Folk Play Learning System Based on Word Games

Foolproof: A Sampling of Mathematical Folk HumorPaul Renteln and Alan DundesI n the discipline known as folkloristics [D1] (thestudy of folklore), a folk is defined as any group whatsoever that shares at least one common linking factor. The factor could be na-tionality, ethnicity, religion, or occupation. Members of a profession would also qualify as a folk group. Hence, mathematicians constitute a folk. And, like all folk groups, mathematicians have their own folk speech (slang), proverbs, limericks, and jokes, among other forms of folklore. It is pre-cisely the folklore of a group that defines that group. So, mathematicians as a group share a com-mon core of mathematical folklore. Some of this folklore tends to be quite esoteric and intelligible only to members of the group. Outsiders not pos-sessing the requisite mathematical vocabulary and knowledge rarely know such esoteric material, and even if they did they would probably not under-stand it. But there is also exoteric mathematical folklore that is known to a limited number of out-siders, for example, physicists, chemists, and en-gineers. Much of this exoteric folklore consists of classic jokes contrasting members of different but related academic disciplines. We intend to offer a brief sampling of both esoteric and exoteric math-ematical folklore, concentrating on humorous gen-res such as jokes. We are persuaded that these data not only serve as a basis for identity among mathematicians but also provide a unique window on mathematical culture in general, and even a clue as to the nature of mathematical thinking.As the readership of the Notices is principally professional mathematicians, we do not feel it is necessary to explain all the data we present. Most readers will already be familiar with many if not all of the examples given here. That is to be ex-pected. All folklore exists in multiple forms and demonstrates variation. There are often many ver-sions of a classic joke or legend. The same absent-minded professor joke, for example, may be at-tached to various historical individuals, for example, Norbert Wiener and John von Neumann. So it is likely that each reader will recognize a par-ticular item even though he or she might have heard it (orally) in a slightly different form.1Making Fun of MathematicsMany English-language mathematical jokes are based on word play involving standard mathe-matical concepts and terminology. In fact, many of the jokes involve food items, which may be a re-flection of the fact that some mathematical con-cepts are hard to digest, or difficult to swallow:Q:What’s purple and commutes?A:An abelian grape.Paul Renteln is professor and chair of the Department of Physics at California State University, San Bernardino, and visiting associate in the Department of Mathematics at the California Institute of Technology. His email address is prenteln@.Alan Dundes is professor of anthropology and folklore at the University of California, Berkeley. His email address is carolynb@.There are several other variants of this joke:Q:What’s purple, commutes, and is worshipped by a limited number of people?A:A finitely-venerated abelian grape.Q:What is lavender and commutes?A:An abelian semigrape.Q:What is purple and all of its offspring have been committed to institutions?A:A simple grape: it has no normal subgrapes.Q:What’s purple, round, and doesn’t get much for Christmas?A:A finitely presented grape.Q:What’s an abelian group under addition, closed, associative, distributive, and bears a curse?A:The Ring of the Nibelung.Q:What’s nutritious and commutes?A:An abelian soup.Q:What’s hot, chunky, and acts on a polygon?A:Dihedral soup.Q:What’s sour, yellow, and equivalent to the Axiom of Choice?A:Zorn’s Lemon.There are thousands of lemmas in mathemat-ics, so in principle the same word play would work on any one of them, yet this joke based on Zorn’s lemma continues to be popular. Why should this be so? The reason for this becomes clearer when we recall that the Axiom of Choice can be used to prove many counterintuitive results in set theory, such as the Banach-Tarski paradox. This leads some mathematicians to reject the Axiom of Choice. Yet because it is so useful (and because it seems so innocuous at first glance), many mathemati-cians accept it, but with strong reservations. They find it distasteful, or unpalatable, or, if you will, a bitter fruit.Here is another one, even more revealing:Q:What is brown, furry, runs to the sea, and is equiv-alent to the Axiom of Choice?A:Zorn’s lemming.Lemmings are known to follow each other blindly into the sea, where they drown. The impli-cation of this version seems to be that mathe-maticians who rely upon the Axiom of Choice are all lemmings who blindly follow one another and who may all be headed for intellectual death.Q:What is green and homeomorphic to the open unit interval?A:The real lime.Q:What’s yellow, linear, normed, and complete? A:A Bananach space.Q:What do you call a young eigensheep?A:A lamb, duh!Q:What’s the value of a contour integral around Western Europe?A:Zero, because all the Poles are in Eastern Europe. Addendum: Actually, there ARE some Poles in West-ern Europe, but they are removable!2Q:Why did the mathematician name his dog “Cauchy”?A:Because he left a residue at every pole.Q:What is a topologist?A:Someone who cannot distinguish between a doughnut and a coffee cup.Q:Why didn’t Newton discover group theory?A:Because he wasn’t Abel.Q:What do you get if you cross an elephant and a banana?A:|elephant|*|banana|*sin(theta).2There are several shaggy dog stories based on wordplay in complex analysis. For example:A bunch of Polish scientists decided to flee their re-pressive government by hijacking an airliner and forcing the pilot to fly them to a western country. They drove to the airport, forced their way on board a large passenger jet, and found there was no pilot on board. Terrified, they listened as the sirens got louder. Finally, one of the sci-entists suggested that since he was an experimentalist, he would try to fly the aircraft. He sat down at the con-trols and tried to figure them out. The sirens got louder and louder. Armed men surrounded the jet. The would-be pilot’s friends cried out, “Please, please take off now!!! Hurry!!!” The experimentalist calmly replied, “Have pa-tience. I’m just a simple pole in a complex plane.”A group of Polish tourists is flying on a small airplane through the Grand Canyon on a sightseeing tour. The tour guide announces: “On the right of the airplane, you can see the famous Bright Angle Falls.” The tourists leap out of their seats and crowd to the windows on the right side. This causes a dynamic imbalance, and the plane violently rolls to the side and crashes into the canyon wall. All aboard are lost. The moral of this episode is: always keep your poles off the right side of the plane.For a classification of shaggy dog stories, including ones with punch lines based on “an Axiom of Science”, see [B].Q:What do you get if you cross a mosquito with a mountain climber?A:You can’t cross a vector with a scalar.or this variationQ:What do you get when you cross a mountain goat and a mountain climber?A:Nothing—you can’t cross two scalars.3Q:What is a compact city?A:It’s a city that can be guarded by finitely many nearsighted policemen.4Q:What’s a dilemma?A:A lemma that produces two results.Q:What’s a polar bear?A:A rectangular bear after a coordinate transform.Q:What goes “Pieces of seven! Pieces of seven!”A:A parroty error.Q:Why