(Lecture at the 4 th IHVO-Symposium, May 5 th , 2007)
by Klaudia Kruszynski
This is an account of the project „Time“, which I conducted in the years 2004 and 2005. It was a project for the advancement of mathematical talent of gifted children.
I have been working at a Catholic kindergarten in Münsterland as a supplement kindergarten teacher since 1989, before that I worked at a Polish elementary school. I was well aware of the phenomenon of giftedness at that time and my approach to fostering the talents of gifted children was to instruct them individually, so that the weaker students as well as the stronger students would be taught at their own pace.
…in a nutshell…
A comprehensive and multifaceted project demonstrates that mathematics is omnipresent in the children’s world of experience. The mathematics inherent in the project are being explored, which shows how children capture mathematics holistically with all their senses. It is not about pure mathematical training, it is always an approach that addresses different areas of the children’s development simultaneously. Mathematics that come along not as an effort but as play and fun.
The activities to be described here were directed at the in-depth interests of gifted pre-school children.
In order to achieve this goal the following character traits were necessary:
- Competence and knowledge with regard to the subject.
- The ability to present this knowledge in an interesting and appealing manner.
- Equal concern about all students, especially the weaker ones and the gifted students.
- Creative and unconventional approaches to the subject matters.
These attributes turned out to be of great value in the effort of advancing gifted pre-school children as well. It was quite exciting for me, and honestly speaking, also rather strenuous to work with these children. Often they were quite annoying, wouldn’t abide by the rules, revolted during morning circles and had fits of fear or anger for no apparent reasons.
Many times I wondered what might be the cause for this conduct. I realised, that these children and their parents as well experienced little sympathy. All children were to be looked at alike and special needs or wishes were not to be granted. This attitude did not seem to be in line with my personal idea of treating the children as individuals. I wanted to help these children, and for that matter myself, too.
Four years ago I attended a symposium conducted by Hanna Vock and immediately signed in for a seminar at her institute. There I met colleagues who understood my concerns and encouraged me to keep lobbying for gifted children.
My Way of Working in Projects
I have learned to observe children closely to find out about their inclinations, abilities and knowledge. Based on these observations I would develop ideas for suitable activities. These were at first designed for children whom I suspected of having special talents. Close observation, conversation and questionnaires on individual interests were then be the basis for the activities. The starting point of such ventures would often be a question raised by the child, a statement given by the child, an observed behaviour or an incidence of some sort.
I learned rather soon that gifted children cannot be advanced separately. They have friends who want to join in, and, of course, other children, too, will show interest in special activities. Singular activities at first, soon developed into full-blown projects which were carried on over longer periods of time and were open to everybody who was interested in participating.
It was my aim to cover the entire spectrum of child development, starting with the just acquired ability to speak in whole sentences, to handle a pen correctly, to show their age by holding up fingers going on to such skills as reading, calculating with numbers exceeding 20 – including multiplication -, pondering life, death and the transitoriness of time …
At the same time the projects were to cover the broad range of childlike interests as they were dinosaurs, planets, automobiles all the way to numbers, books, mechanics and others.
These projects were practical assignments in the course of my further training.
I decided to look at the world from a mathematical point of view together with the children.
With this I meant to pick up on the interest of all children with regard to numbers and their skills in arithmetic. Simultaneously I intended to meet the needs of the gifted children. I meant to help them find answers to their further questions and pick up on their deeper interests.
The project as a whole comprised several subordinate projects which were worked on in different ways:
- activities for a single child,
- activities for a part of our group,
- activities for the whole group or,
- activities for members of different groups.
The project was designed to address a number of different areas of child development such as: cognition, fine and gross motor skills, speech, thinking. This meant: The range of activities covered the most rudimentary sensory experience as well as purposeful action and even philosophical reasoning.
At this point I would like to describe my way of going about advancing gifted children. These are the essential steps:
- Determine a possible talent, most of the time one may barely have a hunch that this child might be somewhat different from its peers in age.
- Spend some time observing – interests, knowledge, inclinations and stage of development.
- Contemplate suitable activities, which are not exclusively based on the foregoing observations but also on current events, festivities and other projects.
- Start the project while observing the children and reflecting the observations made. At this stage it is important to also watch the other children and find out whether the project has been arousing the attention of any of them, sometimes they have hidden talents.
- Enhancement – experiences made, knowledge accumulated and questions raised along the way find their way into the project, a constant updating of the project’s contents.
- Realisation of further activities, observation and reflexion.
This is how a project may stretch over months, being fed with ever new input, yet never losing its thread.
At this point I will only describe the project at large, because the focus here is on the mathematical contents of the project.
An in-depth discussion of these activities can be found in the article Project Time, right here in the manual. The article also informs you about the methodology I applied in the different activities.
The Project „Time“
In the years 2004 and 2005 I was so lucky as to have more than one talented child in my group. (I cannot speak of giftedness here, since the children had not been tested at the time.)
Among them were two boys: Tim, 5 years, and Jan, 4 years old. They got along great and a wonderful friendship developed between the two. In the course of the project another child joined in, which turned out to be rather talented, too.
Tim was interested especially in cars. He had a deck of cards with all types of cars of different brands and he knew all the manufacturers, the technical data, the top speeds and the prices.
