Grass Roots Learning Revolution

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I have a feeling that the stage is being set up for a grassroots revolution in the field of education. Bill Gates is backing Khan Academy to develop internet based course content that is relatively easy to grasp. Ivy League institutions like Harvard, M.I.T., Yale and Stanford are making their course materials available online.

However, a lot more needs to be done beyond just providing the course content to get a revolution going in education. There are already reports like The Online Education Revolution Drifts Off Course. We need more than just an education revolution. We need a learning revolution at the grass roots level.

The following factors are required for the success of such a revolution.

  1. Generating a passion for learning in the young and old alike.

  2. Rapidly addressing past failures in one’s education.

  3. Providing a practical course in learning how to learn.

  4. Making course content available that is easy to assimilate.

  5. Setting up the student to start learning on his or her own.

It is the product in (5) that will ignite the learning revolution. By filling up the blanks in understanding we may very well handle the past failures in education and restore the passion for learning. Courses on learning how to learn may very simply be organized along the lines of Subject Clearing. Easily graspable course content is already being made available on Internet.

I envision setting up a Tutoring and Coaching Center where service is available to carry out the above functions. A person will be assisted on a one-on-one basis to clear up the obstacles encountered in learning. The person shall then be coached in the best practices of learning until he starts to learn on his own. After that he shall learn under supervision on a personal program that is customized for him.

I am close to retirement.  I have enough experience from running a Math Club, and otherwise, to set up such a center and run it full time. I shall be happy to provide such services free. I shall also be willing to train other volunteers on providing these services. I believe that such services should be provided free on a broad basis; only then we may be able to spark a learning revolution.

I would like to see this center having a place of its own where computers, or docking stations for computers, are provided. The expenses shall be minimal and it would be possible to run the center with community support. Once a passion for learning takes hold the progress will accelerate. Then only the sky will be the limit.

I welcome suggestions on this thread to make this project become a reality. I shall keep you informed of the progress.

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Going Beyond Counting

Going Beyond Counting

A counting number is based on the idea of a unit. A unit could be a goat, a house, a cookie or anything.

A UNIT is what we count one at a time. A COUNTING NUMBER is how many units we have counted.

The term “fraction” comes from the concept of a “broken piece.” A unit may be broken into pieces and these pieces may then be counted. For example, we may break a cookie into 4 equal parts. Each part is called a quarter. We may then count 3 of those parts as three-quarters of the cookie.

A FRACTION is part of a unit, which, in its turn, may act as a smaller unit. Thus, both numbers and fractions are based on the idea of counting.

The Brotherhood established by Pythagoras believed that by understanding the relationships between numbers they could uncover the spiritual secrets of the universe and bring themselves closer to the gods. Today that basic search continues in terms of finding that one equation that would explain all universal phenomena.

In particular the Brotherhood focused on the study of rational numbers as described in the essay Numbers & Consciousness. Rational numbers depend on the idea of ratio. A ratio tells you how many times a number is to another number in terms of the same unit. If Johnny is 10 years old and his father is 40 years old, then his father is four times as old as Johnny. The ratio of father’s age to Johnny’s age is 4 to 1. If Johnny’s mother is 30 years old, then his mother is three times as old as Johnny. The ratio of mother’s age to Johnny’s age is 3 to 1.

Since the mother is 3 times Johnny’s age, and the father is 4 time’s Johnny’s age, the ratio of his mother’s age to his father’s age may be expressed as 3 to 4 using Johnny’s age as the common measure or common “unit”.

To summarize, if a and b represent counting numbers then a/b represents a rational number. Here both a and b are multiples of some indivisible common unit.

A RATIONAL NUMBER is a number that can be expressed exactly by a ratio of two counting numbers based on some indivisible common unit.

It seemed at that point in time that rational numbers represented all possible numbers that could ever exist.  A unit could be broken into smaller and smaller units making it possible to represent any quantity as a ratio of its mutiples. Therefore, it came as a big surprise when numbers, such as √2, were discovered that could not be written down as a ratio based on some indivisible common unit. It meant that no small enough common unit could be found for such numbers. The idea of an ultimate indivisible unit came under intense doubt.

It was a discovery so illogical that it was rejected outright by Pythagoras. The following is the earliest proof available: (see Irrational number – Wikipedia, the free encyclopedia).

The first proof of the existence of irrational numbers is usually attributed to a Pythagorean (possibly Hippasus of Metapontum.) The Pythagorean method at that time would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into the hypotenuse just as well as into the arms of an isoscles right angle triangle.  However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction. He demonstrated clearly that there may exist numbers, such as √2, that cannot be expressed as ratios of two counting numbers. Hence, they are not based on any unit that can be counted. It is said that Pythagoras was so enraged that he ordered Hippasus to be drowned.

These are irrational numbers. They defy the sanctity of the idea of a permanent indivisible unit. If you attempt to express an irrational number as a decimal you end up with a number that continues forever with no regular or consistent pattern. There can be two rational numbers that are infinitesimally close to each other, and yet there can be infinity of  irrational numbers betweem them. There is no limit to how small the difference between two numbers can be.

There was no going around this new consciousness that could not be disproven even when Hippasus was drowned. Today, the most famous irrational number is π (pi), which represents the ratio of the circumference of a circle to its diameter.

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ALGEBRA – Common Graphs

ALGEBRA – Common Graphs

(Paul’s Online Math Notes)

Lines, Circles and Piecewise Functions

[Practice Problems] [Assignment Problems]

Parabolas

[Practice Problems] [Assignment Problems]

Ellipses

[Practice Problems] [Assignment Problems]

Hyperbolas

[Practice Problems] [Assignment Problems]

Miscellaneous Functions

[Practice Problems] [Assignment Problems]

Transformations

[Practice Problems] [Assignment Problems]

Symmetry

[Practice Problems] [Assignment Problems]

Rational Functions

[Practice Problems] [Assignment Problems]

ALGEBRA – Graphing and Functions

ALGEBRA – Graphing and Functions

(Paul’s Online Math Notes)

Graphing

[Practice Problems] [Assignment Problems]

Lines

[Practice Problems] [Assignment Problems]

Circles

[Practice Problems] [Assignment Problems]

The Definition of a Function

[Practice Problems] [Assignment Problems]

Graphing Functions

[Practice Problems] [Assignment Problems]

Combining functions

[Practice Problems] [Assignment Problems]

Inverse Functions

[Practice Problems] [Assignment Problems]

ALGEBRA – Solving Equations

ALGEBRA – Solving Equations

(Paul’s Online Math Notes)

Solutions and Solution Sets

[Practice Problems] [Assignment Problems]

Linear Equations

[Practice Problems] [Assignment Problems]

Applications of Linear Equations

[Practice Problems] [Assignment Problems]

Equations With More Than One Variable

[Practice Problems] [Assignment Problems]

Quadratic Equations, Part I

[Practice Problems] [Assignment Problems]

Quadratic Equations, Part II

[Practice Problems] [Assignment Problems]

Quadratic Equations : A Summary

[Practice Problems] [Assignment Problems]

Applications of Quadratic Equations

[Practice Problems] [Assignment Problems]

Equations Reducible to Quadratic Form

[Practice Problems] [Assignment Problems]

Equations with Radicals

[Practice Problems] [Assignment Problems]

Linear Inequalities

[Practice Problems] [Assignment Problems]

Polynomial Inequalities

[Practice Problems] [Assignment Problems]

Rational Inequalities

[Practice Problems] [Assignment Problems]

Absolute Value Equations

[Practice Problems] [Assignment Problems]

Absolute Value Inequalities

[Practice Problems] [Assignment Problems]