The paper proposes a rewrite of first-year calculus. It treats differentials like algebraic objects students can manipulate directly, pushes formal limits to the end of the course, and argues that many standard rules for derivatives are really the same operation in disguise. The author’s concrete complaint is that introductory courses force students to memorize several separate procedures, hide what symbols like dy and dx are supposed to mean, and introduce second-derivative notation that breaks the algebraic story the rest of the course is trying to tell. In the comments, the author said this came out of teaching small homeschool calculus classes and later writing a textbook, not from a large institutional rollout. He also said he avoids forcing his alternative higher-derivative notation in the main text because students still have to survive conventional later courses.
The strongest reaction was that the diagnosis is right even if the exact cure is debatable. People broadly agreed that limits are the hardest conceptual jump in beginner calculus, and that the standard path often produces students who can pass tests without really grasping rates of change, approximation, or why the notation works. Several commenters pointed out that calculus itself historically ran on intuition long before
epsilon-delta rigor arrived, so teaching intuition first is not some betrayal of mathematics. What mattered more than allegiance to any one foundation was sequencing. Many thought introductory courses try to do too much, too late, and with the wrong emphasis. Suggestions included shrinking the scope of “intro calculus,” treating common limit rules as givens instead of making epsilon-delta the front door, pairing calculus with physics or statistics to keep the ideas grounded, and rethinking the bloated “precalculus” pipeline.
Where the conversation got more technical was over what to do with dy, dx, and infinitesimals if you want the notation to mean what it looks like it means. One camp said the usual classroom treatment leaves students in an awkward half-state where dy and dx act like quantities when convenient and vanish into limit language when they become inconvenient. That group argued that
nonstandard analysis,
hyperreals,
dual numbers, or related infinitesimal formalisms make the notation more honest and often simpler. Another camp pushed back that this swaps one pedagogical burden for another. Hyperreals drop familiar properties of the real numbers.
Smooth infinitesimal analysis demands
intuitionistic logic and strong smoothness assumptions. For a first course, that can be more machinery than students need. The practical center of gravity was simpler than the formal debate. Visual explanations help some students. Others said visuals are a warm-up, not a substitute for worked examples and repetition. On that narrower point, there was little disagreement. Whatever conceptual frame you choose, students still need lots of exercises, feedback, and contact with real applications for the ideas to stick.
arxiv.org
Calculus from the Ground Up