Advice for Undergraduates on Special Aspects of Writing Mathematics

Transcription

1 Advice for Undergraduates on Special Aspects of Writing Matematics Abstract Tere are several guides to good matematical writing for professionals, but few for undergraduates. Yet undergraduates wo write matematics papers need special guidance. For instance, professionals may need elp writing clear definitions, but at least tey know wy explicit definitions are needed and know te basic format. In general, matematics as many special formats tat are not mere tecnical conventions but instead serve important purposes. Often students are not conscious of tese conventions, and tey rarely know teir purposes. Tis paper contains a guide for students written in ligt of tese observations. Interest in good matematics writing bot by professors and by students is on te increase. Tis interest as led to two recent publications on ow to write matematics, Gillman [3] and Knut et al. [5], tat nicely complement te classics in te field, suc as te AMS Manual for Autors [1] and Steenrod et al. [9]. But all of tese publications are aimed at professional matematicians. Undergraduate students need oter advice because tey lack background knowledge presupposed in te guides for professionals. For instance, wile some of tese guides discuss ow to state definitions effectively, none of tem discuss wy definitions are needed at all or wat sould be defined. Working matematicians know answers to tese questions, but undergraduate students often don t. So students need matematics writing guides, too. Tere are now many essays about student writing (see te compilations [2] and [10]), but tese are discussions for teacers about different types of writing programs. I know of just one publised ow-to guide for students, te article [7] by Price. I igly recommend it. However, [7] is not sufficient, because it is geared toward just one of te many different types of student writing assignments te traditional assignment of writing up solutions to omework. Muc more difficult is te sort of matematics writing we professionals most often do, articles and reports. Until recently, few matematics students were asked to write suc papers, except peraps senior majors, but more recently paper writing as become more common at all levels. Now tat matematics courses often ave associated (computer) labs, one can also assign lab reports [4]. Finally, all of te types of writing described so far are formal, in tat (ideally) tey are polised end-products giving te final state of a student s understanding. Te current empasis in writing instruction is to favor informal, expressive writing frequent, sort, unpolised writing tat accompanies and tus elps actively

2 Writing Matematics: Special Aspects page 2 engage te student in te process of coming to understand te matematics at and. See te introductory article by Connolly in [2]. Te situation at Swartmore requires me to be particularly concerned about traditional papers. By faculty vote in 1985, all Swartmore departments must give at least one fresman/sopomore Primary Distribution Course (PDC), and a student must take six PDCs (over two years) in different departments. Among oter tings, PDCs empasize writing, and students are asked to write full-fledged papers in te style of te course s discipline. A full description and rationale are given in [6]; suffice it to say ere tat te writing is not only intended to elp students compreend te course material but also to give tem a broad perspective on different writing styles. Because I knew of no guides for undergraduates writing matematics papers, I wrote my own, one for eac PDC I gave, and I revise tem eac year as I see additional writing problems in te student papers I read. Wat follows is basically te latest version of my guide for a calculus course, wit items specific to Swartmore deleted. I will be pleased if readers make use of tis publication wit teir students, and PRIMUS as allowed me to state tat faculty may reproduce tis article for tat purpose witout requesting permission. If instead tis article gives you ideas for producing your own andout peraps wit some advice directly contrary to mine tat s fine too. In fact, I imagine many faculty ave produced andouts for students on writing matematics. I ope readers will sare tem wit me; peraps oters will be publised and a bibliograpy can be produced. Te one oter substantial essay of tis sort I already know about is by Steen [8] and was written for matematics majors. Some readers of earlier versions of tis paper ave been critical. Tey felt it addressed te wrong issues for its student audience. Specifically, tey felt: 1) My paper is about formal writing, but informal writing done in te process of learning is muc more effective for students. 2) Even if one does assign formal end-product writing, my article says too little about big issues (e.g., ow to plan out wat to write) and too muc about arbitrary tecnical conventions tat only professionals need to learn. I ave tougt about tese criticisms, and tey convinced me to delete some material. But mostly I decided tat I disagree and tus need to say wy, bot in tis preface and in te student guide itself. I ave revised accordingly. To respond to 1), informal process writing is surely very effective wit many students and deserves wide use. If te only goal is to get students to learn matematics (ence te prase writing to learn ), informal writing may be sufficient. However, if anoter goal is to elp students communicate teir newly gained knowledge to oters, ten writing wit time-

