Basic Ideas of Mathematics

by Hanna Vock

Rotational symmetry in all its marvel is visible in mandalas. Rotational symmetry is a generating principle. By painting mandalas the children intuitively understand the inherent rotational symmetries.

So, does that mean it is possible to identify mathematically talented children by their fondness of painting mandalas? Not at all – unless you catch the narrow time span between two points: The point in time, when the child is intellectually ready to grasp the principle at all, and the point in time, when its understanding of the matter reaches certainty and firmness. This time span will, as matter of course, be rather short if a child is talented or even gifted in the mathematical area.

So even though there is plenty mathematics in mandalas the simple colouring of a mandala is hardly ever satisfying for a gifted child. Beware, though, that oftentimes the principle of symmetry itself will be taken as much more interesting than the artistry in actually painting the mandala.

Children are discovering and seriously experimenting with symmetries when they suddenly begin building symmetric objects or when they string beads onto a thread. This is where mirror and rotational symmetry can be combined:

If a gifted child has created large symmetric buildings with coloured building bricks at the age of 2 – and has thereby worked out the problem of mirror symmetry – a pronounced interest in stringing beads is not likely to occur at the age of 4 or 5, if it does, it will be ephemeral, unless the thrill of it is in the shared activity and the chatting with other children who are stringing beads. But symmetry is not a new discovery then (consequently we cannot make an imagined ‘tick’ for ‘cognitive advancement’).

Children also experience mirror symmetry when perceiving their own bodies as consisting of two equal but mirror inverted parts. Many rhythmic moves and dances are symmetric. Symmetry is one major principle of construction in the world, especially in living beings, and hence is a basic idea of mathematics. And finally, it is also an aesthetic concept.

In physics as well symmetry is a basic principle. Physicists when discovering a particle or a force will immediately look out for the anti-particle or the counter-force, respectively.

One way of helping children appreciate symmetry is, for example, creating a symmetry book: taking photographs of symmetric things, then drawing the axes of symmetry onto the photographs and sticking them into a (paper-) notebook or onto a big poster. We play “symmetry detectives” with the children for a while.

There are many more possibilities …

There is a very nice piece of gaming material about this, see: ONLY A BOOK !!!

Interesting Games: „Spiegeln mit dem Spiegelbuch“ (Mirroring with the Mirror Book)

2) The Notion of Numbers and Their Positional Notation

No, being able to write numbers is not so important at first.

Even without this motor skill many children are fondling around with numbers in their minds, they meditate on numbers, sets and mathematical structures.

Recommended Reading: Mittring, Gert: What’s Happening Inside When We Do Arithmetic?

Writing numbers and the corresponding motor skills should not be an impediment to mathematical activities with the children. (We do not need letters for speaking either.)

If children want to write numbers, they will practice them tenaciously, as long as we provide the appropriate materials: a number panel, numbers made of cardboard, wood, foam rubber, a number puzzle, magnetic numbers, number stamps – which of these we offer does not make much of a difference, sometimes less can be more here. Younger children will however prefer numbers that are not only printed or drawn but numbers that come as little physical objects so they can experience them haptically.

Children acquire the notion of numbers early on and independently of the ability to read and write numbers, they understand that each number-word signifies a specific amount of things.

Beware: this is not the same as being able to say number-words in the correct order by heart. In order to really count one must have understood the correlation of a given number and the idea of a set of so many individual objects as signified by that number and this ability is eventually a rhythmic one (!). Very young children will oftentimes quickly lose their rhythm when counting objects, piece by piece, so that they lose track and the correct outcome becomes a matter of chance at best.

The notion of numbers also includes the ability to see beyond the ducks, teddy bears or candies and apprehend the quantity signified by the number. There are many pre-school exercise books on this topic … , however, many gifted children enter kindergarten with an operational understanding of numbers already. If they, for example, are able to convert the number on the tossed dice into the correct number of moves in a board game, it is obvious that they dispose of an abstract notion of numbers.

So then what is interesting for these children during their long time spent in kindergarten?

First of all: ever increasing numbers. Quite a few children quickly learn to count further and they are interested in the structure of the system of numbers (configuration in the decimal system) and they contemplate on this.

Do we want to leave them alone with this or do we want to accompany them in the venture as their partners?

