mathematics assignment calligraphy

Old Pappus' Book of Mathematical Calligraphy

20th February 2014

1. Introduction

2. Calligraphy

3. The Fraktur Letter

4. Script Capitals

5. Mathematical Calligraphy

A. Chalkboard Fraktur

B. Script Capitals

C. Numerical Typeface

D. Drawing the Fraktur Font

At time of writing, the author was a professional mathematician and amateur calligrapher. His mathematical statements, of which there are none, should be viewed as reliable whereas his calligraphic comments, of which there are many, should not.

It is a truth universally acknowledged, that a mathematician in possession of a good theorem must be in want of notation.

1 Introduction

It is almost as universally acknowledged that an alphabet of merely 26 letters is insufficient to express the vast richness of mathematical ideas. Whilst there may not be many articles that truly use more than 26 symbols, other factors weigh in to ensure that it is desirable to have a wide range to choose from. In particular, it is good style to use notation consistent with that of other authors in ones field, and to avoid notation that clashes. Notation should be used to aid comprehension, not to hinder it. Whilst there are times when it is appropriate for an author to say

I am sure that most readers would rather that he didn't.

A common practice in mathematical writing is to use similar letters for similar things. I find that my default assumption for the English alphabet is that of Table  1 . ("Locally constant" means that when used then they should be considered as constant, but the constant is arbitrary.)

Table 1: The author's standard interpretation of letters

Your assumptions may be different; nonetheless, I would expect to see the same sort of grouping in that similar letters represent similar things.

This grouping is something that we have come to expect and is something that a good notation system can exploit. Using similar letters for things is a way of saying, "These are related.". But herein lies the problem for there are many types of relationship.

Let us consider two groups, G and H . Now for pairs of elements, g 1 , g 2 ∈ G and h 1 , h 2 ∈ H , we can ask whether or not there exists a homomorphism γ : G → H which takes g 1 to h 1 and g 2 to h 2 .

Here, we have several types of relationship. The two things designated G and H are similar things (they are both groups ). The elements g 1 and g 2 are elements of G , whereas h 1 and h 2 are elements of H . The homomorphism is a third thing and so we have brought in a third alphabet and used the thing that most looks like a g from that alphabet.

This illustrates a common strategy: when we have more than two types of relationship, we need to use more alphabets. The difficulty with this is twofold. Firstly, there may not be a clear correspondence between the new alphabet and the old; it is standard that α represents a and γ represents g , but what represents c ? Secondly, by using symbols from an unfamiliar alphabet we lose the quickness of association that familiar symbols bring.

An alternative is to use the same letters but from different typefaces. The choice of g for an element of G is actually an example of this. A more extreme example is the common notation for the Lie algebra of a Lie group : 𝔤 of G .

In days of yore, this was self-limiting. Articles with fancy typefaces required careful handling, often with the symbols written by hand. In modern times, such archaic methods seem … archaic. With the plethora of typefaces available, it is possible to have a whole panoply of symbols all related and all in different typefaces.

This freedom does have a negative side. It is now possible to use obscure typefaces in articles and presentations. However, from time to time it is desirable to actually write a symbol by hand, either on a chalkboard or, dare I say it, a piece of paper. The temptation, when faced with a complicated symbol in a complicated typeface, is to replace them all by just "squiggle" (unicode character U+21DC ). This is less than perfect as it breaks the relationship that the choice of notation was designed to preserve.

One can easily imagine an extreme case. A mathematician writes an article involving some concepts that he chooses to express using letters from a variety of typefaces; say g , G , 𝔤 , and 𝒢 . This mathematician is then asked to speak on that article and decides, either willingly or unwillingly, to present the material on a chalkboard. He then has to write these symbols on the board. One of the listeners to this talk likes to take notes. She then has to copy these symbols from the board to her pad of paper. Unfortunately, the speaker is a little indistinct, both in speech and writing, and the symbol 𝔤 is never clearly explained. So the listener writes down some approximation of it, but it is such a poor approximation that when she looks back at these notes six months later 1 then it isn't clear if this symbol was 𝔤 , ℘ , or something else entirely.

1 She's a very keen mathematician and actually reads again the notes that she makes.

In conclusion, to enjoy the freedom of the wide variety of typefaces available, mathematicians need to learn how to write these symbols both on the chalkboard and paper.

