3 Modes Of Thinking: Lateral, Divergent & Convergent Thought

3 Modes Of Thinking: Lateral, Divergent & Convergent Thought

by TeachThought Staff

TeachThought is about, more than anything else, human improvement.

A core tenet of humanity is our ability to think critically and with imagination and creativity. Therefore, it makes sense that our ability–and the decision to–do this consistently in some ways defines us as a species. Critical thinking, in part, involves simply avoiding cognitive biases.

See also What It Means To Think Critically

Further, it’s not a huge leap to say that the ability and tendency to think critically and carefully and creatively supersedes content knowledge in importance, but that’s a discussion for another day. In general, it is our position that critical thinking is of huge importance for students, and as such is a big part of our content and mission at TeachThought.

benefits of asking questions

In pursuit, the sketch note above from Sylvia Duckworth is a nice addition to that index of content. Sylvia has consistently done a great job converting ideas into simple visuals–on our 12 Rules Of Great Teaching, for example.

You can follow Sylvia on twitter here.

We’ve taken the visual and fleshed it out with some commentary from Wikipedia (a resource we love, by the way).

3 Modes Of Thinking: Lateral, Divergent & Convergent Thought

1. Convergent Thinking

Summary: Using logic

Also called: Critical Thinking, Vertical Thinking, Analytical Thinking, Linear Thinking

Wikipedia Excerpt & Overview

‘Convergent thinking is a term coined by Joy Paul Guilford’ (who also coined the term for the ‘opposite’ way of thinking, ‘Divergent Thinking’).

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‘It generally means the ability to give the “correct” answer to standard questions that do not require significant creativity, for instance in most tasks in school and on standardized multiple-choice tests for intelligence.

Convergent thinking is often used in conjunction with divergent thinking. Convergent thinking is the type of thinking that focuses on coming up with the single, well-established answer to a problem.[1] Convergent thinking is used as a tool in creative problem-solving. When an individual is using critical thinking to solve a problem they consciously use standards or probabilities to make judgments.[2] This contrasts with divergent thinking where judgment is deferred while looking for and accepting many possible solutions.’

2. Divergent Thinking

Summary: Using imagination

Also called: Creative Thinking or Horizontal Thinking

Wikipedia Excerpt & Overview

‘Divergent thinking is a thought process or method used to generate creative ideas by exploring many possible solutions. It is often used in conjunction with its cognitive colleague, convergent thinking, which follows a particular set of logical steps to arrive at one solution, which in some cases is a ‘correct’ solution. By contrast, divergent thinking typically occurs in a spontaneous, free-flowing, ‘

By contrast, divergent thinking typically occurs in a spontaneous, free-flowing, ‘non-linear’ manner, such that many ideas are generated in an emergent cognitive fashion. Many possible solutions are explored in a short amount of time, and unexpected connections are drawn. After the process of divergent thinking has been completed, ideas and information are organized and structured using convergent thinking.’

3. Lateral Thinking

Summary: Using both Convergent and Divergent Thinking

Also called: ‘Thinking Outside the Box’

Wikipedia Excerpt & Overview

‘Lateral thinking is solving problems through an indirect and creative approach, using reasoning that is not immediately obvious and involving ideas that may not be obtainable by using only traditional step-by-step logic.[1]

To understand lateral thinking, it is necessary to compare lateral thinking and critical thinking. Critical thinking is primarily concerned with judging the truth value of statements and seeking errors. Lateral thinking is more concerned with the “movement value” of statements and ideas. A person uses lateral thinking to move from one known idea to creating new ideas.’

3 Modes Of Thinking: Lateral, Divergent & Convergent Thought

Can You Get a Negative out of a Square Root?

Can You Get a Negative out of a Square Root?

 

Unmatched math delimiters. Adding final one for you.

The simple answer is: yes you can get negative numbers out of square roots. In fact, should you wish to find the square root of any positive real numbers, you will get two results: the positive and negative versions of the same number.

Writing a Square Root Equation for Positive and Negative Results

Consider the following:

16 = 4 * 4 = (-4) * (-4)

In the equation above, you can either multiply 4 by itself or multiply (-4) by itself to get the result of 16. Thus, the square root of 16 would be:

The √ symbol is called the radical symbol, while the number or expression inside the symbol—in this case, 16—is called a radicand.

Why would we need a ± symbol in front of 4? Well, as we have discussed before, the square roots of 16 can be either 4 or (-4)​.

Most people would simply write this equation simply as . While it is technically true and there’s nothing wrong with it, it doesn’t tell the whole story.

Instead, you can also write the equation in such a way that it explicitly indicates that you want both the positive and negative square root’s results:

This way, other people can easily tell that the one who writes the equation wishes to have positive and negative numbers as the result.

Perfect and Imperfect squares

As you may know, 16 is a perfect square. Perfect squares are radicands in the form of an integer, or a whole number, that has a square root of another integer. In the example above, 16 is a perfect square because it has the number 4 as its square root.

Positive real numbers are not always perfect squares. There are also other numbers such as 3, 5, or 13 that are referred to as imperfect squares. If a radicand is not a perfect square, then the square root of the radicand won’t result is an integer. Take a look at the equation below.

5 is not a perfect square. Therefore, its square root won’t be an integer. Additionally, the square root is not even a rational number. Rational numbers are numbers that can be expressed as fractions composed of two integers, e.g. 7/2​, 50/4​, and 100/3​.

The square root of 5 is an irrational number since it can’t be expressed as fractions. The numbers right of the decimal 2.236067 … would continue on endlessly without any repeating pattern. Still, both irrational and rational numbers are part of real numbers, meaning they have tangible values and exist on the number line.

Square Roots of Zero and Negative Numbers

We mentioned earlier that any positive real numbers have two square roots, the positive one and the negative one. What about negative numbers and zero?

For zero, it only has one square root, which is itself, 0.

On the other hand, negative numbers don’t have any real square roots. Any real number—whether it’s positive or negative—that is multiplied by itself is always equal to a positive number, except for 0. Instead, the square root of all negative numbers is an imaginary number.

or

By definition, the square root of (-1)​ is i​, which is an imaginary unit. As a side note, imaginary numbers do not have a tangible value. They are not part of real numbers in the sense that they can’t be quantified on the number line. However, they are still used in math and the study of sciences including quantum mechanics, electricity, and more.

To get a better understanding, let’s take a look at an example. For instance, let’s say we want to identify the square root of (-9)​, what would it be?