Some children at pre-school age enjoy (and all the while learn a lot) counting in steps of 2, 3, 4, 5 and so forth. They try to count backwards, they ponder the relatedness of numbers, for example the relation ‘smaller – bigger’ or ‘even – odd’ (only even numbers can be divided by 2), they break numbers down into smaller numbers, in other words: they begin doing arithmetic.

Some children will make use of aids (stones or whatever), others perform theses mental operations in their minds only, abstractly.

Our number system (which most people in the world use) is based on the number 10 (decimal system). This is most probably due to the fact that we have 10 fingers. Basically we keep counting to 10 over and over again while we simply raise the digit in front by one every time we have counted to 10.

Some time in pre-school life the mathematically talented child discovers something else (or at least develops an intuition of it): When in the course of counting the first digit goes beyond 9 the entire number increases in length by one digit (from 9 to 10, from 99 to 100, from 999 to 1000). This is how the child closes in on the question of ‘place values’.

The insight is near: It is important to understand the place value of each digit in larger numbers. The 3 in 432 stands for 30 while the 4 signifies 400.

That is pretty much it.

It is not surprising, that mathematically gifted (pre-school) children will soon turn to more difficult considerations, for example: What is it with the points in numbers? What does 17.8 degrees really mean? What is 3.98 Euros? What is -7? What is an eighth? How does multiplication work?

See also the example of Lena in the article: Giftedness A Definition

Very well, then do play with numbers in kindergarten and involve those children who are interested.

3) Assignment and Function

In kindergarten there are all kinds of games in which entities have to be assigned to one another and at a certain developmental stage of their cognition children are constantly assigning things to other things: this is mine (not in the sense of ownership, but in the sense that ‘this thing and I belong together’), this is yours (correspondingly), that is Sven’s jacket, this is Lina’s hat…, they practise the mathematical operation of assignment, and it is fun to them.

So, whenever we offer or observe games being played in which things are to be assigned, games of whatever kind, the children are putting their knowledge into order, all the while building a basic mathematical understanding on the side. Many popular tabletop games are also based on the ability to assign objects correctly to another (and they are an exercise of this operation).

All this appears simple and easy to us; however, children have yet to develop these mental structures in their first years of life.

Gifted children will often do so at a very early stage. For example, there was a girl of one year and nine months, who had finished her own and now wanted her grandmother’s chocolate Santa, too. She was caught speaking to herself: “Naah, that’s grammy’s dollyman.”

Another girl, exact same age, uttered: “That’s Anna’s hammer, no, it’s Anna’s mother’s hammer.“

Assignment is a prerequisite for understanding mathematical functions. These are the often overwhelming terms, loaded with many x’s and y’s and latin or greek letters alongside many additional mathematical symbols, which will remain unintelligible to us non-mathematicians.

But there are simple functions as well, which children meditate and work out for themselves even as early as in their pre-school years, without any knowledge of their mathematical notation.

The Wikipedia definition of the mathematical term ‘function’ reads:

A function assigns exactly one value to each input of a specified type. The argument(input) and the (assigned)value may be real numbers, but they can also be elements from any given sets : the domain and the codomain(range) of the function.

At first sight this may look awfully complicated to us laymen (and indeed, anyone will receive undivided support on any TV talk show in Germany by declaring categorically: “No, mathematics has never been for me!”)

No doubt, there are extremely intricate mathematical functions which 99.99% of all people would lose their minds over – but luckily we are dealing with elementary education, we are talking about important basic knowledge .

Every function is basically an equation. This means you can do things on both sides of the equal sign, you might, for example, add something, but you must do it on both sides so that the equal sign isn’t lying.

Letters are used in equations to represent certain things or to signify values that are yet to be computed.

Implementing the above (Wikipedia) definition of a function the popular kindergarten game “Shoes on a Pile!” would be described mathematically:

There are 25 children in our group – they are all participating in the game. In the language of mathematics they are the elements of the set we are presently looking at, the so-called domain (“x-value”).

Now exactly one pair of shoes (the right pair, of course) is to be assigned to every child, to every element , that is. The 25 pairs of shoes are the elements of the range (“y-value ”) . Both sets have the same number of elements – 25 children own 25 pairs of shoes. Therefore there is exactly one corresponding “partner” for each element of both sets, a one-to-one mapping is possible and this gives us the simplest equation for a function there is:

y = x.

