1 A Math cian ’s Lam ent emati Paul Lockha by rt musician wakes from a terrible night mare. In his dream he finds himself in a society where A has been made manda music educ stude nts becom e more tory. “We are helping ation our und - ators, school systems, and the state are petitive in an increasingl com y so filled world.” Educ d, com es are com mittees are formed, and in charge of this vital project. Studi put missione ce or participation decisions e working musician or the advi all without — are made of a singl com pos er . n their ideas in the form of sheet music, these curious Since musicians are know n to set dow ge of music.” It is imperative that stude nts black dot lines must cons s and titute the “langua usical com petence; inde ed, it degree of m e fluent in this langua becom ge if they are to attain any to exp ect a child to sing a song or play an instrum ent without having a woul d be ludi crous groundi ng in music not theory. Playing and listening to music, let alone thor ough ation and an original piece, are con sidered nced topi cs and are generally put off unt il pos very adva com ing . te school college, and more often gradua s, their mission is to train stude nts to use this As for the primary and second ary school to jiggl e sym bol s around according to a fixed set of rules: “Music class is where we langua ge — them or teacher put staff paper, our es on the bo ard, and we copy s som take out our e not key sign atures e them into a different key. We have to make sure to get the clefs and transpos our teacher is very picky about makin g sure we fill in our qua rter - not es com pletely. right , and em and One time we had a chrom , but the teacher gave me no credit atic scale probl I did it right the wrong because I had the stems poi way.” nting realize that even very young , educ In their wisdom children can be given this kind ators soon ’s third grader hasn’t te shameful if one - idered qui In fact it is cons of musical instruction. com pletely memorized his circle of fifths. “I’ll have to get my son a music tutor. He simply ework. He says it’s bor ing. He just sits there staring out t appl y himself to his music h won’ om tune s to himself and making up silly songs the window , hum ming .” nts must be prepared for the In the highe r grades the pressure is really on. After all, the stude adm s exams. Stude nts must take cour ses in Scales and standa ission rdized tests and college Mode erpoi nt. “It’s a lot for them to learn, but later in college s, Meter, Harmony, and Count eciate all the work they did in high when they finally get to hear all this stuff, they’ll really appr many stude school to conc entrate in music, so onl y a few will .” Of course, not nts actually go on that the black dot tant that every ever get to hear the sounds s represent. Nevertheless, it is impor ation member of society be able to recogni l passage, regardless of the fact or a ze a modul fuga nts just aren’t very good the truth, most stude that they will never hear one . “To tell you at music. ework is barely legible. Most of They are bor ed in class, their skills are terrible, and their hom are less about how impor tant music is in toda y’s world; they just want to take the dn’ them coul t c ber of music cour ses and be done minimum ss there are just music peopl e and num with it. I gue - e. I had this one kid, though, man was she sensationa l! Her sh eets were non music peopl every not place, perfect calligraphy, — sharps, flats, just beautiful. impeccable e in the right ng hell of a musician som eday.” She’s goi to make one
2 2 Waking up in a cold sweat, the musician realizes, gratefully, that it was all just a crazy se!” he reassures himself, “No society woul drea d ever reduc m. “Of cour e such a beautiful and so mindl trivial; no culture coul d be so cruel to its ething meaningf ess and ul art form to som expr ession. How children as to deprive them of such a natural, satisfying means of hum an absurd!” n, a painter has just awakened from a similar hile, on the other side of tow Meanw night mare... ar school classroom — no I was surprised to find tube s of paint. myself in a regul easels, no t actually appl il h igh school ,” I was told by the stude nts. “In seventh “Oh we don’ y paint unt appl ed me a worksheet. On one side were colors and icators.” They show grade we mostly study swatches of color with blank spaces next to them. They were told to write in the names. “I like one of them remarked, “they tell me what to do and I do painting, ” it. It’s easy!” e with the teacher. “So your stude nts don’ t actually do any painting? ” I After class I spok asked. “Well, next year they take Pre Paint - by - Num bers. That prepares them for the main - Paint by - Num bers seque nce in high school . So they’ll get to use what they’ve learned here and - y it to real - situations — dippi ng the brush into paint, wiping it off, stuff like that. appl life painting the one stude ability. The really excellent painters — se we track our s who know Of cour nts by brushes backw ards and forwards — they get to the actual painting a little soone r, their colors and som anced Placement classes for college credit. But mostly and e of them even take the Adv to give the se kids a good founda tion in what painting is all about , so when they we’re just trying there in the real world and t make a total mess of it.” get out paint their kitchen they don’ school classes you mentione d...” “Um, these high bers? We’re seeing “You by - Num - much highe r enrollments lately. I think it’s mean Paint mostly com ing from parents wanting to make sure their kid gets into a good college. Nothing looks better than Adva - by - Num bers on a high school transcript.” nced Paint do colleges care if you ber ed regions with the correspondi ng color?” “Why can fill in num nt is , it show - headed logi know ng. And of cour se if a stude s clear “Oh, well, you cal thinki ng to major in one of the visual sciences, like fashion or interior decorating, then it’s really planni idea to get your painting requi rements out of the way in high school a good .” “I see. And when do nts get to paint freely, on a blank canva s?” stude sound of my professors! They were always goi ng on about expressing “You like one self and your feelings your things like t hat — really way - out - there abstract stuff. I’ve got a and degree in Painting myself, but I’ve never really worked much with blank canva sses. I just use the Paint - - Num bers kits suppl ied by the school boa rd.” by *** present system of mathematics educ mare. In is precisely this kind of night Sadly, our ation fact, if I had to design pos e of destroying a child’s natural a mechanism for the express pur curiosity and love of pattern - making, I coul dn’ t pos sibly do as good a job as is currently being - done dn’ t have the imagination to com e up with the kind of senseless, soul I simply woul — crushing ideas that cons titute cont empor ary mathematics educ ation. Everyone know s that som ething is wrong. The politicians say, “we need highe r standa rds.” pm The school e mone y and equi s say, “we need mor ent.” Educ ators say one thing, and teachers
3 3 on her. They are all wrong. e who unde rstand what is goi ng y peopl are the one s say anot The onl nts. They say, “math class is stupi d and most often blamed and least often heard: the stude bor ,” and they are right . ing Mathematics an d Culture to unde rstand is that mathematics is an art. The difference between math and he first thing T is that our culture doe recogni ze it as such. the other arts, such as music and painting, s not that poe unde are expr essing ts, painters, and Everyone rstands musicians create works of art, and es to when it com themselves in word, image, and sound. In fact, our society is rather generous directors are c ons idered to be working ession; architects, chefs, and creative expr even television not artists. So why mathematicians? Part of the probl em is that nobody has the faintest idea what it is that mathematicians do. seems to be that mathematicians are som ehow conne cted with The com mon perception help the scientists with — big num bers into science perhaps they their formulas, or feed e reason question that if the world had to be divided into ers for som or other. There is no com put l thinke rs” most peopl e woul d place mathematicians i n the the “poe tic dreamers” and the “rationa y. latter categor Nevertheless, the fact is that there is not hing as dreamy and poe tic, not hing as radical, psychedelic, as mathematics. It is every bit as mind blow ing as cosmology or subve rsive, and ers actually found conc e s long before astronom of black hol any) , ics (mathematicians phys eived s more freedom of expr ession than poetry, art, or music (which depend heavily on and allow ical uni verse). Mathematics is the pur prope rties of the phys est of the arts, as well as the most rstood. misunde So let me try to expl I can hardly do ain what mathematics is, and what mathematicians do. better than to begin with G.H. Hardy’s excellent description: t, is a maker A mathematician, like a painter or poe of patterns. If his patterns are more permanent than . th eirs, it is because they are made with ideas making So mathematicians sit around patterns of ideas. What sort of patterns? What sort of e we leave to the biologi eros? No, thos langua ge the rhinoc ideas? Ideas about sts. Ideas about not usually. These things are all far too com plicated for most mathematicians’ and culture? No, like a uni aesthetic principle in mathematics, it is this: hing simple is taste. If there is anyt fying ngs . Mathematicians enjoy about the simplest pos sible thi ng , and the simplest iful beaut thinki are imagi nar y . pos sible things to think about — and I often am — I might imagine a For example, if I’m in the mood shapes e inside a rectangul triangl ar box:
4 4 r how much of the box e takes up? Two - thirds maybe ? The impor tant I wonde the triangl rstand is that I’m not about this draw ing of a triangl e in a box . Nor am I talking thing to unde e metal triangl e forming part of a girder system for a bridge . There’s no som about talking e here. I’m just playing . That’s what math is — wonde ring, playing, ulterior practical pur pos your self with your For one thing, the que stion of how much of the box the amusing imagination. doe ical obj sense for real, phys e takes up ects. Even the most carefully triangl sn’t even make any ing ngl lessly com plicated collection of jiggl ical tria atom s; it change s made phys e is still a hope one its size from e to the next. That is, unl ess you want to talk about som e sort of minut appr measurements. Well, that’s where the aesthetic com es in. That’s just no t simple, oximate cons eque y que stion which depends on all sorts of real - world details. Let’s and ntly it is an ugl que stion is about an imaginary triangl e inside an leave that to the scientists. The mathematical The edge t them to be — that is the sort of obj ect I imaginary box. s are perfect because I wan about . This is a major theme in mathematics: things are what you want them to prefer to think be. You have endl ess choi reality to get in your way. ces; there is no on have made your cho i ces (for example I might choos e to make On the other hand, ce you e sym metrical, or not ) then your my triangl do what they do, whether you like it or new creations not . This is the amazing thing about making imaginary patterns: they talk back! The triangl e takes up a certain amoun t of its box, and I don’ t have any cont rol ove r what that amount is. There is a num ber out it’s two - thirds, maybe it isn’t, but I don’ t get to say what it there, maybe find out is. I have to what it is. make patterns and stion s about imagine whatever we want and So we get to play and ask que we answer these que stions ? It’s not at all like science. There’s no them. But how do riment I can do with test tube s and equi pm ent and whatnot that will tell me the truth about a expe ent of my imagination. y way to get at the truth about onl figm our imaginations is to use our The , and imaginations that is hard work. e in its box, I do see som ething simple and pretty: In the case of the triangl the rectangl e into two pieces like this, I can see that each piece is cut diagona lly in If I chop half by the sides of the triangl e. So there is just as much space inside the triangl e as out side. That means that the triangl e must take up ! exactly half the box and feels This is what a piece of mathematics looks like. That little narrative is an example elegant que stions simple and our imaginary creations , of the mathematician’s art: asking about crafting satisfying and beautiful expl anations . There is really not hing else qui te like this and realm of pur e idea; it’s fascinating, it’s fun, and it’s free! Now where did this idea of mine com e from ? How did I know to draw that line? How doe s a painter know where to put expe rience, trial and error, dum b luck. his brush? Inspiration, these beautiful little poems of thought , these sonne ts of pur e reason. That’s the art of it, creating There is som ething so wonde rfully transformationa l about this art form. The relations hip between the triangl e and the rectangl e was a mystery, and then that one little line made it
5 5 bvi ous dn’t see, and then all of a sudde n I coul d. Som ehow , I was able to create a o . I coul of not hing, and profound simple beauty out change myself in the process. Isn’t that what art is all about ? to mathematics in school . This it is so heartbreaking to see what is bein This is why g done nture of the imagination has been reduc ed to a sterile set of “facts” to be adve rich and fascinating es to be follow ed. In place of a simple and natural que stion about memorized and procedur rewarding process of inve a creative and and discove ry, stude nts are treated to shapes, and ntion this: Trian gle Area Formula: = 1/2 A h h b b e is equa l to on - half its base times its height .” Stude nts are asked to “The area of a triangl e y” it ov ove r in the “exercises.” Gone is the thrill, then “appl er and memorize this formula and frustration of the creative act. There is not even a probl em the joy, ore. even the pain and anym stion has been asked and answer ed at the same time — there is not hing left for the The que stude nt to do. Now what I’m obj ecting to. It’s not about formulas, or memorizing let me be clear about facts. That’s fine in cont it has its place just as learning a vo cabul ary doe s — interesting ext, and to create richer, more nua the helps you fact that triangl es take up nced works of art. But it’s not that matters. What matters is the beautiful idea of choppi ng it with the line, and half their box how that might inspire other beautiful ideas and lead to creative breakthroughs in other probl ems som ething a mere statement of fact can never give you . — ng the creative process and onl y the results of that process, you virtually By removi leaving one saying real enga gement with the subj ect. It is like rantee that no that gua will have any letting me see it. How am I suppo sed to be Michelange lo created a beautiful sculpture, without that? (And of cour se it’s actually much worse than this — at least it’s unde rstood that inspired by there is at I am being prevented from appr eciating) . an art of sculpture th entrating why what , and leaving out By conc , mathematics is reduc ed to an empty shell. on in the “truth” but in the expl anation, the argum ent. It is the argum ent itself which The art is not ont ext, and determines what is really being said and meant. Mathematics is gives the truth its c the art of explanat ion . If you deny stude nts the oppor tuni ty to enga ge in this activity — to pos e their ow n probl n conj ectures and discove ries, to be wrong, to be creatively ems, make their ow e together their ow proof anations and frustrated, to have an inspiration, and to cobbl s — you n expl them mathematics itself. So no, I’m not com plaining about the presence of facts and deny formulas in our mathematics classes, I’m com plaining abo ut the lack of mathematics in our mathematics classes. f your art teacher were to tell you that painting is all about in num bered regions , you filling I that som was wrong d know . The culture informs you — there are museum s and woul ething l as the art in your n hom e. Painting is well unde rstood by society as a galleries, as wel ow
6 6 of hum an expr Likewise, if yo ur science teacher tried to convi nce you that medium ession. predicting astronom their date of birth, you woul d kn ow she a person’ y is about s future based on know s was crazy — science has seeped into the culture to such an extent that almost everyone galaxies and s and math teacher gives you the impression, atom about laws of nature. But if your or by default, that mathematics is about fo rmulas and either expr and essly definitions algor straight ? memorizing ithms, who will set you - math from mons ter: stude nts learn about em is a self their The cultural probl perpetuating it from their teachers, so this lack of unde rstandi ng and teachers, and teachers learn about eciation for mathematics in our culture replicates itself inde finitely. Worse, the perpetuation appr of this “pseudo - sis on the accurate yet mindl ess manipul ation of mathematics,” this empha bol s, creates its ow its ow n set of values. T hos e who have becom e adept at it sym n culture and esteem from they want to hear is that - their success. The last thing derive a great deal of self aesthetic sensitivity. Many a gradua te stude math is really about e raw creativity and nt has com r, after a decade of being told they were “good at math,” that in fact to grief when they discove real mathematical talent and are just very good at follow they have no directions . Math is not ing about ing directions , it’s about making new directions . follow I haven’t even mention . At no time are And ed the lack of mathematical criticism in school nts let in on the secret that mathematics, like any literature, is created by hum stude for an beings their ow n amusement; that works of mathematics are subj ect to critical appr aisal; that one can have and develop mathematical taste . A piece of mathematics is like a poe m, and we can ask if it satisfies our aesthetic criteria: Is this argum Does it make sense? Is it simple and ent sound? Of cour se there’s no criticism goi elegant? Does it get me closer to the heart of the matter? ng on there’s no done to criticize! — art being in school t we want our children to learn to do mathematics? Is it that we don’ Why don’ t trust them, it’s too hard? We seem to feel that they are capable o f making argum ents and that we think ing to their ow n conc lusions about Napol eon, why com about triangl es? I think it’s simply not that we as a culture don’ what mathematics is. The impression we are given is of t know - ething y techni cal, that no o ne coul d pos sibly unde som — a self very cold and highl rstand fulfilling prophe sy if there ever was one . It woul d be bad enough if the culture were merely ignor ant of mathematics, but what is far worse is that peopl e actually think they do know what math is about — and are appa rent ly unde r the gross misconc eption ehow useful to society! This is already a hug e that mathematics is som the other arts. Mathematics is viewed by difference between mathematics and the culture as e sort of tool for science and technol ogy. Everyone know s that poe try and music are for pure som ing ent and ifting and ennobl for upl the hum an spirit (hence their virtual elimination from enjoym the publ ic school curriculum ) but no, math is important . rs no useful or SIMPLICIO: Are you really trying to claim that mathematics offe practical applications to society? Of course not. I ’ m merely suggesting that just because something SALVIATI: happens to have practical consequences, doesn ’ t mean that ’ s what it is about ’ s not why people . Music can lead armies into battle, but that write symphonies. Michelangelo decorated a ceiling, but I ’ m sure he had loftier things on his mind.
