Types of Thinking

Until now, we have used several terms related to thinking - Critical Thinking, Algebraic Thinking, Structured Thinking, Logical Thinking and Computational Thinking. Let us try to understand the difference between these types of thinking, particularly in relation to math.

I am not an expert on these topics. Please take it as a simpler explanation for further investigation.

The ideas in many of these types of thinking overlap. The list is mainly for our convenience.

Critical Thinking is the most comprehensive type of thinking. Other kinds of thinking are aspects of critical thinking. It is about checking the truth of the facts presented and to see if they are relevant to the issue to be decided. It is also about checking the logic of the steps by which known facts are taken to a conclusion. It is applicable to any field of knowledge based on reason.

Critical Thinking has to be practiced in the context of a subject or discipline with a certain amount of content knowledge. The processes of weighing options, prioritizing them, understanding the interconnections, relating them logically will require this content knowledge. In math, two of the most important aspects of critical thinking would be logical & algebraic thinking.

At the primary school, the ability to read a word problem, understand the math embedded in the language, understand the math metaphor it is portraying and locate the relevant & irrelevant information are some aspects of critical thinking.

This process is illustrated with a framework at the end of this chapter.

Algebraic thinking, as we have seen earlier, is the abstraction of general principles from specific instances that we know. Starting from the additive relations between number, we abstract the additive relations between classes of numbers, such as Odd & Even. It is one aspect of critical thinking where such an abstracting skill is used.

It is an understanding of the algebraic structure of the place value system. It is an understanding that an equation like 3 + 4 = 7 contains within itself an infinite variety of life situations.

Logical thinking starts with a few propositions which are accepted to be true. Then by a series of arguments, a conclusion is reached, through a series of intermediary propositions. Each step should be such that "it is the only conclusion" that can be reached from the given propositions.

Euclid's development of Plane Geometry is an excellent example of logical thinking.

In logical thinking, the starting propositions are accepted to be true. In critical thinking these may not be accepted as true and some other evidence demanded.

Algebraic and logical thinking overlap in certain areas. The principle of abstraction from the particular to the general may not follow principles of logic. Certain logical processes may not use any algebraic thinking.

Using Venn Diagrams in the classroom in an excellent way to promote logical thinking. They can be used to sharpen thinking about similarities and contrasts between two concepts. For example all numbers can be classified into either Odd & Even or Prime & Composite. Can these concepts be represented with a Venn Diagram in an integrated manner so as to bring our these interconnections?

Computational Thinking- In the last 5 decades, after the advent of computer algorithms and software, the idea of computational thinking has emerged. It is about analyzing a problem to be solved in terms of specific steps which can be translated into an algorithm which a computer software or a layperson can understand and work out the solution. With the increasing importance of ML (Machine Learning) and AI (Artificial Intelligence), computational thinking is gaining importance.

Since computational thinking is a recent arrival, after the advent of computer science, we will deal with it in detail in the next chapter.

Structured Thinking - All these types of thinking may involve understanding of substantial amounts of data. Structured thinking is a way of organising the data in a way that the patterns & relations hidden in the data are visible.

Tree diagrams, Venn diagrams, tables & charts are various structures used in math.

Solving Word Problems is an excellent way of developing critical thinking with a focus on problem-solving. Let us look at the various steps involved as detailed in Chapter 15.3.

1. Language & Life Experience

   1. Understanding the language in which the problem is presented

   2. Understand the meaning of the words in the context of the problem

   3. Identify the information which is relevant to the problem. Discard the irrelevant information

   4. Abstracting from the specific details of the problem using prior knowledge & life experiences

2. Metaphor

   1. Abstracting from the problem to the operation metaphor it is related to

3. Structured Thinking

   1. Work out the steps starting from the given information to the required solution in logical steps

4. Numbers & Operations

   1. Identify the type of numbers & units and recall the computational & procedural rules for performing the computations

5. Review & Recheck

   1. Review the logic of the steps in the solution

6. Presentation

   1. Present the steps in a logical manner which is comprehensible to the reader

   2. Use appropriate language for precision and clarity.

We see that all the different thinking skills are untilized.