can’t you grow wheat in Z/6Z?A:It’s not a field.Q:What’s big, grey, and proves the uncountability of the reals?A:Cantor’s diagonal elephant.5Q:What is gray and huge and has integer coeffi-cients?A:An elephantine equation.Q:What is very old, used by farmers, and obeys the fundamental theorem of arithmetic?A:An antique tractorization domain.Q:What is hallucinogenic and exists for every group with order divisible by p k?A:A psilocybin p-subgroup.Q:What is often used by Canadians to help solve cer-tain differential equations?A:The Lacross transform.Q:What is clear and used by trendy sophisticated engineers to solve other differential equations? A:The Perrier transform.Q:Who knows everything there is to be known about vector analysis?A:The Oracle of del phi!Q:Why can fish from the United States enter Cana-dian waters without a passport?A:This is permitted by the Law of Aquatic reci-procity!Q:Why are topologists especially prone to malaria? A:This disease comes from the Tietze fly!!6Q:Why do truncated Maclaurin series fit the origi-nal function so well?A:Because they are “Taylor” made.Q:What is locally like a ring and very evil?A:A devilish scheme.Q:What is a proof?A:One-half percent of alcohol.Q:Can you prove Lagrange’s Identity?A:Are you kidding? It’s really hard to prove the iden-tity of someone who’s been dead for over 150 years!Q:What is black and white ivory and fills space? A:A piano curve.Q:What’s polite and works for the phone company? A:A deferential operator.Q:What does an analytic number theorist say when he is drowning?A:Log-log, log-log, log-log, ….3For further discussion of cross breed riddles, see [AH]. 4Some mathematicians are so concerned about precisionthat they feel compelled to correct the mathematics in the jokes themselves. The following email comment on this joke appeared at http://www.xs4all.nl/~jcdverha/ scijokes/1_6.html: From: http://wheierman# (Will Heierman): Re-cently, I read the following riddle on a math joke website. It was attributed to Peter Lax.“What is a compact city?”“A city that can be guarded by a finite number of near-sighted policemen.”However, I doubt that he would make such a mis-take. Moreover, he was my advisor when I was a graduate student, and I actually recall a con-versation with him regarding this anecdote. It did not go exactly like this (but this makes a better story):“Dr. Lax, wouldn’t it be better to say that a compact city is one that can be guarded by a finite number of police-men, no matter how nearsighted they are?”“That’s not any better, really, for if the nth policeman could only see a distance of 1/2(n+2), no finite number of them could guard even [0,1]!”“Wow! How do we handle this?”“I might reword it slightly: A compact city is one which canbe guarded by a finite number of policemen, no matter how nearsighted a policeman is.”5For a discussion of the significance of other jokes in the elephant cycle, see [AD].6This joke contains an epidemiological error and should more correctly read “Why are topologists especially prone to sleeping sickness?”, as it is sleeping sickness, not malaria, that is spread by the Tsetse fly.Q:What does a topologist call a virgin?A:Simply connected.7Some joking questions are adaptations of stan-dard folkloristic forms. The question “How many xxxx’s does it take to change a lightbulb” has sev-eral answers [D2], including the following:Q:How many topologists does it take to change a lightbulb?A:Just one, but what will you do with the dough-nut?Q:How many number theorists does it take to change a lightbulb?A:This is not known, but it is conjectured to be an elegant prime.Q:How many geometers does it take to screw in a lightbulb?A:None. You can’t do it with a straightedge and com-pass.Q:How many analysts does it take to screw in a light-bulb?A:Three. One to prove existence, one to prove uniqueness, and one to derive a nonconstructive algorithm to do it.Q:How many Bourbakists does it take to replace a lightbulb?A:Changing a lightbulb is a special case of a more general theorem concerning the maintenance and repair of an electrical system. To establish upper and lower bounds for the number of personnel required, we must determine whether the sufficient conditions of Lemma 2.1 (Availability of personnel) and those of Corollary 2.3.55 (Motivation of personnel) apply. If and only if these conditions are met, we derive the result by an application of the theorems in Section 3.1123. The resulting upper bound is, of course, a result in an abstract measure space, in the weak-* topology.Q:How many mathematicians does it take to screw in a lightbulb?A:0.999999….In addition to making fun of beginning mathe-matics students who have difficulty grasping the equality 0.9999···=1, this joke also shows how mathematics is essentially a solitary endeavor, as it only takes one mathematician to do it. This can be contrasted with many of the other standard an-swers to this question, which demonstrate that groups of people are often required. In fact, the following text shows that mathematicians are so clever that they can imagine a lightbulb that changes itself!Q:How many lightbulbs does it take to change a lightbulb?A:One, if it knows its own Gödel number.This joke also expresses the fear that mathe-matics itself may be inconsistent (and perhaps even lead to paradoxes), as Gödel numbers were used by Gödel to prove his two celebrated incom-pleteness theorems, both of which shook the foun-dations of mathematics. The same anxiety sur-faces in the following parody of the traditional joke: “Why did the chicken cross the road? To get to the other side.”Q:Why did the chicken cross the road?A:Gödel: It cannot be proved whether the chicken crossed the road.Some other answers to this question provided by famous mathematicians include:Q:Why did the chicken cross the road?A:Erdös: It was forced to do so by the chicken-hole principle.Q:Why did the chicken cross the road?A:Riemann: The answer appears in Dirichlet’s lec-tures.Q:Why did the chicken cross the road?A:Fermat: It did not fit on the margin on this side.A good mathematician always looks for possi-ble counterexamples to unproved conjectures. Here is an amusing variant of the chicken crossing joke, demonstrating that the notion of “other side” is not always as straightforward as it seems:Q:Why did the chicken cross the Möbius strip?A:To get to the other–er….If you cannot prove the theorem, prove a dif-ferent one:Q:Why did the chicken cross the Möbius strip?A:To get to the same side.Although puns and one-liners occur orally, mod-ern folklore is transmitted by photocopier, email, and the Internet, frequently in the form of lists [DP]. An example of this form is the following “Top Ten Excuses for Not Doing Homework”:•I accidentally divided by zero and my paper burst into flames.7Here, again, the jokester reveals a lack of understand-ing of either mathematics or anatomy.•Isaac Newton’s birthday.•I could only get arbitrarily close to my textbook.I couldn’t actually reach it.•I have the proof, but there isn’t room to write it in this margin.•I was watching the World Series and got tied up trying to prove that it converged.•I have a solar-powered calculator and it was cloudy.•I locked the paper in my trunk, but a four-dimensional dog got in and ate it.•I couldn’t figure out whether i am the square of negative one or i is the square root of negative one.•I took time out to snack on a doughnut and a cup of coffee [and] I spent the rest of the night try-ing to figure which one to dunk.