Humanistic Mathematics Network Journal
Issue 6 Article 6
5-1-1991 An Historical Approach to Precalculus and Calculus Victor J. Katz University of the District of Columbia
Follow this and additional works at: http://scholarship.claremont.edu/hmnj Part of the Curriculum and Instruction Commons, Intellectual History Commons, Mathematics Commons, Scholarship of Teaching and Learning Commons, and the Science and Mathematics Education Commons
Recommended Citation Katz, Victor J. (1991) "An Historical Approach to Precalculus and Calculus," Humanistic Mathematics Network Journal: Iss. 6, Article 6. Available at: http://scholarship.claremont.edu/hmnj/vol1/iss6/6
This Article is brought to you for free and open access by the Journals at Claremont at Scholarship @ Claremont. It has been accepted for inclusion in Humanistic Mathematics Network Journal by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. AN HISTORICAL APPROACH TO PRECALCULUS AND CALCULUS
VictorJ. Katz Professor of Mathematics Univefs;ry of the District of Columbia
As a college teacher of mathematics I receive many backgrounds in history and literature with their scientific new texts each year in precalculus and calculus, each studies. It also encourages the student at every stage of one trumpeting its virtues and its new ideas. But a study hiS/her studies to explore the ramifications of scientific of the actual material presented shows that not only are workas it relates to the world around them. And it seems there few new ideas but that chapter for chapter and to me that the prospective scientists and engineers I am almost section for section each such book is a repeat 01 teaching will rrcre than ever need a sense of how their every other one . In fact, I amazed my daughter one day own highly technical work fits in with the needs of society by telling her the titles of the first ten chapters in the as they make decisions which wiD affect the fate of the calculustext she was using. without ever having seen the world. bOOk itself. Since all such texts are the same. one could assume that there is a general agreement in the math By an historical approach to malhematics teaching , ematcarccmrrcnny that there is erecorrect way to leach I do not mean simply giving the historical background lor precalculus and calculus. With all the conferences and eaChseparate topic or giving a biographical sketch of the position papers of the past several years, first on the developers of various ideas . I do mean the organizing of desirabi lity of discrete mathematics and more recentlyon the desired topics in essentially their historical order of the lean and lively calculus, it appears, however, that development. I do mean discussing the historical moti large numbers of faculty merrbers are dissatisfied with vations forthe development of each of these topics, both the way these courses are presented. As a mattero'fact. those within mathematics and those from other fields. I the high failure rate in calculus seems to indicate that also mean connecting the development of each of the students too are dissatisfied. mathematical topics with the development of the other sciences and with the other things which were going on As a possible answer to this dissatisfaction, and as in the world . Naturally, I cannol always stick precisely to a new way of organizing these two courses, I have the historical record. Many seemingly good ideas led to experimented overthepast several years with an historical dead ends or to methods which are too difficult for the approach to both precalculus and caJaJlus-considered level of these particular courses. And one should make together as a four or five term sequence. Not only does use of toclay's technology 01 calculators and computers thts approach help integrate the discrete algorithmic to perform tedious cofT1)Utations rather than have the material with the continuous analytic mathematics, since students repeat their ancestors' hand calculations. in faCllTlJch of the formerwasdeveloped alongsideot the Nevertheless, using history as a general guide does latter. but it also helps to introdJce our science and provide an organizing tool for both precalculus and engineering students to the retatcnstrc between math calculus which helpsto motivate the students and shows ematics and the rest ct our culture. As it stands now, them that mathematics has atNays been an important many students are sadly lacking in an awareness of the part of the overall culture, not only inthe West but also in place which mathematics occupies in our culture. They other parts of the work:l. are interested in the mastery of technique to the exdu sian olleaming the reasons that the ideas were developed A reasonable question at this point is how we can do and the use 01 mathematics in the work:l. Andwithout the all this when we can bare ly cover all the material in the intellectual content behind the mathematical techniques, synabus as it is. I do not have a simple answer 10 that it appears that in large measure the students fail to grasp question. My experience is that to a large extent, this even Ihe techniques. An historical approach to these historical approach is more lime-efficient than a more courses helps to provide a solid motivation tor the learning standard approach and that the historical connections Of mathematics as it ties together much of the students' drawn help to motivate and excite the students, enabling
