Learning Trajectories & Subitizing: A Conversation with Dr. Douglas Clements
For over 25 years, Dr. Douglas Clements and his wife, Dr. Julie Sarama, have been dedicated to transforming early childhood math education. What started as a four-year project funded by the National Science Foundation became a lifelong mission—one that has shaped how we understand and teach math to young learners.
Recently, our CEO, Adrianne Meldrum, had the privilege of interviewing Dr. Clements to dive deeper into his groundbreaking research on learning trajectories, subitizing, and why early math education is more critical than ever.
Table of Contents
Why Early Math?
When Dr. Clements first began this work, preschool math curricula were scarce. Many educators underestimated the importance of building strong mathematical foundations in the early years. However, research has since revealed that early math skills are the best predictor of later academic success—even more than early literacy.
But simply introducing math concepts wasn’t enough. Dr. Clements and Dr. Sarama wanted to ensure that every activity, lesson, and teaching strategy was rooted in how children naturally develop mathematical thinking.
The Birth of Learning Trajectories
Building on the work of researcher Marty Simon, Dr. Clements and his team developed learning trajectories, a research-based framework that maps out how children progress through different levels of mathematical thinking.
Check out Dr. Douglas Clements’ Learning Trajectories Website Here!
Over the years, Dr. Clements and his team have tested, refined, and expanded their curriculum to reach more children, educators, and schools. Their research continues to guide teachers in designing lessons that are developmentally appropriate, engaging, and effective.
During the conversation, Adrianne highlighted one of her favorite tools from the Learning Trajectories website: an engaging spaceship game where students must briefly view a pattern of dots, hold it in their memory, and then recall what they saw. This type of activity is particularly powerful for building working memory—a skill that is crucial for students who struggle with math.
Dr. Clements explained that this game, along with others in the Learning Trajectories platform, is designed to be both engaging and impactful while supporting math skills.
Subitizing: Why It’s Essential for Early Math Learning
As part of our conversation with Dr. Douglas Clements, we dove into a fascinating topic: subitizing—the ability to instantly recognize the number of objects in a set without counting. While many people think of subitizing as simply identifying small quantities of dots, Dr. Clements and his wife, Dr. Julie Sarama, have redefined and expanded the concept into two distinct types: perceptual subitizing and conceptual subitizing.
What Is Subitizing?
You may have seen a child glance at a small group of objects and instantly say “three” without counting each one. That’s perceptual subitizing—a built-in neurological ability that allows us to recognize small quantities automatically.
Interestingly, this ability is not unique to humans. Research shows that animals, from bees to crows, use a similar skill to navigate and survive.
However, Dr. Clements highlights that not all individuals develop perceptual subitizing in the same way, particularly those with learning differences. This underscores the importance of explicitly teaching and reinforcing number patterns for children who may struggle with this foundational skill.
Beyond the Basics: Conceptual Subitizing
While perceptual subitizing typically works for groups of up to three or four, conceptual subitizing allows us to recognize and mentally organize larger quantities. Instead of seeing seven dots as individual items, a child might see them as three and four combined—forming a part-whole relationship.
This is a game-changer in early math learning. Instead of laboriously counting each dot, children who develop strong conceptual subitizing skills can quickly synthesize number patterns, leading to stronger number sense, flexible thinking, and a deeper understanding of arithmetic.
Why Does This Matter?
Dr. Clements emphasizes that subitizing is not just about speed, but about building foundational number sense. While math education often focuses on quick recall of facts, the true power of subitizing lies in developing automaticity in recognizing number relationships. When children can see how numbers break apart and come together, they are better equipped to understand addition, subtraction, and more advanced mathematical concepts.
Dr. Clements made this clear: every teacher should be using subitizing—not just in preschool and early elementary grades, but well into later years of math instruction.
Why Does Subitizing Matter Beyond Kindergarten?
While subitizing is often introduced in preschool through second grade, Dr. Clements points to research that shows its benefits extend far beyond those early years. In fact, using quick images to strengthen numerical intuition has been shown to improve math performance across all topics, not just in subitizing itself.
For example, research by Grayson Wheatley found that fifth graders who practiced subitizing routinely throughout the school year scored higher on math achievement tests—not just in number sense, but across the board.
Why? Because subitizing strengthens the fundamental quantity sense and part-whole relationships that underlie mathematical thinking at all levels.
Best Practices for Teaching Subitizing
Dr. Clements emphasizes that effective subitizing instruction should be:
📌 Short and Frequent – Instead of long, drawn-out lessons, subitizing should be a quick daily routine—just a few minutes each day. This helps build automaticity without overwhelming students.
Different Stages of Subitizing
Dr. Douglas Clements walked through how subitizing, in particular, unfolds across different stages. While there are many nuanced levels, he outlined four broad developmental stages:
1️⃣ Approximate Number System (ANS)
This foundational ability is hardwired in most humans (and even some animals!). It allows people to differentiate between large quantities without counting, even when factors like size or spacing change. For example, if you quickly see a group of eight dots and another group of sixteen dots, you can instinctively tell which has more—even if the smaller group takes up more space.
However, some students struggle with this innate skill, and research suggests that difficulty with ANS can be an early indicator of math challenges. For these students, Tier 2 and Tier 3 interventions may be necessary to build foundational number sense.
2️⃣ Number Recognizer
At this stage, children begin to associate a visual quantity with a number word—but it’s not automatic yet. If a child sees four dots, they can figure out that it’s four, but it still takes a moment to process. This stage typically develops in toddlers and preschoolers.
3️⃣ Perceptual Subitizing
By ages 3–4, most children become fluent in recognizing small numbers (up to 4 or 5) instantly. If you flash three dots in front of them, they immediately know it’s three—without counting. This is the foundation of strong number sense.