3 Writing Matematics: Special Aspects page 3 tested conventions is called for. And if te goal is to elp students develop writing skills tat work in a variety of circumstances (learning to write as well as writing to learn), ten andson experience wit conventions of formal writing in several disciplines provides benefits tat informal writing cannot. As for 2), my essay does indeed devote muc space to tecnicalities, but I contend tat tecnical conventions are sometimes far from arbitrary and we (students and faculty) ave a lot to learn by reflecting on tem. Te fact tat some matematics conventions ave been universally adopted around te world suggests tat tey accomplis someting important. I ve tougt about tis (necessary because guides like [1,3,5,9] rarely give underlying reasons for writing conventions), and it seems to me tat many matematics conventions are crucial in making te always complicated progression of tougt in matematics papers muc easier to follow. In any event, tose matematics students wo do ave to write papers need certain advice wic does not seem to be readily available. I am trying to fill tis gap. So, in my andouts I concentrate on aspects of writing tat are special to matematics and for wic I feel tere are good reasons. I state te reasons I perceive. I d be very interested in reader reaction. For instance, do you agree tat te tecnicalities I talk about are important? Wat do you tink of my discussion of te difference in pilosopy of references between matematics and oter umanities? Special Aspects of Writing Matematics Papers 1. Introduction Matematics writing is different from ordinary writing and arder in addition to all te requirement of ordinary good writing, tere are additional constraints and conventions in matematics. An additional constraint is tat matematics follows muc more demanding rules of logic tan ordinary discourse, and you must make your logic clear. Some of te additional conventions are tose for defining new concepts and tose for organizing te material troug teorems and examples. Tis essay is about tese constraints and conventions, especially te conventions. Altoug you ave seen tese constraints and conventions in te matematics texts you ave read over te years, you ave probably not realized wen and wy tey sould be followed. Tey may seem igly tecnical and arbitrary, and tus not wort learning if matematics is a side sow for you. Wy, for instance, sould calculations be displayed in just certain ways, or

4 Writing Matematics: Special Aspects page 4 bibliograpies be organized is just certain ways? Indeed, as far as I can tell some matematics conventions are arbitrary, for instance, te ones for bibliograpies; I won t discuss tem furter or old you to tem wen you write papers. But many of te conventions, including tose for displaying calculations, are not arbitrary. Tere are good reasons beind tem, and once you understand tese reasons, you understand te nature of matematics a little better and you become more perceptive about ow to explain tings in ways tat you can carry over to your oter writing. Te organization of tis essay is simple. Eac section discusses a special aspect of matematics writing tat undergraduates ave ad trouble wit. I try to tell you not only wat you sould do but also wy. It is important to understand wat tis essay is not about. It does not attempt to give you general advice about writing tat will elp in any field. It does not even advise you (except incidentally) about matematics writing in te large, i.e., ow to come to grips wit your topic and outline your approac. It is obvious tat advice on tose two topics is useful. It is probably not so obvious tat advice on matematics conventions is useful; tat s wy I ve written tis. Matematics conventions are not rigid rules, and some autors write very well altoug (peraps because) tey break tem. But before you can decide intelligently to disregard a convention, you ave to know wat te convention is, know wy it is, and ave some practice following it. 2. Wat Kind of Matematics Paper? You will be writing expository matematics papers. You will start wit some issue (e.g., wat is instantaneous cange, ow can derivatives be computed easily, ow to optimize), sow tat tere is some definition, teorem and/or computation metod tat precisely and correctly resolves te issue, and give various examples. Most papers written by working matematicians are researc papers; tese present new discoveries and are usually written in a terse definition-teorem-proof style wit limited intervening commentary. Expository papers are more informal, wit muc more discussion, but tey still sould be carefully and visibly organized carefully because tat s te way matematics is; visibly because matematics is subtle, so readers need many guideposts to follow it. Ideas are made very precise and expressed in symbols as well as words. Key definitions, teorems, examples and formulas are igligted, numbered and referred to by number. Your text is probably a pretty good example of expository style ask your professor. For oter examples, tere sould be some expository matematics journals in te library. I

