That’s not a progression I am familiar with. In trigonometry and precalculus I learned Series then Limits, then in calculus derivative first then integrals. There are definitely new directions to go with Series and Limits after learning integrals, but the concepts are prerequisites to calculus (and arguably trigonometry).
That's the historical progression, not the teaching progression. That's the point. Integrals were conceived over 2000 years ago. Derivatives came Newton-ish. Formalism with limits came last.
There was another step: Leibniz' pre-limits formulation of calculus used hyperreal numbers: infinite and infinitesimals (small-delta) numbers. This version is conceptually simpler and for many people more intuitive than the limits-based explanation. It was the standard approach for many years.
Infinitesimal calculus was discarded in late 19th century pedagogy in favor of limits as no rigorous proofs yet existed of its general correctness, whereas such proofs had been formulated for the limits version. It was suspected that while infinitesimals were a nice intuition, they lacked rigor which might lead to unknown errors.
It was eventually rigorously proven in the 1960s that the infinitesimals approach is indeed as complete and correct as the limits version, but by then standard pedagogy had become entrenched in limits.
I understand that nonstandard analysis is a firm footing on which to build calculus, but is it totally equivalent to real analysis?
I loved real analysis in college, for example I found cantors set kind of crazy. Does that "exist" in nonstandard analysis? Or are there other mind-bending implications?
> I understand that nonstandard analysis is a firm footing on which to build calculus, but is it totally equivalent to real analysis?
This is a bit of a philosophical question which hinges on what the word "equivalent" means. I disagree with the other comment. It is not totally equivalent, or even at all equivalent.
I think about math in terms of set theory. You create sets, like the real numbers, the integers, or otherwise. You can add other elements to those sets. For example:
* You can add a variety of infinities (countable, uncountable, positive, negative, etc.). You have a self-consistent system.
* You can add imaginary numbers, think of the whole thing as a ball, with a zero on one end and infinity on the other. That gives you complex analysis, which is self-consistent as well.
All of those are helpful, useful, and powerful, and generally lead to the same place where they overlap, they're not very compatible with each other, either formally or intuitively.
I view the two formulations of calculus as similar. Even if they lead to the same results, that's not the same as saying they're equivalent. They're different formalisms.
Integral calculus before differential calculus is how it is done in Apostol's "Calculus". Here are the first several chapter titles to give an idea of what is covered when.
Introduction
The Concepts of Integral Calculus
Some Applications of Integration
Continuous Functions
Differential Calculus
The Relation Between Integration and Differentiation
The Logarithm, The Exponential, and The Inverse Trigonometric Functions
Polynomial Approximations to Functions
Introduction to Differential Equations
Complex Numbers
Sequences Infinite Series, Improper Integrals
Sequences and Series of Functions
(the rest of volume I is vector algebra and calculus and linear algebra).
That book was widely used at Caltech, MIT, and several other top schools for decades.
Starting out with integration makes so much more sense, and it's way more graspable from a geometric standpoint (what's the area under this line?) as opposed to the weirdness (and relative abstraction) of the derivative.
Derivatives have nice physical problems to solve like "find the maximum" and "find the relationship between these two variables" which you can find by measuring a linear slope a lot of times. Learning derivatives starts with linear regression, which you do when learning about lines on a Cartesian plane.
By the time you know about drawing lines on planes to find the integral area underneath, you've already figured out the derivative so you know what direction to move your pen in next while drawing the line
> problems to solve like "find the maximum" and "find the relationship between these two variables"
I think these are actually pretty abstract, so it's hard exactly to grasp what a derivative is until maybe Calc II or III -- imo, the main problem is that it just seems like mathematical black magic: you need to define a limit, you need to use that definition to subtract two slopes, the complexity of which magically kind of vanishes, and you end up with a "method" for spitting out derivatives. Whereas finding area under a curve seems like a common sense thing to do: you take smaller and smaller slices of a thing and add it all up.
I do agree that if you really learn anti-derivatives (which, by all accounts, is harder), you basically get derivatives for free.
I totally understand your position, and I completely agree that integration is way more complex, often doesn't work, and very quickly gets you in the Gamma function danger zone, but I still think it should be taught first (maybe just looking at elementary functions) because it's way more intuitive in my opinion.
There's a reason it was developed first, historically speaking.
Freshmen calculus I does not have to immediately jump into mechanically difficult integrals. You can leave those for Calculus II and concentrate on the fundamentals of integration and differentiation in Calc I.
You completely missed my point. I mean the actual mathematical analysis concepts involved in integration (measure theory, Rieman integration, principal value) are far more intricate than the concepts involved in differentiation.
This is the same progression that I learned, but I am not sure it is the best one. I wonder if there has been any rigorous scientific study on the pedagogical benefits one way or another?
Why do you think we don't? Different teachers, institutions, and educators have used differing curricula and systems and styles education for all of history. In modern times in civilized countries (and states) they are even required to update their knowledge over time and interact with other educators and experts to improve their techniques, knowledge, and systems. Large government education bodies regularly try to put a scientific spin on the process but it's often clear they aren't being rigorous or honest with themselves.
If your local educators are not required to participate in this system of continuing education and improving techniques, that's a local government decision and you should seek to fix that.
Fixing it is often hard in the states because there is a not insignificant amount of Americans that see schools as undermining their authority, see education as a useless and undesirable endeavor, often as a "radicalizing" and "socialist" influence and institution, and so they fight back against any improvement of schooling. There are states in the union that allow pretty much anyone, with no qualifications or certifications, to become "educators", and other states in the union that require a masters degree and 12 credits of class work every few years to continue to be an educator. Outcomes often imply a correlation between those requirements and the result, and yet the states with much lower bars of entry do not increase them. Education in the states is a politics issue, for some stupid reason. There are people who believe for whatever reason that middle schoolers are being taught "Critical Race Theory", as if that isn't a college level course that the vast majority of humanities majors don't even experience.
That’s not a progression I am familiar with. In trigonometry and precalculus I learned Series then Limits, then in calculus derivative first then integrals. There are definitely new directions to go with Series and Limits after learning integrals, but the concepts are prerequisites to calculus (and arguably trigonometry).