HMN Newslener #6 21 them to dO more wor1( independently. Of course, we CUriousty, there is no biographical data available on cannot neglect the development of techn~1 pl'of~ency the autrcr of the wond's most farrous mathematics text or of problem solv ing skills . But again. both of these except a few storie s dating from some 700 years later aspects can be set into the histolical context. And if, in than the time of its writing. Since the te xt is a compilation fact, this approach forces us to drop a few topics in the of several different strands of Greek mathematics and current syllabus , this will only be in line with current since it was written in Ale xandria, a city whose inhabit recomme ndations in any case . ants came from many diHerent backgrounds, some his torians have speeulatedthat it was written by a committee. Let me now describe the mathematics course I have Arrong the possibilities lor members 01 this committee in mind . In the context of the school at which I teach, it is would be native Africans,Jews . and even women.Though necessary to beg in with preca lculus material. But it this particular idea on the authorship of the Elements is woukl not be difficu lt to beg in this course at a later point. pure speculation , it should be madeci ear to studems that As will be clear, vinually all of the standard precalaJlus though certain groups have traditionally been excluded and calculus topics will be dea~ with, but often in an order from mathematical knowledge, there were always indio and a conte xt differing trom the standard ones . The viduals who somehow managed to bud<. the trend and course description will erroeasee the nove l aspects of make their own contributions. Unfortunately, sometimes my approach and how it better serves the students than the history books ignorethesecontributions or possibUities, the usual method . denying a sense of 'own ership" to large pans of our population. The course beg ins with a description of the common mathematical knowledge of various ancient civilizations, Euclid's wor1( is followed in the course by an introduc namely, basic algebra through quadratic equations, ba tion to conic sectons in terms of sect ions of a cone and sic geometric formulas, the Pythagorean theorem, and in the context of the problem of doubling the cube . I the calculation of square roots. We discuss the nature of generalty stretch the historical record a bit here and these ancient societies, the Babylonian, the Egyptian, interpret Apollonius' w or1( in tenns of coordinate geom- the Chinese, and what it means for a certain society to etry as Idevebptheequations and some of the elementary "know~ mathematical ideas. Who in the society knew propernes of these imponant curves. I also introduce the these ideas? Why did they know them? What kinds of beginn ings of mathematical physics in the work of problems dkj the y need to solve? In panicutar, we Archimedes, emphasizing in particular the idea of a discuss the value of pi. Wha t does it mean to approx imate mathematical mod el. the ratio of the circumference of a circle to its diameter and how good an approx imation is necessary?In the The idea of a model is further devel oped with the same mode , we also discuss the square root algorithm study of trigonometry, since that subject originated as a and its geometric origins. In this context, an introductory mathematical tool for astronomy, which in tum, in the discussion of the nature of an algorithm is warranted Greek wOnd, was based on the two-sphere model of 1he along with a discussion 01 accuracy. A review of the universe. Thus I introduce the students to some ideas in quadratic formu la is also useful along with the Chinese astronomy. using the model having the earth fixed in the numerical method of solving quadratic equations. center of the heavens. It is interesting , in fact. to ask the students for any evidence that the earth is not stationary. The major change from th ese ancient civilizations to I introdu ce trigonometry itself in essence as pt olemy the classical Greek in terms of mathematics was in the treated it, though rathe r than deal with his chords I give introduction ot deductive proof. Th.Js we deal brielly with the modem ratio definitions of the sine. cosine, and the nature 01 Greek civiliz ation and its differences from tangent. But the sum, diHerence, and half·angl e Icrmrlas those of Egypt and Babylon ia. Then . even though the are oene following ptolemy'Sgeometric proofs and these students have generally had some course in geometry, a are used to begin the actual calculation of values 01the discussion 01 some salient point s of Euclid'S