5 Writing Matematics: Special Aspects page 5 recommend you browse at Te College Matematics Journal, Matematics Magazine and Te American Matematical Montly. 3. Know Your Reader In any writing it is always important to ask yourself: For wom am I writing? In matematics especially, te amount of explanation called for varies greatly wit te reader s background. Assume you are writing for a student wo as te same background as you, except tat tis student does not know te particular topic in your paper. Tus points tat ave been ard for you will be ard for your reader, and you sould explain tem carefully, using watever approac finally worked for you. Just trowing down some cryptic calculations won t work! Anoter important question is: How will my reader use my paper? If your reader really wants to learn te topic from you, e or se will frequently ave to refer back to earlier definitions, teorems, explanations and examples people ave to go over matematical items several times before tey sink in. Terefore, key elements of your paper sould be marked so tat tey can be found easily. Tis requires igligting and numbering, wic we will discuss later. 4. Titles Every paper sould ave one. It sould be informative witout being too long. Coosing a good title in a matematics paper is not so easy. Often a paper inges on a concept tat is defined only witin te paper itself, so using use te name of tat concept in te title will convey no meaning at all. 5. Introduction Again, every paper sould ave one. In sort papers it need only be a paragrap. Matematics papers are ard to read, and for encouragement your reader deserves to be informed of wat se is getting into and wy se sould care. Giving a good introduction is difficult in te same way giving a good title is: a satisfactory explanation may inge on concepts you aven t introduced yet. But in a paragrap you can give a roug idea of te key concepts (admit tat it s roug) and wat you will do wit tem. 6. Division into Sections Break papers of longer tan 4 or 5 pages into sections. Your paper will ave longer and fewer sections tan tis essay, because your paper will ave a single topic. In a paper wit

6 Writing Matematics: Special Aspects page 6 only a few sections, te introduction sould probably include a brief description of wat eac section will cover. At te start of eac section remind te reader wat te section is about. How does it fit into your sceme of tings? Wy is it tere at all? Like te introductory section, tese reminders will give te reader encouragement and guideposts. Tey will also elp you. Writing tem will force you to tink ard about (and ten maybe revise!) wat you are trying to do and te order in wic you are trying to do it. 7. Teorems Any teorem in your paper sould be igligted I suggest indenting it or putting extra vertical space around it. For instance, Teorem. For any constant p, te function f(x) =x p is differentiable, and d dx xp = px p 1. Oter formats are allowed; browse and see. Wy igligt teorems (and oter key items suc as examples)? Because matematical arguments can be so complex. Formal statements of teorems (and oter key items) serve as toucstones, like aving an outline witin te text body. Tey also makes it easy for te reader to refer back to key specifics. As furter toucstones, if you ave any proofs in your paper, you sould indicate clearly were eac proof begins and ends. Levels of Confirmation (prove, verify, sow, illustrate). In ordinary discourse, to prove someting means to give any sort of fairly convincing evidence. In matematics, prove as a muc stricter meaning; you ave proved someting only if you ave given a airtigt argument airtigt because it arks back to definitions or to oter teorems tat ark back to definitions. Lesser levels of confirmation are often quite useful, but tey are referred to wit different words suc as sow and illustrate. For instance, consider te constant multiple rule in calculus: (cf) = cf. If you state tis rule and ten say For instance, (d/dx)4x 2 =4(d/dx)x 2,youavemerelyillustrated ow to use te rule. If you actually prove from te definition tat (d/dx)4x 2 =4(d/dx)x 2,you ave sown ow te proof goes by proving a special case. Similarly, if you state te power rule (d/dx)x p = px p 1 and ten prove (d/dx)x 2 =2x, you ave illustrated te rule by proving a special case (and tis time te special case doesn t really suggest ow te complete proof would go). Matematicians use verify somewat like prove. If you say you are going to verify a teorem, ten you must prove it. But you can also verify a numerical claim; tis is a muc less

7 Writing Matematics: Special Aspects page 7 demanding type of verification. Suppose I claim tat, if my distance at time t is s(t) =t 3,ten my speed at t = 2 is 12. I can verify tis by te following simple computation: s (t) =3t 2,so s (2) = 3 4 = 12. Suc a computation would not be called a proof. You could say I ave sown tat te velocity is 12. Suppose you give an example of a definition. For instance, suppose you ave just given te definition of derivative and now use tat definition to find tat (d/dx)x 2 =2x. Youave not proved te definition, because definitions are conventions and cannot be proved (see next section). Rater, you ave illustrated te definition by proving te formula for (d/dx)x 2. In summary: prove as a very strict meaning; sow is looser, and illustrate is looser still and refers to examples. Altoug illustration is te loosest, it is very important. Sometimes good examples will do more to elp te reader understand and believe a result tan a complete proof will. 8. Definitions Definitions are a major way tat matematics writing differs from general writing. Most disciplines don t need to make definitions explicit nearly so often as matematics does tey don t need to be so precise nor do tey deal so regularly wit situations outside common experience. Matematical writing involves defining bot words (e.g., derivative) and notation ( d dx ). Notation is important because, if you use a new concept frequently, you need a sortand way to refer to it or you will tie yourself in verbal knots. Wen you give a definition, you can do it in-line (witin a paragrap), but te word or prase being defined sould be igligted, by underlining, byitalic print, orbyboldface. Boldface is now te most common format, since underlining is not common in typeset material and italic already as a use in bot matematical and ordinary writing to indicate empasis. Here s an example of a definition: A prime number is a positive integer wit no positive integer divisors oter tan 1 and itself. Anoter format is to display definitions just like teorems are displayed. Te display format sould be reserved for te most important definitions. For instance, in calculus te definition of derivative migt be displayed, but te definition of polynomial (if you need it at all) can be done in-line. Once you define a word w, tere are two tings you souldn t do. First, if anoter word w is a synonym of w in ordinary Englis, don t use w as if it as te same precise matematical meaning as w. Second, don t use w itself in a loose ordinary Englis way tat is in disagreement wit its precise matematical definition. As an example of te first rule, if you ave defined slope at a point on te grap of f to mean te value of te derivative, don t afterwards sometimes say steepness wen you mean derivative (unless you ave explicitly said tat