This is how even so early, at the age of 4, he was not only able to read but also to compare numbers of 5 to 6 digits length.
His friend was a tinkerer. He constructed several objects that were supposed to be able to move. He was the most successful at this when working on grandpa’s computer at home. He still tried to do with what we had at kindergarten and built his spaceships, his yet unpractised fingers struggling quite a bit.
Tim and Jan always complained that they never had enough time to spend playing together, the best playing corners were always occupied, when they wanted to play there.
So we tried to find out how that could be. It was a matter of fact: they didn’t have enough time. So I got them a time measuring device, a clock. They usually spent most of their time at the breakfast table. Now every morning they would put the clock on the table before breakfast and then, when they were finished having breakfast, check the time and write it down. And they did so all by themselves, no need for further encouragement or reminding. It was not long before they noticed that these measured time spans differed quite a bit. They also made little drawings of what they had had for breakfast.
They set a time limit for themselves and tried to finish before the timer. That is what caught the other children’s attention. Now many of them wanted to measure the time they took for breakfast and enter the race against the timer. Everybody was given their own sheets of paper to make notes of the measured times and they kept track in writing. One day I brought paper straps with a grid of 270 squares in one row, representing the minutes of the entire morning until lunch. I asked them to colour in the number of squares they had measured and that’s how they saw how those, who had a shorter part of the strap coloured, were earlier off the breakfast table to make a dash for the desired playing corners.
The children used the clock for all kinds of other measurements, too.
How Long Is a Minute?
The first thing we measured was the duration of a song when sung fast or slow. All measurements were made in the unit of “minutes”.
But then, how long is a minute really?
The children were to keep their eyes closed for what they thought was a minute. After a few seconds some of them opened their eyes. Too early!
So they tried again. It wasn’t easy, we used the timer.
They watched the numbers change, getting smaller and smaller. Upon zero the alarm went off.
Then they went on to measure other activities, for example how long it takes to read a picture book. We set off the timer and … all of a sudden Tim started reading the book out loud. With some effort, spelling out the longer words to himself first, he struggled through the book. It took him about nine minutes.
We were speechless, none of us was aware, he could read. His younger friend said sadly, he really wished he too could read.
These activities comprised the following mathematical contents:
- Perceiving the duration of one minute
- Estimating the duration of one minute
- Grasping the notion of “amount” (here: of time) in numbers (here: of minutes)
- Conceiving of numbers as signifying position / order
- The growing of numbers
- Minutes as a unit for the measurement of time
- A minute as the equivalent of 60 seconds
- A minute as 1/60 of an hour
- Sensing rhythm and tempo
- Handling of a clock
The gifted children were pursuing the whole issue with much more zeal. They would ask questions, look for answers themselves, experiment and implement their newly acquired knowledge in their games. For example, when they were playing “spaceport” they would start their rockets with a countdown, certainly starting no lower than at 20.
Advancement of gifted children:
Understanding of a minute as a unit for the measurement of time
Comparing the analogue and the digital method of time measurement
Elaborating on the way a countdown works and using it
Estimating the duration of different activities
Performing different measurements and taking down the results
Seconds, hours and other units of time measurement
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.
- 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.
Advancement of gifted children:
Measuring time and writing it down
Coming up with ideas and suggestions
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?
- 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.
Advancement of gifted children:
Understanding the working principle of a sundial
Looking for a good place for the sundial
Writing down times
Reporting, telling the other children how the sundial works
Comparing the performance of a sundial with that of other kinds of clocks
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.
Advancement of gifted children:
Reading the manual
Organizing the work
Explaining the way it works
Understanding mechanics – why do the hands move?
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”.
- 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
- Meaning of numbers on a clock face
- Crafting / shaping numbers
- Correct order of numbers on the clock face
Advancement of gifted children:
Active participation in the discussion
Recognizing of states and processes
Developing own ideas
Trying to explain: What was the basic idea here?
Fine motor skills trained (putty)
Participation in creative activities
Explaining, reporting to an audience
The finished painting was exhibited, displayed in a special place in the room and explained to the parents by the artists.
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.
- Recognizing single steps in a complex procedure
- Correct order of steps leads to success
- Every single step is necessary in order to succeed
Advancement of gifted children:
Defining a sequence:
Cause -> status -> action -> effect
Defining the course of actions in a complex activity
Deliberations: „What happens if I skip one part in the sequence?“
Deliberations: „What happens if I change the order?“
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.
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.
- 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
Advancement of gifted children:
Documentation of the experiment
Completing observation forms
Responsibility for the progress of the experiment
Counting great numbers / amounts
Adding and subtracting and their notation
Coordinating activities in creating the picture book
Understanding food production and agriculture
Understanding trade as a form of exchange
Understanding commercial ties
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.
- 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
Advancement of gifted children:
Contributing knowledge of the phenomenon of time
Contributing knowledge of the universe
Contributing knowledge about the history of earth
Understanding of large numbers
Understanding changes in time
Understanding the irreversibility of time
Understanding the laws of nature
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:
- a sense of rhythm and order,
- enjoying doing arithmetic,
- playful approach to numbers and mathematical operations,
- 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.