Calculus without The formal definition of limits Limits C. K. Raju Introduction ◮ On present-day mathematics, the symbol d dx is The size of calculus texts defined for a function f , using another symbol lim h → 0 . The difficulty of limits ◮ The difficulty of f ( x + h ) − f ( x ) df defining R dx = lim . Imitating the European experience h h → 0 The difficulty of set ◮ lim h → 0 is formally defined as follows. theory The integral x → a g ( x ) = l lim The difficulty of defining functions What the student if and only if ∀ ǫ > 0 , ∃ δ > 0 such that takes away Calculus with limits: why teach 0 < | x − a | < ǫ ⇒ | g ( x ) − l | < δ, ∀ x ∈ R . it? Conclusions
Calculus without The missing element Limits C. K. Raju Introduction The size of calculus texts The difficulty of limits The difficulty of defining R ◮ The texts of Thomas and Stewart both have a Imitating the European experience section called “precise definition of limits”. 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 The missing element Limits C. K. Raju Introduction The size of calculus texts The difficulty of limits The difficulty of defining R ◮ The texts of Thomas and Stewart both have a Imitating the European experience section called “precise definition of limits”. The difficulty of set ◮ But the definitions given are not precise. theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without The missing element Limits C. K. Raju Introduction The size of calculus texts The difficulty of limits The difficulty of defining R ◮ The texts of Thomas and Stewart both have a Imitating the European experience section called “precise definition of limits”. The difficulty of set ◮ But the definitions given are not precise. theory The integral ◮ They have the ǫ ’s and δ ’s. The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without The missing element Limits C. K. Raju Introduction The size of calculus texts The difficulty of limits The difficulty of defining R ◮ The texts of Thomas and Stewart both have a Imitating the European experience section called “precise definition of limits”. The difficulty of set ◮ But the definitions given are not precise. theory The integral ◮ They have the ǫ ’s and δ ’s. The difficulty of defining functions ◮ But are missing one key element: R . What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without The formal reals Limits Dedekind cuts C. K. Raju Introduction The size of calculus texts The difficulty of limits The difficulty of ◮ Formal reals R often built using Dedekind cuts. 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 The formal reals Limits Dedekind cuts C. K. Raju Introduction The size of calculus texts The difficulty of limits The difficulty of ◮ Formal reals R often built using Dedekind cuts. defining R Imitating the European experience ◮ Set theory provides a model for formal natural The difficulty of set numbers N , which provide a model for Peano theory arithmetic. The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without The formal reals Limits Dedekind cuts C. K. Raju Introduction The size of calculus texts The difficulty of limits The difficulty of ◮ Formal reals R often built using Dedekind cuts. defining R Imitating the European experience ◮ Set theory provides a model for formal natural The difficulty of set numbers N , which provide a model for Peano theory arithmetic. The integral The difficulty of ◮ N can be extended to the integers Z . defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without The formal reals Limits Dedekind cuts C. K. Raju Introduction The size of calculus texts The difficulty of limits The difficulty of ◮ Formal reals R often built using Dedekind cuts. defining R Imitating the European experience ◮ Set theory provides a model for formal natural The difficulty of set numbers N , which provide a model for Peano theory arithmetic. The integral The difficulty of ◮ N can be extended to the integers Z . defining functions ◮ This integral domain Z can be embedded in a field of What the student takes away rationals Q . Calculus with limits: why teach it? Conclusions
Calculus without Dedekind cuts Limits contd. C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Finally, Q can be used to construct cuts. 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 Dedekind cuts Limits contd. C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Finally, Q can be used to construct cuts. The difficulty of ◮ α ⊂ Q is called a cut if 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 Dedekind cuts Limits contd. C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Finally, Q can be used to construct cuts. The difficulty of ◮ α ⊂ Q is called a cut if defining R Imitating the European 1. α � = ∅ , and α � = Q . 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 Dedekind cuts Limits contd. C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Finally, Q can be used to construct cuts. The difficulty of ◮ α ⊂ Q is called a cut if defining R Imitating the European 1. α � = ∅ , and α � = Q . experience 2. p ∈ α and q < p ⇒ q ∈ α 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 Dedekind cuts Limits contd. C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Finally, Q can be used to construct cuts. The difficulty of ◮ α ⊂ Q is called a cut if defining R Imitating the European 1. α � = ∅ , and α � = Q . experience 2. p ∈ α and q < p ⇒ q ∈ α The difficulty of set theory 3. � ∃ m ∈ α such that p ≤ m ∀ p ∈ α The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without Dedekind cuts Limits contd. C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Finally, Q