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Calculus without Limits C. K. Raju Introduction The size of calculus texts Calculus without Limits: The difficulty of limits the Theory The difficulty of defining R Current pedagogy of the calculus: a critique Imitating the European

slide-1
SLIDE 1

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Calculus without Limits: the Theory

Current pedagogy of the calculus: a critique

  • C. K. Raju

Inmantec, Ghaziabad and Centre for Studies in Civilizations, New Delhi

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SLIDE 2

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Calculus without Limits: the Theory

Current pedagogy of the calculus: a critique

  • C. K. Raju

Inmantec, Ghaziabad and Centre for Studies in Civilizations, New Delhi

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SLIDE 3

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Outline

Introduction The size of calculus texts The difficulty of limits The difficulty of defining R Imitating the European experience The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

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SLIDE 4

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The size of calculus texts

◮ Typical early calculus texts (e.g., Thomas1, Stewart2)

today have over 1300 pages in large pages (and small type).

  • 1G. B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano,

Thomas’ Calculus, Dorling Kindersley, 11th ed., 2008

2James Stewart, Calculus: early Transcendentals, Thomson

books, 5th ed, 2007.

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SLIDE 5

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The size of calculus texts

◮ Typical early calculus texts (e.g., Thomas1, Stewart2)

today have over 1300 pages in large pages (and small type).

◮ Thomas = 1228 + 34 +80 + 14 + 6 + 6 + xvi (=1384)

pages; size 11 × 8.5 inches.

  • 1G. B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano,

Thomas’ Calculus, Dorling Kindersley, 11th ed., 2008

2James Stewart, Calculus: early Transcendentals, Thomson

books, 5th ed, 2007.

slide-6
SLIDE 6

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The size of calculus texts

◮ Typical early calculus texts (e.g., Thomas1, Stewart2)

today have over 1300 pages in large pages (and small type).

◮ Thomas = 1228 + 34 +80 + 14 + 6 + 6 + xvi (=1384)

pages; size 11 × 8.5 inches.

◮ Stewart = 1168 + 134 + xxv pp. (= 1327) pages; size

10 × 8.5 inches + CD).

  • 1G. B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano,

Thomas’ Calculus, Dorling Kindersley, 11th ed., 2008

2James Stewart, Calculus: early Transcendentals, Thomson

books, 5th ed, 2007.

slide-7
SLIDE 7

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The size of calculus texts

◮ Typical early calculus texts (e.g., Thomas1, Stewart2)

today have over 1300 pages in large pages (and small type).

◮ Thomas = 1228 + 34 +80 + 14 + 6 + 6 + xvi (=1384)

pages; size 11 × 8.5 inches.

◮ Stewart = 1168 + 134 + xxv pp. (= 1327) pages; size

10 × 8.5 inches + CD).

◮ At an average of 10 pages per day (and 45 days per

term) a student will take about 3 terms to read these texts!

  • 1G. B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano,

Thomas’ Calculus, Dorling Kindersley, 11th ed., 2008

2James Stewart, Calculus: early Transcendentals, Thomson

books, 5th ed, 2007.

slide-8
SLIDE 8

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The size of calculus texts

◮ Typical early calculus texts (e.g., Thomas1, Stewart2)

today have over 1300 pages in large pages (and small type).

◮ Thomas = 1228 + 34 +80 + 14 + 6 + 6 + xvi (=1384)

pages; size 11 × 8.5 inches.

◮ Stewart = 1168 + 134 + xxv pp. (= 1327) pages; size

10 × 8.5 inches + CD).

◮ At an average of 10 pages per day (and 45 days per

term) a student will take about 3 terms to read these texts!

◮ At the end what does the student learn?

  • 1G. B. Thomas, Maurice D. Weir, Joel Hass, Frank R. Giordano,

Thomas’ Calculus, Dorling Kindersley, 11th ed., 2008

2James Stewart, Calculus: early Transcendentals, Thomson

books, 5th ed, 2007.

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SLIDE 9

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The difficulty of defining limits

◮ Surprisingly little!

  • 3J. V. Narlikar et al. Mathematics: Textbook for Class XI, NCERT,

New Delhi, 2006, chp. 13 “Limits and Derivatives”, p. 281.

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SLIDE 10

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The difficulty of defining limits

◮ Surprisingly little! ◮ Understanding a simple calculus statement

d dx sin(x) = cos(x),

  • 3J. V. Narlikar et al. Mathematics: Textbook for Class XI, NCERT,

New Delhi, 2006, chp. 13 “Limits and Derivatives”, p. 281.

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SLIDE 11

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The difficulty of defining limits

◮ Surprisingly little! ◮ Understanding a simple calculus statement

d dx sin(x) = cos(x),

◮ needs a definition of d dx .

  • 3J. V. Narlikar et al. Mathematics: Textbook for Class XI, NCERT,

New Delhi, 2006, chp. 13 “Limits and Derivatives”, p. 281.

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SLIDE 12

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The difficulty of defining limits

◮ Surprisingly little! ◮ Understanding a simple calculus statement

d dx sin(x) = cos(x),

◮ needs a definition of d dx . ◮ However, Indian NCERT class XI text says:

First, we give an intuitive idea of derivative (without actually defining it). Then we give a naive definition of limit and study some algebra of limits3

  • 3J. V. Narlikar et al. Mathematics: Textbook for Class XI, NCERT,

New Delhi, 2006, chp. 13 “Limits and Derivatives”, p. 281.