7 7 But don t we need people to learn those useful consequences of math? SIMPLICIO: ’ t we need accountants and carpenters and such? Don ’ How many people actually use any of this practical math ” they “ SALVIATI: supposedly learn in school? Do you think carpenters are out there using trigonometry? How many adults remember how to divide fractions, or solve a quadratic equation? Obviously the current p t working, and for good reason: it is ractical training program isn ’ excruciatingly boring, and nobody ever uses it anyway. So why do s so im portant? I don ’ t see how it ’ s doing society any people think it ’ memories of good to have its members walking around with vague algebraic formulas and geometric diagrams, and clear memories of hating them. It might do some good, though, to show them something beautiful and give them an opportunity to enjoy being creative, flexible, open minded thinkers — the kind of th ing a real - mathematical education might provide. SIMPLICIO: But people need to be able to balance their checkbooks, don ’ t they? I ’ SALVIATI: m sure most people use a calculator for everyday arithmetic. And why not? It s certainly easier and more reliab le. But my point is not ’ ’ just that the current system is so terribly bad, it ’ s missing s that what it is so wonderfully good! Mathematics should be taught as art for art ’ s sake. These mundane “ useful ” aspects would follow naturally as a trivial by - t. Beethoven could easily write an advertising jingle, produc but his motivation for learning music was to create something beautiful. SIMPLICIO: But not everyone is cut out to be an artist. W hat about the kids who ’ t “ math people? ” How would they fit i nto your scheme? aren SALVIATI: If everyone were exposed to mathematics in its natural state, with all the challenging fun and surprises that that entails, I think we would see a dramatic change both in the attitude of students toward onception of what it means to be “ good at mathematics, and in our c math. W e are losing so many potentially gifted mathematicians — ” creative, intelligent people who rightly reject what appears to be a meaningless and sterile subject. They are simply too smart to waste their tim e on such piffle. SIMPLICIO: But don ’ t you think that if math class were made more like art class ’ t learn anything? that a lot of kids just wouldn SALVIATI: They ’ re not learning anything now! Better to not have math classes at all than to do what is cu rrently being done. At least some people might have a chance to discover something beautiful on their own. SIMPLICIO: So you would remove mathematics from the school curriculum? SALVIATI: The mathematics has already been removed! The only question is w hat to do with the vapid, hollow shell that remains. Of course I would prefer to replace it with an active and joyful engagement with mathematical ideas.
8 8 SIMPLICIO: But how many math teachers know enough about their subject to teach it that way? I: Very few. And that ’ s just the tip of the iceberg... SALVIAT Mathematics in School more reliable way to kill enthus iasm and interest in a subj ect than to make here is surely no T part of the school curriculum it as a major com pone nt of it a manda tory . Include you virtually gua ation establishm ent will suck the rdized testing standa and rantee that the educ not boa what math is, neither do educ ators, textbook of it. School life out unde rds do rstand panies, and most of our math teachers. The scop e of com ishing s, publ author sadly, neither do where to begin. the probl em is so enor mous , I hardly know Let’s start with the “math reform” debacle. For many years there has been a grow ing Studi es h ave been ething ation. awareness that som is rotten in the state of mathematics educ count less com mittees of teachers, textbook d, conf missione com erences assembled, and ators (whatever they are) have been formed to “fix the probl em.” Quite publ ishers, and educ serving - the textbook ind ustry (which profits from apart from the self interest paid to reform by itical fluctuation minut “new” editions of their unr eadable mons trosities), any by offering e pol up doe sn’t the entire reform move ment has always missed the poi nt. The mathematics curriculum need to be reformed, it needs to be d appe scr . about which “topi cs” shoul All this fussing in what order, or the use and primping d be taught instead of that not or which make and mode l of calculator to use, for god ’s of this not ation ation, ng the deck chairs on the Tita nic! Mathematics is the music of reason . — sake it’s like rearrangi ge in an act of discovery and conj ecture, intuition and inspiration; To do mathematics is to enga — it because it makes no sense to you, but because you gav e usion to be in a state of conf not still don sense and what your creation is up to; to have a breakthrough idea; to ’t unde you rstand an almost painful beauty; to be ov be frustrated as an artist; to be awed and alive erwhelmed by , can have all the conf like ; it won’ t you erences you mathematics and this from damn it. Remove matter. Operate all you ient is already dead . want, doc tors: your pat interesting” and The sadde st part of all this “reform” are the attempts to “make math don’ to make math interesting — it’s already more “relevant to kids’ lives.” You t need than we can handl e! And the glory of it is its com irrelevanc e to our lives. plete in teresting it’s so fun! That’s why life inevitably appe ar forced and Attempts to present mathematics as relevant to daily see kids, if you cont algebra then you can figur e out how old Maria is if we rived: “You know ” (As if anyone woul know that she is two years older than twice her age seven years ago! d ever not about daily of information, and her age.) Algebra is not have access to that ridiculous kind sym metry life, it’s about and this is a valid pur suit in and of itself: num bers and — and difference of two num bers. How Suppos e I am given the sum what the numbers are themselves? can I figure out ade appe it requi res no effort to be m and aling. The elegant que stion, Here is a simple and ans enjoye ancient Babyl on such probl ems, and so do our stude nts. (And I hope oni d working
9 9 will enjoy you it too! ) We don’ t need to bend ove r backw ards to give thinki ng about s: that of being art doe a mathematics relevance. It has relevance in the same way tha t any meaningf ul hum an expe rience. really think kids even want something that is relevant to their daily lives? In any case, do you ng ething com pound interest is goi practical like to get them excited? Peopl e som think You fant asy , and that is just what mathematics can provi de — a relief from daily life, an enjo y to the practical workaday world. anodyne b to “cutesyne em occurs when teachers or textbooks ss.” This is A similar probl succum called “ where, in an attempt to com ety” (one of the panopl y of diseases which bat so - math anxi ed by school ), math is made to seem “friendl y.” To help your stude nts are actually caus ference of a circle, for example, you inve nt this might circum memorize formulas for the area and “ Mr. C,” who drives around “Mrs. A” and tells her how nice his “two pies whol e story about 2 are” ( π r ) and how her “pies are squa re” ( A = π r C ) or som e such nons ense. But what about = 2 the story? The on e about manki nd’ s struggl e with the probl em of measuring curves; about real and Archimedes and of exha ustion; about the transcende nc e of pi? Which Eudoxus the method measuring paper, using dimensions of a circular piece of graph — a is more interesting the rough (and hande without expl anation eone made you memorize and practice formula that som d you r and ove r) or hearing the story of one of the most b eautiful, fascinating ove ems, and one of probl the most brilliant and erful ideas in hum an history? We’re killing peopl e’s interest in circles pow ’s sake! for god aren’t we giving our stude nts a chance to even hear about these thing s, let alone giving Why an oppor tuni ty to actually do som e mathematics, and to com e up with their ow n ideas, them opi nions , and reactions ? What other subj ect is rout inely taught without any mention of its history, phi losophy, ent, aesthetic criteria, and current sta tus? What other thematic developm ect shuns its primary sour — beautiful works of art by som e of the most creative minds in subj ces in favor - rate textbook bastardizations ? — history of third mathematics is that there are no ems. em with school Oh, I know what The main probl probl for probl ems in math classes, these insipid “exercises.” “Here is a type of probl em. Here pas ses to solve it. Yes it will be on the test. Do exercises 1 - 35 odd for hom ework.” What a sad is how way to learn mathematics: to be a trained chimpanzee. que em, a genui - to - goodne ss natural hum an st stion — that’s anot her thing. But a probl ne hone long is the diagona l of a cube ? Do prime num How ng on forever? Is infinity a bers keep goi num many ways can I sym metrically tile a surface? The history of mathematics is the ber? How nd’ history of manki gement with que stions like these, not the mindl ess regur gitation of s enga formulas and algor ithm s (toge ther with cont rived exercises designe d to make use of them). A good probl ething you don’ t know how t o solve. That’s what makes it a good em is som zle, and serves as oppor tuni ty. A good probl em doe s not just sit there in isolation, but puz a good rd to stions interesting que a springboa . A triangl e takes up half its box. What about a other amid inside its three - dimensiona l box? Can we handl e this probl em in a similar way? pyr I can unde rstand the idea of training stud ents to master certain techni que s — I do that too. But not as an end que in mathematics, as in any art, shoul d be learned in cont ex t. in itself. Techni ems, their history, the creative process — that is the prope r setting. Give your The great probl stude nts a good probl em, let them struggl e and get frustrated. See what they com e up with. e tech Wait unt ng for an idea, then give them som il they are dyi nique . But not too much.