2 Calligraphy

The art of writing beautifully is called calligraphy . By using a little of this art of letter-forming, a mathematician can ensure that these strange symbols that she uses are clear both to others and, six months later on, to herself. There are many, many books that explain how to do calligraphy and which explain in detail how to form the letters. However, calligraphy for mathematicians is slightly different to calligraphy in general.

The major difference is in the materials used. Either this will be chalk on a board, or a normal pen on paper. Whilst an experienced calligrapher can produce beautiful works of art using driftwood for a pen, it is more usual to start with a special calligraphic pen. These come in two sorts: thick pens and thin pens. Thick pens are usually held at a constant angle and drawn across the page, never being pushed or turned. A thin pen (or a copperplate pen) is usually used in a more fluid manner and it is common to vary the pressure to change the thickness of the line. Examples of the type of stroke produced with each are in Figure  1 .

Figure 1: Thick and thin calligraphic pens

The peculiarity of mathematical calligraphy is that whilst the typefaces are often designed to be written with a thick pen, the tools are closer in output to a thin pen as they produce a stroke whose width does not depend on the angle at which the implement is held. There is a further difference between chalk and a normal writing pen in that chalk is less amenable to being pushed: drawing a circle with chalk will usually require two strokes whilst with a pen only the one. A third aspect of chalkboard mathematical calligraphy is that it should, by design, be clear at a distance. Thus intricacies are to be severely frowned upon.

The problems are most acute for the chalk-wielder. She not only has to write an unusual symbol, it has to be clear from a distance (clear enough that the inexperienced note-taker can make a reasonable stab at copying it), and it has to look like a symbol from a typeface designed for a wholly different type of pen. No wonder many resort to squiggles!

3 The Fraktur Letter

The fraktur style, of which 𝔤 is an example, is one of the more common of the "fancy alphabets" in use in mathematics. It is therefore a reasonable choice for an example of showing how to go from a proper calligraphic letter to something suitable for a chalkboard. Let us begin with the letter itself, as displayed in Figure  2 next to a standard one for comparison.

Figure 2: The letter "g" in fraktur and ordinary mathematics typeface.

What is clear from this is the angularity of the letter. Although individual lines may curve (the lowest stroke being the most obvious), major direction changes are sharp. When converting to a chalkboard style, these are the characteristics that we wish to preserve (even to exaggerate).

Let us begin by replicating the character in a calligraphic manner as in Figure  3 . The left-hand rendering is the original character from the typeface, the middle is as if drawn with a calligraphic pen, and the right-hand version follows the same strokes but with an "ordinary" pen.

Figure 3: The fraktur "g": original font, as if with a calligraphic pen, and the "skeleton"

Now, while this is reasonably faithful, it is not suitable for a chalkboard. The thin line will be lost, and the angles could easily be smoothed out. The fact that the top left corner is continuous is a coincidence brought on by the thickness of the lines (this can be seen in the "skeleton" rendering). So we need to modify it to exaggerate the angularity, to make it feasible to draw on a chalkboard, and ensure that it remains distinct from the ordinary g .

The top of the letter has a nice dip rising to a point at the right. We can emphasise that. This is matched by the thin line at the bottom of the main part of the letter, but the thinness will be lost when rendered by chalk so we replicate (and mirror) the dip at the top. The side lines may as well be vertical. The top of the letter is now sufficiently angular, but the descender has no distinguishing features. A simple remedy for this is to make it sweep up instead of down. This also means that there is plenty of space for an exaggerated swash if so desired. The result is in Figure  4 .

Figure 4: A more angular letter

Whilst this can no longer be called fraktur , it has the same angularity and thus will be clear on a chalkboard.

Seeing and writing are two different things. Fortunately, the construction of the letter allows us to see how it should be written as in Figure  5 .

Figure 5: How to write a "g"

This is a letter which is clearly distinguishable from an ordinary " g " and which is simple to write. It is therefore suitable for using on a chalkboard.

The characteristics of this letter are obvious. The verticals are decidedly vertical and the horizontals have a definite arc which goes in the obverse direction to the expected one. These combine to create the angular effect. We can apply these principles to the entire alphabet to produce a legible "mathematicians fraktur" wherein the letters are close enough to their "true" forms that their meaning is apparent, and are simple enough that this meaning is conveyed even when they are drawn in chalk on a chalkboard. Only the "x" proves disappointing, due to its lack of any horizontal or vertical lines.