•I could have sworn I put the home work inside a Klein bottle, but this morning I couldn’t find it.Every mathematician relies on the truth of prior propositions proved by other mathematicians, and it is ordinarily impossible for a person to check the validity of every proof upon which his work is based. For this reason, mathematicians place tremendous emphasis on the correctness of mathematical proofs. The following list of proof techniques hints at the anxieties felt by many mathematicians regarding the degree to which mathematical truth is dependent upon the trust-worthiness of previous results. This anxiety is ex-acerbated by the fact that some mathematicians have a less rigorous proof style than other math-ematicians.How to prove it. Guide for lecturers.Proof by vigorous handwaving:Works well in a classroom or seminar setting.Proof by forward reference:Reference is usually to a forthcoming paper of the author, which is often not as forthcoming as at first.Proof by funding:How could three different government agencies be wrong?Proof by example:The author gives only the case n=2and suggests that it contains most of the ideas of the general proof.Proof by omission:“The reader may easily supply the details.”“The other 253 cases are analogous.”Proof by deferral:“We’ll prove this later in the course.”Proof by picture:A more convincing form of proof by example. Com-bines well with proof by omission.Proof by intimidation:“Trivial.”Proof by seduction:“Convince yourself that this is true!”Proof by cumbersome notation:Best done with access to at least four alphabets and special symbols.Proof by exhaustion:An issue or two of a journal devoted to your proof is useful.Proof by obfuscation:A long plotless sequence of true and/or meaning-less syntactically related statements.Proof by wishful citation:The author cites the negation, converse, or gener-alization of a theorem from the literature to sup-port his claims.Proof by eminent authority:“I saw Karp in the elevator and he said it was prob-ably NP-complete.”Proof by personal communication:“Eight-dimensional colored cycle stripping is NP-complete [Karp, personal communication].”Proof by reduction to the wrong problem:“To see that infinite-dimensional colored cycle stripping is decidable, we reduce it to the halting problem.”Proof by reference to inaccessible literature:The author cites a simple corollary of a theorem to be found in a privately circulated memoir of the Slovenian Philological Society, 1883.Proof by importance:A large body of useful consequences all follow from the proposition in question.Proof by accumulated evidence:Long and diligent search has not revealed a coun-terexample.Proof by cosmology:The negation of the proposition is unimaginable or meaningless. Popular for proofs of the existence of God.Proof by mutual reference:In reference A, Theorem 5 is said to follow from Theorem 3 in reference B, which is shown to fol-low from Corollary 6.2 in reference C, which is an easy consequence of Theorem 5 in reference A.Proof by metaproof:A method is given to construct the desired proof. The correctness of the method is proved by any of these techniques.Proof by vehement assertion:It is useful to have some kind of authority relation to the audience.Proof by ghost reference:Nothing even remotely resembling the cited theo-rem appears in the reference given.Proof by semantic shift:Some of the standard but inconvenient definitions are changed for the statement of the result.Proof by appeal to intuition:Cloud-shaped drawings frequently help here.In common parlance, if you ask someone to prove something, it can be considered a challenge, which might lead to conflict. Even among mathe-maticians, there are those who insist on a high de-gree of rigor and those who are more willing to ac-cept less rigorous standards:What is the difference between an ar-gument and a proof? An argument willconvince a reasonable man, but a proofis needed to convince an unreasonableone.There are also parodies of the logic utilized in mathematical proofs:Theorem.A cat has nine tails.Proof.No cat has eight tails. A cat has one more tail than no cat. Therefore a cat has nine tails.and a variant:Theorem.All dogs have nine legs.Proof.Would you agree that no dog has five legs? Would you agree that a dog has four legs more than no dog? 4+5=?Theorem.All positive integers are interesting.Proof.Assume the contrary. Then there is a low-est noninteresting positive integer. But, hey, that’s pretty interesting! A contradiction.The following class of joke is often told by math-ematicians when commenting on the common mis-takes of nonmathematicians in attempting to con-struct valid proofs. But these jokes are also a reflection of the worry that perhaps one’s own work may be based upon false premises. There are many fallacious proofs of the type purporting to prove that 1=2. Here is just one example using the ever-popular trick of dividing by zero.Theorem.3=4.Proof.Supposea+b=c.This can also be written as:4a−3a+4b−3b=4c−3c.After reorganizing:4a+4b−4c=3a+3b−3c.Take the constants out of the brackets:4(a+b−c)=3(a+b−c).Remove the same term left and right:4=3.The next joke originates from the time before calculators, when scientists used slide rules or ta-bles of logarithms to carry out complex calculations:You know that during the Great Flood,Noah brought along two of every speciesfor reproductive purposes. Well, after afew weeks on the ark, all the coupleswere getting along fine, except for thesetwo snakes. Day and night, Noah wor-ried that this was going to mean the endof this species. Finally when the floodended and the ark hit ground, the twosnakes darted out of the ship and headedto the nearest picnic table where theystarted to “go at it”. It was then thatNoah realized that…Adders can’t mul-tiply without their log tables.Here is another version:The Flood is over and the ark has landed.Noah lets all the animals out and says,“Go forth and multiply.” A few monthslater, Noah decides to take a stroll andsee how the animals are doing. Every-where he looks he finds baby animals.Everyone is doing fine except for onepair of little snakes. “What’s the prob-lem?” says Noah. “Cut down some treesand let us live there,” say the snakes.Noah follows their advice. Several moreweeks pass. Noah checks on the snakesagain. Lots of little snakes, everybody ishappy. Noah asks, “Want to tell me howthe trees helped?” “Certainly”, say thesnakes. “We’re adders, so we need logsto multiply.”There are many mathematical limericks and songs, but in the interests of brevity we include only the following. It is based upon a traditional song “One Hundred Bottles of Beer on the Wall” which could be written in mathematical notation:N bottles of beer on the wall,N bottles of beer,You take one down, and pass it around,N−1bottles of beer on the wall.The song begins at N=100and then repeats until there are no bottles of beer remaining. It is often sung by children on long car rides, as a way of keep-ing them busy. The following mathematical variant is an amusing way to keep the children occupied a bit longer:Aleph-null bottles of beer on the wall,Aleph-null bottles of beer,You take one down, and pass itaround,Aleph-null bottles of beer on the wall.Making Fun of MathematiciansMost of the above texts would not be likely to cir-culate among nonmathematicians, as they rely on specialized knowledge of the field. However, there is a substantial body of humor involving mathe-maticians themselves, typically contrasted with other scientists, which is more accessible to a gen-eral audience [G]. These jokes are often told by