Mathematical reasoning helps students understand not just how to solve a problem, but why a strategy works. When students explain their thinking, compare different approaches, and examine strategies that don't work, they build understanding they can apply to new problems.
Consider two eighth-grade students who solve the same problem and arrive at the same answer. One student can explain how they solved it. The other can also examine a classmate's work, identify where the reasoning breaks down, and explain why the approach doesn't work.
Both students found the answer, but the second student is taking an additional step toward deeper mathematical understanding.
This kind of reasoning is central to Connected Mathematics® 4 (CMP4), the problem-based middle school mathematics curriculum developed at Michigan State University. Throughout the curriculum, students explore different strategies, compare their own thinking with others' approaches, and identify connections between the mathematical ideas behind those strategies. These discussions help students understand how different approaches can represent the same concept and how that understanding applies to new problems.
Getting the correct answer is important, but it doesn't always show how well a student understands the mathematical principles behind it. A student might follow a familiar procedure successfully but struggle to apply that knowledge when a problem looks different.
Asking students to explain their reasoning helps teachers see more of what students understand. It also gives students opportunities to make connections, test their ideas, and consider different ways to approach a problem.
When students examine multiple strategies or why one strategy works and another doesn't, they have to think beyond the steps used to find an answer. They must understand the mathematical ideas behind those steps and use that understanding to evaluate whether a solution makes sense. It deepens and expands their understanding to discuss a variety of strategies.
This is one of the goals of problem-based learning. In CMP4, problems offer multiple entry points, giving students opportunities to approach the same mathematical situation in different ways. As the CMP4 Quick Start Guide explains, students "learn mathematics and learn how to learn mathematics."
Working through problems this way helps students make sense of mathematical ideas and develop understanding they can recall and apply in future learning.
When we spoke with the CMP4 authors, one point stood out: they focus on student thinking and strategies rather than simply labeling an answer as a “mistake”.
That distinction can make a difference in the classroom. When a strategy doesn't work, students have an opportunity to examine the strategy, identify what needs to change, and learn from the process. These discussions can also help create an environment where students feel more comfortable sharing their thinking, even when they aren't sure their approach is correct.
Mathematical understanding develops over time. Early in a unit, students may use informal strategies or have only a partial understanding of a concept. As they work through connected problems, they revisit their ideas, refine their approaches, and build toward more formal mathematical understanding.
CMP4 supports this progression through its Arc of Learning ® Framework, which traces how students move from informal ideas toward more formal mathematics.
The curriculum also incorporates student thinking drawn from CMP classrooms and conversations between the authors and teachers. Students can see how other students approach a problem, consider the reasoning behind those approaches, and compare that thinking with their own.
Examples of student conjectures, reasoning, and strategies appear throughout the curriculum. The goal is to help students see themselves as doers, creators, and knowers of mathematics.
When students examine another student's approach, they have more than an answer to consider. They have an opportunity to explain, compare, question, and build on mathematical ideas.
Research on Connected Mathematics ® provides compelling evidence for giving students regular opportunities to explain their thinking, compare strategies, and learn from one another.
Since the Connected Mathematics Project ® began at Michigan State University in 1990, researchers have produced more than 150 research articles, 70 dissertations and theses, and 25 evaluation studies related to the curriculum.
Research comparing CMP students with non-CMP students has reported stronger performance in problem solving, conceptual understanding, and mathematical reasoning among CMP students, with some advantages continuing into high school. Teachers involved in CMP4 field testing have also reported greater student engagement in deep mathematics.
As stated in A Guide to Connected Mathematics ®: Understanding, Implementing, and Teaching:
“Effective teachers ask open-ended questions to elicit student thinking and ask students to explain their thinking and comment on one another’s work (NCTM, 2014; Staples, 2007; Webb et al., 2019).”
These practices give teachers opportunities to learn more about students' understanding while helping students develop the skills to explain and evaluate mathematical reasoning.
For educators interested in bringing problem-based learning into their classrooms, research highlights the value of giving students time to explore problems, try different strategies, and discuss their approaches. These experiences help students build a deeper understanding of mathematical concepts and learn from one another.
The Pool Problem from Problem 1.1, Tiling Pools, in Say It With Symbols: Reasoning with Equivalent Expressions and Equations, Grade 8 Unit 6, offers a practical example of how students can develop mathematical reasoning. It also demonstrates the CMP4 STEM Problem Format in action.
In the Pool Problem, students explore how to represent the number of border tiles needed to surround a square swimming pool. Each tile is one foot square, and the pool has side lengths of s feet.
Students write an expression for the number of tiles needed, then develop a different expression that represents the same situation. This gives them an opportunity to explore equivalent expressions and explain why different expressions can produce the same result.
The problem also introduces expressions developed by students in a fictional classroom. Students examine the expressions and determine which ones are equivalent.
For example, consider these expressions:
Although they look different, all three expressions represent the same number of border tiles. Students can use the Distributive Property to explain why they are equivalent.
Another expression, 4(s+2), counts too many tiles because it counts each corner twice. Recognizing this difference requires students to think about what each part of the expression represents in the actual situation.
The problem then gives students opportunities to examine how other students analyze the expressions. They use different approaches, including substituting a number, comparing the expression to a diagram, applying the Distributive Property, examining graphs, and using tables. Each approach offers a different way to investigate the same mathematical relationship.
By comparing these strategies, students can identify connections between different representations and understand how each approach reveals something about the expressions. For example, the Distributive Property helps students recognize why the expressions are equivalent algebraically, while substituting a number allows them to check whether the expressions produce the same result. Looking at these approaches together helps students connect the calculations to the situation they represent.
Students can then use these connections to explain which expressions are equivalent and why 4(s+2) does not accurately represent the number of border tiles needed. Rather than simply identifying the incorrect expression, they can examine how the reasoning differs and explain what the comparison reveals about the mathematical relationship.
This is where student thinking becomes a valuable learning tool. Students aren't simply checking whether an answer is correct. They're examining the reasoning behind it, making connections between mathematical ideas, and using those connections to strengthen their understanding.
This is what makes the Pool Problem a valuable example of mathematical reasoning in action. Students aren't just learning how to write equivalent expressions. They're connecting mathematical representations to a real situation, evaluating different approaches, and building an understanding they can apply to other problems.
Getting the right answer is important, but understanding the reasoning behind it helps students apply what they've learned to new problems. When students compare strategies, explain their thinking, and examine why an approach works or doesn't, they build connections between mathematical ideas.
Published by Lab-Aids and developed by Michigan State University, Connected Mathematics ® 4 (CMP4) brings this approach to life through problem-based investigations that encourage students to think critically, share their reasoning, and build mathematical understanding. Explore CMP4 to learn more about how it could support meaningful mathematics learning in your classroom.
Mathematical reasoning is the ability to use prior understandings to make sense of a problem, explain an approach, and justify why a solution works. It helps students move beyond finding the correct answer to understanding the mathematical ideas behind it.
Comparing strategies helps students identify connections, consider different approaches, and explain their reasoning. Examining a strategy that doesn't work can also help students recognize misconceptions and develop a stronger understanding of a concept.
CMP4 incorporates student strategies and reasoning from CMP classrooms throughout its problems. Through the STEM Problem Format and the Launch, Explore, Summarize lesson structure, students develop their own approaches, compare strategies, and discuss the mathematical ideas behind their solutions.