More Than Just Answers: Mathematical Communication in Secondary Classrooms

By Constance Hallemeier

Welcome to the first in our three-part series on the Standards for Mathematical Practice (SMPs) in the secondary classroom! The SMPs are the heartbeat of math instruction; they describe the ways of knowing and habits of mind that we want to cultivate in our students as they grow into mature mathematical thinkers. While the content standards tell us what students need to know, the practices tell us how they should engage with and do mathematics.

Over the next few posts, we will explore bundles of these practices to help focus our instruction. This series is for secondary educators, and a parallel series for the elementary level is being posted separately to provide a full K-12 perspective. We begin our journey with the powerful pair that forms the foundation of mathematical discourse: the Communicating Reasoning bundle, featuring SMP 3 and SMP 6. 

As secondary math educators, we know the power of a correct answer. But true mathematical proficiency goes far deeper. It lies in the process—in the ability to clearly articulate reasoning, evaluate solutions, and use mathematical language accurately. This is the heart of Mathematical Practice Standards 3 and 6, and focusing on these in our classrooms can transform students from passive equation-solvers into confident, articulate mathematical thinkers.

Standard 3: The Art of Communicating Reasoning

Construct viable arguments and critique the reasoning of others.

Oh, this is the Geometry Proof standard – isn’t that what viable arguments entail? However, MP3 is so much more than just proofs. This standard is about building a classroom culture where mathematical thinking is a shared, visible practice. It moves students beyond simply “getting the right answer” to understanding why that answer is correct and being able to defend their approach—or respectfully challenge a peer’s. It’s about choosing a method for solving a proportion, calculating the slope, finding zeros of a polynomial, or factoring a quadratic and justifying that method. It’s also understanding that there is more than one way to approach these problems. From early years in math education, students should be encouraged to talk about their problem solving and thinking as well as evaluating or solving problems using a variety of methods.

MP3 in Action

Students who master constructing arguments and critiquing others can:

• explain their thinking using a variety of tools: graphs, tables, digital graphing tools, objects, drawings, actions, examples/non-examples, and context.
• provide multiple representations, approaches, or error analysis for problems. 
• compare strategies by actively listening, asking clarifying questions, and identifying the mathematical connections between different methods.
• give feedback and ask questions of each other about solutions.

Practical Classroom Strategies:

Strategy	Goal
5 Whys	Deepen the explanation of why a step was taken, moving beyond surface-level descriptions.
Error Analysis	Develop critical thinking as students identify, correct, and explain the mathematical flaw in a sample problem.
Critique, Correct, and Clarify	Analyze other student’s work, critique the methods, fix any error, and then clearly explain the correct process or alternative process.
Decide and Defend	Presents students with two or more solutions and requires them to choose the correct one and write a justification.
Counter-examples	Encourage students to provide counterexamples to illustrate flawed logic.
Sentence Stems	Provides scaffolds for constructive dialogue (e.g., “I agree with ____ because…”, “A step I’d like to about is…”, “I did the problem this way _____ and you did it like ______…”).

By incorporating these routines, students are empowered to not only understand their own strategy but to meaningfully engage with the strategies of their peers.

Standard 6: The Power of Precision

Attend to precision.

Oh, this is the standard about labeling our answers, right? However, precision is so much more than labels. Precision is the hallmark of a careful and clear mathematician, including usage of vocabulary and presenting the solution to a problem in terms of the context. In secondary math, this means more than just accurate calculation—it involves the exact use of language, symbols, and labels. Math is a language, and MP6 requires students to use it fluently and correctly.

MP6 in Action

Careful and clear mathematicians consistently:

• use precise math vocabulary.
• employ symbols that have meaning (e.g., the difference between = and ≈).
• include labels and units of measure in their answers.
• embed expectations of how precise a solution needs to be based on the context of the problem.
• ensure their calculations are accurate and efficient.

Practical Classroom Strategies:

Strategy	Goal
Three Reads	A reading strategy to focus on the problem context, then the mathematical structure, and finally the precise question being asked.
Word Walls	Clearly display essential vocabulary so students can reference terminology and expect them to use the vocabulary (e.g. “I used the thing with the parentheses” vs. “I used the distributive property.”
Collaborative Vocabulary Use	Create opportunities (like paired problem-solving or gallery walks) that require students to use the specific vocabulary in their discussion or written responses.
Required Identification	Consistently require students to identify variables and appropriate units or labels (e.g., “Let t be the time in seconds, and h(t) be the height in meters”).
Stronger or Clearer Each Time	To provide a structured and interactive opportunity for students to revise and refine both their ideas and their verbal and written output.

Building Mathematical Communication Together

Teachers

How can you plan questions that push students beyond “what” they did to “why” it works? In what ways can you explicitly model precise mathematical language and hold students accountable for using it with each other?

Coaches

During classroom observations, what evidence do you see of students building on, questioning, or refining each other’s ideas? What talk structures or tasks could you suggest to teachers to create more opportunities for this kind of discourse?

Administrators

How do you foster a school-wide culture where reasoning and precision are valued in every math classroom? How can you support teachers with high-quality instructional materials that are designed to promote the Standards for Mathematical Practice?

Secondary math teachers can build on the strategies mentioned in the Elementary article about mastering communication in the math classroom to help students continue to grow as mathematicians. Mastering communication standards is essential for success beyond the math classroom. When we prioritize clear articulation by “Constructing viable arguments and critiquing the reasoning of others.” and meticulous detail by “Attending to precision”, we are preparing students for college-level work, professional environments, and analytical thinking in all aspects of life.

How will you adjust your lesson plans this week to prioritize clear communication over just the final answer?

Our next post will explore Modeling and Data Analysis (SMPs 2, 4, and 5), followed by a final look at Problem Solving (SMPs 1, 7, and 8).

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