# Group 5: Towards Abstraction

Passage to Abstraction – (Group V)

Since beginning Mathematics as a sensorial exploration of Multiples of Variables and Groups, the child has handled successively more abstract representations of quantity using the Golden Beads, Stamps and Dots.  She has applied the operations to these representations of quantities, manipulating them to perform dynamic shifts from one category to another, applying the laws of the decimal system.  After sufficient repetition using different material to carry out these operations the child internalises the process in her mind.  To support her handling of numbers she has been asked to use the operations to memorise the combinations of basic addition, subtraction and multiplication with the Snake Games, Strip Boards and Charts.  Thorough knowledge of the multiplication combinations allows her to complete the blank Division Chart.  The child has also been completing ‘Word Problems’ where she solves basic maths problems mentally, deciding which operation she must use and holding the numbers in her mind and manipulating them in abstract.  Along with other number games, the ‘Memory Game of Numbers’ and the ‘Zero Activity’ expand her ability to hold numbers in her memory as she analyses them.

The materials of Group V, especially the Bead Frames and the Division Racks and Tubes encourage the child to put together her abilities to perform the operations and her knowledge of combinations.  She is introduced to numbers up to one million, supplied with concrete representations which are quickly be represented symbolically on the large bead frame. These materials further limit the child’s handling of the materials, they are now beads on wires or in tubes, their categories differentiated by position and colour not size or number, so they are further abstracted. She uses special notation paper to represent the numbers in near abstract and performs the work with multiplication mentally based on her knowledge of the combinations before representing them symbolically, revealing an independence from the material. As the child gains confidence with these materials they act as a ‘bridge’ to pure abstraction where the child realises she can perform the operations without any material support.  Now she is ready to continue to solve more complex ‘Word Problems’ with figures up to one million in the abstract.

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Note:

Until now the child has had little practice writing large numbers, to prepare to handle large numbers and carry out abstract operations the following material is given.  Here the child works with numbers she has been exposed to in the Dot Game, the procedure is similar to that of the Stamp Game

Material Description:

Two Chowkis beside each other

• A frame with four horizontal wires, each with ten beads: 10 green for units, 10 blue for tens, 10 red for hundreds and 10 green for thousands.  On the left side of the frame categories are marked.
• Notation paper with vertical lines in the colours of the categories.
• Pencil, eraser and ruler, squared paper for the introduction, a coloured pencil may be used to underline each completed answer.

Presentation:

Introduction – Frame only

• Bring the Small Bead Frame and Golden Bead material to the chowkis and link them by asking the child to identify a unit in the Golden Bead material and saying, “This is also one unit”, pointing to the green beads of single units of the Small Bead Frame.  Do the same for the other categories.
• Move a few Beads from one category to the far side of the frame, ask the child to identify the amount, then ask her to form a given number from any single category.  Give sufficient practice using multiple categories, then begin recording the numbers on paper.

Introduction – Frame and Notation Paper

• Count the number of beads on each wire, starting with units; 1,2,3…10.  Having reached ten show that 10 units are equivalent to 1 ten by sliding a blue bead to the right and sliding all the green beads back to the left.  Continue counting; 20, 30, 40…100 and push a red bead to the right and all the blue beads back to the left.  Continue, finally marking 1,000 with the first green bead.
• Relate the columns on the notation paper to the frame, counting as above while marking the figure in single digits on the corresponding coloured lines of the notation paper.  The child’s attention is thus drawn the hierarchical value associated with the wires of the frame and the lines on the paper.
• At the child’s request fill in the paper as above using ‘0’ to hold the place value, for the tens beads asking, “How many units are there” and marking a ‘0’ on the green unit line.  Continue letting the child fill the numbers for 200 to 900

Formation of Numbers

Note:  No writing is done for the first one or two examples, for your example only use 1 thousand bead.

