This week, I’m thrilled to deliver on a big promise I made to you last time. I will introduce a systematic approach to help students comprehend story problems in math called Structures of Equality.

Last week, I explored four common approaches to solving story problems and assessed their strengths and limitations. Even though numberless word problems and bar models had a myriad of advantages, they all fell short of the ultimate goal – for students to deeply understand word problems.

By the end of this post, you’ll understand the inspiration and the why behind the strategy I developed. And you’ll be on your way to helping students solve story problems with success.

A bit about my journey

Like you, I was frustrated that there was no comprehensive solution that addressed the struggle with reading comprehension in math word problems. When I became a math coach, my teachers expressed the same sentiment. No matter what they tried, there wasn’t a structure to effectively teach all students how to tackle word problems.

It became my mission to figure out a solution. A way that all kids could access story problems, internalize them, and have a strategy to conceptualize and make sense of them. 

The research

It all started with bar models. When I began consistently using these to help students problem solve, I saw an increase in their overall understanding of problems. They were able to make visual representations of quantities and began to critically think about what was happening in the math stories. 

But there was one huge limitation. I noticed it was possible for a student to draw a bar diagram and still not completely understand the relationships in a story.

So I started to do some research and came across George Polya’s book, How to Solve It. (Polya, George. How to Solve It. 2nd ed., Princeton University Press, 2004) I was shocked when I noticed the first edition of this book was written in 1945. For 80 years, educators have been struggling with how to help students approach word problems. Everything he mentioned is still relevant today.

He identified 4 key phases for problem solving. “First, we have to understand the problem; we have to see clearly what is required. Second, we have to see how the various items are connected, how the unknown is linked to the data, in order to obtain the idea of the solution, to make a plan. Third, we carry out our plan. Fourth, we look back at the completed solution, we review and discuss it.”

How Structures of Equality came to be

In addition to using bar models, I was also exploring Cognitively Guided Instruction (CGI). I knew that if I could find a way to merge the two, I could come up with a structure to deepen comprehension. A representation that would be impossible to draw unless students understood the relationships occurring in the story.

The result? I came up with 3 Structures of Equality that could be used to represent any problem type in the elementary Common Core Standards. Simple structures with key features that allowed all students to access and successfully work through word problems. 

When explicitly taught and consistently used, over time students could internalize the process and independently apply it to solve problems. In a couple of weeks, I’ll share some data with you around student growth and proficiency measures.

What are teachers saying?

“For years, I saw the lack of understanding about the relationships between numbers in a given problem blocking their growth.

Structures of Equality prevents this – students must actually see the relationship between the numbers and represent it in consistent, taught, visual structures. From there, the logic of how to solve the problem often becomes apparent. 

Students are excited to understand why the calculation makes sense and enjoy telling the reasoning behind their solution.” 

– Cissy McKissick, WCPSS SPED teacher

So what exactly are the 3 Structures of Equality?

Come back next week where you’ll get an introduction to this reading comprehension tool for solving math problems! I’ll share the Structures of Equality, an overview of how they work, and example problems. Make sure you’ve subscribed so you don’t miss it.

Here’s a little sneak peek.

Conclusion

Fueled by frustration and on a mission to find a better way, I began a journey that led to the innovation of Structures of Equality (SoEs). Born from the fusion of bar models and Cognitively Guided Instruction (CGI), this reading comprehension tool can be applied to solve any problem type in the elementary standards. Testimonials for educators like Cissy McKissick further reinforce the impact of SoEs in fostering reasoning and problem-solving skills.