## Counting-on-and-back stage

The counting-on-and-back stage involves students using the names of numbers as being equivalent to completed counts. That is, to find the total of six and three a student can take six as the result of a count that has already occurred and say: “Six, … seven, eight, nine, … nine!”. The essential feature of this strategy is that the student counts on from “six”. This way of counting on to find the total is sometimes described as counting up from a number. To successfully count up from a number the student needs a way of keeping track of the number of counts to know when to stop. This may involve sequentially raising three fingers in coordination with saying the number words from “seven” to “nine”. Alternatively the student may simply recognise saying “seven, eight, nine” is three words.

Counting on can also be used to determine how many are missing with questions such as “6 plus some equals 9. How many?” Here the student may say, “Six, … seven, eight, nine, … three!”. As before, the student counts on from “six” and keeps track of counts, but does not know in advance the number of counts. Rather, the student initially knows only the number to which he or she is counting. This strategy of counting up to a number to find the difference between two numbers is often used when solving questions where the addend is missing. The counting may be accompanied by the student sequentially raising three fingers (seven, eight, nine).

Counting up to 11

Counting down is also used at this stage. To answer a question such as, “I have 9 and I remove 3, how many remain?” the student says “Nine, …eight, seven, six,… six!”. This strategy is described as counting down from a number. A similar use of counting backwards, or counting down, often appears when students solve questions where the subtrahend (the number being taken away) is missing. When responding to the question, “ I have 9 and I remove some, 6 remain. How many did I remove?” the student knows in advance the number to which he or she is counting. The student may say, “Nine,…eight, seven, six,…three!” This strategy requires keeping track of the backward counts and is described as counting down to a number.