A full lowercase rendition of "blackboard fraktur" can be found in Appendix  A .

4 Script Capitals

The other typeface favoured by chalk artists is the script typeface, primarily in uppercase. The difficulty with this typeface is that there is both a danger of not making it sufficiently ornate and of making it too ornate. If we use LaTeX fonts, we can see in Figure  6 that the letters produced by \mathcal are not sufficiently distinguishable from ordinary letters to be usable on a chalkboard. At the other extreme we have \mathscr where, as shown in Figure  7 , some letters are so ornate that it is hard to tell what they are.

Figure 6: Ordinary and mathcal letters.

Figure 7: Identify these mathscr letters.

As with the fraktur typeface, the goal is to find a characteristic of such letters that clearly identifies the typeface and can be replicated easily whilst not obscuring the letter. With fraktur letters, it is the angularity that is distinctive. Let us look at a few characters from the \mathscr font where we can tell what the letter should be, see Figure  8 .

Figure 8: "ABC" in mathscr.

The exaggerated lean of the characters is certainly characteristic of this typeface, but is also hard to clearly reproduce. There are few chalk aficionados who attempt to use the distinction between x and x (or even × ) on a chalkboard so using the lean to distinguish between 𝒜 and A is probably insufficient. More visually robust is the loop that accompanies most characters (the "M" and "N" are the notable outliers here: ℳ and 𝒩 ). Thus our solution is to add a loop to all the letters. Again, the goal is to make a typeface that is instantly recognisable and easy to reproduce so we make the loop as uniform as possible across the alphabet. For most letters we can choose to make it either the initial upright or, for the curved letters, part of the top. For a few, such as the "H", then practicality weighs in and the loop is slightly altered from what one might expect. The letter "A" is seen in Figure  9 and a full uppercase alphabet 2 in Appendix  B .

2 Created as a font using iFontMaker .

Figure 9: A simplified script "A".

5 Mathematical Calligraphy

Thus far the term mathematical calligraphy has meant "calligraphy for mathematicians". That is, how can a little knowledge of calligraphy aid the generic mathematician. Another interpretation, and probably the one that initially comes to mind, would be of mathematically inspired calligraphy. There are, perhaps surprisingly, few examples of this. Whilst it is not hard to devise a typeface with some connection to mathematics 3 , to qualify as mathematical calligraphy then it should be a reasonable task to form the letters using a pen.

3 There is, apparantly, a phase every typeface designer goes through wherein they design a typeface based on ruler-and-compass constructions. A modern (and beautiful) version of this can be found in the online book A Constructed Roman Alphabet . I've also experimented with mathematical lettering as can be seen in the documentation of the braids and TQFT packages. Perhaps one day the definition of a braid will be written as a braid itself. But I digress.

Given the dearth of such examples, we can begin at the begining with numbers. Here, then, we present a mathematical typeface suitable for calligraphy designed with numbers in mind. The full alphabet 4 is presented in Appendix  C . The inspiration behind it came from writing the word:

4 Created using iFontMaker . Unfortunately, I have lost my original construction of this typeface and I'm not sure I remembered all of the assignments correctly. Any suggestions for improvements will be gratefully received.

This does give a whole new dimension to the idea of encoding formal arithmetic as numbers. As an example consider the following statement (taken from Wikipedia ).

In set theory, which is capable of acting as an axiomatic foundation for modern mathematics, natural numbers can be represented by classes of equivalent sets.

The Holy Grail for this typeface would be to find a statement of the form:

This sentence is its own Godel encoding.

A Chalkboard Fraktur

B Script Capitals

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

C Numerical Typeface

a b c d e f g h i j k l m n o p q r s t u v w x y z

D Drawing the Fraktur Font

Here's an animation of how to draw the Fraktur Font.

Writing for a Math Class

The union of the mathematician with the poet,

fervor with measure, passion with correctness,

this surely is the ideal.