If we suppose every child had 2 pairs of shoes stored at kindergarten, we would have to assign 2 pairs of shoes (elements of the range) to each child (element of the domain) – no doubt, a somewhat more difficult task. The equation would then read: y = 2x (spoken: “y equals two x”). In spoken language this means: “How many pairs of shoes (y) are there if all children (x) each have two of them?” With 25 children this would amount to 50. We would not have enough space for so many shoes in our kindergarten, but some of our children would probably still be able to match each pair correctly with the right owner.

The advantage of such equations with x and y is that they allow for universally valid solutions to a given problem of assignment: “y = 2x” therefore does not only work for 25 children but also for 27 children, even if we wanted to determine the sum total of the children’s ears.

By the way, many a mathematically gifted and predisposed child will be rather irritated if even one child has thrown two pairs instead of the requested one pair of shoes on the pile. That ruins the function (this is a breach of the exact correspondence of one element from the domain with no more than one element from the codomain), and all the beauty, which the gifted child experiences in any kind of order, is spoiled.

Such functions are evident all over kindergarten. Every child has its little icon by its coat hook, every child has its own little locker, and so forth.

If we – just for the thrill of it and for mathematics’ sake – were to ask how many elements we came up with if we allowed the following categories of possessions owned by the children (every child has exactly):

1 last name (B), 2 first names (C), 1 icon at the coat hook (D), 1 coat (E) and 2 gloves (F),

Then the equation representing our function would read:

y = 1x B + 2x C + 1x D + 1x E + 2x F = 7x = 175

(with “ y “ being the sum total of possessions)

As we know that there are 25 children in the group, x=25 (children), this equation renders a value of y=175 possessions.

If three gloves were lost and five children complain because they only have one first name, we have to alter the equation:

y = 1x B + 2x C – 5 + 1x D + 1x E + 2x F – 3 summed up we get:

y = 7x – 8 = 175 – 8 and that comes to:

y = 167 (possessions).

Yeees, sure, but the point is that these obvious correlations in our everyday life represent just what is called a function in mathematics, and this is just to show how these ordinary facts of life can very well be translated into mathematical notation, in other words they are in and of themselves mathematical by nature, even if most of the time we don’t think of them as arranged in mathematically representable order. And children’s thinking cannot help following mathematically valid principles.

Carolin, 4 years of age, was working on the function y = a + b + c when counting cars. The sum total of cars driving by the house was the sum of ‘a’ (number of red cars), ‘b’ (number of white cars) and ‘c’ (number of all other cars).

She did so by first assigning every red car to one box on her sheet of paper, each white car to another box and all ‘other cars’ to a third box. Next she counted the cars of each of the subsets and finally added these numbers up.

You can find a detailed account of this episode in the article Custom-fit Cognitive Advancement.

An exciting application of ‘assignment‘ is inventing and using a cipher . In a simple version this can be done by assigning a certain symbol to every letter in the alphabet. This could be another letter – or a number or a little icon…

Highly gifted children, once they have tried this for a little while, often come up with complicated ciphers: The symbol to represent the original letter is determined by a specific general rule, as for example: A becomes D, B becomes E, C becomes F … (rule: replacement by third successor in the alphabet). One day, a gifted girl, five years old, was thrilled and delighted when discovering that she didn’t need a rule at all and could just as well write the lower row entirely at random, as, for example, like this:

she said. What really tickled her was that she had set the F for the F. A truly clever idea for a five-year-old!

Some gifted children will even conceive of much more sophisticated systems. One six years old boy in kindergarten once let me in on the cipher he had just made up:

In a first step every letter is transformed into a number: A is 1, B is 2 …
Now all these numbers are raised to the power of 2, i.e. multiplied by themselves.

His name, Thomas, would then read: 400 – 64 – 225 – 169 – 1 – 361.

However, he still was not quite happy with the system:

He thought it would be safer to substitute the smaller numbers by prime numbers. Prime numbers are numbers that divide only by 1 and by themselves. He set the 37 for A, the 83 for B, the 131 for the C and the 137 for the very frequent E. With this, he reckoned, he had done enough to conceal the principle behind his cipher.

