Numeracy has its foundations in our practical living world. The world of nature is a living example of mathematical principles and equations formulating growth and beauty. Our earliest experiences of Numeracy are perceived through sound and light frequencies; that high-light our awareness of:-
- structure and shape;
- repetition and pattern;
- rhythms of music, song and speech
Structure and shape mayinitially be experienced as sides on a 3D shape. The sphere has one surface and the spiral is one continuous line. A saucer shaped disc has two identical surfaces; a cone has two very different surface shapes.
One of the earliest formal associations with numeracy is presented as auditory counting. This is initially a rote recall of what we call number values, as they move up one unit at a time. The spoken words after twenty present a pattern of repeated sounds e.g. twenty one twenty two….. etc. until one gets to thirty one, thirty two, thirty three etc. The counting process embraces a sense of rhythm and repetition.
When counting, individual items can be categorised as a collective number value; such as six bananas and two oranges. Alternatively objects can be placed in patterns that visually illustrate a numerical pattern such as one white bead then one black bead then one white bead, and then another black etc. On the dice or dominoes the dots are placed in a specific visual pattern that is directly related to the number of dots. Thus, the simplest presentation of number value can be expressed as either ‘patterns of colour’ and ‘patterns of sound’ and ‘designed patterns of placement and structure’.
The first numerical concept of understanding is that of ‘conservation’. This means quite simple, that six oranges will remain six oranges until one or more of the six are eaten or given away.
One
Spoken prose and poetry can be coordinated with body movements that enhance a person’s feeling for the rhythm, rhyme and meaning of the words. Just as dance movements can enhance a person’s relationship with musical sounds, movement can also be choreographed to enhance the expression of feelings associated with spoken words. Early awareness of number value is associated with physical body awareness and movement related to objects. This aspect of rhythm can be illustrated through body movement. Hopping on one leg offers a whole body experience of the quantity of one unit. Similarly the following activity uses one single rod:-
The following is an example of spoken words coordinated with the movement of a single copper (or wooden) rod:-
The tide rises, the tide falls.
The wind swirls, the wind stills.
In daring adventures we rush forward.
Then gently roll home again and rest.
We balance our body and mind
And focus on a place of playful grace.
| For the 1st line the rod is thrown horizontally from both hands upwards and as it falls it is caught at about waist level.For the 2nd line the rod is twirled around in circles vertically in front of the body. In the 3rd line the rod is rolled on the arms from the shoulders to the hands, In the 4th forth line the rod is rolled from the hands back into bent elbows where it is held still for a few moments. In the 5th line the rod is balanced upright from the palm of the hand or fingers. In the 6th line the hand releases from the rod and catches it quickly before it falls from its upright position. |
Stamping alternate feet or jumping with two feet together offers a whole body experience of the quantity of two. Marching, Jumping, walking, running and skipping, or bouncing or catching a ball, are all activities that can be coordinated with verbal counting. When these activities are specified to a chosen number value e.g. walk seven steps, they become a whole body discipline related to counting and a rhythmic association with different number values.
Waltz music and dance movements present a whole body experience of the value of three. Traditionally a drum would be used alongside the counting and movement The one-two rhythm on a twin Bongo drum can be repeated twice to produce a four beat rhythm. Alternatively a shaker can be moved to create the one, two, three, four rhythm of body movement and sound.
- A skipping rhyme:-Kittens in the cupboard 1 2 3, 4 5 and 6
Watch them play their funny tricks
Kittens in the cupboard 1 2 3 4 5
What a the picture, what a surprise!
Kittens in the cupboard 1 2 3 4
One has fallen on the floor.
Kittens in the cupboard 1 2 3
You can see them just like me.
Kittens in the cupboard 1 and 2
Yes, but is it really true?
Kittens in the cupboard only 1
Within our number system the concept that adding one more unit makes the next named number is the first and most prominant learning. Young children who learn to count may not be able to say the value of a number if one more is added. When asked to follow a practical demonstration e.g. you have six buttons (presented on the table) and then I give you one more button (one button is added to the group of six) now how many buttons do you have?; young children usually have to recount the buttons to reach the answer seven. Similarly young children who can count confidently may not have an understanding of the conservation of number. This child has to recount the same objects when they are moved into a different pattern, or a different place or they just look different because something else has change. For this child adding one more is a new counting task rather than an auditory request for what number comes next. Older children who are working with number sums between 1 and 50 and can identify and count up to 100 and more, often cannot answer the questions of one more such as 44+1= ? or 89+1= ? If a person has secured this one more aspect of our numerical counting structure then any question even those with very high numbers such as – a million and four add one, and fifteen thousand six hundred and eighty three add one, will be answered immediately [providing they can correctly recall the original number that was said]. This is a good memory game to play and if participants are also asked to record the question as a sum, they can get good practice transcribing large numbers from their spoken form into their visual (written) numerical formate. This illusrates how this simple number, one, can be explored throughout a person’s life of learning about numbers.
The use of any form of counting frame is based on this concept of one more. The standard base ten abacus illustrates the more complex aspects of place value however they can be used successfully even without any numerical understanding if the process of adding one more is maintained accurately when moving the beads on the abacus.
The numerical information presented as linear counting activities addresses the identification of a numerical amount. Counting as such cannot be extended into operations of addition and subtraction. Linear counting activities encourage the child to believe that every mathematical operation is dependent upon an ever increasing ability to calculate using a one by one counting system and associated verbal recall of numerical counting.
