# Subitizing, Part 2

In my last blog I mentioned that there are two distinct types of subitizing – perceptual and conceptual. I am fascinated by the subtle differences that students show and what that means about their thinking.

Perceptual subitizing is the ability to recognize a number without using other mathematical processes (Clements 1999) and there are four sub-categories within this stage: initial perceptual subitizing, perceptual subgroup subitizing, perceptual ascending, and perceptual descending.
Conceptual subitizing is where a child begins to attend to the subgroups based on how the items are clustered or symmetrically arranged (MacDonald 2013) and has two sub-categories: rigid conceptual and flexible conceptual subitizing.

Over the last month I have had the opportunity to interview preschoolers between the ages of 3 and 4 on their ability to subitize, specifically trying to identify where they are along this progression. In my interviews I not only seek to see if they give the correct number word response to a small collection of objects, but I also try to see into their little black boxes (their heads) to understand how they saw it. Questions I use to get a look at their thinking are, “How do you know?” “What did you see?” “How did you see it?” “Can you make what you saw (using counters or on paper)?”

Just a few weeks ago, I had the chance to visit with my friend’s grandchildren. Maria is 4 years, 4 months and this was our conversation about this first card:

Me: “What did you see?”
Maria: “I saw 5.”
Me: “How did you see it?”
Maria: “I saw two, two, and one.”
Me: “I also saw five, but in a different way. Do you want to look at it again?”
Maria: “Ok…” I proceeded to show the card again and again she said, “I saw 5.”
Me: “How did you see it?”
Maria: “I saw two, two, and one.”

Omar, who just turned 4, when shown the same configuration said this:

Me: “What did you see?”
Omar: “I saw a square with a dot in the middle.”
Me: “How many dots did you see?”
Omar: “Five. I saw four and one in the inside.
Me: “I see how you saw that.”
Omar: “Hey, I see two on the top, one in the middle, and two on the bottom, too.”
Me: “Yeah, I can see it that way too.”

I then showed him another card, and this was his response:

Omar: “Hey, that’s five too.”
Me: “How do you know?”
Omar: “I saw three on top and two on the bottom.”
Me: “I see how you saw that.”

I showed him one more card and here is his response:

Omar: “That’s five, too.”
Me: “How did you see it?”
Omar: “Two and three. It’s the same as the other one… You turned it.”

Both children fall in the category of a Conceptual Subitizer, with Maria being a Rigid Conceptual Subitizer based on her ability to describe what she saw as a composite unit (five) followed with one set of subgroups (two, two, and one). Omar on the other hand can be classified as a Flexible Conceptual Subitizer based on his ability to describe the composite unit (five) and provide three sets of subgroups.

Coming up in my next blog… I will discuss why a child’s ability to subitize is important to developing a deep understanding of mathematics and offer some strategies to help move students along the trajectory toward flexible conceptual subitizing.

### 3 Responses to Subitizing, Part 2

1. It is valuable information to know that kids can and do move in and out of perceptual and conceptual subitizing. The ability to recognize a number without counting is a skill that is needed before they learn to count. An important part of knowing where a child lies in ability is to ask how they got the answer. It is very interesting that students can see the same pattern of dots , yet form it differently when subitizing.

2. Leslie Smith says:

Learning about perceptual and conceptual subitizing in our research class brought up many questions for me. Looking at student’s ability and what category they fall in posed the questions for me. While observing student’s subitizng some student seem to fluctuate between stages. One thing I did learn is that some students could subitize three’s but then struggled with five’s and couldn’t subitize seven’s at all. The bigger numbers definitely play a role in their ability.

• Liz Gamino says:

Yes, I too have seen students who appear to be between stages depending on the numbers in which we are working within. If students are unable to subitize numbers within five then numbers such as seven (as you mentioned) can not be subitized.