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From Concrete to Abstract
There are several ways of presenting the place value idea from the concrete to abstract. We presented 2 most concrete methods using bundles & sticks and sheets, strips & pieces.
Let us now see other methods which can help children to proceed from concrete to the abstract representations.
Idea of Bundle & Stick Banks
Initially children should practice counting ten sticks and then grouping them into a bundle. Similarly they need to experience breaking up a bundle to count ten sticks.
Once they are familiar with the idea of bundles & sticks, they can stop making & breaking up bundles. Instead they can use the idea of a bank of "bundles" & a bank of "sticks".
The bundles can be prepared in advance and kept in a "bundle bank".
Students can deposit a bundle into a "bundle bank" and take out ten sticks from the "stick bank". Conversely, they can deposit ten sticks into the "sticks" bank and take a bundle from the "bundle" bank.
From Bundle of Ten to Just a Bundle
As the children get familiar with this idea they can move to another level of abstraction.
The bundles (having actually ten sticks) can be replaced with a bundle (with any number of sticks) which is just identified by a rubber band around a bundle, which may contain any number of sticks.
Proportional Shapes to Just Shapes
The shift from bundles & sticks to sheets, strips & pieces is itself a step towards abstraction.
First, we can start with a strip which is 10 times a piece and a sheet which is 10 times a strip in size.
They can even be cross hatched so as to bring this relation out. Firstly the hatching can be dropped.
Later the idea of proportion itself can be abandoned.
The Hundred and One are represented by a large square and a small square. The ten can be represented by a rectangle.
Now a 3 digit number can be represented by a large squares & small squares and long rectangles!
Shapes & bundles to Coloured Tokens
At a still later stage, the sheets, strips and pieces as well as the bundles can replaced by tokens which can either be coloured differently or have labels H, T & O written on them.
At this stage the manipulatives may resemble the currency notes of different denomination that we use.
The Manipulatives are Scaled as per the Number Base
The most important concept to understand is the relation between the various materials.
They are all in the ratio of the "base" of the number system which we use. The number system that we use in daily life has a Base of Ten.
This means one Flat is equivalent to ten Longs and one Long is equivalent to ten Pieces. Looked at in reverse, a Piece is one tenth of a Long and a Long is one tenth of the Flat.
For representing 4-digit numbers, we can think of a Cube (to represent a thousand) which is equivalent to ten Flats.
In any system using a Base other than ten, we just have to replace the ten in the above sentences with the base of that system
Manipulatives for representing a number in the Base 8 or Octal System will be scaled by 8!
Any Manipulative Can Represent One
The other important concept is that 1 can be represented by any of the manipulatives, as per our need.
The other manipulatives will automatically acquire the "relative" values.
Hence if we have an arrangement of 1 Flat, 2 Longs & 3 Pieces -
It will represent 123 (One Hundred & Twenty Three) if we take the Piece as representing 1.
It will represent 1230 (One thousand Two Hundred Thirty) is we take the Piece as representing 10.
It will represent 12.3 (Twelve point Three) is we take the Long as 1.
Place Value System is an Algebraic Concept
If the base of a number system is x, then we can represent an arrangement of 1 Flat, 2 Longs & 3 Pieces as x^2 + 2x + 3, which is an algebraic expression of 2nd degree.
Hence a clear understanding of the place value system for representing numbers, helps in understanding algebraic expressions.
Abacus
Here all tokens are of the same shape & size. Their value depends of the location of the spike into which they are inserted. The spikes represent various place values and the number of tokens on the spikes represents the numerals which make up the number.
Numerals
From the abacus to numeral representation is a very simple one. The numerals represent the number of tokens at each place value.
Words
The word representation of numbers combines the difficulties of both math & language. We will take this up in a subsequent Section (14) on Math & Language
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