Inculcating Reasoning Skills

Development of logical thinking (or reasoning) is a vital part of cognitive development. Development psychologists like Piaget have conducted many experiments with growing children and extensively studied the development of reasoning in children. The term “reasoning” has wide connotations. In the chapters in this section, we will use the term “thinking” to mean “thinking with the help of math to develop reasoning”.

One of the important objectives of math is to develop the ability to “apply & use” to “quantify, mathematize & understand” real world problems. Objects & events in the world keep on changing. Understanding “how” they change, the ability to quantify this change and predict future change are important life skills which math helps in developing.

Up to now we have studied various kinds of numbers and operations that can be performed with these numbers. We have also explored various life situations, classified them into “math metaphors” and related them to various arithmetic operations.

Based on our understanding of these mathematical operations, we are now ready for understanding the process of logical thinking. At the basic level, there are two modes of logical thinking in math– additive & multiplicative.

Thinking which is closely modelled by the operations of addition & subtraction is called “additive thinking.” Additive thinking can be of 2 types.

Thinking closely modelled by the operations of multiplication and division is called “multiplicative thinking”. Multiplicative thinking can be of 3 types.

We will study each of these in detail in subsequent chapters. We need to remember that these represent simplified versions of actual life experiences, meant mainly to understand the underlying concepts.

Developmental psychologists say that additive thinking is easier for children to understand. The development of multiplicative thinking takes more time and higher level of cognitive skills. It happens typically when they are in 5thor 6thgrade. This a crucial stage when children transition from “concrete operations” to “formal operations” stage.

Multiplicative thinking also throws up the idea of a ratio, which in simple terms is a quotient of 2 numbers. Ratio itself is an important kind of number and we will study it in detail in chapters 18.4 & 18.5.

We have limited our study to areas where the 4 basic arithmetic operations can be applied. There are other modes of mathematical thinking which require understanding of more complex mathematical concepts & processes. They will be studied in higher classes.

The processes of thinking are mental. They are both qualitative & quantitative. Qualitative in the sense of understanding events & processes in the environment and deducing the nature of relations. They cannot be directly taught to students. Students need to be exposed explicitly to a variety of life experiences familiar to them, and in which these concepts are embedded. Thinking abilities have to be constructed in individual minds in ways that are unique to the individual. Students can think about the similarity of patterns in these situations and understand the underlying concepts.

With this understanding they can translate qualitative relations into quantitative ones. They can also apply these concepts to other life situations. Hence development of thinking takes time, experience & introspection.

It is not a good pedagogical strategy to go too rapidly into mathematical formulas and computational methods. Children should be helped to gain a qualitative understanding of the above processes and to predict outcomes in situations with simple mental computation. Only then should the formal methods be introduced. The mathematics of reasoning is simple but the underlying thinking process is not.