Some of you may be very familiar with the Roman art of rhetoric, which included public speech making or oratory. We still use some of these same techniques today. Plus the Romans probably inherited their teachings from yet older sources. The tradition is called "hermetic".
What techniques am I talking about? The "memory palace" for one. In order to keep your knowledge organized, learn to visualize a building, perhaps a public one you'll be able to visit for real, or perhaps an imaginary one. Store mental icons in the different rooms (like pigeon holes) to remind yourself of various topics, and what links to what. When you have a speech to deliver, scatter the icons in sequential order and mentally tour the rooms: your speech will sound more coherent and better prepared.
Schoolish math comes with some simple mnemonics such as "a red indian thought he might eat tobacco in church" (of course he did, given the sacredness of tobacco) -- a way to remember "arithmetic". Not politically correct, but what I learned in a British school in the 1960s. I'm sure you will think of others. The Web is packed with such memory tricks.
However, more important than these simple rules of thumb are mnemonic structures of a more general sort. We speak metaphorically, yet also concretely, about trees and networks. An outline might be in tree form, with headings, subheadings, and sub- subheadings. The "document object model" (DOM) is such a nested tree structure, of nodes within nodes.
Or consider a VRML file, or a POV-Ray file in scene description language.... we're talking about structure and grammar, as much as we're talking about mathematics.
When it comes to networks, also think about polyhedrons, their nodes as web pages, with hyperlinks for edges. Exercise: create four web pages that all link to each other bidirectionally along six edges. Consider that a "mnenomic structure" of sorts (or semantic web). The concepts of "web" "network" and "graph" all share a lot in common. A public building of linked rooms is early hypertext, especially if said building is mental, i.e. is a memory palace.
In getting students to visualize the interconnectedness of ideas in pure principle, as networks or even polyhedrons, we're getting them to think about knowledge as a terrain. Lou Talman uses a similar metaphor, of trails criss- crossing every which way. There's not a single "right way" to traverse this terrain. However, one's sense of how it all connects is reinforced as one keeps returning to the same "nodes" time after time, from different directions. Pascal's Triangle is like a Grand Central Station. Here we are again, fresh from "tetrahedral numbers" or "combinatorics" or "binomial theorem"...
As we train up our students on the humanities side, we will be using math texts as objects of literary criticism. Rather than be intimidated or made to feel inferior by the cryptic notations therein, we will apply our tools of the trade to rank them according to various criteria. We will think more like publishers. Why did this work get published and what interests did it serve? What was the agenda of this curriculum? Why was this topic left out, why was this one included?
We don't leave such discussion to math teachers. On the contrary, the process of learning a math is likewise the process of investigating its location in whatever phase spaces.
Why was Hermann Grassmann's universal algebra considered ground-breaking? How did Cantor's work go over at first? Are quaternions making a come-back, thanks to game engines?
These are legitimate essay questions and would take mathematical research, as well as historical research, to answer. And yes, go ahead and use the free Web as a resource. The free Web contains many excellent reference materials, literary criticism, non-fictive accounts, that you will not find anywhere else.
Other recent writings by this author on math-teach: