Early Numeracy and Arithmetic - MCI (London)

How do you prepare the child for early numeracy in a Montessori prepared environment?

Numbers are abstract. They do not exist in physical form.  They symbolize the quantity, size, weight, distance, speed, temperature, clock time and calendar.  We think of numbers when we want to know about mathematical properties of objects.  The language we have developed for basic mathematical queries is how many, how much, how big, how deep, how hot, how fast, how far, what time, at what time, how early, etc.

Numbers are represented by numerals.  Numerals could be words, letters, symbols, figures or anything that expresses numbers.  For example, one, two (in words), 3, 100 (in digits), iii, xv (in Roman), 1/2, 3/4 (in fractions), 3x, 5a (in Algebra), 3+6 = 9, an abacus, tally marks, ten fingers, ten toes, even the round or oval shape for zero, which is an Arabic invention to stand for 'nothing' and is not a natural number, are all numerals.

Arithmetic is a branch of mathematics which provides acquiescence to basic mathematical skills, such as, counting, addition, subtraction, multiplication, division, fraction, estimation and patterns.  It streamlines a mind for dexterity in further branches of mathematics in higher levels, like geometry, Algebra, trigonometry, mensuration, calculus, statistic, etc.

Arithmetic could be as simple as counting the objects from 1 to 5 and say how many they are.  It could be a complex activity like comparing among objects, such as, more than, fewer than, how deep, how high, how hot or how cold, etc.  It can also be a complicated transaction, such as, to calculate the volume of water in a tank or to change a fraction into a decimal figure or express as a percentage.

 Aims and Objectives

Aim: Introducing early numeracy to children in a convenient orderly manner through Montessori material during the three-year Montessori period.

Objectives:
The objectives should be to help the child,
1- understand one on one correspondence between numbers/numerals and objects (preparing mind for problem solving skills)
Knobbed Cylinders, Cards and Counters, Large Number Rods, Spindle Box, etc.

2 - learn the language pertaining to Math, for example, more, less, heavy, light, right, left, deep, long, tall, short, large, small, add, take away, half, quarter, measure, etc. (language skills)
      Pink Tower, Broad Stairs, Short Bead Stairs, Seguin Board, etc.

3 - do counting (development from concrete to abstract)
     Short Bead Stairs, Seguin Board, Large Number Cards, etc.

4 - identify geometrical shapes and learn their dimensions (preparation for geometry)
      Geometric Cabinet, Constructive Triangles, Geometric Solids, etc.

5 - distinguish the objects by their properties (preparing mind for problem solving skills)
     The entire Sensorial material

6 - learn the concept of more and less (preparation for addition and subtraction)
      Golden Bead Material, material in the area of Activities of Everyday Living

7 - do matching and grouping (problem solving skills)
      Puzzles, Geometric Cabinet, Binomial Cubes, Trinomial Cubes, etc.

8 - learn cardinal (quantity), ordinal (position) and nominal (number tags/codes) numbers/numerals
   
9 - identify numerals (visualization)
      Sandpaper numerals, Spindle Box, Large Number Cards, etc.

10 - draw basic strokes for writing numerals (develop writing skills)
        Painting, Scribbling on writing board, tracing around geometric trays, Sandpaper Numerals, etc.

Aim: Pre-requisite for the above mentioned aim of this area
         Preparing child for early numeracy and arithmetic

Objectives:
To obtain the above mentioned objectives, it is necessary to...
1 - to develop child's interest for counting orally through poems and nursery rhymes
2 - to develop the basic principles of conservation, reversibility and one-on-one correspondence
3 - to initiate the process of counting from body parts
4 - to develop and refine fine motor skills as a requisite for pre-writing faculty
5 - to develop the recognition of numerals through finger tracing and scribbling on a slate

Objective # 2

Conservation: It is an ability to conserve actions, order of actions, mathematical properties and concepts - the two equals remain equal unless a change occurs in their physical or spiritual manner.  The child constantly absorbs millions of impressions since his/her birth and moderately discovers the relation, pattern and order among the objects in his/her surroundings.

