The Sand Reckoner is an enjoyable read for But the charm of the book is that it will convey the same sense of awe and excitement to everyone. tenth standard mathematics. It may be easier to It will place mathematical discovery and its A Review of appreciateanyone fifteen this and novel up, if with one isbasic already knowledge immersed of applications in a historical and social context. It in the world of math and math history, but it will is the ideal way to illustrate the story-like quality certainly also appeal to readers whose knowledge of the course of math history to even the most Math! Encounters of the subject does not go beyond the very basic. reluctant and intimidated disciples of the subject. with High School Students DAKSHAYINI SURESH is an alumnus of Centre for Learning, Bangalore, and has just begun an honours course in English at Lady Shri Ram College. She likes to read, write and cook. She has a fraught history with Dialogue and Mathematics—Serge Lang Style! mathematics but occasionally finds herself looking back fondly on the days when math meant more than review grappling with the week's accounts! She can be reached at: [email protected].
he notion of dialogue and mathematics may at first seem Shashidhar Jagadeeshan a strange combination, but if one thinks about it, often Tin a lively interactive classroom this is exactly what is transpiring. According to the late physicist David Bohm, the root of the word dialogue comes from the Greek dialogus. The word logos in turn can be interpreted as ‘meaning of the word’ and dia means ‘through’. So dialogue can then be seen as a process where there is a flow of meaning. All teachers would agree that this is what they would like in their classrooms.
The book under review, Math! Encounters with High School Students by Serge Lang, is an old one, published in 1985, but well worth bringing to the notice of students and teachers of mathematics. It is a series of seven dialogues on mathematics with school students and a postscript discussing mathematics teaching.
Apart from the content, which I will discuss later, the book is unique in its style of delivery. Even though we are not in the audience, we can sense the energy and excitement of the exchange. One wonders (without being completely reductionist), what are the ingredients needed for such a flow of energy and meaning to take place between teacher and taught? The obvious
Keywords: Dialogue, facilitation, pedagogy, creativity
98 At Right Angles | Vol. 4, No. 3, November 2015 Vol. 4, No. 3, November 2015 | At Right Angles 99 4 SHASHIDHAR JAGADEESHAN
‘He can’t talk now, he is writing a book. I will put you on hold’.” He was a prodigious author and wrote more than 61 books (some feel this is an underestimate) and 120 research articles. Most famous amongst his textbooks is Algebra, which is a classic in the area. For school teachers, apart from the book under review, I would recommend they refer to [2] and [3].
Lang could not have had a better mathematical lineage. He wrote his PhD thesis under the famous algebraist Emil Artin and did postdoctoral work with André Weil. He won the Cole Prize (1959) and the Steele Prize (1999) of the American Mathematical Society. He was elected to the National Academy of Sciences in 1985.
He was a deeply committed teacher with a great passion for communicating mathematics and ones are a mastery of the subject on the part Serge Lang: What’s a generalization of what I Serge Lang was born in 1927 in Paris and died soon fail, because he seems to be rather short devoted a considerable part of his life to teaching. In recognition for his commitment he was of the teacher, an ability to gauge the level of have just done there? I started in 2 dimensions, in 2005 in Berkeley, California. Anyone who has tempered and confrontational, but at the same awarded the Dylon Hixon Prize for teaching in Yale College. His passions included mathematics, the students and begin from where they are, a then I went to 3 dimensions . . . . studied higher mathematics would be familiar time kind and generous, especially to young music and politics (‘trouble making’ in Lang’s words). Jorgenson and Krantz pay him the greatest sense of humour, encouraging students to think with his name as the author of mathematics people and his students. He was driven by strong Serge: [Interrupts.] Four dimensions. OK. It’s the compliment (from my point of view) that a person can receive: “Serge Lang’s greatest passion in on their feet, generating a creative tension and books on almost every topic under the sun! On his convictions and fought several public battles next product. I see. It’s rst whatever. life was learning” [1]. He demonstrated this by writing books and teaching courses in new areas of finally pulling it all off. retirement from Yale University in 2005, where he based on these convictions. It is best to quote Lang mathematics, because he believed that that was the best way to learn. He was famous for cajoling Serge Lang: Ah, rst whatever. That’s right. So was a faculty member from 1972, Yale president on this! The excerpts below illustrate these points well. young mathematicians to teach him new mathematics. suppose I have a solid in four dimensions. You see Richard C. Levin shares a joke about this. the four dimensions? Now I can’t draw it. “Someone calls the Yale Mathematics Department, Excerpts from page 20 Reading the article [1] on Lang by JorgensonI personally and prefer Krantz, to wherlive ine a several society famous mathematicians and asks for Serge Lang. The assistant who Serge Lang: . . . 