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How to Reach Diverse Learners with Differentiated Math Instruction
In a math class, no two students are the same. Teachers see a wide range of skills and abilities in their classrooms,...
Fostering mathematical discourse in the classroom is key to helping students understand and enjoy math.
It can instill the concepts, skills, and thought processes necessary for a student’s long-term mathematical success.
By encouraging open conversations, teachers can create a lively learning environment where students share ideas, solve problems together, and see how math connects to their daily lives.
This blog answers what mathematical discourse is, why it matters, and different strategies for the classroom.
Math discourse is the communication students and teachers use to process and explain mathematical concepts.
This intentional communication prompts students to think deeply and reason their way through problems. Far from general classroom conversation, math discourse is focused discussion that helps students explore, understand, and articulate mathematical ideas.
“When students participate in math discourse, they learn to exercise that communication muscle, explore and defend ideas, and retain concepts that otherwise would have slipped through the cracks.”
Picture a classroom of students learning how to calculate the area of rectilinear shapes.
The teacher asks them to determine the area of an L-shaped figure and explain their process. One student, Maya, shows the class how she separated the L-shape into two separate rectangles, multiplied the width and height of each rectangle, then added them together.
Another student, Jessie, raises his hand and asks how she decided where to split the figure into two rectangles. He notes that his two rectangles look different than how Maya separated hers.
Mateo joins the discussion to say that he thinks as long as you find a way to separate the L-shape into rectangles without leaving any gaps or counting the same space twice, you will get the correct answer. He concludes there can be more than one way to divide the L-shape and still get the right answer.
In this example, students have the freedom to express their thoughts, ask questions, and approach the problem in their own ways.
This natural flow of discussion not only bolsters their mathematical understanding, it also sharpens their critical thinking and problem-solving skills.
Research confirms what math teachers know intuitively: students need to be able to effectively communicate about math in order to fully understand it.
When students participate in math discourse, they learn to exercise that communication muscle, explore and defend ideas, and retain concepts that otherwise would have slipped through the cracks.
Even complex math vocabulary, when applied to the conversational context of math discourse, can resonate and “stick” with students.
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“Students who learn to articulate and justify their own mathematical ideas, reason through their own and others’ mathematical explanations, and provide a rationale for their answers develop a deep understanding that is critical to their future success in mathematics and related fields.” –Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary Schools |
Aside from the well-known benefits of auditory learning, math discourse also facilitates active learning. When students listen thoughtfully to peers and form their own responses, they’re engaging more deeply with lessons and learning how to be positive participants in their class.
Not only does this yield better in-class results, it also builds a foundation for a more confident math identity that will serve them for years to come.
One well-documented article from the Proceedings of the National Academy of Sciences (PNAS) supports the importance of active learning by analyzing 225 studies focusing on teaching methods for undergraduate STEM students.
The study compared how course performance changed with traditional lecturing versus active learning methods. Results demonstrated that on average, students in traditional lecture courses were 1.5 times more likely to fail than students in courses with active learning. For math teachers, this means incorporating intentional discourse into the lesson plan is well worth the time, and may lead to greater success than stand-alone lectures.
While most teachers understand and support math discourse abstractly, putting it into practice in the classroom can feel daunting. Productive discourse does require intentional planning and practice, but a few simple strategies are often all it takes to get conversations rolling. Try these tips in your next class:
If students struggle to vocalize ideas, a few thoughtful questions can be the subtle support they need. These could come from several angles. Here’s a quick table to reference with some ideas for strategic questions to ask to get the conversation going in your math class:
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Strategic question type |
Example |
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Questions that prompt reasoning and defense |
How did you know to invert the fraction to multiply?” |
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Questions that scaffold instruction |
“How is this problem similar to what we practiced last week?” |
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Questions for hypothesizing |
“Will the pattern for solving this equation change if we switch numbers?” |
When in doubt, there are some extra questions to always have at the ready to keep the momentum of the conversation going.
The following back-pocket questions can be used anytime to help guide discussion and student thinking:
Productive math discourse can’t occur if students don’t have sufficient time to think. When introducing a question or problem for discussion, give students quiet reflection time to process their thoughts and form questions.
This leads to richer discussion and helps students retain concepts. This could be as little as two minutes, or longer depending on the complexity of the topic. Teachers can also use this time to anticipate student responses and formulate questions.
The best math discourse often happens in pairs and small groups, where students have the time and space to converse without the intimidation of the whole class.
One-on-one peer discussions enable advanced students to delve deeper into the subject, while quieter students or those struggling with the material are encouraged to step up in a more intimate setting. Similarly, small groups provide an excellent opportunity for differentiation, as teachers can assign groups according to diverse student needs.
Teachers can circulate while students discuss, observe conversational patterns, and interject when necessary.