4) Part-Whole Relation

Three-year-olds and younger children often do not yet understand what is meant when somebody calls: “All children come here!”. Either they don’t react at all or, if they already have a notion of the word “all”, they may ask “Me too?” They are not aware that “all children” includes every single child, they still aren’t sure about the ‘part-whole-relation’. This is another example of a pattern of thought which every child has to develop and that has a lot to do with mathematics.

In order to understand the more advanced ideas of set theory or geometry this basic knowledge must be developed.

Even the easiest addition, as in 3 + 4 = ?, can only be performed on the basis of an understanding that objects, planes, sets can be divided into single parts and reassembled into a whole.

A child working on a puzzle is dealing with the part-whole relation, just like a child building a tower or taking apart an old alarm clock. Children discover that there are things which are made of parts that are the same (for example pieces of a pie – the whole is only a “more” of the same) while other things are composed of quite different parts which still add up to a whole, which is then of an entirely new quality (alarm clock).

Complex analytical and synthetisizing thought in mathematics has its origins in these early discoveries in childhood.

5) Algorithms

There is no need to fear this Arabic word.

An algorithm is a finite list of well-defined instructions by which to solve a given problem.

Recipes are algorithms. The interesting thing is, that it is not easy to say of how many steps an algorithm must be composed. In everyday life, for example in the case of recipes, much is assumed: The recipe neither says that the pot has to be put on the stove nor that the stove has to be turned on or that one has to watch when the food is hot and the stove has to be turned down – all the recipe says is: “heat up”.

This is one of the basic problems in recipes, manuals and pedagogics: How much can be assumed, how much must be explained?

Do I have to say, that the plug has to be stuck into the socket in order to operate an electric device? Or can I presume it? Or does the unexperienced cook, when told “heat up” possibly wonder: “Well, how? In a pan, in a pot, in the oven, and how do I know when it’s hot enough?”

Children have this problem all the time: “Wipe the table clean!” But what, if the child does not know the algorithm “wiping the table clean”? It has no concept and may sluggishly wipe across the table with its sleeve, while our idea (algorithm) was quite different: get a cloth (where?), wet the cloth (where and how?) and squeeze it (sufficiently), clear the table of all objects (where to put them?), wipe the table entirely (not all that easy to do) and wipe it dry (again the entire table top), put the cloth away (where?).

It’s not a bad thing to begin developing the routine “how to wipe the table so that it is clean afterwards” at the age of 3. It may come in handy when at the age of 11 one wants to do some cooking and not have the parents prohibit any further attempt because one has left the kitchen in a mess.

The more a person knows, the shorter an algorithm will do. It’s the same with computer programs. “If the computer “knows” a lot of helpful programs, if it disposes of an operating system and a number of subroutines, a word processing program can build on it. Mathematical algorithms, too, can be just the shorter the more mathematics the calculator (be it man or machine) understands already.

In order to act autonomously (in whatever area) the child must first build and “hard-wire” many, many basic algorithms in its brain, which it can then compile to more complex operations (“I’ll set the table, and I’ll wipe it clean first!”).

Having skills and abilities is just that: disposing of such algorithms (well-defined procedures). The actual acting out of such procedures reinforces the algorithm itself. Mistakes, misleads and detours are being eliminated (just like the so-called ‘Banana Softwares’ for computers, which got the name because, like bananas, they mature when already in the hands of the user). Then again, once clean of all bugs, an algorithm will guarantee a quick success, oftentimes to become increasingly mechanized; that frees the mind for new strategies and creative ideas.

Unfortunately in many families there is way too little being done with respect to such cognitive developmental processes. As a result there are children who lack all independence and who “don’t really know” how to do anything. On the other hand there are well fostered adepts, who seem to be managing things at ease, who make their own sandwiches or plant a flower at the age of three, who know how to cross a street safely, how to settle a dispute or how to agree on the rules of a game when they are five years of age.

Deliberate action is best learned by acting deliberately. This is where kindergarten is at an advantage: The children are requested to be self-reliant, self-reliance is valued highly, the children learn, for example, to do handicraft works with more and more deliberation and consideration for the limitations set by the current situation…, the more children are encouraged to act upon good thought and planning, which includes readjusting strategies and checking results, the better.