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SLIDE 13

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The formal definition of limits

◮ On present-day mathematics, the symbol d dx is

defined for a function f, using another symbol limh→0.

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SLIDE 14

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The formal definition of limits

◮ On present-day mathematics, the symbol d dx is

defined for a function f, using another symbol limh→0.

◮

df dx = lim

h→0

f(x + h) − f(x) h .

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SLIDE 15

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The formal definition of limits

◮ On present-day mathematics, the symbol d dx is

defined for a function f, using another symbol limh→0.

◮

df dx = lim

h→0

f(x + h) − f(x) h .

◮ limh→0 is formally defined as follows.

lim

x→a g(x) = l

if and only if ∀ ǫ > 0, ∃ δ > 0 such that 0 < |x − a| < ǫ ⇒ |g(x) − l| < δ, ∀x ∈ R.

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SLIDE 16

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The missing element

◮ The texts of Thomas and Stewart both have a

section called “precise definition of limits”.

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SLIDE 17

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The missing element

◮ The texts of Thomas and Stewart both have a

section called “precise definition of limits”.

◮ But the definitions given are not precise.

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SLIDE 18

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The missing element

◮ The texts of Thomas and Stewart both have a

section called “precise definition of limits”.

◮ But the definitions given are not precise. ◮ They have the ǫ’s and δ’s.

slide-19
SLIDE 19

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The missing element

◮ The texts of Thomas and Stewart both have a

section called “precise definition of limits”.

◮ But the definitions given are not precise. ◮ They have the ǫ’s and δ’s. ◮ But are missing one key element: R.

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SLIDE 20

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The formal reals

Dedekind cuts

◮ Formal reals R often built using Dedekind cuts.

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SLIDE 21

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The formal reals

Dedekind cuts

◮ Formal reals R often built using Dedekind cuts. ◮ Set theory provides a model for formal natural

numbers N, which provide a model for Peano arithmetic.

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SLIDE 22

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The formal reals

Dedekind cuts

◮ Formal reals R often built using Dedekind cuts. ◮ Set theory provides a model for formal natural

numbers N, which provide a model for Peano arithmetic.

◮ N can be extended to the integers Z.

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SLIDE 23

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The formal reals

Dedekind cuts

◮ Formal reals R often built using Dedekind cuts. ◮ Set theory provides a model for formal natural

numbers N, which provide a model for Peano arithmetic.

◮ N can be extended to the integers Z. ◮ This integral domain Z can be embedded in a field of

rationals Q.

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SLIDE 24

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

contd.

◮ Finally, Q can be used to construct cuts.

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SLIDE 25

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

contd.

◮ Finally, Q can be used to construct cuts. ◮ α ⊂ Q is called a cut if

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SLIDE 26

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

contd.

◮ Finally, Q can be used to construct cuts. ◮ α ⊂ Q is called a cut if

  • 1. α = ∅, and α = Q.
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SLIDE 27

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

contd.

◮ Finally, Q can be used to construct cuts. ◮ α ⊂ Q is called a cut if

  • 1. α = ∅, and α = Q.
  • 2. p ∈ α and q < p ⇒ q ∈ α
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SLIDE 28

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

contd.

◮ Finally, Q can be used to construct cuts. ◮ α ⊂ Q is called a cut if

  • 1. α = ∅, and α = Q.
  • 2. p ∈ α and q < p ⇒ q ∈ α
  • 3. ∃ m ∈ α such that p ≤ m

∀p ∈ α

slide-29
SLIDE 29

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

contd.

◮ Finally, Q can be used to construct cuts. ◮ α ⊂ Q is called a cut if

  • 1. α = ∅, and α = Q.
  • 2. p ∈ α and q < p ⇒ q ∈ α
  • 3. ∃ m ∈ α such that p ≤ m

∀p ∈ α

◮ +, . , and < among cuts defined in the obvious way.

slide-30
SLIDE 30

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

contd.

◮ Finally, Q can be used to construct cuts. ◮ α ⊂ Q is called a cut if

  • 1. α = ∅, and α = Q.
  • 2. p ∈ α and q < p ⇒ q ∈ α
  • 3. ∃ m ∈ α such that p ≤ m

∀p ∈ α

◮ +, . , and < among cuts defined in the obvious way. ◮ May be readily shown that the cuts form an ordered

field, viz., R.

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SLIDE 31

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

contd.

◮ Finally, Q can be used to construct cuts. ◮ α ⊂ Q is called a cut if

  • 1. α = ∅, and α = Q.
  • 2. p ∈ α and q < p ⇒ q ∈ α
  • 3. ∃ m ∈ α such that p ≤ m

∀p ∈ α

◮ +, . , and < among cuts defined in the obvious way. ◮ May be readily shown that the cuts form an ordered

field, viz., R.

◮ Called “cuts” since Dedekind’s intuitive idea

  • riginated from Elements 1.1.
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SLIDE 32

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Elements 1.1

The fish figure

W E N S

Figure: The fish figure.

