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 key idea I haven’t talked about yet, the critical learning phases and their related understandings as foundational for math success.

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.” They “are not skills that can be directly taught to children”. 

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.”

As students progress through the development of number concepts, there are varying levels of understanding within each one. In her book, How Children Learn Number Concepts, Kathy Richardson breaks down the phases for each of the following 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 are critical learning phases so important?

If you’re an early childhood educator, or you work with older children who are struggling in math, 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 they still need to understand regarding a particular concept. Teachers will be able to recognize the difference between getting their students to do or say something that gives the appearance of knowledge and evidence that shows what they really know.”

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

How do critical learning phrases and word problems (number stories) relate?

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

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 into play. 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. I knew students needed a way to visually represent the situations that were occurring in number stories. To help make sense of them and interpret what was happening.

How the critical learning phases impact the ability to interpret mathematical situations

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.”

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

“For example, if a child is not yet able to answer the question, “How many more in the red train than the blue train?” they will not be able to understand a word problem that asks the same question such as “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. Students have no way to access this problem because they haven’t yet developed the ability to describe the relationship between the numbers.

How SoE can help students develop the understandings necessary to move through the critical learning phases

Since the easiest types of number stories to solve can be represented with a Parts Equal Total structure, I’d start there. The question stems I developed can help with your classroom discussion.

One of the most important parts of teaching students how to use the structures successfully is how you introduce them. It’s essential that you engage students in discussion around the context of the situation in a number story. 

If the reason your students are struggling is because they haven’t yet developed the underlying concepts necessary for them to understand counting or number relationships, this will be especially important. 

While they might not yet be able to solve independently, they can engage in conversation around number stories. This is actually one of the most effective ways to help build the concepts they need to progress through the critical learning phases.

Conclusion

There are lots of factors to consider when trying to figure out why your students struggle with number stories. While reading comprehension is often the culprit, this is not always the case. We can use the critical learning phases as a tool to guide our observations of what our students know and can do. This helps us tease out whether the challenges being faced are a result of mathematical understanding or reading comprehension.

If you haven’t read How Children Learn Number Concepts, it’s one I’d highly recommend. (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.