Elements is trigonometric funct ions . The values for 30, 45, and 60 wa rranted. Among the topics included are the nature of degrees come from simple right triangle ge ometry, while logical argument and the idea of an axiomatic system, the those for 36 and 72 degrees come from Euclid's penta basic triangle and parallelogram theorems of Book I, the gon construction. I think it is importan1for 1he students circle theorems of Book Ill , 1he construction of the pen actually to pertcrrn some of these calculations them tagon in Book IV(lor its later use in trigonometry), the selves , so that they learn that the sine function on their similarity results of Book VI, and finally, Euclid'Sresult on calculator is not mag ically generated nor does it come the area 01 a circl e from Book XII. from actually measuring triangles. Arrong the be nefits of, the se carcnatere is an appreciation of interpolation and
22 HMNNewsletter #6 approximation as well as the realization that the sine can introduce these numbers, as Bontlelli did late in the function is nearty urear for small angles. In fact, it was sixteenthcentury, and use them in solving notonly cubic that realization which enabled ptolemy to complete the but also quadratic equations. calculation of his table. I even recalculate the sine using radian measure, since, in eHeet, radian measure was It isnow worthwhile to continuethe studyof solutions used both in Greece and in India eartyon . Itwas realized of ecaatcns from other points of view.once the COftlllex. that for small angles, using this measure, sine x is very ity of the cubic and. perhaps, the quartic 10fTnJias are nearly equal to x itself. understood. For exaftllle. I usethe work of Descartes10 develop the factor and remaindertheorems asweD asthe The students naturally use their calculators rather methods of finding rational solutions to polynomial than the calculated Iab'e in applying trigonometry to equations. It no rational solutions exist, the Chinese solve problems. These problems irclJde not onty plane method already disaJsse The next major topic, as we move out 01 the Greek Having already considered the binomial theorem in period.isthat of equationsolving. Ibeginthis sectionwith the cases n • 2 and n • 3, it is now time to give a more a geometric justification of the quadratic formula taken detailedtreatment of combinatoricsconcentratingon the from the work 01 the Arab algebraist al-Khwarizmi. (That worK of Pascal and his medieval Chinese. Arabic, and is, completing the square means exactly what it says.) European predecessors. First, the binomial theorem for This justification needs a geometric version of the bino any positive integral exponent needs to be developed. mial theorem (a+b)2. a2+ 2ab + b2. COntinuing in this Naturally, this is the place to introduce and study the vein, I ask the natural question of how to solve a cubic notion of mathematical induction. A considerationof the equation. There are several medieval Arab works which background and motivation for Pascal'striangle is useful seek to answer this question, although the answers they as an introduction to the ideas of probability and the give are probably not what students expect. We there calculations of permutations and combinations. A sec fore diSQJSS what it means to Msolve- a cubic (or any) ond major result is the fcrmuta forthe sums of powers of equation. The Chinese inthe thirteenth century certainly integers. This is often associated with Bernoulli,but the could solve such an equation numerically, while the basic ideas date lrom somewhat eartier. A third impcr Arabs of the same time period knew how to do it geo tant ideatreated here is the general ideaof arithmeticand metrically. One interesting method to discuss is that ot geometric sequences, These latter provide a gentle at-Just, who used techniques related to the calculus to introduction to the idea of an infinite series and its sum, decide what types of solutions to a cubic were possible. anideaalreadyunderstoodinsomesensebyArchimedes. To find an algebraicsolution.however.one fT'IJst tum Once we have geometric and arithmetic sequences. to sixteenth century Italy and the story of Tartaglia and it is time to investigate logarithms. Idiscuss the scientific Cardano, a story that shouldcertainly be sharedwith the needfor logarithms as an aid in the tediouscomputations students. The studentsshouldalso be madeawareof the necessaryfor the preparation of astronomicaltables. So Renaissancebackground here and learn why there was I also need to diSOJss the need for suCh tables in the age arenewedurgein Europetooo mathematics. Inpartio.Jtar. of European exploration and discovery. In anyevent.the it was otten the merchants who brought mathematics relationshipbetweengeometricandarihmeticseqJences backto Europethroughtheirtravels to Aftica and Asia. In and the desire to convert fTlJltiplication to addition by any case. the algebraic solution of the OJbic begins with using this relationship leads to the necessity for the the binomial theorem in degree