2πr = C. There is your formula. Serge: Well, you could not draw it either in three recall their interactions with him,where a picture people emerges do think of independently an extremely colourfuland clearly. and energetic per- answers says, ‘He can’t talk now, he is writing a Do you agree that’s a proof? [Serge Lang points to dimensions! sonality, not always the easiest ofOne persons of my to principal relate to. goals Any isattempt therefore to categorize to make him would soon book. I will put you on hold’.” He was a prodigious Rachel.] fail, because he seems to be ratherpeople short think. tempered When andfaced confro withntational, persons who but atfudge the same time kind Serge Lang: That’s a very good remark. You are author and wrote more than 61 books (some and generous, especially to youngthe people issues, and or his cover students. up, or attempt He was drivento rewrite by strong convictions Rachel: Yes. [Her tone is uncertain.] absolutely right. So the truth of what I am saying feel this is an underestimate) and 120 research and fought several public battles basedhistory, on the these process conviction of clarifyings. It is bestthe issues to quote does Lang on this! does not depend on my ability to draw the picture! articles. Most famous amongst his textbooks is Serge Lang: You do? lead to confrontation, it creates tension, and it . . . Algebra, which is a classic in the area. For school may be interpreted as carrying out a ‘personal Rachel: Yes. [Laughing a little.] teachers, apart from the book under review, I And suppose I have a solid in n dimensions, and I I personally prefer to livevendetta’ in a society . . . . I whereregard people such an do interpretation think independently as and would recommend they refer to [2] and [3]. Serge Lang: What do you mean ‘yes’? Is it yes by make the dilation by factor of r in all directions, in clearly. One of my principalvery goals unfortunate, is therefore and to make I reject peopl it totally.e think. When faced intimidation or a yes by conviction? Or a little bit all n dimensions. How does the volume change? Lang could not have had a betterwith mathematical persons who fudge the issues, or cover up, or attempt to rewrite history, the of both? lineage. He wrote his PhD thesis processunder the of famous clarifying theLet issues us turn does our lead attention to confrontation to the contents, it creates of tension, and Serge: r to the power n. algebraist Emil Artin and did postdoctoralit may be work interpreted asthe carrying book. The out intention a ‘personal is to vendetta’ make beautiful . . . . I regard such an Rachel: A little bit of both. [Laughter.] Serge Lang: rn, and that’s how it is in n with André Weil. He won the Coleinterpretation Prize (1959) as very unfortunate,mathematics and accessible I reject itto totally students. of roughly Serge Lang: Well, where is the intimidation? dimensions. OK? Any problems? Sandra. and the Steele Prize (1999) of the American Mathematical Society. He was elected to the is pi?” It is extremely important that high school Rachel: I don’t know. Sandra: No. [The other students nod, and seem class 8 to 10. The first dialogue is called “What National Academy of SciencesLet in us 1985. turn our attention to thestudents contents get ofa good the book. understanding The intention of this is well- to show beautiful math- perfectly at ease.] Serge Lang: You don’t know? [Laughter.] All right, ematics accessible to students ofknown roughly constant class 8of to nature. 10. The The fi rstmisconceptions dialogue is called “What is pi?” He was a deeply committed teacher with a great let’s make it all conviction. Look, where do I start Serge Lang: . . .But I think it’s remarkable how you It is extremely important that highabout school π that students I encounter get aamong good teachers understanding and of this well-known passion for communicating mathematics and from? . . . react to the possibility in n dimensions. constant of nature. The misconceptionsstudents about oftenπ alarmthat I me! encounter They remember among teachers it as and students often devoted a considerable part of his life to teaching. alarm me! They remember it asthe the fraction fraction 22 ,, oror 33.14,.14, and veryvery fewfew are awareawarethat it is irrational, [ ] I am slightly taken aback at the way In recognition for his commitment he was 7 Excerpts from pages 34 and 35 Laughter. that it is irrational, let alone transcendental. Lang awarded the Dylon Hixon Prize for teaching in you just went along with it. actually deals with the subtle point as to why the Serge Lang: . . .Do you accept all that? [Students Yale College. His passions included mathematics, same constant π shows up both in the formula for approve . . . ] So we can make a general result: music and politics (‘trouble making’ in Lang’s Excerpts from page 120 the circumference and area of a circle. Under dilation by a factor of r, s, t in the three Student: So far you used different methods; you dimensions, the volume of a solid changes by greatest compliment (from my point of view) Dialogues 2 to 5 deal with derivations of the words). Jorgenson and Krantz pay him the the factor of the product, rst. method; probably a different method would do it that a person can receive: “Serge Lang’s greatest formulae for the volume of a pyramid, cone and first used one method, then you changed the passion in life was learning” [1]. He demonstrated sphere and the formulae for the circumference 2 for all numbers. r this by writing books and teaching courses in new of the circle and the surface area of the sphere. if we