Category Archives: Education

The Stages of Learning


Reference: Critical Thinking in Education


There are several stages of learning as follows:

  1. Being tutored on a one-on-one basis

  2. Self-learning under supervision in a tutorial classroom

  3. Tutoring under supervision in a tutorial classroom

  4. Practicing mindfulness, which is seeing things as they are with full awareness of all assumptions.


In the first stage of being tutored, the best place to start is with Mathematics. Mathematics means “tools for learning”. It helps the student build a sense of logic slowly and deliberately.

Mathematics starts with counting. This is a very simple activity, which involves the student in his environment.  Counting extroverts the student’s attention. Throughout the study of mathematics, the attention should be kept extroverted and on the mathematical rules as they apply to the environment.

The rules of mathematics should be learned in the sequence in which they come about naturally—numbers, digits, reading and writing numbers, large numbers, addition, subtraction, multiplication, division and so on.

These rules should be learned slowly and deliberately to understand their logic. The student should never go by something that he or she does not understand. Memory has its uses but memorization defeats the whole purpose of learning. Mathematics puts the student in a frame of mind to learn other subjects with understanding.


In the second stage, you get them to start studying materials by themselves. You check their understanding randomly as they go along. Correct their understanding per the materials as necessary. Teach them how to look up words in a dictionary to understand the lessons better.  So you train them how to self-learn given good materials.

Self-learning is an activity that a Tutorial class supervisor has to make sure that every student has it down pat. If a person is having difficulty with mathematics later then this skill in self-learning must be examined thoroughly.

It is a good understanding of the basics that makes a good self-learner. So, if the skill of self-learning is missing then the student should be thoroughly examined for his understanding of the basic concepts in mathematics.


In the third stage, an accomplished self-learner can start tutoring new students who need one-on-one tutoring. He puts them on the road to become self-learners.


In the fourth stage the person practices “seeing things as they are with full awareness of all assumptions”. This practice is called mindfulness.

In practicing mindfulness, a person learns to recognize anomalies and resolve them. Anomalies are discontinuities, disharmonies and inconsistencies in mental perception. Now he can learn anything that he wants to learn on his own by going to the Internet, or from experiences in life.


Self-Learning Diagnostic #2

Reference: Critical Thinking in Education


The above is the second “Self-Learning” Diagnostic Test for students in middle school and above

Can you find the illegible numbers represented by the *’s?

The purpose of this diagnostics is to assess the following:

  • Is the student able to move beyond patterned thinking?
  • Can the student think in novel ways to resolve problems?
  • Is the student’s mental math techniques up to par?

Calculators are very useful. They speed up the ability to calculate. But they should augment and not replace a person’s ability to calculate mentally and on paper with pencil. The person should not get so addicted to calculators that he loses his number sense and the gut feeling when computation go wrong.

This exercise may be timed. If the student can do this exercise rapidly and accurately then his attention and self-learning potential are in good shape. No remedy is needed at this level.

If the student is unable to move beyond patterned thinking then he should review the following documents to learn that different ways of thinking are possible.

Mental Math Techniques for Subtraction

Mental Math Techniques for Multiplication

This diagnostic helps locate and fill some of the early holes in the understanding of math. Filling of such holes in a subject restores student’s eagerness to learn.

With eagerness comes the ability to self-learn.


Self-Learning Diagnostic #1


Reference: Critical Thinking in Education


The above is the first “Self-Learning” Diagnostic Test for students in middle school and above

Can you compute the three addition problems above on paper with pencil?

The purpose of this diagnostics is to assess the following:

  • Is the student’s attention well focused?
  • Is the student confident of his/her answer?
  • Is the student’s mental addition techniques up to par?

In this diagnostics, the student

  1. Adds the numbers from top to bottom to get the sum.
  2. Adds the numbers from bottom to top to verify the sum.
  3. Checks his answers against those provided on the right.

This exercise may be timed. If the student can do this exercise rapidly and accurately then his attention and self-learning potential are in good shape. No remedy is needed at this level.

If the student’s focus and confidence in math needs improvement then he should practice mental addition on a gradient as follows.

  1. Practice adding two single-digit numbers.
  2. Practice adding a single-digit number to a double-digit number.
  3. Practice adding two double-digit numbers

It all boils down to knowing the sum of two single-digit numbers. And, for that, there are limited numbers of combinations. The rest is attention and technique.

The techniques for mental addition help develop basic number sense. The student is then able to rapidly add two numbers, while also verifying the sum at the same time. This skill is then carried forward to rest of the basic math operations. This builds up a confidence that is hard to shake.

This exercise develops the fundamental thinking skill on which subsequent math skills are built. It fills an early hole in the understanding of math.

The following document provides basic mental addition techniques and exercises. After learning these techniques, the student may develop his own techniques.

Mental Math Techniques for Addition

This diagnostic helps locate and fill one of the early holes in the understanding of math. Filling of such holes in a subject restores student’s eagerness to learn.

With eagerness comes the ability to self-learn.


Subject Clearing


Reference: Critical Thinking in Education


Since additional information on a subject may easily be accessed through Internet these days, the purpose of study is not to memorize but to improve one’s critical thinking.

The true purpose of study is to resolve inconsistencies (things that do not make sense) as you come across them in a subject. This develops clarity of mind and the ability to think fast on your feet.

When one is studying the basic postulates and fundamentals concepts of a subject it is very important to be able to detect hidden assumptions and subjective opinion. Such erroneous ideas can be very pervasive, and may even enter the definitions of words provided in dictionaries.

