A Progression of Counting and Quantity
As a former secondary math teacher and intermediate math coach, my new role as K-12 math consultant has led to a wealth of knowledge that I wish I had during my years spent in the classroom. My conversations about student learning needs with intermediate and senior math teachers always seems to come down to gaps in student understanding, however rarely were we able to dig back far enough in the math continuum of learning to determine exactly where those gaps began.
Recently, our Math Strategy Team focused on planning professional development for our math leads around composing and decomposing numbers all the way to addition and subtraction strategies. With this, Sharon Johnson shared her knowledge with the group around Basic Counting Principles that students must obtain for them to be successful composing and decomposing numbers. After taking some time to dig in and read more about counting principles and their importance in developing student sense of number and quantity, I realized that some of my intermediate students could still be struggling with some of these basic ideas.
Although researchers might differ in the number, naming or description of some of these counting principles, I find that this list of 9 Principles of Counting seem reasonable and resonate with me.
1. Stable Order
The first principle of counting involves the student using a list of words to count in a repeatable order. This “stable list” must be at least as long as the number of items to be counted.
For example, if a student wants to count 20 items, their stable list of numbers must be to at least 20.
2. Order Irrelevance
The order in which items are counted is irrelevant.
Students have an understanding of order irrelevance when they are able to count a group of items starting from different places. For example, counting from the left-most item to the right-most and visa versa.
Understanding that the count for a set group of objects stays the same no matter whether they are spread out or close together.
If a student counts a group of items that are close together and then needs to recount after you spread them out, they may not have developed an understanding of the principle of conservation.
Abstraction requires an understanding that we can count any collection of objects, whether tangible or not.
For example, the quantity of five large items is the same count as a quantity of five small items or a mixed group of five small and large things.
Another example may include a student being able to count linking cubes that represent some other set of objects like cars, dogs, or bikes.
5. One-to-One Correspondence
Understanding that each object in a group can be counted once and only once. It is useful in the early stages for children to actually tag or touch each item being counted and to move it out of the way as it is counted.
Understanding that the last number used to count a group of objects represents how many are in the group.
A student who must recount when asked how many candies are in the set that they just counted, may not understand the cardinality principle.
The ability to “see” or visualize a small amount of objects and know how many there are without counting.
Since it becomes increasingly difficult to subitize as the number of items increases, you’ll notice that five- and ten-frames are common in early years mathematics education.
8. Movement is Magnitude
Understanding that as you move up the counting sequence (or forwards), the quantity increases by one and as you move down (or backwards), the quantity decreases by one or whatever quantity you are going up/down by.
Unitizing involves taking a set of items and counting by equal groups (i.e.: skip counting).
For example, if there is a large group of candies on a table, one might choose to create groups of five (often doing this by subitizing these groups) and skip counting up by five.
Unitizing is also important for students to understand that objects are grouped into tens in our base-ten number system. For example, once a count exceeds 9, this is indicated by a 1 in the tens place of a number.
As we move through the counting principles and get to unitizing, I quickly see how important this understanding is for students when we explore place value, fractions, unit rates, and other big ideas connected to proportional reasoning.
Have I missed any? Have some great insight to add to these descriptions? Please share in the comments!