8 Writing Matematics: Special Aspects page 8 steepness means te same as slope ). As an example of te second rule, if you ave defined critical point to be one at wic f (x) = 0, don t afterwards say points were concavity canges are critical to graping a curve. Tis would be perfectly good ordinary Englis, wit critical aving its general meaning of important, but it s not acceptable in your paper because critical as become a defined term wit a more narrow meaning. Wy tese two special restrictions? Te wole point of matematics definitions is to give ordinary words extra precision needed for a matematical discussion. But tere may be several different precise meanings tat are matematically useful and tat one can attac to te same ordinary concept. An experienced matematical reader, upon seeing steepness in your paper after you ave defined slope, will assume you are using bot words because you need two different refinements of te ordinary slope concept. (Maybe you are using steepness to mean average slope.) Suc a reader will ten attempt to find were you defined steepness and will get frustrated. Similarly, once te reader as gotten straigt te restricted meaning you ave given words, e will get very confused if you use tose words in ways tat contradict your definitions. Local Definitions. So far I ave been talking about global definitions, tose tat apply trougout your paper. Most global definitions introduce words or notation. In contrast, local definitions apply only briefly, say, to te current example or current paragrap. Usually local definitions are definitions of symbols, for instance, Let f(x) =x 2. A few paragraps later it is acceptable to say Now let f(x) =e x. Local definitions don t require special igligting, altoug tey sould be displayed if tey involve complicated formulas (see Section 14, Displays). In any event, te item being defined sould always come on te left-and side of te equation; e.g., Let x 2 = f(x) is wrong. Wy? Because wen an equal sign is used wit let to make a definition, te equal sign as a special meaning: te quantity on te rigt is being assigned to te quantity on te left. Tis is quite different from regular equality, were it makes no difference if you write a = b or b = a. 9. Examples Examples really elp to make abstract concepts clear, so a good expository paper contains many more examples tan definitions and teorems. Examples are like definitions, in tat tey can appear in-line or be igligted by indentation and extra spacing. For a very brief example, te in-line metod is fine. However, a lengtier example sould be displayed and numbered, especially if it is a key item of your work. Not only does tis format draw attention to your example, but is also makes it easy to refer back to later (e.g., you can say see Example 3). If an example is a sample problem, make clear were te solution begins and ends.

9 Writing Matematics: Special Aspects page Figures Figures can be extremely elpful in an expository paper, just as tey are in books. Eac figure sould be numbered (for easy reference later) and inserted sortly after te first reference to it. (Anoter convention is to put all figures at te end. If you use tis alternative, please say so te first time you reference a figure.) Usually, eac figure sould ave a caption as well; e.g., Te steeper te line, te greater te slope. If you know ow to use computer software to produce a good figure and place it into your paper, great, but it s fine if you insert your figures by and. You ll probably need more space tan you first tink. 11. Big Little Words (let, tus, so) Te tree words above are common words in ordinary Englis, and many people use tem casually. But tey are big words in matematics because tey set fort te logic of your argument. Tey ave precise matematical meanings and sould be used properly. I ave already discussed let under definitions. Let sets fort a convention, usually temporary, usually for a symbol. A related but not matematically synonymous word is suppose. You could say Suppose f(x) =x 2, but you really souldn t. Suppose is best used for temporary ypoteses, not temporary definitions. For example, imagine you want to sow tat, if f (x) canges sign at x = a, tenf as a local extremum at a. You could argue as follows: Suppose f (x) > 0forx<a. Ten te grap of f is increasing to te left of a. Sincef canges sign at a, weavef (x) < 0forx>a. So te grap of f is decreasing to te rigt of a. Sincetegrapoff first increases and ten decreases, it as a local maximum at a. On te oter and, suppose f (x) < 0forx<a.Ten... f as a local minimum at a. Te point is, to say f (x) canges sign at a involves two cases: it canges from positive to negative, or vice versa. Eac case is argued separately. So we suppose one case and ten te oter. It wouldn t really be correct to say instead Let f (x) > 0forx<a because we don t really ave control over te sign of f (x), and te word let means tat te matter is a convention tat is up to us. As for tus and so, tey mean tat te next sentence or clause is a logical consequence of te previous sentence or clause. Terefore, if te next sentence would still make sense and be true even if you ad not included te previous sentence, ten te next sentence may not begin wit tus or so.

10 Writing Matematics: Special Aspects page 10 For example, consider te sentence Let x =1andy =2,sox + y =3. Here te use of so is correct. But now consider Let f(t) be te temperature at time t. Tusf (t) = lim 0 [f(t+) f(t)]/. Here te use of tus is incorrect, because te fact tat f(t) represents temperature as notingtodowitwyf (t) is defined te way it is. 12. Wen to Give Credit Te conventions for giving credit to oters are somewat different in matematics tan in general academic writing. Briefly put, direct quotes and almost direct quotes must be credited, but paraprase generally need not be. Furtermore, matematics papers rarely include direct quotes. Tus, wile a umanities paper is full of footnotes giving credit, in a matematics paper te credits are less frequent (and tey typically appear in text). Tis is all elaborated below. Pay close attention, since failure to give proper credit is plagiarism, a major academic sin. Rigtly or wrongly, matematics is regarded as aving an existence independent of te words used to describe it. Tus your text may describe te Cain Rule teorem in sligtly different words tan anyone else, but tat doesn t give its autors any special credit. If you use te Cain Rule, and you learned about it from your text, you sould not reference your text at te point were you introduce te Cain Rule te autors would never claim credit for te Cain Rule temselves. Tis same principle applies to definitions; don t reference your text for te definition of derivative. Note. Teorems ave two sorts of names: descriptive names, suc as Cain Rule ; and sequential names, suc as Teorem 6 and Limit Rule II. A text gives a descriptive name only if it is widely used by oters, so you can use te name Cain Rule too. But sequential names are specific to an individual text. Tus, if you must talk about Limit Rule II in your text, you ave to reference your text and give a page number oterwise readers won t ave a clue wat rule you are referring to. But wy make your readers look tis up wen you could simply state te rule in your paper? Te independent existence perspective also explains wy matematicians rarely quote eac oter directly your original words are generally not perceived to ave an advantage over my paraprase. Tis attitude is quite different from tat in umanities, were te nuances in somebody s verbalization of an idea may make all te difference.

11 Writing Matematics: Special Aspects page 11 Te original discoverers of matematical results are given credit, and if a result is fairly recent, te paper in wic it is first publised must be referenced. But all te teorems in your calculus course are classical (some of tem over 300 years old). Tey are so muc considered a common eritage tat people s names are only occasionally associated wit individual results, and references to original publications are never made (except in a istory of matematics paper). I ve discussed giving credit for definitions and teorems. In your papers you are more likely to use examples (for instance, a max-min problem) taken from or based on your text. Sould tese be credited? Again, te answer is usually no. If you use te exact words or even just te exact numbers of an example from some book, ten (and only ten) give credit; but in general you souldn t be using problems verbatim anyway. Wy? Because in order to write about an example wit understanding you sould internalize it and make it your own before using it. And if you ave internalized it, you can recreate it, no doubt wit different numbers. Finally, if your wole line of tinking comes from one source, it is appropriate to give one reference to it. Somewere near te beginning of your paper, say someting like In writing tis paper, I ave drawn eavily on Goldstein et al. [1] (were Goldstein is reference 1 in your bibliograpy) or Muc of my material is adapted from lectures by Smit [2]. 13. Complicated Matematical Expressions Publisers call matematics penalty copy because it is so ard to set on te page. Matematics involves (among oter tings) subscripts and superscripts, numerous styles of type, special and/or large symbols (te integral sign), and expressions tat must be stacked vertically (fractions). Traditional publising involves setting type across simple lines, and many matematical features interrupt tat flow. Typewriters and most computer word processing systems also are set up to go across lines, so you will ave te same problem publisers do. Even if you ave special matematics publising software (e.g., te TEX program I use and tis journal uses), suc software takes a long time to learn. And you re not trying to get a job as a matematics typesetter; you re just trying to learn ow to write. So wat sould you do about complicated mat? Answer: Write in by and any (parts of a) matematical expression tat would take too long to do well using te typewriter or computer programs you know. On te oter and, some aspects of setting matematics are easy to master, and oters can be avoided. Almost every word processing system allows easy access to subscripts and superscripts, and you sould use tem. Te next two subsections tell you ow to deal wit