◮ With W as centre and WE as radius two arcs are

drawn, and they intersect at N and S.

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SLIDE 33

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Elements 1.1

The fish figure

W E N S

Figure: The fish figure.

◮ With W as centre and WE as radius two arcs are

drawn, and they intersect at N and S.

◮ Used in India to construct a perpendicular bisector to

the EW line and thus determine NS.

slide-34
SLIDE 34

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

Rejection of empirical methods of proof

◮ Elements, I.1 uses this figure to construct the

equilateral triangle WNE on the given segment WE.

slide-35
SLIDE 35

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

Rejection of empirical methods of proof

◮ Elements, I.1 uses this figure to construct the

equilateral triangle WNE on the given segment WE.

◮ Empirically manifest that the two arcs must intersect

at a point.

slide-36
SLIDE 36

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

Rejection of empirical methods of proof

◮ Elements, I.1 uses this figure to construct the

equilateral triangle WNE on the given segment WE.

◮ Empirically manifest that the two arcs must intersect

at a point.

◮ This appeal to empirical methods of proof was

rejected in the West.

◮ R required for formal proof.

slide-37
SLIDE 37

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

Rejection of empirical methods of proof

◮ Elements, I.1 uses this figure to construct the

equilateral triangle WNE on the given segment WE.

◮ Empirically manifest that the two arcs must intersect

at a point.

◮ This appeal to empirical methods of proof was

rejected in the West.

◮ R required for formal proof. ◮ If arcs are drawn in Q × Q they may “pass through”

each other, without there being any (exact) point at which they intersect,

slide-38
SLIDE 38

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

Rejection of empirical methods of proof

◮ Elements, I.1 uses this figure to construct the

equilateral triangle WNE on the given segment WE.

◮ Empirically manifest that the two arcs must intersect

at a point.

◮ This appeal to empirical methods of proof was

rejected in the West.

◮ R required for formal proof. ◮ If arcs are drawn in Q × Q they may “pass through”

each other, without there being any (exact) point at which they intersect,

◮ since there may be “gaps” in the arcs, corresponding

to the “gaps” in rational numbers.

slide-39
SLIDE 39

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

Key properties of R

◮ With cuts, wherever the arc is “cut” there is always a

point (number), and never a gap.

slide-40
SLIDE 40

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

Key properties of R

◮ With cuts, wherever the arc is “cut” there is always a

point (number), and never a gap.

◮ Formally, corresponds to the least upper bound (lub)

property of R: if A ⊂ R is bounded above, then it has a lub in R (that is, ∃m ∈ R such that a ≤ m, ∀a ∈ A and if a ≤ m1, ∀a ∈ A then m ≤ m1.

slide-41
SLIDE 41

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Dedekind cuts

Key properties of R

◮ With cuts, wherever the arc is “cut” there is always a

point (number), and never a gap.

◮ Formally, corresponds to the least upper bound (lub)

property of R: if A ⊂ R is bounded above, then it has a lub in R (that is, ∃m ∈ R such that a ≤ m, ∀a ∈ A and if a ≤ m1, ∀a ∈ A then m ≤ m1.

◮ And other similar properties.

slide-42
SLIDE 42

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Cauchy sequences

Alternative construction of R

◮ Alternative approach via equivalence classes of

Cauchy sequences in Q.

slide-43
SLIDE 43

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Cauchy sequences

Alternative construction of R

◮ Alternative approach via equivalence classes of

Cauchy sequences in Q.

◮ {an} is called Cauchy sequence if ∀ǫ > 0, ∃N such

that |an − am| < ǫ, ∀n, m > N.

slide-44
SLIDE 44

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Cauchy sequences

Alternative construction of R

◮ Alternative approach via equivalence classes of

Cauchy sequences in Q.

◮ {an} is called Cauchy sequence if ∀ǫ > 0, ∃N such

that |an − am| < ǫ, ∀n, m > N.

◮ The decimal expansion of a real number is an

example of such a Cauchy sequence.

slide-45
SLIDE 45

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Cauchy sequences

Alternative construction of R

◮ Alternative approach via equivalence classes of

Cauchy sequences in Q.

◮ {an} is called Cauchy sequence if ∀ǫ > 0, ∃N such

that |an − am| < ǫ, ∀n, m > N.

◮ The decimal expansion of a real number is an

example of such a Cauchy sequence.

◮ For xinR, x ∈ Q, the decimal expansion neither

terminates nor recurs.

slide-46
SLIDE 46

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Alternative construction of R

contd

◮ Let {an}, {bn} be Cauchy sequences. We say

{an} ~{bn}, if an − bn → 0. That is, ∀ǫ > 0, ∃N such that |an − bn| < ǫ, ∀n > N.

slide-47
SLIDE 47

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Alternative construction of R

contd

◮ Let {an}, {bn} be Cauchy sequences. We say

{an} ~{bn}, if an − bn → 0. That is, ∀ǫ > 0, ∃N such that |an − bn| < ǫ, ∀n > N.

◮ ~ is an equivalence relation, and we define +, . , and

< in Q/~in the obvious ways to get R

slide-48
SLIDE 48

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Alternative construction of R

contd

◮ Let {an}, {bn} be Cauchy sequences. We say

{an} ~{bn}, if an − bn → 0. That is, ∀ǫ > 0, ∃N such that |an − bn| < ǫ, ∀n > N.

