What Makes a Good Math Images Page
From Math Images
As a page developer, it's easy to get totally consumed in technical issues: debugging coding of equations, getting images and equations to align properly, etc. It's easy to forget about the equally important task of writing well.
This page is meant to help with this task. It contains an overview of aspects of writing and layout that help make a page interesting, informative, accessible, and aesthetic. The Checklist for writing pages describes the same ideas, but as a list of bullet points that are more specific than the big ideas expessed in this page
Aspects of a Good Math Image Page
The many facets of good writing and layout identified in this page are divided into 6 categories, and are explained below.
The goal is for pages to be interesting and appealing, not just informative. Especially for readers who don't already love math, it helps to explicitly address the question, "So what?" or "Who cares?", or, less belligerently, "Why is this interesting?"
A few ways to set context include answering any of the following questions:
- Can the topic of a page be applied to other fields or other areas of mathematics?
- Why did people study this topic in the first place?
- Does the main image represent anything physical?
- What are some surprising or counter-intuitive ideas that are part of your topic?
Different topics lend themselves to different types of context, and no way of setting a context will make a page appealing to everyone. Still, every page can be enriched by setting some kind of explicit context that is clearly related to the main content of the page.
- Parabolic Reflector (describes and proves how a light or sound collector shaped like a parabola focuses the light or sound at a single point)
- Three Cottage Problem (describes the problem of connecting each of three wells to each of three houses without intersecting pipes)
- Vector Fields (describes the class of functions known as vector fields and what they represent as well as describing one way in which they are useful)
Quality of Prose and Page Structuring
Even though this is writing about math, pages should still adhere to basic tenets of good writing in order to be informative (or even comprehensible). Simple statements of facts all in a row do not make a page. It is important for pages to flow and have all of the ideas be connected to one another.
Developed and flush out your points. The reader should be able to understand why you have included each piece of information and each point.
Where possible, keep your sentences short: probably 2 clauses per sentences at most. Also, grammar and usage do, in fact, affect page readability.
A final special challenge with this project is not scaring the less mathematically advanced readers. Put heavy terminology and equations and especially difficult concepts, as late in the page as is feasible, because some readers will go running as soon as they see the heavy machinery.
Math topics are all interconnected, and explaining connections between the topic of your page and other branches of math helps the reader access prior knowledge as they read and also helps them understand how the content of your page fits into a bigger picture of mathematics as a whole.
Wherever it makes sense and is not forced, include in your pages connections to other topics with explanations that make the relevance of these connections clear. Take advantage of the wiki format by linking to other pages on the Math Images site. If you feel your page should link to another Math Image page that doesn't yet exist, you are encouraged to create that other page!
Examples, calculations, applications, and proofs
Including examples and calculations can help readers understand the concepts in a page. If you describe a property shared by all Fibonacci Numbers, give an example and explain how your example works.
Including applications and proofs helps make a page much deeper and richer. Describe applications of your topic to engineering, economics, or mathematics. If you present an interesting theorem, present a proof if possible.
Different pages can benefit in different ways, but almost all pages can benefit from examples or calculations, and can benefit either from applications or proofs (or maybe both).
Mathematical Accuracy and Precision of Language
Clearly, the goal is always for terminology and calculations to be free of errors, as far as you know. And where you don't know, put a note on the discussion page for others' attention!
It's helpful to readers to be clear about the background knowledge expected for pages and to define all terms that readers with that background knowledge may not know. Definitions are most useful if they are precise; don't just say "rectangle" when you mean "square". It's better to define too many terms than to define too few.
Readers are also more likely to read definitions if they are
- heralded as definitions by using the accepted mathematical definition heralding notation, namely, the word/phrase being defined is boldfaced; and
- well-explained, but concise and free from too many dense symbols (again, take background knowledge into consideration). Pictures can help a lot with this.
To help make definitions feel shorter, it can also help to use mouse-overs for shorter definitions, and to use Helper Pages, rather than in-page explanations, for more involved definitions. (These techniques won't result in boldfacing the defined word, but they will result in a different color and/or underlining, which is about as good.)
