Monthly Archives: August 2011

MILESTONE 3: Subtraction

The idea of subtraction as “reverse addition” is explored in greater detail in this document.

MATH MILESTONE #A3: SUBTRACTION

Subtraction is taking a quantity away from another quantity. For example, suppose you have 15 marbles; and you give 6 marbles to your friend. You can find the remaining marbles in two different ways:

  1. Take 6 away from 15:  start from 15 and count back 6 to see the number you end up with.
  2. Find 6 and “what” is 15: start from 6 and see how many counts it takes to arrive at 15.

The method in (a) requires a new skill of counting backwards. The method in (b) utilizes the already learned skill of addition. Both these methods work because

Subtraction is the opposite of addition.

The better one can do addition, the more skillful one gets at subtraction. In subtraction, one also learns the concept of regrouping the place values. The following videos from Khan Academy demonstrate subtraction.

Basic Subtraction

Subtraction 2

Subtraction 3: Introduction to Borrowing or Regrouping

Alternate mental subtraction method

Level 4 Subtraction

Why borrowing works

When we take away all the units, which are there, we are left with nothing. Thus, subtraction provides us with the idea of “nothing” as a “number.” We call it zero.

When we need to take away more than what is there, we are faced with a definite shortage. Thus, subtraction also provides us with a measure of “shortage.” We call it a negative number.

Calculators may be used to save time with subtraction; but still mental awareness is necessary to know if the calculated answer is correct. This awareness comes from the practice with mental math as outlined in the document above.

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MILESTONE 2: Addition

One may easily learn to add numbers mentally up to one hundred using the procedures given here.

MATH MILESTONE #A2: ADDITION

Addition is an extension of counting. For example, by counting you know that you have 3 marbles. By counting again you know that you got 4 more marbles from your friend. You put all these marbles in a box and count them together. You find that you now have 7 marbles.

Addition is counting together.

In the initial stages one learns to add small numbers by counting on one’s fingers. For larger numbers, one may use abacus. The first concept that one learns here is “carry-over” across the place values. After gaining sufficient familiarity with objects, one may add on paper using digits.

It is easy to visualize addition with objects. We may also visualize addition using a number line. Basic addition is demonstrated in the following video from Khan Academy.

Basic Addition

When we have larger numbers we may add them more easily by column. See the demonstration in the following videos from Khan Academy

Addition 2

Level 2 Addition

Addition 3

Addition 4

Here are some comments on addition.

(1) Addition depends on the concept of the same units. For example, we may add 2 cats and 3 cats to come up with the sum of 5 cats. However, we cannot express the sum of 2 cats and 3 dogs in terms of a single unit, unless we change the unit to “animal”, which is inclusive of both cats and dogs.

(2) Mathematical units may be the same, but units in the real world are never exactly the same. For example, any two oranges would never be exactly be the same in all aspects including the number of atoms they contain.

Thus, mathematics may come very close to describing the physical universe, but it is never exact.

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Future of Education: Khan Academy

Go to KHAN ACADEMY FOR THE WORLD

It is a lot of fun to learn maths here; and it is absolutely free. 🙂

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The Tathagata

The Tathagata
Beyond all coming and going: the Tathagata

Here is an excellent Bachelor’s Thesis
Thinking in Buddhism: Nagarjuna’s Middle Way

From Wikipedia’s article Tathagata:

Thus in this interpretation Tathāgata means literally either, “The one who has gone to suchness” or, “The one who has arrived at suchness”.

The article a little further says,

The aggregates of form, feeling, perception, mental formations, and cognizance that comprise personal identity have been seen to be dukkha (a burden), and an enlightened individual is one with “burden dropped”. The Buddha explains “that for which a monk has a latent tendency, by that is he reckoned, what he does not have a latent tendency for, by that is he not reckoned.

From what happens to the Sensory input, the mind seems to be structured as follows,

1. Perception
2. Experience
3. Information
4. Hypothesis
5. Theory
6. Principles
7. Axioms
8. Self

From this perspective, SELF appears to be

(a) the sum total of what the sensory input has ultimately reduced to

(b) the core of the mind

(c) the repository of all latent tendencies.

In my view the above model supports Buddha’s insight. The self is introduced (added) through sensory input. If the SELF is fixed then it acts as the ultimate inconsistency.

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Inconsistency

There is no absolute truth or untruth. Truth is found only to be relative. Please see the essay The Nature of Truth.

What appears to be true simply dissolves into a sense of harmony. This harmony forms the very background of what one is. There is nothing specific about this background to be asserted. This background may only highlight things that are not in harmony.

Thus, as far as specifics are concerned, there are only things that are not in harmony with this background. We may call them inconsistencies. An inconsistency may manifest itself as uncertainty, incoherence, confusion, contradiction, dissonance, illogic, discord, conflict, unwanted condition, etc.

It is only when all inconsistencies are resolved that one may go back to one’s natural equilibrium. Only then the ultimate truth, if there is one, may be grasped.