◮ ~ is an equivalence relation, and we define +, . , and

< in Q/~in the obvious ways to get R

◮ R is complete: every Cauchy sequence in R

converges.

slide-49
SLIDE 49

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Alternative construction of R

contd

◮ Let {an}, {bn} be Cauchy sequences. We say

{an} ~{bn}, if an − bn → 0. That is, ∀ǫ > 0, ∃N such that |an − bn| < ǫ, ∀n > N.

◮ ~ is an equivalence relation, and we define +, . , and

< in Q/~in the obvious ways to get R

◮ R is complete: every Cauchy sequence in R

converges.

◮ This is equivalent to the lub property of R.

slide-50
SLIDE 50

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Imitating the European experience

◮ Teaching R is regarded as too complicated and is

postponed to texts on advanced calculus4 or mathematical analysis.5

4e.g. D. V. Widder, Advanced Calculus, 2nd ed., Prentice Hall, New

Delhi, 1999.

5e.g. W. Rudin, Principles of Mathematical Analysis, McGraw Hill,

New York, 1964.

slide-51
SLIDE 51

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Imitating the European experience

◮ Teaching R is regarded as too complicated and is

postponed to texts on advanced calculus4 or mathematical analysis.5

◮ Notice that this repeats the European historical

experience.

4e.g. D. V. Widder, Advanced Calculus, 2nd ed., Prentice Hall, New

Delhi, 1999.

5e.g. W. Rudin, Principles of Mathematical Analysis, McGraw Hill,

New York, 1964.

slide-52
SLIDE 52

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Imitating the European experience

◮ Teaching R is regarded as too complicated and is

postponed to texts on advanced calculus4 or mathematical analysis.5

◮ Notice that this repeats the European historical

experience.

◮ Calculus came first, the ǫ–δ definition of limits

followed, and then R was constructed.

4e.g. D. V. Widder, Advanced Calculus, 2nd ed., Prentice Hall, New

Delhi, 1999.

5e.g. W. Rudin, Principles of Mathematical Analysis, McGraw Hill,

New York, 1964.

slide-53
SLIDE 53

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Imitating the European experience

◮ Teaching R is regarded as too complicated and is

postponed to texts on advanced calculus4 or mathematical analysis.5

◮ Notice that this repeats the European historical

experience.

◮ Calculus came first, the ǫ–δ definition of limits

followed, and then R was constructed.

◮ (Cauchy 1789-1857, Dedekind 1831-1916)

4e.g. D. V. Widder, Advanced Calculus, 2nd ed., Prentice Hall, New

Delhi, 1999.

5e.g. W. Rudin, Principles of Mathematical Analysis, McGraw Hill,

New York, 1964.

slide-54
SLIDE 54

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The problem of set theory

◮ The construction of R requires set theory.

slide-55
SLIDE 55

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The problem of set theory

◮ The construction of R requires set theory. ◮ Students are not taught the definition of a set.

slide-56
SLIDE 56

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The problem of set theory

◮ The construction of R requires set theory. ◮ Students are not taught the definition of a set. ◮ What the student learns about set theory is

slide-57
SLIDE 57

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The problem of set theory

◮ The construction of R requires set theory. ◮ Students are not taught the definition of a set. ◮ What the student learns about set theory is ◮ What the student typically learns is something as

follows. “A set is a collection of objects”

  • r

“A set is a well-defined collection of objects”

slide-58
SLIDE 58

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Differing ideas of sets

◮ Students have differing ideas of what “collection”

means.

slide-59
SLIDE 59

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Differing ideas of sets

◮ Students have differing ideas of what “collection”

means.

◮ They often think elements in a set must somehow be

related.

slide-60
SLIDE 60

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Differing ideas of sets

◮ Students have differing ideas of what “collection”

means.

◮ They often think elements in a set must somehow be

related.

◮ Can a human being and an animal can be included

in a set?

slide-61
SLIDE 61

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Differing ideas of sets

◮ Students have differing ideas of what “collection”

means.

◮ They often think elements in a set must somehow be

related.

◮ Can a human being and an animal can be included

in a set?

◮ As a beginning teacher I tried explaining that it was

possible to put Indira Gandhi and a cow in a set.

slide-62
SLIDE 62

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Differing ideas of sets

◮ Students have differing ideas of what “collection”

means.

◮ They often think elements in a set must somehow be

related.

◮ Can a human being and an animal can be included

in a set?

◮ As a beginning teacher I tried explaining that it was

possible to put Indira Gandhi and a cow in a set.

◮ The students disagreed.

slide-63
SLIDE 63

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Differing ideas of sets

◮ Students have differing ideas of what “collection”

means.

◮ They often think elements in a set must somehow be

related.

◮ Can a human being and an animal can be included

in a set?

◮ As a beginning teacher I tried explaining that it was

possible to put Indira Gandhi and a cow in a set.

◮ The students disagreed. ◮ The management was even more unhappy.

slide-64
SLIDE 64

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Differing ideas of sets

contd

◮ The management told me not to teach wrong things.

slide-65
SLIDE 65

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Differing ideas of sets

contd

◮ The management told me not to teach wrong things. ◮ They cited the authority of a professor in Bombay

University.

slide-66
SLIDE 66

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Differing ideas of sets

contd

◮ The management told me not to teach wrong things. ◮ They cited the authority of a professor in Bombay

University.

◮ I cited the authority of Bourbaki.

slide-67
SLIDE 67

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Differing ideas of sets

contd

◮ The management told me not to teach wrong things. ◮ They cited the authority of a professor in Bombay

University.

◮ I cited the authority of Bourbaki. ◮ (They had not heard of Bourbaki, so I resigned!)

slide-68
SLIDE 68

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

What set theory the student learns

◮ With such a loose definition it is not possible to

escape things like Russell’s paradox.

6e.g. P

. R. Halmos, Naive Set Theory, East-West Press, New Delhi, 1972.

slide-69
SLIDE 69

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

What set theory the student learns

◮ With such a loose definition it is not possible to

escape things like Russell’s paradox.

◮ Let

R = {x|x / ∈ x} .

6e.g. P

. R. Halmos, Naive Set Theory, East-West Press, New Delhi, 1972.

slide-70
SLIDE 70

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

What set theory the student learns

◮ With such a loose definition it is not possible to

escape things like Russell’s paradox.

◮ Let

R = {x|x / ∈ x} .

◮ If R ∈ R then, by definition, R /

∈ R so we have a contradiction.

6e.g. P

. R. Halmos, Naive Set Theory, East-West Press, New Delhi, 1972.

slide-71
SLIDE 71

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

What set theory the student learns

◮ With such a loose definition it is not possible to

escape things like Russell’s paradox.

◮ Let

R = {x|x / ∈ x} .

◮ If R ∈ R then, by definition, R /

∈ R so we have a contradiction.

◮ On the other hand if R /

∈ R then, again by the definition of R, we must have R ∈ R, which is again a

  • contradiction. So either way we have a contradiction.

6e.g. P

. R. Halmos, Naive Set Theory, East-West Press, New Delhi, 1972.

slide-72
SLIDE 72

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

What set theory the student learns

◮ With such a loose definition it is not possible to

escape things like Russell’s paradox.

◮ Let

R = {x|x / ∈ x} .

◮ If R ∈ R then, by definition, R /

∈ R so we have a contradiction.

◮ On the other hand if R /

∈ R then, again by the definition of R, we must have R ∈ R, which is again a

  • contradiction. So either way we have a contradiction.

◮ Paradox is supposedly resolved by axiomatic set

theory, but even among professional mathematicians, few learn axiomatic set theory.

6e.g. P

. R. Halmos, Naive Set Theory, East-West Press, New Delhi, 1972.

slide-73
SLIDE 73

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

What set theory the student learns

◮ With such a loose definition it is not possible to

escape things like Russell’s paradox.

◮ Let

R = {x|x / ∈ x} .

◮ If R ∈ R then, by definition, R /

∈ R so we have a contradiction.

◮ On the other hand if R /

∈ R then, again by the definition of R, we must have R ∈ R, which is again a

  • contradiction. So either way we have a contradiction.

◮ Paradox is supposedly resolved by axiomatic set

theory, but even among professional mathematicians, few learn axiomatic set theory.

◮ Most make do with naive set theory.6

6e.g. P

. R. Halmos, Naive Set Theory, East-West Press, New Delhi, 1972.

slide-74
SLIDE 74

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The integral

◮ what about

  • f(x)dx?
slide-75
SLIDE 75

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The integral

◮ what about

  • f(x)dx?

◮ Most calculus courses define the integral as the

anti-derivative, with an unsatisfying constant of integration. if d dx f(x) = g(x) then

  • g(x)dx = f(x) + c
slide-76
SLIDE 76

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The integral

◮ what about

  • f(x)dx?

◮ Most calculus courses define the integral as the

anti-derivative, with an unsatisfying constant of integration. if d dx f(x) = g(x) then

  • g(x)dx = f(x) + c

◮ It is believed that some clarity can be brought about

by teaching the Riemann integral obtained as a limit

  • f sums.

b

a

f(x)dx = lim

µ(P)→0 n

  • i=i

f(ti)∆xi

slide-77
SLIDE 77

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The integral

◮ what about

  • f(x)dx?

◮ Most calculus courses define the integral as the

anti-derivative, with an unsatisfying constant of integration. if d dx f(x) = g(x) then

  • g(x)dx = f(x) + c

◮ It is believed that some clarity can be brought about

by teaching the Riemann integral obtained as a limit

  • f sums.

b

a

f(x)dx = lim

µ(P)→0 n

  • i=i

f(ti)∆xi

◮ Here the set P = {x0, x1, x2, . . . xn} is a partition of

the interval [a, b], and ti ∈ [xi, xi−1].

slide-78
SLIDE 78

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The integral

contd

◮ Once more defining the Riemann integral requires a

definition of the limits.

slide-79
SLIDE 79

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The integral

contd

◮ Once more defining the Riemann integral requires a

definition of the limits.

◮ and a proof of the fundamental theorem of calculus.

slide-80
SLIDE 80

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The integral

contd

◮ Once more defining the Riemann integral requires a

definition of the limits.

