# Cognitive Mediums Some great links are here: * [Using spaced repetition systems to see through a piece of mathematics](https://cognitivemedium.com/srs-mathematics) * [Cognitive Medium](https://cognitivemedium.com/) * [Reinventing Explanation](https://michaelnielsen.org/reinventing_explanation/index.html) * [Toward an exploratory medium for mathematics](https://cognitivemedium.com/emm/emm.html) * [Augmenting Long-term Memory](http://augmentingcognition.com/ltm.html) > Typically, my mathematical work begins with paper-and-pen and messing about, often in a rather _ad hoc_ way. But over time if I really get into something my thinking starts to change. I gradually internalize the mathematical objects I’m dealing with. It becomes easier and easier to conduct (most of) my work in my head. I will go on long walks, and simply think intensively about the objects of concern. Those are no longer symbolic or verbal or visual in the conventional way, though they have some secondary aspects of this nature. Rather, the sense is somehow of working directly with the objects of concern, without any direct symbolic or verbal or visual referents. Furthermore, as my understanding of the objects change – as I learn more about their nature, and correct my own misconceptions – my sense of what I can do with the objects changes as well. It’s as though they sprout new affordances, in the language of user interface design, and I get much practice in learning to fluidly apply those affordances in multiple ways. This is a very difficult experience to describe in a way that I’m confident others will understand, but it really is central to my experience of mathematics – at least, of mathematics that I understand well. I must admit I’ve shared it with some trepidation; it seems to be rather unusual for someone to describe their inner mathematical experiences in these terms (or, more broadly, in the terms used in this essay). > In retrospect, I think that what’s going on is what psychologists call [chunking](http://augmentingcognition.com/assets/Simon1974.pdf). People who intensively study a subject gradually start to build mental libraries of “chunks” – large-scale patterns that they recognize and use to reason. This is why some grandmaster chess players can remember thousands of games move for move. They’re not remembering the individual moves – they’re remembering the ideas those games express, in terms of larger patterns. And they’ve studied chess so much that those ideas and patterns are deeply meaningful, much as the phrases in a lover’s letter may be meaningful. It’s why [top basketball players](https://www.youtube.com/watch?v=eNVJFRl6f6s) have extraordinary recall of games. Experts begin to think, perhaps only semi-consciously, using such chunks. The conventional representations – words or symbols in mathematics, or moves on a chessboard – are still there, but they are somehow secondary. > Now, the only way I’ve reliably found to get to this point is to get obsessed with some mathematical problem. I will start out thinking symbolically about the problem as I become familiar with the relevant ideas, but eventually I internalize those ideas and their patterns of use, and can carry out a lot (not all) of operations inside my head. --- Date: 20220507 Links to: Tags: References: * []()