How should these approaches be introduced: Jump, split or both?
Klein and Beishuizen (1994) summarized studies investigating mental computation strategies and whether those strategies should be taught. Based on those studies, Klein and Beishuizen suggested that ‘‘the didactic order should then be: first N10 to enhance mental (noncolumn) arithmetic and, much later, 1010 as a transition to written (column) arithmetic’’ (p. 127, 1994). That is, the jump method of incrementing by tens from the number (N10) is introduced before the split method of separating and collecting tens (1010).
The transition to the written algorithm relies on confidence at regrouping numbers. To avoid relying on counting by ones students currently need to develop part-whole knowledge of number combinations to at least twenty. Using trading with the traditional subtraction algorithm, subtraction problems such as 53 – 27 are transformed into problems involving subtraction within 20(13 – 7 in this example). This is because the subtraction algorithm with trading leads to a standard way of regrouping the 53 as 40 and 13.When this standard regrouping is used students need part-whole knowledge of numbers to 20.
Developing a rich multi-unit understanding of place value means that the student also treats hundreds as composite units. One hundred is at the same time one hundred “ones”, ten “tens” and one unit of a “hundred”. This allows a flexible and efficient reorganisation of units to occur. For example, a student can quickly determine the number of tens in 732 by reorganising the units and does not need to reconstruct the units of one hundred by counting by ten. A multi-unit place value concept creates a sound basis for mental computation. Regrouping 1256 into 1200 + 56 simplifies 1256 ÷ 6 far more than the standard (canonical) decomposition.