dilate by r in each direction; a factor of rs if Serge Lang: That is a very weak argument. Lang uses essentially Archimedes’ method of Just like yesterday: area changes by a factor of areas of mathematics, because he believed that we dilate by a factor of r in one dimension and s [Laughter.] Because the argument is based on that was the best way to learn. He was famous for ‘exhaustion’ for these derivations. As far as I am in the other dimension; and now volume changes psychology, and I am asking you to deal with cajoling young mathematicians to teach him new aware, standard school mathematics textbooks by a factor of rst if you dilate by a factor of r in one mathematical problems. Not psychological mathematics. rarely derive these formulas. Perhaps there is a dimension, s in the another and t in the third. ones. [Laughter.] So if you start basing your mathematical intuition on my psychology, that they are best done using integral calculus. And the three dimensions are in perpendicular Krantz, where several famous mathematicians [Laughter.] you’re going to have a hard time with But,feeling as Langthat these so aptly derivations demonstrates, are too they difficult, are very or directions. Now I will deal mostly in three recallReading their the interactions article [1] on with Lang him, by aJorgenson picture and it. That’s dangerous. Think again. accessible to younger students, and in fact if done dimensions, but what would be a natural emerges of an extremely colourful and energetic before the students see integral calculus, it serves generalization of this? Serge. [Students talk among each other.] personality, not always the easiest of persons to to show not only the power of calculus, but also relate to. Any attempt to categorize him would Serge: (a student): I don’t know. the limitations of the method of exhaustion.
100 At Right Angles | Vol. 4, No. 3, November 2015 Vol. 4, No. 3, November 2015 | At Right Angles 101 4 SHASHIDHAR JAGADEESHAN
‘He can’t talk now, he is writing a book. I will put you on hold’.” He was a prodigious author and wrote more than 61 books (some feel this is an underestimate) and 120 research articles. Most famous amongst his textbooks is Algebra, which is a classic in the area. For school teachers, apart from the book under review, I would recommend they refer to [2] and [3].
Lang could not have had a better mathematical lineage. He wrote his PhD thesis under the famous algebraist Emil Artin and did postdoctoral work with André Weil. He won the Cole Prize (1959) and the Steele Prize (1999) of the American Mathematical Society. He was elected to the National Academy of Sciences in 1985.
He was a deeply committed teacher with a great passion for communicating mathematics and ones are a mastery of the subject on the part Serge Lang: What’s a generalization of what I Serge Lang was born in 1927 in Paris and died soon fail, because he seems to be rather short devoted a considerable part of his life to teaching. In recognition for his commitment he was of the teacher, an ability to gauge the level of have just done there? I started in 2 dimensions, in 2005 in Berkeley, California. Anyone who has tempered and confrontational, but at the same awarded the Dylon Hixon Prize for teaching in Yale College. His passions included mathematics, the students and begin from where they are, a then I went to 3 dimensions . . . . studied higher mathematics would be familiar time kind and generous, especially to young music and politics (‘trouble making’ in Lang’s words). Jorgenson and Krantz pay him the greatest sense of humour, encouraging students to think with his name as the author of mathematics people and his students. He was driven by strong Serge: [Interrupts.] Four dimensions. OK. It’s the compliment (from my point of view) that a person can receive: “Serge Lang’s greatest passion in on their feet, generating a creative tension and books on almost every topic under the sun! On his convictions and fought several public battles next product. I see. It’s rst whatever. life was learning” [1]. He demonstrated this by writing books and teaching courses in new areas of finally pulling it all off. retirement from Yale University in 2005, where he based on these convictions. It is best to quote Lang mathematics, because he believed that that was the best way to learn. He was famous for cajoling Serge Lang: Ah, rst whatever. That’s right. So was a faculty member from 1972, Yale president on this! The excerpts below illustrate these points well. young mathematicians to teach him new mathematics. suppose I have a solid in four dimensions. You see Richard C. Levin shares a joke about this. the four dimensions? Now I can’t draw it. “Someone calls the Yale Mathematics Department, Excerpts from page 20 Reading the article [1] on Lang by JorgensonI personally and prefer Krantz, to wherlive ine a several society famous mathematicians and asks for Serge Lang. The assistant who Serge Lang: . . . 