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sums will set you free

how to teach your child numbers arithmetic mathematics

subtraction and more counting

New translation, the Magna Carta

how to teach your child number arithmetic mathematics - subtraction and more counting is part of the series of documents about fundamental education at These pages are a sub-set of sums will set you free.
how to teach a person number, arithmetic, mathematics on teaching reading
methodology and examples
counting in the real world
category - logic blocks maths educational counter
end notes

The approach to teaching mathematics and other languages is profoundly unsound, but the habits are deeply ingrained.

I have already provided documents explaining where the problems, are on a technical and adult level. To dig into this, start at why Aristotelian logic does not work and laying the foundations for sound education.

This sub-section of is designed to lay out a rational and logical base for teaching arithmetic and mathematics from basics. I shall not always justify the methods in this section as I go along, but the methods are very relevant and purposefully structured. Throughout this section, many of the yellow links take you to a more advanced, or technical, explanation.

It is vital to understand that there is no fundamental or logical difference between the symbolism of teaching English and teaching arithmetic/mathematics. This congruence becomes part of the learner’s understanding. It is a deep and dangerous pedagogical error to allow the learner to imbibe the erroneous concept that mathematics and English are different “subjects”.

methodology and examples
On these pages, you will be given basic methodology and necessary examples. You will not be provided with hundreds of examples, those you make up as you work with the learner, adjusting those examples according to the person’s problems. Some examples should be interspersed which are easy for the learner, in order to reinforce and to give experience of success, while others should be aimed at specific difficulties.

As you will see, there’s a great deal of stuff to absorb here, especially if you are three, four or five, or even eight, nine or ten. Human understanding evolves, the human absorbs and gradually organises the vast streams of information coming from those small holes called ears and eyes.

For example, getting used to and seeing clocks of different types lays grounds for understanding what the shapes and numbers mean. Trying to rush these processes leads to stress and often to confusion - not good.

Letting the child run wild, without any help or guidance, leaves them struggling to adapt to a civilisation and culture that has taken thousands of years to develop, and which is now running in over-drive. It is every bit as foolish to leave a person in confusion as it is to feed information too quickly and hammer it in with a mallet.

The purpose of mathematics is to understand patterns and logic, to help you organise the filing system in your head. Mathematics is not something esoteric, but there is rather a lot of it! The sane objective of learning is not to memorise enormous lists by rote, it is to teach organised ways of thinking about problems, and where and how to research for relevant information in the ever growing data banks of human experience (knowledge).

counting in the real world

It is widespread common sense that a child learning to read has their attention drawn to varying sources of written text. Likewise, it is useful to help a child to gain numerical fluency as they explore the world.

Keep in mind that this page is to help you in teaching young children the basics of counting and, here in particular, subtraction. This section shows various numbers you can use in the real wore ld to increase the child’s awareness. There is not the slightest intention or expectation that the learner is going to gain a comprehensive grasp of all these various wonders at this point. The purpose here instead is to generate familiarity and to take opportunities to engage the person with this modern civilisation.

  • to count fence posts
  • to recognise the use of numbers as indices for bus numbers
  • to learn methods of counting space by judging distance using floor tiles in precincts. How far do you usually step? How far can you step?
    Tiles in a shopping precinct. Insert shows measuring tile width/length.
    Tiles in a shopping precinct.
    Insert shows measuring tile width/length.
  • by learning to judge distance by pacing
  • learning to interpret the numbers and positions on analogue or digital clocks
  • remember that there are counters in a car for mileage and speed
  • teach the child to count while skipping or bouncing a ball
  • car number plates
  • road numbers
  • traffic speed signs
    30km speed zone
  • house or shop numbers
    House number
  • dates: 21/05/2009
  • weights
  • calorie contents on packaging
  • mobile phone
    mobile phone

An analogue clock with Roman numerals

Clock with no numerals

Clock with modern numerals

Another clock with no numbers

Car digital display - time, temperature, mile per gallon

  • ducks on a pond
    One duck with nineteen ducklings

These are things which are interesting to chatter about while out, gradually introducing an understanding of the different way numbers can be used.

For technical background, see the error of ‘zero’, ... and sections comparing predicates, relational strengths.

It is important that the child understands that numbers have many different uses. Counting apples, or trees, of different sizes has differences from counting inches (or centimetres), where the aim is to have a fairly constant size unit.


Subtraction is like addition in reverse, taking an object, or several objects, from a group of objects. Take for instance a group of three blocks on cloths already drawn together.
Three blocks
one two three
three minus two
Now draw the cloths apart, so separating one block from the other two
One block; two, three blocks
one two three
1 2, 3
one one two
 one two
1 2
one and two
When the cloths are separated so one cloth is out of sight, the subtraction is complete. Two blocks are taken away from three blocks, leaving one block. Three minus two equals one.
Similarly, we have three blocks and five blocks.
Eight blocks Separate five blocks  from eight blocks Three and five blocks
      eight             three                 five

Notice that this can also be looked at backwards - as eight minus three, by turning round the pictures, or by going round the table. And the cloths can be drawn back together again as an addition. Thus, ideas of reversal are introduced naturally.