can be used to construct cuts. The difficulty of ◮ α ⊂ Q is called a cut if defining R Imitating the European 1. α � = ∅ , and α � = Q . experience 2. p ∈ α and q < p ⇒ q ∈ α The difficulty of set theory 3. � ∃ m ∈ α such that p ≤ m ∀ p ∈ α The integral ◮ +, . , and < among cuts defined in the obvious way. The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without Dedekind cuts Limits contd. C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Finally, Q can be used to construct cuts. The difficulty of ◮ α ⊂ Q is called a cut if defining R Imitating the European 1. α � = ∅ , and α � = Q . experience 2. p ∈ α and q < p ⇒ q ∈ α The difficulty of set theory 3. � ∃ m ∈ α such that p ≤ m ∀ p ∈ α The integral ◮ +, . , and < among cuts defined in the obvious way. The difficulty of defining functions ◮ May be readily shown that the cuts form an ordered What the student field, viz., R . takes away Calculus with limits: why teach it? Conclusions
Calculus without Dedekind cuts Limits contd. C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Finally, Q can be used to construct cuts. The difficulty of ◮ α ⊂ Q is called a cut if defining R Imitating the European 1. α � = ∅ , and α � = Q . experience 2. p ∈ α and q < p ⇒ q ∈ α The difficulty of set theory 3. � ∃ m ∈ α such that p ≤ m ∀ p ∈ α The integral ◮ +, . , and < among cuts defined in the obvious way. The difficulty of defining functions ◮ May be readily shown that the cuts form an ordered What the student field, viz., R . takes away Calculus with ◮ Called “cuts” since Dedekind’s intuitive idea limits: why teach it? originated from Elements 1.1. Conclusions
Calculus without Elements 1.1 Limits The fish figure C. K. Raju N Introduction The size of calculus texts The difficulty of limits The difficulty of defining R E W Imitating the European experience The difficulty of set theory The integral S The difficulty of defining functions Figure: The fish figure. What the student takes away Calculus with limits: why teach it? ◮ With W as centre and WE as radius two arcs are Conclusions drawn, and they intersect at N and S.
Calculus without Elements 1.1 Limits The fish figure C. K. Raju N Introduction The size of calculus texts The difficulty of limits The difficulty of defining R E W Imitating the European experience The difficulty of set theory The integral S The difficulty of defining functions Figure: The fish figure. What the student takes away Calculus with limits: why teach it? ◮ With W as centre and WE as radius two arcs are Conclusions drawn, and they intersect at N and S. ◮ Used in India to construct a perpendicular bisector to the EW line and thus determine NS.
Calculus without Dedekind cuts Limits Rejection of empirical methods of proof C. K. Raju Introduction The size of calculus texts ◮ Elements , I.1 uses this figure to construct the The difficulty of limits equilateral triangle WNE on the given segment WE. 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 Dedekind cuts Limits Rejection of empirical methods of proof C. K. Raju Introduction The size of calculus texts ◮ Elements , I.1 uses this figure to construct the The difficulty of limits equilateral triangle WNE on the given segment WE. The difficulty of ◮ Empirically manifest that the two arcs must intersect defining R Imitating the European at a point. 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 Dedekind cuts Limits Rejection of empirical methods of proof C. K. Raju Introduction The size of calculus texts ◮ Elements , I.1 uses this figure to construct the The difficulty of limits equilateral triangle WNE on the given segment WE. The difficulty of ◮ Empirically manifest that the two arcs must intersect defining R Imitating the European at a point. experience The difficulty of set ◮ This appeal to empirical methods of proof was theory rejected in the West. The integral The difficulty of ◮ R required for formal proof. defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without Dedekind cuts Limits Rejection of empirical methods of proof C. K. Raju Introduction The size of calculus texts ◮ Elements , I.1 uses this figure to construct the The difficulty of limits equilateral triangle WNE on the given segment WE. The difficulty of ◮ Empirically manifest that the two arcs must intersect defining R Imitating the European at a point. experience The difficulty of set ◮ This appeal to empirical methods of proof was theory rejected in the West. The integral The difficulty of ◮ R required for formal proof. defining functions ◮ If arcs are drawn in Q × Q they may “pass through” What the student takes away each other, without there being any (exact) point at Calculus with limits: why teach which they intersect, it? Conclusions
Calculus without Dedekind cuts Limits Rejection of empirical methods of proof C. K. Raju Introduction The size of calculus texts ◮ Elements , I.1 uses this figure to construct the The difficulty of limits equilateral triangle WNE on the given segment WE. The difficulty of ◮ Empirically manifest that the two arcs must intersect defining R Imitating the European at a point. experience The difficulty of set ◮ This appeal to empirical methods of proof was theory rejected in the West. The integral The difficulty of ◮ R required for formal proof. defining functions ◮ If arcs are drawn in Q × Q they may “pass through” What the student takes away each other, without there being any (exact) point at Calculus with limits: why teach which they intersect, it? ◮ since there may be “gaps” in the arcs, corresponding Conclusions to the “gaps” in rational numbers.
