Teaching numbers by counting can create confusion in the student between order and quantity (Please see order before reading this).  Quantity is not a sequential pattern but it is taught that way when using the same sequence as ordering. The confusion is further exacerbated because the symbols used for order are the same as the symbols used for quantity.

It is not too uncommon to go into any high school in the United States and find students who use their fingers to add five and seven.  It is possible for those of us who do not need our fingers to solve this addition problem to get a glimpse of what might be happening inside these students’ minds. To do this, all we need is to use a set of labels that we learned sequentially. It is common throughout the literate world to teach children to count at the same time that they are taught the alphabet. The alphabet is also sequential.  In fact, there are some languages that use the same word for their letters as their numbers but that is not the case in English or Spanish, the two dominant languages in the United States. So let’s ask our addition question using the alphabet. What is E plus G? Most people have to revert to counting E letters passed G.  It is probable that the solution to 5 + 7 being 12 might be memorization because, unlike Chinese or Korean, Americans use new words for the quantities of 11 through 20, 30, 40, 50, 60, 70, 80, and 90. Therefore, if a student was using some initial number sense for understanding 5 and 7, it is thrown out as 12 is not ten and two but a new word, twelve. To truly appreciate this task in global perspective, imagine a race between two students to label (i.e. learn the words for) the symbols 1 to 99.  The first student has to learn ten words for the numbers 1 to 10, then that student can reuse these words to label the numbers up to 99.  For example 37 would be labeled three-tens-seven.  The pattern after ten is identical for every decade. The second student also has to learn the same ten words for the numbers 1 to 10 but, unlike the first student, this student has to learn an additional ten words for the numbers 11 to 20 and an additional seven more words for the decades thirty to ninety. Ready, set, go!  Learn and manipulate 10 words or 27.  And if one really wants to exacerbate the difference further then consider the effect that multiplication and division will have on these two students.

This problem is well known. Mandarin and Korean are a couple of languages that only need to know ten words to number up to ninety-nine items.  The difficulty of retraining over 200 million American citizens new names for numbers is inconceivable.  However, if counting is the only basis for our number sense then we are out of options.  Please return to quantity and then explore subquan.