three and, at least in the choice of agood base. Adaptirv the ideas of Napier and Simple cases, is not difficult to understand. I do not give Briggs. I show the students why -nensar logarithms as a complete treatment of OJbics. but only enough for the weD ascommon k:garithms were both developed. Infact. ~tude nts to get the general idea and see why, in the the question of the -naturalness· of natural Iogarittvns lfTeducible case,complexnurrtlers are necessary. Ithen leads via the worX of Napier himsell to some imponant HMN NeWSletter #6 23 ideas of differential caculus as well as to the idea of the Of the problems the calcu lus enabled Newton and others exponential function. to solve. Another idea being developed in the 17th century, It is at this point in the syllabuS that I believe the through the wol1( of Fermat and Descartes, was that of historical aporoacn makes its most important contribu analytic geomet ry. We therefore return to Apollonius' tion 10the study of calculus. namely the earty introcluction conic sections and show algebraically that any quadratic of the notion of a power series . Power series were one eqoaton in two variables leads to a conic seocn. The of Newton's ea r1iest discoveries in calculus and one important tangent and focal properties of these curves which he used constantly. In fact .I also note that for the are then treated as a 1urther introduction to ideas of sine and cosine, power series had been deve loped even calculus. Kepler's laws are then discussed as a con ear1 ier in India. It is qu ite effective pedagogically to tinuation of our eerter notion of a mathematical rrodel. introduc e series atthe ear1iest possible moment, namely, Of course, whene ver one discusses Kepler, one also as soon as the basic derivative and integral algorithms must discuss Galilee and the idea that the scientific are know n and the fundamental theorem is proved. success of a particUlar mathematical mdel does not Power series can then be used as a theme throughout the always mean its acceptance. But we also deal with rest of the course. They provide examples of algorithms. Galileo as a mathematical physicist as we treat the explicit calculations of certain integrals, ideas on the motion 01 projectiles and the general idea of a 'unction. relation ships among various functions. a further intro Though functions have been discussed earner on an ad duction 10 the fundamental idea of convergence, and a hoc basis, it is here tnat Jmake the first attempt to devek)p prime exarrple of the melhod of discovery throug h the idea in detail, first in terms of physical phenomena analogy. and then as a purely mathematical idea. In particular, we consider and graph polynomial funct ions and some easy As a beginning 10 the study of power series, we rational funct ions, including one we will use often later, simply deal with them as generalized polynomialSwhich y . 1/(x+1). Andagain the idea of a mathematical model we can add , subtract , rrultiply, and divide , We then occur s, this time in earthly terms rather than in the discuss convergence by trying to use power series to heavens. Finally,the graphs of the logarithm, exponential, represent funct ions. In particular, we can think 01 power and trigonometric1unctions are briefly discussed, leaving series as generalizing infinite decimal expansions 01 more details to the development of the calaJlus of these numbers. As a first nice exarrple of a power series. I funct ions. show the generalization of the binomial theorem to negative and fractional exponents.I then use this to Certain calculus ideas having been introduced ear calculate square and cube roots. for example. I also note lier, it is nowtime for a detailed discussion, using the work that , since the power series can be considered as gen 01 Fermat, of the two basic problems of calculus, areas eralized polynomial functions,we can take the derivative and extrema (or tangents). I first solve these two prob and anti-derivative term by term . lems for the curves y • x2 and y • x3 and then proceed to generalize to v>xn. For derivatives, I use the binomial We do need other techniques in caicnus. So we theorem, and lor integrals, I use either the material on proceed tctne bas icapproximalion theorem ,f(x+h). f{xl Bernoulli numbers or the work on geometric sequences. + 1'(x)h and then the ideas of leibniz and his followers, in In both cases, we need Ihe basic idea of a limit, so an particular the idea of a differential. The appro ximation intuiHve discussion of limits is warranted here. The theorem gives us straightforward proofs of the product elementary resuns on power funct ions can then be ex rule, quotient rule, and cha in rule. We can then dea l tended to a procedure for find ing derivatives and integrals easily with integrat ion by parts and by substitut ion. On of polynomials. Now anhough Fermat essentia lly had the other hand , we can integrate Y1 - )(2 and ~ by these results, he never noticed what Newton did, the using power series since the other methods do not