Counting Principles – Counting and Cardinality

A Progression of Counting and Quantity

Counting Principles - Principles of Counting and Quantity Featured Image

Having spent the majority of my professional life teaching secondary math and mentoring intermediate (grades 7 to 10) math teachers, 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 10 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.

Counting Principles - 1 - Stable Order

You can see this in video format here.

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.

Counting Principles - 2 - Order Irrelevance

You can see this in video format here.

3. Conservation

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.

Counting Principles - 3 - Conservation

You can see this in video format here.

4. Abstraction

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.

Counting Principles - 4 - Abstraction

You can see this in video format here.

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.

Counting Principles - 5 - One to One Correspondence

You can see this in video format here.

6. Cardinality

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.

Counting Principles - 6 - Cardinality

You can see this in video format here.

7. Subitizing

In general, subitizing is the ability to “see” or visualize a small amount of objects and know how many there are without counting. While this idea may seem simple on the surface, subitizing is actually quite complex. If we dig deeper, we can see that there are two types of subitizing that could be going on in our mind when we are learning to count called perceptual subitizing and conceptual subitizing.

Perceptual Subitizing

Perceptual subitizing takes place when you are able to look at a group of objects and know how many objects there are without having to do any thinking. Often times, when we look at groups of 5 objects or less, we are subitizing perceptually.

Examples of perceptual subitizing could include:

  • knowing there are 3 candies on a table without counting the candies
  • knowing you rolled 5 with a single die without counting the dots
  • knowing there are 2 cars in your driveway without counting the cars

Perceptual Subitizing - Counting Principles

Conceptual Subitizing

Conceptual subitizing takes place when you are still able to “see” how many objects are in group, but the number of objects is too large to subitize without decomposing into two or more smaller groups.

We often shift from perceptual subitizing to conceptual subitizing when the number of objects in a group is larger than 5.

You may find that you are able to perceptually subitize groups of more than 5 items when the items are organized in familiar ways. For example, most “know” they have rolled 6 on a single die because of the familiar arrangement of the dots. However, you may struggle to perceptually subitize those 6 dots if they were arranged in an unfamiliar way and resort to conceptually subitizing by breaking up the 6 dots into two groups of 3 in your mind without even realizing it!

While many may believe that using 5- and 10-frames in early mathematics is simply because 5’s and 10’s are very friendly numbers in our base 10 system, we can see that the organization of items in a 5-frame is an important tool to help students shift from perceptual subitizing to conceptual subitizing.

Perceptual and Conceptual Subitizing - Counting and Quantity Principles

You can see this in video format here.

8. Hierarchical Inclusion

Understanding that all numbers preceding a number can be or are systematically included in the value of another selected number.

For example, knowing that within a group of 5 items, there is also a group of 4 items within that group; 3 items within that group; 2 items… and so on.

Hierarchical Inclusion - Counting and Quantity Principles

You can see this in video format here.

9. 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.

Counting Principles - 8 - Movement is Magnitude

You can see this in video format here.

10. Unitizing

Unitizing refers to the understanding that you can count a large group of items by decomposing the group into smaller, equal groups of items and then count those.

For example, if there is a large group of candies on a table, one might choose to create groups (or “units”) of 2 (often doing this by perceptually subitizing these groups) and skip counting up by 2’s. Some may choose to create “units” of 3 and skip count up by 3’s.

Unitizing Groups of 2 and 3 - Counting and Quantity Principles

I like to connect unitizing back to 1-to-1 correspondence by also calling this “2-to-1 correspondence” in the situation where we are counting 2 candies for every 1 group or “3-to-1 correspondence” in the 3 candies for every 1 group situation. I find by thinking of unitizing this way, it can be helpful to see connections to multiplication and the underlying ratios that exist whenever we count anything. Complex, but interesting to think about!

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.

Unitizing Groups of 10 for Place Value - Counting and Quantity Principles

You can see this in video format here.

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, multiplication/division (i.e.: groups of), fractions, unit rates, and other big ideas connected to number sense, measurement and proportional reasoning.

Have I missed any? Have some great insight to add to these descriptions? Please share in the comments!

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