—William James

Writing for a math class strikes many students, and teachers too, as an odd idea to say the least. However, an increasing number of educators have recognized the importance of written composition, especially in lower division and survey courses, for helping students to master and express mathematical ideas. When a math class consists, as it too often does, of nothing more than a collection of techniques to be learnt by rote and regurgitated on exams, then certainly writing about those techniques is superfluous. But when a mathematics course, as it ought, becomes a journey of discovery—of mathematical ideas and the importance of those ideas in our appreciation of the world—then writing about mathematics can become a powerful component of the learning process.

An increasing number of educators have begun to incorporate a modest amount of composition into their syllabi, with surprising results. The reason for their success is not difficult to understand. Most instructors will readily admit that they “never truly understood” the basic mathematical ideas that constitute algebra, trigonometry, and calculus until they had to teach those ideas to others. What this fact illustrates is that until an idea is communicated to others it has not really made itself at home in our own minds. Communicating ideas brings them alive, pulling them off the flat page of our memory and giving them solid shape in the dynamic space of the intellect. When students communicate through written composition about the ideas they are learning, they can achieve the same result for themselves.

A second reason for writing in the mathematics classroom is to bring context and background to students’ appreciation of their subject. Mathematics is rich with its own history, a history moreover peopled by extraordinary characters and punctuated by the steady advance of powerful ideas. Sadly, most students learn the basic math curriculum—right up through college—largely in ignorance of and untouched by this history. Little wonder that to many students mathematics remains a dead subject, wholly abstract and removed from their lives and interests.

If the Fundamental Theorem of Calculus is arguably one of the profoundest creations of the human intellect and human civilization, should not students learn of the people and history surrounding its inception? But where will they learn of this, if not in their calculus class? Is there no value or interest in learning that Pascal developed combinatorics so as to win at the gambling table? Or what the response of the Pythagoreans was when the irrationality of the square root of two was demonstrated? Or how Alan Turing perhaps saved England during World War II by cracking the German Enigma code, and that he did so only a few years after describing, for the first time, the mathematical idea of a programmable computer, an event that has transformed all our lives? By providing the opportunity to explore these events and learn about these people, even the occasional small writing assignment can transform a student’s outlook on mathematics.

Admittedly, for both the student and the instructor, incorporating writing into the math curriculum presents real challenges, not the least of which is the pressure of time. These issues are examined in the next section, and guidance is provided (for students and instructors) for making the writing experience in the math classroom a successful and rewarding one.

For the Instructor

I am a math teacher, not an English teacher, we hear you cry. Quite right. You are not trained to teach composition, nor are you probably much inclined that way. Nor should you be. But consider: Should the science teacher teach math? Probably not. Nonetheless, you would quite understand the science teacher’s expectation that his or her students use math—correctly—when called upon to do so in the science classroom. Our disciplines do not exist in isolation from one another, and expecting your students to be able to compose standard written English on a mathematical topic is entirely within your purview as a math instructor.

There remains the question of how to implement writing assignments in a math-class setting. The issues to be addressed include the structure of writing assignments, the composition standard required of students, and assessment of written work. We will review these in turn.

The Structure of Writing Assignments

The first thing to realize is that you have complete freedom here. Writing assignments can be as short or long, simple or complex, as you wish. However, particularly if giving and grading writing assignments is a new venture for you, we recommend keeping things as simple as possible at the outset. When we first tried it ourselves, we began by assigning one or two page biographical sketches of famous mathematicians as extra credit assignments. This is an excellent way to develop a feel for what it is like to assess such assignments. From this beginning, we have experimented with incorporating writing in the math class more thoroughly, with assessment of writing assignments forming an integral part of student assessment for the course. Starting small and simple helps your students also, as they can then become accustomed to the standard you will require of them. Additionally, the simpler the assignment, the less time must be devoted during class (and by your students outside of class) to getting the assignment organized, understood, and completed.

The Required Format

Although you may be tempted not to require any consistency of style or format from your students, we strongly advise that you do so, for many reasons. First, the (admittedly bland) uniformity of the required style permits the reader (i.e., the grader) to focus on the content and substance of a paper without being influenced (or distracted) by its formatting. Second, the required format promotes clarity, and makes it easy to find important information, such as the author of the paper, the references used, and so on. Third, it lets students know what is expected, and permits the instructor to apply a uniform grading standard vis-à-vis the purely structural aspects of the paper. In short, keeping in mind that we are not trained to teach composition, setting a minimum standard for the purely formal compositional elements frees us to focus assessment of our students’ efforts on the content rather than the form of their papers.

