Using counting as a problem solving process
The progression of solution strategies where students make use of counting as a problem solving process can be outlined through an example.
Teacher: How many jellybeans?
The student’s response to this question where all of the items can be seen will give an indication of his or her current counting procedures. Children learn the forward sequence of number words initially in the same way as they learn the alphabet, as a continuous string. To find the answer to the teacher’s question, young students need to know the correct sequence of number words, to match the count with the objects and to recognise that the last number stated signifies the total.
Teacher: Now I have added some more jelly beans. How many altogether?
A student who counts them all but starts again from one is using a less sophisticated strategy than a student who counts-on from nine. To count on from nine, students need to be able to start the forward sequence of number words from nine.
Typically, students progress along a pathway that begins with developing knowledge of the sequence of number words, and moves to combining and counting all the objects where they can see the objects, then to counting objects that they cannot see or touch, next to countingon from the larger number and eventually to using addition facts. This sequence of development can also be described as a progression from counting by ones to a facility with using part-whole knowledge of numbers.
When students count on, they no longer create the initial number as a count. A number word like ‘nine’ is taken to stand in place of carrying out a count to nine. The process of counting to nine is taken as having been completed by saying the word nine.
Teacher: If I have nine jelly beans and I add some more to make twelve jelly beans, how many jelly beans did I add?
Student: …9, 10, 11, 12
When approaching addition questions using a strategy of counting-on, an efficient approach is always to count on from the larger number. Alternatively, in answering 3 + 9, students can first “build to ten” saying, “One plus nine is ten, and I have two more, making twelve”. Separating three into one and two combined with recognising what is needed to form ten (sometimes called ‘bridging to ten’), shows a flexible part-whole knowledge of numbers that does not rely on counting by ones.