Numbers, with feet on the ground


one, two, three, four, …, ninetynine, …

These are the names of the numbers, they are words.

1, 2, 3, 4, …, 99, …

These are signs or symbols for numbers.  they are NOT numbers, and they are not unique –

          ten is  1010  in the binary system

                         A   in the hexadecimal system

                         X   in the Roman system

                and   10  only in the decimal system.

Of course, sooner or later we treat the symbolic form as “the number” , and it works, but we should never forget that symbols are not numbers.


The basic numbers, or natural numbers (as mathematicians call them).

These are one, two, three  etc, and including zero as it is very useful, and are for COUNTING and COMPARING quantities of things.

“How many peas in this pod?”

“How many sweets in this bowl?”

Also for resolving issues like “You have got more sweets than I have!”

Operations with the natural numbers.

Combining quantities leads to “addition”.

Comparing quantities leads to “subtraction”.

Combining like sized groups of things leads to “multiplication”.

Sharing leads to “division”.

Each of these processes needs to be thoroughly explored and understood in words before inflicting symbolic notation on the students.


In passing I have to comment on the term “word problem”, which I have encountered very frequently.  To me, teaching engineers, computer scientists and others a problem is a problem.  A much more satisfactory classification is “real problems” and “symbolic problems”, and mathematics is about both of these.  The term “word problem” is at best misleading and at worst downright stupid.


It should be noticed that with addition and subtraction of natural numbers that the objects concerned must be of the same type, but this is NOT the case with multiplication and division.  Consequently the order is irrelevant with addition, but this is definitely not obvious with multiplication.

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