Introducing integral in textbooks

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Discussion Overview

The discussion revolves around the teaching of integral approximations in introductory calculus courses, specifically comparing the trapezoidal rule and the midpoint Riemann sum. Participants reflect on their educational experiences and question the prevalence of certain methods in textbooks, particularly referencing Needham's "Visual Complex Analysis."

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Meta-discussion

Main Points Raised

  • Some participants express surprise at the emphasis on the trapezoidal rule in calculus courses, noting that their own education focused more on the midpoint approximation.
  • Others mention that they learned about Darboux sums, which they referred to as Riemann sums, and did not encounter the trapezoidal rule until a numerical methods course.
  • One participant argues for the inclusion of Lebesgue integration in introductory courses, suggesting that it provides a deeper understanding of volumes compared to traditional methods.
  • Concerns are raised about the accuracy of Needham's claim that the trapezoidal rule is universally taught, with participants questioning the validity of such sweeping statements.
  • Some participants criticize Needham for not distinguishing between calculus and numerical mathematics, suggesting that this oversight diminishes the relevance of his claims.
  • A later reply questions the nature of "geometric" interpretations in the context of teaching integrals.

Areas of Agreement / Disagreement

Participants generally express disagreement regarding the teaching methods and the claims made by Needham. There is no consensus on the best approach to teaching integrals or the accuracy of Needham's statements.

Contextual Notes

Participants note limitations in verifying Needham's claims due to the lack of context provided in the discussion. There is also an acknowledgment of the variability in educational experiences across different regions and institutions.

Who May Find This Useful

Readers interested in educational methodologies in mathematics, particularly in calculus and integration, may find this discussion relevant.

Hill
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I was very surprised to read the following in Needham, Visual Complex Analysis:

"It is therefore doubly puzzling that the Trapezoidal formula is taught in every introductory calculus course, while it appears that the midpoint Riemann sum RM is seldom even mentioned."

I was surprised because I clearly remember that in my school (long time ago, in a country far away) the midpoint approximation of a curve was the main visualization for integral while the trapezoidal one has been mentioned but deemed unnecessary.

I wonder if Needham is right and if so, why is it different?
 
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We had Darboux, called it Riemann, and no trapezoid - and Lebesgue should have been in my opinion.
 
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fresh_42 said:
We had Darboux, called it Riemann, and no trapezoid - and Lebesgue should have been in my opinion.
Why Lebesgue in an introductory calculus course of all places? Fwiw we did Darboux and called it Riemann too (trapezoid was part of a numerical methods course that mostly discussed better methods) but I don't think it helped me much...except in terms of gaining an appreciation for how these things are formalized I guess.
 
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haider said:
Why Lebesgue in an introductory calculus course of all places?
Well, firstly for the pun with yet another name. And I regret that I didn't add Archimedes to the row.

Secondly, I am of the opinion that we should start to teach mathematics at school, nowadays more than ever. WA can do all that silly algorithmic stuff we use to torture kids with. It has never been mathematics, it is algorithmics, maybe computing, or calculating, but definitely not mathematics. Darboux or Riemann or the trapezoid rule are all adjustments of Archimedes, 300 B.C. Just as if we hadn't developed mathematics in the meantime. Archimedes can be taught in 1 or 2 hours. It is the application of volumes. Lebesgue requires the understanding of volumes.

haider said:
Fwiw we did Darboux and called it Riemann too (trapezoid was part of a numerical methods course that mostly discussed better methods) but I don't think it helped me much...except in terms of gaining an appreciation for how these things are formalized I guess.
 
Does anybody here possess Needham's book? I would like to see the entire quotation plus context plus chapter plus page.

"... the Trapezoidal formula is taught in every introductory calculus course ..." is right away wrong as two out of two answers show. And by the way: every and all are never true in real life. I really dislike such claims. So maybe we do Needham wrong here as we cannot verify the claim. And maybe Needham just used it as a rhetorical means for what he really wanted to say.
 
fresh_42 said:
Does anybody here possess Needham's book? I would like to see the entire quotation plus context plus chapter plus page.

"... the Trapezoidal formula is taught in every introductory calculus course ..." is right away wrong as two out of two answers show. And by the way: every and all are never true in real life. I really dislike such claims. So maybe we do Needham wrong here as we cannot verify the claim. And maybe Needham just used it as a rhetorical means for what he really wanted to say.
IMG_0112.jpeg
IMG_0111.jpeg
 
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So Needham is indeed to blame for that casual statement. I would have thought that a British would know that the world is a bit bigger than San Fransisco. Disappointing. Particularly disappointing is the fact that he does not distinguish between calculus and numerical mathematics. The methods that are used in a calculus book are completely irrelevant as both are simply ##O(x).##
 
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fresh_42 said:
So Needham is indeed to blame for that casual statement.
Yes.
I enjoy reading his book, but he makes such statements from time to time.
How right is the following one?

1699733735524.png
 
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What constitutes a "geometric" interpretation, one wonders?
 

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