By making it a point to habitually reflect and evaluate each step in a work process together with the children, when they are not yet entirely familiar with it, we make them aware of the underlying algorithms and promote their autonomy as well as their basic understanding of mathematics. Children who are being involved in the household chores, at home too , in gardening and DIY activities around the house or any other result-oriented tasks as early as possible are clearly at an advantage. They also lay the necessary groundwork for a later ability to conceive of capable algorithms like “how to plan our wedding” – or by the same token to design computer programs, create recipes or write manuals (further important skills and talents presumed).

6) Measuring

What are all the things that can be measured? And how is it done? And why measure things at all?

IHVO course participants who worked on the topic of measurement with kindergarten children were thrilled to see how intensely, intelligently and persistently these children occupied themselves with these questions.

Starting with the simple measurement of length, width and depth (for young children this is, of course, new), going on to measuring weight, temperature, time and volume. Their activities ranged from making measurements to creating technical constructions on to considering philosophical questions (What is time? What is infinity?), then even raising health concerns (regularly check the temperature inside the refrigerator, take everybody’s temperature before they are allowed to enter kindergarten).

Here too, it is a good method to get as many children as possible acquainted with the simple principles of measurement and then to observe: Which children want / need more input? However, it is not necessarily a good idea to start with all children and take it siiiiiimply and sloooowly and trust that the brighter and more curious ones will await the slower ones’ catching on, and that the brighter ones will still want to continue. Oftentimes they will turn away either inwardly or outwardly.

Starting with the children, who are not so strong (in this area), and watching how far they are interested and willing to go with this, taking all the necessary time even for the simple aspects has proven to be the better idea. One can take those children aside, who are stronger (in this area), the ones that are more talented, capable, knowledgeable and motivated, and go about the issue separately with them, rush through or even skip the easy stuff and go straight to the more involved questions. For example: How can we measure which of two stones of similar size is really the bigger one? What exactly does “bigger” mean: taller, wider, greater mass or volume? (Hint: check the displacement of water.) And which one is heavier? (If we compare both, volume and weight, this will lead to the question of density.)

An understanding of measurements and methods of measuring broadens the horizon with regard to mathematical questions and scientific research.

Acquiring instruments of measurement and familiarizing children with them, always keeping them in reach for the children, has proved its worth. Ruler, measuring tape, folding rule, measuring cup, thermometer (without mercury), different clocks including self-made clocks (sundial, water meter), different kinds of scales (kitchen scales, letter scales, traditional scales with two pans and balance weights).

All this fascinates children.

See also: Number Detectives Are Taking Measurements.

7) Approximation, Estimates

At summer festivals there are sometimes these interesting guessing games: How many peas or marbles are there in this jug? Whoever comes closest with his estimate wins the prize. Often it is not until then that we realize how poor our ability to give an estimate really is. Many people do not really know what they are being told, when someone gives them directions: “It’s about 800 meters to the station.”

Anything, that can be counted or measured, can also be estimated: number of, weight, speed, wind force, temperature, length and distance, height and depth, brightness, loudness, solidity of materials, time span, shape (this is just about spherical)…

What does this have to do with mathematics, and why should children learn it?

It makes sense to acquire a sense of speed, of amounts and other things as early as possible. It makes for a sense of orientation in the world and gives a person security when acting. If I knew that the person giving me directions (800 meters to the station) knew the area and was able to give a safe estimate, and if, in addition, I knew how many meters I cover per minute when walking, I could easily assess whether I can take my good old time or whether I better hurry up to make it in time for my train.

It is also rather helpful to be able to estimate the bill at the restaurant in order to detect possible errors on the bill right away. Being able to judge orders of magnitudes is often important and helpful in everyday life. Scientific mathematics, economic mathematics and statistics make use of estimates, too, which often saves time and money while still rendering sufficiently precise results.

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Some more things that interest gifted pre-school children early on:

Maps (also city maps, blue-prints …):

Just about anything that is an abstract representation of the real world and thereby helps to orient oneself in reality. Of course, creating and drawing maps and sketches is also highly interesting and instructive for gifted children.

More about this: Plans, Drawings, Sketches, Mind-Maps.

Pondering Probabilty and Chance.

