Arithmetic Explained

Here are some things you may want to know about arithmetic and Math Path's approach to tutoring it:

• Arithmetic uses numbers.
• Each number occupies an invisible place.
• Each place has an invisible multiplier.
• The invisible places and invisible multipliers form an invisible structure.
• We call this invisible structure Place Value.


• Place Value allows us to use shorthand notation, where we can express 300 + 30 + 3 + 3/10 + 3/100 as 333.33.


• It is this shorthand notation that makes arithmetic (as we know it) possible.
• Math Path is successful because it makes the invisible structure of place value visible. By doing so, it eliminates having to introduce the concept of place value to small children. They will work with place value. They will swim in it. But they won't have to think about it any more than a fish devotes thought to its environment. "Only one number to a square" rules this newly visible environment.

• It's a simple but extremely effective concept, and it saves a world of confusion for tutor and child. It's a shame that hardly any educators take advantage of it.

There are three major components to arithmetic operations: Arabic Numerals, Place Value, and Algorithms.

• You can read about this in much greater detail by clicking on "Why Your Child Can't Understand Math".

Arabic Numerals (0  1  2  3  4  5  6  7  8  9)

• We call them "Arabic" because they arrived in Europe via the Muslim (Arab) Empire. But the credit belongs to Hindus from the Indian sub-continent. The correct term is Hindu Arabic numerals.
• Briefly, Arabic numerals enable us to express intended quantities (from 0 to 9) by writing a single symbol instead of several. The quantity three, for example, was expressed in Roman numerals as "III." That required three individual symbols. It's necessary to "add" the three symbols to arrive at the intended quantity. Arabic numerals express the same quantity with one symbol, "3." In other words, the Hindus devised a system of numerals in which the addition already has been done. It seems a simple thing, but it changed the world.


Place Value
• Numbering systems need operating systems. The Romans used counting boards.The Hindus used Place Value and algorithms (see below) to manipulate their "Arabic" numerals. Place value is a system in which the actual value (denomination) assigned to a digit within a number is determined by an invisible multiplier assigned to the place the digit occupies. Place value is what makes arithmetic possible.



1. The first place invisible multiplier is '1.'  Thus we have a ones column.
2. The second place invisible multiplier is '10.' Thus we have a tens column.
3. The third place invisible multiplier is '100.'  Thus we have a hundreds column.
4. The fourth place invisible multiplier is '1000.'  Thus we have a thousands column.

• What we have done above is illustrate the process of using invisible multipliers in base 10, place value arithmetic.


• Another thing we have done is make visible the invisible structure that contains the individual places.


• That's the Math Path secret to helping little children maintain comfort while working with numbers and places and invisible multipliers. We make the invisible structures containing all that information visible to them.


• And that's the Math Path secret to helping inexperienced parents and tutors stay in their own comfort zone while working with children and arithmetic.


• It may seem simple, and it is, but it works, and small children are quite comfortable with the format.


• When the time comes to abandon this format, the change will be quick and simple.


Algorithms
• Credit the Hindus once again. Algorithms are sets of instructions that tell us how to solve certain categories of arithmetic problems. For example, "Add the numbers in the first column" is part of the algorithm for solving addition problems.


• Using algorithms (and handling such items as 'carryover' and 'borrowing') becomes much easier when the invisible structure of 'place value' is made visible.