4️⃣ Conceptual Subitizing
After mastering perceptual subitizing, children begin seeing relationships between numbers. For example, they might recognize that an arrangement of 8 dots is made up of two groups of 4, or that 7 can be seen as 5 + 2.
This stage progresses in steps:
How Ten-Frames Support Addition & Subtraction
One of the key takeaways from our discussion was the power of structured arrangements—such as five-frames and ten-frames—for developing addition and subtraction fluency. Instead of relying on flashcards or timed drills, students can build a visual and conceptual understanding of numbers.
🔹 Example: Suppose a child sees 8 dots on a ten-frame and 7 dots on another. With practice, they learn to move 2 dots from the 7 to the 8 to make a full 10, instantly seeing that 8 + 7 = 15.
This process eventually becomes automatic, allowing students to mentally decompose and recompose numbers without relying on rote memorization. When they need a reminder, they can visualize the structure to make sense of the problem, rather than feeling lost.
💡 This is the kind of fluency we should aim for—not just speed, but flexible and deep number sense.
Why Counting is a Critical Part of Learning Trajectories
Why is counting such an essential part of the learning process? Dr. Clements’ answer provided a fascinating look at how these two skills work together to build a strong foundation in number sense.
Counting and Subitizing: Two Powerful Quantifiers
Dr. Clements describes both counting and subitizing as quantifiers—each helping children assign a number to a quantity, but from different directions. When combined, they create a powerful system for understanding numbers.
Here’s how they work together:
✅ Subitizing builds a foundation for recognizing small quantities instantly.
✅ Counting extends far beyond subitizing, allowing students to work with larger numbers and more complex arithmetic.
✅ Together, they reinforce each other, helping students develop stronger number sense, arithmetic skills, and even algebraic thinking.
How Subitizing Strengthens Counting
One common misconception is that kids need to learn to count before they can recognize numbers—but research shows that’s not true!
Many children can recognize four dots as “four” before they fully understand what counting means. However, they may not yet grasp the idea that the last number counted tells how many are in the set.
🟢 Example:
This happens because they haven’t yet connected counting to quantity. But when subitizing is introduced first, kids start to make that connection naturally.
💡 How to Teach It Effectively:
Instead of simply pointing at objects one at a time while counting, Dr. Clements suggests using a more structured approach.
For example:
🖐 Hold four small cubes in your hand and pull them out one at a time while counting aloud:
1️⃣ “One” (child sees 1 cube)
2️⃣ “Two” (child sees 2 cubes)
3️⃣ “Three” (child sees 3 cubes)
4️⃣ “Four” (child sees all 4 cubes together)
💡 Key Insight: Because they have already learned to recognize four dots instantly, they can now connect that visual pattern to the verbal counting sequence—reinforcing that the last number counted tells them how many are in the set.
How Counting Supports Subitizing
Counting also plays a crucial role in checking subitizing accuracy and expanding number understanding.
🔹 Example 1: Verifying a Number
Imagine a child sees seven dots and tries to subitize, but different kids give different answers—one says six, another says eight. How do they check?
👉 By counting! Counting helps them confirm their answer and strengthens their ability to recognize patterns over time.
🔹 Example 2: Adding with a Combination of Counting & Subitizing
Let’s say a child is solving 5 + 4.
✅ If they can subitize 5, they can start from 5 and count up: 6, 7, 8, 9.
✅ This approach is much faster and more efficient than counting everything from 1.
Counting alone can be inefficient, especially when solving problems like 8 + 5:
🚫 Counting from 1: The child counts all the way to 13 but loses track of the original numbers.
✅ Using subitizing and counting together:
This reduces cognitive load and makes mental math easier, setting students up for stronger arithmetic skills.
Exploring LearningTrajectories.org: A Free Resource for Teachers & Families
1️⃣ Professional Development for Teachers
Under the “Resources” tab, there’s a section dedicated to teachers filled with:
✅ Videos that break down essential math concepts like subitizing, counting, shapes, measurement, and patterns.
✅ Insights from Dr. Clements & Dr. Julie Sarama, capturing the same ideas they would share in a live professional development session.
✅ A clear progression of learning levels, helping teachers understand where students are and how to support their next steps.
Bonus: When you open a specific topic (like counting, subitizing, or patterning), look in the upper right corner—there’s a “Learn About” section with a quick video of Dr. Clements explaining why that topic is important.
2️⃣ Resources for Families
LearningTrajectories.org isn’t just for educators—it also includes a section specifically for families and parents!
🔹 Click on Resources → Families/Parents to find:
📌 Short videos & documents that explain key early math concepts.
📌 Simple, research-backed activities that families can use at home.
📌 Ways to support children’s math learning through play—no worksheets required!
This is a great tool for parents who want to reinforce math skills in a fun and meaningful way. Whether you’re an educator or a family member, these resources help make math learning feel natural and engaging.
The Bajillions Project—More Games, More Resources, More Learning!
During the conversation with Dr. Clements, he shared an exciting upcoming project that educators and families will want to keep on their radar: Bajillions!
What is Bajillions?
🔹 A new software product designed to provide even more resources for teachers.
🔹 Includes a wider range of games at multiple levels, helping kids build math skills in a fun, engaging way.
🔹 Features games that are more diagnostic, allowing teachers to better understand where students are in their mathematical development.
Want to Stay in the Loop?
If you’re interested in updates about Bajillions, Dr. Clements invites you to reach out directly via email at douglas.clements@du.edu to be added to their list.
This is an exciting development in math education, and we can’t wait to see how Bajillions helps students build strong conceptual foundations in math!
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Jennie Miller
Marketing Assistant
is our Marketing Assistant and content creator here at Made for Math. Jennie loves being part of a company that is working to make mathematics accessible to children with dyscalculia.