The Advancement of Mathematical Talent in Kindergarten

 

Ever since that day our clock has not been resting in its place; measurements had to be taken, alarms went off, more and more children hooked on to the topic.

At the Gym

When all children had learned to handle the clock and write down times, they decided to take the clock to the gym.

As ‘time detectives’ they measured time, wrote it down, gave instructions and all the while got an idea how difficult it is to keep a grip on the group.

For the gymnastic game we distributed animal symbols: snail, hedgehog, hare. There were little cards with animal symbols on them that were distributed randomly among the children later on.

Mathematical contents:

• Measuring time
• Writing down time
• Comparing results
• Different speeds: slow, moderate, fast
• Matching the speeds with the animals
• Combining speeds and distances
• Deliberately moving at specific speeds
• Different distances: near and far
• Comparing the measured times

I would like to use this activity as an example by which to demonstrate to you how I went about laying mathematical groundwork with the children. As you are aware children need sensory experience in order to later on be able to perform abstract operations.

Even before they can count they are able to experience amounts in the sense of magnitudes by their own actions. They notice when they make more steps or that they are running faster. They feel their breath, feel that their legs getting tired. They experience the change in place when moving, they see the distance they have covered. At first they move spontaneously and at random, but with time their moves become more targeted and controlled by their will.

At the same time children grow aware that the world, too, moves around them: everybody else, animals, vehicles. Here too, children recognize different speeds and they experience them in connection with different distances covered.

That is how they know that a snail is very slow and needs a lot of time to cover a distance. The hedgehog does much better and the hare is the fastest.

By imitating the animals the children learn to change their speed intentionally. They also experience how as a snail it takes them forever to make it to the other side of the gym.

The gifted children have long been able to recognize speeds and match them with the right animals. Their task is much more abstract and complex, and allows for no more making use of their own physical sensations; they have to rely solely on their experience and knowledge. In this game they are to pick one child from the crowd, watch it for a while and match it with the animal symbol. They can even count the many steps, measure the time, write it down and compare it.

Sundial

Well, the story of time wasn‘t all told, it seemed, it unraveled and revealed some more. The question, how we know what time it is, was quickly answered – we see it on the clock. But how does the clock know? Or: what happens if there is no clock? This was getting tricky. I got some books about time and about clocks. From these books the children learned how time was measured in former times. The sundial mystified them.

It was surprisingly simple to build a sundial, all you needed was a stick. Somehow the children also knew that the sun was involved, because the stick’s shadow served as the ‘clock’s hand’. On a sunny morning they went outside into the yard and stuck a stick between the stone slabs. The sundial was operational at once – just, what time was it showing?

Mathematical contents:

• Realisation of the changes in the position and the length of the shadow
• Relation: sun + stick = shadow
• No sun: no shadow
• Motion of the sun – time passing
• Thinking ahead
• Drawing conclusions
• Writing down time

In this project all children were able to understand practically a mathematical relation:
sun + stick in the ground = shadow, and they could experience what happens if one part of the equation is missing.

They understood that the moving of the shadow meant a change in time.

The gifted children understood the way the sundial worked so well that they were able to estimate the position of the shadow, even when the sun was not shining or where it would be at later point in time. They also recognized the down sides of measuring time this way.

That’s how they came to understand that a sundial is not a reliable way of telling the time.
Therefore we decided to use modern clocks to measure time.

Engineering a clock

How does a clock work?

That is a question some children will raise. When taking an old alarm clock apart, all kinds of axles and wheels emerge, when you shake the clock they move. But they are much too delicate for little children’s hands.

From a mail-order catalogue I ordered an alarm clock kit. The parts were made of plastic, big enough, and I was confident this would be great fun for my “tinkerer”.

One afternoon the time had come to put it together. He unpacked all parts and checked if the kit was complete. He examined each part and then unfolded the manual. He proceeded quite systematically and with great patience he put the parts together, step by step. Then he closed the case, wound the clock and waited. The clock worked for a few seconds then it stood still. So he took it apart and reassembled it, meticulously following the manual. With great confidence he wound the clock again, but with the same result. He took it home to ask his grandfather for help.

However, what was positive even about this failed attempt was: the boy was able to conceptualize exactly how the parts fit together, how the wheels and pinions would mesh, how they would drive the hands and what the pendulum was good for. He showed great patience, wanted to solve the problem himself, checked his procedure several times and knew where he could get help.

What is the mathematics in assembling a clock? That’s what some might ask. The numbers on the face, the gear teeth on the wheels, different speeds of the wheels, these are mathematical contents of this activity.

Further mathematical contents are:

• Matching parts and sketches in the manual
• Comparing the number of teeth with the size if the wheels
• Understanding a manual and the necessity of doing things in the right order
• Different rotational speeds of the different size wheels

Gifted children are able to understand the manual, to follow its instructions and come up with the right results. They are not discouraged by set-backs, but try to find the causes for failures and try again. They understand the interplay of the gear wheels and explain the way a clock works to other children.

Art-Project

One day I brought my collection of clocks to kindergarten. I spread them on a blanket and we listened to the Pink Floyd CD with the song “Time” on it. On the wall we had the famous Salvador Dali painting “The Persistence of Memory” with the dangling watches.

I gave the children some time to look around and appreciate everything around them. They held the clocks to their ears, turned the little wheels.

This is some of the „philosophical discourse“ with the children:

„What do the clocks feel like?“

“Solid, cold, hard.”

“Does time feel hard, too?”

“Time isn’t hard, it’s soft.”

“Can time be held in the hand?”

“No …”, everybody was laughing, one boy made circular motions and said “Time always goes like this.”

“True, there was a painter who thought the same way, that time couldn’t be held in the hand because it always moves on and he painted it in his picture.”

So the children looked at the picture.

What on the picture is hard?”

“The mountains, the table, the tree…, no, the tree isn’t hard, it looks dead, but if the clock goes on ticking, spring will come and the tree lives again.”

“And what on the picture is soft?“

“The watches, they’re showing the time, and it is not dead, it’s always moving on.”

The children had really understood the nature of time, they now also distinguished between time and the instrument for measuring it, the clock; a difference that at the onset of the project was not all that clear.

We decided to paint our own copies of the picture by Salvador Dali.

This project was open to all children of the group. Everybody contributed their skills and knowledge. For some of the children it was quite an achievement to read the time on the clock and understand its use to measure time. Others were able to align the numbers in the order of their value. Along the way they realized that the hands on the clock keep covering the same ‘route’ over and over again, that after two rounds a new day begins and that this never ends because time itself keeps moving on and doesn’t end either. Just like a circle doesn’t have a beginning or an end.

The children all experienced several different sensory inputs. They felt how hard the surface of a clock was and how soft the putty, they listened to music, looked at the painting by Salvador Dali, mixed up the colours when painting themselves, made use of different materials for their handicraft works.

 

The gifted children recognized the meaning of time through their sensory experiences – its soft and ever moving nature. They realized that something in that motion is repetitive, yet at the same time changes, that the tree will have leaves again and again when spring comes around, but that after many years that tree will die. Just like it is with people.

What feels hard, though, doesn’t change, isn’t alive, like the clocks and the mountains. Their time can become “infinite”.

Mathematical contents:

• Perception of the transitoriness of time
• Perception of the changes rendered in time
• Rate of change: fast, slow, „no“ change
• Matching speeds and objects: the clock ticks fast, the tree changes in a yearly cycle, the mountains don’t seem to change at all
• The circle as an appropriate geometric form to represent time – the hands moving in a circle, at different paces
• Configuration of a clock
• Numbers
• Meaning of numbers on a clock face
• Crafting / shaping numbers
• Correct order of numbers on the clock face

The finished painting was exhibited, displayed in a special place in the room and explained to the parents by the artists.

 

Brushing Teeth

In this project we were dealing with algorithms. The term algorithm is pivotal in mathematics. See also: Basic Ideas of Mathematics . Its meaning is: a sequence of instructions (steps) which define a procedure, beginning with the input of certain data and always rendering a definite result. Any purposeful activity can be broken down into its algorithmic steps. For example: brushing teeth.

On day I asked some of the children: „What are all the things that need to be done to clean one’s teeth?” I got all kinds of answers, quite a jumble.