◮ and a proof of the fundamental theorem of calculus. ◮ This is not done. Instead, the focus is on mastering

techniques.

slide-81
SLIDE 81

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The integral

contd

◮ Once more defining the Riemann integral requires a

definition of the limits.

◮ and a proof of the fundamental theorem of calculus. ◮ This is not done. Instead, the focus is on mastering

techniques.

◮ the two key techniques of (symbolic) integration are

slide-82
SLIDE 82

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The integral

contd

◮ Once more defining the Riemann integral requires a

definition of the limits.

◮ and a proof of the fundamental theorem of calculus. ◮ This is not done. Instead, the focus is on mastering

techniques.

◮ the two key techniques of (symbolic) integration are

◮ integration by parts (inverse of Lebiniz rule), and

slide-83
SLIDE 83

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The integral

contd

◮ Once more defining the Riemann integral requires a

definition of the limits.

◮ and a proof of the fundamental theorem of calculus. ◮ This is not done. Instead, the focus is on mastering

techniques.

◮ the two key techniques of (symbolic) integration are

◮ integration by parts (inverse of Lebiniz rule), and ◮ integration by substitution (inverse of chain rule)

slide-84
SLIDE 84

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The integral

contd

◮ Once more defining the Riemann integral requires a

definition of the limits.

◮ and a proof of the fundamental theorem of calculus. ◮ This is not done. Instead, the focus is on mastering

techniques.

◮ the two key techniques of (symbolic) integration are

◮ integration by parts (inverse of Lebiniz rule), and ◮ integration by substitution (inverse of chain rule)

◮ since integration techniques are more difficult to

learn than differentiation techniques.

slide-85
SLIDE 85

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The difficulty in defining functions

◮ Thus the student learns differentiation and

integration as a bunch of rules.

slide-86
SLIDE 86

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The difficulty in defining functions

◮ Thus the student learns differentiation and

integration as a bunch of rules.

◮ To make these rules seem plausible, it is necessary

to define functions, such as sin(x)

slide-87
SLIDE 87

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The difficulty in defining functions

◮ Thus the student learns differentiation and

integration as a bunch of rules.

◮ To make these rules seem plausible, it is necessary

to define functions, such as sin(x)

◮ However, the student does not learn the definitions of

sin(x) etc.

slide-88
SLIDE 88

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The difficulty in defining functions

◮ Thus the student learns differentiation and

integration as a bunch of rules.

◮ To make these rules seem plausible, it is necessary

to define functions, such as sin(x)

◮ However, the student does not learn the definitions of

sin(x) etc.

◮ since the definition of transcendental functions

involve infinite series and notions of uniform convergence. ex =

∞

  • n=0

xn n! .

slide-89
SLIDE 89

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

The difficulty in defining functions

◮ Thus the student learns differentiation and

integration as a bunch of rules.

◮ To make these rules seem plausible, it is necessary

to define functions, such as sin(x)

◮ However, the student does not learn the definitions of

sin(x) etc.

◮ since the definition of transcendental functions

involve infinite series and notions of uniform convergence. ex =

∞

  • n=0

xn n! .

◮ The student hence cannot define ex, and thinks

sin(x) relates to triangles.

slide-90
SLIDE 90

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

What the student takes away

◮ Thus, the best that a good calculus text can do is to

trick the student into a state of psychological satisfaction of having “understood” matters.

slide-91
SLIDE 91

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

What the student takes away

◮ Thus, the best that a good calculus text can do is to

trick the student into a state of psychological satisfaction of having “understood” matters.

◮ The trick is to make the concepts and rules seem

intuitively plausible

slide-92
SLIDE 92

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

What the student takes away

◮ Thus, the best that a good calculus text can do is to

trick the student into a state of psychological satisfaction of having “understood” matters.

◮ The trick is to make the concepts and rules seem

intuitively plausible

◮ by appealing to visual (geometric) intuition, or

physical intuition etc.

slide-93
SLIDE 93

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

What the student takes away

contd

◮ Thus, apart from a bunch of rules, the student carries

away the following images: function = graph derivative = slope of tangent to graph integral = area under the curve.

slide-94
SLIDE 94

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

What the student takes away

contd

◮ Thus, apart from a bunch of rules, the student carries

away the following images: function = graph derivative = slope of tangent to graph integral = area under the curve.

◮ The student is unable to relate the images to the

rules.

slide-95
SLIDE 95

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

What the student takes away

contd

◮ Thus, apart from a bunch of rules, the student carries

away the following images: function = graph derivative = slope of tangent to graph integral = area under the curve.

◮ The student is unable to relate the images to the

rules.

◮ Ironically, the whole point of teaching limits is the

belief that such visual intuition may be deceptive.

slide-96
SLIDE 96

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Belief that visual intuition may deceive

◮ Recall that Dedekind cuts were motivated by the

doubt that the “fish figure” (Elements 1.1) is deceptive.

W E N S

Figure: The fish figure. Figure: Dedekind’s doubt was that the two arcs which visually seem to intersect need not intersect since there are gaps in Q.

slide-97
SLIDE 97

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

More misconceptions

◮ Students in practice have more misconceptions.

slide-98
SLIDE 98

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

More misconceptions

◮ Students in practice have more misconceptions. ◮ They say the derivative is the slope of the tangent

line to a curve.

slide-99
SLIDE 99

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

More misconceptions

◮ Students in practice have more misconceptions. ◮ They say the derivative is the slope of the tangent

line to a curve.

◮ And define a tangent as a line which meets the curve

at only one point.

slide-100
SLIDE 100

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

More misconceptions

◮ Students in practice have more misconceptions. ◮ They say the derivative is the slope of the tangent

line to a curve.

◮ And define a tangent as a line which meets the curve

at only one point.

◮ When pressed they see that a tangent line may meet

a curve at more than one point.

slide-101
SLIDE 101

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

More misconceptions

◮ Students in practice have more misconceptions. ◮ They say the derivative is the slope of the tangent

line to a curve.

◮ And define a tangent as a line which meets the curve

at only one point.

◮ When pressed they see that a tangent line may meet

a curve at more than one point.

◮ But are unable to offer a different definition of the

tangent.

slide-102
SLIDE 102

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Misconceptions about rates of change

◮ Such half-baked appeals to intuition confound the

student also from the perspective of physics.

slide-103
SLIDE 103

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Misconceptions about rates of change

◮ Such half-baked appeals to intuition confound the

student also from the perspective of physics.

◮ In physical terms, the derivative is usually explained

as “the rate of change”.

slide-104
SLIDE 104

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Misconceptions about rates of change

◮ Such half-baked appeals to intuition confound the

student also from the perspective of physics.

◮ In physical terms, the derivative is usually explained

as “the rate of change”.

◮ But consider Popper’s argument.

slide-105
SLIDE 105

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Popper’s argument about rates of change

◮ Velocity v = ∆x ∆t , then

slide-106
SLIDE 106

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Popper’s argument about rates of change

◮ Velocity v = ∆x ∆t , then ◮ Choosing a large value of ∆t will mean v is the

average velocity over the time period ∆t,

slide-107
SLIDE 107

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Popper’s argument about rates of change

◮ Velocity v = ∆x ∆t , then ◮ Choosing a large value of ∆t will mean v is the

average velocity over the time period ∆t,

◮ this could differ substantially from the

alertinstantaneous velocity at any instant in that interval.

slide-108
SLIDE 108

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Popper’s argument about rates of change

◮ Velocity v = ∆x ∆t , then ◮ Choosing a large value of ∆t will mean v is the

average velocity over the time period ∆t,

◮ this could differ substantially from the

alertinstantaneous velocity at any instant in that interval.

◮ But choosing small ∆t will increase the relative error

  • f measurement.
slide-109
SLIDE 109

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Popper’s argument about rates of change

◮ Velocity v = ∆x ∆t , then ◮ Choosing a large value of ∆t will mean v is the

average velocity over the time period ∆t,

◮ this could differ substantially from the

alertinstantaneous velocity at any instant in that interval.

◮ But choosing small ∆t will increase the relative error

  • f measurement.

◮ Hence there must be an optimum value of ∆t neither

too large nor too small.

slide-110
SLIDE 110

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Popper’s argument about rates of change

◮ Velocity v = ∆x ∆t , then ◮ Choosing a large value of ∆t will mean v is the

average velocity over the time period ∆t,

◮ this could differ substantially from the

alertinstantaneous velocity at any instant in that interval.

◮ But choosing small ∆t will increase the relative error

  • f measurement.

◮ Hence there must be an optimum value of ∆t neither

too large nor too small.

◮ This is quite different from taking limits, and not at all

what calculus texts have in mind.

slide-111
SLIDE 111

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Why teach limits?

◮ If one is ultimately going to rely on (possibly faulty)

visual and physical intuition

slide-112
SLIDE 112

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Why teach limits?

◮ If one is ultimately going to rely on (possibly faulty)

visual and physical intuition

◮ why teach calculus with limits at all?

slide-113
SLIDE 113

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Why teach limits?

◮ If one is ultimately going to rely on (possibly faulty)

visual and physical intuition

◮ why teach calculus with limits at all? ◮ Why teach students to manipulate symbols they

don’t clearly understand?

slide-114
SLIDE 114

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Why teach limits?

◮ If one is ultimately going to rely on (possibly faulty)

visual and physical intuition

◮ why teach calculus with limits at all? ◮ Why teach students to manipulate symbols they

don’t clearly understand?

◮ The human mind revolts at the thought of syntax

devoid of semantics (as in the difficulty of assembly-language programming.

slide-115
SLIDE 115

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Why teach limits?

◮ If one is ultimately going to rely on (possibly faulty)

visual and physical intuition

◮ why teach calculus with limits at all? ◮ Why teach students to manipulate symbols they

don’t clearly understand?

◮ The human mind revolts at the thought of syntax

devoid of semantics (as in the difficulty of assembly-language programming.

◮ This job of symbolic manipulation can be done more

easily by symbolic manipulation programs running on low-cost computers.

slide-116
SLIDE 116

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Why teach limits?

◮ If one is ultimately going to rely on (possibly faulty)

visual and physical intuition

◮ why teach calculus with limits at all? ◮ Why teach students to manipulate symbols they

don’t clearly understand?

◮ The human mind revolts at the thought of syntax

devoid of semantics (as in the difficulty of assembly-language programming.