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THEORY

Sensory input, as it is processed, flows down as follows:

  1. Sensation
  2. Perception
  3. Experience
  4. Information
  5. Hypothesis
  6. Theory
  7. Principles
  8. Axioms
  9. Self

Each succeeding layer represents greater integration of input. The degree of integration depends upon the harmony among the elements. A greater integration and harmony means increased rationality.

A greater harmony brings about increased rationality.

An inconsistency may enter at any point when this processing is interrupted or corrupted.  The inconsistencies just float around since they are not integrated. They influence the thinking in unpredictable ways, making it irrational.

Inconsistencies introduce irrationality into thinking.

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Thinking & Memory

Thinking involves retrieval of required memory from the mind. The data exists in the mind in the form of the layers as described above. There is no ready made memory. At the time of retrieval, memory is created newly from the indexing of data among these layers. 

It would be difficult to re-create memory when the data is not properly indexed. Lack of indexing occurs due to a lack of integration of input, which, in turn, is caused by the presence of inconsistencies.

The problem with some memory is not that it cannot be accessed. The problem boils down to the presence of unresolved  inconsistencies in the mind. If a memory in certain area is “difficult to access” then the approach should be to identify and resolve all inconsistencies in that area.

Underlying any difficulty in accessing memory there are inconsistencies that needs to be identified and resolved.

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Thinking & Visualization

Thinking also involves the visualization of new possibilities and solutions. These visualizations then feed back as input into the layers above. When these visualizations are in harmony with the processing of the sensory input they speed up the integration greatly. On the other hand, when these visualizations are not in harmony, they may contribute to inconsistencies themselves.

For example, a belief may come about in an effort to restrain some confusion. This belief may then appear as an inconsistency at the level of hypothesis when not in harmony with the sensory input.

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Nature of Inconsistency

Inconsistencies on various levels may be named as follows:

  1. Perception …..(Engram)
  2. Experience …..(Unwanted feelings & emotions)
  3. Information …..(Indoctrination)
  4. Hypothesis …..(Beliefs)
  5. Theory ……….(Doctrines)
  6. Principles …….(Fixed ideas)
  7. Axioms ……….(Fixed viewpoints)
  8. Self …………..(Fixed identity)

The inconsistency may be defined as the variance between the way something appears and the harmony that should be there. Therefore, the method for dissolving any inconsistency would be to look at the area of inconsistency non-judgmentally and without resistance as described in KHTK 1 & 2. This will help restore the harmony.

APPLICATION

In the absence of a guide these exercises may be done by oneself. The guide may help the student as follows.

(A) Go over the theory section with the student.

  1. Answer any questions as best as you can.
  2. Discuss the theory materials until no questions remain.
  3. Make sure the student understand the main points highlighted in bold in theory section.

(B) Have the student do the exercises in sequence.

  1. Guide the student through the exercises.
  2. Maintain an open and friendly communication about student’s experience on the exercise.

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EXERCISE

THE PURPOSE OF THESE EXERCISES IS TO PRACTICE SPOTTING THE INCONSISTENCIES AT VARIOUS LEVELS. 

Preliminary steps:

  1. Find a place where you can sit comfortably for a while without being disturbed or distracted.

  2. Simply observe with your eyes open. If you find your mind adding to what is plainly visible, simply notice that fact and continue.

  3. Once you can comfortably notice what is there, you may close your eyes.

 

Step 1

Pay close attention to what comes up in the mind. Continue until some confusion or thought appears, which does not seem to make sense. 

 

Step 2

Observe the layer on which that confusion or thought seems to belong to. Then approximate the possible nature of the underlying inconsistency from the list below.

  1. Engram = Inconsistency on the layer of perception
  2. Unwanted feelings & emotions = Inconsistency on the layer of experience
  3. Indoctrination = Inconsistency on the layer of information
  4. Beliefs = Inconsistency on the layer of hypothesis
  5. Doctrines = Inconsistency on the layer of theory
  6. Fixed ideas = Inconsistency on the layer of principles
  7. Fixed viewpoints = Inconsistency on the layer of axioms
  8. Fixed identity = Inconsistency on the layer of self

 

Step 3

Examine that inconsistency more closely by applying KHTK principles.

  1. Simply perceive what is there without adding anything to it.

  2. Experience any emotions as they come up without putting any resistance.

  3. Acknowledge any ideas, thoughts, and considerations that come up without being judgmental.

  4. Continue until your attention starts to move by itself to another subject.

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Glossary

Inconsistency

  1. The inconsistency may be defined as the variance between the way something appears and the harmony that should be there.
  2. There are things that are not in harmony with their background. We may call them inconsistencies. An inconsistency may manifest itself as uncertainty, incoherence, confusion, contradiction, dissonance, illogic, discord, conflict, unwanted condition, etc. It is only when all inconsistencies are resolved that one may go back to one’s natural equilibrium.
  3. An inconsistency is arbitrary to some degree. Yet it is accepted because it seems to explain some confusion. Examples of this would be unreal beliefs that people carry around, and strange rituals that they engage in.
  4. Inconsistency exists where consistency is missing and contradictions exist. For example, Bill claims to be a successful businessman, but he is often filing for bankruptcies.

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