Because there is not a universally accepted style for mathematical writing (and to the extent that there is one, it addresses exposition at the research level, which is irrevelent here), we have always implemented the Modern Language Association (MLA) academic standard for writing assignments. This is the standard that is taught in freshman composition courses, and the primary reference text is always available in college bookstores. (For flexibility’s sake, we generally permit students trained in the APA standard to use it instead—provided they do so properly .) We recommend that you choose the standards for format and references you are most comfortable with, and require your students to use it.

As regards other qualities of their compositions, such as sentence structure, paragraphing, and so forth, we allow students more latitude than a composition teacher would do, because such issues are not central to the purpose of a writing assignment in a math-class setting. We do, however, mark down a paper for failure to follow the prescribed format, sloppy presentation, poor grammar, and rampant spelling errors. This simply reinforces a standard that should be common to all academic work.

It is natural to divide assessment of students’ written work into two parts—form and content. By setting a minimum standard for the form of the paper (as outlined above, and as described below in the For The Student section), assessment of the form of the paper is facilitated and made uniform. How content is assessed will vary greatly depending on the assignment. The guiding principle is to ensure that students know what is expected of them, and that assessment of their work is applied as uniformly as possible. The challenge for the instructor is that some students will be very good at written composition, whereas others will be very poor at it. By focusing on content rather than style, you can help to level the playing field in such a way that all your students will get the greatest benefit possible from the assignment.

For the Student

When you are told that you will be writing compositions for your math class, you may be surprised, and perhaps even a little resentful. “Isn’t math class for doing math?” Of course it is. But you should look upon your writing assignments as opportunities. They will give you a chance to talk about the math you are learning, which is an excellent way to enhance your knowledge. Your efforts will go more smoothly and be more successful if you approach these assignments the same way you should approach doing mathematics; methodically, and with a spirit of play and discovery. The information which follows is intended to make writing assignments more manageable for you, and to give you an edge in writing the best papers possible.

When writing a paper, concentrate first on the writing itself and save formatting and proofing till last—but then be thorough. Give yourself time to write and rewrite without being rushed. In an academic paper, searching for what you want to say can take time, and then saying it clearly can take several attempts. The biggest part of your grade comes from organizing your ideas and presenting them clearly. The perfect grade is apt to go to that paper which is free of glaring spelling and grammar errors, is in the proper format, and presents its material clearly, supporting it well from the sources. Study the information in this page thoroughly, and refer back to it each time you write a paper for your mathematics course.

Basic Requirements

• Papers should always be typed, double-spaced, using either courier or times roman typeface in black 12 point type. DO NOT use any other typeface, color, or size.
• The paper used should be stock white, and a standard size (8 1/2×11 in. in North America, A4 elsewhere).
• Every page should have a 1 inch (2.5 cm) margin on all sides. This is critical to allow the grader to write comments in the margins.
• When references are used, a works cited page must be prepared.

Detailed Requirements

• Citations are made in accordance with the standard specified by your instructor. In the United States, this will typically be either the MLA (Modern Language Association) standard or the APA (American Psychological Association) standard. You will find many online resources to help you.
• Every paper should have a title.
• The student author’s name should appear, together with the page number, in the top right hand corner of every page (this element only may violate the 1 inch margin rule).
• The first page should have the student’s name, the instructor’s name, the class, and the date in the top left-hand corner of the page (see illustration below).
• Titles of books, titles of web pages, and any non-English text must either be underlined or italicized.
• DO NOT fully justify text (left justify only). In other words, the right edge of the text should be “ragged.”
• When a complete paragraph or block of text has a single source, place the citation at the end of the paragraph or block of text.
• Quotations longer than three lines of text must be indented.
• Use of graphs and other images is encouraged. Images that cannot be printed inline with the text (such as hand-drawn graphs, etc.) should be attached on separate sheets, with titles such as “Figure 1,” and referenced by title in the text. Note that such extra sheets will ordinarily not count towards length requirements.
• Alphabetize the works cited page, and ensure that the references are listed the same way they are cited in the text.
• The second and following lines of references are indented on the works cited page, not the first line.