• The Directress composes a number with more than one category of the frame and shows the child how to write it on one horizontal line of the notation paper by, asking; “How many thousand beads are there?” and writing the digit on the left green line. Proceed in the same way for the other categories in descending order.
• Now leave the child to form and write numbers of her own choice.  After sometime reverse the procedure, writing the symbols, then forming the quantities.
• Ask the child how to make ‘One hundred and twenty’ and then ‘twelve tens’, asking her to read back the number shown.  This draws her attention to the different ways to show and describe the same thing; give many examples, e.g. ‘fifteen hundreds’, ‘twenty five units’ and so on.
• Later form numbers with ‘0’ as the place holder

Operations:

• The adult dictates two addends written on the notation paper, e.g. 2156 + 1243.  The child is asked to form the first addend on the frame
• Either record the answer at each stage or when the sum is found

• The adult dictates two addends written on the notation paper, e.g. 2483 + 6238.  The child is asked to form the first addend on the frame
• Beginning with the unit ask her to add the digits of the second addend, when no beads remain continuously chant the figure reached in the second addend while taking a bead from the next higher category and sliding back the ten used units, continue counting out loud immediately while sliding across one bead at a time until the second addend is reached.
• Continue till the addition is complete, either recording the answer at each stage or when the sum is found
• When the child has understood invite her to write her own addends, there can be more than two.
• Later ask her to use addends which involve many changes such as ‘999+1=’, to show the effect of a single unit
• Show her that when handling many addends it is easier to work with all of the digits of that category first and record the answer.  As the child’s capacity to abstract builds she will begin to do this mentally.

Subtraction: Static

• The adult dictates two numbers written on the notation paper, e.g. 8396 – 2313.  The child is asked to form the minuend on the frame
• Beginning with the unit ask her to subtract the digits of the subtrahend, counting out loud the figure being taken
• Either record the answer at each stage or when the difference is found

Subtraction: Dynamic

• The adult dictates two numbers written on the notation paper, e.g.7493 -2548.  The child is asked to form the minuend on the frame
• Beginning with the unit ask her to subtract the digits of the subtrahend, when no beads remain continuously chant the figure reached in the subtrahend while removing a bead from the minuend’s next higher category and taking the ten used units, continue counting out loud immediately while sliding across one bead at a time until the subtrahend is reached.
• Continue till the subtraction is complete, either recording the answer at each stage or when the difference is found

Multiplication: Static

Note: The child must know many multiplications to do this

• The adult dictates two multiplicands written on the notation paper, e.g. 321 x 3.  Beginning with the unit, verbalise; ‘one taken three times is?’, the child provides the answer and the Director represents the product on the frame.
• Continue for the tens, hundreds and thousands
• Record the answer at each stage or when the product is found

Multiplication: Dynamic

• The adult dictates two multiplicands written on the notation paper, e.g. 1412 x 6.  Beginning with the unit, verbalise; ‘two taken six times is?’, the child provides the answer and the Director represents the product on the frame.
• Represent the dynamic product with beads of two categories
• Continue till the multiplication is complete, recording the answer when the product is found

Note: If the child has difficulties show the child how to analyse the multiplicand on the left side of the paper on the respective lines

1412 x 6 =                2

10            x 6

400

1000

Criteria of Perfection (Control of Error):

• The child may do the operations of addition and subtraction in reverse.

Direct Aim:

• The material shows clearly the steps for addition and subtraction with large numbers done on paper.  The child has the opportunity to apply all that she has learnt previously and therefore the Bead Frame acts as a bridge to work without concrete materials in the abstract.
• The child sees again the laws of the decimal system, where;
• Ten of one category can be exchanged for a unit of the next higher one
• Each category can hold no more than nine units
• The value of the numbers is determined by the place they hold
• Zero serves as a placeholder

Age at Presentation:

Five and a half to six years

Wooden Hierarchical Material

Material Description:

Seven wooden solids representing the following hierarchal values:

1: a green cube of 0.5 cm3

10: a blue prism 0.5 cm x 0.5 cm x 5 cm, divided into equal parts by green lines, so that each section represents the preceding value of one unit.

100: a red prism 0.5 cm x 5 cm x 5 cm, divided into equal parts by blue lines, so that each section represents the preceding value of ten units.

1,000: a green cube of 5 cm3, divided into equal parts by red lines, so that each section represents the preceding value of one hundred units.

10,000: a blue prism 5 cm x 5 cm x 50 cm, divided into equal parts by green lines, so that each section represents the preceding value of one thousand units.

100,000: a red prism 5 cm x 50 cm x 50 cm, divided into equal parts by blue lines,

so that each section represents the preceding value of ten thousand units.

1,000, 000: a green cube of 50 cm3, divided into equal parts by red lines,

so that each section represents the preceding value of one hundreds of thousand units.

