Pedagogy - Concrete to abstract: What does it mean?

Another aspect of pedagogy is related to the role and nature of materials in the classroom. We generally believe that abstract concepts are acquired through a process of creating, experiencing and analyzing concrete situations. There has been an increasing stress on putting in more and more concrete materials in the Mathematics classrooms. The idea of so-called Math lab has been supported and advocated widely. The feeling is that children understand concepts through the experiences in Mathematics laboratory. This needs to be considered carefully. It is evident that the idea of using concrete materials and contexts for helping children learn is important. These serve as a temporary model to represent abstract concepts. For example 5 stones are a concrete model for 5 and so 5 chairs. A triangle cut out from card board is a model for triangle as it can portray some key properties of the triangle. It must be recognized that these artifacts do not fully represent the concepts of 5 or the triangle. They are only scaffolds for us to communicate what these terms mean in the initial stages. Gradually learners have to move away from these concrete scaffolds and be able to deal with mathematical entities as abstract ideas that do not lend themselves to concrete representations.

A quadrilateral is closed figure bounded by 4 straight lines. A line is a one dimension infinite string that has no

thickness. The point is that an actual line and hence a quadrilateral cannot be represented by even a drawing on the board leave alone by a concrete representation. So while it is important to begin with concrete experiences, gradually the child must articulate using her own language and move on. Mathematics going through the stage of using pictures and then tally marks etc. has to transit to symbols. This is an essential component of learning to do Math. The learning of Mathematics has to culminate in  being able to deal with mathematical ideas on their own without any scaffolds. Therefore, when we advocate the Math laboratory for senior schools there is both a pedagogic as well as an epistemic question about whether this is the appropriate direction to proceed in.

The idea of laboratory in Science is to have the students explore some phenomena. She would make observations related to it and then based on the observations attempt to deduce some kind of causal connections. Utilizing many such experiments and data from earlier experiences, the student can attempt generalization and building hypothesis that can be checked by further experimentation. The epistemological touch stone for ideas in Science can be arguably experimental observations and validations. This unfortunately is not true for Mathematics and therefore using the Math lab to have children deduce or prove

mathematical statements by measurements or through models, is an epistemic and also a pedagogic error. 

The attempt at this stage has to be to enable the child to deal with abstract ideas.Unlike the rich experience of language that the child comes to school with, ideas of Mathematics are not so richly experience based. All children are able to deal with numbers and arithmetic that they need in daily life. They are also able to organize the space around them and carry out spatial transformation to the extent they need. This knowledge is profound and complex. It shows the innate capability of the child to acquire these ideas. All children in any society are able to deal with these ideas. The problem comes when we attempt to transact Mathematics and want them to de-contextualize and abstract the number, shapes transformation, operations and why all these work. The discipline of Math is to be able to talk about abstractions and how relations between abstract quantities can be understood and developed. In the primary classes Social Science and Science are also largely experience based and there is recognition that abstract concepts should not be imposed at this stage. Even in the upper primary classes it is possible to make Science replete with concrete experiences and use the available experiences of the child as well as the experiences provided in the classroom to help her construct a framework of concepts. Mathematics does not allow this easily.

A lot of Mathematics pedagogy depends upon how the teacher engages with children. The classroom atmosphere has to be such that children can participate, articulate their ideas, make mistakes and talk about them without fear. Such an atmosphere will determine the relation children have with Mathematics. There is no one method or one technique that we can recommend for teachers to follow. She has to follow the classroom and create processes that facilitate engagement and dialogue that move forward gradually but can also return to an earlier point and develop again in a different way. The key aspect of Math classroom has to be the recognition that children will develop mathematical ideas and concepts through assimilation with their own previous ideas and experiences and modify them in the process of interactions. Each of us develop our own way of solving problems. It may require exposure to a lot of algorithm and methods but with an openness to create and examine more. They should be able to absorb available ideas and accommodate them in their conceptual framework. The models that anyone of us use or the artifacts a student constructs can help her understand the problem and develop a strategy but would not help everyone. They will be different for each of us. You cannot help a person learn Mathematics by giving her short-cuts or imposing on her your way of solving problem. Your way may appear very simple, neat and elegant to you but that may not be so for her. We categorize and use ideas in our own ways and use steps that we can think of. It is a doubly difficult task to understand the problem and then also discover the underlying logic of the process you have used to construct the solution.

This will help us derive specific expectations and purposes for different class and age groups. This is what constitutes the syllabus. The first two components have to be informed by the so-called subject, its nature, purpose for human society and for the students for whom the transaction program is being developed. One has to keep in mind the person who is going to transact the learning so as to understand what the aims, expectations and learner backgrounds demand from her. The third : Is there any specific understanding that we need about how this subject is learnt? This will help us construct classrooms that aid learning. The fourth is the prevalent attitude in society about Math- be it teaches, students or parents. All these contribute critically to the pedagogy of the subject.