apply. Fundamental Theorem . In any case, we have now Ofcourse ,physicatproblems, partcuianytbcse centered reached the central figure in the scientific revolut ion. A around Newton's laws ot motion, must be dealt with here statement and at least a sket ch of a proof of the Fun nowthat thebasictools are available. Maximum-minirrum damental Theorem can now be given, prObably in terms problems are among the more important of these types 01 velocity and distance . In fact , with the right prepara of problems. tion, the students can "discove r" this theorem lor them- selves. But I also discuss to some extent the scientific As in any ca lculus course, it is now necessary to deal revolution itself and Newton's part in it. In particular, with the transcendental functions. The natural logarithm throughOu1the remainder of the course I deal with some 24 HMN Newsletter #5 can be developed as is standard, but is also historically selves , however, since power series methods are avail justified, via the integral of 1/(1+x). But by considering able to give results and since new computer algebra the power series for that fundion, we can also get the systems will generate these results in any case. But we series for the logarithm. And this, 01 course, enables us do need to deal with such ideas as arc length, volume. to calculate values. The series for the exponential anc:l center of gravity and see what integrals are neces function can then be developed and disaJssedby inverting sary to solve these problems. At the same time, some of the series for the logarithm, by Euler's method of using the elementary ideas of differential equations. including the binomial theorem, or, pertlaps, by solving the differ· the separation of variables and the integrahon of exact ential equa tion v> kyoessentially introduced in the earlier equations, are covered inthe context of the physieal and treatment ot these functions. There are many interesting mathematical problemswhich led to their study in the first problems involving the exponential and logarithm func place. lions in earty calculus textbooks. It is interesting for today's students to see what kinds of problems eartier It is at the end of the one-variable section of the students solved, so I show them some from the calculus calculus course that I give a detailed treatment of the text of Maria Agnesi. probably the best of those before the notions of limit and convergence. tt is only after the works of Euler and evidence that women could and did experience of dealing with these ideas on an intuitive study mathematics. basis for many months that J can expect the students to understand their theoretical underpinnings and the use The calculus of the trigonometric functions isdevel- of epsilons and dertas. Atter aD. the work of Cauchy and oped via Newton's idea of relating them 10arcs of circles . BolzanoocO,medful1y 150 years afterthat of Newton and This leads again to the "naturalness· of radian measure. Leibna. Thus I begin with arc length and the definition ot the inverse functions. Again, these are done in terms of An historical approach to the study of precalculUS power series. The series for sine, cosine, and tangent and calculuscan providevaluable insights to the students. can easily be developed by various methods as can the With appropriate examples, it can also serve to show thaI basic rules for derivatives. In fact. the latter is best done women and minorities have been involved in mathematics by a geometric argument using differentials rather than in the past and can certainly expect to make further via the limit argument commonly used. The powerseries contributions in the future . I am convinced that tNs representations of eX. sin x, and cos x then provide a approach to this ilJllOrtant segment of mathematics is natural question of how these functions are related. The betterjhan the OJrrentmethod which, in catculus at least. answer leads into a renewed discussion of corrorex begins ~ite unhistoncally, and quite unsuccessfully, oombers and also into a treatment of the hyperbolic with the abstract notion of a limit. Unfortunately, the functions. Finally, the notion of simple harmonic m:>tion publishers of mathematics texts hesitate to publish a text and its associated differential equation y" • -ky is dealt which deviates in any major respect from the current with and shown to result in the trigonometric functions. model. It istherefore not very easy to get new curriculum Thus the basic periodicity of these functions can be ideasoUl into the mathematicscommunity, lithe readers understood in terms of an important physical idea. of this essay try out some of the ideas expressed in it, however, and exchange their findings, the mathematical Other types of physical problems are treated in the comrrunity as a whokt may eventuaUysee the benefits 01 process of dea ling withvarious techn~esofintegration. this approachto the leaching of the mostpoputarcourses Not too much time is spent on the techniques them- in the mathematics curriculum HMN Newslenef #6 25