You put on your socks and then your shoes. Will it work if you put your shoes on first?

You lay out your plate, and then your spoon. If you lay out your spoon first, does it still work?

In technical language: Are these operations commutative? Do they commute? The sock example does not, addition and subtraction do. Often young children like to learn long and fancy words. In more ordinary language, are the processes (operations) reversible? It is also useful to pick up the jargon, as specialists are often desperately attached to their ‘special’ languages.

Help on writing down subtraction sums.









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Such actions as the following are also useful practice for dexterity, and comparison and harmony.

Can you put one more brick on?                 Without the pile toppling?

  Can you take one off?

Make the two stacks the same height.

This introduces the notion of equality - equalising - balance.

  Each stack has five bricks.
Bring the child’s attention to other forms of balance - balance weighing machines, the see-saw in the local park, walking across a room and back. Keep in mind that no two things are the same. The idea of sameness/equals is just a very useful fiction. From time to time, keep the learner aware that every block in the tower is different, made out of different wood, by different acts, at different times. It is very important for a person to fully internalise this understanding of reality. Each duck on the pond is different, each step taken is different. As I regularly point out, the world is changing all the time. Even the blocks are under continual change, as the child breathes and grows.

category - logic blocks

Here is a list of words meaning category:
category, universal, predicate, property, class, set, attribute, type, collection.

The purpose of this section is to encourage and develop an understanding of the formation of collections. It will be soon needed in the next stage (multiplication). If you do not have logic blocks (yet?), onions and stones and insects and flowers and boxes of detergent will serve.

Logic blocks, also known as attribute blocks, are made as a set of plastic shapes. This is a four attribute set - shape, size, thickness and colour. The blocks can be sorted into groups of a specific attribute or several attributes.

Sorted according to thickness (and by shape)

Sorted by size (and by shape)

Sorted by colour

sorted by shape (and size and thickness)

Logic blocks can also be laid out around an infant to provide bright stimulus of shapes. Logic blocks are also handy for teething. maths educational counter

[This counter functions with javascript, you need to ensure that javascript is enabled for the counter to work.]

On this page is a more concise version of the Brilliant educational maths counter. The full version with more detailed instructions, go to the introduction page.

So, to practise doing subtractions,

  • Reset Counter Value to 2;
  • Change Step to 1;
  • Switch Direction to Decreasing;
  • Now click on the Manual Step button.

The counter counts down (decreasing): 2, 1, 0, -1, -2 and so on.
Then you could change the Step size to 3! Encourage the learner to try many starting values (Counter Value) and numbers to subtract (Decreasing Steps).

[This counter functions with javascript, you need to ensure that javascript is enabled for the counter to work.]
Is the counter Manual or Automatic? :

You have done manual steps since the last reset

Decimal Places
[between 0 and 5]:
the counter is displayed up to decimal places
Reset Counter Value:

[enter number in base 10]

Change Step:
Enter step size:
[enter: step size in base 10]

change step size:

is added or subtracted on each update


Counting up/counting down

Base [between 2 and 32]:
the counter is displayed in base
Change Speed:
the counter changes every seconds.

You are welcome to reproduce this configurable counter. However, all pages that include this counter must display a prominent and visible link to:, with the following text:
“The Configurable Practice Counter was developed by the auroran sunset on behalf of and is copyright to © 2009”.
This text and the code, including all comments, must not be altered.

© 2009


end notes

  1. Estimating distance by pacing: remember a child’s pace length will change as they grow . Bring this to their attention, have a height-measuring point - a door jamb is one useful position. Understanding change and movement is an important basic concept.

  2. Analogue and digital
    A typical analogue device is a clock in which the hands move continuously around the face. Such a clock is capable of indicating every possible time of day. In contrast, a digital clock is capable of representing only a finite number of times (every tenth of a second, for example).

    An arrow flying through the air is continuous movement. Counting the number of arrows you have is digital. For thousands of years, through using numbers to describe the flight of the arrow, humans have become confused by these two different uses of numbers.

    For more, see also Words.

  3. Reversing an operation is often loosely referred to as ‘doing the opposite’. For example, the opposite of riding your bicycle from Oxford to London, in clement weather, when you’re fit and fresh, is riding back at night, in pouring rain, after a hard day’s partying.

  4. Logic or attribute blocks are available from several sources. A reliable source is from or An attribute blocks class set, also called a giant or jumbo set, similar to the set illustrated above, costs $25.95 [at, as at 05/2013] or £30.99 [, as at 05/2013]. Its shipping weight is 6lb/4kg.
sums will set you free includes the series of documents about economics and money at
moneybookers information e-gold information fiat money and inflation
calculating moving averages the arithmetic of fractional banking

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