Westfield State College Writer's Guide

This section highlights some of the main issues facing inexperienced writers of mathematics.

Like other sections of this site, this remains under construction and will continue to expand.

• Justifying Claims, Results and Remarks, and the ways one accomplishes such justification, forms a fundamental distinction between communication in mathematics and in any other field, and therefore this area is often the most critical problem area for beginning writers of mathematics. Any claim, result or remark must be justified by i) providing a deductive proof, either a formal proof or a collection of equations that together provide a proof, ii) by citing a known result that provides direct deductive justification, iii) by providing precise details how a known result you cite can be used indirectly to provide deductive justification, or iv) by providing an outline that the reader can use to establish a deductive proof. If you cannot establish a claim, result or remark deductively you must clearly inform the reader that it is only a conjecture. In this case you should explain whether you used inductive or inferential reasoning, what evidence you have, if there are heuristic reasons to support the conclusion, etc.

• Context and Foreshadowing are critical given the emphasis on justifying claims, results and remarks described above. A perponderance of any mathematical work is generally devoted to the appropriate justification of the claims, results and remarks that make up the paper. Because of this it is critical to provide the reader with direction that will help her/him see the theme from the details ("forest from the trees"). This should happen not just via an abstract or introduction, but as the written work progresses it is critical to give the reader a clear indication of the papers direction and goals. One clear rule of thumb is to state results before you prove them. It is also often helpful to remind the reader why the result that will be proven is important and what heuristic, empirical, or ancedotal evidence there might be to suggest such a result.
  In short, before you do anything in a mathematical paper you should briefly inform the reader what you are going to do, how you are going to do it, and why you are going to do it.

• Details and Documentation necessary in a mathematical paper vary depending on the type and level of the paper. Keep your audience in mind, and, when unsure, provide extra details and/or documentation. (Nb. the term documentation here does not mean citation or reference, it is a broader term.)
  EXAMPLE involving the maximum of the function f(x) = (1/3)x³ - x² - 3x + 5 on the interval (3,9]:
  "The function f(x) is increasing, so it is biggest at x = 9."
  This is a poor example regardless of context. It provides neither significant justification nor significant details. Moreover, the language is awkward. A function might have a maximum when x = 9, but when referring to a maximum we are interested in the maximum value of the function, in this case the value of f(9). In different contexts, the following examples would be much more appropriate.
  • For a labratory report the following would be appropriate: "Using the standard rules for differentiation we showed that f'(x) = x² - 2x - 3. Factoring the derivative we found f'(x)=(x + 1)(x - 3), so that f'(x)>0 for all x in the function's domain (3,9]. Since the derivative is positive the function must be increasing throughout the interval (3,9] [Hughes-Hallett, et. al., p. 106]. Hence, the maximum of the function f on the interval (3,9] must occur x=9 when the value of the function is f(9)=(1/3)9³ - 9² - 3(9) + 5 = 140."
  • For a research paper the audience would certainly know how to find the maximum of a polynomial function on an interval. Finding the extremum of this function is apparently a necessary part of some larger undertaking, so it is better just to get on with it: "Using standard techniques from differential calculus one shows the function f on the interval (3,9] attains a maximum of 140 at x = 9, thus..."
  • For a survey or expository paper the following passage would be appropriate. "As noted in our discussion of tangents to graphs above, the derivative can be used to determine when a function is increasing or decreasing. To find the maximum of our function f(x) = (1/3)x³ - x² - 3x + 5 on the interval (3,9] we can employ the derivative. What is the derivative of our function? A "calculus" is literally a collection of algorithms for computing, and the differential calculus is certainly aptly named. There are rules for differentiating all of the standard elementary functions, including polynomials like our function f. Using these rules it is easily shown that the derivative of our function f is f'(x) = x² - 2x - 3. The positivity or negativity of the derivative is not yet apparent, but a bit of algebra helps: f'(x)= x² - 2x - 3 = (x + 1)(x - 3). For any x in the domain (3,9] both of the factors (x + 1) and (x - 3) are clearly positive, so our derivative is positive throughout its domain. This means that the function is continually increasing throughout its domain. Because of this it must attain its maximum value at its right-most endpoint, when x = 9. We can then readily compute the maximum, f(9) = (1/3)9³ - 9² - 3(9) + 5 = 140."

• Clearly Defining all Constants, Variables, and Functions is critical. Any imprecision could undermine all of your efforts.
  • Example: "Let t = years." The context of this statement might allow the reader to work out what the variable t represents, but this should never be left to chance. Much more appropriate phrases would be: "Let t denote the year the record was broken." or "We will denote the year the record was broken with the varialble t, letting 1900 correspond to t=0, 1910 correspond to t=10, etc."

• Graphs provide a critical vehicle for displaying and visualizing information and data. In a mathematical paper each graph should have a caption which contains a brief description of the information displayed as well as a name so it can be referred to later. For example: "Figure 2. Graph of population (P) versus time (t)"; "Graph C. Area (A) as a function of the number of partitions (n)"; for several graphs displayed together "Fig. 5 a) distance versus time, b) velocity versus time, and c) acceleration versus time". The axes of a graph should be clearly labeled and should include numberical scales with an indication of units where appropriate. Scales of graphs should be chosen so that the space is used effectively. Graphs or data should generally take up most of the graph if it is scaled properly. If the axes is broken to accomodate data away from the origin one should indicate a broken axis as follows: ---/ /--- . Graphs of functions should be clearly labeled with the name and/or algebraic representation of the function.
  Tables are quite useful for displaying information and one should consider carefully the kind of information being presented when deciding whether a graph or table is most appropriate.

• Tables, like graphs, provide a critical vehicle for displaying and visualizing information and data. Tables, like graphs, should each have a caption which contains a brief description of the information displayed as well as a name so it can be referred to later. (See above under "Graphs" for examples.) Tables have rows of data running horizontally and columns of data which run vertically. The quantities being recorded in a given row or column must be clearly indicated and appropriate units must be given as appropriate. For example: "Distance (m.)"; Velocity (v) in ft./sec."; "n=number of partitions" (note that there are no units since n is a dimensionless number). Tables should be arranged to utilize space well. For example, a table recording a single measure quantity over 20 years should run horizontally to best use available space on the page.
  

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