10 10 put away lesson plans and your ove rhead projectors, your full - color textbook So your circus freak , your ROMs and the whol e rest of the traveling - show of inations abom CD ary educ ation, and simply do mathematics with your s tude nts! Art teachers don’ t cont empor rote training in specific techni que s. They do what is natural waste their time with textbooks and — easel to easel, making They go around from ect to their subj they get the kids painting. and offering gui dance: sugge stions was thinki “I about our triangl e probl em, and I not iced som ething. If the triangl e is really ng slanted then it sn’t take up half it’s box! See, look: doe “Excellent obs ervation! ng argument assum es that the tip of the triangl e lies Our choppi ly ove r the base. Now we need a new idea.” direct d I try choppi “Shoul ng it a different way?” what you e up with!” “Absolutely. Try all sorts of ideas. Let me know com do o how stude nts to do mathematics? By choo sing enga ging and natural we teach our S ems suitable to their tastes, persona level of expe rience. By giving them time obl pr lities, and formulate conj ectures. By helping them to refine their argum ents and to make discove ries and re of healthy and . By being flexible and creating an atmosphe vibrant mathematical criticism s in direction to which their curiosity may lead. In shor n to sudde t, by having ope n change an nts and stude subj ect. hip with our st intellectual relations hone our is impos sible for a num ber of reason s. Even put ting aside the Of cour se what I’m sugge sting y, I doubt standa fact that statewide curricula and rdized tests virtually eliminate teacher autonom hip with their stude res to have such an intense relations nts. It requi want that most teachers even too much respons too — in shor t, it’s too much work! much vul nerability and ibility e publ isher’s “materials” and to follow the It is far easier to be a passive condui t of som bot deeply and thought “lecture, test, repeat” than to think - fully about the shampoo tle instruction ’ s subj how best to conv ey that meaning directly and hone stly to one ’s meaning ect and of one the difficult task of making decision s based on our aged to forego stude nts. We are encour l wisdom and cons cience, and to “get with the progr am.” It is simply the path of least indi vidua resistance: TEXTBOOK PUBLISHERS : TEACHERS :: A) pha rmaceutical com panies : doc tors panies : disk jockeys B) record com ations essmen C) corpor : congr D) all of the above
11 11 e is that math, like painting or poe har d creative w ork . That makes it very The troubl try, is emplative process. It takes time to produc difficult to teach. Mathematics is a slow e a work , cont se it’s easier to pos of art, and t a set of rules it takes a skilled teacher to recogni ze one. Of cour artists, and it’s easier to write a VCR manua than to gui de aspiring young l than to write an nt of view. with a poi actual book , and art shoul d be taught by working artists, or if not , at least by peopl e Mathematics is an art eciate the art form and who ze it when they see it. It is not necessary that yo u appr can recogni l com want your er, but woul d you a professiona self or your child to be learn music from pos som eone who doe sn’t even play an instrum ent, and has never listened to a piece of taught by d you a ccept as an art teacher som eone who has never picked up a music in their lives? Woul pencil or steppe d foot ? Why is it that we accept math teachers who have never in a museum ed an original piece of mathematics, know not of the history and phi losophy of the produc hing hi hing about recent developm ents, not ect, not in fact beyond what they are expe cted to ng subj te stude nts? What kind of a teacher is that? How can som eone teach ortuna present to their unf ething that they themselves don’ t do? I can’t dance, and cons eque som d never ntly I woul presum that I coul d teach a dance class (I coul d try, but it woul dn’ t be pretty). The e to think know just t have anyone telling me I’m good at dancing difference is I I can’t dance. I don’ a bunc because I know h of dance words. I’m not saying Now at math teachers need to be professiona l mathematicians — far from it. th But shoul dn’ t they at least unde rstand what mathematics is, be good at it, and enjoy doi ng it? If teaching ed to mere data transmission, if there is no sharing of excitement and is reduc r, if teachers themselves are passive recipients of information and creators of new wonde not nts? If adding is to the teacher an arbitrary set is there for their stude fractions ideas, what hope the out com e of a creative process and t he result of aesthetic choi of rules, and desires, not ces and of cour se it will feel that way to the poor students. then is not about information. It’s about having an hone st intellectual relations Teaching hip with your nts. It requi res no method , no tool s, and no training. Just the ability to be real. And if stude can’t be real, then you to inflict your right you self upon innoc ent children. have no In particular, can’ t teach teaching . School s of educ ation are a com plete crock. Oh, you you can take classes in early c hildhood developm ent and whatnot , and you can be trained to use a blackboa rd “effectively” and plan” (which, by the way, insures to prepare an organized “lesson you lesson d , and therefore false), but planne will never be a real teacher if you that your will be illing to be a real person. are unw means ope nne ss and hone sty, an ability to share Teaching excitement, and a love of learning. Without these, all the educ ation degrees in the world won’t help you, and with them they are com pletely unne cessary. It ’s perfectly simple. Stude aliens. They respond to beauty and pattern, and are nts are not naturally curious else. Just talk to them! And more impor tantly, listen to them! like anyone SIMPLICIO: All right, I understand that there is an art to mathemat ics and that we are not doing a good job of exposing people to it. But isn t this a ’ rather esoteric, highbrow sort of thing to expect from our school system? W e ’ re not trying to create philosophers here, we just want people to have a reasonable command o f basic arithmetic so they can function in society.
12 12 But that s not true! School mathematics concerns itself with many SALVIATI: ’ things that have nothing to do with the ability to get along in society — algebra and trigonometry, for instance. These studi es are utterly ply suggesting that if we are going to m sim ’ irrelevant to daily life. I ’ basic education, that we include such things as part of most students do it in an organic and natural way. Also, as I said before, just because a subject happens to ha ve some mundane practical use does not mean that we have to make that use the focus of our teaching and learning. It may be true that you have to be able to read in order to fill out s not why we teach children to read. W e forms at the DMV, but that ’ h them to read for the higher purpose of allowing them access to teac t only would it be cruel to teach beautiful and meaningful ideas. No reading in such a way — to force third graders to fill out purchase — it wouldn ’ orders and tax forms e learn thing s because t work! W they interest us now, not because they might be useful later. But this is exactly what we are asking children to do with math. SIMPLICIO: But don ’ t we need third graders to be able to do arithmetic? W hy? Yo u want to train them to c alculate 427 plus 389? It ’ SALVIATI: s just not a question that very many eight year - olds are asking. Fo r that matter, - most don ’ t fully understand decimal place - value arithmetic, and adults you expect third graders to have a clear conception? Or do you not care if they understand it? It is sim ply too early for that kind of technical training. Of course it can be done, but I think it ultim ately does more harm than good. Much better to wait until their own natural curiosity about numbers kicks in. what should we do with young children in math class? Then SIMPLICIO: Play games! Teach them Chess and Go , Hex and Backgammon, SALVIATI: m, whatever. Make up a game. Do puzzles. Expose Sprouts and Ni ’ them to situations where deductive reasoning is necessary. Don t rry about notation and technique, help them to become active and wo creative mathematical thinkers. SIMPLICIO: ’ d be taking an awful risk. W hat if we de - emphasize It seems like we arithmetic so much that our students end up not being able to add and subt ract? SALVIATI: I think the far greater risk is that of creating schools devoid of creative expression of any kind, where the function of the students is to memorize dates, formulas, and vocabulary lists, and then regurgitate — them on standardized tests P reparing tomorrow ’ s workforce today! ” “ SIMPLICIO: But surely there is some body of mathematical facts of which an educated person should be cognizant. SALVIATI: Ye s, the most im portant of which is that mathematics is an art form done by human beings for p leasure! Alright, yes, it would be nice if people knew a few basic things about numbers and shapes, for instance. But this will never come from rote memorization, drills, lectures, and exercises. Yo u learn things by doing them and you
