in reply to Cesar A. Hidalgo at https://x.com/realAnthonyDean/status/1844409919161684366

via Anthony Dean @realAnthonyDean

Willard, Stephen. General Topology. Addison-Wesley Series in Mathematics. Addison-Wesley Publishing Company, Inc., 1970.

Annotation url: urn:x-pdf:eb874067e68a720ea298d18714fcd41e

Alternate annotations at: https://jonudell.info/h/facet/?user=chrisaldrich&max=100&exactTagSearch=true&expanded=true&url=urn%3Ax-pdf%3Aeb874067e68a720ea298d18714fcd41e

Gemignani, Michael C. Elementary Topology. 2nd ed. Addison-Wesley Series in Mathematics. Reading, MA: Addison-Wesley Publishing Company, 1971.

Annotations URL<br /> Alternate annotations URL

It Only Takes Two Weeks by [[The Math Sorcerer]]

Within a particular class versus their peers, a dedicated student can usually catch up to the best students in 2 weeks.

Undergraduate Topology by Kasriel by [[The Internet Sorcerer]]

Undergraduate Topology by Robert Herman Kasriel

Mary E. Rudin: "Set theory and General Topology" by [[UM-Milwaukee Department of Mathematical Sciences]]

Fancy Topology Book by [[The Math Sorcerer]]

1974 text on topology by Murray Eisenberg

Topology Without Tears by [[Sidney A. Morris]]

The Best Topology Book For Beginners is Free by [[The Math Sorcerer]]

Topology Without Tears by Sidney A. Morris<br /> https://www.topologywithouttears.net/

Topology: A Short Introduction by [[badbettybooks]]

only the most surface level review here from the perspective of a relatively unsophisticated undergraduate

Best Books for Learning Topology by [[The Math Sorcerer]]

Some heavy math shaming at the start of this video... ugh.

Maybe a democracy can work, if it's voting system works well enough.

Tao, Terence. “What Is Good Mathematics?,” February 13, 2007. http://arxiv.org/abs/math/0702396.

Variations of this can also be applied to other fields, like history. What makes good history, good historians, good history teachers, etc.?

https://www.jonmsterling.com/index.xml

ᔥRichard in A forest of evergreen notes at 2024-06-02<br /> (accessed:: 2024-06-07 11:14:53)

A forest of evergreen notes by Richard

See also comments at cross-posting at https://www.reddit.com/r/Zettelkasten/comments/1d69inw/a_forest_of_evergreen_notes/

reply to u/scihuy at https://www.reddit.com/r/Zettelkasten/comments/1c2b2d6/note_taking_in_the_past/kzcg3qa/

I posed your question to my own card index:

Generally scientists haven't spent the time to talk about their methods the way those in the social sciences and humanities are apt to do. This being said, their methods are unsurprisingly all the same.

If you want to look up examples, you can delve into the nachlass (digitized or not) of most of the famous scientists and mathematicians out there to verify this. Ramon Llull certainly wrote, but broadly memorized all of his work; Newton had his wastebooks; Leibnitz used Thomas Harrison's Ark of Studies cabinet; Carl Linnaeus "invented" index cards for his work (search for the work of Staffan Müller-Wille and Isabelle Charmantier); Erasmus Darwin and Charles Darwin both used commonplace books; physicist Mario Bunge had a significant zettelkasten practice; Richard Feynman used notebooks; engineer Ross Ashby used a combination of notebooks which he indexed using a card index.

For historical reasons, most used a commonplace book method in which they indexed against keywords rather than Luhmann's variation, but broadly the results are the same either way.

Computer scientist Gerald Weinberg is one of the few I'm aware of within the sciences who's written a note taking manual, but again, his method is broadly the same as that described by other writers for centuries:

Weinberg, Gerald M. Weinberg on Writing: The Fieldstone Method. New York, N.Y: Dorset House, 2005.

