What surprised me in reading the articles in the Maths Extra was that there was no mention of the properties of natural numbers. In terms of primary children, the associative property of addition and multiplication means that they can be done upwards, downwards, laterally or even selectively, which gives young children plenty of choice or strategies.
The commutative property of addition and multiplication halves the necessity for learning all the basic number-facts. The distributive property allows children to "chop up" numbers they cannot deal with mentally, operate upon the pieces and put them back again to obtain the correct result, thus permitting the children to go mentally well outside the normal range of number-facts. The reciprocal property of subtraction and division is also very useful in calculating complements.
Ian Thompson tentatively mentions the Dienes blocks for two-digit place value but then consigns them to the cupboard in favour of the Categno cards. In fact, during those "wicked Sixties" we used the Dienes blocks (and others) with great success to show place value up to four figures.
GORDON PEMBERTON 25 Hunters Rise Pogmoor Barnsley, South Yorkshire