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you either die a mathematician or live long enough to see yourself become a computer scientist

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bad news, euclid fans! i just disproved his assertation that parallel lines don't intersect for their entire infinite length. i walked along them for approximately 8.6 x 10⁸ km and you know what i found? that's right, a tangle.

they don't touch for the rest of their infinite length though, i did check. in both directions.

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Proof by you'd really think so wouldn't you?

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Kindergarten-level math research papers:

Field: Combinatorial Number Theory

Journal: Communications in Addition and Subtraction

Title: A Lower Bound on the Largest Natural number

Abbreviated Abstract: We prove a lower bound on the largest number. The proof proceeds in two steps: we begin with 1, and proceed by induction as until we lose count. We then add that number to itself. The main advance in the first step is to get a big number, and the second step notably avoids using multiplication (they don't teach that until 3rd grade).

Field: Topological Geometry

Journal: Advances in Nonlines

Title: The Four-Color Scribbles

Abbreviated Abstract: In this work we show a zoo of examples of nonlines (curves) with the unique property that they are either red, blue, orange, or purple, or some combination therein. The key idea is to use less colors rather than more, creating a clear and easy to follow proof. This provides a clear basis for simplifications to further work, such as scribbling with 5 colors.

Field: Playground Analysis

Journal: Slide Dynamics

Title: Sufficient Conditions to Yell Weeeeeee on Spiral Slides

Abbreviated Abstract: We identify sufficient conditions for a slide to cause joy. We identify a notion of a "fun slide," and prove that fun slides are a sufficient condition to make someone go Weeeee on a slide. We then verify a spiral slide is a fun slide, and provide numerous examples and non-examples (notably, a ramp is really not fun to slide down).

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A circle inscribed in a square covers 79% of the square.

A ball inscribed in a cube fills 52% of the cube.

A 4-ball inscribed in a hypercube fills 31% of the hypercube.

A 5-ball inscribed in a 5-cube fills 16% of the 5-cube.

⋮

A 9-ball inscribed in a 9-cube fills it up less than 1% of the way, yet there’s no room to fit a second ball of the same size without intersecting the first.

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In dimensions 23 and up, you can fit a little cube in the corner of the diagram, such that the cube has a larger volume than the ball!

That's especially disconcerting because the 23-cube has 8,388,608 corners. Even if you inscribe the little red cubes in all eight million corners, your big cube will still be 99.5% empty space.

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please let me merge please please please please please please please please please please please please please please

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gradually forgetting a lot of early school knowledge over the years but never forgetting that in x > y the x is greater because the crocodile wants to eat the bigger number

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i appreciated this study: "They Can't Read Very Well: A Study of the Reading Comprehension Skills Of English Majors At Two Midwestern Universities"

essentially, a pair of professors set out to test their intuitive sense that students at the college level were struggling with complex text. they recruited 85 students, a mix of english majors and english education majors - so, theoretically, people focusing on literature, and people preparing to teach adolescents how to read literature - and had them read-while-summarizing the first seven paragraphs of dickens's bleak house (or as much as they made it through in the 20 minute session). they provided dictionaries and also said students could use their phones to look up whatever they wanted, including any unfamiliar words or references. they found that the majority of the students - 58%, or 49 out of the 85 students - functionally could not understand dickens at all, and only 5% - a mere 4 out of the 85 students - proved themselves proficient readers (leaving the remaining 38%, or 32 students, as what the study authors deemed "competent" students, most of whom could understand about half the literal meaning - pretty low bar for competence - although a few of whom, they note, did much better than the rest in this group if not quite well enough to be considered proficient).

what i really appreciated about this study was its qualitative descriptions of the challenges and reading behaviors of what the authors call "problematic readers" (that bottom 58%), which resonated strongly with my own experiences of students who struggle with reading. here's their blunt big picture overview of these 49 students:

The majority of these subjects could understand very little of Bleak House and did not have effective reading tactics. All had so much trouble comprehending concrete detail in consecutive clauses and phrases that they could not link the meaning of one sentence to the next. Although it was clear that these subjects did try to use various tactics while they read the passage, they were not able to use those tactics successfully. For example, 43 percent of the problematic readers tried to look up words they did not understand, but only five percent were able to look up the meaning of a word and place it back correctly into a sentence. The subjects frequently looked up a word they did not know, realized that they did not understand the sentence the word had come from, and skipped translating the sentence altogether.

the idea that they had so many trouble with every small piece of a text that they could not connect ideas on a sentence by sentence basis is very familiar to me from teaching and tutoring, as was the habit of thought seen in the example of the student who gloms on to the word "whiskers" in a sea of confusion and guesses incorrectly that a cat is present - struggling readers, in my experience, seem to use familiar nouns as stepping stones in a flood of overwhelm, hopping as best they can from one seemingly familiar image to the next. so was this observation, building off the example of a student who misses the fact that dickens is being figurative when he imagines a megalodon stalking the streets of london:

She first guesses that the dinosaur is just “bones” and then is stuck stating that the bones are “waddling, um, all up the hill” because she can see that Dickens has the dinosaur moving. Because she cannot logically tie the ideas together, she just leaves her interpretation as is and goes on to the next sentence. Like this subject, most of the problematic readers were not concerned if their literal translations of Bleak House were not coherent, so obvious logical errors never seemed to affect them. In fact, none of the readers in this category ever questioned their own interpretations of figures of speech, no matter how irrational the results. Worse, their inability to understand figurative language was constant, even though most of the subjects had spent at least two years in literature classes that discussed figures of speech. Some could correctly identify a figure of speech, and even explain its use in a sentence, but correct responses were inconsistent and haphazard. None of the problematic readers showed any evidence that they could read recursively or fix previous errors in comprehension. They would stick to their reading tactics even if they were unhappy with the results.

i have seen this repeatedly, too - actually i was particularly taken with how similar this is to the behavior of struggling readers at much younger ages - and would summarize the hypothesis i have forged over time as: struggling readers do not expect what they read to make sense. my hypothesis for why this is the case is that their reading deficits were not attended to or remediated adequately early enough, and so, in their formative years - the early to mid elementary grades - they spent a lot of time "reading" things that did not make sense to them - in fact they spent much more time doing this than they ever did reading things that did make sense to them - and so they did not internalize a meaningful subjective sense of what it feels like to actually read things.

like, i've said this before, but the year i taught third grade i had multiple students who told me they loved reading and then when i asked them about a book they were reading revealed that they had absolutely no idea what was going on - on a really basic literal level like "didn't know who said which lines of dialogue" and "couldn't identify which things or characters given pronouns referred to" - and were as best as i could tell sort of constructing their own story along the way using these little bits of things they thought they understood. that's what "reading" was, in their heads. and they were, in the curriculum/model that we used at the private school where i taught, receiving basically no support to clarify that that was not what reading was, nor any instruction that would actually help them with what they needed to do to improve (understand sentences) - and i realized over the course of that year that the master's program that had certified me in teaching elementary school had provided me with very little understanding of how to help these kids (with perhaps the sole exception of the class i took on communications disorders, not because these kids had communications disorders but because that was the only class where we ever talked, even briefly, about things like sentence structures that students may need instruction in and practice with to comprehend independently). when it comes to the literal, basic understanding of a text, the model of reading pedagogy i was taught has about 6 million little "tools" that all boil down to telling kids who functionally can't read to try harder to read. this is not productive, in my experience and opinion, for kids whose maximum effort persistently yields confusion. but things are so dysfunctional all the way up and down the ladder that you can be a senior in college majoring in english without anyone but a pair of professors with a strong work ethic noticing that you can't actually read.

couple other notes:

obviously it's a small study but i'm not sure i see a reason to believe these are particularly outlierish results (ACT scores - an imperfect metric but not a meritless one IMO for reading specifically, where the task mostly really is to read a set of texts written for the educated layperson and answer factual questions about them - were a little bit above the national average)

the study was published last year, but the research was conducted january to april 2015. so there's no pandemic influence, no AI issue - these are millennials who now would span roughly ages 28-32 (i guess it's possible one of the four first-year students was one of the very first members of gen z lol). if you're in your late 20s or early 30s, we are talking about people your age, and whatever the culprit is here, it was happening when you were in school.

i think some people might want to blame this on NCLB but i find this unconvincing for a variety of reasons. first of all, NCLB did not pass because everyone in 2001 agreed that education was super hunky-dory; in fact, the sold a story podcast outlines how an explicit goal of NCLB was to train teachers in systematic phonics instruction, because that was not the norm when NCLB was passed, and an unfortunate outcome was that phonics became politicized in ed world. second, anyone who understands anything about reading should need about ten minutes max to spend some time on standardized test prep and recognize that if your goal is truly to maximize scores... then the vast majority of your instructional time should be spent on improving actual reading skills because you actually can't meaningfully game these tests by "practicing main idea questions" (timothy shanahan addresses this briefly near the top of this post). so i find it very difficult to believe that any school that pivoted to multiple choice drill time in an attempt to boost reading scores was teaching reading effectively pre-NCLB, because no set of competent literacy professionals would think that would work even for the goal of raising test scores. third, NCLB mandated yearly testing in grades 3-8 but only one test year in high school; kansas set its reading and math test year in high school as tenth grade. so theoretically these kids all had two years of sweet sweet freedom from NCLB in which their teachers could have done whatever the fuck they wanted to teach these kids to actually read. the fact that they didn't suggests perhaps there were other problems afoot. fourth, and maybe most saliently for this particular study, the sample text was the first seven paragraphs of a novel - in other words, the exact kind of short incomplete text that NCLB allegedly demanded excessive time spent on. i'm not really sure what universe it makes sense in that students who can't read the first seven paragraphs of a novel would have become much better reader if everything else had been the same but they had been making completely wack associations based on nonsense guesses for all 300 pages instead. (if you read the study it's really clear that for problematic readers, things go off the rails immediately, in a way that a good program targeted at teaching mastery of text of 500 words or less would have done something about.)

all but 3 of the students reported A's and B's in their english classes and, again, 69% of them are juniors and seniors, so like... i mean idk kudos to these professors for being like "hold up can these kids actually read?" but clearly something is wack at the college level too [in 2015] if you can make your way through nearly an entire english major without being able to read the first seven paragraphs of a dickens novel. (once again i really do encourage you to look at the qualitative samples in the study, lest you think i am being uncharitable by summarizing understandable misunderstandings or areas of confusion that may resolve themselves with further exposure to the text as "can't read.") not to mention the fact that most students could not what they had learned in previous or current english classes and when asked to name british and american authors and/or works of the nineteenth century, roughly half the sample at each college could name at most one.

the authors of the study are struck by the fact that students who cannot parse the first 3 sentences of bleak house feel very confident about their ability to read the entire novel, and discover that this seeming disconnect is resolved by the fact that these students seem to conceptualize "reading" as "skimming and then reading sparknotes." i think it's really tempting to Kids These Days this phenomenon (although again these are people who in some cases have now been in the workforce for a decade) and categorize it as laziness or a lack of effort, but i think that there is, as i described above, a real and sincere confusion over what "reading" is in which this makes a certain logical sense because it's not like they have some store of actual reading experiences to compare it to. i also think it's pretty obvious looking at just how wildly severed from actual textual comprehension their readings are that these are not - or at least not entirely - students who could just work harder and master the entirety of bleak house all on their own. like i don't think you get from "charles dickens is describing a bunch of dinosaur bones actually walking the streets of london" to comfortably reading nineteenth century literature by just trying harder. i really just don't (and i say that acknowledging i personally have had students who like... were good readers if i was forcing them to work at it constantly... but i have also had students, including ones getting ready to enter college, who were clearly giving me everything they had and what they had was at the present moment insufficient). i think that speaks to a missing skillset that they don't know are missing, because they don't have any other experience of "reading" to compare it to.

just wanna highlight again that although they don't give the breakdown some of these students are not just english majors but english education majors a.k.a. the high school english teachers of tomorrow. some of them may be teaching high school english right now, in case anyone wishes to consider whether "maybe some high school english teachers can't read the first seven paragraphs of bleak house?" should be kept in mind when we discuss present-day educational ills.

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a is the opposite of z and m is the opposite of n, so you may be tempted to assume that the opposite of the nth letter is the (26-n)th letter, but the pattern doesn't hold and there are actually more exceptions than examples. for example b is the opposite of d and x is the opposite of o. h, j, k, p, q, and r seem to have some kind of oppositeness relationships to each other except that r is obviously the opposite of l. further research is needed

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I think math teacher don't focus enough on history. Like, yeah, Abel results on the convergence of series are great and all, but did you know Cauchy hated Abel and is indirectly responsible for his death ?

Galois was a math wizard and died in a duel. But did you know he also was kind of a leftist, got thrown out of École Normale Supérieure because of that, joked about killing Louis Philippe and probably died because of that ?

Gauss is cool, but did you know he was saved by Sophie Germain, discovered at the same time Sophie Germain was a woman, and wrote one of the coolest letter blaming "our customs and prejudices" for not allowing women in math ?

Urysohn's lemma is cool, but did you did you know he tragically died while swimming with Alexandrov ? Oh, also they were both gay and in love. And later Alexandrov had a relation with Kolmogorov

There's tons of other stories, even minor ones (like how Italian mathematicians argued against some notations in vector calculus because they were used by Germans). Focusing only on how "abstract" and "pure" math is, is pedantic. Mathematicians, on the other side, are way funnier and not just lost in their abstract and complicated world

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Louisiana students Ne’Kiya Jackson and Calcea Johnson wowed their teachers in 2022 when they discovered a new way to prove the 2000-year-old Pythagorean theorem in response to a bonus question in a high school math contest. But that was only the beginning.

A volunteer at their former school, New Orleans’ St. Mary’s Academy, encouraged them to submit their work on the famous mathematical theory to a professional conference, and in March 2023 they became the youngest people to present at the American Mathematical Society’s Southeastern Sectional conference in Atlanta. Their appearance elicited a wave of media coverage, including a spot on “60 Minutes.” The pair also received symbolic keys to the city of New Orleans and a shout-out from Michelle Obama.

Now Jackson and Johnson, who started college last year, have notched another achievement: authoring an academic paper detailing their original proof — plus nine more. Their work published Monday in the scientific journal American Mathematical Monthly.

Bonus:

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i love knowing exactly enough graph theory to be able to tell if a stupid "go through every door once" puzzle is impossible