2πr = C. There is your formula. Serge: Well, you could not draw it either in three recall their interactions with him,where a picture people emerges do think of independently an extremely colourfuland clearly. and energetic per- answers says, ‘He can’t talk now, he is writing a Do you agree that’s a proof? [Serge Lang points to dimensions! sonality, not always the easiest ofOne persons of my to principal relate to. goals Any isattempt therefore to categorize to make him would soon book. I will put you on hold’.” He was a prodigious Rachel.] fail, because he seems to be ratherpeople short think. tempered When andfaced confro withntational, persons who but atfudge the same time kind Serge Lang: That’s a very good remark. You are author and wrote more than 61 books (some and generous, especially to youngthe people issues, and or his cover students. up, or attempt He was drivento rewrite by strong convictions Rachel: Yes. [Her tone is uncertain.] absolutely right. So the truth of what I am saying feel this is an underestimate) and 120 research and fought several public battles basedhistory, on the these process conviction of clarifyings. It is bestthe issues to quote does Lang on this! does not depend on my ability to draw the picture! articles. Most famous amongst his textbooks is Serge Lang: You do? lead to confrontation, it creates tension, and it . . . Algebra, which is a classic in the area. For school may be interpreted as carrying out a ‘personal Rachel: Yes. [Laughing a little.] teachers, apart from the book under review, I And suppose I have a solid in n dimensions, and I I personally prefer to livevendetta’ in a society . . . . I whereregard people such an do interpretation think independently as and would recommend they refer to [2] and [3]. Serge Lang: What do you mean ‘yes’? Is it yes by make the dilation by factor of r in all directions, in clearly. One of my principalvery goals unfortunate, is therefore and to make I reject peopl it totally.e think. When faced intimidation or a yes by conviction? Or a little bit all n dimensions. How does the volume change? Lang could not have had a betterwith mathematical persons who fudge the issues, or cover up, or attempt to rewrite history, the of both? lineage. He wrote his PhD thesis processunder the of famous clarifying theLet issues us turn does our lead attention to confrontation to the contents, it creates of tension, and Serge: r to the power n. algebraist Emil Artin and did postdoctoralit may be work interpreted asthe carrying book. The out intention a ‘personal is to vendetta’ make beautiful . . . . I regard such an Rachel: A little bit of both. [Laughter.] Serge Lang: rn, and that’s how it is in n with André Weil. He won the Coleinterpretation Prize (1959) as very unfortunate,mathematics and accessible I reject itto totally students. of roughly Serge Lang: Well, where is the intimidation? dimensions. OK? Any problems? Sandra. and the Steele Prize (1999) of the American Mathematical Society. He was elected to the is pi?” It is extremely important that high school Rachel: I don’t know. Sandra: No. [The other students nod, and seem class 8 to 10. The first dialogue is called “What National Academy of SciencesLet in us 1985. turn our attention to thestudents contents get ofa good the book. understanding The intention of this is well- to show beautiful math- perfectly at ease.] Serge Lang: You don’t know? [Laughter.] All right, ematics accessible to students ofknown roughly constant class 8of to nature. 10. The The fi rstmisconceptions dialogue is called “What is pi?” He was a deeply committed teacher with a great let’s make it all conviction. Look, where do I start Serge Lang: . . .But I think it’s remarkable how you It is extremely important that highabout school π that students I encounter get aamong good teachers understanding and of this well-known passion for communicating mathematics and from? . . . react to the possibility in n dimensions. constant of nature. The misconceptionsstudents about oftenπ alarmthat I me! encounter They remember among teachers it as and students often devoted a considerable part of his life to teaching. alarm me! They remember it asthe the fraction fraction 22 ,, oror 33.14,.14, and veryvery fewfew are awareawarethat it is irrational, [ ] I am slightly taken aback at the way In recognition for his commitment he was 7 Excerpts from pages 34 and 35 Laughter. that it is irrational, let alone transcendental. Lang awarded the Dylon Hixon Prize for teaching in you just went along with it. actually deals with the subtle point as to why the Serge Lang: . . .Do you accept all that? [Students Yale College. His passions included mathematics, same constant π shows up both in the formula for approve . . . ] So we can make a general result: music and politics (‘trouble making’ in Lang’s Excerpts from page 120 the circumference and area of a circle. Under dilation by a factor of r, s, t in the three Student: So far you used different methods; you dimensions, the volume of a solid changes by greatest compliment (from my point of view) Dialogues 2 to 5 deal with derivations of the words). Jorgenson and Krantz pay him the the factor of the product, rst. method; probably a different method would do it that a person can receive: “Serge Lang’s greatest formulae for the volume of a pyramid, cone and first used one method, then you changed the passion in life was learning” [1]. He demonstrated sphere and the formulae for the circumference 2 for all numbers. r this by writing books and teaching courses in new of the circle and the surface area of the sphere. if we dilate by r in each direction; a factor of rs if Serge Lang: That is a very weak argument. Lang uses essentially Archimedes’ method of Just like yesterday: area changes by a factor of areas of mathematics, because he believed that we dilate by a factor of r in one dimension and s [Laughter.] Because the argument is based on that was the best way to learn. He was famous for ‘exhaustion’ for these derivations. As far as I am in the other dimension; and now volume changes psychology, and I am asking you to deal with cajoling young mathematicians to teach him new aware, standard school mathematics textbooks by a factor