“The only way to do great work is to love what you do.”

I use this quote from Steve Jobs as my title because this is how I feel about the place I work and the work that I do. Everyday I am tasked with learning more about how children acquire their mathematical knowledge; how do children come to know? Does this sound like a daunting task?  Does this even sound plausible?

What could be more exciting than ultimately affecting the lives of children in the Central Valley here in California? Well, perhaps sitting in a cabana in Tahiti or Costa Rica looking out onto the ocean. However, at this point of my life this is not feasible, so for now I come into work and do what I love to do.

The AIMS Center for Math & Science is dedicated to studying existing research, translating that research, and making it applicable for teachers to use in their classrooms. My task is to do this in the realm of early learning, specifically with children 3 – 5 years of age.  Thus far in my journey as a research associate, I have learned a lot through the research of Karen Fuson, Douglas Clements, Julie Sarama, Beth MacDonald, Ernst VonGlaserfeld, Les Steffe, and Jean Piaget.

By definition, subitizing is “just seeing” how many objects there are in a group, a quick attention toward numerosity when viewing a small set of items. (Clements 1999). For years this was my working definition, however it wasn’t until recently that I had fully grasped the complexity of the developmental progress one goes through in their learning trajectory of number and recognition (one’s ability to subitize).  

img_7791There are two distinct types of subitizing: perceptual and conceptual. Perceptual subitizing, is the ability to recognize a number without using other mathematical processes (Clements 1999) and consists of four different sub-categories: initial perceptual subitizing, perceptual subgroup subitizing, perceptual ascending, and perceptual descending. Conceptual subitizing is where children begin 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. Refer below to Table 1 for a detailed explanation.  

Every day I read and/or analyze student video trying to answer the following questions:

I do not have the answer to these questions, but it is my hope to have answers in the near future. From the research I have read so far, there appears to be a strong relationship between subitizing and mathematical achievement.

Next month I will further discuss subitizng and share a few student stories. In the meantime, please view my subitizing PowerPoint and think about how you recognized the subitizable patterns.


Table 1: (adapted from MacDonald February 2016)

Perceptual Subitizing



Initial perceptual subitzing (IPS)

Child describes the visual motion of the dots.

A child typically says the shape they saw. “I saw a triangle.”

Perceptual subgroup subitizing (PSS)

Child numerically subitize small subgroups of two or three but cannot subitize the entire composite group.

A child typically responds with “I saw two and two.”

Perceptual ascending subitizing (PAS)

Child describes the subgroups and then the composite group.

A child typically responds with “I saw two and two, four.”

Perceptual descending subitizing (PDS)

Child describes the composite groups and then describes the subgroups.

A child typically responds with” I saw five because I saw two and three.”

Conceptual Subitizing

Rigid conceptual subitizer

Child describes seeing the composite unit and then one set of subgroups that always maintains the same regardless of orientation or color.

Typically, a child at this stage will say, “I saw four because I saw two and two.”

Flexible conceptual subitzier

Child describes seeing the composite unit and then two or more sets of subgroups in different task regardless of orientation or color.

Typically, a child at this stage will say, “I saw five because I saw two and three” but in a previous task the child said, “I saw five because I saw two, two and one” for a group of 5.



6 Responses to “The only way to do great work is to love what you do.”

  1. In my work with children learning math, ages 6-16, I have found a few children who are age 6 who have not yet become numerate. They may not relate a numeral such as 4 with the abstract quantity of 4. THey may not be able to give a number before or after a given single digit number. To work with these children, I use large dice, and dominoes. We play many games with these tools that utilize and develop subitizing, with the aim of becoming more numerate.

    • Hi Teresa,

      In working with our preschool children (3 and 4 year olds) we too see that many are not able to coordinate a number word “four” with the numeric symbol “4” with the quantity of 4 items in a collection. In fact they may not be able to do so until they reach the age of 5 or 6, possibly even later.

      Using dice and dominoes are excellent tools to help students recognize spatial patterns, which will ultimately lead to assigning numerosity to a collection of items. As students become familiar with patterns, they will move from being a perceptual subitizer (having to count pips on a die) to being a conceptual subitizer, who is able to describe what they saw when the patterns are flashed, by combining or partitioning.

      I recently was googling subitizing games online and came upon this site that you may want to explore. https://mindfull.wordpress.com/2015/01/24/find-it-a-subitizing-bingo-game/. I am thinking of taking this and adapting to include finger patterns in addition to the dot patterns.

  2. I really liked reading this Blog. As a student in the research class at FPU , this is a great chart to refer to when observing students. I printed this to use as a reference, and added it to my journal. I went back and watch several of the videos of 3-4 year-old subitizing. I was then able to more clearly identify their stage of development.

    • I am so glad that you enjoyed reading this blog and found the chart helpful in identifying where students are in the continuum. In my upcoming blog “Subitizing Part 3” I will share some subitzing games/activities that might be helpful in moving students along the subitizing continuum.

        • Hi again,

          Have you had an opportunity to read part 3 of my series? If you find you need more games/activities than what was shared please send me an email. I would be happy to continue this conversation.

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