Calculus without Dedekind cuts Limits Key properties of R C. K. Raju Introduction The size of calculus texts The difficulty of limits The difficulty of ◮ With cuts, wherever the arc is “cut” there is always a defining R Imitating the European point (number), and never a gap. 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 Dedekind cuts Limits Key properties of R C. K. Raju Introduction The size of calculus texts The difficulty of limits The difficulty of ◮ With cuts, wherever the arc is “cut” there is always a defining R Imitating the European point (number), and never a gap. experience The difficulty of set ◮ Formally, corresponds to the least upper bound (lub) theory property of R : if A ⊂ R is bounded above, then it has The integral The difficulty of a lub in R (that is, ∃ m ∈ R such that a ≤ m , ∀ a ∈ A defining functions and if a ≤ m 1 , ∀ a ∈ A then m ≤ m 1 . What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without Dedekind cuts Limits Key properties of R C. K. Raju Introduction The size of calculus texts The difficulty of limits The difficulty of ◮ With cuts, wherever the arc is “cut” there is always a defining R Imitating the European point (number), and never a gap. experience The difficulty of set ◮ Formally, corresponds to the least upper bound (lub) theory property of R : if A ⊂ R is bounded above, then it has The integral The difficulty of a lub in R (that is, ∃ m ∈ R such that a ≤ m , ∀ a ∈ A defining functions and if a ≤ m 1 , ∀ a ∈ A then m ≤ m 1 . What the student takes away ◮ And other similar properties. Calculus with limits: why teach it? Conclusions
Calculus without Cauchy sequences Limits Alternative construction of R C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Alternative approach via equivalence classes of The difficulty of defining R Cauchy sequences in Q . 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 Cauchy sequences Limits Alternative construction of R C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Alternative approach via equivalence classes of The difficulty of defining R Cauchy sequences in Q . Imitating the European experience ◮ { a n } is called Cauchy sequence if ∀ ǫ > 0 , ∃ N such The difficulty of set theory that | a n − a m | < ǫ, ∀ n , m > N . The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without Cauchy sequences Limits Alternative construction of R C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Alternative approach via equivalence classes of The difficulty of defining R Cauchy sequences in Q . Imitating the European experience ◮ { a n } is called Cauchy sequence if ∀ ǫ > 0 , ∃ N such The difficulty of set theory that | a n − a m | < ǫ, ∀ n , m > N . The integral ◮ The decimal expansion of a real number is an The difficulty of defining functions example of such a Cauchy sequence. What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without Cauchy sequences Limits Alternative construction of R C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Alternative approach via equivalence classes of The difficulty of defining R Cauchy sequences in Q . Imitating the European experience ◮ { a n } is called Cauchy sequence if ∀ ǫ > 0 , ∃ N such The difficulty of set theory that | a n − a m | < ǫ, ∀ n , m > N . The integral ◮ The decimal expansion of a real number is an The difficulty of defining functions example of such a Cauchy sequence. What the student takes away ◮ For xin R , x �∈ Q , the decimal expansion neither Calculus with terminates nor recurs. limits: why teach it? Conclusions
Calculus without Alternative construction of R Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Let { a n } , { b n } be Cauchy sequences. We say The difficulty of defining R { a n } ~ { b n } , if a n − b n → 0. That is, ∀ ǫ > 0 , ∃ N such Imitating the European experience that | a n − b n | < ǫ, ∀ n > N . 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 Alternative construction of R Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Let { a n } , { b n } be Cauchy sequences. We say The difficulty of defining R { a n } ~ { b n } , if a n − b n → 0. That is, ∀ ǫ > 0 , ∃ N such Imitating the European experience that | a n − b n | < ǫ, ∀ n > N . The difficulty of set theory ◮ ~ is an equivalence relation, and we define +, . , and The integral < in Q /~in the obvious ways to get R The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without Alternative construction of R Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Let { a n } , { b n } be Cauchy sequences. We say The difficulty of defining R { a n } ~ { b n } , if a n − b n → 0. That is, ∀ ǫ > 0 , ∃ N such Imitating the European experience that | a n − b n | < ǫ, ∀ n > N . The difficulty of set theory ◮ ~ is an equivalence relation, and we define +, . , and The integral < in Q /~in the obvious ways to get R The difficulty of defining functions ◮ R is complete: every Cauchy sequence in R What the student converges. takes away Calculus with limits: why teach it? Conclusions
Calculus without Alternative construction of R Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Let { a n } , { b n } be Cauchy sequences. We say The difficulty of defining R { a n } ~ { b n } , if a n − b n → 0. That is, ∀ ǫ > 0 , ∃ N such Imitating the European experience that | a n − b n | < ǫ, ∀ n > N . The difficulty of set theory ◮ ~ is an equivalence relation, and we define +, . , and The integral < in Q /~in the obvious ways to get R The difficulty of defining functions ◮ R is complete: every Cauchy sequence in R What the student converges. takes away Calculus with ◮ This is equivalent to the lub property of R . limits: why teach it? Conclusions
Calculus without Imitating the European experience Limits C. K. Raju Introduction The size of calculus texts ◮ Teaching R is regarded as too complicated and is The difficulty of postponed to texts on advanced calculus 4 or limits The difficulty of mathematical analysis. 5 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 4 e.g. D. V. Widder, Advanced Calculus, 2 nd ed., Prentice Hall, New Delhi, 1999. 5 e.g. W. Rudin, Principles of Mathematical Analysis , McGraw Hill, New York, 1964.
Calculus without Imitating the European experience Limits C. K. Raju Introduction The size of calculus texts ◮ Teaching R is regarded as too complicated and is The difficulty of postponed to texts on advanced calculus 4 or limits The difficulty of mathematical analysis. 5 defining R Imitating the European ◮ Notice that this repeats the European historical experience The difficulty of set experience. theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions 4 e.g. D. V. Widder, Advanced Calculus, 2 nd ed., Prentice Hall, New Delhi, 1999. 5 e.g. W. Rudin, Principles of Mathematical Analysis , McGraw Hill, New York, 1964.
Calculus without Imitating the European experience Limits C. K. Raju Introduction The size of calculus texts ◮ Teaching R is regarded as too complicated and is The difficulty of postponed to texts on advanced calculus 4 or limits The difficulty of mathematical analysis. 5 defining R Imitating the European ◮ Notice that this repeats the European historical experience The difficulty of set experience. theory ◮ Calculus came first, the ǫ – δ definition of limits The integral The difficulty of followed, and then R was constructed. defining functions What the student takes away Calculus with limits: why teach it? Conclusions 4 e.g. D. V. Widder, Advanced Calculus, 2 nd ed., Prentice Hall, New Delhi, 1999. 5 e.g. W. Rudin, Principles of Mathematical Analysis , McGraw Hill, New York, 1964.
Calculus without Imitating the European experience Limits C. K. Raju Introduction The size of calculus texts ◮ Teaching R is regarded as too complicated and is The difficulty of postponed to texts on advanced calculus 4 or limits The difficulty of mathematical analysis. 5 defining R Imitating the European ◮ Notice that this repeats the European historical experience The difficulty of set experience. theory ◮ Calculus came first, the ǫ – δ definition of limits The integral The difficulty of followed, and then R was constructed. defining functions ◮ (Cauchy 1789-1857, Dedekind 1831-1916) What the student takes away Calculus with limits: why teach it? Conclusions 4 e.g. D. V. Widder, Advanced Calculus, 2 nd ed., Prentice Hall, New Delhi, 1999. 5 e.g. W. Rudin, Principles of Mathematical Analysis , McGraw Hill, New York, 1964.
Calculus without The problem of set theory Limits C. K. Raju Introduction The size of calculus texts The difficulty of ◮ The construction of R requires set theory. 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 The problem of set theory Limits C. K. Raju Introduction The size of calculus texts The difficulty of ◮ The construction of R requires set theory. limits The difficulty of ◮ Students are not taught the definition of a set. 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 The problem of set theory Limits C. K. Raju Introduction The size of calculus texts The difficulty of ◮ The construction of R requires set theory. limits The difficulty of ◮ Students are not taught the definition of a set. defining R Imitating the European ◮ What the student learns about set theory is 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 The problem of set theory Limits C. K. Raju Introduction The size of calculus texts The difficulty of ◮ The construction of R requires set theory. limits The difficulty of ◮ Students are not taught the definition of a set. defining R Imitating the European ◮ What the student learns about set theory is experience The difficulty of set ◮ What the student typically learns is something as theory The integral follows. The difficulty of “A set is a collection of objects” defining functions What the student or takes away “A set is a well-defined collection of objects” Calculus with limits: why teach it? Conclusions
Calculus without Differing ideas of sets Limits C. K. Raju Introduction The size of calculus texts ◮ Students have differing ideas of what “collection” The difficulty of limits means. 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 Differing ideas of sets Limits C. K. Raju Introduction The size of calculus texts ◮ Students have differing ideas of what “collection” The difficulty of limits means. The difficulty of defining R ◮ They often think elements in a set must somehow be Imitating the European experience related. 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 Differing ideas of sets Limits C. K. Raju Introduction The size of calculus texts ◮ Students have differing ideas of what “collection” The difficulty of limits means. The difficulty of defining R ◮ They often think elements in a set must somehow be Imitating the European experience related. The difficulty of set theory ◮ Can a human being and an animal can be included The integral in a set? The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without Differing ideas of sets Limits C. K. Raju Introduction The size of calculus texts ◮ Students have differing ideas of what “collection” The difficulty of limits means. The difficulty of defining R ◮ They often think elements in a set must somehow be Imitating the European experience related. The difficulty of set theory ◮ Can a human being and an animal can be included The integral in a set? The difficulty of ◮ As a beginning teacher I tried explaining that it was defining functions What the student possible to put Indira Gandhi and a cow in a set. takes away Calculus with limits: why teach it? Conclusions
Calculus without Differing ideas of sets Limits C. K. Raju Introduction The size of calculus texts ◮ Students have differing ideas of what “collection” The difficulty of limits means. The difficulty of defining R ◮ They often think elements in a set must somehow be Imitating the European experience related. The difficulty of set theory ◮ Can a human being and an animal can be included The integral in a set? The difficulty of ◮ As a beginning teacher I tried explaining that it was defining functions What the student possible to put Indira Gandhi and a cow in a set. takes away ◮ The students disagreed. Calculus with limits: why teach it? Conclusions
Calculus without Differing ideas of sets Limits C. K. Raju Introduction The size of calculus texts ◮ Students have differing ideas of what “collection” The difficulty of limits means. The difficulty of defining R ◮ They often think elements in a set must somehow be Imitating the European experience related. The difficulty of set theory ◮ Can a human being and an animal can be included The integral in a set? The difficulty of ◮ As a beginning teacher I tried explaining that it was defining functions What the student possible to put Indira Gandhi and a cow in a set. takes away ◮ The students disagreed. Calculus with limits: why teach ◮ The management was even more unhappy. it? Conclusions
Calculus without Differing ideas of sets Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of limits The difficulty of defining R ◮ The management told me not to teach wrong things. 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 Differing ideas of sets Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of limits The difficulty of defining R ◮ The management told me not to teach wrong things. Imitating the European experience The difficulty of set ◮ They cited the authority of a professor in Bombay theory University. The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without Differing ideas of sets Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of limits The difficulty of defining R ◮ The management told me not to teach wrong things. Imitating the European experience The difficulty of set ◮ They cited the authority of a professor in Bombay theory University. The integral ◮ I cited the authority of Bourbaki. The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without Differing ideas of sets Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of limits The difficulty of defining R ◮ The management told me not to teach wrong things. Imitating the European experience The difficulty of set ◮ They cited the authority of a professor in Bombay theory University. The integral ◮ I cited the authority of Bourbaki. The difficulty of defining functions ◮ (They had not heard of Bourbaki, so I resigned!) What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without What set theory the student learns Limits C. K. Raju ◮ With such a loose definition it is not possible to Introduction escape things like Russell’s paradox. 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 6 e.g. P . R. Halmos, Naive Set Theory , East-West Press, New Delhi, 1972.
Calculus without What set theory the student learns Limits C. K. Raju ◮ With such a loose definition it is not possible to Introduction escape things like Russell’s paradox. The size of calculus texts The difficulty of ◮ Let limits R = { x | x / ∈ x } . 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 6 e.g. P . R. Halmos, Naive Set Theory , East-West Press, New Delhi, 1972.
Calculus without What set theory the student learns Limits C. K. Raju ◮ With such a loose definition it is not possible to Introduction escape things like Russell’s paradox. The size of calculus texts The difficulty of ◮ Let limits R = { x | x / ∈ x } . The difficulty of defining R Imitating the European ◮ If R ∈ R then, by definition, R / ∈ R so we have a experience The difficulty of set contradiction. theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions 6 e.g. P . R. Halmos, Naive Set Theory , East-West Press, New Delhi, 1972.
Calculus without What set theory the student learns Limits C. K. Raju ◮ With such a loose definition it is not possible to Introduction escape things like Russell’s paradox. The size of calculus texts The difficulty of ◮ Let limits R = { x | x / ∈ x } . The difficulty of defining R Imitating the European ◮ If R ∈ R then, by definition, R / ∈ R so we have a experience The difficulty of set contradiction. theory ◮ On the other hand if R / ∈ R then, again by the The integral definition of R , we must have R ∈ R , which is again a The difficulty of defining functions contradiction. So either way we have a contradiction. What the student takes away Calculus with limits: why teach it? Conclusions 6 e.g. P . R. Halmos, Naive Set Theory , East-West Press, New Delhi, 1972.
Calculus without What set theory the student learns Limits C. K. Raju ◮ With such a loose definition it is not possible to Introduction escape things like Russell’s paradox. The size of calculus texts The difficulty of ◮ Let limits R = { x | x / ∈ x } . The difficulty of defining R Imitating the European ◮ If R ∈ R then, by definition, R / ∈ R so we have a experience The difficulty of set contradiction. theory ◮ On the other hand if R / ∈ R then, again by the The integral definition of R , we must have R ∈ R , which is again a The difficulty of defining functions contradiction. So either way we have a contradiction. What the student takes away ◮ Paradox is supposedly resolved by axiomatic set Calculus with theory, but even among professional limits: why teach it? mathematicians, few learn axiomatic set theory. Conclusions 6 e.g. P . R. Halmos, Naive Set Theory , East-West Press, New Delhi, 1972.
Calculus without What set theory the student learns Limits C. K. Raju ◮ With such a loose definition it is not possible to Introduction escape things like Russell’s paradox. The size of calculus texts The difficulty of ◮ Let limits R = { x | x / ∈ x } . The difficulty of defining R Imitating the European ◮ If R ∈ R then, by definition, R / ∈ R so we have a experience The difficulty of set contradiction. theory ◮ On the other hand if R / ∈ R then, again by the The integral definition of R , we must have R ∈ R , which is again a The difficulty of defining functions contradiction. So either way we have a contradiction. What the student takes away ◮ Paradox is supposedly resolved by axiomatic set Calculus with theory, but even among professional limits: why teach it? mathematicians, few learn axiomatic set theory. Conclusions ◮ Most make do with naive set theory. 6 6 e.g. P . R. Halmos, Naive Set Theory , East-West Press, New Delhi, 1972.
Calculus without The integral Limits C. K. Raju ◮ what about � f ( x ) dx ? 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 The integral Limits C. K. Raju ◮ what about � f ( x ) dx ? Introduction ◮ Most calculus courses define the integral as the The size of calculus texts The difficulty of anti-derivative, with an unsatisfying constant of limits integration. The difficulty of defining R Imitating the European if d � experience dx f ( x ) = g ( x ) then g ( x ) dx = f ( x ) + c 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 The integral Limits C. K. Raju ◮ what about � f ( x ) dx ? Introduction ◮ Most calculus courses define the integral as the The size of calculus texts The difficulty of anti-derivative, with an unsatisfying constant of limits integration. The difficulty of defining R Imitating the European if d � experience dx f ( x ) = g ( x ) then g ( x ) dx = f ( x ) + c The difficulty of set theory The integral ◮ It is believed that some clarity can be brought about The difficulty of by teaching the Riemann integral obtained as a limit defining functions of sums. What the student takes away � b n Calculus with limits: why teach � f ( x ) dx = lim f ( t i )∆ x i it? µ ( P ) → 0 a i = i Conclusions
Calculus without The integral Limits C. K. Raju ◮ what about � f ( x ) dx ? Introduction ◮ Most calculus courses define the integral as the The size of calculus texts The difficulty of anti-derivative, with an unsatisfying constant of limits integration. The difficulty of defining R Imitating the European if d � experience dx f ( x ) = g ( x ) then g ( x ) dx = f ( x ) + c The difficulty of set theory The integral ◮ It is believed that some clarity can be brought about The difficulty of by teaching the Riemann integral obtained as a limit defining functions of sums. What the student takes away � b n Calculus with limits: why teach � f ( x ) dx = lim f ( t i )∆ x i it? µ ( P ) → 0 a i = i Conclusions ◮ Here the set P = { x 0 , x 1 , x 2 , . . . x n } is a partition of the interval [ a , b ] , and t i ∈ [ x i , x i − 1 ] .
Calculus without The integral Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Once more defining the Riemann integral requires a The difficulty of definition of the limits. 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 The integral Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Once more defining the Riemann integral requires a The difficulty of definition of the limits. defining R Imitating the European ◮ and a proof of the fundamental theorem of calculus. 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 The integral Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Once more defining the Riemann integral requires a The difficulty of definition of the limits. defining R Imitating the European ◮ and a proof of the fundamental theorem of calculus. experience The difficulty of set ◮ This is not done. Instead, the focus is on mastering theory techniques. The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without The integral Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Once more defining the Riemann integral requires a The difficulty of definition of the limits. defining R Imitating the European ◮ and a proof of the fundamental theorem of calculus. experience The difficulty of set ◮ This is not done. Instead, the focus is on mastering theory techniques. The integral ◮ the two key techniques of (symbolic) integration are The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without The integral Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Once more defining the Riemann integral requires a The difficulty of definition of the limits. defining R Imitating the European ◮ and a proof of the fundamental theorem of calculus. experience The difficulty of set ◮ This is not done. Instead, the focus is on mastering theory techniques. The integral ◮ the two key techniques of (symbolic) integration are The difficulty of defining functions ◮ integration by parts (inverse of Lebiniz rule), and What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without The integral Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Once more defining the Riemann integral requires a The difficulty of definition of the limits. defining R Imitating the European ◮ and a proof of the fundamental theorem of calculus. experience The difficulty of set ◮ This is not done. Instead, the focus is on mastering theory techniques. The integral ◮ the two key techniques of (symbolic) integration are The difficulty of defining functions ◮ integration by parts (inverse of Lebiniz rule), and What the student ◮ integration by substitution (inverse of chain rule) takes away Calculus with limits: why teach it? Conclusions
Calculus without The integral Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Once more defining the Riemann integral requires a The difficulty of definition of the limits. defining R Imitating the European ◮ and a proof of the fundamental theorem of calculus. experience The difficulty of set ◮ This is not done. Instead, the focus is on mastering theory techniques. The integral ◮ the two key techniques of (symbolic) integration are The difficulty of defining functions ◮ integration by parts (inverse of Lebiniz rule), and What the student ◮ integration by substitution (inverse of chain rule) takes away Calculus with ◮ since integration techniques are more difficult to limits: why teach it? learn than differentiation techniques. Conclusions
Calculus without The difficulty in defining functions Limits C. K. Raju ◮ Thus the student learns differentiation and Introduction The size of calculus texts integration as a bunch of rules. 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 The difficulty in defining functions Limits C. K. Raju ◮ Thus the student learns differentiation and Introduction The size of calculus texts integration as a bunch of rules. The difficulty of limits ◮ To make these rules seem plausible, it is necessary The difficulty of to define functions, such as sin ( x ) 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 The difficulty in defining functions Limits C. K. Raju ◮ Thus the student learns differentiation and Introduction The size of calculus texts integration as a bunch of rules. The difficulty of limits ◮ To make these rules seem plausible, it is necessary The difficulty of to define functions, such as sin ( x ) defining R Imitating the European ◮ However, the student does not learn the definitions of experience The difficulty of set sin ( x ) etc. theory The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without The difficulty in defining functions Limits C. K. Raju ◮ Thus the student learns differentiation and Introduction The size of calculus texts integration as a bunch of rules. The difficulty of limits ◮ To make these rules seem plausible, it is necessary The difficulty of to define functions, such as sin ( x ) defining R Imitating the European ◮ However, the student does not learn the definitions of experience The difficulty of set sin ( x ) etc. theory ◮ since the definition of transcendental functions The integral The difficulty of involve infinite series and notions of uniform defining functions convergence. What the student takes away ∞ x n e x = � n ! . Calculus with limits: why teach it? n = 0 Conclusions
Calculus without The difficulty in defining functions Limits C. K. Raju ◮ Thus the student learns differentiation and Introduction The size of calculus texts integration as a bunch of rules. The difficulty of limits ◮ To make these rules seem plausible, it is necessary The difficulty of to define functions, such as sin ( x ) defining R Imitating the European ◮ However, the student does not learn the definitions of experience The difficulty of set sin ( x ) etc. theory ◮ since the definition of transcendental functions The integral The difficulty of involve infinite series and notions of uniform defining functions convergence. What the student takes away ∞ x n e x = � n ! . Calculus with limits: why teach it? n = 0 Conclusions ◮ The student hence cannot define e x , and thinks sin ( x ) relates to triangles.
Calculus without What the student takes away Limits C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Thus, the best that a good calculus text can do is to The difficulty of defining R trick the student into a state of psychological Imitating the European experience satisfaction of having “understood” matters. 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 What the student takes away Limits C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Thus, the best that a good calculus text can do is to The difficulty of defining R trick the student into a state of psychological Imitating the European experience satisfaction of having “understood” matters. The difficulty of set theory ◮ The trick is to make the concepts and rules seem The integral intuitively plausible The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without What the student takes away Limits C. K. Raju Introduction The size of calculus texts The difficulty of limits ◮ Thus, the best that a good calculus text can do is to The difficulty of defining R trick the student into a state of psychological Imitating the European experience satisfaction of having “understood” matters. The difficulty of set theory ◮ The trick is to make the concepts and rules seem The integral intuitively plausible The difficulty of defining functions ◮ by appealing to visual (geometric) intuition, or What the student physical intuition etc. takes away Calculus with limits: why teach it? Conclusions
Calculus without What the student takes away Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of ◮ Thus, apart from a bunch of rules, the student carries limits away the following images: The difficulty of defining R Imitating the European experience function = graph The difficulty of set theory derivative = slope of tangent to graph The integral integral = area under the curve. The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without What the student takes away Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of ◮ Thus, apart from a bunch of rules, the student carries limits away the following images: The difficulty of defining R Imitating the European experience function = graph The difficulty of set theory derivative = slope of tangent to graph The integral integral = area under the curve. The difficulty of defining functions ◮ The student is unable to relate the images to the What the student takes away rules. Calculus with limits: why teach it? Conclusions
Calculus without What the student takes away Limits contd C. K. Raju Introduction The size of calculus texts The difficulty of ◮ Thus, apart from a bunch of rules, the student carries limits away the following images: The difficulty of defining R Imitating the European experience function = graph The difficulty of set theory derivative = slope of tangent to graph The integral integral = area under the curve. The difficulty of defining functions ◮ The student is unable to relate the images to the What the student takes away rules. Calculus with limits: why teach ◮ Ironically, the whole point of teaching limits is the it? belief that such visual intuition may be deceptive. Conclusions
Calculus without Belief that visual intuition may deceive Limits ◮ Recall that Dedekind cuts were motivated by the C. K. Raju doubt that the “fish figure” (Elements 1.1) is Introduction deceptive. The size of calculus texts The difficulty of N limits The difficulty of defining R Imitating the European experience The difficulty of set E W theory The integral The difficulty of defining functions What the student takes away S Calculus with Figure: The fish figure. limits: why teach it? Conclusions Figure: Dedekind’s doubt was that the two arcs which visually seem to intersect need not intersect since there are gaps in Q .
Calculus without More misconceptions Limits C. K. Raju Introduction The size of calculus texts The difficulty of ◮ Students in practice have more misconceptions. 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 More misconceptions Limits C. K. Raju Introduction The size of calculus texts The difficulty of ◮ Students in practice have more misconceptions. limits The difficulty of ◮ They say the derivative is the slope of the tangent defining R Imitating the European line to a curve. 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 More misconceptions Limits C. K. Raju Introduction The size of calculus texts The difficulty of ◮ Students in practice have more misconceptions. limits The difficulty of ◮ They say the derivative is the slope of the tangent defining R Imitating the European line to a curve. experience The difficulty of set ◮ And define a tangent as a line which meets the curve theory at only one point. The integral The difficulty of defining functions What the student takes away Calculus with limits: why teach it? Conclusions
Calculus without More misconceptions Limits C. K. Raju Introduction The size of calculus texts The difficulty of ◮ Students in practice have more misconceptions. limits The difficulty of ◮ They say the derivative is the slope of the tangent defining R Imitating the European line to a curve. experience The difficulty of set ◮ And define a tangent as a line which meets the curve theory at only one point. The integral ◮ When pressed they see that a tangent line may meet The difficulty of defining functions a curve at more than one point. What the student takes away Calculus with limits: why teach it? Conclusions
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