of rst if you dilate by a factor of r in one mathematical problems. Not psychological mathematics. rarely derive these formulas. Perhaps there is a dimension, s in the another and t in the third. ones. [Laughter.] So if you start basing your mathematical intuition on my psychology, that they are best done using integral calculus. And the three dimensions are in perpendicular Krantz, where several famous mathematicians [Laughter.] you’re going to have a hard time with But,feeling as Langthat these so aptly derivations demonstrates, are too they difficult, are very or directions. Now I will deal mostly in three recallReading their the interactions article [1] on with Lang him, by aJorgenson picture and it. That’s dangerous. Think again. accessible to younger students, and in fact if done dimensions, but what would be a natural emerges of an extremely colourful and energetic before the students see integral calculus, it serves generalization of this? Serge. [Students talk among each other.] personality, not always the easiest of persons to to show not only the power of calculus, but also relate to. Any attempt to categorize him would Serge: (a student): I don’t know. the limitations of the method of exhaustion.
100 At Right Angles | Vol. 4, No. 3, November 2015 Vol. 4, No. 3, November 2015 | At Right Angles 101 DIALOGUE AND MATHEMATICS 5 let alone transcendental. Lang actually deals with the subtle point as to why the same constant π shows up both in the formula for the circumference and area of a circle.
Dialogues 2 to 5 deal with derivations of the formulae for the volume of a pyramid, cone and sphere and the formulae for the circumference of the circle and the surface area of the sphere. Lang uses essentially Archimedes’ method of ‘exhaustion’ for these derivations. As far as I am aware, standard school mathematics textbooks rarely derive these formulas. Perhaps there is a feeling that these derivations are too difficult, or that they are best done using integral calculus. But, as Lang so aptly demonstrates, they are very accessible to younger students, and in fact if done before the students see integral calculus, it serves to show not only the power of calculus, but also the limitations of the method of exhaustion.
One of my favourite parts is the derivation of the volume of a pyramid. Lang and his students stumble upon the special case of the cube, which can actuallyDIALOGUEbe AND divided MATHEMATICS into three congruent 5 pyramids (see Figure 1). This helps us to understand where the 1 factor comes in. It is also a nice let alone transcendental. Lang actually deals3 with the subtle point as to why the same constant π activity to get students to make netsshows of these up both pyramids, in the formula as for ma theny circumference surprises and await area of thea circle. student in doing I would like to end the review with comments and References so! excerpts illustrating Lang’s views on mathematics Dialogues 2 to 5 deal with derivations of the formulae for the volume of a pyramid, cone and 1. Serge Lang, 1927-2005. Notices of the AMS, Volume 53, Number 5, May 2006, pages education from the preface and from the sphere and the formulae for the circumference of the circle and the surface area of the sphere. Lang 536-553. Postscript, which is also a dialogue among Lang, Jorgenson, Jay and Krantz, Steven G. uses essentially Archimedes’ method of ‘exhaustion’ for these derivations. As far as I am aware, educators and a student. Lang has strong views on 2. Lang, Serge. The Beauty of Doing Mathematics: Three Public Dialogues. Springer-Verlag, 1985. standard school mathematics textbooks rarely derive these formulas. Perhaps there is a feeling the curriculum: to quote him from the preface, 3. Lang, Serge and Murrow, Gene. Geometry: A High School Course. Springer-Verlag, 1983. DIALOGUEthat AND these MATHEMATICS derivations are too difficult, or that they5 are best done using integral calculus. But, as Lang so aptly demonstrates, they are very accessible to younger students, and in fact if done before A lot of the curriculum of elementary and let alone transcendental. Lang actually deals with the subtle point as to why the same constant π the students see integral calculus, it serves to showhigh not onl schoolsy the is power very dry. of You calculus, may never but also have the shows up both in the formula for the circumference and area of a circle. limitations of the method of exhaustion. had a chance to see what beautiful mathematics is like. . . . I have many objections to the high Dialogues 2 to 5 deal with derivations of theOne formulae of my favourite for the volume parts is of the a pyramid, derivation cone of and the volume of a pyramid. Lang and his students SHASHIDHAR JAGADEESHAN has been teaching mathematics for the last 25 years. He is the author of sphere and the formulae for the circumference of the circle and the surface area of the sphere. Langschool curriculum. Perhaps the main one is the Math Alive!, a resource book for teachers, and has written articles in education journals sharing his interests stumble upon the special case of the cube, which can actually be divided into three congruent and insights. He may be contacted at [email protected]. uses essentially Archimedes’ method of ‘exhaustion’ for these derivations. As far as I am aware, incoherence of what is done there, pyramids (see Figure 1). This helps us to understand where the 1 factor comes in. It is also a nice standard school mathematics textbooks rarely derive these formulas. Perhaps there is a feeling the lack of3 sweep, the little exercises