13 13 ers to you. W e have millions of adults wandering remember what matt negative “ b squared b around with plus or minus the square root of 2a in their heads, and absolutely no idea whatsoever all over minus 4ac ” what it means. And the reason is that they were never given the c hance to discover or invent such things for themselves. They never had an engaging problem to think about, to be frustrated by, and to create in them the desire for technique or method. They were never s relationship with numb ers — no ancient told the history of mankind ’ Liber Abaci Ars Babylonian problem tablets, no Rhind Papyrus, no , no . More im agna M portantly, no chance for them to even get curious about a question; it was answered before they could ask it. SIMPLICIO: But we don ’ t have tim e for every student to invent mathematics for themselves! It took centuries for people to discover the Pythagorean Theorem. How can you expect the average child to do it? ’ I don t. Let ’ SALVIATI: ’ m complaining about the complete s be clear about this. I art and invention, history and philosophy, context and absence of ’ perspective from the mathematics curriculum. That doesn t mean that notation, technique, and the development of a knowledge base have no e should have both. If I object t place. Of course they do. W o a pendulum being too far to one side, it doesn ’ t mean I want it to be all the way on the other side. But the fact is, people learn better when the product comes out of the process. A real appreciation for poetry does not come from memorizing a bunch of poems, it comes from writing your own. SIMPLICIO: Ye s, but before you can write your own poems you need to learn the alphabet. The process has to begin somewhere. Yo u have to walk before you can run. No , you have to have something you want t o run SALVIATI: . Children can toward write poems and stories as they learn to read and write. A piece of - year - old is a wonderful thing, and the spelling and writing by a six ’ punctuation errors don t make it less so. Even very young children can they haven t a clue what key it is in or what type of invent songs, and ’ meter they are using. SIMPLICIO: But isn ’ t math different? Isn ’ t math a language of its own, with all sorts of symbols that have to be learned before you can use it? SALVIATI: No t at all. Mathematic s is not a language, it ’ s an adventure. Do musicians speak another language ” sim ply because they choose to “ abbreviate their ideas with little black dots? If so, it ’ s no obstacle to the toddler and her song. Ye s, a certain amount of mathematical shortha nd has evolved over the centuries, but it is in no way essential. Most mathematics is done with a friend over a cup of coffee, with a diagram scribbled on a napkin. Mathematics is and always has been about ideas, and a valuable idea transcends the symbol s with which you choose to represent it. As Ga uss once remarked, “ W hat we need are no tions , not notations. ”
14 14 But isn t one of the purposes of mathematics education to help SIMPLICIO: ’ students think in a more precise and logical way, and to develop their quantitative reasoning skills? Don ’ t all of these definitions and “ ” formulas sharpen the minds of our students? No they don ’ t. If anything, the current system has the opposite effect SALVIATI: of dulling the mind. Mental acuity of any kind comes from s olving problems yourself, not from being told how to solve them. Fa SIMPLICIO: ir enough. But what about those students who are interested in pursuing a career in science or engineering? Don ’ t they need the es? Isn training that the traditional curriculum provid t that why we ’ teach mathematics in school? SALVIATI: How many students taking literature classes will one day be writers? That is not why we teach literature, nor why students take it. W e teach to enlighten everyone, not to train only the future professionals. In any case, the most valuable skill for a scientist or engineer is being able to think creatively and independently. The last thing anyone traine d . needs is to be The Mathematics Curriculum about the way mathe matics is taught he truly painful thing is not what is missing — in school T done in our — but the fact that there is no actual mathematics being mathematics classes heap know n as “the used what is there in its place: the conf of destructive disinformation to take a closer look at exactly what our stude nts are up mathematics curriculum .” I t is time now what they are being expos how they are being against — ed to in the name of mathematics, and harmed in the process. about The most striking thing - called mathematics curriculum is its rigidity. This is this so to school , city to city, and state to state, the same especially true in the later grades. From school said and done in the same exact order. Far exact things are being in the same exact way and ups t his Orwellian state of affairs, most peopl e have simply disturbed and from et by being as being synonym ous with math itself. accepted this “standa rd mode l” math curriculum the idea that mathematics can cted to what I call the “ladder myth” This is intimately conne — ects” each being in som be arrange nced, or “highe r” nce of “subj d as a seque e way more adva mathematics into a race — som e stude nts are than the previous . The effect is to make school behind. ” And es “ahead” of others, and parents worry that their child is “falling where exactly do you’ ve at the finish line? It’s a sad race to now this race lead? What is waiting here. In the end ation, and you don’ t even know it. been cheated out of a mathematical educ e in a can there is no such thing as an Algebra I I idea . sn’t com Real mathematics doe — to where they take you . Art is not a race . The ladde r myth is a false image of Probl ems lead you a teacher’s ow the standa rd curriculum reinforces this myth and ect, and the subj n path through mathematics as an orga nic whol e. As a result, we have a math prevents him or her from seeing rence, a fragm with no historical perspective or thematic cohe ented collection of curriculum techni que s, uni ted onl y by the ease in which they can be reduc assorted topi - by - cs and ed to step es. step procedur
15 15 discove ry and expl we have rules and regul ations . We never hear a stude nt In place of oration, d make any sense to raise a num er, and I saying, “I wanted to see if it coul ber to a negative pow choo ocal.” Instead we have get a really neat pattern if you se it to mean the recipr found that you the “negative expone nt rule” as a fait d’accom pli teachers and textbooks presenting with no this choi ce, or even that it is a choi ce. of the aesthetics behind mention ems, which might lead to a synt hesis of diverse ideas, to In place of meaningf ul probl and to a feeling harted territories of discussion of thematic uni ty and harmony in unc debate, and redund que unde r ess and ant exercises, specific to the techni mathematics, we have instead joyl so disconne cted from each o ther and from mathematics as a whol e that neither discussion, and e their teacher have the foggi or why such a thing might have com est idea how nts nor the stude in the first place. up em cont ext in which students can make decisions about what t hey In place of a natural probl want their words to mean, and what not they wish to codi fy, they are instead subj ected to an ions ess sequ ence of unm a priori “definitions .” The curriculum is obs essed with endl otivated and nom provi y for no other pur pos e than t o and de teachers with jargon enclature, seemingl ething to test the stude No mathematician in the world woul d bot her making these nts on. som ber,” while 5/2 is an “imprope r fraction. ” They’re : 2 1/2 is a “mixed num senseless distinctions for crying out loud . They are the s ame exact num equal have the same exact prope rties. bers, and Who side of four th grade? uses such words out se it is far easier to test som ’s know ledge of a poi ntless definition than to inspire Of cour eone ething beautiful and to find their ow n meaning. Even if we agree that a basic them to create som mon com abul ary for mathematics is valuable, this isn’t it. How sad that fifth - graders are voc to say “qua drilateral” instead of “four - sided shape,” but are never given a reason to use taught words like “conj ecture,” and “count erexample.” High school stude nts must learn to use the secant func tion, x ,’ as an abbr eviation for the reciprocal of the cosine func tion, ‘1 / cos x ,’ ‘sec with as much intellectual weight as the decision nd. ” ) That (a definition to use ‘&’ in place of “a a hol r from fifteenth century nautical tables, is still with us thand, dove this particular shor (whereas others, such as the “versine” have died out) is mere historical accident, and is of utterly precise shipb oard com put ation no longe r an issue. Thus we value in an era when rapid and is no math classes with poi ntless nom enclature for its ow n sake. clutter our is not even so much a seque nce of topi cs, or ideas, as it is a In practice, the curriculum seque nce of not . App arently mathemati cs cons ists of a secret list of mystical sym bol s and ations ation. Young are given ‘+’ and ‘÷.’ Only later can they be rules for their manipul children √ ̄,’ and then ‘ x ’ and ‘ y entrusted with ‘ the alchemy of parentheses. Finally, they are ’ and indoc ’ ‘ f(x) ,’ and if they are deemed worthy, ‘ trinated in the use of ‘sin,’ ‘log, ’ and ‘ ∫ .’ All d without having h ad a singl e meaningf ul mathematical expe rience. This progr am is so firmly fixed in place that teachers and author s can reliably textbook adva to what stude nts will be doi ng, dow n predict, years in the very page of nce, exactly at all u ncom mon to find second - year algebra stude nts being asked to calculate exercises. It is not [ f ( x + h ) – f ( x ) ] / h for various func tions f , so that they will have “seen” this when they take calculus a few years later. Naturally no motivation cted) for why su ch a is given (nor expe y random seemingl bination of ope rations would be of interest, although I’m sure there are com many teachers who try to expl ain what such a thing might mean, and think they are doi ng their ten ove stude , when in fact to them it is just one more bor in g math probl em to be got nts a favor r with. “What do they want me to do? Oh, just plug it in? OK.”
