Counting and Number Relationships: As easy as 1, 2, 3…or is it?

Recently, Graham Fletcher (@gfletchy) released a new video in his progressions series titled The Progression of Early Number & Counting.  Having seen his progression videos on multiple other mathematical domains, I was eager to see his portrayal of this work. Graham certainly didn’t disappoint!  In a matter of a little more than 7 minutes, he was able to capture the bird’s eye view of the progression of learning to count and how the work of building number relationships supports this learning.  Watching this video reminded me of the rigor that is teaching PreK and Kindergarten.

A common frustration among this group of teachers is the perception that the math they teach is “easy”.  While it is true that these foundations of mathematics have gained automaticity in most adults, we must not forget the challenge of learning a new skill for the first time. Not only that, but we must realize that counting with meaning is not simply uttering the count sequence without error.  Indeed, it is much, much more complex than that!  In the words of David Foster, “Your child can count to 100.  Great!  What other songs can she sing?”  While learning the count sequence is absolutely foundational to the work of counting with meaning, it simply doesn’t stop there.  As children move forward in their learning, they begin to make meaning of their count sequence by corresponding one object to one number and one number to one object.  This is referred to as one-to-one correspondence.  Given a set of objects to count, a student would likely look at/touch/move each object as they utter the sequence of numbers to match.  Students must have an accurate count sequence to be successful in this work!  But they must also be building the skill of matching number to object.

Beyond this is the skill of cardinality, or the ability to state that the last number uttered in the count is the total amount of objects in the collection.  A good test for determining whether a student is counting at the level of one-to-one correspondence or cardinality is to ask them at the end of their count “So…how many are there?”  The student who returns to the collection to count again (“1, 2, 3, 4…” all while looking at you thinking, “Didn’t I just do this?”) is a student working in the capacity of one-to-one correspondence.  The student who simply states the final number of their count as the total is working in the capacity of cardinality.   Having worked with hundreds of young mathematicians over the course of my years in teaching, I have seen evidence of this sequence over and over again and believe strongly in using the progression to support students counting with meaning.  Here is an image of the counting progression, as suggested by Graham’s video.

Image result for progression of counting

What I really grappled with after watching the video was the placement of subitizing, or the ability to immediatley recoginze the quantity of a set of objects.  My original thinking was that subitizing was the end game of counting.  That is to say that once I made it to the summit of my counting sequence progression, I would bury my flag in the sand of subitizing.  It seemed so sensical.  If I had worked and worked to learn how to count a set of objects and truly know the meaning of the numbers I was counting, I should be rewarded with “just knowing” how many were in the set.  So when this video showed subitizing as the jumping point of the progression, you can imagine the cognitive dissonance this caused.  Could it be possible that subitizing was not the end game, but rather the foundation for it all?  How could I have been so off base?

I was fortunate enough to talk to Graham about this progression and I gained some clarity of the progression.  There are two types of subitizing, perceptual and conceptual.  Perceptual subitizing is just as I defined earlier-The immediate recognition of the quantity of a set of objects.


Image result for subitizing

For example, I can look at 6 pips a die and just know that it is 6 without counting.  This skill is often observed in students who have yet to acquire the learning of the counting progression and is therefore used as a foundation for this work.  However, with conceptual subitizing, smaller sets of objects are recogized within the larger set, allowing students to use their understanding of number relationships to find the total amount of a larger set.

Image result for subitizing

In this case, I might see the 6 in this image, just as it appears in the die image above, and the 3 at the top of the image.  Combining the 6 and 3 would give me the total number of dots, 9.  While I could not perceive all the dots as one collection, I could see the subsets and quickly combine them to find the total amount.  My understanding of the number 9 and all the numbers contained in 9 (called Heirarchial Inclusion) supported my ability to conceptually subitize this collection.

The clarity that I walked away with from my conversation with Graham was that while perceptual subitizing serves as a foundation for counting with meaning, conceptual subitizing is the culmination of the progression.  It’s like a subitizing sandwich.  We start with the perceptual subitizing base, layer in counting with meaning, and end with the ability to conceptually subitize.  Obviously, counting is really not as easy as 1, 2, 3 and we need to be mindful of the progression as we facilitate students’ learning in this domain.

One idea that is still swirling in my mind remains after talking with Graham.  During our conversation, he shared this Number Sense Trajectory document that caused me pause.

K-2 Learning Trajectories

As an A-type person (and teacher!), I would really like to believe that learning is linear. Truly, I would live and die by checking boxes off the to-do list of learning if I could.  Alas, my experience has taught me that learning looks a lot more like the image on the right:

Image result for learning not linear

So I began to wonder what a true representation of the counting progression would look like.  During a resent presentation, Tracy Zager (@tzager) spoke about mathematical intuition and shared this image:


The image really struck a chord with me as it was somewhat linear, but also represented the interconnected cycle of many mathematical ideas.  My current wondering is what the counting trajectory would look like if not represented linearly, as it is above.  I worry that a linear representation might lead a teacher to expect mastery of one concept before moving on to the next, as the linear model suggests.  While a student may not have yet mastered the count sequence to 30, for example, I believe strongly that the same student can begin to think about one-to-one correspondence and cardinality within the set of numbers they do know how to count.

By shear virtue of models, they will all be imperfect representations of the true work. However, in the spirit of constantly pushing our thinking forward, is there a model that is closer to the true work?




Published by: jgarner05

I am a math consultant with the Stanislaus County Office of Education and a preservice math content instructor at California State University, Stanislaus. It is my passion to work with teachers to improve math content understanding and instruction.


2 thoughts on “Counting and Number Relationships: As easy as 1, 2, 3…or is it?”

  1. Wouldn’t it make life a bit simpler if the linear checklist trajectory described all learning for all individuals. But I think life and learning would be less interesting too! We don’t all learn the same way. We can see that in the ways young children develop language and learn to walk. But the trajectories are still helpful. We can think of a “hypothetical learning trajectory” as a model that helps us see what misunderstandings or lack of prior knowledge may lead to struggles and helps us develop activities to help students learn .
    As stated in this summary of Cathy Fosnot’s work on trajectories, we can think of “a “hypothetical learning trajectory” from Simon (1995). Learning trajectories are claimed to be “hypothetical” since until students actually attempt a problem, “we can never be sure what they will do or whether and how they will construct new interpretations, ideas and strategies” (Fosnot & Dolk, 2002, p.22).
    In this way, Fosnot & Dolk (2002) do not reject the idea of a learning trajectory, but rather emphasize that a linear model does not account for “real” learning since real learning “is messy” (p.23). Instead, they suggest the metaphor of a landscape. In the landscape of learning, “knowledge of models, strategies, and big ideas,” are essential to organizing a coherent hypothetical learning trajectory. In other words, teachers and curriculum developers still need to have a clear and organized idea of what they want their students to know.

    I think the landscape model can be closer to the true work. It certainly isn’t linear. But I think the linear model can help us in many ways as long as we remember that is is just a guide to help us think about how ideas build.


    1. I can’t agree with you more! One challenge I see is the use of the Landscape of Learning to determine the progress of a student, especially in the area of counting, where the progression from one step to the next has been supported time and again by research. The challenge for me with the linear model is that is doesn’t represent the fact that the progression often starts all over again with each successive number. I seek to find/create a tool that is a responsible balance between these two tools.


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