The Four Stages Of Number Sequences

I stumbled across this while trying to figure out why skip counting is considered a good way to help student thinking switch from additive to multiplicative.  I'll explain why I was researching that topic another time but in the article, they discussed the four stages of number sequences.  I'd never heard of it because it explains how students acquire certain numerical concepts in elementary school.

The first stage of number sequences is the initial number sequence.  If a student has not hit the initial number sequence, they are considered pre-numerical and who do not understand cardinality.  When a child reaches the initial number sequence, they are able to understand a number such as 5, explain their cardinality, and can count them in order as a result but not as an finial unit.  In other words if they counted the five objects in front of them and arrived at five, on could add four more objects and most children would just continue counting till they arrived at 9.  They often rely on touching the item as they count or use their fingers to keep track of the total number. 

The second stage is known as the tacitly nested number sequence (TNS) in which students develop the understanding that there are subsequences contained within larger sequences such as 1 to 5 is a subsequence within the sequence of 1 to 10.  They can count the numbers of the subsequence regardless of where it occurs within a larger sequence such as. counting beginning at 5 to 12 inside of the sequence of 1 to 20 and this is referred to as a composite unit.  

If a student learns to count using skip counting, they are learning to use a composite unit such as when they count 3, 6, 9..., they are using a composite unit of 3.  In this stage, students use the composite units while counting by may not be aware of which composite unit they will use prior to counting. In addition, they understand that 8 is 8 ones or 8 times a single unit at this point so they don't count by ones all the time.  

The third stage is the explicitly nested number sequence (ENS). In this stage students can recognize that six ones is the same as one six.  They also understand that four sevens is the same as four composite units and the composite unit is made up of seven objects.  So in this stage, they understand multiplicative situations.  Furthermore, they have an abstract understanding of composite units and that six items is the same as six ones.  They also see that five ones is the same as one five and vice versa, so they understand they are referable.  In addition they can figure out how many groups of four are in 24 but they might struggle with  how many equal groups of six can 24 be divided into.

The final stage is the generalized nested number sequence (GNS) are students who are fluent in the multiplicative situations.  They understand composite units and how smaller units can fit into larger ones such as with a composite unit of 3 used eight  times to make 24 or as eight composite units containing 3 items in each unit.  A child who is in the GNS stage will begin building exponential structures, and begin to understand other things such as Lowest Common Multiples and Greatest Common Factors.

Skip counting is one step along this journey of number sequences.  Let me know what you think, I'd love to hear.  Have a great day.