Reversibility: It is an ability to understand that the possibility remains for an action to be undone or re-done.

Jean Piaget developed the theory of cognitive learning and set conservation and reversibility as it's two principles as a sign of normal intellectual progress of a child.

For example, an infant conserves the idea of flipping on his/her shoulder to reach an object, later on, he/she figures out the reverse action of rolling back.  Similarly, in his/her infancy, the child opens his/her mouth for the spoon and sucks and swallows the food, later, he/she learns to spit out the food after having enough meal.  So, in simple words, flipping and sucking food is conservation while rolling back and spitting out is reversibility.  

One-on-One Correspondence: This is an ability of pairing objects to their corresponding numbers and/or for completing a set.

In a Montessori environment, for example, a child removes the cylinders which is conservation, he/she puts them back in the holes which is reversibility while trying to put them back in their respective holes (through control of error) is one-on-one correspondence.  

Numeracy in Pakistan

Pakistani culture is comparatively more privileged with cultivating numerical skills in children's absorbing mind.  The calendar is used frequently for cultural purposes than just for learning months and days.

Both, Gregorian and Lunar (Hijri) calendars are the part of people's daily life in Pakistan.  Children's date of birth is marked and remembered in both calendars.  Since their birth, children get to hear the conversation concerning the lunar dates for celebrating their birthdays, national days and Islamic festivals; 1st of Muharram (the beginning of Islamic year), 10th of Muharram (martyrdom of the grandson of the Prophet Muhammad (pbuh), 12th of Rabi-ul-Awwal (birth of Prophet Muhammad pbuh), 27th of Rajab (Prophet Muhammad pbuh's ascention to the heavens), 14th of Sha'aban (the Night of Destiny), 1st of Ramadan (beginning of the fasting month), 21st/23rd/25th/27th/29th of Ramadan (the Night of Power), 1st of Shawwal (Eid-ul-Fitr), 9th of Dhul-Hijjah (Hajj/pilgrimage to Makkah) and 10th of Dhul-Hijjah (Eid-ul-Adha).  On Gregorian calendar, 23rd March, 14th August, 9th November, 25th December are the most significant national celebrations.

On the Independence Day, 14th of August, children draw or make national flags, focusing on it's rectangular shape, the crescent and a five-cornered star in the middle and it's green and white colour.  The national flower of Pakistan, jasmine, has five petals.  There are four main provinces of Pakistan.  The five main rivers of Pakistan merge at a point to flow into River Indus.

Pakistani wedding ceremonies provide plenty of opportunities for children to learn numerical skills.  children love to decorate, count and hold the clay bowls of different sizes full of 'mehndi' for pre-nuptial ceremonies.  Girls love counting their bangles and make sure there are dozen in each hand and they match with their dresses which are in green, yellow and orange combination.

Children hear 'Adhan' (call for prayer from mosques) five times a day and gradually learn the times of the day and clock timings through adhan.  During the month of Ramadan, 29 or 30 days of fasting, children hear the conversation regarding 'sehri' (waking up and eating before dawn) and 'iftar' (breaking fast at sunset) and the countdown for fasting days.  On Eid day, they count 'eidi' (money given to them by elders) and plan how many things they can buy with this amount.  Most children are aware of the significance of Friday because of the Friday congregational prayer and they know it is offered at Zuhr (noon).

Satisfying Parents' Concerns

While preparing the curriculum for numeracy, the milieu in reference to children's social and cultural background cannot be disregarded.  Generally, in Pakistan, parents from all ethnic backgrounds and sects expect and admire the result-oriented system of education.  They do appreciate the inclusion of new techniques and modern approach in teaching but do not withdraw from their demand of traditional mode of repercussion.  As far as the freedom of choice and freedom of movement is concerned, parents will acquiesce with these terms but with ambiguities and lack of trust unless the daily progress of their child accords their anticipation.  

Vertical grouping is also unacceptable for parents as they don't allow their child to sit with the children or older or younger age and learning from each other.  Considering this a waste of time and money, they expect teachers to be there to educate and train their child.   