physicists and engineers, but they are also related by mathematicians, despite the fact that they are typically the object of ridicule. One reason for this may be that the mathematician is more concerned with ideas than with reality:Engineers think that equations approx-imate the real world. Physicists thinkthat the real world approximates equa-tions. Mathematicians are unable tomake the connection.Another variant of this reads:An engineer thinks that his equations arean approximation to reality. A physicistthinks reality is an approximation to hisequations. A mathematician doesn’tcare.The stereotypical mathematician is depicted as being impractical, abstract, and not concerned about everyday affairs:An engineer, a physicist, and a mathe-matician are staying in a hotel. The en-gineer wakes up and smells smoke. Hegoes out into the hallway and sees a fire,so he fills a trash can from his roomwith water and douses the fire. He goesback to bed. Later, the physicist wakesup and smells the smoke. He opens hisdoor and sees a fire in the hallway. Hewalks down the hall to a fire hose andafter calculating the flame velocity, dis-tance, water pressure, trajectory, andso forth, extinguishes the fire with theminimum amount of water and energyneeded. Later the mathematician wakesup and smells smoke. He goes to the hall,sees the fire and then the fire hose. Hethinks for a moment and then exclaims,“Ah, a solution exists!” and then goesback to bed.Here is another version:A physicist and a mathematician aresitting in a faculty lounge. Suddenly, thecoffee machine catches on fire. Thephysicist grabs a bucket and leaps to-ward the sink, fills the bucket with water,and puts out the fire. Second day, thesame two sit in the same lounge. Againthe coffee machine catches on fire. Thistime, the mathematician stands up, getsa bucket, and hands the bucket to thephysicist, thus reducing the problem toa previously solved one.The principle of reducing a theorem to a previ-ously proved theorem is often misinterpreted by nonmathematicians as that of reducing a “problem”to a previously solved one. The result is some-times a bit unexpected:A mathematician and an engineer areon a desert island. They find two palmtrees with one coconut each. The engi-neer shinnies up one tree, gets the co-conut, and eats it. The mathematicianshinnies up the other tree, gets the co-conut, climbs the other tree and puts itthere. “Now we’ve reduced it to a prob-lem we know how to solve.”In the following story the mathematician is so wrapped up in his own mathematical world that he appears heartless:There are a mathematician and a physi-cist and a burning building with peopleinside. There are a fire hydrant and ahose on the sidewalk. The physicist hasto put the fire out…so, he attaches thehose to the hydrant, puts the fire out, andsaves the house and the family. Thenthey put the people back in the house, setit on fire, and ask the mathematicianto solve the problem. So, he takes thehose off the hydrant and lays it on thesidewalk. “Now I’ve reduced it to apreviously solved problem,” and walksaway.The mathematical solution to a problem is often the most elegant but does not solve the original problem in an effective manner:One day a farmer called up an engi-neer, a physicist, and a mathematicianand asked them to fence in the largestpossible area with the least amount offence. The engineer made the fence ina circle and proclaimed that he had themost efficient design. The physicist madea long, straight line and proclaimed “Wecan assume the length is infinite…” andpointed out that fencing off half of theEarth was certainly a more efficient wayto do it. The mathematician just laughedat them. He built a tiny fence aroundhimself and said, “I declare myself to beon the outside.”Some mathematical constructions, when applied to the real world, give rather absurd results:A biologist, a physicist, and a mathe-matician were sitting in a street caféwatching the crowd. Across the streetthey saw a man and a woman enteringa building. Ten minutes later they reap-peared together with a third person.“They have multiplied,” said the biologist.“Oh no, an error in measurement,” thephysicist sighed. “If exactly one personenters the building now, it will be emptyagain,” the mathematician concluded.Three men with degrees in mathematics,physics, and biology are locked up in darkrooms for research reasons. A week laterthe researchers open the door and the bi-ologist steps out and reports: “Well, I sataround until I started to get bored, thenI searched the room and found a tin whichI smashed on the floor. There was foodin it which I ate when I got hungry. That’sit.” Then they free the man with the de-gree in physics and he says: “I walkedalong the walls to get an image of theroom’s geometry, then I searched it. Therewas a metal cylinder at five feet into theroom and two feet left of the door. It feltlike a tin and I threw it at the left wall atthe right angle and velocity for it to crackopen.” Finally, the researchers open thethird door and hear a faint voice out ofthe darkness: “Let C be an open can.”Here is a variant with a more dire consequence:There was a mad scientist (a mad …social…scientist) who kidnapped threecolleagues, an engineer, a physicist, anda mathematician, and locked each ofthem in separate cells with plenty ofcanned food and water but no canopener. A month later, returning, themad scientist went to the engineer’s celland found it long empty. The engineerhad constructed a can opener frompocket trash, used aluminum shavingsand dried sugar to make an explosive,and escaped. The physicist had workedout the angle necessary to knock the lidsoff the tin cans by throwing them againstthe wall. She was developing a goodpitching arm and a new quantum the-ory. The mathematician had stacked theunopened cans into a surprising solutionto the kissing problem; his desiccatedcorpse was propped calmly against awall, and this was inscribed on the floorin blood: Theorem: If I can’t open thesecans, I’ll die. Proof: Assume the oppo-site.…In the following story the mathematician ap-plies flawless logic, but shows an unwillingness to generalize from a single case, a procedure that holds many mathematical pitfalls, but which is often valid in the real world:A mathematician, a physicist, and anengineer were traveling through Scot-land when they saw a black sheepthrough the window of the train. “Aha,”says the engineer, “I see that Scottishsheep are black.” “Hmm,” says the physi-cist, “You mean that some Scottish sheepare black.” “No,” says the mathemati-cian, “All we know is that there is atleast one sheep in Scotland, and that atleast one side of that one sheep is black!”Here, the mathematician again gives impracti-cal answers to real-world questions:Three men are in a hot-air balloon. Soon,they find themselves lost in a canyonsomewhere. One of the three men says,“I’ve got an idea. We can call for help inthis canyon and the echo will carry ourvoices far.” So he leans over the basketand yells out, “Helloooooo! Where arewe?” (They hear the echo several times.)Fifteen minutes later, they hear this echo-ing voice: “Hellooooo! You’re lost!!” Oneof the men says, “That must have beena mathematician.” Puzzled, one of theother men asks, “Why do you say that?”The reply: “For three reasons: (1) He tooka long time to answer, (2) he wasabsolutely correct, and (3) his answerwas absolutely useless.”Although the three points made in the man’s reply do apply to many mathematicians, there is more that is unspoken. After all, there is something。

