Updated May 20, 2026

There are lots of reasons why your students might struggle with number stories (word problems). 

So far, some of the challenges we’ve explored in previous blogs are:

• Reading comprehension and ELL students
• Fixed math mindsets
• The problems with procedural approaches to number stories

But there’s one foundational idea I haven’t covered yet: the critical learning phases. 

What are critical learning phases?

Developed by Kathy Richardson, the critical learning phases “are the understandings that must be in place if children are going to be successful in the study of mathematics.” (Richardson, 2012) They aren’t skills that can be taught directly; they develop through experience and meaningful interaction with numbers. 

For example, one stage of counting is that students “know how many” after counting. Have you ever watched a student count a collection, then when asked how many, they count again? It’s not because they forgot; it’s because the last number they said doesn’t yet hold meaning for them. They don’t understand that it represents the total amount of objects counted.

I think of each phase as a brick in the mathematical foundation necessary to build number sense. As with any structure, cracks will eventually show if we don’t start with a solid foundation. Students may “appear to be successful in the short term but their lack of meaningful learning eventually shows up.” (Richardson, 2012)

Within each phase, there are varying levels of understanding. In her book How Children Learn Number Concepts, Kathy Richardson breaks down the phases for each of these areas: :

• Understanding counting
• Understanding number relationships
• Understanding addition and subtraction: parts of numbers
• Understanding place value: tens and ones
• Understanding place value: numbers as hundreds, tens, and ones
• Understanding multiplication and division

Why critical learning phases matter more than what students appear to know  

If you’re an early childhood educator, or you work with older children who don’t yet have foundational number sense, the development of number concepts is crucial for success. 

As educators (and parents), it’s easy to mistake what students do with what they know and understand. The critical learning phases address conceptual understandings that may not be immediately apparent. 

If you’ve ever heard a little kid sing the ABC song, you’ve likely heard them sing “lmnop” all lumped together. Yes, they can sing the alphabet song. But do they know that l, m, n, o, and p are five different symbolic representations for distinct letters that compose the alphabet? It’s the same with counting. A child may be able to count to 20, or 100, without understanding that the numbers they say each represent a distinct value.

Instead of working with what we think students know, the critical learning phases help us figure out what students still need to understand. They give teachers a way to tell the difference between performance, getting students to do or say the ‘right’ thing, and actual understanding. 

Now, let’s explore how these phases directly impact the way students approach number stories.

Why some students can read the word problem but still can’t solve it 

So what does any of this have to do with word problems? According to Kathy Richardson, there are 4 things students need to do to solve a number story:

1. Interpret the mathematical situation described in the problem
2. Be able to arrive at the answer in efficient ways, given the numbers presented in the problem
3. Be able to record the problem and their answer
4. Reflect on the answer to be sure that it makes sense

Interpreting situations is exactly where the Structures of Equality (SoE) come in. We both know reading comprehension is often the biggest barrier to students’ success with number stories. This struggle is what led to the creation of SoE. 

To make sense of what’s happening in the story, students needed a way to identify the math main idea and visually represent it with accuracy.

When the issue isn’t reading but the math concept itself 

If you’re like I was as a classroom teacher, you’ve noticed that reading comprehension is an issue in math and have tried lots of different strategies:

• Focusing on vocabulary and context
• Reading the problem aloud to students
• Having students draw or act out what’s happening
• Asking them to identify what the problem is asking them to do

And for some, those strategies work. But for many, they don’t. 

“If children are asked to think about numbers at levels they have not reached, they simply will not be able to comprehend what they are being asked to do.” – Kathy Richardson

(Wow, I read that line twice when I first saw it.)

Richardson gives this example: if a child can’t yet answer “How many more in the red train than the blue train?” they won’t be able to understand a word problem that asks the same thing, like “Emma picks up 4 shells on the beach. Jamie has 6 shells. How many more shells does Jamie have than Emma?” 

In this situation, the inability to work through the story is not about the context. It’s about the mathematical concept. The student has no way to access this problem because they haven’t yet developed the ability to describe the comparison relationship between the numbers, to think about how the two quantities relate to each other.

This matters for how we approach the problem. A teacher might look at the shells question and think the student needs more practice with subtraction. But subtraction isn’t the issue. The student is being asked to compare two quantities and describe how they relate and that’s a different kind of thinking than taking something away or finding what’s left. Labeling stories with comparison relationships as subtraction problems can actually get in the way, because it pushes students toward an operation before they’ve made sense of the relationship. 

Where to start: using PET to build the missing foundations 

If you’re working with students who are still developing these foundational understandings, I’d start with Parts Equal Total (PET). PET is the foundational structure for parts and total thinking. It gives students a way to see how a total can be decomposed into parts, and how parts come together to make a total. That foundation supports everything that comes later, including the kind of comparison thinking the shells problem requires. 

One of the most important parts of teaching students how to use the structures is how you introduce them. Engage students in discussion around the context of the story before you do anything with numbers. The question stems I developed can help guide that conversation.  

This matters even more if your students haven’t yet developed the underlying concepts for counting or number relationships. The conversation itself is the work.  

A student might not be able to solve the problem independently yet. But they can engage in conversation about it, and that engagement is one of the most effective ways to build the concepts they need to move through the critical learning phases. 

What to do with this in your classroom 

There are lots of reasons why number stories are challenging for students. Reading comprehension is often the culprit, but not always. The critical learning phases give you a tool to observe what students actually know and can do, so you can tell whether the challenge is mathematical understanding, reading comprehension, or both.

If you haven’t read How Children Learn Number Concepts, I highly recommend it. (I’m not an affiliate in any way. This is just a personal plug for a great resource.)

Richardson, K. (2012). How Children Learn Number Concepts: A Guide to the Critical Learning Phases. Math Perspectives Teacher Development Center.