Trying not to be heedless of parents' expectations and avoiding raising contentions regarding the advantages and disadvantages of Montessori system of education should be top priority of an educationist in Pakistan.  I affirm that parents wouldn't mind at all the shelves full of Montessori material and will only value it as an additional  source to the traditional methodology.

During the last two decades, the early childhood period in Pakistan has started to be from age 2.5 till 4.5.  A 5 year old must be ready for 1st Grade, otherwise, the child is considered to be left behind.  The child has to pass an entry test for an admission in 1st Grade.  So, by the age of 5, the child must know to read and write Urdu and English alphabets properly on and between lines, reading and writing counting 1 to 100, single digit addition, table of 2 and to write sentences in Urdu and English.  The child must recognize all the colours and most shapes, should know how to spell the name of birds, farm animals, wild animals, colours, shapes, etc.  This is the least to be mentioned.  This is the ground reality of the system here in Pakistan and at least I wouldn't consider myself 'proficient' if I fail to present an environment fulfilling the requirements from both sides - the parents of children and the Montessori philosophy.

The pre-school educational system should make parents realize why they want to put their child into a Montessori environment.  They should be explained in brief the importance of three-year Montessori period and of child working on his/her own instincts for a while before jumping to the conclusions.  Parents need to forebear with child's moods and should institute a learning environment at home for a stead progress.  They need to stand parallel to teachers/facilitators for building a concrete foundation for their child's learning.  

Planning

"It is certain that mathematics organizes the abstract path of the mind, so we must offer it at an early age, in a clear and very accessible manner, as a stimulus to the child whose mind is yet to be organized."

                                                                           Maria Montessori - The Discovery of the Child

Montessori classroom provides a comprehensive learning environment which is composed of five basic areas namely; 1) Activities of Everyday Living  2) Sensorial  3) Early Literacy  4) Early Numeracy and Arithmetic and 5) Knowledge and Understanding of the World.  The entire Montessori environment is actually lubricated with mathematical concepts as all theses areas incorporate to impart the knowledge of basic mathematical features in some way.  The mathematical characteristics are present in a Montessori environment concurrently.  The flash cards in the language area are rectangular and their corners are curved, not pointy.  The globe in the cultural area is sphere in shape.  The pouring and transferring exercises develop equilibrium sense for measurement.  Thus, the teacher/facilitator must be mindful of highlighting mathematical characteristics in each of these areas while preparing the classroom and the lesson plan as well.  An integrated curriculum would serve the best to develop connection among all five areas in a Montessori environment.

The area of 'Activities of Everyday Living' provides numerous activities to develop mathematical skills indirectly.  During presentations, this area can be used to encourage counting objects, such as, large beads and sorting them by colour before threading.  The child can be encouraged to count buttons, hooks and bows on dressing frames along with emphasizing on the sequence of work from top to bottom  and one-on-one correspondence.  The pouring and transferring exercises help the child visualizing the amount poured or transferred  into containers and the child develops the concept of 'conservation and reversibility'.

The sensorial activities are designed to focus on sharpening five basic sense and six sub-senses and to refine the concepts of 'conservation and reversibility' and one-on-one correspondence in different ways.  This area mainly consists of the material based on geometrical shapes and colours, which helps refining stereognostic and other senses and numerating flat surfaces, edges and corners of solid geometrical shapes.

Solid shapes of sphere, oval and ovoid can be used to introduce the concept of 'zero' as they have no edges and corners.  'Zero' is usually written in round or oval.

The area of 'Early Literacy' is enormously enriched with numerical skills and is an effective source of building vocabulary pertaining to numeracy.  The words in stories present the concept of time and distance which helps the child to form images of the events taking place from the very beginning.  The story usually begins with,

'Once upon a time, in a far away land.....'
'Long time ago.....'
'It was calm and cloudy.....'
'It was a stormy night....'

Children hear the words that show quantity, such as,
'Oncce there was a king who had four sons.....'
'Once there were two boys who were very good friends...'
'Once there was a little girl who had seven dolls....'
'The old man had not eaten anything for three days....'
'The young prince enjoyed the pleasant sight of hundreds of colourful flowers....'