名词解释nguage: language is a system of arbitrary vocal symbols used for human communication.2.Design features of language(语言的区别性特征) :i.Arbitrariness:the forms of linguistic signs bear no natural relationship to theirmeaning=language is arbitrary in the sense that there is no intrinsic connection between a word(e.g.pen) and the object it refers to .ii.Duality:the property of having two levels of structures, such that units of the primary level are composed of elements of the secondary level. By duality we mean that language system has two sets of structures, one of sounds and the other of meanings.指拥有两层结构的这种特性，底层结构是上层结构的组成成分。

指语言是声音和意义双重结构组成的系统。

举例：Sounds > syllables > words > phrases > clauses > sentences> texts/discoursesiii.Productivity： Language can be used to create new meanings because of its duality .语言可以理解并创造无限数量的新句子，是由双层结构造成的结果（Understand and create unlimited number with sentences）iv.Displacement：Human languages enable their users to symbolize something which are not present at the moment of communication.v.Cultural transmission: language is passed on from generation to generation through teaching and learning rather than instinct.3.Linguistics is the scientific study of language. It is a scientific study because it is based on thesystematic investigation of linguistic data, conducted with reference to some general theory of language structure.4.Psycholinguistics: It is the study of how language is acquired, understood and produced.ngue:F. de Saussure. Langue refers to the abstract linguistic system shared by all themembers of a speech community.语言指语言系统的整体，这个整体相对是比较稳定的。