16 16 her example is the training of stude ess information in an unn ecessarily Anot nts to expr e distant future period com Does any it will h plicated form, merely because at som ave meaning. algebra teacher have the slight he is asking his stude nts to rephr ase e school middl est clue why x lies between three and seven” as | x - 5| < 2 ? Do these hope lessly inept textbook “the num ber stude author preparing them for a pos sible day, years s really believe they are helping nts by etry or an rating ext of a highe r be ope dimensiona l geom within the cont hence, when they might - it. I exp ect they are simply copyi ng each other decade after abstract metric space? I doubt c hangi ng the font s or the highl decade, maybe colors, and beaming with pride when an ight school s their book, and becom es their unw itting accom plice. system adopt Mathematics is about probl ems must be made the focus of a stude nts probl ems, and creatively frustrating as it may be, stude nts and their teachers mathematical life. Painful and d at all times be eng aged in the process — having ideas, not having ideas, discov ering shoul patterns, making conj structing examples and count erexamples, devising argum ents, ectures, con d critiqui each other’s work. Specific techni an ques and methods will arise naturally out of this ng isolated from , but organically conne cted to, and as an process, as they did historically: not out grow th of, their probl em - backgr ound. Engl ish teachers know at spelling and pronunc iation are best learned in a cont ext of reading th writing. History teachers know dates are uni nteresting when remove d from and that names and remain backstory doe s mathematics educ ation olding stuck in the the unf of events. Why eenth century? Com pare your ow n expe ninet algebra with Bertrand Russell’s rience of learning recollection: heart: ‘The squa of two “I was made to learn by re of the sum bers is equa l to the sum of their squa res increased by twice num t.’ I had not th e vague st idea what this meant and their produc when I coul d not remember the words, my tutor threw the book at my head, which did not stimulate my intellect in any way.” Are things different toda y? really any I don ’ t think that ’ s very fair. Surely teach ing methods have im proved SIMPLICIO: since then. SALVIATI: Yo u mean training methods. Teaching is a messy human relationship; it does not require a method. Or rather I should say, if you need a method you ’ re probably not a very good teacher. If you don ’ t have en ough of a feeling for your subject to be able to talk about it in your own voice, in a natural and spontaneous way, how well could you understand it? And speaking of being stuck in the nineteenth century, isn ’ t it shocking how the curriculum itself is stu ck in the seventeenth? To think of all the amazing discoveries and profound revolutions in mathematical thought that have occurred in the last three centuries! There is no more mention of these than if they had never happened. asking an awful lot from our math teachers? Yo SIMPLICIO: But aren ’ t you u expect them to provide individual attention to dozens of students, guiding them on their own paths toward discovery and enlightenment, and to be up on recent mathematical history as well?
17 17 SALVIATI: expect your art teacher to be able to give you individualized, Do you knowledgeable advice about your painting? Do you expect her to know anything about the last three hundred years of art history? But it were so. ’ seriously, I don t expect anything of the kind, I only wish So you blame the math teachers? SIMPLICIO: No SALVIATI: , I blame the culture that produces them. The poor devils are ’ trying their best, and are only doing what they ve been trained to do. m sure most of them love their students and hate what they are being ’ I forced to put them through. They know in their hearts that it is meaningless and degrading. They can sense that they have been made cogs in a great soul - crushing machine, but they lack the perspective ght against it. They only know they needed to understand it, or to fi “ have to get the students ” ready for next year. SIMPLICIO: Do you really think that most students are capable of operating on such a high level as to create their own mathematics? If we honestly believe th at creative reasoning is too “ high ” for our SALVIATI: ’ students, and that they can t handle it, why do we allow them to write history papers or essays about Shakespeare? The problem is not that the students can ’ t handle it, it s that none of the teachers can. They ’ ve ’ never proved anything themselves, so how could they possibly advise a student? In any case, there would obviously be a range of student interest and ability, as there is in any subject, but at least students would like or dislike mathematics for what i t really is, and not for this perverse mockery of it. But surely we want all of our students to learn a basic set of facts and SIMPLICIO: skills. That s what a curriculum is for, and that ’ s why it is so ’ — there are certain tim eless, cold hard fact uniform s we need our students to know: one plus one is two, and the angles of a triangle add up to 180 degrees. These are not opinions, or mushy artistic feelings. SALVIATI: On the contrary. Mathematical structures, useful or not, are invented and developed wi thin a problem context, and derive their meaning from that context. Sometim es we want one plus one to equal zero ( as in so - called ‘ mod 2 ’ arithmetic ) and on the surface of a sphere the angles of a triangle add up to more than 180 degrees. There are no “ acts ” per se; everything is relative and relational. It is the story that f matters, not just the ending. I ’ m getting tired of all your mystical mumbo - jumbo! Basic arithmetic, SIMPLICIO: all right? Do you or do you not agree that students should learn it ? If you mean having an That depends on what you mean by “ it. ” SALVIATI: appreciation for the problems of counting and arranging, the advantages of grouping and naming, the distinction between a representation and the thing itself, and some idea of the h istorical development of number systems, then yes, I do think our students should be exposed to such things. If you mean the rote memorization
18 18 of arithmetic facts without any underlying conceptual framework, then no. If you mean exploring the not at all obvious fact that five groups of seven is the same as seven groups of five, then yes. If you mean making a rule that 5 x 7 = 7 x 5, then no. Doing mathematics should always mean discovering patterns and crafting beautiful and meaningful explanations. S hat about geometry? Don ’ t students prove things there? Isn ’ t High IMPLICIO: W ometry a perfect example of what you want math classes to School Ge be? School Geom etry: Instrument of the Devil High hing here is not te so vexing to the author of a scat hing indi ctment as having the primary qui target of his venom offered up in his suppor t. And never was a wolf in sheep’s clothing as T a false friend as treacherous Geom etry. It is precisely becau se it insidious , nor , as High School e stude ’s attempt to intr rous . is school nts to the art of argum oduc ent that makes it so very dange nts will finally enga stude ge in true mathematical Posing as the arena in get to which the very essence of creative ng ng, reasoni this virus attacks mathematics at its heart, destroyi ent, poi soni ng the stude nts’ enjoym ent of this fascinating and beautiful subj rat iona l argum ect, them from thinki intuitive way. and permanently disabling ng about math in a natural and devious. The stude - victim is fir st stunne d and le and nt this is subt The mechanism behind of poi ntless definitions paralyzed by itions , and not ations , and is then slow ly an ons laught , propos y weaned away from any natural curiosity or intuition about shapes and their and painstakingl trination into the ge and artificial format of so - called patterns by a systematic indoc stilted langua .” etric proof “formal geom All metaphor aside, geom etry class is by far the most mentally and emotiona lly destructive nt of the entire K - 12 mathematics curriculum . Other math cour ses may hide the com pone it in a cage, but in geometry class it is ope cruelly tortured. b eautiful bird, or put nly and aside.) ting all metaphor (Appa rently I am incapable of put rmining of the stude nt’s intuition. A proof , that is, What is happe ning is the systematic unde m. Its goa l is to satisfy . A beautiful proof ent, is a work of fiction, a poe a mathem atical argum ain, and it shoul d expl ain clearly, deeply, and elegantly. A well - written, well - crafted shoul d expl d feel like a splash of cool be a beacon of light — it shoul d refresh the ent shoul argum water, and And it shoul d be char ming . spirit and illuminate the mind. charming what passes for proof in geom etry class. Stude nts are hing There is not about atic format in which their so - called “proof s” are to be conduc ted — a presented a rigid and dogm inappr iate as insisting cessary and that children who wish to plant a garden format as unne opr and ers by genus refer to their flow species. e specific instances of this insanity. We’ll begin with the example of two Let’s look at som cr ossed lines:
