; they collaborate, ask questions to each other and learn to see connections from
other fields into math through other students’ feedback. A studying group is being
created. While many students may continue to study like isolated islands, carrying doubts
and struggling to solve homework by themselves, the students who perceive the
community as valuable will maintain the student community and will extend learning
past classroom and campus. Home learning will also be integrated into metacognitive
learning and higher level thinking skills.
Conclusions and Implications for Future Research
In conclusion, interaction and encouragement may contribute to positive changes
in behavior, emotion and perception which may also motivate students to increasingly
participate in math content activities. As a result, students become more confident in their
ability to do mathematics. The higher confidence in turn may result in positive changes in
attitude, emotion and perception. Students usually begin to operate together and rely
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more on each other’s input. They begin to expect interactions and want to participate
more in the groups. A student community is being built.
Some practical strategies might include encouraging students to form a study
group which meets once or twice a week. Follow up on the groups after the first week.
Sometimes, listening to one pair of students who met, or interacted beyond the
mathematics class is sufficient to trigger the classroom community. The routine of the
meetings usually establishes a pattern and a support group is formed. Meeting means
interaction, so that students may also use the phone, email and technology associated
with webct courses to meet. With the help of the instructor, the students need to log in the
course at the same time and chat during that time both on academic and non academic
issues. Instant messaging is a great technology for providing support for tests but solving
more complicated mathematical problems may require a face to face meeting too.
Eventually students will physically spend time with each other and develop studying
groups. During these meetings students should go over problems they had trouble with.
Either some group member will help or the student will soon discover that they are all
stuck on the same boat. Then, it is the ideal time to request help from the instructor. If the
instructor can interact at that time then there is great likelihood of dealing with the zone
of proximal development (Vygostsky, 1978; 1985). Learning is likely to occur to all
members of this study group community.
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. Woven activities which
successfully connect mathematics to other disciplines have positive impact in student
learning because they integrate and extend learning outside the classroom. Each activity
helps form a student-centered environment which nurtures a student community. This
community engages students to use real-world applications and it integrates cross-cultural
competencies across different areas of the curriculum.
Further research linking self contained student communities in mathematics
classes is still needed. Furthermore, as the number of second language learners in U.S.
classroom increases, so do the possibilities for content-area instructors to integrate
collaborative efforts with various subjects other than mathematics. It is essential to move
away from the political forces that continue to isolate students in their own communities.
Such a diversity body of students in our college ought to be in itself an attractive for the
formation of new student communities.
Referentes
1. Bransford, J. et al. 2000. How People Learn: Brain, Mind, Experience, and School.
National Research Council: National Academy Press.
2. Edwards, B., 2006. College Physics Majors’ Mathematical Reasoning. Proceedings of
the Third International Conference o the Teaching of Mathematics (ICTM3).
3. Kalkowski, P., 1995. Peer and Cross-Age Tutoring.
http://www.nwrel.org/scpd/sirs/9/c018.html
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4. Keane, Eliane, 2006. Flying Geometrical Kites in French. Proceedings of the Third
International Conference o the Teaching of Mathematics (ICTM3).
5. Leigh Smith, Barbara, 2002. Realizing the Potential of Learning Communitites.
htttp://www.mcli.dist.maricopa.edu/events/lcc02/presents/smith.html.
6. Mandler, G., 1989. Affect and Learning: Causes and Consequences of emotional
interactions. In Douglas B. McLeod and Verna M. Adams, Ed. Affect and Mathematical
Problem solving: A new Perspective. Springer-Verlag New. 3- 20.
7. Menezes, Vera Lúcia e Oliveira . 1997. Fractal Model of Language Acquisition.
http://www.veramenezes.com/model.htm
8. Mitchell & Sackney, 2001. Canadian Journal of Educational Administration and
Policy, Issue #19, February 24.
9. Papert, S. 1980. Mindstorms: Children, computers, and powerful ideas. New York,
Basic Books.
10. Phillips, J. D. 2006. The Mathematics Curriculum, Broadly Conceived. Proceedings
of the Third International Conference o the Teaching of Mathematics (ICTM3).
11. Reilly, I. 2006. Peer-tutors and collaborative tutorials. Proceedings of the Third
International Conference o the Teaching of Mathematics (ICTM3).
12. Roueche, J. E., Milliron, M. & Roueche, S, 2003. Practical Magic. Community
College Press.
13. Schoenfield, A. 1992. Learning to think Mathmeatically:Problem solving,
metacognition, and sense-making in mathematics. Handbook for Research on
Mathemeatics Teaching and Learning: New York: MacMillan.
14. Teppo, A. 1991. Van Hiele Levels of Geometric Thought Revisited. Mathematics
Teacher, March, 210-221.
15. Tinto, Vincent. 1998. Learning Communities: Building Gateways to Student Success.
http: //www.ntlf.com/html/lib/suppmat/74tinto.htm.
16. Varsavsky, C. 2006 . Managing student expectations and improving student
engagement in a first year calculus course. Proceedings of the Third International
Conference o the Teaching of Mathematics (ICTM3).
17. Vygotsky, L. S. 1985. Thought and Language. The MIT Press.
18. Vygotsky, L. S. 1978. Mind in Society. Boston, Harvard University Press.
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19. Webb, N. 1991. Task-Related Verbal Interaction and Mathematics Learning in Small
Groups. Journal for Research in Mathematics Education, Vol. 22, No. 5. pp. 366-389.
20. Webb, N. 1989. Peer Interaction and Learning in Small Groups. In Peer Interaction,
Problem-Solving And Cognition: Multidisciplinary Perspectives, edited by N. M. Webb.
New York: Pergamon Press.
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