Forget formatting— until you have finished doing your research, compiling your sources, and preparing your draft paper. When the paper is essentially finished, then is the time to carefully format it. Finally, proof the completed paper for spelling (use the spell checker!) and grammar. Sloppiness with respect to proper formatting, correct spelling, and good grammar will cause the paper to be marked down.

The example given above uses the MLA standard. Whichever standard your instructor specifies, follow the assigned format for references as closely as you can. The guiding concern here is that the reader of your paper be able to find your source and locate the information you used as conveniently as possible.

As for any class, the papers you write for your math class will invariably reflect the time and effort you put into them. Although we are all familiar with the somewhat romanticized image of the disorganized student burning the midnight oil to finish a paper only a few hours before it is due, it should be remembered that such students are rarely successful. Start working on your paper early, give yourself time to come back to it, and aim to have your draft completed at least a day or two early. This way, you can “pretty it up” at your leisure. If writing well is a particular challenge for you, we strongly recommend you get a copy of Strunk & White’s The Elements of Style. This brief book, easily absorbed in an hour or two over a cup of something warm, will change how you write forever, and very much for the better.

We will leave you with some excellent advice from a fine writer:

Don’t use no double negatives.

Proofread carefully to see if you any words out.

Take the bull by the hand and avoid mixed metaphors.

If I’ve told you once, I’ve told you a thousand times, resist hyperbole.

“Avoid over use of ‘quotation “marks.” ’ ”

Avoid commas, that are not necessary.

If you reread your work, you will find on rereading that a great deal of repetition can be avoided by rereading and editing.

Avoid clichés like the plague.

Never use a long word when a diminutive one will do.

Avoid colloquial stuff.

—William Saffire

Contributors

• B. Sidney Smith, author

Citation Info

• [MLA] Smith, B. Sidney. "Writing for a Math Class." Platonic Realms Minitexts. Platonic Realms, 13 Mar 2014. Web. 13 Mar 2014.
• [APA] Smith, B. Sidney (13 Mar 2014). Writing for a Math Class. Retrieved 13 Mar 2014 from Platonic Realms Minitexts: http://platonicrealms.com/minitexts/Writing-For-A-Math-Class/

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Assignments on writing

Examples of short assignments, term papers, designing assignments that enable students to write well.

Writing well requires mastery of writing principles at a variety of different scales, from the sentence and paragraph scale (e.g., ordering information within sentences so content flows logically ) to the section and paper scale (e.g., larger-scale structure ). To simplify teaching, you can begin the term with shorter assignments to address the smaller-scale issues so you can more easily focus on the larger-scale issues when you assign longer assignments later in the term. At all scales, students best learn to communicate as mathematicians if the assignments are as authentic as possible: if the genre and rhetorical context are as similar as possible to those encountered by mathematicians.

Many of the following ideas are currently implemented in M.I.T.’s communication-intensive offerings of Real Analysis and Principles of Applied Mathematics .

• Require that at least one question on each problem set be typed up and written in the style of an expository paper (rather than the usually terse and sometimes scattered style of a homework solution).
• Assign short exposition tasks such as summarizing the proof of a theorem done in class or filling in the gaps in an explanation given briefly in class.
• To help students learn LaTeX or how to use equation editors, have an assignment requiring at least basic math formatting due early in the semester so students aren’t required to learn it as they’re researching and writing their term papers. Begin with simple math formatting exercises, building to more complex: e.g., see the assignments for M.I.T.’s Real Analysis recitations 1 (text with math) , 2 (table and figure) and 13 (slides containing a figure with LaTeXed labels) .
• Begin with communicating simple arguments, building to more complex (e.g., having students explain the heapsort algorithm and then revise the explanation based on feedback provides a rich opportunity for teaching about writing clear definitions, giving conceptual explanations as well as rigorous details, and presenting information in an order that is helpful to readers.) See the sequence of assignments from M.I.T.’s Principles of Applied Mathematics .
• Have students revise part of a concise textbook such as Rudin’s, Principles of Mathematical Analysis in the style of a more-thorough lecture note.
• Before an exam, have students formulate and submit to you a list of 2+ questions they have about the material. Students have a hard time formulating precise questions, yet this is an important communication and learning skill. Some students may feel they understand the course material, so permit questions that go beyond the scope of the course. You can use the questions to focus a review session. More detail about this assignment is given in this lesson plan from M.I.T.’s communication-intensive offering of Real Analysis.