that that these derivations are too difficult, oractivity that they to are get bes studentst done using to make integral netsof calculus. these pyramids, But, as as many surprisesdon’t mean await anything. the student in doing so! Figure 1. Lang so aptly demonstrates, they are very accessible to younFIGUREger students,1. and in fact if done before the students see integral calculus, it servesOne toof showmy favourite not onl yparts the is power the derivation of calculus, of but also theIn reaction to the feeling that school students are limitations of the method of exhaustion. the volume of a pyramid. Lang and his students not mature enough to see proofs, Dialogue 6 deals with Pythagoreanstumble triplets.upon the special Here case students of the cube, a whichre introduced to the problem and the completeOne solutionof my favourite is demonstrated parts is the derivationcan actually with of the the be volume divided help of into a of pyramid. three the paramcongruent Lang andetric his students representationThere is the ofscandal! the unitThose circleproofs are very stumble1−t2 upon the special case2t of the cube,pyramids which (see can Figure actually 1). Thisbe divided helps us into to three congruentbeautiful, it’s real mathematics. They allow you to x(t)= 2 and y(t)= 2 , explainingunderstand where the geometricthe 1 factor comes significance in. It is of the parameter t. Here t is pyramids1+t (see Figure 1). This1+ helpst us to understand where the 3 factor comes in. It is also a niceappreciate mathematics, to see why something is the slopeactivity of to get a studentsspecial to makeline, nets and of these onealso pyramids,a gets nice activity a as very ma tony get elegant surprises students await connecto make the student netstion in to doing doubletrue angle by using formulae arguments fromwhich are so! of these pyramids, as many surprises await the quite understandable. trigonometry. student in doing so!
Dialogue 6 deals with Pythagorean triplets. Here But in the course of the same dialogue, almost FIGURE 1. The last mathematical dialoguestudents deals are with introduced infinities, to the problem takin andg studentsthe contradicting from the veryhimself, basics he insists all that the memorization way to the result that real numberscomplete areDialogue notsolution denumerable. 6 deals is demonstrated with Pythagorean with the triplets. help Hereof students formula is are essential! introduced to the problem and the of completethe parametric solution representation is demonstrated of the withunit circle the help of the parametric representation of the unit circle 1−t2 2t There is no way to avoid this, so you must ask x(t)= 2 and y(t)= 2 ,, explaining the geometric significance of the parameter t. Here t is 1+t 1+t kids to repeat the formula ten times. . . . It must I have used this little book in manythe slope ways of a as special a teacher: line, and as one a gets refe arence very elegant book, connec as ation model to double to conduct angle formulae from be driven into their ears like music. You shouldn’t trigonometry. mathematical dialogues and alsoparameter a source t. Here for t students is the slope of to a reaspeciald on line, their own andask every make time presentations. why the formula is true. andexplainingF IGUREone gets the1. a geometricvery elegant significance connection of to the double In short, I would highly recommendThe it to last students mathematical and dialogue teacher dealss ofwith secondary infinities, takin andg students high fromschool. the very basics all the angle formulae from trigonometry. One may not agree with Lang’s philosophy or Dialogue 6 deals with Pythagorean triplets.way Here to the students result thatare introduced real numbers to the are problem not denumerable. and the ideas all the time, but he does force you to think complete solution is demonstrated with theThe helplast mathematical of the parametric dialogue representation deals with of the unit circle 1−t2 2t about what we are doing as teachers. He ends the x(t)= 2 and y(t)= 2 , explaining the geometricI have used significance this little book of the in parameter many wayst. as Here a teacher:t is as a reference book, as a model to conduct 1+t 1+t dialogue on a more humane note and we will leave the slope of a special line, and one getsthe amathematical very way elegant to the result connec dialogues thattion real to and doublenumbers also anglea sourceare formulaenot for students from to read on their own and make presentations. infinities, taking students from the very basics all the readers with that. trigonometry. denumerable.In short, I would highly recommend it to students and teachers of secondary and high school. I have used this little book in many ways as The last mathematical dialogue deals with infinities, taking students from the very basics all theEach teacher must do according to his own way, a teacher: as a reference book, as a model to way to the result that real numbers are not denumerable. his own taste. Each one must use their own means conduct mathematical dialogues and also a source to excite the students. One needs everything for students to read on their own and make I have used this little book in many ways as a teacher: as a reference book, as a model to conduct without exclusivity. presentations. In short, I would highly recommend mathematical dialogues and also a source for students to read on their own and make presentations. it to students and teachers of secondary and high In short, I would highly recommend it to students and teachers of secondary and high school. school.