Structured prompts steer conversations in the right direction and help students express ideas clearly and confidently. After presenting a problem to solve, give pairs or groups of students sentence starters to use alongside their discussion. Consider questions such as:
One of the most important lessons math students can learn is how to approach problems in different ways. In math discourse, students see how others solve problems and how different methods complement each other.
They also learn how to defend their own methods while being open to new ideas.
One way to do this is to present a challenge to the class, then divide students into groups based on how many solution pathways the problem has.
For example, tell students you have 24 feet of fencing to build a rectangular garden. Ask them what dimensions the garden would need to be in order to give you the largest area.
Divide students into three groups and give each a starting point: one team draws a diagram, another lists dimensions in a table, and another writes equations. Have students discuss their methods in their groups, then select a leader from each group to present their findings to the larger class.
Ask students how their methods were different from others, what made them similar, and which method made the most sense to them.
In order to reach a deeper level of mathematical understanding, students need to do more than simply memorize processes. They need to explain why their methods work.
By explaining their reasoning, students clarify their thinking and more easily uncover gaps in their understanding. Teachers can help students justify their answers by asking questions like:
Discourse is an invaluable part of math education in any grade–beginning as early as kindergarten and advancing through high school and beyond. Teachers from each grade can adapt their discourse strategies to meet the skills and developmental stages of their students.
This is the stage where students explore different methods for solving problems and solidify their own understanding of key concepts. Teachers should take time to reinforce each student’s position out loud, help them fill gaps, and clarify what they struggle to articulate. Sentence frames are especially useful in elementary math discourse.
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Example: Ask the class to compare ¾ and ⅝ and determine which is bigger. Student A does this by drawing two circles and shading in pieces, while another student writes out the fractions and changes ¾ to 6/8. Ask another student to articulate what student C did to make the fractions the same, and how a visual drawing helped determine the size of the larger fraction. |
Students can now shift from general method exploration towards more critical assessment. Help students justify their answers by asking intentional questions that promote critical thinking about their own and others’ reasoning. Offer sufficient time for 1:1 and small group discussion, so every student has the time and space to make their voice heard.
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Example: Show students three tile designs: Design 1 uses five tiles, Design 2 uses eight tiles, and Design 3 uses eleven tiles. Ask them to identify a rule for Design n. One student believes the rule should be 3n, since the second and third designs each increase by three, but another student disproves this by applying the rule to Design 1. Another member of the group figures out that adding two to 3n works for each design. The group collectively decides that “3n+2” is the rule for Design n. Wrap up this discussion by asking the group to identify what the first student focused on (the rate of change), and what the other two students focused on (the starting value). |
Discourse now takes on a more formal argumentation structure, allowing students to deepen their reasoning, generalize, and make predictions. Students at this stage explore whether methods work for every case, or only specific ones. Teachers can facilitate meaningful discourse by asking students to both defend and critique their reasoning, and provide more formal presentation opportunities.
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Example: Have students debate whether a quadrilateral is a rhombus if it has perpendicular diagonals Student A claims the statement is true based on standard drawings, but Student B disproves this by drawing a kite as a counterexample. Student C notes that the statement is only true if the diagonals also bisect each other. At this point, you can give the rest of the class the opportunity to respond to Student C’s conjecture using triangle congruency theorems. |
As student needs change and new demands arise, there may be seasons when math discourse becomes more challenging than at other times. Here are some common obstacles to productive discourse and how to tackle them:
Limited time: Keep discourse brief and predictable with 5-10 minute daily routines. Quick exercises like What Doesn’t Belong and Number Talks give students the opportunity to express and share ideas without taking up too much time.
Uneven participation: Put students in pairs and ask each person to synthesize the other’s idea. This eliminates whole-class hand raising, which typically favors talkative students, and gives less enthusiastic students the constructive “pressure” to participate.
Reluctant speakers: Provide enough processing time for the challenge and give students a few minutes to write out their responses before articulating them to the class or small group.
Varied readiness: Scaffold discourse by offering tiered entry points that gradually build to deeper conversations. Start with simple, accessible questions: What information do we have, and what do we still need? What visual patterns do you see? Change sentence frames according to readiness levels, starting with “I notice that…” and building to “my method is more effective because…”
Students talking in class, even when on topic, lacks the structure and depth of authentic math discourse. Guided by the teacher and supported by classroom strategies, math discourse encourages students to process, understand, and articulate mathematical reasoning.
Students need time and practice to become confident in math discourse. Offer short, consistent opportunities and accessible questions that help reluctant participants become familiar and confident in the process.
High-quality math discourse reveals new ideas, brings thoughtful questions to the surface, and encourages even reluctant speakers to use their voice. Pay close attention to small-group discussions and notice how students respond to the problem or the sentence structures. Are their answers short and uninterested, or are they thoughtful and productive to the rest of the group?
Math discourse is an invaluable tool in transforming a group of passive students into active lifelong math learners.
Giving students the opportunity to express ideas and think critically about math concepts deepens their mathematical understanding, builds their confidence, and makes for a more enjoyable classroom experience.
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