The following books, articles, and websites contain short writing assignments.

• Stephen Maurer’s Undergraduate Guide to Writing Mathematics has an extensive appendix of writing exercises designed to target various aspects of writing mathematics.
• Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go , by A Crannell et al . [link goes to MAA review] This 119 page book from the MAA contains “writing projects suitable for use in a wide range of undergraduate mathematics courses, from a survey of mathematics to differential equations.” Each prompt is written in the form of an (often amusing) letter from someone who needs help with a “real-world” problem that requires math expertise. Students must solve the problem and write a letter of response. On his website, Tommy Ratliff (one of the co-authors) gives a brief account of using such projects in his calculus course.
• Annalisa Crannell’s Writing in Mathematics website has writing assignment for Calculus I, II, and III as well as links to colleagues’ websites that have further writing assignments.
• Quantitative Writing from Pedagogy in Action, the SERC Portal for Educators, has many examples of short and long writing assignments based on “ill-structured problems,” which are “open-ended, ambiguous, data-rich problems requiring the thinker to understand principles and concepts rather than simply applying formulae. Assignments ask students to produce a claim with supporting reasons and evidence rather than ‘the answer.'”
• The Nuts and Bolts of Proofs by Antonella Cupillari includes exercises for an introductory proof-writing course. Proof topics include calculus and linear algebra.
• Platt, M. L.. (1993). Short essay topics for calculus. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies 03.1 , 42-46.

Additional information about journal-writing assignments and other writing-to-learn assignments can be found on the page about using writing to help students learn math .

For each assignment, indicate your expectations about audience and length, so students know how much explanation to include. An appropriate audience is often other students in the class who are unfamiliar with the specific topic of the assignment, or other math majors not in the class.

Term papers enable students to pursue areas of their own interest and so can be among the most rewarding assignments for students. To help students succeed, give students guidance for choosing a sufficiently focused topic, for finding helpful sources, and for using sources appropriately. See this assignment for proposing a term paper topic , from M.I.T.’s Principles of Applied Mathematics –it includes guidance for how to choose a good paper topic.

One of the (interesting) challenges of assigning a term paper is generating a list of possible paper topics. Ideally, each topic should have well-defined scope and have at least two or three available resources accessible to students in the course. You may want to emphasize to the students that they are not expected to do original mathematics research. However, the paper must be their own — they cannot paraphrase and closely follow a published survey paper.

One of your institution’s librarians may be happy to collaborate with you to show students how to find useful sources.

To provide students with an authentic rhetorical context for their term papers, consider showing them samples of expository papers and suggesting that they write for a journal that publishes expository papers (e.g., The American Mathematical Monthly , Math Horizons , Mathematics Magazine , and The College Mathematics Journal .

Don’t assign a term paper unless a variety of topics exist at an appropriate level. For example, a term paper may not be appropriate for an introductory class in analysis.

Be aware that plagiarism may be an issue particularly in large classes on subjects for which a wealth of material is available online. In such classes, you may find it to be helpful to tightly specify the paper topics or to supply a specific slant to the papers (e.g., apply such-and-such method to an application of your choice). Vary the assignments from year to year. These precautions may be less important in small classes.

In some classes (e.g., applied mathematics classes), it may be necessary to carefully guide students to choose topics that contain sufficient mathematical content. For that reason, using caution when approving unfamiliar topics.

A poorly focused assignment will leave students confused about what is expected of them and is likely to result in poor writing. Students are likely to write their best if the assignment is interesting and if students are told (or are able to confidently identify for themselves) the following:

• educational objectives of the assignment
• audience knowledge and interest, and author’s relationship to the audience
• purpose of the text to be written (e.g., to convince, to entertain mathematically, to teach, to spark interest)
• content to be addressed
• details of the genre ( proof ? research paper? funding proposal?)
• how the writing will be graded
• an effective writing process (you can provide support by assigning intermediate due dates or revision )

The following resources explain these points and give further guidance for designing effective assignments:

• Bahls, P., Student Writing in the Quantitative Disciplines: A Guide for College Faculty , Jossy-Bass 2012, pp. 36-46, contains sections on structuring writing assignments (includes sample prompts), sequencing assignments throughout a course, and sequencing writing from course to course.