Seven white cards, each with one of the following numbers written in black: 1; 10;

100; 1,000; 10,000; 100,000; 1,000,000

A large Floor Mat

Notes:

The Wooden Hierarchal Material provides a sensorial introduction to numbers beyond the thousands.  The coloured lines are an added sensorial element which link one category to another.  The Geometric shape of the material clarifies the fact that the three elements of each family (simple, thousand and million) share the same name despite their increase in dimension and value.

The child may have encountered an operation which resulted in 10,000 in her own workings with the decimal system, stamp game or bead frame.  She will have seen the symbol for 10,000 in the presentation for the Dot Game

Presentation:

• Unroll the mat, bring the Wooden Hierarchal Material and Golden Bead Material
• Link the Golden Bead Material by showing a Golden Bead unit and, indicating to the simple unit of the Wooden Hierarchal Material, saying, “This is also a unit of one”.
• Ask the child to name the 10,100 and 1,000 of the Wooden Hierarchal Material
• Replace the Golden Bead Material

Introducing the Wooden Hierarchal Material

• Ask the child to select the simple unit and the unit of ten from the Wooden Hierarchal Material and ask her to count the cubes of simple units in the prism of ten, then, having identified ten ask her to count the units of ten in the prism of one hundred, then show one unit of a thousand and ask her to identify the hundreds therein.
• Bring forward the other three pieces of the Wooden Hierarchal Material, asking her to count their values and give their name
• Give a Three Period Lesson with the three new pieces, involving the others during the second and third periods, finally ordering them and counting them in sequence forwards and backwards

Grouping the Wooden Hierarchal Material

• Later group the pieces of the Wooden Hierarchal Materials according to their shapes and colours.
• Ask the child to name each piece within the group, without giving further details simply allow a sensorial impression to form

Symbols – Cards only

• Bring the cards to a chowki or mat, isolate those of the 1, 10,100 and 1,000 and ask the child to name them
• Bring forward the other symbols of the other three numbers, asking her to count their zeros and give their name in a Three Period Lesson, which emphases the zeros

Symbols and Quantities

• Place the Wooden Hierarchal Material at random on a mat.
• Show the child the cards and ask the child to read them and match them to their corresponding quantities

Exercise:

• Arrange the Wooden Hierarchal Material in their families, tell the child, “There are three groups 1; 10 and 100 is the first, 1,000; 10,000 and 100,000 and the second and the third is 1,000,000.
• Tell the child that each group is called a ‘family’, 1; 10 and 100 are called ‘simple’, 1,000; 10,000 and 100,000 are called ‘thousand’ and  1,000,000 is called a million
• Ask her to match each card with each pice of material, read the ‘simple’ family the ‘thousand’ family and the ‘million’ family card.  Point out that the number of groups of zero indicate the name of the family

Language:

• The names of the units beyond 1,000
• The names of the families of numbers

Criteria of Perfection (Control of Error):

• In the child’s ability

Direct Aim:

• To familiarise the child with categories beyond 1,000 and to give the child practice in reading and writing these numbers

Age at Presentation:

Five and a half to six years

Material Description:

• Two chowkis side by side
• Pencil, ruler, coloured pencil and eraser if necessary
• A Large Bead Frame, similar to that of the Small Bed Frame, but with seven horizontal wires, representing seven categories up to units of one million
• Notation paper, similar to that of the Small Bead Frame, but with vertical lines capable of representing all the categories
• Wooden Hierarchical Material for the Introduction

Presentation:

Introductions and Formation of Numbers

• The same procedure as for that of the Small Bead Frame is followed, using the Wooden Hierarchal Material rather than the Golden Beads material.
• The Child will realise that the frame and paper go up to one million

Operations

• The same procedure as for that of the Small Bead Frame is followed.

Note: The child works independently with the material, moving into abstraction

Criteria of Perfection (Control of Error):

• The child may do the operations of addition and subtraction in reverse.