◮ This job of symbolic manipulation can be done more

easily by symbolic manipulation programs running on low-cost computers.

◮ Why teach human minds to think like low-cost

machines?

slide-117
SLIDE 117

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Why teach limits?

contd.

◮ Thus, what the student learns in a calculus course

(manipulating unclearly defined symbols)

slide-118
SLIDE 118

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Why teach limits?

contd.

◮ Thus, what the student learns in a calculus course

(manipulating unclearly defined symbols)

◮ is a skill which has become obsolete.

slide-119
SLIDE 119

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Why teach limits?

contd.

◮ Thus, what the student learns in a calculus course

(manipulating unclearly defined symbols)

◮ is a skill which has become obsolete. ◮ it is today a useless skill.

slide-120
SLIDE 120

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Why teach limits?

contd.

◮ Thus, what the student learns in a calculus course

(manipulating unclearly defined symbols)

◮ is a skill which has become obsolete. ◮ it is today a useless skill. ◮ However, teaching the student to obey rules he does

not understand

slide-121
SLIDE 121

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Why teach limits?

contd.

◮ Thus, what the student learns in a calculus course

(manipulating unclearly defined symbols)

◮ is a skill which has become obsolete. ◮ it is today a useless skill. ◮ However, teaching the student to obey rules he does

not understand

◮ teaches blind obedience to mathematical authority

(which lies in the West).

slide-122
SLIDE 122

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Why teach limits?

contd.

◮ This sort of teaching started during colonialism.

slide-123
SLIDE 123

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Why teach limits?

contd.

◮ This sort of teaching started during colonialism. ◮ But why should it continue today in a free society?

slide-124
SLIDE 124

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Conclusions

  • 1. The present-day course on calculus at the school

(K-12) or beginning undergraduate level does NOT teach the following.

slide-125
SLIDE 125

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Conclusions

  • 1. The present-day course on calculus at the school

(K-12) or beginning undergraduate level does NOT teach the following.

◮ The definition of the derivative (which depends upon

limits).

slide-126
SLIDE 126

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Conclusions

  • 1. The present-day course on calculus at the school

(K-12) or beginning undergraduate level does NOT teach the following.

◮ The definition of the derivative (which depends upon

limits).

◮ The definition of the limit (which depends upon R).

slide-127
SLIDE 127

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Conclusions

  • 1. The present-day course on calculus at the school

(K-12) or beginning undergraduate level does NOT teach the following.

◮ The definition of the derivative (which depends upon

limits).

◮ The definition of the limit (which depends upon R). ◮ The definition of R (which depends upon set theory).

slide-128
SLIDE 128

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Conclusions

  • 1. The present-day course on calculus at the school

(K-12) or beginning undergraduate level does NOT teach the following.

◮ The definition of the derivative (which depends upon

limits).

◮ The definition of the limit (which depends upon R). ◮ The definition of R (which depends upon set theory). ◮ The definition of a set (which depends upon

axiomatic set theory).

slide-129
SLIDE 129

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Conclusions

  • 1. The present-day course on calculus at the school

(K-12) or beginning undergraduate level does NOT teach the following.

◮ The definition of the derivative (which depends upon

limits).

◮ The definition of the limit (which depends upon R). ◮ The definition of R (which depends upon set theory). ◮ The definition of a set (which depends upon

axiomatic set theory).

◮ The definition of the integral (which is defined only as

an anti-derivative).

slide-130
SLIDE 130

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Conclusions

  • 1. The present-day course on calculus at the school

(K-12) or beginning undergraduate level does NOT teach the following.

◮ The definition of the derivative (which depends upon

limits).

◮ The definition of the limit (which depends upon R). ◮ The definition of R (which depends upon set theory). ◮ The definition of a set (which depends upon

axiomatic set theory).

◮ The definition of the integral (which is defined only as

an anti-derivative).

◮ The definition of functions, such as ex.

slide-131
SLIDE 131

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Conclusions

  • 1. The present-day course on calculus at the school

(K-12) or beginning undergraduate level does NOT teach the following.

◮ The definition of the derivative (which depends upon

limits).

◮ The definition of the limit (which depends upon R). ◮ The definition of R (which depends upon set theory). ◮ The definition of a set (which depends upon

axiomatic set theory).

◮ The definition of the integral (which is defined only as

an anti-derivative).

◮ The definition of functions, such as ex. ◮ How to correlate the derivative with the calculation of

rates of change useful in physics.

slide-132
SLIDE 132

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Conclusions

contd.

  • 2. The present-day course does teach how to

manipulate the symbols d

dx , and

  • without knowing

their definition. This is a task which can be easily performed on a low-cost computer, using freely available programs.

slide-133
SLIDE 133

Calculus without Limits

  • C. K. Raju

Introduction

The size of calculus texts

The difficulty of limits The difficulty of defining R

Imitating the European experience

The difficulty of set theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions

Conclusions

contd.

  • 2. The present-day course does teach how to

manipulate the symbols d

dx , and

  • without knowing

their definition. This is a task which can be easily performed on a low-cost computer, using freely available programs.

  • 3. By forcing a student to learn a subject without proper

understanding, the present-day calculus course, also teaches a student subordination to mathematical authority (which lies in the West).