General resources (not specific to mathematics)

• How can I avoid getting lousy student writing?
• What makes a good writing assignment?
• The webpage Integrating Writing and Speaking Into Your Subject , provided by MIT’s Writing Across the Curriculum, has several subpages about writing assignments.
• Creating Writing Assignments , MIT’s Writing Center

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12 February 2014

Doing your homework in latex.

It is a common occurrence for other students to comment on my homework whenever I turn it in for one of my classes.

The complete LaTeX file (and the pdf output) can be found in my repository, latex-homework-template .

Below are a few screenshots of problems that I’ve done in the past:

If I didn’t know how easy it was and the benefits that I get from typesetting my homework, I’d probably ask as well. However, I’d argue that using LaTeX to type up homework has made me a far better student than when I used to handwrite my homeworks.

And that is something that I care a lot about.

The Benefits

I can summarize the benefits like so:

It can be kept in Source Control. Handwriting can’t be stored in a version control system; once you erase something, it’s gone.

You can see your homework materialize in front of you. Seeing the results and the equations in their complete LaTeX-glory is a very powerful way to conceptualize things. There’s just something different about the way things look so perfect that makes the subject easier to understand.

You’ll do better in your classes. This one goes with the previous point, but having the ability to see your homework helps you understand it. By understanding it well, you’ll do better on tests. You will maximize how much you can learn as well as maximize your grade (if that matters to you).

It’s very neat & tidy. Although my handwriting has improved quite a bit, I still find myself slipping back into a rushed, messy script from the past. LaTeX gives zero doubt that the professor/TA will be able to read my solutions.

About LaTeX

A very short history.

Donald Knuth , a legendary Computer Scientist as well as one of my favorites, is well known for the system that he created called just TeX .

It is a piece of typesetting software that aids in writing documents and formulas. The power comes from the fact that the document that you write is plain source code.

The code that you write is then “typeset” into the final document in whatever form you wish.

Here’s an example of some basic LaTeX code:

With the output looking like below:

Using the Template

I’ve created a GitHub repository, latex-homework-template , just for my homework template that I’ve been using ever since I started. I found it online and used it as a base to start my template.

To use it, just download the homework.tex file and start editing. Once you need to typeset it, you’ll need LaTeX here .

After that, you just need to compile it and you’ll get your output. There are tons of different resources that I’ve found useful in learning LaTeX:

TeX StackExchange

LaTeX Wikibook

Effect on Performance

I have a solid set of anecdotal evidence in favor of using LaTeX for writing up my homework.

In all the classes that I’ve used LaTeX, I’ve come out of the class with a very strong understanding of the material as well as a good grade. Although I’m not a big fan of grades (like at all), I know it matters to some people.

This might have to do with the fact that doing the homework in LaTeX takes longer. It might have to do with the fact that I perfect the appearance and spend a lot more time looking at the subject.

The most likely reason is a combination of all that I previously mentioned plus other factors. I’m usually one to always want to quantify something, but in this case, I know it helps; that’s all I need.

Learning Curve

There definitely is a learning curve when it comes to trying to use LaTeX for homework. I felt that it was definitely worth the effort unlike how it might seem to some students.

I reasoned that when I go to graduate school, I will want to use it there. I also know how pervasive it is in textbooks. Since I love to read textbooks so much, I wanted to see what it took to write them so elegantly. I may even want to write one in the future; we’ll have to see =]

To me it seemed like a small tradeoff for the great benefits that it provided.

I cannot recommend using LaTeX for your homework enough.

The benefits go a long way. It helps you learn the material and in a way that isn’t as easily achieved when just using pencil and paper.

LaTeX is also widely used in academia and learning about the tool is almost essential if you wish to go to graduate school.

Once I graduate from university, I plan on releasing all my code for the last three semesters as open source. It includes all my LaTeX code which has really accumulated over the last year. It should provide a nice resource for others.

In the meantime, hopefully if you start using LaTeX for your homework, you won’t be able to resist doing it early because of how fun it is. Well, at least it was fun for me =]

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Back to Assignments or Home . Updated 13 March 2024 .