19 19 the first thing Now ng of the waters with that usually happe ns is the unne cessary muddyi arently, one simply speak of two crossed lines; on e must give App cannot excessive not ation. simple names like ‘line 1’ and ‘line 2,’ or even ‘ a ’ and ‘ elabor .’ ate names to them. And not b School Geom etry) select random and irrelevant poi nts on these to High We must (according the special “line not ation. ” lines, and then refer to the lines using D A B C see, now we get to call them and CD . And God forbid you AB You d om it the little bars shoul AB (at least I think that’s how it works). Never top — ‘ AB ’ refers to the on h of the line lengt es the mind how poi ntlessly com plicated it is, this is the way one must learn to do it. Now com som nt, usually referred to by actual stateme e absurd name like PROPOSITION 2.1.1. A D ∠ APC ≅ ∠ BPD . A B and Let C D intersect at P . Then P C B igur ation of two h sides are the same. Well, duh bot es on In other words, the angl ! The conf this patently obvi symm crossed lines is as if this wasn’t bad enough, ous for crissake. And etrical statement about lines and angl es must then be “proved.” : Proof Statement Reason 1. Angl e Addi tion Postulate 180 1. m ∠ APC + m ∠ APD = m ∠ BPD + m ∠ APD = 180 2. Subs rty Prope titution ∠ APD ∠ APD = m ∠ m APC + ∠ m 2. BPD + m lity 3. Reflexive Prope rty of Equa APD = m ∠ 3. APD m ∠ 4. Subt rty of Equa raction Prope lity m ∠ APC = m ∠ BPD 4. e Measurement Postulate 5. Angl ∠ BPD 5. ∠ APC ≅ ten by enjoy able argum ent writ Instead of a witty and an actual hum an being, and conduc ted - in one of the world’s many natural langua ges, we get this sullen, soul less, bur eaucratic form ain being made of a molehill! Do we really want to sugge st letter of a proof what a mount . And st: did forward obs ervation like this requi res such an extensive preamble? Be hone that a straight you actually even read it? Of cour se not . Who would want to? being r som ething made ove so simple is to make peopl tion The effect of such a produc e the obvi ly that it be “rigor insisting , by ous ous stion into que ling their ow doubt n intuition. Cal
20 20 d” (as if the above even cons ) is to say to a stude nt, “Your prove titutes a legitimate formal proof ideas are suspect. You need to think feelings way.” and and speak our ce for formal proof que stion. But that place is not a there is a pla Now in mathematics, no tion to mathematical argum ent. At least let peopl e get familiar with som e stude nt’s first introduc learn mathematical obj expe ct from them, before you start formaliz ing ects, and what to Rigor formal proof onl y becom es impor tant when there is a crisis — when you everything. ous r that your discove ects behave in a count erintuitive way; when there is a parado x imaginary obj of som But such excessive preventative hygi ene is com plete ly unne cessary here — e kind. s got se if a logical crisis shoul d arise at som e poi nt, then obvi ous ly nobody’ ten sick yet! Of cour d be inve stigated, and the argum ent made more clear, but that process can be carried out it shoul informally as well. I of mathematics to carry out intuitively and such a n fact it is the soul with one n proof . dialogue ’s ow onl So not used by this pedantry — not hing is more mystifying y are most kids utterly conf than a proof of the obvi ous — but even thos e few whos e intuition remains intact m ust then retranslate their excellent, beautiful ideas back into this absurd hierogl yphi c framework in order ehow for their teacher to call it “correct.” The teacher then flatters himself that he is som nts’ minds sharpening . his stude mple, let’s take the case of a triangl e inside a semicircle: As a more serious exa Now this pattern is that no matter where on the circle you place the the beautiful truth about e, it always forms a nice right e. (I have no obj tip of the triangl to a ter m like “right angl ection angl e” if it is relevant to the problem and makes it easier to discuss. It’s not terminol ogy itself that I obj ect to, it’s poi ntless unne cessary terminol ogy. In any case, I woul d be happy to use “corner” or even “pigpe n” if a stude red.) nt prefer intuition Here is a case where our ewhat in doubt . It’s not at all clear that this shoul d is som be true; it even seems unl ikely — shoul dn’ t the angl e change if I move the tip? What we have here is a fantastic math probl e? If so, why is it true? What a great project! What a em! Is it tru terrific oppor tunity to exercise one ’s inge nui ty and imagination! Of cour se no such oppor tuni ty is given to the stude nts, whos e curiosity and interest is immediately deflated by:
21 21 B EM 9.5. Let ABC be inscribed in a semicircle with diameter THEOR ∆ C . A is a right angl e. ABC ∠ Then C A O : Proof Reason Statement 1. Draw radius OB = OC = OA 1. Given OB Then . e Theorem 2. Isosceles Triangl BCA 2. ∠ OBC = m ∠ m OBA = m ∠ ∠ m BAC 3. Angl e Sum Postulate m ∠ 3. ∠ OBA + m ∠ OBC ABC = m 4. The sum of the angl es of a triangl e is 180 180 BAC = ∠ BCA + m ∠ ABC + m m 4. ∠ 5. Subs titution (line 2) ∠ 5. m ABC + m ∠ OBC + m ∠ OBA = 180 6. Subs titution (line 3) m ∠ ABC = 180 6. 2 of Equa lity 7. Division Prope rty m ∠ ABC = 90 7. 8. Definition of Right e Angl ABC 8. is a right e ∠ angl any argum ent be more Coul d anyt hing be more una ttractive and inelegant? Coul d the from obf uscatory and unr eadable? This isn’t mathematics! A proof shoul d be an epipha ny , not a code d message from the Pentagon . This is what com es from a misplaced sense of Gods r a heap of conf using ent has been bur . The spirit of the argum iness ugl : cal rigor logi ied unde formalism. No mathematician works this way. No mathematician has ever worked thi s way. This is a ng of the mathematical enterprise. Mathematics is not rstandi about com plete and utter misunde plicated. erecting barriers between our selves and our intuition, and making simple things com int simple. simple things and keeping uition, tacles to our obs removi Mathematics is about ng ent devised by one of Com pare this una ppe tizing mess of a proof with the follow ing argum my seventh - graders: rotate it around e and - “Take the triangl so it makes a four e got turned e triangl inside the circle. Since th sided box the sides of the box pletely around, com must be parallel, am. But it can’t be a slanted box so it makes a parallelogr because bot ls are diameters of the circle, so h of its diagona angl e. l, which means it must be an actual rect they’re equa angl That’s why the corner is always a right e.” ent is any Isn’t that just delight ful? And the poi nt isn’t whether this argum better than the , the poi nt is that the idea com es across. (As a matter of fact, the idea of the idea other one as an a glass, darkly.) te pretty, albeit seen as through rst proof is qui fi
22 22 tantly, the idea was the stude nt’s n . The class had a nice probl em to work on, More impor ow s were attempted, and this is what one nt came up with. Of conj ectures were made, proof stude several days, and nce of failures. seque se it took c was the end result of a long our iderably. The original was qui te a bit more cons To be fair, I did paraphr ase the proof cont ained a lot of unne cessary verbiage (as well as spelling and gra mmatical convol uted, and the feeling of it across. And they errors). But I think I got these defects were all to the good; to do as a teacher. I was able to poi logi cal ething gave me som nt out several stylistic and the stude nt was then able to improve the argum ent. For instance, I wasn’t probl ems, and — with the bit about h diagon als being diameters bot I didn’ t think that was com pletely happy ous — but that onl y meant there was more to think about and more unde rstandi ng to entirely obvi the situation. And be gained from nt was able to fill in this gap qui te nicely: in fact the stude “Since the triangl the circle, the tip e got rotated halfway around exactly oppo must end site from up where it started. That’s why is a diameter.” l of the box the diagona So a great projec sure who was more proud, t and a beautiful piece of mathematics. I’m not nt or myself. This is exactly the kind of expe rience I want my stude nts to have. the stude em with the standa rd geom etry curriculum is that the private, persona l exp erience The probl of being ing artist has virtually been eliminated. The art of proof has been replaced by a a struggl - step pattern of uni nspired formal deduc tions . The textbook presents a set of rigid step by , theorems, and proof s, the teacher copi es them ont o the blackbo definitions the stude nts ard, and copy ebook s. They are then asked to mimic them in the exercises. Thos e that them into their not to the pattern qui ckly are the “good” students. catch on The result is that the stude nt becom es a passive participant in th e creative act. Stude nts are making statements to fit a preexisting - pattern, not because they mean them. They are proof trained to ape argum ents, not intend them. So not onl y do they have no idea what their being to what they themselves are saying . e no idea teacher is saying, they hav l way in which definitions are presented is a lie. In an effort to create an Even the traditiona of “clarity” before embarking on the typi cal cascade of propos itions and theorems, a set illusion are provi so that statements and of definitions their proof s can be made as succinct as ded sible. On the surface this seems fairly innoc ; why not make som e abbr eviations so that pos uous can be said more econom ically? The probl em is that definitions things . They com e from matter aesth etic decisions about what distinctions you as an artist cons ider impor tant. And they are probl em - generated . To make a definition is to highl ight and call attention to a feature or structural prope rty. Historically this com of working on a probl em, not as a prelude to it. es out nt is you ever had an t start with definitions , you start with problems. Nobody The poi don’ ber being al of a l” unt il Pythagoras attempted to measure the diagon idea of a num “irrationa re and squa ered that it coul d not be repr esented as a fraction. Definitions make sense discov when a poi nt is reached in your argum ent which makes the distinction necessary. To make definitions is more likely to cause conf usion. motivation without This is yet anot her example of the way that students are shielded and exclude d from the mathematical process. Stude nts need to be able to make their ow n definitions as the need arises to frame the debate themselves. I do n’t want stude nts saying, “the definition, the — ,” I want them sayi ng, “my definition, my theorem, my proof .” theorem, the proof
