Teaching calculus in a student-friendly manner involves structuring the curriculum in a way that builds upon students’ prior knowledge and fosters a deeper understanding of the concepts. Reordering calculus topics to start with integration, followed by differentiation, sequences and series, and finally limits can indeed offer several advantages:

**Concrete Applications**: Starting with integration allows students to immediately see the practical applications of calculus, such as finding areas, volumes, and accumulated quantities. This approach provides tangible examples that can help students grasp the relevance and usefulness of calculus concepts from the outset.**Visual Understanding**: Integration often involves geometric interpretations, such as finding the area under curves. Teaching integration first can leverage visual aids and intuitive geometric reasoning, which can make the concepts more accessible and engaging for students.**Natural Progression**: After students have a solid understanding of integration, introducing differentiation builds naturally upon this foundation. Many differentiation rules and techniques, such as the power rule or the chain rule, have direct applications in finding antiderivatives and evaluating integrals.**Problem-solving Skills**: Both integration and differentiation involve problem-solving skills, but they often require different approaches. Starting with integration can help students develop problem-solving strategies and analytical thinking, which they can then apply to differentiation and other calculus concepts.**Sequences and Series**: Introducing sequences and series after differentiation and integration allows students to explore more advanced topics while still building upon their knowledge of calculus fundamentals. Sequences and series involve concepts of limits and convergence, which students can better appreciate after learning about derivatives and integrals.**Limits as a Culminating Concept**: Ending with limits provides a cohesive framework for understanding the underlying principles of calculus. By this point, students have encountered various calculus concepts that naturally lead to the concept of limits, which serves as the foundation for calculus and connects the different branches of the subject.

Overall, reordering calculus topics to start with integration, followed by differentiation, sequences and series, and finally limits can provide a coherent and student-friendly approach to teaching calculus, which also is associated with the historical development of calculus topics. This approach emphasizes practical applications, visual understanding, and a logical progression of concepts, ultimately fostering a deeper comprehension and appreciation of calculus for students. It is great to see a network of cutting-edge and growing schools like Chesterton Academy taking advantage of teaching the crown jewel of mathematics in a pedagogically favorable approach to high school students.