I identify as both a mathematician and an engineer, and I have a paper-based zettelkasten for these areas, primarily as I prefer writing out equations versus attempting to write everything out as LaTeX. I'm sure others here could add their experiences as well. I've previously written about zettelkasten from the framing of set theory, topology, dense sets, and have even touched on it with respect to the ideas of equivalence classes and category theory, though I haven't published much in depth here as most don't have the mathematical sophistication to appreciate the structures and analogies.

Ideally, when classifying, one should draw up classes which are mutually exclusive so as to better differentiate the items under consideration. This is not always possible and there are some cases where having overlap my be beneficial in one's work.

He here is A. J. Ellis

https://www.mccoys-kecatalogs.com/SpecialSubjects/Zlotnian_Calculator/zlotnian.htm

https://www.mccoys-kecatalogs.com/SpecialSubjects/Fingers_Calculator/fingers_calculator.htm

Kells, Lyman M., Willis F. Kern, and James R. Bland. K+E Slide Rules: A Self Instruction Manual. New York: Keuffel & Esser Co., 1955.

https://mastodon.social/@christianp@mathstodon.xyz/111759984202211741

Is there a way to mathematically encode colors, similar to RGB perhaps, such that the colors in nearby neighborhoods all have values close to each other?

Silverman, Joseph H., and John T. Tate. Rational Points on Elliptic Curves. 2nd ed. Undergraduate Texts in Mathematics. 1992. Reprint, New York: Springer, 2015.

urn:x-pdf:e59b1cd55350541e359fbb11896edd0c

annotations at: https://jonudell.info/h/facet/?user=chrisaldrich&max=100&exactTagSearch=true&expanded=true&url=urn%3Ax-pdf%3Ae59b1cd55350541e359fbb11896edd0c

https://handwiki.org/wiki/Termial

How to fold and cut a Christmas star<br /> Christian Lawson-Perfect https://www.youtube.com/watch?v=S90WPkgxvas

What a great simple example with some interesting complexity.

For teachers trying this with students, when one is done making some five pointed stars, the next questions a curious mathematician might ask are: how might I generalize this new knowledge to make a 6 pointed star? A 7 pointed star? a 1,729 pointed star? Is there a maximum number of points possible? Is there a minimum? Can any star be made without a cut? What happens if we make more than one cut? Are there certain numbers for which a star can't be made? Is there a relationship between the number of folds made and the number of points? What does all this have to do with our basic definition of what a paper star might look like? What other questions might we ask to extend this little idea of cutting paper stars?

Recalling some results from my third grade origami days, based on the thickness of most standard office paper, a typical sheet of paper can only be folded in half at most 7 times. This number can go up a bit if the thickness of the paper is reduced, but having a maximum number of potential folds suggests there is an upper bound for how many points a star might have using this method of construction.

Professor Richard Schwartz proves 50-year-old Möbius Strip conjecture in preprint paper

Paper: https://arxiv.org/pdf/2308.12641.pdf

reply to u/Apprehensive_Net5630 and u/marco89lcdm at https://www.reddit.com/r/antinet/comments/17m7ggz/comment/k83bou9/?utm_source=reddit&utm_medium=web2x&context=3

Don't sweat the difference as there is a one-to-one and onto (or bijective) relationship between what you're doing and what u/Apprehensive_Net5630 suggests. Mathematicians would call the relationship homomorphic (ie: of the same shape), so other than the make-work exercise, you'd end up with the same exact thing with the same ordering in the end.

reply to u/acosmicjoke at https://www.reddit.com/r/Zettelkasten/comments/17bqztm/applying_zettelkasten_for_math_heavy_subjects/

Most of my math section in my ZK is primarily very basic definitions and theorems. I have very few proofs of basic things outside of my own personal work. There is an occasional useful example or two or lists of various lists of things that fit certain structures (lists of groups, rings, fields, categories, etc.)

Digital notes just don't work for me at all with respect to math, so I'm all-in on index cards and simply typeset all the necessary parts when I'm done with something if I intend to publish or share with others. Paper also makes it much easier to shuffle things around and reshape pieces as necessary to try out different structural approaches.

I also have a short stack of method cards (not dissimilar to Eno & Schmidt's "Oblique Strategies", but with a mathematical bent) to remind me of different approaches to try out when I get stuck which has been pretty beneficial.

I also take a fairly segmented approach between the writing I do for understanding a math text as I'm reading it and the permanent sort of notes I specifically make after-the-fact. My goal is never to recreate entire textbooks within the main section of my own zettelkasten, but create material for new works I might be writing for others to read.

Depending on what level a writer stipulates their usage, they may come to some drastically bad conclusions. One should watch out for these sorts of biases.

Compare with the results of accepting certain axioms within mathematics and how that changes/shifts one's framework of truth.

• for: unreasonable effectiveness of mathematics, emptiness, emptiness - unreasonable effectiveness of mathematics

• comment

  • everything is an expression of emptiness

https://math.ucr.edu/home/baez/act_course/

https://www.uclaextension.edu/sciences-math/math-statistics/course/topics-ring-theory-and-modules-math-900

category theory + zettelkasten... hmmm... feels a bit like Leibniz chasing a universal language

what is a zettelkasten monad?

Might we coin "zettelkastenly" as a function of (box) composed with (manually+cerebrally)?

hand:manually::brain:cerebrally::box:zettelkastenly

I would use the LaTeX \boxtimes symbol for this composition I think....

I love the idea here of analogizing the abstract nature of category theory in math with the abstract nature of metaphor in writing!

Good job Dale Keiger!

https://hub.jhu.edu/magazine/2021/winter/emily-riehl-category-theory/

original source?

creative mathematics versus computational or calculational mathematics...

we need more of the creative in early education

partial quote from Emily Riehl

How does one concretely define "inside" and "outside"? This definition is part of the missing space between the intuition and the mathematical proof.

Too many non-mathematicians view numbers as solid, concrete things which are meant to make definite sense and quite often their only experience with it is just that. Add two numbers up and always get the same thing. Calculate something in physics with an equation and get an exact, "true" answer. But somehow to be an actual mathematician, one must not see it as a "solid area" (using these words in their non-mathematical senses), but a wholly abstract field of abstraction built upon abstraction. While each abstraction has a sense of "trueness", it will need to be abstracted over and over while still maintaining that sense of "trueness". For many, this is close to being impossible because of the sense of solidity and gravity given to early mathematics.

How can we add more exploration for younger students?

Sardarli, Arzu, and Ida Swan. Cree Dictionary of Mathematical Terms with Visual Examples. University of Regina, 2022. https://opentextbooks.uregina.ca/creemathdictionary/.

via Tagged Cooltech: Cree Dictionary of Mathematical Terms with Visual Examples by Adam Levine

https://kleinbottle.com/

https://www.amazon.com/Vaughn-Cube-Multiplication-Petersons/dp/0768941776

Dean Vaughn has apparently renamed the method of loci as a commodity to be able to market a method for memorizing the multiplication tables.

Trying to follow an argument given here: https://youtu.be/mFZs7uGwNBo?t=3413

The sequence A002267 is claimed by Haris Neophytou to be the 1st 15 "super singular prime numbers" (ie, the primes that divides the order of the Monster Group). The order is the number of elements in the group.

Note that the last 3 elements [47, 59, 71] multiply to give the number of dimensions in which the Monster group exists: 196,883.

Neophytou believes A002267 gives a different way of looking at the monster group \(M\).

Around 1:02:45, Neophytou says he'll start from A002822...

(a list of numbers, \(m\text{,}\) such that \(6m - 1\) and \(6m + 1\) are twin primes)

... and construct "the minimal order of the monster" (what?)

To Solve the Rubik’s Cube, You Have to Understand the Amazing Math Inside<br /> by Dave Linkletter

suggested by Matt Maldre's annotations

We've got solutions for the number of configurations there are to solve a Rubic's cube with from 1 move up to 15, but we don't know how many cube configurations there are that can be solved with 16-20 moves.

• Example: the number of positions that require exactly one move solve them is 18, which is counted by multiplying the six faces and each of the three ways they can be twisted.

AMS Open Math Notes

Resources and inspiration for math instruction and learning

Welcome to AMS Open Math Notes, a repository of freely downloadable mathematical works hosted by the American Mathematical Society as a service to researchers, faculty and students. Open Math Notes includes: - Draft works including course notes, textbooks, and research expositions. These have not been published elsewhere and are subject to revision. - Items previously published in the Journal of Inquiry-Based Learning in Mathematics, a refereed journal - Refereed publications at the AMS

Visitors are encouraged to download and use any of these materials as teaching and research aids, and to send constructive comments and suggestions to the authors.

What tensor products taught me about living my life<br /> by Cathy O'Neil aka mathbabe

Zettelkasten workflow<br /> by pqnelson

What was the specific change in mathematical teaching instituted by Whewell, Peacock, and Herschel in An Elementary Treatise on Mechanics (1819)?

https://math.stackexchange.com/questions/160039/contradiction-any-symbol-for

Woit, Peter. Quantum Theory, Groups and Representations: An Introduction. Revised and Expanded version [2022]. Springer, 2017. https://www.math.columbia.edu/~woit/QM/qmbook.pdf.

https://ncase.me/polygons/

An interesting interactive model for segregation here. See also https://www.bloomberg.com/news/articles/2014-12-10/an-immersive-game-shows-how-easily-segregation-arises-and-how-we-might-fix-it for press coverage.

reply to u/phil98f at https://www.reddit.com/r/Zettelkasten/comments/10dbza7/equations_and_formulas_in_mathematics_physics_and/

Perhaps you've not gotten far enough in your studies to see the pattern yet, but most advanced mathematics texts (Hungerford's Algebra or Rudin's Real Analysis for example) and many physics texts are written as if they were pre-numbered zettelkasten. Generally every definition, theorem, corollary, proposition, and lemma in the text will be written out succinctly with its own unique number and arranged in some sort of branching order. Most of these texts you could generally cut up and paste onto cards and have something zettelkasten-like without any additional work.

I generally follow this same pattern and usually separate proofs on cards behind their associated theorems (typically only for those I've written out myself). Individual equations can be numbered, but I rarely give an equation its own card and instead reference them by card number with a particular equation line in parenthesis when necessary. I can then cross reference definitions, theorems, etc. easily as I continue building things up. Over time you can eventually cross reference various branches of math, physics, chemistry, biology, etc. For example I've got a dozen different proofs and uses of Schwartz' inequality in six different branches of mathematics which goes toward showing how closely knit various disparate branches of math can be.

In most cases I might also suggest against too heavy a focus on the equations, but on what they may say/mean, and what you can use it for. Alternately drawing diagrams or pictures of the relationships can be valuable. Understand it first then write it down. As an example you can look at Boyle's Law, Charles' law, Avogadro's Law and Gay-Lussac's law, but if you know and understand them properly then you should be able to write down and understand the more generic Ideal Gas Law, which shows how they're interlinked. Later in your studies you might also then be able to derive it from microscopic kinetic theory with statistical mechanics as well, then you'll be able to link up those concepts at that time.

Dramatically important in mathematics, but also in every other area.

The idea of place value is thought to have been a Sumerian invention (d'Errico et al., 2017), but the example of <Y> in the work of B. Bacon, et al (2023) may push the date of the idea of place value back significantly.

McCoy, Neal Henry. The Theory of Rings. 1964. Reprint, The Bronx, New York: Chelsea Publishing Company, 1973.

Consider that: 1. Sine and cosine are orthogonal to each other 2. Hence, you can rewrite -sin(Θ) = cos(Θ + π/2) cos(Θ) = sin(Θ + π/2) 3. Therefore, the angle between the standard basis vectors, and their orientation, are preserved!

My internal mathematician wishes there was more substance in this particular portion.

• Use of \(LaTeX\)
• annotating the breakdown of logic in problems
• providing missing context
• filling in details of problems left as an exercise for the student
• others?

Introduction to Daniel Rosiak's spectacular "Sheaf Theory through Examples" available open access from MIT Direct Press: https://doi.org/10.7551/mitpress/12581.003.0003

https://www.reddit.com/r/Zettelkasten/comments/yg2g8l/a_thought_experiment_what_if_luhmann_had_been_a/

reply (unsent)<br /> I appreciate where you're coming from, and it's an excellent thought experiment. However, knowing that there was a clear older prior zettelkasten tradition for several hundred years prior to Luhmann which also included a number of mathematician practitioners including not only Leibnitz but also Newton, who incidentally invented his version of calculus in his waste book (also a part of that tradition). (See also: https://www.newtonproject.ox.ac.uk/texts/notebooks?sort=date&order=desc).

If you give a title to your notes, "claim notes" are simply notes with a verb.<br><br>They invite you to say: "Prove it!"<br><br>- "The positive impact of PKM" (not a claim)<br>- "PKM has a positive impact in improving writer's block" (claim)<br><br>A small change with positive mindset consequences

— Bianca Pereira | PKM Coach and Researcher (@bianca_oli_per) October 6, 2022

Bianca Pereira coins the ideas of "concept notes" versus "claim notes". Claim notes are framings similar to the theorem or claim portion of the mathematical framing of definition/theorem(claim)/proof. This set up provides the driving impetus of most of mathematics. One defines objects about which one then advances claims for which proofs are provided to make them theorems.

Framing one's notes as claims invites one to provide supporting proof for them to determine how strong they may or may not be. Otherwise, ideas may just state concepts which are far less interesting or active. What is one to do with them? They require more active work to advance or improve upon in more passive framings.

link to: - Maggie Delano's reading framing: https://hypothes.is/a/4xBvpE2TEe2ZmWfoCX_HyQ

Jungius worked for a long time on various calendar systems. In his letters one can see the problems it caused at the time that some areas followed the Gregorian calendar and others, including Hamburg, still followed the Julian.

Joachim Jungius worked on various calendar systems at a time when the Gregorian calendar was coming into use while his home city of Hamburg still followed the Julian calendar. Jungius used the Julian calendar as well, but generally only dated his notes with the month and the last two digits of the year and excluded the century.

https://www.uclaextension.edu/sciences-math/math-statistics/course/theory-and-applications-continued-fractions-math-x-45150

Mike Miller relayed to me on 2022-09-27 on the first meeting of his class Theory and Applications of Continued Fractions that he met a professor colleague walking on campus who mistakenly thought that he was teaching an 8th grade class on basic fractions for the UCLA math department.

Khinchin, Aleksandr Yakovlevich. Continued Fractions. 3rd ed. Chicago: University of Chicago Press, 1964.

https://www.science.org/doi/10.1126/science.145.3631.478.b

Originality has three dimensions: new sources, new arguments, and/or new audiences.

Originality has many facets including:<br /> - new arguments - new sources - new audiences

What other facets may exist? Think about communication theory to explore this.

I particularly like the new audiences aspect as there are dramatically different audiences for different pieces. (Eg: academic articles, newspapers, magazines, books, blog posts, social media, etc.) A plethora of audiences may be needed to reach the right audiences.

Compare this with Terry Tao's list of mathematical talents which includes communication.

Allosso points out some useful critiques of numbering systems, but doesn't seem to get to the two core ideas that underpin them (and let's be honest, most other sources don't either). As a result most of the controversies are based on a variety of opinions from users, many of whom don't have long enough term practices to see the potential value.

The important things about numbers (or even titles) within zettelkasten or even commonplace book systems is that they be unique to immediately and irrevocably identify ideas within a system.

The other important piece is that ideas be linked to at least one other idea, so they're less likely to get lost.

Once these are dealt with there's little other controversy to be had.

The issue with date/time-stamped numbering systems in digital contexts is that users make notes using them, but wholly fail to link them to anything much less one other idea within their system, thus creating orphaned ideas. (This is fine in the early days, but ultimately one should strive to have nothing orphaned).

The benefit of Luhmann's analog method was that by putting one idea behind its most closely related idea was that it immediately created that minimum of one link (to the thing it sits behind). It's only at this point once it's situated that it can be given it's unique number (and not before).

Luhmann's numbering system, similar to those seen in Viennese contexts for conscription numbers/house numbers and early library call numbers, allows one to infinitely add new ideas to a pre-existing set no matter how packed the collection may become. This idea is very similar to the idea of dense sets in mathematics settings in which one can get arbitrarily close to any member of a set.

link to: - https://hypothes.is/a/YMZ-hofbEeyvXyf1gjXZCg (Vienna library catalogue system) - https://hypothes.is/a/Jlnn3IfSEey_-3uboxHsOA (Vienna conscription numbers)

Within mathematical contexts one of the major factors often at play is the idea of abstraction: how can one use a basic idea and then abstract it to other situations to see what results.

The idea of abstraction in mathematics is analogous to analogy and metaphor in literature.

https://news.yahoo.com/mathematician-tiktok-gives-example-insane-192823997.html

A sad, but subtle bit of math shaming going on here. Worse it's indicating that math is hard for even the elite without providing proper context.

http://www.tricki.org/

archive copy: https://web.archive.org/web/20220326200254/http://www.tricki.org/

a Wiki-style site with a large store of useful mathematical problem-solving techniques.

https://www.youtube.com/watch?v=s6OmqXCsYt8

It's often said that, 'The hand is the visible part of our brain.'

see also: https://en.wikipedia.org/wiki/Soroban

S. Harris cartoon "I think you should be more explicit here in step two."

https://www.aeaweb.org/articles?id=10.1257/app.3.3.158

https://algebra.org/wp/

https://www.nytimes.com/2001/01/07/education/algebra-project-bob-moses-empowers-students.html

The five step philosophy of the Algebra Project: - physical experience - pictorial representation - people talk (explain it in your own words) - feature talk (put it into proper English) - symbolic representation

"people talk" within the Algebra project is an example of the Feynman technique at work

Link this to Sonke Ahrens' method for improving understanding. Are there research links to this within their work?

https://www.nytimes.com/2001/01/07/education/algebra-project-bob-moses-empowers-students.html

https://www.theescalanteprogram.org/

https://thecalculusproject.org/

Mathematical education programs that are attempting to decrease educational gaps: - Bob Moses' Algebra Project - Adrian Mims' Calculus Project - Jaime Escalante math program - Harlem Children's Zone

https://windowsontheory.org/2022/04/27/a-personal-faq-on-the-math-education-controversies/

In querying math professors at many universities, I've discovered that many feel as if they're spending all their time and energy preparing students in the sciences and engineering and very little of their time supporting students in the math department. If one left things up to them, then it's likely that STEM and CS would die on the vine.

<small><cite class='h-cite via'>ᔥ <span class='p-author h-card'>Boaz Barak</span> in Brian Conrad takes down the CMF – Windows On Theory (<time class='dt-published'>06/01/2022 21:35:06</time>)</cite></small>

https://scottaaronson.blog/?p=6389

Evidence for Benjamin Franklin encouraging the education of women in mathematics.

This same analogy also works in advanced mathematics courses where students are often learning dense and technical vocabulary and then moments later applying it directly to even more technical ideas and proofs.

How might this sort of solution from law school be applied to abstract mathematics?

Adam Kucharski. (2020, December 13). I’ve turned down a lot of COVID-related interviews/events this year because topic was outside my main expertise and/or I thought there were others who were better placed to comment. Science communication isn’t just about what you take part in – it’s also about what you decline. [Tweet]. @AdamJKucharski. https://twitter.com/AdamJKucharski/status/1338079300097077250

https://www.sfchronicle.com/bayarea/article/California-math-wars-get-ugly-Accusations-of-17060072.php

clickbait mostly

https://www.linkedin.com/pulse/incorrect-use-information-theory-rafael-garc%C3%ADa/

A fascinating little problem. The bigger question is how can one abstract this problem into a more general theory?

How many questions can one ask? How many groups could things be broken up into? What is the effect on the number of objects?

Semasiography are iconic and abstract symbols and languages not based on spoken words, but which carry information.

Mathematical formulas, musical notation, computer icons, emoji, buttons on washing machines, and quipu are considered semasiographic systems which communicate information without speech as an intermediary.

semasiography from - Greek: σημασία (semasia) "signification, meaning" - Greek: γραφία (graphia) "writing") is "writing with signs"

Math teacher Brendan Jeffreys from the Auburn school district in Auburn, WA created simple hand gestures to accompany or replace mathematical terms. Examples included making a Pac-Man shape with one's arms to describe an acute angle, spreading one's arms wide as if to hug someone to indicate an obtuse angle, or flexing your arm to show your muscles to indicate a right triangle. Other examples included a karate chop to indicate division or a motion imitating using a knife to butter bread to indicate finding the area of a shape.

Mathematics can often be more difficult compared to other subjects as students learning new concepts are forced not only to understand entirely new concepts, but simultaneously are required to know new vocabulary to describe those concepts. Utilizing gestures to help lighten the cognitive load of the new vocabulary to allow students to focus on the concepts and operations can be invaluable.

The value of gesture is sometimes disparaged with the phrase "handy waving".

Some of this statement is misleading as a hand waving argument relies solely on the movement of the hands as "proof" of something which is neither communicated well with the use of either words or the physical hand movements. The communication fails on both axes, but the blame is placed on the gestural portion of the communication, perhaps because it may have been the more important of the two?

Link this to the example of the Riverside teacher who used both verbal and visual gestures and acting to cement the trigonometry ideas of soh, cah, toa to her students and got fired for it. In her example, the gauche behaviour was overamplified by extreme exaggeration as well as racist expression.

https://thinkzone.wlonk.com/Numbers/NumberSets.htm

A relatively clear explanation of the types of numbers and their properties.

I particularly like this diagram:

https://linear.axler.net/LADRvideos.html

I've seen Gilbert Strang's excellent lecture series. I'm curious how these rank in relation.

https://www.theguardian.com/us-news/2015/oct/17/berkeley-math-professor-alexander-coward-campus-battle

A mathematics lecturer at UC Berkeley went against the grain and got fired for it. I'm curious what his teaching secrets were.

In modern literate contexts, it is easier to establish doubletalk in oral contexts than it is in written contexts as the written is more easily reviewed for clarity and concreteness. Verbal ticks like "you know what I mean", "it's easy to see/show", and other versions of similar hand-waving arguments that indicate gaps in thinking and arguments are far easier to identify in writing than they are in speech where social pressure may cause the audience to agree without actually following the thread of the argument. Writing certainly allows for timeshiting, but it explicitly also expands time frames for grasping and understanding a full argument in a way not commonly seen in oral settings.

Note that this may not be the case in primarily oral cultures which may take specific steps to mitigate these patterns.

Link this to the anthropology example from Scott M. Lacy of the (Malian?) tribe that made group decisions by repeating a statement from the lowest to the highest and back again to ensure understanding and agreement.

This difference in communication between oral and literate is one which leaders can take advantage of in leading their followers astray. An example is Donald Trump who actively eschewed written communication or even reading in general in favor of oral and highly emotional speech. This generally freed him from the need to make coherent and useful arguments.

Successful problem solving requires having the ability to adaptively and flexibly focus one's attention with respect to the demands of the work. Having a toolbelt of potential methods and combinatorially working through them can be incredibly helpful and we too often forget to explicitly think about doing or how to do that.

This is particularly important in mathematics where students forget to look over at their toolbox of methods. What are the different means of proof? Some mathematicians will use direct proof during the day and indirect forms of proof at night. Look for examples and counter-examples. Why not look at a problem from disparate areas of mathematical thought? If topology isn't revealing any results, why not look at an algebraic or combinatoric approach?

How can you put a problem into a different context and leverage that to your benefit?