12 Writing Matematics: Special Aspects page 12 two major problems in setting matematics: styles of type and fractions. Fonts for Mat. A font is a style of type, e.g., roman or italic. In books and journals, letters representing matematical quantities, suc as a, y and f(x), are set in italic, as just done, to distinguis tem from ordinary roman text. (Some matematics is set in oter fonts, e.g., Greek letters for angles in trigonometry and boldface for vectors in linear algebra.) Te reason for special fonts is tat, witout tem, sometimes it would not be clear wat is intended, a matematical expression or an ordinary word. For instance, consider te sentence So te answer is 4pm. Only te italics makes it certain tat we mean 4 times p times m instead of 4 o clock. Similarly, witout italics a could be confused wit te Englis word a. Unfortunately, mat italic is different from ordinary italic, because only te letters are slanted, not te punctuation (parenteses, brackets, exclamation points). So, unless you ave mat italic fonts (most word processors don t), you ave to go back and fort from roman to italic even to write simple expressions. So, te best ting to do is avoid italics entirely, and use an alternative convention usually reserved for typewritten documents: put extra blank space around matematical letters and expressions. For instance, Consider te variables x and y in te equation y = 2x+3 for a line. Weter you use italic or not, you ave to tink about spacing witin matematics expressions too. Tigt spacing like y=f(x+) f(x) makes already complicated expressions even arder to grasp. Wide spacing like y = f(x + ) f(x) is muc preferred see te difference? My personal preference is for tigt spacing witin inner groups (e.g., expressions witin parenteses) and wide spacing oterwise: y = f(x+) f(x). Te point is, matematical expressions often ave several layers; you can always use parenteses and brackets alone to indicate te layers, but sometimes te use of spacing as well makes reader compreension easier. Watever font you use for a matematical symbol, make sure you use tat font all te time. For instance, if you are talking about velocity, do not use v sometimes and v oter times. Wy? Matematics is font-sensitive. Trained readers expect te same letter in different fonts (or wit different accent symbols) to mean different tings. Examples of te same letter appearing in different fonts are not common in calculus papers, but tey occurs frequently in oter fields of matematics. For instance, a statistics paper migt well use x for a data value, x for a sequence of data values, and x for te average value of te data. Matematics is case-sensitive too, tat is, upper and lower case mean different tings. For instance, tat statistics paper probably uses X for te random variable of wic x is a value. So, wat do you do if you want to start a sentence wit a variable you ave named a?

13 Writing Matematics: Special Aspects page 13 Answer: Don t write A. Start wit lower case, or (better) rewrite te sentence. For instance, instead of starting a > 0 because, start Quantity a is positive because. Caution: Some computer software for matematics is not case-sensitive. For instance, most versions of BASIC will treat te variables B and b as te same. Fractions. Fractions in built-up form like y 2 y 1 x 2 x 1 (1) are especially ard to produce witout special software. About te best you can do is use tree separate lines, for numerator, fraction bar, and denominator. Even ten it will be ard to center te parts properly. (You migt tink tat underlining wen you type te numerator will take care of te bar, but try it for display (1) and see wat appens.) So wat sould you do? Witin a paragrap, for simple fractions you can avoid te problem by using silling style, tat is, a/b instead of a b. But as soon as te numerator or denominator involves sums and differences, te silling approac requires parenteses. For instance, in silling form display (1) must be written as (y 2 y 1 ) / (x 2 x 1 ); wy? If a fraction already involves parenteses in built-up form, it elps te reader if you use brackets on te outside in silling form: [f(x+) f(x)]/. Alas, if a fraction is sufficiently complicated, people find it almost impossible to make sense of silling form, no matter wat; consider [f(x+)(g(x+) g(x)) + g(x)(f(x+) f(x))]/[(x+) ]. Don t writesucanexpressioninapaper,eventougitiscorrect. Insuccasesyoumust use built-up form. Suggestion: draw in at least te fraction bar by and. Handwriting Matematics. If an expression is going to take a lot of time to set (even wit te advice above), ten leave a blank space in your paper and write te wole ting in by and. At first, you will probably grossly underestimate te amount of space you need for and insertion. But if te rest of your paper is written on a computer, it will be easy to output anoter copy wit more space. 14. Displays Any long expression wit matematical symbols is displayed centered on a line by itself, wit extra vertical space around it. Tere are tree reasons for displays. First, if an expression is particularly important, you draw attention to it by displaying it. Second, longer 5 dx matematical expressions tend to be tall. To fit someting like x 2 in-line requires +1 2