Another issue to keep in mind is that you should avoid glossing over finer points in a way that is actually misleading to the reader. If you use the words obviously and clearly (eg "We can clearly see that this implies....), it's quite possible that the reader might convince themselves that it's "obviously" true because of wrong reasons they conjure up on their own.
We all know as users of the web that we're much more likely to engage with pages that are aesthetically pleasing and feel easy to navigate. In your pages, keep paragraphs short. Images should fit logically into your pages. Sometimes, for different browser window sizes, images can end up chopping up text, and you need to manually add in additional breaks.
Taking the time to adjust the formatting of a page can feel time consuming and tedious, but it's really worth it.
A good page should lay out text, images, and applets so that the page is uncluttered and easy to look at. Additional images should be placed appropriately near the text that refers to them, and should actually inform the accompany text. Text should be in small paragraphs, mouse overs should provide short definitions and reduce clutter on the page, and all equations should be easy to read.
Special Concerns for Different Types of Pages
Math Images pages can be about almost any math topic so long as you can find an image to go along with it. Because this is about as open as can be, there is a very wide range of pages on this site. So far, four main types of pages have been identified, and each of these types of pages has some particular issues writers should be aware of. These types are certainly not mutually exclusive, and some examples are given for each type.
Not all of these examples are complete, but they should be helpful in understanding the types.
If you think you've identified another type of page, please let us know on the discussion page.
Pages that describe particular problems or proofs
Pages that describe particular problems or proofs often immediately provide the reader with context. Since there is a problem to be solved, or theorem to be proved, there is immediately some motivation to the page. It's always better to add in details about why the theorem or problem is important in mathematics, and how people first came to ask the question involved. It's also useful to include some information about similar problems, or questions that a theorem is used to solve.
It can be difficult to include a sufficient number of examples, applications, calculations and proofs in these pages. Try to solve the problem or prove the theorem in at least two different ways. This not only provides you with more content in these area, but it also gives the reader different ways of approaching the problem, which in turn may help them understand the content better.
Pages that describe special topics
Special topics include topics that are not covered in introductory mathematics courses, and might not be covered in advanced classes either. For the post part, these pages are entirely centered around a beautiful images that may have obscure of complicated math behind them. A very large portion of pages on this site fall into this category.
For these pages, it is often difficult to find examples, applications, calculations, or proofs to go along with the context. In these cases, it is possible to include a simplified diagram or applet to explain the construction of the image. You can also describe similar constructions as examples.
These pages often lack a straight forward context or connections to other areas of math. To address these issues, it's often best to start out trying to figure out *why* someone made the image in the first place. The artist/mathematician's motivation for creating the image can be the reader's motivation for reading your page.
Also, look through the Math Images site to see if there are other images that are made using similar principles as the one you're describing. Linking between pages that describe similar methods can help the reader.
Pages that describe general topics in mathematics
Some very valuable pages on this site deal with topics that are frequently seen in introductory level mathematics courses. For these pages, it is often possible to create your own image to go along with these pages.
These pages should provide rather basic information about a topic--going into too much depth can result in a very long page that intimidates readers. Often, these are topics on which you can find whole chapters of books or whole books. Try to pick and choose what information is most important and easiest to explain.
These pages can often lend themselves to straight forward inclusion of examples, just like those in textbooks. Even when you are writing a helper page, using as three or more examples can be very instructive.
- Vector Fields
- Volume of Revolution
- Completing the Square
- Parametric Equations
- Divergence Theorem
- Probability Distributions
Pages that describe an application of a general topic
In between special topics and general topics lies applications of general topics. These are more specific pages that use a particular topic that is commonly taught in introductory mathematics classes to describe a very specific topic that may not ever appear in an mathematics course.
Often it's easier to include connections and examples in these pages than in special topics pages. The challenges are to provide a strong, motivating context and to work on connecting the specific topic to areas of mathematics beyond the general topic that the page is based on.
Abram was heavily involved in providing feedback to student writers during the summer of 2010. He drafted the bullet point version in the Checklist for Writing Pages near the end of the summer. This version departed slightly from the 2009 version, based on experiences of that summer.
In the fall of 2010, this page was edited to better match the Checklist for Writing Pages, and the Checklist was updated to incorporate any elements from this page it didn't already capture.