2023年上海市宝山区高三二模英语试卷（本试卷满分140分，考试时间120分钟）I. Listening ComprehensionSection ADirections: In Section A, you will hear ten short conversations between two speakers. At the end of each conversation, a question will be asked about what was said. The conversations and the questions will be spoken only once. After you hear a conversation and the question about it, read the four possible answers on your paper, and decide which one is the best answer to the question you have heard.1. A. In the library. B. In the bookstore. C. In the museum. D. In the gallery.2. A. $200. B. $250. C. $300. D. $350.3. A. Accountant. B. Secretary. C. Mechanic. D. Manager.4. A. Customer and tailor. B. Customer and bank clerk.C. Customer and waitress.D. Customer and repairman.5. A. The big one. B. The one on the other side.C. The middle one.D. The one on the far right.6. A. Bring flowers. B. Take a pot plant. C. Buy postcards. D. Send fruits.7. A. Night flights are available to him.B.He can't get any direct flights.C.He has missed the flight he wants.D.Direct flights have been delayed.8. A. She hopes he can stop reading. B. She will turn it off at once.C.She is reluctant to turn it down.D. She thinks reading is not important.9. A. Listen to folk music at home. B. Watch a live performance.C. Appreciate a dance show on TV.D. Go to an exhibition of works of art.10. A. The electronic toy is unique. B. The toy is safe for all users.C. Plastic toys are of low quality.D. The toy isn't environmentally friendly. Section BDirections:In Section B, you will hear two passages and one longer conversation. After each passage or conversation, you will be asked several questions. The passages and conversation will be read twice, but the questions will be spoken only once. When you hear a question, read the four possible answers on your paper and decide which one is the best answer to the question you have heard.Questions 11 through 13 are based on the following passage.11. A. Admiration for one's own appearance. B. A need for popularity.C. A need to belong to a group.D. A love for photography.12. A. Women are more likely to be famous.B. Men are more eager to be seen as popular.C. Men tend to post selfies more often.D. Women use editing software more frequently.13. A. Photos have to be looked at.B. Cameras are mostly used for selfies.C. Clicking "like" doesn't necessarily mean photos have been seen.D. It is the same experience as traditional one.Questions 14 through 16 are based on the following passage.14. A. Suggestions on how to think up great ideas.B.Proposals of how to set up your own business.C.Tips on how to be a successful businessman.D.Ideas about how to treat your customers.15. A. Because your chances of success will be increased.B.Because your business idea will be more creative.C.Because it will help decide whether to further pursue the education.D.Because you can't do your work without business learning.16. A. Because they enter the market with their eyes open.B.Because they believe the more failures, the better.C.Because they were born to be great risk takers.D.Because they can do a careful study to ensure their success.Questions 17 through 20 are based on the following conversation.17. A. A fortune. B. Nothing at all. C. Twice the fortune. D. A little.18. A. The company won't accept it. B. It is partly broken.C. The date on the card isn't correct.D. It is in use today.19. A. Answer the password, dial his dad's number and the number on the card.B. Dial the number on the card and his dad's, then input the password.C. Dial the number on the card, answer the password, and dial his dad's number.D. Dial his dad's number then input the password and the number on the card.20. A. 3:30 p.m. B. 3:30 a.m. C. 6:30 p.m. D. 6:30 a.m. II. Grammar and VocabularySection ADirections: After reading the passage below, fill in the blanks to make the passage coherent and grammatically correct. For the blanks with a given word, fill in each blank with the proper form of the given word; for the other blanks, use one word that best fits each blank."Wow ... what a beautiful girl!" you think to yourself. Today, you logged into Facebook and saw a friend request from a lovely stranger. You think that she must be a model ( 21 ) _____ all of her photos look very professional. She doesn't have many friends, but maybe she's just new to Facebook. You may ( 22 ) ______ ( tempt ) to accept her request and say hello, but you'd better think twice because you might get catfished!Catfishing is an online cheat ( 23 ) ______ someone on a social media platform pretends to be someone they're not. The word was made popular by a 2010 documentary film with the same name. Just like in the movie, most catfishing cheats start with the catfish ( 24 ) ______ ( try ) to sweet talk their victim into falling in love with them. Why? There are a variety of reasons people catfish.Some catfish attempt ( 25 ) ______ ( charm ) their victim into trusting them, and then ask them for money. Others will ask for sensitive information or private pictures ( 26 ) ______ ______ they can then use them to blackmail (敲诈) their victims into sending them money. But in some situations, catfish are just lonely people, ( 27 ) ______ ( dissatisfy ) with their lives and using social media to live in an alternate reality.On social media, it's possible to remain anonymous (匿名的) even after you ( 28 ) ______ ( start ) building a relationship with someone. According to the Better Business Bureau in the U.S., around 85 percent of catfishing schemes start on Facebook and the Facebook Dating service. In 2021 as the pandemic pushed many to search for love online, romance schemes, such as catfishing, rose ( 29 ) ______ 80 percent compared to 2020. These cheats cost victims around $1.3 billion in losses and left many with psychological and emotional damage.The internet is a wonderful tool that allows users to learn new things, play games and meet interesting people. However, everyone (30) ______ be careful of the catfish who hide in the shadows. If someone seems too good to be true, they probably are!Section BDirections: After reading the passage below, fill in each blank with a proper word given in the31one thing to look at. Instead, it surrounds them with images and sounds. This usually takes the form of rooms where music plays and images are 32 onto walls, changing when visitors interact with them. Sometimes pleasant smells are added to draw people in through yet another sense. These rooms are known as immersive(沉浸式的) exhibits because they seek to involve viewers completely.The most famous company 33 immersive exhibits is called teamLab. What this company does is draw on people with a huge range of skill sets: artists, engineers, 34 andeven mathematicians. All of these skills are necessary to produce the complex immersive exhibits teamLab is known for. Much of the company's art is based on nature. This is a(n) 35 choice as their goal is to help people feel more connected to the rest of the world. This goal reflects some traditional Asian philosophies, which makes sense because teamLab is a Japanese company. TeamLab's philosophy states that it seeks to transcend (超越) the 36 that people have set up to separate them from other things.Other companies are also developing immersive exhibitions 37 at providing entertainment, education or both. For example, American company Imagine Exhibitions has designed over 40 exhibitions, which combine art, lighting, textures, scents and more. Physical objects are carefully selected and placed throughout the exhibits to add to their realism. The topics of their exhibits 38 from the history of ancient Egypt to nature to TV shows and games such as Angry Birds.Immersive exhibits maybe a new development, but they are 39 by older art forms. Some immersive exhibits are even apparently about the work of artists who worked in other media, such as the painter Vincent van Gogh. Even those who do not reference other artists so clearly are often influenced by older art styles. And many of the immersive exhibits also 40 some form of music.By combining skills and technologies in creative ways, teams are developing new ways for people to experience art.III. Reading ComprehensionSection ADirections: For each blank in the following passage there are four words or phrases marked A, B, C and D. Fill in each blank with the word or phrase that best fits the context.You have to make a speech. You have done your research and now have plenty of ideas bouncing around in your head that you want to get across. The big 41 you face at this point is how to organize those thoughts. Or perhaps you are planning to write a paper, and all you have is the main topic. You need to brainstorm some supporting ideas and then 42 how those ideas fit together.These are just two cases that many people deal with 43 . Some use outlining to flesh out and 44 their ideas, but is that the best strategy? If everyone thought in a linear(线性的) manner, then the answer would be yes, but most people don't.Enter mind mapping, a 45 that shows how different ideas and facts relate to one another. This two-dimensional (二维的) 46 is designed to help you remember information because its format is easy for your mind to 47 . Mind maps have proven to increase not only productivity and 48 but also memory. More than 250 million people worldwide use them, so why don't you have a try and see what they are all about?What is a mind map? Simply put, it is one's ideas put into the form of a 49 diagram that starts with a central idea. Put this idea in a bubble(气泡) in the middle of a page with lines radiating out to other bubbles that contain related themes or concepts. It should look like a many-legged spider. Use words, short phrases or 50 to express your ideas. These associations often lead to other ideas that can be added at the same level or in smaller branches where 51 . Use color-coded bubbles for a visual representation of each branch's 52 . You are now mapping your thoughts and ideas in the way your brain thinks.Studies have shown that this technique of using pictures, colors and visual arrangement has improved people's recall and memory of information by 10 to 15 percent. But it doesn't stop there. Creativity is aroused because of the spatial arrangement, 53 people to make more links and associations on topics of any kind. 54 , it increases one's productivity because mind mapping makes it easier to learn faster while communicating more effectively.So whether you're 55 a project or writing a research paper, make this technique a core part of your planning process!41.A. purpose B. change C. question D. project42.A. turn out B. take out C. give out D. figure out43.A. consequently B. frequently C. entirely D. barely44.A. organize B. express C. share D. spread45.A. process B. foundation C. technique D. result46.A. route B. structure C. objective D. direction47.A. remember B. confirm C. draw D. design48.A. confidence B. capacity C. efficiency D. creativity49.A. vivid B. special C. visual D. normal50.A. images B. sentences C. passages D. languages51.A. obvious B. appropriate C. meaningful D. empty52.A. function B. extension C. connection D. determination53.A. encouraging B. enabling C. reminding D. informing54.A. Additionally B. Sincerely C. Briefly D. Generally55.A. in love with B. in terms of C. in favour of D. in charge of Section BDirections: Read the following three passages. Each passage is followed by several questions or unfinished statements. For each of them there are four choices marked A, B, C and D. Choose the one that fits best according to the information given in the passage you have just read.