14 Writing Matematics: Special Aspects page 14 adding disconcerting interline space (see it?), or else making te symbols disconcertingly small, like tis: 5 dx. Tird, if you put te expression inside a paragrap, it migt come at te end 2 x 2 +1 of a line and ave to be broken across two lines. You can ypenate words, but matematical expressions are usually longer tan words and don t bear line breaks well. So if an expression would ave to be broken if it appeared in te middle of a paragrap, it sould instead be put in a display. For instance, if you wis to say y = d dx (x2 +3x) witin a line, it would be bad form to break tis after te + and terrible form to break it after d. According to some writers, it is even bad form to break it after =. dx A display sould be numbered if you find tat you refer to it anywere oter tan witin a few lines of it. Te numbering can appear eiter on te left, as in (2) y = d dx (x2 +3x), or on te rigt, as in y = d dx (x2 +3x), (3) but be consistent. Just below a display you can refer to it as te previous display, but two pages later it s too long-winded to refer to te tird display on page 7. Besides, if you modify your paper, canging te page breaks, you will ave to rewrite suc a reference completely. Tat s wy we use numbering. Multiline Displays. In most cases, tese sould be lined up so tat te main connectives (usually an =, but maybe or = ) line up vertically: 2x +1 = 2+3, 2x = 4, x = 2. (4) Here, lining up te equal signs empasizes tat te same ting as been done to bot sides in getting from one line to te next. Similarly, one writes x 2 +2x < x 2 +2x +1 = (x+1) 2. (5) Notice tat display (4) as expressions on bot sides of te equal sign on eac line, but display (5) as only a rigt-and side after te first line. Wen a display could ave been

15 Writing Matematics: Special Aspects page 15 written as one long line, e.g., x 2 +2x <x 2 +2x+1 = (x+1) 2, but is written in several lines for empasis or because it won t all fit on one line, ten tis rigt side only format is appropriate. Tis format empasizes tat eac expression (not eac line) is obtained by doing someting to te previous expression. Notice te commas and te period in display (4). Matematics is written in sentences, and tis display is a sentence consisting of a sequence of clauses, te equations. Tus te clauses are separated by commas (some writers would use semicolons) until te sentence ends wit a period. However, tere is an alternative convention (followed by about alf te publisers in te US) wic leaves off punctuation at te end of displays. Personally, I m not so concerned about te commas (te separation into different lines tells me to pause) but I do feel strongly about te period. Sometimes a sentence does not end at te end of te display; tat is, I am supposed to keep reading in order to understand wy te display makes sense. On te oter and, if te display ends in a period, tat is a signal tat I am supposed to figure out for myself wy te display is legitimate. In oter words, te presence or lack of a period at te end of a display tells me weter tis is a point at wic I need to stop and digest wat as just been said. Since it is very important for te autor to give te reader cues like tat, te punctuation at te end of a display is crucial. In display (5) tere is no comma after te first line and tere never sould be. Wy? Explaining your displays. Te fact tat calculations can be written as one long display witout words doesn t mean tat tey sould be written tis way. Matematics does not consist of calculations alone! Very elementary algebra, as in (4), need not be explained, but more complicated calculations sould be, especially if te calculations are justified by matematics tat te reader is just learning. Tere are two ways to provide te explanation. One way is to put it before, between and after te lines of calculation. For instance, suppose you ave defined f(x) =x 2, and want to sow tat [f(x+) f(x)]/ =2x+. You could write: Substituting te definition f(x) =x 2, f(x+) f(x) = (x+)2 x 2. Ten expanding te first square, combining like terms, and finally canceling te common factor, weobtain f(x+) f(x) = (x2 +2x + 2 ) x 2 2x + 2 = = 2x +.

