improving math – A veteran math educator in New York argues that students often master procedures while hesitating to explain why they work, and that classrooms—from Italy to the U.S.—improve when structure, real-world meaning, collaboration, and inquiry are taught together.
A strong student once handled complex equations “flawlessly” — right up until a simple prompt landed: “Why does this method work?” The pause that followed wasn’t about accuracy. It was about reasoning.
After more than 30 years teaching mathematics and physics in Italian scientific high schools. and now working in an international school in New York. Francesco D’Anna. Ph.D.. says the gap he sees isn’t tied to a single country or curriculum.. In his view. it stems from how learning is structured. how concepts are connected. and whether students are pushed to think rather than just produce answers.
Working with approximately 100 students from diverse academic backgrounds, he argues that improving math instruction isn’t about choosing one teaching model over another. Instead, it’s about blending what different approaches do well.
“Start with structure–but make connections visible” is where he places the first lever.. He points to the Italian system’s emphasis on logical progression. where concepts are introduced in sequence and each topic builds on the previous one.. But he adds that structure alone doesn’t automatically help students understand — unless learners can clearly see how ideas connect.. In his own lessons, before introducing limits in calculus, he revisits functions, graph behavior, and algebraic transformations.. He often opens with a prompt: “What do we already know that can help us here?” For him. that shift reduces the feeling that each topic is isolated and helps students approach new material with confidence.
He then turns to the way many U.S.. classrooms emphasize relevance and application.. D’Anna says that when this emphasis is done well. it can support learning noticeably — especially in topics like exponential functions.. Instead of treating them as abstract exercises. he asks students to model real-world situations such as population growth or compound interest.. He describes a visible change in the room: students ask more questions. participate more actively. and persist longer when challenges appear.
But he stresses a boundary: application must reinforce conceptual understanding rather than replace it.. Real-world examples, in his account, work best when they help students grasp why the math process works.. He is careful to frame student difficulty in a specific way — students do not struggle because mathematics is inherently too difficult.. They struggle when it feels disconnected from meaning.
Collaboration is the next tool D’Anna highlights for making thinking visible.. In his classes, he regularly uses group-based problem solving.. Students work in groups of three, with each student 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.. He says the goal isn’t only engagement.. Misconceptions surface earlier, and understanding deepens when students must explain reasoning to others instead of simply arriving at an answer.. He links this approach especially to international classrooms, where students bring different levels of preparation and diverse learning styles.
That diversity leads to another tension he believes educators must plan for: not all students start from the same place.. Differences in prior knowledge, language, and learning habits require flexibility.. For improving math instruction. he says he combines structured introduction of core concepts. guided practice to build confidence. and open-ended problems with multiple entry points.. The intent, he writes, is to keep academic rigor while ensuring all students can meaningfully engage.
Perhaps the most direct shift in his approach is moving inquiry into the center of instruction.. D’Anna says one of his biggest changes has been from answer-focused teaching to inquiry-based learning.. Instead of focusing only on solutions. he asks students: “Is there another way to solve this?” “Which method is more efficient. and why?” and “How can we verify that this result makes sense?” Even brief discussions built around these questions. he says. change how students perceive mathematics — from a set of procedures to memorize into a process of reasoning.. Over time, he adds, that builds both confidence and independence.
One paragraph’s worth of facts runs through the full approach: D’Anna says Italian-style logical progression helps learners develop a coherent framework. U.S.-style real-world application boosts engagement when it reinforces why the process works. and group-based problem solving makes thinking visible by requiring explanation and critique.
For educators looking for something they can do tomorrow, D’Anna argues the change doesn’t have to be systemic.. He points to small classroom strategies that he says can be applied consistently: 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; and ask students to compare methods. not just produce answers.
He concludes with the common thread he says international classrooms bring into focus: improving math instruction isn’t about choosing between structure and flexibility. or between rigor and engagement.. In his telling. it is about integrating frameworks. meaningful contexts. and collaborative opportunities to think — so students develop deeper. more durable understanding.
The message ends where it began, returning to that earlier moment of hesitation over “why.” D’Anna’s final point is that improving math instruction is not about changing everything; it is about connecting what already works.
math instruction inquiry-based learning collaboration curriculum education policy international classrooms STEM teaching methods