Teaching Basic Concepts of Information handling for relational understanding of mathematics

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 in informal speech. We say that a person ‘understands’ something, we speak 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 word and 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 studnets also 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.