sums will set you free
how to teach a person numbers, arithmetic, mathematics
First, a word of warning: I have no doubt that there are literally thousands of web pages and sites aimed at teaching basic arithmetic. This series of pages is something entirely different. It is designed to lay down empirically sound foundations and to emphasise the widespread weaknesses in teaching mathematics abroad in this society. These pages are are not designed to duplicate what may be easily obtained elsewhere in books, in class rooms and on the Net. They are designed as a corrective to poor instruction.
Children can be taught to count before they can read, just as they learn learn to talk before they can read or write. Thus, I am starting these guides to basic arithmetic in the context of a pre-literate person, and treating the reading and writing of numbers separately.
Most seriously numerate people can do a great deal with just the four basic operations of addition, subtraction, multiplication and division, drawing up necessary methods (formulae, or even tricks). It is often common that such people manage with a very basic cheap calculator, rather than a fancy device with all manner of almost random formulae; just as in days of yore any self-respecting engineer would carry a slide-rule as a badge of office, just as a medic carries a stethoscope.
With anything complicated, a mathematician will now tend to start with scribbling on a table napkin and then transfer an outline to a computer program. If the task is at all complicated, that program will tend to undergo continual modification.
So these basic operations are a very great deal of the foundation of numeracy. It is important to make these foundations secure and rational. This requires practice and the gradual learning of many short cuts. It is also vital that understandings are founded securely in the real world at every step. A very great deal of maths teaching is not founded in this manner and is taught in vague and even incorrect abstractions.
I abhor the common way of teaching mathematics, as a meaningless dogma with parroted rote rules (or formulae, in the jargon). My view of mathematics is that it should be used to teach useful mental methods that can be applied profitably to the real world. Otherwise put, mathematics is about ways to think and, thus, is highly related to reasoning (‘logic’).
Of course, no five-year-old will be expected to follow this with the mind of an experienced adult, but the more you know and the more background you have, the more easily you will be able to understand the concepts that the child is learning. The important idea here is to grow the experience, which is the foundation for understanding the different ideas involved.
The correctives are often difficult for people who have been taught in a sub-optimal manner. These problems are dealt with in great detail around abelard.org. Where I think it appropriate, I have put links to relevant discussion. It may not be necessary for your purposes to dig in this deep, unless you become interested, or are teaching as a profession. Here are a couple of useful places to start if you wish to have a look at that area: Laying the foundations for sound education and Why Aristotelian logic does not work.
You will find some things, such as the educational maths counter, on more than one page. This is because I intend that the pages stand alone as far as is efficient. More generalised information can be found on this introductory page.
Note that yellow, underlined links will take you to associated
pages at abelard.org, while blue links
are to other parts of the same page.
Be patient and remember the brain is a physical object which actually changes during the child’s learning. That process involves nerves making new physical connections, and this process takes time. Thus, constantly repeated, fairly short learning sessions, followed by time to let the brain get on with its growing, are an integral part of learning.
In this context, it is also important that the child learns to concentrate. Without concentration, memories will tend not to be laid down permanently (physically); only a short-term retention will occur, such as when you memorise a phone number before transcribing it to paper or organiser (only to be forgotten by the morrow).
In doing long and complex calculations, some learners can take delight (and satisfaction) in performing the multiplications and subtractions needed to complete the sub-sums. Other learners may become frustrated or overwhelmed. It is normal for humans to make regular errors when transcribing in detailed calculations, just as you will find mis-spellings even in the most rigorously edited books.
Good teaching is not simple.
The difference between maths as often taught and how maths is taught at abelard.org is understanding the language and learning the language by rote, that is like learning Latin responses to priests.
Learning chess, cricket, snooker and other games is common. It’s a matter of learning arbitrary definitions and rules, and such learning is often enthusiastic.
Maths is not much different. Much of it is a clear way of communicating about realities.
The way maths is widely taught is as a language without context. It would be like learning English without meanings - learning ‘the cat sat on the mat’ with no idea what a mat, a cat, or a sat meant.
So ‘add five oranges to three oranges’ is much better than ‘what is 5 + 3?’.
There are many forms of table. A child learns to form a statistical universal (called ‘table’) from their experience of language usage.
‘Two’ is ‘it’ and ‘it’. ‘One’ becomes a synonym for ‘it’. Dopes babble about ‘abstracts’ instead of explaining ‘it’ clearly.
‘Oranges’ is an abstract, and so this is something to introduce while teaching.
Having determined by experiment that 5 oranges added to 3 oranges give 8 oranges, the learner can add 5 apples to 3 apples and observe that the sum is 8 apples. That is, by trying it out and seeing if it works; by counting, using the ordered grunts known as numbers.
In time, the learner will understand that adding any five objects to any three objects will result in a collection of eight objects, just as they ‘know’ a table when they see one. This is in the same way we know apples fall from trees and call it ‘gravity’.
As a pragmatist, after seeing enough incidences, I am prepared to treat it as a belief until empiricism shows me a situation that breaches the belief.
I translate ‘the same’ as ‘similar enough that I don’t care about the difference right now’. Numbers are real in the individual’s head, and they are a pragmatic means of communication.
Every use of ‘one’or ‘apple’ is new in the real world. These are formed/learned statistical universals.
What we will do on these pages, is show various methods of doing the same sort of sum, so that the learner will start to understand what is involved with each sort of sum, rather than just learning rules by heart.
Teaching in parrot lockstep, with rigid rules, does the opposite from showing a learner how to understand different situations. It may let learners go through doing sums like a parrot and achieve the right answer, but the result is that the learners do not understand what is going on, why they have the answers they do, and cannot apply the mathematical tools to other situations.
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.
It is simple to make up examples appropriate to your current teaching objectives, checking the answers with a calculator.
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, becoming 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).
Other good advice will be interspersed in the text of the succeeding pages.
Allow the learner to play with this wonderful counter. If you press the Change Mode button to start the counting, it will count up slowly in ones. The counter is added to other arithmetic pages, and will be configured to illustrate the relevant sums. These settings may be changed by the user on any page.
Play around and you’ll find you can do some pretty complicated and weird sums, if that sort of thing gives you joy. If you’re really determined, you can even make the counter say “NaN”, or “Not a Number”. That’s an error message telling you that you’ve done something silly. ☺
The operator may
Any further extensions to this counter would make it into a calculator, see below.
Once a learner has understood enough from playing the counter, it is time to introduce the learner to handling a hand-held or screen-based calculator. Screen-based calculators are generally packaged with computer operating systems, such as Windows. Google also allows much calculating and unit conversion. For instance, try entering “What is 2.6 acres in hectares?” into the Google query box.
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