Children take interest in counting body parts of animals; the cat has four legs, two ears and a tail, birds have two wings, snakes have no legs and arms.

Children don't just listen to the story but they do concentrate on the number of people, animals or objects.  Try telling a story of two kittens, mentioning the activities of one kitten for couple of minutes and the child will inquire about what happened to the other one.

Lubricating Little Minds for Numeracy

Montessori environment is not a place for imposing lessons.  The child either learns by presentations offered by the facilitator or whenever he/she is fascinated with the material.  That is why the area of Early Numeracy and Arithmetic in a Montessori environment should be well-equipped with the material sufficient for developing mathematical agility.  The facilitator must be attentive to serve the child in either case.

Beside the preparation of spider chart and the weekly lesson plan, I personally would prefer to maintain a record of the material that the child has been through and of the ones which he/she needs to be attracted to.  As we know that each material is designed with a specific purpose, the child's avoidance to any of the material can't be neglected.  A good facilitator must investigate the reasons of that avoidance and make sure that the child must somehow be introduced to all materials during his/her Montessori period.

To implement lesson plans according to the sequence of the material for Numeracy and Arithmetic and it's presentation as it is being taught during Montessori training is hypothetical.  We know that children are unpredictable and they don't always end up according to our expectations.  Therefore, a multiple planning with appropriate techniques and precise vocabulary and language must be contemplated.  Besides waiting for children to get attracted to the material or offering presentations on sandpaper numerals, number rods, spindle box, cards and counters, short bead stairs, Seguin Board and Golden Beads material - to initiate the process of counting and developing interest in numeracy, I would rather moisten their minds with some old-fashioned techniques.

For example, singing nursery rhymes and poems is the most entertaining part of the curriculum.  This is one activity which substantiate child's vocal, physical, educational, psychological and cognitive development at the same time.  Some of the rhymes are a reinforcement for counting straight and backwards.

'One two, buckle my shoe...'

'This old man, he plays one, he plays one...'

'One two three four five, once I caught a fish alive...'

'Five little ducks went out one day...'

'Five little monkeys were jumping on the bed...'

'the ants go marching one by one hoorah, hoorah.....'

Counting body parts has also proved to be one of the best techniques for learning counting.  Most children love counting ten fingers and ten toes and then count them together from 1 to 20.

One of the practice that I learned at home in my childhood and have tried on many children is to count on finger cubes.  It is very simple and very effective too.  The child learns to count in order, starting from right hand pinky towards thumb from 1 to 15 (every time starting from the top cube) and then from left hand pinky from 16 to 30.  It builds a memory of correspondence between numbers and finger cubes.  The child in few weeks identifies the cubes by their numbers and can do addition and subtraction later on.

Counting steps on a marked distance is also fun.  Children count their steps moving forward on the marks and them coming back either straight or backward.

Another old time approach for learning numbers and writing is to let the child scribble on a slate.  The child learns to draw strokes through scribbling and finally starts forming shapes and numerals.

Children in my time were more interested in collecting their favourite objects as their precious possessions, such as, dolls, coins, rocks and pebbles of specific shapes and colours, seashells, pieces of broken colourful bangles, their art work, etc.

The question 'how to prepare a child for numeracy' requires an explanation on a facilitator's understanding of the area of early numeracy and arithmetic, his/her preparation for numeracy and arithmetic for early stages and planning and methodology for developing the child's interest for learning mathematical faculty.  It is understood that once the child's interest is developed, nothing can resist his/her curiosity from putting hands on material and absorb the concepts behind each activity.

http://www.amontessorimusingplace.org/2010/10/childs-mathematical-mind-part-1.html
http://www.infomontessori.com/mathematics/introduction.htm
http://montessoricompass.com/mathematics
http://tothelesson.blogspot.com/2011/04/mathematical-mind.html
http://www.studymode.com/essays/Montessori-1226629.html
http://www.montessorianswers.com/myth-fantasy.html
http://www.simplypsychology.org/piaget.html