Subitizing: More Than Meets the Eye

Subitizing is a relatively new concept for me. Sadly, it is not an integrated part of the mathematics curriculum yet. Therefore, children often have little experience with subitizing. When I first learned about this concept, I thought it pertained only to kindergarten, first grade, and struggling students beyond those years. However, I have now seen that children of all ages benefit from subitizing.

What is subitizing, and why is it important? Clements and Sarama (2009) define two types of subitizing. The first, perceptual subitizing, pertains to the ability to both perceive intuitively and simultaneously the amount in small number sets. No counting is neccessary, you just know the amount when you see it. Children develop the prerequisite skills for perceptual subitization at a young age. According to Clements and Sarama, children begin naming collections of 1, 2, and 3 from ages 1-2. By age three, children can also create collections made of 1-3 objects, sometimes 4. Perceptual subitization up to a collection of 4 occurs at age four, and the recognition of sets of 5 develops at age five. To encourage the development of perceptual subitizing, parents and teachers should play snapshot games where children see an organized picture of dots, squares, and other simple geometrical shapes organized in a linear fashion for couple of seconds, then determine the number.

The second type, conceptual subitizing, relates to the ability to instantly see the parts, and join them together to make a whole. For example, given the picture of five arranged with three and two dots, a conceptual subitizer would see 3 and 2, and know that makes 5. Again, no mathematical operations may be consciously enacted, but rather, an instant recognition that the parts make that whole. As children progress, subitizing helps with the visualization of operations and mental math. Conceptual subitizing to five and ten begins at age five. By age six, children are able to conceptually subitize to 20. At this age, five and ten-frames are helpful organizers for promoting subitizing at these higher levels. Skip and counting and place value with subitization begin at age seven, and by age eight, children see multiplicative relationships, such as 5 groups of 10, and 4 groups of 3, which makes 50 and 12, so 62 dots. As you can see, children benefit from subitizing activities well into third grade and beyond. Below is a table to organize these milestones in subitizing.

Trajectory of Subitizing

Age	Type of Subitizing	Example
1-2	Precursor to subitizing: Name small collections up to 3	I see two grandmas.
3	Precursor to subitizing: Create small collections up to 3 or 4	I can count three crackers.
4	Perceptual up to 4	**** I see four stars.
5	Perceptual to 5	***   I see five stars.    **
5	Conceptual to 5	***  I see three and two **   stars. There are five stars
5	Conceptual to 10	* * * * * * * I see 3, 3, and 1, which makes 7.
6	Conceptual to 20	* * * * * * * * * * * * * I see 5, 5, and 3, so that makes 13.
7	Conceptual with place value and skip counting	I saw tens and twos, so 10, 20, 30, 32, 34.
8	Conceptual with place value and multiplication	I saw groups of tens and fours, so 5 tens makes 50, 4 fours makes 16, so 66.

All information adapted from Clements and Sarama (2009)

I encourage you to try out subitizing with your students or children. Where are they in their development? I have seen eight year olds that are conceptual with place value and multiplication, and 14-year olds who are at perceptual to 4. Experiences with numbers, both counting and visualizing, are crucial for number development, regardless of the age. Once foundations are set, subitizing can be used to teach more complex topics. I have used both subitizing and arrays to develop understanding of multiplication, algebraic properties, and division. If you are new to teaching Common Core, then my new unit on teaching multiplication using subitizing and array cards may be just for you! These cards are intended to be part of a program that also includes problem solving. You can check out my 17 page free download to see for yourself!

Clements, D.H., Sarama, J. Learning and Teaching Early Math The Learning Trajectories Approach (2009). Routledge: New York, NY.