## Using abstract units of ten

When students work with abstract units of ten, where it is both a unit of ten, and ten ones at the same time, we sometimes describe the ten as an iterable unit. If ten is treated by the student as an iterable unit, he or she can count by tens and ones in the absence of any representation of composite and single units, with each act of counting by ten (forward or backward) being the same as the addition (or subtraction) of ten ones. Iterable units of ten are formed when the units of ten are constructed * during* the counting process and each unit increments the total by ten.

Another way of dealing with abstract units of ten in the absence of materials is to treat tens and ones using collection-based strategies. That is, when dealing with 46 + 57 using a collections-based strategy, rather than building on one total with jumps of tens and ones, the tens and the ones in each number are collected and combined. This approach is known as the split method where the numbers are split into tens and ones before these are collected. Ten is treated as an abstract unit that allows the collection of ones (6 + 7) to be redistributed as one ten and three ones. Traditional use of multi-base arithmetic blocks (MAB) is representative of a collections-based sense of tens and ones.

In recognising that multi-unit place value concepts are difficult for students, we need to differentiate between the difficulty that students have in dealing with representing numbers and the difficulty they have in coordinating abstract units of ones, tens and hundreds (Chandler & Kamii, 2009). The abstraction of the number 24 as a quantity (24 single units, 2 units of ten and 4 ones or even 1 unit of twenty and 4 ones) rather than the number word ‘twenty-four’ is also different from numeral representations ‘24’ or XXIV. The errors students make in using numeral representations make it clear that the link between the numerals and the abstract quantities they represent is often missing (Figure 2).

*Figure 2. Representing six lots of 402 as 252.*