Hierarchical structures-how mathematical ideas grow, Mathematics

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Hierarchical Structures :  As the abstractions from concrete objects and materials become more and more general, they represent wider and wider ideas. If we put down each step of the process of generalisation, we would have a series of ideas, each contained in the generalised idea following it.

An hierarchy of ideas is a system of ideas in which they are organised into different grades, ranked one above another.

For example, consider the number system.

i) From the counting of concrete objects, we abstract the set of natural numbers, namely, N = {1,2,3,4...}.

ii) If, in this set, we include zero, we get the set of whole numbers, namely, W={0,1,2,3,4...}.

iii) This set can be further enlarged to include negative numbers, and we get Z = {...-3, -2, -1, 0, 1; 2, 3...} as the set of integers.

iv) To the set of integers, we can add positive and negative fractions to get the set of rational numbers, Q, and so on. We have pictorially shown this in Figure.

Now, if I don't understand what the natural numbers mean, I will certainly not be comfortable with whole numbers. Similarly, if I don't grasp what negative numbers mean, I doubt if I will understand what a rational number is. So, to understand each of these abstract concepts, I need to understand every concept that comes before it in the stepwise build-u, or hierarchy, of ideas.

Now, look closely at the mathematical ideas that you are familiar with, and try the following exercise.

E1) Write down three hierarchical chains in mathematics. You can look for examples related to number operations, geometry and algebra.

The hierarchy of concepts also has implications for the way concepts are learnt. If you look at the historical development of any concept, hierarchically lower concepts usually came before hierarchically higher concepts. The learning of concepts by children is also, broadly, on the same pattern. Therefore, it is usually better to introduce a child to an hierarchy of ideas the way they developed.

Unfortunately, this does not always happen.

For example, a square is a particular type of rectangle, and a rectangle is a particular case of a parallelogram. But many children in Class 2 are taught these concepts at the same time, without even relating them to each other. What is the result? Even two years later many of them will say that a square is not a parallelogram.

So, to really understand a new mathematical idea a person requires a proper understanding of mathematical concepts that come before it. This is what we mean when we say mathematics is an hierarchically structured discipline. In the following exercise we ask you to consider the implications of this fact for teaching.

E2) The hierarchical structure of mathematics is one reason that it is considered a difficult subject to learn/teach. Do you agree? Why?

Now that we have seen some ways in which mathematical ideas grow, and are acquired, let us consider the special nature of mathematics.

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