This post discusses some ideas I have for improving abstractmath.org.
Handbook of mathematical discourse
The Handbook was kind of a false start on abmath, and is the source of much of the material in abmath. It still contains some material not in abmath, particularly the citations.
By citations I mean lexicographical citations: examples of the usage from texts and scholarly articles.
I published the Handbook of mathematical discourse in 2003. The first link below takes you to an article that describes what the Handbook does in some detail. Briefly, the Handbook surveys the use of language in math (and some other things) with an emphasis on the problems it causes students. Its collection of citations of usage could some day could be the start of an academic survey of mathematical language. But don’t expect me to do it.
The Handbook exists as a book and as two different web versions. I lost the TeX source of the Handbook a few years after I published the book, so none of the different web versions are perfect. Version 2 below is probably the most useful.
- Handbook of mathematical discourse. Description.
- Handbook of mathematical discourse. Hypertext version without pictures but with active internal links. Some links don’t work, but they won’t be repaired because I have lost the TeX input files.
- Handbook of mathematical discourse. Paperback.
- Handbook of mathematical discourse. PDF version of the printed book, including illustrations and citations but without hyperlinks.
- Citations for the paperback version of the Handbook. (The hypertext version and the PDF version include the citations.)
Soon after the Handbook was published, I started work on abstractmath.org, which I abbreviate as abmath. It is intended specifically for people beginning to study abstract math, which means roughly post-calculus. I hope their teachers will read it, too. I had noticed when I was teaching that many students hit a big bump when faced with abstraction, and many of them never recovered. They would typically move into another field, often far away from STEM stuff.
These abmath articles give more detail about the purpose of this website and the thinking behind the way it is presented:
Presentation of abmath
Abmath is written for students of abstract math and other beginners to tell them about the obstacles they may meet up with in learning abstract math. It is not a scholarly work and is not written in the style of a scholarly work. There is more detail about its style in my rant in Attitude.
Scholarly works should not be written in the style of a scholarly work, either.
- The article in the Handbook about the mathematical register.
Every time I revise an article I find myself rewriting overly formal parts. Fifty years of writing research papers has taken its toll. I must say that I am not giving this informalization stuff very high priority, but I will continue doing it.
One major difference concerns the citations in the Handbook. I collected these in the late nineties by spending many hours at Jstor and looking through physical books. When I started abmath I decided that the website would be informal and aimed at students, and would contain few or no citations, simply because of the time it took to find them.
Boxouts and small screens
The Handbook had both sidebars on every page of the paper version containing a reference index to words on that page, and also on many pages boxouts with comments. It was written in TeX. I had great difficulty using TeX to control the placement of both the sidebars and especially the boxouts. Also, you couldn’t use TeX to let the text expand or contract as needed by the width of the user’s screen.
Abmath uses boxouts but not sidebars. I wrote Abmath using HTML, which allows it to be presented on large or small screens and to have extensive hyperlinks.
HTML also makes boxouts easy.
The arrival of tablets and i-pods has made it desirable to allow an abmath page to be made quite narrow while still readable. This makes boxouts hard to deal with. Also I have gotten into the habit of posting revisions to articles on Gyre&Gimble, whose editor converts boxouts into inline boxes. That can probably be avoided.
I have to decide whether to turn all boxouts into inline small-print paragraphs the was you see them in this article. That would make the situation easier for people reading small screens. But in-line small-print paragraphs are harder to associate to the location you want them to refer, in contrast to boxouts.
For the first few years, I used Microsoft Word with MathType, but was plagued with problems described in the link below. Then I switched to writing directly in HTML. The articles of abmath labeled “abstractmath.org 2.0” are written in this new way. This makes the articles much, much easier to update. Unfortunately, Word produces HTML that is extraordinarily complicated, so transforming them into abmath 2.0 form takes a lot of effort.
Abmath does not have enough illustrations and diagrams. Gyre&Gimble has many posts with static illustrations, some of them innovative. It also has some posts with interactive demos created with Mathematica. These demos require the reader to download the CDF Player, which is free. Unfortunately, it is available only for Windows, Mac and Linux, which precludes using them on many small devices.
- Monads for high school III: algebras. Innovative illustrations.
- Showing categorical diagrams in 3D. Interactive demo and innovative illustration.
- Demos for graph and cograph of calculus functions. Interactive demo and innovative illustration.
- A visualization of a computation in tree form. Interactive demo.
- Inverse image revisited. Interactive graphs of functions showing the inverse image of an interval. The interval can be changed with a slider.
- Create new illustrations where they might be useful, and mine Gyre&Gimble and other sources.
- There are many animated GIFs out there in the math cloud. I expect many of them are licensed under Creative Commons so that I can use them.
- I expect to experiment with converting some of the interactive CFD diagrams that are in Gyre&Gimble into animated GIFs or AVIs, which as far as I know will run on most machines. This will be a considerable improvement over static diagrams, but it is not as good as interactive diagrams, where you can have several sliders controlling different variables, move them back and forth, and so on. Look at Inverse image revisited. and “quintic with three parameters” in Demos for graph and cograph of calculus functions.