Working with Basic Numeracy also involves an ability to perform a progression of logical thinking that orchestrates a notable path of activity. The aim to progress from (A) to arrive at (B) has to accommodate a predetermined structure of possibilities which are directly related to ones arrival at the desired destination (answer/conclusion). Traditional ‘maze’ puzzles are an excellent example of this type of application and completion of a logical task and they present valuable preparation for numerical thinking skills.
The abacus is designed to combine rhythms of movement with placement of beads to record number values. The abacus introduces the concept of ‘adding on’ or inversely ‘taking away. Multiplication is a repetition of a specified act of addition and division is a repetition of a specific act of subtraction.
Fortunately, our numeracy is based upon a base ten system of place value; and therefore it is not limited to how far one can verbally count or how well one can read numbers.
These rods develop the concept that all number operations relate to number bonds to ten: 1+9=10; 2+8=10; 3+7=10; 4+6=10; 5+5=10.
[Note: the full set of ‘Eastwood Number Rods’ are presented in Numeracy Folder5 Files 4th(5c) and 4th(5d)]
There are many different ways of presenting any one numerical definition. However, for example number bonds to ten represent the total value of ten i.e. ‘1 and 9’ and ‘2 and 8’ and ‘3 and 7’ and ‘4 and 6’ and ‘5 and 5’. Five plus five can also be written as 2×5. Thus numeracy presents many different way of presenting a specified process of calculation and its conclusion as a specific numerical value.
Acquiring a Working Knowledge of the Multiplication Tables
The young learner first learns to relate the concrete 3D presentation of number value with the related four operations i.e. add, subtract, multiplication and division. A working knowledge and automatic recall of Number Bonds to ten facilitate mathematical operations involving addition and subtraction. A working knowledge and automatic recall of the multiplication tables facilitate mathematical operations involving multiplication and division.
The earlier these skills are established the easier it is to embrace the practical application of further mathematical skills and related problem solving techniques. However, learning the multiplication tables remains an awesome task for many learners who are unable to secure the automatic accuracy required to aid every day and higher mathematical operations. The author has therefore given below some carefully described descriptions of some of the multi-sensory learning techniques she has invented alongside some of the more recognised supportive scaffolding.
Traditionally the multiplication tables were taught through the dominantly auditory mode of verbal repetition. However the ability to automatically chant a multiplication table, e.g. the two times tables, does not necessarily mean that the learner can automatically recall random elements such as 7×2 or 2×3. For some the answer to 7×2 can only be found by chanting the two times tables form the beginning through to the required seventh element. 1×2=2, 2×2=4, 3×2=6, 4×2=8, 5×2=10, 6×2=12, and 7×2=14! This extended process of recall is time consuming and likely to distract the learner from the mathematical understanding required for successful completion of the task in hand.
Steiner schools place a focus on rote learning of the multiplication tables once the children have entered main school class one at seven plus years. In Steiner schools repetitive rote learning is accompanied by rhythmic movements such as stamping or clapping, catching and throwing together with random participation. This encourages individual children to recall a specific answer without recalling the whole of the multiplication table from the beginning. This combination of repetitive chanting and random recall within a rhythmic structure of movement is considered more beneficial than standard rote learning through repetitive recall of the tables from one times through to ten or twelve times.
The importance of learning is superseded by the nature of enjoyable participation, the development of understanding and later application within a wide range of associated intellectual activity. Multisensory activities are designed to meet the widest possible range of learning and future application. Ideally they may need to present both auditory information and visual information within structural and rhythmic movements and actions with associated kinaesthetic sensory perception. The wide range of activities for learning multiplication tables presented in this section are designed to support enjoyable learning and thereby help every learner gain an instant recall of the multiplication tables and a wide spectrum of understanding. Later development of mathematical skills involves recording information related to multiplication and division as a working part of mathematical calculations when completing sums, measurements, mathematical investigation, problem solving formulas, algebra systems.
A Rhyming Multiplication game
Example:- Three times table (The words in the brackets are optional choices)
One, two, three, (once three is three or 1×3=3), go climb a tree!¬
Four, five, six, (two threes are six) pick up the sticks/now you’re in a fix
Seven, eight, nine, (three threes are nine) lets go out to dine/ the weather it is fine
Ten, eleven, twelve, (four threes are twelve) lets dig and delve.
Thirteen, fourteen, fifteen, (five threes are fifteen) now where have you been!?
Sixteen, seventeen, eighteen, (six threes are eighteen) and what have you seen?
Nineteen, twenty, twenty one, (seven threes are twenty one) did you see Mary’s son?
Twenty two, twenty three, twenty four, (eight threes are twenty four) someone is knocking at the door.
Twenty five, twenty six, twenty seven, (nine threes are twenty seven) is there time to go to heaven?
Twenty eight, twenty nine, thirty, (ten threes are thirty), certainly not you are much too dirty!
There are many creative rhyming answers e.g. either a single word (three/tree) – this could also be a nonsense rhyming word, a short phrase (What can you see?) or a story (It’s time for your tea.). The table rhymes can be created by an individual person or a group and recited either as a single piece of prose or spontaneously created by a group who take it in turns to add the next line when their turn comes around. The rhyming tables can be spoken and/or recorded in numerical form 3 x 3 = 9 (nine socks on the line) with drawn pictures to represent the rhyming word/phrase at the end of each line. For those who know their tables and can easily recall any element of each table, this activity presents a creative verbal challenge, for those that are not expert at their tables this activity also challenges the mind to address automatic recall of the mathematical tables from a new light-hearted perspective.