“How are you going to explain it to somebody, who has never brushed his teeth before?”

So we imagined we were in the bathroom wanting to brush our teeth.

I starts with getting the mug. Then fill it with water and rinse the mouth. Next thing open the toothpaste, put some on the brush, brush the teeth: from all sides, back and forth, circling motions, spitting and rinsing the mouth again. Clean the mug and the brush, put them back, dry the mouth, done.

Mathematical contents:

• Algorithms
• Recognizing single steps in a complex procedure
• Correct order of steps leads to success
• Every single step is necessary in order to succeed

Algorithms regulate the proceedings and lead us safely to the end if we take all steps in the correct order. Mathematical operations are algorithms, too. One of the most elementary is: how to write a number; later: how to add or perform other operations, how to figure out an unknown value or how to determine the equation of a function.

The gifted children had great fun determining different algorithms. We called them recipes. For the “brushing teeth recipe” we created a sequence of photographs. We glued the pictures on a strap and hung it up on the wall in the bathroom.

Growing Beans

Around the turn of the year the children sing a lot of songs, and one is especially popular – it is called “The Yearly Clock”, with all the months in their correct order, and then it all starts all over again!

This is how the children understand that time can be measured, for example, human life or the life of plants. Human life can be rather long, even the children know that, but the life of a bean plant can easily be measured.

So we started a gardening project. The children exposed the bean seeds to different conditions:

• In the water / without water
• In an open bowl / in the dark in a jar
• In the soil / without soil
• In an open container / in a closed container

The beans were being observed from day to day, one child documented the changes in a table, and at the end the beans were photographed with a number to show how many days the experiment was going on. This is how the children saw life develop in time, and waste away again.

When the beans were reaped, the children made a diagram showing how many “baby beans” one “mother bean” had. For this every child was examining one plant. Then we dared to predict how many “grandchildren beans” we would get next year.

But we didn’t wait that long, we made a picture book about a dwarf, who wanted to invite his friends for a bean stew in autumn. We did some cooking ourselves, too, that’s how the beans ended up in our stomachs.

In this project the children did not only see the growth of beans but also growing numbers.

In their daily documentations they laid the number corresponding to the day of the experiment next to the documented objects. It started with “1” and the next day it was the “2” and the “3” on the third day and so on until the day of the harvest.

It was really exciting when the number went beyond ten – how do you write “11”? Then the next obstacle. How do you write “21”? „1“ and then „2“ or the other way around? Some children knew exactly how to write the big numbers, the others learned from them and had their own suggestions to make. They understood that the number is written “backwards”.

[Note from the translator: In Germany, two-digit numbers are written just as anywhere else: from left to right with the tens in the first position and the units in the next position to the right. However, they are spoken backwards, so that 21 is spoken one-and-twenty (einundzwanzig) in German. Since the children learn to speak the numbers before they write them, they think they speak them in the normal order, consequently the way the numbers are written will then appear “backwards” to them.]

And there was another kind of growing numbers the children got to see just after the harvest. From one bean came 15 “baby beans”. If we planted them next year we would get 15 times 15 “grandchildren beans”, and if we did the same thing the following year we would have 15 times 15 times 15 “great-grandchildren beans”. The gifted children tried to figure that number. I suggested they make a drawing of it. That was a great many beans. This way one could acumulate riches over the years. But who wants to eat beans all the time?

What do you do with all those beans? You could trade them for something else, for example for potatoes or carrots. You could sell the beans and buy some clothes.

This is how the gifted children understood how the economy works. Same thing with the crop: the farmer can sell the crop he doesn’t need for himself and buy other things for the money.

Mathematical contents:

• Perception of change in time / growth / death
• Writing numbers in ascending order
• Change of amounts
• Adding up (bean-children)
• Powers – and the phenomenon of „multiplication“
• Exchange / trade / price
• Prediction of future results based on experience

Time-Roll

The many activities and talks about time also raised the question how long time goes back, when did it begin? Jan had the answer: with the big bang. B illions of years ago. And how long ago was this?

In order to demonstrate this to the children I used a roll of wallpaper about 20 meters long and rolled up.