16 Writing Matematics: Special Aspects page 16 An alternative approac is to put all te reasons in comments on te rigt: f(x+) f(x) = (x+)2 x 2 [def. of f(x)] = (x2 +2x + 2 ) x 2 [expand] 2x + 2 = [combine terms] = 2x +. [divide] Actually, most of tese steps are simple enoug tat you could skip te last two side comments. But to display te four lines of algebra just above witout any explanation is unacceptable to me. If a line in a commented display it quite long, put te comment one line below it: [ f(x+) f(x) g(x+) g(x) ] [ f(x+) f(x) ] [ g(x+) g(x) ] lim + = lim + lim [Teorem on limit of a sum] It is best if your displays temselves ave verbs, like =. Compare te following. Version 1: And so we conclude tat te derivative f (x) sould be defined as Version 2: f(x+) f(x) lim. (6) 0 And so we conclude tat te derivative sould be defined by Version 3: And so we arrive at f f(x+) f(x) (x) = lim. (7) 0 Definition 1. Te derivative of f(x), denoted f (x), is f f(x+) f(x) (x) = lim. (8) 0 Version 1 is poor, 2 is pretty good and 3 is best. Wy? Because at a later point you may well want to remind readers of te definition. If you refer tem back to display (6), tey ave got to go back into te text above te display to find out wat te display is defining. If

17 Writing Matematics: Special Aspects page 17 you refer tem back to Eq. (7), te display itself is enoug, unless tey ave forgotten wat te symbolism f (x) means. If you refer tem back to Eq. (8), tey can t elp but catc te boldface word Definition just above, and so tey can t elp but see tat Eq. (8) is not just an equation but actually te definition. 15. Two Common Mistakes Te first mistake is not to rely on matematical symbols enoug. If, for instance, you refer to te same function more tan once, give it a name (let f(x) =x 2 ). Even if you are just discussing functions in general, you need to give te generic function a name if you are going to say anyting about it. (e.g., for any function f(t), f (t) measures te rate at wic...). More generally, if you are working towards te precise formula in some definition or teorem, it may be wise to state te formula early on, before you ave even explained all te symbols tat occur in it. Tis way at least you ave someting concrete to refer to as you work troug your explanation. Oterwise you bog down in vague verbiage. Te second mistake is to rely on matematical symbols too muc. Tis appens wen you present several lines of computation witout any commentary. 16. Miscellaneous Please number all your pages. Tis is especially elpful if you sould staple tem out of order (wic appens) or if I get tem out of order wile reading (tis also appens). If you ave any questions or comments on te above, don t esitate to ceck wit me. If your question/comment is a good one (and most of tem are), I will send it and my reply to all your classmates via computer (unless you ask me to keep your inquiry and my response private). Finally, my writing isn t perfect eiter. If you find fault wit te writing above, please point tis out to me. We can all benefit from eac oter s criticism. References 1. American Matematical Society, A Manual for Autors of Matematical Papers, 8ted., pamplet, Providence, R.I., P. Connolly and T. Vilardi, eds., Writing to Learn Matematics & Science Teacers College Press, New York, 1989.

18 Writing Matematics: Special Aspects page L. Gillman, Writing Matematics Well, Matematical Association of America, Wasington, D.C., G. D. Gopen and D. A. Smit, Wat s an Assignment Like You Doing in a Course Like Tis Writing to Learn Matematics, College Matematics Journal 21 (1990) 2 19, reprinted from [2]. 5. D. E. Knut, T Larrabee and P. M. Roberts, Matematical Writing, MAA Notes #14, Matematical Association of America, Wasington, D.C., S. B. Maurer, Writing in Matematics at Swartmore: PDCs, in [10]. 7. J. J. Price, Learning Matematics Troug Writing: Some Guidelines, College Matematics Journal 20 (1989) L. A. Steen, Some Elementary Principles of Matematical Exposition, unpublised notes (revised version), St. Olaf College, May, N. E. Steenrod, P. R. Halmos, M. M. Sciffer and J. E. Dieudonne, How to Write Matematics, pamplet, 3rd printing, American Matematical Society, Providence, R.I., A. Sterrett, ed., Using Writing to Teac Matematics, Matematical Association of America, Wasington, D.C., Biograpical Sketc Stepen Maurer (B.A. Swartmore 1967, P.D. Princeton 1972) is a Professor of Matematics at Swartmore College. Previously e taugt at te Pillips Exeter Academy ( ), te University of Waterloo in Ontario (73-74), and Princeton (74-79). His researc as been in combinatorics, wit forays (sometimes continuous) into matematical biology, economics and antropology. As for curricular activities, e as written and spoken widely on discrete matematics. During , e was a Program Officer at te Sloan Foundation, working primarily on quantitative education. From 1981 to 1987, e caired te MAA committee on ig scool contests (AHSME, AIME, USAMO). His fresman-sopomore text, Discrete Algoritmic Matematics (wit Antony Ralston) appeared in Fall 1990.