102 At Right Angles | Vol. 4, No. 3, November 2015 Vol. 4, No. 3, November 2015 | At Right Angles 103 DIALOGUE AND MATHEMATICS 5 let alone transcendental. Lang actually deals with the subtle point as to why the same constant π shows up both in the formula for the circumference and area of a circle.
Dialogues 2 to 5 deal with derivations of the formulae for the volume of a pyramid, cone and sphere and the formulae for the circumference of the circle and the surface area of the sphere. Lang uses essentially Archimedes’ method of ‘exhaustion’ for these derivations. As far as I am aware, standard school mathematics textbooks rarely derive these formulas. Perhaps there is a feeling that these derivations are too difficult, or that they are best done using integral calculus. But, as Lang so aptly demonstrates, they are very accessible to younger students, and in fact if done before the students see integral calculus, it serves to show not only the power of calculus, but also the limitations of the method of exhaustion.
One of my favourite parts is the derivation of the volume of a pyramid. Lang and his students stumble upon the special case of the cube, which can actuallyDIALOGUEbe AND divided MATHEMATICS into three congruent 5 pyramids (see Figure 1). This helps us to understand where the 1 factor comes in. It is also a nice let alone transcendental. Lang actually deals3 with the subtle point as to why the same constant π activity to get students to make netsshows of these up both pyramids, in the formula as for ma theny circumference surprises and await area of thea circle. student in doing I would like to end the review with comments and References so! excerpts illustrating Lang’s views on mathematics Dialogues 2 to 5 deal with derivations of the formulae for the volume of a pyramid, cone and 1. Serge Lang, 1927-2005. Notices of the AMS, Volume 53, Number 5, May 2006, pages education from the preface and from the sphere and the formulae for the circumference of the circle and the surface area of the sphere. Lang 536-553. Postscript, which is also a dialogue among Lang, Jorgenson, Jay and Krantz, Steven G. uses essentially Archimedes’ method of ‘exhaustion’ for these derivations. As far as I am aware, educators and a student. Lang has strong views on 2. Lang, Serge. The Beauty of Doing Mathematics: Three Public Dialogues. Springer-Verlag, 1985. standard school mathematics textbooks rarely derive these formulas. Perhaps there is a feeling the curriculum: to quote him from the preface, 3. Lang, Serge and Murrow, Gene. Geometry: A High School Course. Springer-Verlag, 1983. DIALOGUEthat AND these MATHEMATICS derivations are too difficult, or that they5 are best done using integral calculus. But, as Lang so aptly demonstrates, they are very accessible to younger students, and in fact if done before A lot of the curriculum of elementary and let alone transcendental. Lang actually deals with the subtle point as to why the same constant π the students see integral calculus, it serves to showhigh not onl schoolsy the is power very dry. of You calculus, may never but also have the shows up both in the formula for the circumference and area of a circle. limitations of the method of exhaustion. had a chance to see what beautiful mathematics is like. . . . I have many objections to the high Dialogues 2 to 5 deal with derivations of theOne formulae of my favourite for the volume parts is of the a pyramid, derivation cone of and the volume of a pyramid. Lang and his students SHASHIDHAR JAGADEESHAN has been teaching mathematics for the last 25 years. He is the author of sphere and the formulae for the circumference of the circle and the surface area of the sphere. Langschool curriculum. Perhaps the main one is the Math Alive!, a resource book for teachers, and has written articles in education journals sharing his interests stumble upon the special case of the cube, which can actually be divided into three congruent and insights. He may be contacted at [email protected]. uses essentially Archimedes’ method of ‘exhaustion’ for these derivations. As far as I am aware, incoherence of what is done there, pyramids (see Figure 1). This helps us to understand where the 1 factor comes in. It is also a nice standard school mathematics textbooks rarely derive these formulas. Perhaps there is a feeling the lack of3 sweep, the little exercises that that these derivations are too difficult, oractivity that they to are get bes studentst done using to make integral netsof calculus. these pyramids, But, as as many surprisesdon’t mean await anything. the student in doing so! Figure 1. Lang so aptly demonstrates, they are very accessible to younFIGUREger students,1. and in fact if done before the students see integral calculus, it servesOne toof showmy favourite not onl yparts the is power the derivation of calculus, of but also theIn reaction to the feeling that school students are limitations of the method of exhaustion. the volume of a pyramid. Lang and his students not mature enough to see proofs, Dialogue 6 deals with Pythagoreanstumble triplets.upon the special Here case students of the cube, a whichre introduced to the problem and the completeOne solutionof my favourite is demonstrated parts is the derivationcan actually with of the the be volume divided help of into a of pyramid. three the paramcongruent Lang andetric his students representationThere is the ofscandal! the unitThose circleproofs are very stumble1−t2 upon the special case2t of the cube,pyramids which (see can Figure actually 1). Thisbe divided helps us into to three congruentbeautiful, it’s real mathematics. They allow you to x(t)= 2 and y(t)= 2 , explainingunderstand where the geometricthe 1 factor comes significance in. It is of the parameter t. Here t is pyramids1+t (see Figure 1). This1+ helpst us to understand where the 3 factor comes in. It is also a niceappreciate mathematics, to see why something is the slopeactivity of to get a studentsspecial to makeline, nets and of these onealso pyramids,a gets nice activity a as very ma tony get elegant surprises students await connecto make the student netstion in to doing doubletrue angle by using formulae arguments fromwhich are so! of these pyramids, as many surprises await the quite understandable. trigonometry. student in doing so!
Dialogue 6 deals with Pythagorean triplets. Here But in the course of the same dialogue, almost FIGURE 1. The last mathematical dialoguestudents deals are with introduced infinities, to the problem takin andg studentsthe contradicting from the veryhimself, basics he insists all that the memorization way to the result that real numberscomplete areDialogue notsolution denumerable. 6 deals is demonstrated with Pythagorean with the triplets. help Hereof students formula is are essential! introduced to the problem and the of completethe parametric solution representation is demonstrated of the withunit circle the help of the parametric representation of the unit circle 1−t2 2t There is no way to avoid this, so you must ask x(t)= 2 and y(t)= 2 ,, explaining the geometric significance of the parameter t. Here t is 1+t 1+t kids to repeat the formula ten times. . . . It must I have used this little book in manythe slope ways of a as special a teacher: line, and as one a gets refe arence very elegant book, connec as ation model to double to conduct angle formulae from be driven into their ears like music. You shouldn’t trigonometry. mathematical dialogues and alsoparameter a source t. Here for t students is the slope of to a reaspeciald on line, their own andask every make time presentations. why the formula is true. andexplainingF IGUREone gets the1. a geometricvery elegant significance connection of to the double In short, I would highly recommendThe it to last students mathematical and dialogue teacher dealss ofwith secondary infinities, takin andg students high fromschool. the very basics all the angle formulae from trigonometry. One may not agree with Lang’s philosophy or Dialogue 6 deals with Pythagorean triplets.way Here to the students result thatare introduced real numbers to the are problem not denumerable. and the ideas all the time, but he does force you to think complete solution is demonstrated with theThe helplast mathematical of the parametric dialogue representation deals with of the unit circle 1−t2 2t about what we are doing as teachers. He ends the x(t)= 2 and y(t)= 2 , explaining the geometricI have used significance this little book of the in parameter many wayst. as Here a teacher:t is as a reference book, as a model to conduct 1+t 1+t dialogue on a more humane note and we will leave the slope of a special line, and one getsthe amathematical very way elegant to the result connec dialogues thattion real to and doublenumbers also anglea sourceare formulaenot for students from to read on their own and make presentations. infinities, taking students from the very basics all the readers with that. trigonometry. denumerable.In short, I would highly recommend it to students and teachers of secondary and high school. I have used this little book in many ways as The last mathematical dialogue deals with infinities, taking students from the very basics all theEach teacher must do according to his own way, a teacher: as a reference book, as a model to way to the result that real numbers are not denumerable. his own taste. Each one must use their own means conduct mathematical dialogues and also a source to excite the students. One needs everything for students to read on their own and make I have used this little book in many ways as a teacher: as a reference book, as a model to conduct without exclusivity. presentations. In short, I would highly recommend mathematical dialogues and also a source for students to read on their own and make presentations. it to students and teachers of secondary and high In short, I would highly recommend it to students and teachers of secondary and high school. school.
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