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Integrals -> Derivatives -> Series -> Limits? That’s not a progression I am familiar with. In trigonometry and precalculus I learned Series then Limits, then i

Integrals -> Derivatives -> Series -> Limits? 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).

blagie 4y ago
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.
- pjlegato 4y ago
  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. More on this at https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infinitesimal_Approach https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini... , the modern textbook that re-introduced infinitesimals.
  - makr17 4y ago
    For the formal underpinnings of the infinitesimal approach, the phrase to look for is "Nonstandard Analysis": https://en.wikipedia.org/wiki/Nonstandard_analysis https://en.wikipedia.org/wiki/Nonstandard_analysis Fun stuff.
    - drewrv 4y ago
      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?
      - hansvm 4y ago
        Standard math is a subset of nonstandard, and you can prove the existence of NSA within ZFC, so they are exaxtly equivalent.
      - blagie 4y ago
        > 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.
    - zozbot234 4y ago
      Edward Nelson's Internal Set Theory is a somewhat cleaner and more intuitive foundation that's practically equivalent to non-standard analysis.
mhuffman 4y ago
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?
- hammock 4y ago
  Yeah why don’t we do more a/b testing or controlled testing in education?
  - stevofolife 4y ago
    Start up idea
  - mrguyorama 4y ago
    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.
hirvi74 4y ago
I remember when I took calculus we studied those topics in the order of: Limits -> Derivatives -> Integrals -> Series.
- whimsicalism 4y ago
  This is the standard AP calculus approach although you might be familiar with non infinite series from before.
tzs 4y ago
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.
- dvt 4y ago
  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.
  - 77pt77 4y ago
    I completely disagree. Also, there's basically one way to differentiate (not really, but kind of) wheres integration is so tricky there are many, many ways to do it.
    - dvt 4y ago
      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.
    - abdullahkhalids 4y ago
      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.
deleted 4y ago
[deleted]

blagie 4y ago

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.

pjlegato 4y ago
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. More on this at https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infinitesimal_Approach https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini... , the modern textbook that re-introduced infinitesimals.
- makr17 4y ago
  For the formal underpinnings of the infinitesimal approach, the phrase to look for is "Nonstandard Analysis": https://en.wikipedia.org/wiki/Nonstandard_analysis https://en.wikipedia.org/wiki/Nonstandard_analysis Fun stuff.
  - drewrv 4y ago
    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?
    - hansvm 4y ago
      Standard math is a subset of nonstandard, and you can prove the existence of NSA within ZFC, so they are exaxtly equivalent.
    - blagie 4y ago
      > 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.
  - zozbot234 4y ago
    Edward Nelson's Internal Set Theory is a somewhat cleaner and more intuitive foundation that's practically equivalent to non-standard analysis.

mhuffman 4y ago

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?

hammock 4y ago
Yeah why don’t we do more a/b testing or controlled testing in education?
- stevofolife 4y ago
  Start up idea
- mrguyorama 4y ago
  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.

hirvi74 4y ago

I remember when I took calculus we studied those topics in the order of: Limits -> Derivatives -> Integrals -> Series.

whimsicalism 4y ago
This is the standard AP calculus approach although you might be familiar with non infinite series from before.

tzs 4y ago

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.

dvt 4y ago
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.
- 77pt77 4y ago
  I completely disagree. Also, there's basically one way to differentiate (not really, but kind of) wheres integration is so tricky there are many, many ways to do it.
  - dvt 4y ago
    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.
  - abdullahkhalids 4y ago
    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.

deleted 4y ago

[deleted]