The concept of „random chance“ is hard to grasp for many young children. Have they not just spent considerable mental effort in understanding that there is always a cause for everything, and that all events taking place in the world are composed of cause and effect. Consequently they will at first tend to hold someone “responsible” even when an event was purely accidental. Eventually their intellectual development will enter the next level and they will come to see that some things happen by chance, which also means that these things are hard to foresee and impossible to control. Random events can only be awaited. See also: Examples of Great Interest in Systems and Logical Relations. (The example of Daniel and the Christmas Calendar, publicized October 30 th , 2008). But children of high intelligence are never satisfied with what they have understood, they are quick to conquer new concepts and constructs, preferably the concept of probability comes next. Even the advent of a random occurrence can be estimated and calculated.

By attaining an ever deepening understanding of this concept, the child will develop the program “risk assessment”, which is being refined over time and will eventually become a routine running in the background at all times.

If a child reaches this intellectual state very early on, say at an age of 3 or 4 years, it may happen that this child will be considered extraordinarily fearful. More details on this in Timidity and Apprehension in Gifted Children.

The Number Zero.

The number zero is still young. The peoples of India, Babylon, China, Greece and other peoples developed sophisticated numeric systems and mathematics as far back as several thousand years ago. Yet, the number zero was only conceived of some 500 years into our common era. No earlier than in the 11 th century was it introduced in Europe.

What is so difficult about the zero can hardly be observed with gifted pre-school children: They deal with it at ease. Only when it comes to dividing by zero, the issue becomes mysterious, because dividing by zero is “not possible”. There is never a definite result, it always comes to “infinite”.

Hint:

At the Museum „Arithmeum“ in Bonn (offering good guided tours for children, even for kindergarten children if they have an interest in numbers and calculating) much can be learned about the history and the techniques of calculation.

http://www.arithmeum.uni-bonn.de/

Picture Books / Special Interest Books on Numbers and Mathematics.

Zahlenmärchen

(Fairy Tale of Numbers)

By Ida Fleiß and Gert Mittring. Wagner Publishing.

[Note by the translator: The English titles in brackets are merely translations, they do not represent actual titles of English Publications of the books, which to our knowledge there are none.]

Today there are quite a few picture books dealing with the numbers ranging from 1 to 10, which gifted children often tackle as early as in their 2 nd or 3 rd year of life. This book even goes beyond and is therefore interesting for the math-aficionados among the elder kindergarten children, too.

It is a fairy tale taking place in Numberland. Gert, the world-champion of mental arithmetic, takes them on an interesting tour. They visit the Number-Clinic, the Symbol-Factory, the café with its digit-coffee and root-cake, they go to Niners-City and check the Number-Museum, finally they reach the Cave of Numbers.

On their trip by train they listen to suspenseful fairy tales:

* The Number Spider
* How the Three Found True Friends
* A Dino in Love
* The Tale of King Divisibility II
* The Secret of the Dancing 8
* The Dreamdealer
* The Devil and the Number-Witch
* The Tale of the Smart 5

… and it’s always about numbers.

Each fairy tale as well as each stop on the journey can be read out of sequence, so that the children may discover the contents of the book step by step.

A book for kindergarten and elementary school children who consider numbers their friends and enjoy dealing with them.

Das Bilderbuch von Zahlen

(The Picture Book of Numbers)

by Rolf and Margret Rettich, Ravensburger Publishing House
This is a picture book on numbers ranging 1 to 10, maybe the book of choice for very young gifted pre-school children.

Kindergartenspaß mit Willi Wiesel. Zählen und Tüfteln.

(Fun at the Kindergarten with Willi Wiesel. Counting and Puzzling.)

Ensslin Publishing.
A workbook for counting and colouring. Also covers only the numbers from 1 to 10, so don’t offer it too late!

Komm mit ins Zahlenland. Eine spielerische Entdeckungsreise in die Welt der Mathematik

(Come Along to Numberland. A Playful Quest into the World of Mathematics)

by Gerhard Friedrich and Viola de Galgóczy.
A didactic book of games and stories on the numbers 1 to 10. Appropriate for kindergarten. Interesting for gifted children between ages 2 and 4, while they are in the process of discovering the numbers up to 10, not so thrilling once that number range has been mastered.

Was ist Was, Band 12: Mathematik

(What’s What, Vol. 12: Mathematics)

Tessloff Publishing
The book comprises many good illustrations and explanations, good reference reading for teachers as well.

Copyright © Hanna Vock 2007, see Imprint.
Translation: Arno Zucknick
Date of publication in German: September 4 th , 2007