23 23 plaints aside, the real probl em with this kind is that it is All of these com of presentation . Efficiency and y simply do not make good pedagogy. I have a hard time ing econom bor d appr ov e of this; I kno w Archimedes woul dn’ t. believing that Euclid woul No w hold on a minute. I don ’ t know about you, but I actually enjoyed SIMPLICIO: my high school geometry class. I liked the structure, and I enjoyed working within the rigid proof format. I ’ m sure you did SALVIATI: u probably even got to work on some nice . Yo problems occasionally. Lot s of people enjoy geometry class ( although ’ ) lots more hate it . But this is not a point in favor of the current regim e. Rather, it is powerful testimony to the allure of mathematics itself. It ’ s hard to completely ruin something so beautiful; even this faint shadow of mathematics can still be engaging and satisfying. - by - numbers as well; it is a relaxing and Many people enjoy paint ’ t make it the real thing, though. colorful manual activity. That doesn it. But I ’ SIMPLICIO: liked m telling you, I SALVIATI: And if you had had a more natural mathematical experience you would have liked it even more. SIMPLICIO: So we ’ re supposed to just set off on some free - form mathematical , and the students will learn whatever they happen to learn? excursion Precisely. Problems will lead to other problems, technique will be SALVIATI: developed as it becomes necessary, and new topics will arise naturally. in thirteen years of And if some issue never happens to come up schooling, how interesting or im portant could it be? u SIMPLICIO: ’ ve gone completely mad. Yo SALVIATI: Perhaps I have. But even working within the conventional framework ems so as a good teacher can guide the discussion and the flow of probl to allow the students to discover and invent mathematics for themselves. The real problem is that the bureaucracy does not allow an individual teacher to do that. W ith a set curriculum to follow, a teacher cannot lead. There should be no stand ards, and no curriculum. Just individuals doing what they think best for their students. SIMPLICIO: But then how can schools guarantee that their students will all have the same basic knowledge? How will we accurately measure their relative worth? ALVIATI: They can ’ S ’ t. Just like in real life. Ultim ately you have to t, and we won face the fact that people are all different, and that ’ s just fine. In any case, there ’ s no urgency. So a person graduates from high school not knowing the half angle form ulas ( as if they do now! ) So what? At least - that person would come away with some sort of an idea of what the subject is really about, and would get to see something beautiful. In Con clusion ...
24 24 o put the finishing touc my critique of the standa rd curriculum , and as a service to the hes on com muni ty, I now present the first ever completely hone st cour se catalog for K - 12 T mathematics: The Standa Mathematics Curriculum rd School trination t begins. Stude ics is no LOWER SCHOOL MATH. The indoc nts learn that mathemat sitting but that is done to you. Empha sis is placed on ething still, filling do, som ething you som directions . Children are expe cted to master a com plex set of out worksheets, and follow ing Hindi ating bol s, unrelated t o any real desire or curiosity on their part, algor ithm s for manipul sym difficult for the average adul t. Multiplication tables y a few centuries ago regarded onl and as too are stressed, as are parents, teachers, and the kids themselves. t o view mathematics as a set of procedur MIDDLE SCHOOL MATH. Stude nts are taught es, . The holy tablets, or “Math Books rites, which are eternal and akin to religious ,” are set in stone nts learn to addr ch elders as “they” (as in “What do they hande ess the chur the stude , and d out ems” will be ey want me to divide?”) Cont want here? Do th artificial “word probl rived and ry of arithm etic seem enjoya ble by com introduc ed in order to make the mindl ess drudge parison. a wide array of unne cal terms, such as ‘whol e num ber’ Stude nts will be tested on cessary techni r fraction,’ without est rationale for making such distinctions . Excellent ‘prope and the slight preparation for Algebra I. ng about num bers and their patterns, this ALGEBRA I. So as not to waste valuable time thinki sym ols and rules for their manipul ation. The smoot h narrative thread se instead focuses on cour b amian tablet probl ems to the high art of the Renaissance that leads from ancient Mesopot favor of a disturbingly t - mode rn retelling with no algebraists is discarded in fractured, pos expr s be put into various bers and characte rs, plot, or theme. The insistence that all num ession l conf usion as to the meaning of identity and equa lity. standa rd forms will provide addi tiona e reason. dratic formula for som Stude nts must also memorize the qua the rest of the curriculum se will raise the hope s of OMETRY. Isolated from , this cour GE stude ul mathematical activity, and then dash them. Clum sy nts who wish to enga ge in meaningf not ation will be introduc ed, and no pains will be spared to make the simple seem and distracting l of this cour last remaining vestiges of natural plicated. This goa com se is to eradicate any for Algebra II. mathematical intuition, in preparation ect of this cour inappr opr iate use of coor dinate se is the unm ALGEBRA II. The subj otivated and s are introduc ed in a coo rdinate framework so as to avoi d the aesthetic geom etry. Coni c section . Stude nts will learn to rewrite qua dratic forms in a variety s and simplicity of cone their sections reason whatsoever. Expone ntial and of standa rithm ic func tions are also rd formats for no loga ed in Algebra II, despite not algebraic obj ects, simply because they have to be introduc being rently. The name of the cour sen to reinforce the ladde r ewhere, appa se is cho stuck in som Why mythol etry occurs in between Algebra I and its seque l remains a mystery. ogy. Geom
25 25 ent are stretched to semester lengt h by TRIGONOMETRY. Two weeks of cont masturbatory rounds definitiona beautiful phe nom ena, such as the way the sides of . Truly interesting l runa and on es, will be given the same empha sis as irrelevant abbr eviations and le depend a triang its angl ationa l conve ntions , in order to prevent stude nts from forming any obs olete not clear idea as to ect is about nts will learn such mnemoni c devices a s “SohC ahT oa” and “All what the subj . Stude ng a natural intuitive feeling for orientation and Stude nts Take Calculus” in lieu of developi metry. The measurement of triangl without mention of the sym es will be discussed ntal nature of the trigonom etric functions , or t he cons transcende nt lingui stic and eque phi cal probl ems inhe rent in making such measurements. Calculator requi red, so as to losophi further blur these issues. CALCULUS. A senseless boui cted topi cs. Mostly a half - baked PRE llabaisse of disconne - late nineteenth - century analytic methods into settings where they are neither attempt to introduc e helpful. Techni cal definitions of ‘limits’ and ‘cont inui necessary nor ty’ are presented in order to obs ion of smoot h change . As the name sugge sts , this cour se cure the intuitively clear not nt for Calculus, where the final pha se in the systematic obf uscation of any prepares the stude natural ideas related to shape and will be com pleted. motion CALCULUS. This cour se will expl ore the mathematics of motion, and the best ways to bu ry it unde r a mount of unne cessary formalism. Despite being an introduc tion to bot h the ain integral calculus, the simple and ideas of Newton and Leibni z will be differential and profound of the more sophi sticated func tion - based appr oach develope d as a respons e to discarded in favor analytic crises which do various really appl y in this setting, and which will of cour se not be not d. To be taken again in college, verbatim. mentione *** And there you have it. A com plete prescription for permanently disabl ing young minds — a prove n cure for curiosity. What have they done to mathematics! depth and beauty in this ancient art form. How There is such breathtaking heartbreaking c that peopl e dismiss mathematics as the antithesis of creativity. They are m issing out on ironi an art form older than any book , more profound than any poe m, and more abstract than any abstract. And it is school that has done this! What a sad endl ess cycle of innoc ent teachers inflicting damage upon ent stude nts. We coul d all be having so much more fun. innoc SIMPLICIO: Alright, I ’ m thoroughly depressed. W hat now? SALVIATI: W ell, I think I have an idea about a pyramid inside a cube...