Math success isn’t about memorizing rules—it’s about thinking mathematically. Research consistently shows that by helping students understand why concepts work, not just how to apply them, educators can unlock deeper comprehension and long-term confidence.
What Is Mathematical Thinking?
Mathematical thinking is about developing habits where students reason, model, and make sense of problems in meaningful ways. When students think mathematically, they learn to:
- Persevere through challenges rather than giving up when problems seem difficult.
- Reason abstractly, connecting numbers and operations to real-world contexts.
- Use mathematics to model and solve real-world problems.
- Strategically apply mathematical and practice tools.
Why Does Mathematical Thinking Matter in Intervention?
In intervention settings, this principle addresses some of the most common barriers to success and lays the foundation for deeper understanding. When students have struggled with math for years, they often approach problems with a sense of defeat or simply wait for someone to show them the steps. This cycle limits growth and keeps students from developing independence.
Mathematical thinking encourages students to grapple with problems, explore multiple strategies, and persist through challenges. Instead of viewing math as a rigid set of rules, students begin to see it as a process of inquiry.
Inquiry-based learning is at the heart of this approach. When students explain their reasoning and engage in mathematical dialogue, they strengthen problem-solving skills and develop the ability to transfer knowledge to new situations. These habits of mind, including perseverance, reasoning, and adaptability, prepare students not only for success in math intervention but for lifelong learning.
The Research Behind Mathematical Thinking
Mathematical thinking is a research-backed approach that strengthens understanding, reasoning, and problem-solving.
Language and Dialogue
Simply memorizing formulas or solving routine problems doesn’t prepare students to apply math in new or complex situations. Seely emphasizes that students need opportunities to explain their reasoning and engage in meaningful mathematical dialogue. When learners articulate their thought process, they deepen understanding and uncover misconceptions that might otherwise go unnoticed.
This process begins with language. Frye et al. states that encouraging students to describe their world mathematically, starting with everyday words before introducing formal vocabulary, creates a bridge from informal ideas to structured concepts. Over time, these conversations evolve into precise mathematical language, which is essential for abstract reasoning and problem solving (Lager). In short, when students learn to express their ideas clearly, they gain the tools to think critically, reason logically, and connect concepts across context.
Models and Representation
Mathematics can come to life when students are able to visualize ideas. Gersten et al. emphasize that external representations, such as diagrams, number lines, and manipulatives, help connect concrete experiences to abstract ideas. They allow students to “see” the math, making invisible relationships tangible and easier to grasp.
Wong highlights that effective instruction often follows a clear progression from concrete to pictorial to abstract. For example, a student learning fractions might begin by physically dividing objects, then move to drawing fraction bars, and to finally representing fractions symbolically. This gradual shift ensures that understanding is built layer by layer.
Singapore’s national mathematics curriculum exemplifies this approach. The Ministry of Education in Singapore positions modeling as a core strategy, not as an optional add-on. By engaging students in concrete, pictorial, and abstract problem-solving, the curriculum fosters deep conceptual understanding and strong problem-solving skills.
Procedural Fluency and Conceptual Understanding
Hierbet and Grouws emphasize that effective teachers create opportunities for students to develop both procedural fluency and conceptual understanding. Procedural fluency is the ability to carry out calculations accurately and efficiently, while conceptual understanding means grasping the underlying principles that make those procedures work. When these two elements work together, students can not only perform calculations but also explain the reasoning behind them, which is critical for solving unfamiliar problems.
The National Research Council outlines four key strands of mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, and adaptive reasoning. These strands ensure that students can move beyond rote learning to flexible thinking, applying knowledge in varied context. Confrey and Krupa argue that achieving this balance requires integrating the Standards for Mathematical Practice into daily instruction.
Why does this matter? When students only learn procedures, they may succeed on routine tasks but struggle when faced with complexity. Conversely, focusing solely on concepts without fluency can lead to frustration and inefficiency. True mathematical strength comes from building a foundation of understanding while practicing the skills that make problem-solving efficient and adaptable.
How Do The Math Embeds Mathematical Thinking
Mathematical thinking is woven into every aspect of Do The Math. Each lesson is designed to help students reason, model, and communicate their ideas while building confidence and independence. Real-world problems are presented in ways that feel accessible and relevant, allowing students to connect abstract concepts to everyday experiences.
Collaboration is a key part of the process. Students share strategies, explain their reasoning, and learn from one another, while also developing the ability to solve problems independently. These experiences foster perseverance, strategic thinking, and adaptability—the foundational habits that lead to long-term mathematical proficiency.
Ready to learn more? Download the full Research Foundations: Evidence and Efficacy Report to explore how mathematical thinking drives success in math intervention and discover how Do The Math can support your students’ growth.
***
This post is part of a series exploring effective strategies for math intervention. Each post highlights a key instructional principle designed to help students thrive in intervention settings.
Read the blog series:
Read the research behind Do The Math’s intervention strategies.