(A)Digital health refers to the system of interconnected devices that can monitor and report important health information. Doctors can then use that information to ensure that your treatment is tailor-made to your situation.The most useful aspect of digital health is the ability to constantly and accurately monitor vital information. Wearable devices can record your blood pressure or the rhythm of your heartbeat. The information is then accessible remotely through the internet, so your doctor has a wealth of information about your daily health. Your doctor can then use that information remotely to make changes to your treatment at any time.A survey conducted by the company Lightico in March 2020, reveals how satisfied consumers are with this type of remote care. Forty-one percent said that the remote treatment they had received by means of digital devices and the internet was either "good" or "excellent." The same positive attitude toward digital health can be seen among investors. In the first quarter of 2020, US$3.1 billion in new investments poured into the digital health industry, setting a new investment record.Much of the excitement is connected close with the use of voice technology in digital health care. Today, around 2.5 billion devices have voice interfaces (界面) like Alexa, which enable the devices to produce and understand speech. As a result, users can communicate with these devices about their health needs by speaking instead of typing on a computer. Seniors, for example, can use Alexa to check their daily schedules and receive reminders to take their medicine.Advances in digital health, however, also bring privacy concerns. Information regarding your health could be used for advertising or marketing, for example. If health records are accessible online, protecting your information from theft and unauthorized use might be difficult. But if managed properly, the benefits of digital health could certainly outweigh the risks.56.What is this article mainly about?A.The current state of health care trainingB.Health conditions that need urgent attentionC.A technological trend in health careD.Rising pressure on health care workers57.According to this article, what happened during the first three months of 2020?A.A growing number of digital devices were recalled.B.Funding for digital health increased considerably.C.Some digital health firms nearly went out of business.D.Many institutions devoted effort to medical research.58.What does this article explain about Alexa's features?A.How they make workers efficientB.How they can benefit the elderlyC.How they save instructors timeD.How they entertain youngsters59. According to this article, what is one concern about digital health?A.Worries about the side effects caused by modern drugsB.Rising manufacturing costs in the health care industryernment regulations limiting the use of digital devicesD.The danger that personal health data could be stolen( B )60. What does Camilla have to say about sales in her region?A.They've improved.B.They've gone down.C.They've stayed the same.D.They've exploded.61.Why does Bob think commercials via social media is a better use of money?A.Because it doesn't take as much effort to use.B.Because fewer people are used to social media.C.Because of how cool the company will appear.D.Because of their target audience's preference.62. According to the passage, what aspect of the ads is especially effective?A. It shows a lot of different emotions.B. It appeals to people's emotions.C. It contains a very simple message.D. It conveys some subtle ideas.(C)Far below every town and city lies an invisible network of pipes crucial to the lives of their residents. Once water disappears down a pipe or toilet, people rarely think about what happens to it. It's out of sight, out of mind. The water flows into pipes that carry excess rainwater, wastewater and solid waste away, keeping the city clean.There are three main types of sewers (下水道): sanitary (卫生的) sewers, storm sewers and combined sewers. Taipei has both sanitary and storm sewers where rainwater and sewage are collected using different pipes.This sewer system functions as a city's veins (静脉). Water containing human and industrial waste is collected in sanitary sewers and conveyed to sewage treatment plants, the heart of the sewer system, where waste is treated. There, the solid waste is separated from the water, which is then cleaned before being released back into the environment. It will be reused for industrial, agricultural or environmental purposes or to produce energy. Furthermore, the mud and gas from sewage treatment can be used to produce energy. They not only contribute to a healthy environment and good quality of life, but also achieve the goal of sustainable development.While wastewater collection and treatment are critical to the health of urban residents, storm sewers also play an important role in the city. Green space has given way to concrete during the island's rapid development and extensive building projects over the past few decades. As a consequence, there has been a loss of the land's natural water as concrete cannot absorb rainwater. When an extreme weather event like a severe storm or typhoon brings heavy rain, storm sewers must cope with the water, so it does not flood the city. However, storm sewers are only part of the measures for water management and flood prevention.To reduce flooding, an integrated urban drainage management plan has been adopted, where all the factors - storm sewers, land planning and economic development - are considered together. Engineering innovations, additional flood discharge facilities and increased green space requirements in new development projects have improved water maintaining in cities.Who knew that sewers help improve urban public health, prevent urban flooding and boost environmental protection? Together they protect vital water resources and maintain the safety of people and property.63.According to the passage, the word "excess" in the first paragraph probably means______.elessB. pollutedC. extraD. processed64.According to the passage, what does the city build two kinds of sewers for?A.To test wastewater timely and to develop rapidly.B.To treat wastewater and to fight against flooding.C.To form a circulation and to clean the city.D.To clean wastewater and to manage water.65.What can we conclude from the passage?A.What is unseen is as important as what is seen.B.There are three types of sewers in every city.C.People build sewer system because of concrete.D.Sewers are usually repaired every few decades.66.What would be the proper title for the passage?A.A City’s Underground ConstructionB.A City’s Hidden NetworkC.How A City Operates WellD.Two Aspects of A City or TownSection CDirections: Complete the following passage by using the sentences in the box. Each sentence can only be used once. Note that there are two sentences more than you need.You're enjoying a delicious bowl of noodles or your favorite rice dish, chatting with your friends. Being the responsible person that you are, when you're finished you clean up your table and throw away your garbage, including your disposable (一次性的) chopsticks. These chopsticks wind up in a landfill along with other chopsticks around the globe. 67 Turns out we can.For those consuming Asian food, using chopsticks as eating tools is almost second nature. It has been estimated that 20 to 33 percent of the world's population uses chopsticks on a daily basis. Not all of those are disposable, but no doubt many are. Estimates indicate that up to 80 billion disposable chopsticks are used worldwide every year. 68Enter Felix Bock, a doctoral student at the time in the forestry program at the University of British Columbia, Canada. Bock was eager to find a way to recycle wood from construction projects, and he had a particular interest in bamboo. One day he found a drawer in his girlfriend's house filled with disposable bamboo chopsticks, and an idea began to form in his mind.Bock's smart idea was to give a second life to disposable chopsticks by making them into home decorations. He tried to convince restaurant owners to install recycling bins in their restaurants. He also designed a method to clean and sanitize(消毒) the chopsticks so that they could be reused. 69 The square pieces were then artistically arranged into household and office products. Bock's company, called ChopValue, has repurposed chopsticks into wall decorations, desktops, cutting boards and even table games such as dominoes (多米诺骨牌).Since 2016 when the company was founded, ChopValue has rescued billions of chopsticks from landfills. Instead of an economy based on the typical take-make-dispose model, ChopValue seeks to create one that transforms waste into a usable resource. Bock states in an interview with Global Shakers, "Frankly, I'm frustrated by everyone trying to be sustainable when they are looking away from the visible things that we can do immediately with the materials right in front of us." 70IV. Summary Writing71. Directions: Read the following passage. Summarize the main idea and the main point(s) of the passage in no more than 60 words. Use your own words as far as possible.Home sweet home - this phrase arouses all sorts of images and feelings. The place you call home might be a house, an apartment, a tent or any of a countless variety of places. But did you know that the condition of your living space can have an impact on your emotions and even on your mental health? By controlling some of these variables, you can help to achieve a more peaceful life.It goes without saying that the lighting in your room or in your house can affect your mood. If the lighting is too weak, it can have a negative effect on your emotions, and your mood may end up matching the dark atmosphere. Artificial light may not do the trick, however. It is better to open your curtains because natural sunlight makes your whole day brighter.You have to do more than just remove poisonous substances from your home. To ensure good mental health, surround yourself with nature. Numerous studies have demonstrated that the presence of green plants in your home can improve your mood.Let's all admit it: We have trouble keeping our space clean. Somehow stuff just tends to pile up. But this problem involves more than just aesthetics(美感). It doesn't just look terrible: it makes you feel terrible. A cluttered room both reveals stress and causes stress. Disorder can indicate that you're too busy to keep things tidy, but then looking at a mess can lead to more stress. Straightening up doesn't have to take all day - just take 10 or 15 minutes and put a few things away. You'll be surprised how much better you'll feel.The color palette(用色风格) of your environment does more than passively reflect your color preferences. It also has an active influence on your mood and even your mental health. Choosing colors that create a calm atmosphere can be a type of therapy.The sky is the limit to ways you can make your environment beneficial to good mental health. You could put up photos of your family and friends or post encouraging notes from people that show how much they love you. Just remember: Your surroundings can make your "home sweet home" even sweeter.V. TranslationDirections: Translate the following sentences into English, using the words given in the brackets.72.听说下周音乐会的歌曲都是爱国歌曲。

下一代个性化学习：生成式人工智能增强智能辅导系统徐　升1 佟佳睿2 胡祥恩1（1. 华中师范大学 心理学院，湖北武汉 430079；2. IEEE 人工智能标准委员会，新泽西州皮斯卡塔维 08854，美国）[摘要]　长期以来，教育技术领域的研究者致力于利用计算机开展个性化自适应教学。

智能辅导系统作为这一领域的核心，其目标是通过计算机技术提供个性化学习体验和支持。

近年来，生成式人工智能，尤其是大语言模型的发展，为智能辅导系统带来了突破性机遇，为个性化学习开辟了新的路径。

本文回顾了研究者为实现个性化学习所作的努力及其成果和局限，探讨了大语言模型在个性化学习中的潜力和影响，介绍了“苏格拉底游乐园”——一个基于对话的智能辅导系统，如何利用大语言模型实施苏格拉底式教学，揭示了生成式人工智能如何革新个性化学习方式。

针对大语言模型的潜在误用，文章提出了相应的建议，并构建了框架，旨在引导研发者在生成式人工智能和大语言模型范式下开发高效的智能教育应用。

[关键词]　个性化学习；智能辅导系统；ChatGPT ；对话教学系统[中图分类号] G459 [文献标识码] A [文章编号] 1007−2179（2024）02−0013−10一、个性化学习：审视过去21世纪，个性化学习已成为推动学习者发展的核心动力。

2016年发布的《美国国家教育技术计划》，将个性化学习定义为一种根据学习者独特需求定制教学策略的方法（Thomas ，2016）。

这种方法强调以学习者的兴趣为中心，通过设定灵活的学习目标和教学方法，实现教育个性化。

个性化学习被美国国家工程院视为21世纪面临的14大工程挑战之一，肩负着与解构大脑等尖端科技并驾齐驱的重要使命，标志着在解决教育差异化和个性化需求上的重要进展。

个性化学习并非新生事物，可追溯至19世纪末20世纪初的教育理论家约翰•杜威和玛丽亚•蒙特梭利等倡导的以学生为中心的教学法，该教学法为个性化学习理念的发展奠定了理论基础。