Introduction
As a
mathematics teacher I began to teach the basic concepts of information handling
during the visits in a private school of Karachi, Pakistan for grade VIII
students. During these visits, I found the students learning with the
instrumental understanding of mathematics while relational understanding was an
issue for students. Instrumental understanding refers to the, where learners,
only know the rules and methods to solve the given problem but are unable to
answer why they are doing so. Relational
understanding where learners not only know what to do but also know why they
are doing so. They connect the known concepts with the new concepts they are learning
and can apply their learning in the real life experience (Skemp, 1991). We group of three members visited the school
to teach grade VIII students in three phases. During these visits I found them
confused initially when I asked to make the groups and also asked to jot down
the numbers of visits to shops to buy daily life requirements and cards were
distributed among the students. During three visits students were taught data,
raw data, processed data, ungrouped data, grouped data, class interval, class
boundaries and frequency distribution table.
The main purpose of school
visits was to explore the students learning mathematics and teaching of myself
for relational understanding and get in-depth understanding of students and
myself as a teacher. I expect to take on the role of a teacher and teacher
educator in my school.
In future I could utilized my findings and with the
support of my school management, could promote student and teacher learning
that might to the student and staff development and school improvement. This
teaching and study would also give me a opportunity to go through the teaching
process in my own context and contribute to the existing literature.
Literature Review
The purpose of this study is to explore how a teacher can
enable students to learn with relational understanding and meaningful teaching
for the teachers. I attempted to discuss available literature which helped me to
develop an understanding of my topic and also provided me a framework for the
study. Before discussing and going into the detail of what is relational
understanding. It is appropriate to see what literature says about the term
‘Understanding’. Supporting this view Sierpinska says that the term
understanding is very ambiguous and states:
The word
understanding is used in very many forms and expressions ininformal speech. We
say that a person ‘understands’ something, wespeak of a person’s
‘understanding’ something, and of the various understanding’ people may have.
We also speak of ‘mutual understanding’,of somebody’s utterance or somebody’s
writing, of “Understanding” a wordand an expression, a concept, a phenomenon.
(P. 1)
Sierpinska further argues that in educational setup the term ‘understanding’ has a meaning different
from the usage of term in everyday language. He states that, understanding can
be thought of an actual or potential mental experience” (P. 1).Relational
understanding seems to be based on constructivist theory of learning. This theory
claims that “conceptual knowledge can’t be communicated directly. It has to be
constructed a new by every learner in his own mind”(Skemp,1991, p.203).
Heibert and Carpenter (1992) state “a useful way of
describing is in terms of a way an individual’s internal situations are
structured” (p. 66). Cockcroft (1993) shares this view and adds that it is an
integral state of mind which has to be achieved individually” (p.69). Dewy
(1971) defines understandings as “grasping the meanings”. Selinger (1994),
Jaworski (1994), Sierpinska (1994) and Ernest (2000) use the term understanding
and learning interchangeably. For example Ernest (2000) argues that for
learning we need to fully understand new ideas. Therefore learning can’t happen
without understanding.
During the teaching of
information handling in the school, I saw the students not responding when I
asked about their monthly visits to shop as a question of prior knowledge
review. Therefore, prior knowledge is most important factor in the construction
of new knowledge. Caprio (1994) suggests that “students coming to us already
have a set of experiences that have let them to develop cognitive structure
through the way they interpret their environment.They have been dealing with
the natural world with their own ideas” (p.210).
Implementation
In the first class, I found
the students unaware with rationale understanding of mathematics. I found by
asking probing questions from the students that they have no constructivist
approach and conceptual understanding which are important for relational
understanding. For second visit I planned a lesson such a way that students
could able to understand the activities easily. For example, linking the
previous lessons that how many their visits to the shop monthly. I allowed the
students to jot down the visits on their notebooks and different number of
visit I observed on their notebooks and I also jotted down on the blackboard.