In the middle of it there wasn’t anything on the paper, just like at the beginning of the universe. The solid core was dark, it was the big bang. Then I coloured the upper end of the paper roll starting with red in the middle, going to orange, yellow and finally green. Our space expert got the idea and explained to the other children: At the beginning there was nothingness, then there was an explosion, the stars and the planets formed, and everything was still very hot, it cooled down and when the earth turned green it was because of the plants and the animals. And, of course, not the kinds we see today, but dinosaurs. But it was a long time ago and they are extinct now.

The children knew this already.

So the other end of the roll is today. When you unravel the roll you go backwards in time until you get to dinosaur-time. That’s what we did, some children drew a few dinos, others looked through books on the history of earth. A few days later there were quite e few dinos inhabiting the time-roll.

In this project it was the gifted children who explained time to the others. It was incredible how much the children knew about this. Especially Jan understood how matter came to life. He knew that it had taken billions of years until earth had cooled down so that the first plants and animals could populate it. He was able to measure time another way, namely by the most important occurrences.

This is how he aroused everybody else‘s interest in the history of earth.

He sought for validation of his knowledge and I reassured him. He brought books for the group, in which one could view the emergence of life. Only shortly before the end the children discovered the first human beings.

In those times earth looked different. There were no houses, no streets nor cars and the people were wearing clothes made out of fur. They learned to light a fire, ate cooked meat for the first time. How many generations have lived until people lived the way we do today?

All of a sudden the children realized that there are very big numbers, they couldn’t name them, but by examining history they got an idea of their order of magnitude. They know that time moves on, nobody knows how long this will go on, so time is endless, growing with every second, minute, with every year.

Time moves only in one direction, but it is possible to think the past – one can’t hunt dinosaurs, but one can draw them on the time-roll.

Time and time again the children went to get the time-roll and made drawings on it. If they wanted to draw dinosaurs, they had to unroll it quite far. For the first locomotive not so far, even less for cars. One of the last inventions of mankind, the mobile phone, would have to be drawn at the open end of the roll.

Mathematical contents:

• Endlessness of time
• Past, present, future
• Moving in time: forwards and backwards
• Time units: second, minute, hour, day, month, year, decade, era
• Rate of change – ever faster
• Number of changes – ever more
• A feel for the relativity of the length of a human life

This project lasted many months.

It took its time to understand time. The only way was, of course, experiencing a chain of many events.

In order to understand the passing of time one can document the events and remember them later when looking at them.

Any topic is fit to show the different kinds of talents the children have and to promote them. Any topic allows for a number of different activities, which address several domains of child development.

Addressing several senses at the same time strengthens focus, elongates the time span of concentration and creates a special atmosphere that promotes motivation.

By embracing the individual talents and strengths the individual’s sense of affiliation with the group is intensified. The group profits from the talents of the individual and is stimulated in its development.

• Mathematical advancement is present in all areas of the work with children at kindergarten.
  
• Activities do not have to emphasize formal aspects as taught in school.
  
• It starts with the most simple functions like sorting, matching pairs and may be expanded indefinitely. The skills and interests of the children will determine how far they want to take it (understanding amounts, reading numbers, writing numbers, doing arithmetic).
  
• Mathematical advancement will be especially successful if all senses are addressed, including motion.
  

There is no necessity for a special training in mathematics in order to advance gifted pre-school children; however, important are:

• Attentiveness,
• a sense of rhythm and order,
• enjoying doing arithmetic,
• playful approach to numbers and mathematical operations,
• creativity,
• courage to do new things,
• exchanging ideas with the colleagues,
• looking at the available games from a mathematical point of view,
• positive reinforcement of the children’s knowledge and interests,
• working in small groups of children with the same interests and skills.

It is all about developing an understanding of numbers, amounts, operations, effects and phenomena.

We must not forget that the children have gathered mathematical experience long before they enter kindergarten.

It is not a new world which the children are introduced to, therefore they should be accompanied by us with ease and without fear of one’s own incompetence. It is quite obvious that kindergarten is not a place of tedious drill in “writing numbers”, that school’s business.

Date of publication in German: May 5th, 2007
Translated by Arno Zucknick
Copyright © Klaudia Kruszynski, see Imprint.