Key points:
Last year, one of my strongest students could solve complex equations flawlessly–but paused when I asked a simple question: “Why does this method work?”
Her hesitation was not unusual. Many students can perform mathematical procedures accurately, yet struggle to explain the reasoning behind them. This gap between execution and understanding remains one of the central challenges in improving math instruction today.
After more than 30 years teaching mathematics and physics in Italian scientific high schools–and now working in an international school in New York–I have come to see that this challenge is not tied to one country or curriculum. Instead, it reflects how we structure learning, connect concepts, and engage students in thinking.
Each year, working with approximately 100 students from diverse academic backgrounds, I have found that improving math instruction is not about choosing one teaching model over another, but about combining the strengths of different approaches.
Start with structure–but make connections visible
One of the strengths of the Italian education system is its emphasis on logical progression. Concepts are introduced in sequence, and each topic builds on the previous one. This helps students develop a coherent framework for mathematics.
However, structure alone is not enough for improving math instruction unless students clearly see how ideas are connected.
In my classroom, I make these links explicit. Before introducing limits in calculus, for example, I revisit functions, graph behavior, and algebraic transformations. I often begin lessons with a simple prompt:
“What do we already know that can help us here?”
This small shift helps students approach new material with confidence and reduces the perception that each topic is isolated.
Make math meaningful through real-world application
In many U.S. classrooms, there is a strong emphasis on relevance and application. When done well, this can significantly support improving math instruction.
For example, when teaching exponential functions, I ask students to model real-world situations such as population growth or compound interest. Engagement changes immediately: students ask more questions, participate more actively, and persist longer through challenges.
However, application must reinforce–not replace–conceptual understanding. Real-world contexts are most effective when they help students understand why the mathematics process works.
Students do not struggle because mathematics is inherently too difficult. They struggle when it feels disconnected from meaning.
Use collaboration to make thinking visible
Another powerful element in improving math instruction is structured collaboration.
In my classes, I regularly use group-based problem solving:
- Students work in groups of three, each responsible for explaining a part of the solution
- Groups solve the same problem using different methods and compare results
- One student presents while others question and critique reasoning
These activities do more than increase participation. They make thinking visible.
Misconceptions emerge earlier, and understanding deepens when students must explain their reasoning to others rather than simply arrive at an answer.
This approach is particularly effective in international classrooms, where students bring different levels of preparation and diverse learning styles.
Balance consistency with flexibility
Teaching in an international environment highlights a key reality: students do not start from the same place.
Differences in prior knowledge, language, and learning habits require a flexible approach.
For improving math instruction, I combine:
- structured introduction of core concepts
- guided practice to build confidence
- open-ended problems with multiple entry points
This balance maintains academic rigor while ensuring that all students can engage meaningfully with the material.
Encourage inquiry, not just answers
One of the most significant shifts I have made in improving math instruction is moving from answer-focused teaching to inquiry-based learning.
Instead of focusing only on solutions, I ask:
- “Is there another way to solve this?”
- “Which method is more efficient, and why?”
- “How can we verify that this result makes sense?”
Even brief discussions like these change how students perceive mathematics. They begin to see it as a process of reasoning rather than a set of procedures to memorize.
Over time, this builds both confidence and independence.
What educators can do tomorrow
Improving math instruction does not require systemic change. It requires consistent, intentional classroom practices.
Here are small but powerful strategies:
- Begin each lesson by activating prior knowledge
- Connect at least one concept per unit to a real-world context
- Use structured group work where students explain reasoning
- Provide problems with multiple solution paths
- Ask students to compare methods, not just produce answers
These are simple shifts, but when applied consistently, they significantly improve student understanding.
Conclusion
International classrooms show that improving math instruction is not about choosing between structure and flexibility, or between rigor and engagement. It is about integrating these elements thoughtfully.
When students are given clear frameworks, meaningful contexts, and opportunities to think collaboratively, their understanding becomes deeper and more durable.
Ultimately, improving math instruction is not about changing everything. It is about connecting what already works.