Abmath includes most of the ideas about language in the Handbook (rewritten and expanded) and adds a lot of new material.
- The languages of math. Article in abmath. Has links to the other articles about language.
- Syntactic and semantic thinkers. Gyre&Gimble post.
- Syntax trees in mathematicians’ brains. Gyre&Gimble post.
- A visualization of a computation in tree form.Gyre&Gimble post.
- Visible algebra I. Gyre&Gimble post.
- Algebra is a difficult foreign language. Gyre&Gimble post.
- Presenting binops as trees. Gyre&Gimble post.
- Moths to the flame of meaning. How linguistics students also have trouble with syntax.
- Varieties of mathematical prose, by Atish Bagchi and Charles Wells.
The language articles would greatly benefit from more illustrations. In particular:
- G&G contains several articles about using syntax trees (items 3, 4, 5 and 7 above) to understand algebraic expressions. A syntax tree makes the meaning of an algebraic expression much more transparent than the usual one-dimensional way of writing it.
- Several items in the abmath article More about the language of math, for example the entries on parenthetic assertions and postconditions could benefit from a diagrammatic representation of the relation between phrases in a sentence and semantics (or how the phrases are spoken).
- The articles on Names and Alphabets could benefit from providing spoken pronunciations of many words. But what am I going to do about my southern accent?
- The boxed example of change in context as you read a proof in More about the language of math could be animated as you click through the proof. *Sigh* The prospect of animating that example makes me tired just thinking about it. That is not how grasshoppers read proofs anyway.
Understanding and doing math
Abmath discusses how we understand math and strategies for doing math in some detail. This part is based on my own observations during 35 years of teaching, as well as extensive reading of the math ed literature. The math ed literature is usually credited in footnotes.
Math objects and math structures
Understanding how to think about mathematical objects is, I believe, one of the most difficult hurdles newbies have to overcome in learning abstract math. This is one area that the math ed community has focused on in depth.
The first two links below are take you to the two places in abmath that discuss this problem. The first article has links to some of the math ed literature.
To do: Everything is a math object
An important point about math objects that needs to be brought out more is that everything in math is a math object. Obviously math structures are math objects. But the symbol “$\leq$” in the statement “$35\leq45$” denotes a math object, too. And a proof is a math object: A proof written on a blackboard during a lecture does not look like it is an instance of a rigorously defined math object, but most mathematicians, including me, believe that in principle such proofs can be transformed into a proof in a formal logical system. Formal logics, such as first order logic, are certainly math objects with precise mathematical definitions. Definitions, math expressions and theorems are math objects, too. This will be spelled out in a later post.
To do: Bring in modern ideas about math structure
Classically, math structures have been presented as sets with structure, with the structure being described in terms of subsets and functions. My chapter on math structures only waves a hand at this. This is a decidedly out-of-date way of doing it, now that we have category theory and type theory. I expect to post about this in order to clarify my thinking about how to introduce categorical and type-theoretical ideas without writing a whole book about it.
Particular math structures
Abmath includes discussions
of the problems students have with certain particular types of structures. These sections talk mostly about how to think about these structure and some particular misunderstandings students have at the most basic levels.
These articles are certainly not proper introductions to the structures. Abmath in general is like that: It tells students about some aspects of math that are known to cause them trouble when they begin studying abstract math. And that is all it does.
- I expect to write similar articles about groups, spaces and categories.
- The idea about groups is to mention a few things that cause trouble at the very beginning, such as cosets, quotients and homomorphisms (which are all obviously related to each other), and perhaps other stumbling blocks.
- With categories the idea is to stomp on misconceptions such as that the arrows have to be functions and to emphasize the role of categories in allowing us to define math structures in terms of their relations with other objects instead of in terms with sets.
- I am going to have more trouble with spaces. Perhaps I will show how you can look at the $\epsilon$-$\delta$ definition of continuous functions on the reals and “discover” that they imply that inverse images of open sets are open, thus paving the way for the family-of-subsets definition of a topoogy.
- I am not ruling out other particular structures.
This chapter covers several aspects of proofs that cause trouble for students, the logical aspects and also the way proofs are written.
It specifically does not make use of any particular symbolic language for logic and proofs. Some math students are not good at learning languages, and I didn’t see any point in introducing a specific language just to do rudimentary discussions about proofs and logic. The second link below discusses linguistic ability in connection with algebra.
I taught logic notation as part of various courses to computer engineering students and was surprised to discover how difficult some students found using (for example) $p+q$ in one course and $p\vee q$ in another. Other students breezed through different notations with total insouciance.
- The abmath head article for proofs is Proofs. It links to the other articles on proofs.
- Mathematics and linguistic ability. Post in Gyre&Gimble.
Much of the chapter on proofs is badly written. When I get around to upgrading it to abmath 2.0 I intend to do a thorough rewrite, which I hope will inspire ideas about how to conceptually improve it.
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