In the same way another activity was rolling the dices in groups on their
table. I received the different numbers and also instructed to jot down the total
numbers written on dices. Students in this school visit took very interest
involving in activities. Students understood the activities and the data
handling was interesting and were related to daily life experiences. They also
realized the information handling is meaningful in their practical life. I felt
a little confusion in our lesson planning and in the third visit of school, I
re-planned making the activities according to the students’ level and linked to
the previous activities. Before going to teach frequency distribution table,
same data was categorize into intervals. For example, how many students are
there visited (5-10) times to shop in a month, (11-15), (16-20) and up to (50-55)
times in a month.In this activity students easily built-up the concept of class
interval and frequency. If the students visit the shop 5 to 10 times are 7,
then 7 will be the frequency of the class interval (5-10). The frequency is
also represented by tally marks in second column. Number of tally marks were
written which are frequencies of different class intervals.
Overall, a frequency table
was drawn by the students. I also draw the frequency table on black board asked
about table from the students and they were confortable at the end of school
visits while doing information handling by activity based teaching and became
familiar with relational understanding of mathematics. Students realized how
and why information handling can be understood in our practical lives.
Findings
During the school visits I
found myself to know about the conceptual, activity based teaching and
relational understanding of mathematics. By the end of the visits studetns and
teachers including me who were teaching in concern school and class were
feeling more improvement in the teaching and learning process of information
handling. In the beginning of school visits, students were reluctant while
asking the questions in the class at the end of the visits studnets were
comfrotabel while asking questions from the teachers.
About teachers of our group,
I was feeling challenges in teaching mathematics relationally but after
conducting activities and conceptual understanding of studnets, I also felt
confortable while teaching in the class. For this, I though in the previous
classes and reflected on the visit and re-planned to improve my teaching. For
the relational understanding of the mathematics there was a lot of hard work
and reflection on the previous visits was required to coop with the challenges
of the relational understanding.
There is need to reduce the
gap between teacher and student to teach mathematics relationally and also give
the concreat material to teach information handling in the class. Students are
required to encourage to raise more questions when they demanded while teaching
information handlings. I used the strategy of student centered and activity
based teaching during the school visits. In one of the school visits, I asked
the studnets to jot the ages of their family members of each student in the
class. I got the information regarding
their family members in terms of ages. The information got was raw data.
Similarly, I again asked them to arrange the data from smallest to largest
number. Students performed this activity easily and they also divided into the
class intervals of (1-10)years, (11-20)years up to (41-50)years. They knew how
information handling can be arranged and why it is meaningful for them.
Students thought information handling is used in our own practical life. For
example, number of scores in different tests of the studnets family represented
by frequency distribution table. This was a relational understanding of
mathematics that studnets came to understand how were they doing and why. In the
beginning of the school visits, I observed that questions raised by sutdnets
were not about the understanding concept, but they were only about how to write
and solve the exercises in their registers. During the second visit of the
school, a mathematics teacher was also present behind the class. When I asked
question before starting the topic as prior knowledge test, some students were
giving answers about the class interterval and frequency according to their own
conception. During that activity, math’s teacher of the school was strict to a
student while answering the question. The attitude of the teacher reflected
that students were used to instrumental understanding not relational
understanding of mathematics. At the end of the visit we took some reflections
of the studnets and their class teachers. Both teacher and students responded
that they had never compare this topic information handling with concrete
material and practical examples. There is need of relational understanding of
mathematics in our schools, particularly when the level goes up to the higher
grades, the relational understanding of mathematics declines for the sake of
completing syllabus and passing exams.
Conclusion
It is concluded that during
the visits of a private school of Karachi, Pakistan, I found some teaching
strategies and innovative methods are needed to plan and implement in schools
when the basic concepts of information handling are taught. Teaching strategies
should be students centered and based on relational understanding of
mathematics. I learnt about the students in the school that there are use to
instrumental understanding and teacher centered class. There was also a method
of teaching by talk and chalk. By the studnets responses it is clear there was
no concept of group work, cooperative learning, teaching by audio visual aids
and conceptual teaching for the relational understanding of the mathematics
while teaching the basic conpts of information handling.
Students were needed teaching
on activity base in information handling because studnets would be able to
retain the fundamental concept of information handling and interest would be
developed. By the visits of school in Karachi during the teaching of basic
concepts of information handling it is said that we required students encourage
and develop the sense of conceptual and relational understanding of
mathematics.
Furthermore, information
handling based on activities and relational understanding of studnetsalso
improve the teaching skills of the teachers. Relational understanding plays a
vitial role in information handling to make meaningful the lives of studnets
because this discipline is very useful and implacable in pratical life.
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