Direct Aim:

• The material shows clearly the steps for addition and subtraction with large numbers done on paper.  The child has the opportunity to apply all that she has learnt previously and therefore the Bead Frame acts as a bridge to work without concrete materials in the abstract.
• The child sees again the laws of the decimal system, where;
• Ten of one category can be exchanged for a unit of the next higher one
• Each category can hold no more than nine units
• The value of the numbers is determined by the place they hold
• Zero serves as a placeholder
• The child has the opportunity to work with very large numbers to perform addition, subtraction and multiplication

Age at Presentation:

Five and a half to six years

Division with Racks and Tubes

Material Description:

A tray with 7 racks of test tubes: 3 white for the simple family, 3 grey for the thousand family and 1 black for the millions family

There are ten beads in each of the ten test tubes in each rack

Each rack has beads: green for the units, blue for the tens, red for the hundreds

Seven cups: painted on the outside to represent the family and on the inside to represent the category

Skittles: 9 green, 9 blue, 9 red

3 division boards with strips corresponding to either the units (green) tens (blue) or hundreds (red)

Presentation:

First Stage – Expressing the quotient on paper without the remainders

Set out a chowki and writing materials and unroll a large mat

• Set out the racks and bowls in order of category and family
• Put out the unit division board and place four green skittles to represent the divisor and the appropriate colour beads in the bowls to represent the dividend, keeping these vertically to the right of the board and put away the unused material
• Write the dividend and dividend on paper, e.g. 9764 – 4 =
• Put the rack and bowl of 1,000 horizontally above the board and start to divide the dividend amongst the skittles (each would receive 2 with a remainder of 1 left in the bowl)
• Record the figure 2 on the paper and clear the board
• The hundred rack and bowl is next placed above the board next to that of the thousands
• The remaining thousand bead in the bowl is exchanged for ten hundred beads ( there are now 17 beads in the hundreds bowl)
• Put the thousand rack and bowl back on the tray and divide the 17 red beads amongst the skittles (each receive 4, 1 remains)
• Record the figure 4 on the paper and clear the board
• Then continue as above until the whole operation is complete

Note: After some practice divide numbers such as  ‘9216 – 3 = ‘, where there will be insufficient beads for a change to take place, resulting in a need for a zero to hold the place value for the hundreds.  To handle this;

• Distribute 9,000 amongst the skittles, giving ‘3,000’, record ‘3’
• The ‘200’ red beads cannot be equally shared amongst 3 so they are placed into the blue bowl, ask the child, “How many beads can be equally shared amongst the skittles?”, as her answer is ‘zero’ record this
• Place the 2 red beads into the thens bowl and change them for 20 blue beads, making a total of 21 blue beads, these can be shared, giving 7 each, record this, and later the 6 units give 2 each, the quotient is 3072.

Second Stage – Moving towards Abstraction (Expressing the quotient and the remainders)

Set out a chowki and writing materials and unroll a large mat

• Set out the racks and bowls in order of category and family
• Put out the unit division board and place four green skittles to represent the divisor and the appropriate colour beads in the bowls to represent the dividend, keeping these vertically to the right of the board and put away the unused material
• Write the dividend and dividend on paper, e.g. 9764 – 4 =
• Put the rack and bowl of 1,000 horizontally above the board and start to divide the dividend amongst the skittles (each would receive 2 with a remainder of 1 left in the bowl)
• Ask the child, ‘How many beads does each skittle receive?” and record the figure ‘2’ on the paper
• Then ask the child, “How many beads remain?”, record the remaining thousand by writing ‘1’ in by the ‘7’ in the divisor in the hundreds column
• Clear the board
• The hundred rack and bowl is next placed above the board next to that of the thousands
• The remaining thousand bead in the bowl is exchanged for ten hundred beads (there are now 17 beads in the hundreds bowl)
• Put the thousand rack and bowl back on the tray and divide the 17 red beads amongst the skittles (each receive 4, 1 remains)
• Ask the child, ‘How many beads does each skittle receive?” and record the figure ‘4’ on the paper
• Then ask the child, “How many beads remain?”, record the remaining thousand by writing ‘2’ in by the ‘6’ in the divisor in the tens column
• Clear the board
• Then continue as above until the whole operation is complete;
• Dividing 26 tens beads by four skittles will give a quotient of ‘6’ and a remainder of ‘2’, t be recorded by the ‘4’ of the dividends unit column
• Dividing the 24 unit beads amongst the four skittles will give a quotient of 6
• The overall quotient recorded will be 2466

Criteria of Perfection (Control of Error):

• The child may check her own work by doing the reverse of the process

Direct Aim:

• To do short division with long numbers
• To lead the child towards abstract division

Age at Presentation:

Around six years