The following procedure helps one become aware of hidden assumptions and subjective opinions related to a subject.

1.    Make a list of key words in that subject.

Skim through the chapter that you are going to study, and make a list of key words. Put that list on an Excel worksheet. This list may grow as your studies get deeper into the subject.

Look up the broad concept and definitions for each key word, as described in Word Clearing, and note it down on the Excel worksheet.

2.    Gradually build upon each concept for that subject.

Study the materials of the subject one paragraph at a time. Reduce a paragraph to its main thought and note it down in an editable document on computer. Do this before proceeding to the next paragraph.

When it is difficult to reduce the paragraph then look for words that may not be clear in that context. If the difficulty persists then note down the confusion with the “main thought” in that paragraph. It may get clarified by something you read later.

Note down any additional conceptual understanding and questions next to the appropriate word on the Excel worksheet.

3.    Arrange the key words with their concepts in proper sequence.

The concepts in a subject always evolve in a sequence. This can easily be seen in Mathematics and Science.

In Excel, you may separate the key words on two different worksheets categorized as “fundamental concepts” and “derived concepts”. Then arrange the concepts in each worksheet in the order they evolved.

This sequence of the evolution of these concepts may not be linear but multi-dimensional . So, to look at the key words in different sequences, you may create “priority columns” in the worksheet with a number assigned to each key word. The whole idea is to arrange these words in different ways to examine the connections among them.

4.    Note any inconsistencies among the concepts and clarify them.

As the study of the subject progresses and better understanding comes about, rearrange the list of key words to examine consistency among the various concepts. You are looking for things that appear inconsistent and do not make sense. There may be holes among these concepts that need to be filled.

Deeper research may be required to discover such holes and fill them. First review your materials to clarify any inconsistency. If it does not clarify easily then note it down on the worksheet and research through other materials in the library or on Internet.

5.    Clarify the fundamentals of the subject as a priority.

The consistency of the fundamentals determines the consistency of the whole subject. Any inconsistency at the fundamental level must be handled as a priority. For example, a unified theory is desperately being looked for in the subject of Physics, which could bring the fundamentals of Newtonian Physics, the Theory of Relativity, and Quantum Mechanics in line.

There are likely to be many contributors to a subject who may use different words for the same concept. This is the case with religious knowledge from different cultures. Group such words together to discover inconsistencies among concepts.

Study of inconsistencies may lead to discovery of arbitrary assumptions that were made in the absence of knowledge, or you may find erroneous observation, or simply some ideas that are taken for granted. This may reveal gaps in the subject itself. Develop your own understanding by seeking consistency among these concepts.

6.    Make the subject as complete as possible.

There are many examples in the subject of religion where gaps in knowledge are hidden under assumptions and dubious explanations. This may be the case with any subject where inconsistencies abound. Follow up on inconsistencies, which may then reveal gaps in the subject. Real progress then becomes possible.

Fill gaps in the subject with wider research. Make the subject as complete as possible through direct experience and experimentation.


Word Clearing


Reference: Critical Thinking in Education


When there is confusion in a subject, the first thing to do is to isolate the area of confusion and check the understanding of the words used to explain that area. Hopefully that would handle the confusion or, at least, narrow it down.

Words and symbols form our basic understanding.

A word can have not only many definitions but also many shades of definitions. It is not only a missing definition but also a missing correct definition that generates confusion. Different contexts may require different definitions, or shades of definitions, for the same word.

It is the tendency of the mind to fill the missing definitions by preconceived notions. This leads to contradictions and conflicts in the mind. Therefore, it is very important to examine the definitions of the words in the context at hand. Correct definitions bring continuity, harmony and consistency to the understanding of the materials.

The following procedure helps determine correct definitions for words in study materials.

1.    Determine the broad concept of the word.

Look up the word that is not quite clear to you. You may find the broad concept listed under ‘history’, ‘origin’, or ‘derivation’ of the word in a dictionary.

The following broad concepts for some words are taken from “Dictionary of Word Origins” by John Ayto

The word STUDY comes from a Latin word meaning “eagerness, intense application”.

The word MATHEMATICS comes from a Greek word meaning “something learned”.

The word ARITHMETIC comes from Greek, ARITHMOS number + TECHNE skill, which means “number skill”.

2.    Look up the definitions of the word.

Next you may look up the definitions provided for that word. A good but simple dictionary is very useful.

Please note that small words like “a”, “of”, “in”, et cetera, have many different definitions. A wrong interpretation of such words may completely alter the idea expressed.

3.    Visualize the definitions.

Visualize the definitions as you look them up. To get a better idea you may bring up the images for that word on the Internet. Make examples of different definitions to understand the differences. Make sentences to get further clarity on a definition. Relate the different definitions to your experiences to make them real.

4.    Look up words in definitions as necessary.

If a definition contains a word that is not quite clear then look it up following this procedure. This may sometime get you in a long word chain. Keep an account of the words that you are looking up. Cross out the words from the list as you clear them up.

NOTE: It is okay to look up the same word as many times as necessary. Each time you look up a word you may pick up a new dimension of its meaning.

5.    Determine the definition that fits the context.

Once you have understood the basic concept, and have adequately examined all the definitions of the word, determine the definition that fits the context the best. Then review that area to see if the confusion has cleared up.


Keep examining the definition of words in that area until the confusion clears up fully. If the confusion persists in spite of your best efforts at “word clearing” then apply the procedure of “subject clearing”.


Reference Dictionaries: