Trainwreck: The Teach to One Math Experiment in Mountain View, CA Is a Cautionary Tale About the Perils of Digital Math Education

in Business, Math, Online Courses | January 6th, 2017

640px-trainwreckacw

Image via Wiki­me­dia Com­mons

I live in Sil­i­con Val­ley, which oper­ates on the assump­tion that there’s no prob­lem that tech­nol­o­gy can’t solve. It suf­fus­es our cul­ture here, and some­times we pay the price for this tech­no­crat­ic utopi­anism. Case in point: Right now, I’m send­ing my kid to a pub­lic school in Moun­tain View, CA–the home of Google–where the admin­is­tra­tors have upend­ed the entire sixth grade math pro­gram. Last August, they abol­ished the tra­di­tion­al math program–you know, where stu­dents get to sit in a class­room and learn from a trained and qual­i­fied math teacher. And instead the admin­is­tra­tors asked stu­dents to learn math main­ly from a com­put­er pro­gram called Teach to One. Run by a ven­ture called New Class­rooms, Teach to One promis­es to let each stu­dent engage in “per­son­al­ized learn­ing,” where a com­put­er pro­gram gauges each stu­den­t’s knowl­edge of math, then con­tin­u­al­ly cus­tomizes the math edu­ca­tion that stu­dents receive. It all sounds like a great con­cept. Bill Gates has sup­pos­ed­ly called it the “Future of Math Edu­ca­tion.” But the rub is this: Teach to One does­n’t seem ready for the present. And our kids are pay­ing the price.

A new arti­cle fea­tured in our local paper, The Moun­tain View Voice, out­lines well the prob­lems that stu­dents and par­ents have expe­ri­enced with the Teach to One pro­gram. I would encour­age any par­ent or edu­ca­tor inter­est­ed in the pit­falls of these “inno­v­a­tive” math pro­grams to give the arti­cle a good look. (Update: The Moun­tain View Voice has done a series of excel­lent arti­cles on the Teach to One exper­i­ment in Moun­tain View and all that went wrong. They’re all list­ed below.)

If you read the arti­cle, here’s what you will learn. The Moun­tain View school dis­trict appar­ent­ly bud­get­ed $521,000 to imple­ment and oper­ate this new-fan­gled math pro­gram in two local schools (Gra­ham and Crit­ten­den Mid­dle Schools). Had they ade­quate­ly beta test­ed the pro­gram before­hand, the school dis­trict might have dis­cov­ered that Teach to One teach­es math–we have observed–in a dis­joint­ed, non-lin­ear and often errat­ic fash­ion that leaves many stu­dents baf­fled and dis­en­chant­ed with math. The pro­gram con­tains errors in the math it teach­es. Par­ents end up hav­ing to teach kids math at home and make up for the pro­gram’s defi­cien­cies. And all the while, the math teach­ers get essen­tial­ly rel­e­gat­ed to “man­ag­ing the [Teach to One] pro­gram rather than to pro­vid­ing direct instruc­tion” them­selves.

By Octo­ber, many par­ents start­ed to reg­is­ter indi­vid­ual com­plaints with the school dis­trict. By Decem­ber, 180 par­ents signed a let­ter metic­u­lous­ly out­lin­ing the many prob­lems they found with the Teach to One pro­gram. (You can read that let­ter here.) When the school lat­er con­duct­ed a sur­vey on Teach to One (review it here), 61% of the par­ents “said they do not believe the pro­gram match­es the needs of their chil­dren,” and test scores show that this crop of sixth graders has mas­tered math con­cepts less well than last year’s. (Note: there was a big decrease in the num­ber of kids who say they love math, and con­verse­ly a 413% increase in the num­ber of kids who say they hate math.) Giv­en the mediocre eval­u­a­tion, the par­ents have asked for one sim­ple thing–the option to let their kids learn math in a tra­di­tion­al set­ting for the remain­der of the year, until it can be demon­strat­ed that Teach to One can deliv­er bet­ter results. (Teach to One would ide­al­ly con­tin­ue as a small­er pilot, where the kinks would get worked out.) So far the school dis­trict, head­ed by Ayindé Rudolph, has con­tin­ued to cham­pi­on the Teach to One pro­gram in fine­ly-spun bureau­crat­ic let­ters that effec­tive­ly dis­re­gard parental con­cerns and actu­al data points. But the schools have now agreed to let stu­dents spend 5o% of their time learn­ing math with Teach to One, and the oth­er 50% learn­ing math from a qual­i­fied teacher. Why the imprac­ti­cal half mea­sure? I can only spec­u­late.

I post­ed this so that inter­est­ed par­ents and edu­ca­tors, wher­ev­er you live, can be pru­dent and thought­ful when it comes to adopt­ing com­put­er-dri­ven math pro­grams. Per­haps you can learn some­thing from our cau­tion­ary tale. Do your research, run a con­trolled pilot, and make sure the prod­uct is actu­al­ly a good fit for your school. Again, I would encour­age you to read the fine arti­cle in The Moun­tain View Voice, the par­ents’ let­ter out­lin­ing the observed defi­cien­cies in the Teach to One pro­gram, and the eye-open­ing sur­vey results on Teach to One.

Update: It was announced on Jan­u­ary 12 that the Moun­tain View will dis­con­tin­ue the Teach to One math pilot effec­tive imme­di­ate­ly.  Patron­iz­ing­ly, New Class­rooms has attrib­uted the scrap­ping of the pilot to a com­mu­ni­ca­tion prob­lem. “There was a sub­set of par­ents of high­er-achiev­ing stu­dents who didn’t ful­ly under­stand how Teach to One oper­at­ed and how much it ben­e­fit­ed their chil­dren,” Joel Rose is quot­ed as say­ing in The Wall Street Jour­nal. Once again, I’d refer you back to the actu­al data col­lect­ed by our schools. It speaks for itself.

Great Arti­cles by The Moun­tain View Voice: Moun­tain View’s local paper has done some excel­lent report­ing on this fias­co. I would encour­age you to read them all.

This sto­ry has also received cov­er­age from The Wall Street Jour­nal and Edsurge. 

Update 2019: It sounds like anoth­er Teach-to-One pilot in Eliz­a­beth, NJ has its own issues. Read here and here.

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Infinity Minus Infinity Equals Pi: This Video Proves It

in Math | December 8th, 2016

It sounds impos­si­ble. But it turns out that infin­i­ty minus infin­i­ty does­n’t nec­es­sar­i­ly equal zero. It can equal Pi, or 3.14159265359. Or so demon­strates the “Math­ologer” in the video fea­tured above.

In real life the Math­ologer goes by the name of Burkard Pol­ster, and he’s a math pro­fes­sor at Monash Uni­ver­si­ty in Mel­bourne, Aus­tralia. You can check out more of his videos on YouTube here.

If you would like to sign up for Open Culture’s free email newslet­ter, please find it here. It’s a great way to see our new posts, all bun­dled in one email, each day.

If you would like to sup­port the mis­sion of Open Cul­ture, con­sid­er mak­ing a dona­tion to our site. It’s hard to rely 100% on ads, and your con­tri­bu­tions will help us con­tin­ue pro­vid­ing the best free cul­tur­al and edu­ca­tion­al mate­ri­als to learn­ers every­where. You can con­tribute through Pay­Pal, Patre­on, and Ven­mo (@openculture). Thanks!

Relat­ed Con­tent:

Free Online Math Cours­es 

Free Math Text­books

Cit­i­zen Maths: A Free Online Course That Teach­es Adults the Math They Missed in High School

The Math in Good Will Hunt­ing is Easy: How Do You Like Them Apples?

 

 

 

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The Essence of Linear Algebra Explained with Animations

in Animation, Math | September 15th, 2016

Fyi: Grant Sander­son has a knack for math and cod­ing. So he cre­at­ed a tool that has helped him explain “the essence of lin­ear alge­bra” in a “visu­al­ly-dri­ven man­ner.” And he post­ed the result, a series of 13 videos, to YouTube. You can watch the col­lec­tion, called “The Essence of Lin­ear Alge­bra,” above. Top­ics cov­ered include: Vec­tors, what even are they?; Matrix mul­ti­pli­ca­tion as com­po­si­tion; Dot prod­ucts and dual­i­ty; and more.

You can also find com­plete uni­ver­si­ty cours­es on Lin­ear Alge­bra in our col­lec­tion of Free Online Math course, a sub­set of our col­lec­tion, 1,700 Free Online Cours­es from Top Uni­ver­si­ties.

If you would like to sign up for Open Culture’s free email newslet­ter, please find it here. It’s a great way to see our new posts, all bun­dled in one email, each day.

If you would like to sup­port the mis­sion of Open Cul­ture, con­sid­er mak­ing a dona­tion to our site. It’s hard to rely 100% on ads, and your con­tri­bu­tions will help us con­tin­ue pro­vid­ing the best free cul­tur­al and edu­ca­tion­al mate­ri­als to learn­ers every­where. You can con­tribute through Pay­Pal, Patre­on, and Ven­mo (@openculture). Thanks!

via Metafil­ter

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Watch M.C. Escher Make His Final Artistic Creation (1971)

in Art, Math | August 31st, 2016

I first encoun­tered the world of Mau­rits Cor­nelis Esch­er where many oth­ers do: in school. A poster of his 22-foot-long Meta­mor­pho­sis III hung along the walls of my fourth-grade class­room, where I spent many an idle minute or ten star­ing at its intri­cate geom­e­try through which squares became birds, birds became lizards, lizards became fish, and it all some­how arrived at the cliff-like edge of a three-dimen­sion­al chess­board. It came as the last of a tril­o­gy of wood­cuts Esch­er made between 1937 and 1968, and a jour­ney through its 1940 pre­de­ces­sor Meta­mor­pho­sis II ends the 1971 doc­u­men­tary above, M.C. Esch­er: Adven­tures in Per­cep­tion.

Esch­er him­self seem­ing­ly had no hap­py class­room mem­o­ries. “I hat­ed school,” the nar­ra­tor quotes him as say­ing. “The only class I liked at all was art. That does­n’t mean I was any good at it.” Though his work has no doubt inspired many young­sters to take up draw­ing, wood­cut­ting, and print­mak­ing them­selves, it’s sure­ly dri­ven even more of them into math­e­mat­ics.

Obsessed with per­spec­tive, geom­e­try, and pat­tern (Esch­er described tes­sel­la­tion as “a real mania to which I have become addict­ed”), his images have, by the count of math­e­mati­cian and Esch­er schol­ar Doris Schattschnei­der, led so far to eleven sep­a­rate strands of math­e­mat­i­cal and sci­en­tif­ic research.

The twen­ty-minute Adven­tures in Per­cep­tion, orig­i­nal­ly com­mis­sioned by the Nether­lands’ Min­istry of For­eign Affairs, offers in its first half a med­i­ta­tion on the mes­mer­iz­ing, often impos­si­ble world Esch­er had cre­at­ed with his art to date. Its sec­ond half cap­tures Esch­er in the last years of his life, still at work in his Laren, North Hol­land stu­dio. It even shows him print­ing one of the three tit­u­lar ser­pents, thread­ed through a set of elab­o­rate­ly inter­lock­ing cir­cles, of his very last print Snakes. He nev­er actu­al­ly fin­ished Snakes, whose pat­terns would have con­tin­ued on to the effect of infin­i­ty, and even says here of his offi­cial­ly com­plete works that none suc­ceed, “because it’s the dream I tried for that can’t be real­ized.” But those unre­al­ized dreams have kept the rest of us dream­ing, and think­ing, ever since.

Adven­tures in Per­cep­tion will be added to our col­lec­tion of Free Doc­u­men­taries, a sub­set of our list, 4,000+ Free Movies Online: Great Clas­sics, Indies, Noir, West­erns, Doc­u­men­taries & More.

via Laugh­ing Squid

Relat­ed Con­tent:

Meta­mor­phose: 1999 Doc­u­men­tary Reveals the Life and Work of Artist M.C. Esch­er

Inspi­ra­tions: A Short Film Cel­e­brat­ing the Math­e­mat­i­cal Art of M.C. Esch­er

M.C. Escher’s Per­pet­u­al Motion Water­fall Brought to Life: Real or Sleight of Hand?

Back to Bed: A New Video Game Inspired by the Sur­re­al Art­work of Esch­er, Dali & Magritte

David Bowie Sings in a Won­der­ful M.C. Esch­er-Inspired Set in Jim Henson’s Labyrinth

Based in Seoul, Col­in Mar­shall writes and broad­casts on cities and cul­ture. He’s at work on a book about Los Ange­les, A Los Ange­les Primer, the video series The City in Cin­e­ma, the crowd­fund­ed jour­nal­ism project Where Is the City of the Future?, and the Los Ange­les Review of Books’ Korea Blog. Fol­low him on Twit­ter at @colinmarshall or on Face­book.

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Can You Solve These Animated Brain Teasers from TED-Ed?

in Animation, Education, Games, K-12, Math | July 8th, 2016

Zom­bies, alien over­lords, sharks, a mad dictator…math is a dan­ger­ous propo­si­tion in the hands of TED Ed script writer Alex Gendler.

The recre­ation­al math­e­mat­ics puz­zles he retro­fits for TED’s edu­ca­tion­al ini­tia­tive have been around for hun­dreds, even thou­sands of years. In the past, sto­ry­lines tend­ed to rely on bias­es 21st-cen­tu­ry puz­zle solvers would find objec­tion­able. As math­e­mati­cian David Singmas­ter told Sci­ence News:

One must be a lit­tle care­ful with some of these prob­lems, as past cul­tures were often bla­tant­ly sex­ist or racist. But such prob­lems also show what the cul­ture was like.… The riv­er cross­ing prob­lem of the jeal­ous hus­bands is quite sex­ist and trans­forms into mas­ters and ser­vants, which is clas­sist, then into mis­sion­ar­ies and can­ni­bals, which is racist. With such prob­lems, you can offend every­body!

Gendler’s updates, ani­mat­ed by Artrake stu­dio, derive their nar­ra­tive urgency from the sort of crowd pleas­ing sci fi predica­ments that fuel sum­mer block­busters.

And for­tu­nate­ly for those of us whose brains are per­ma­nent­ly stuck in beach mode, he nev­er fails to explain how the char­ac­ters pre­vail, out­wit­ting or out­run­ning the afore­men­tioned zom­bies, aliens, sharks, and mad dic­ta­tor.

(No wor­ries if you’re deter­mined to find the solu­tion on your own. Gendler gives plen­ty of fair warn­ing before each reveal.)

Put your brain in gear, pull the skull-embossed lever, and remem­ber, team­work — and induc­tive log­ic — car­ry the day!

The pris­on­er hat rid­dle, above, hinges on a hier­ar­chy of beliefs and the alien overlord’s will­ing­ness to give its nine cap­tives a few min­utes to come up with a game plan.

Go deep­er into this age old puz­zle by view­ing the full les­son.

Gendler’s spin on the green-eyed log­ic puz­zle, above, con­tains two brain teasers, one for the hive mind, and one for an indi­vid­ual act­ing alone, with a strat­e­gy culled from philoso­pher David Lewis’ Com­mon Knowl­edge play­book. Here’s the full les­son.

Rar­ing for more? You’ll find a playlist of TED-Ed puz­zles by Gendler and oth­ers here. The full les­son for the bridge prob­lem at the top of the post is here.

Relat­ed Con­tent:

Free Online Math Cours­es 

200 Free Text­books: A Meta Col­lec­tion 

Dan­ger­ous Knowl­edge: 4 Bril­liant Math­e­mati­cians & Their Drift to Insan­i­ty

Ayun Hal­l­i­day, author, illus­tra­tor, and Chief Pri­ma­tol­o­gist of the East Vil­lage Inky zine, will be lead­ing a free col­lab­o­ra­tive zine work­shop  at the Glue­stick Fest in Indi­anapo­lis Sat­ur­day, July 9. Fol­low her @AyunHalliday

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The Books on Young Alan Turing’s Reading List: From Lewis Carroll to Modern Chromatics

in History, Math, Physics | June 8th, 2016

turing book list

Image via Wiki­me­dia Com­mons

We now regard Alan Tur­ing, the trou­bled and ulti­mate­ly per­se­cut­ed crypt­an­a­lyst (and, intel­lec­tu­al­ly, much more besides)—who cracked the code of the Ger­man Enig­ma machine in World War II—as one of the great minds of his­to­ry. His life and work have drawn a good deal of seri­ous exam­i­na­tion since his ear­ly death in 1954, and recent­ly his lega­cy has even giv­en rise to pop­u­lar por­tray­als such as that by Bene­dict Cum­ber­batch in the film The Imi­ta­tion Game. So what, more and more of us have start­ed to won­der, forms a mind like Tur­ing’s in the first place?

A few years ago, math­e­mat­ics writer Alex Bel­los received, from “an old friend who teach­es at Sher­borne, the school Tur­ing attend­ed between 1928 and 1930,” some “new infor­ma­tion about the com­put­er pio­neer and codebreaker’s school years” in the form of “the list of books Tur­ing took out from the school library while he was a pupil.” Bel­los lists them as fol­lows:

“As you can see, and as you might expect,” writes Bel­los, “heavy on the sci­ences. The AJ Evans, a mem­oir about the author’s escape from impris­on­ment in the First World War, is the only non-sci­en­tif­ic book.” He also notes that “the physics books he took out all look very seri­ous, but the maths ones are light­heart­ed: the Lewis Car­roll and the Rouse Ball, which for decades was the clas­sic text in recre­ation­al maths prob­lems.” Sher­borne archivist Rachel Has­sall, who pro­vid­ed Bel­los with the list, also told him that “the book cho­sen by Tur­ing for his school prize was a copy of the Rouse Ball. Even teenage genius­es like to have fun.”

If you, too, would like to do a bit of the read­ing of a genius — or, depend­ing on how quan­ti­ta­tive­ly your own mind works, just have some fun — you can down­load for free most of these books the young Tur­ing checked out of the school library. Pro­gram­mer and writer John Gra­ham-Cum­ming orig­i­nal­ly found and orga­nized all the links to the texts on his blog; you can fol­low them there or from the list in this post. And if you know any young­sters in whom you see the poten­tial to achieve his­to­ry’s next Tur­ing-lev­el accom­plish­ment, send a few e‑books their way. Why read Har­ry Pot­ter, after all, when you can read A Selec­tion of Pho­tographs of Stars, Star-Clus­ters & Neb­u­lae, togeth­er with infor­ma­tion con­cern­ing the instru­ments & the meth­ods employed in the pur­suit of celes­tial pho­tog­ra­phy?

via Alex Bel­los

Relat­ed Con­tent:

The Enig­ma Machine: How Alan Tur­ing Helped Break the Unbreak­able Nazi Code

Alan Tur­ing, Bril­liant Math­e­mati­cian and Code Break­er, Will Be Final­ly Par­doned by British Gov­ern­ment

Bene­dict Cum­ber­batch Reads a Let­ter Alan Tur­ing Wrote in “Dis­tress” Before His Con­vic­tion For “Gross Inde­cen­cy”

Based in Seoul, Col­in Mar­shall writes and broad­casts on cities, lan­guage, and style. He’s at work on a book about Los Ange­les, A Los Ange­les Primer, the video series The City in Cin­e­ma, the crowd­fund­ed jour­nal­ism project Where Is the City of the Future?, and the Los Ange­les Review of Books’ Korea Blog. Fol­low him on Twit­ter at @colinmarshall or on Face­book.

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Can You Pass This Test Originally Given to 8th Graders Living in Kentucky in 1912?

in Education, History, K-12, Math, Science | May 20th, 2016

bcschoolexam1912sm--1-

Can you spell “con­ceive”?

Of course you can! All it takes is a device with a built-in spelling app, an inno­va­tion of which no eighth grad­er in the far west­ern reach­es of blue­grass area Ken­tucky could have con­ceived back in 1912.

They were, how­ev­er, expect­ed to be able to name the waters though which an Eng­lish ves­sel would pass en route to Mani­la via the Suez Canal.

Can you?

While we’re at it, how much do you real­ly know about the human liv­er? Enough to locate it, iden­ti­fy its secre­tions, and dis­course on its size rel­a­tive to oth­er bod­i­ly glands?

If you answered yes, con­grat­u­la­tions. There’s a good chance you’d be pro­mot­ed to high school back in 1912. Not bad for a kid attend­ing a one-room school in rur­al Bul­lit Coun­ty.

And now for some extra cred­it, name the last bat­tles of the Civ­il War, the War of 1812, and the French and Indi­an War. Com­mand­ing offi­cers, too…

That’s the sort of mul­ti­part ques­tion that await­ed the eighth graders con­verg­ing on the Bul­lit Coun­ty cour­t­house for 1912’s com­mon exam, above. The very same cour­t­house in which the mod­ern day Bul­litt Coun­ty His­to­ry Muse­um is locat­ed. A civic-mind­ed indi­vid­ual donat­ed a copy of the test to this insti­tu­tion, and the staff put it online, think­ing it might be fun for lat­ter-day spec­i­mens like you and me to see how we mea­sure up.

So—just for fun—try typ­ing the phrase “com­mand­ing offi­cer last bat­tle french & indi­an war” into your search engine of choice. For­get instant grat­i­fi­ca­tion. Embrace the anx­i­ety!

Com­mon wis­dom holds that stan­dard­ized tests are a lot hard­er than they used to be. But look­ing at the sort of stuff your aver­age eighth grad­er had to regur­gi­tate two years pri­or to the start of WW1, I’m not so sure…

Thank god the Inter­net was there to define “kalso­min­ing” for me. Even with the aid of a cal­cu­la­tor, math is not my strong suit. That said, I’m usu­al­ly good enough with words to get the nar­ra­tive gist of any sto­ry prob­lem.

Usu­al­ly.

I con­fess, I was so demor­al­ized by my igno­rance, I couldn’t have dreamed of attempt­ing to fig­ure out how much it would cost to “kalsomine” a 20 x 16 x 9 foot room, espe­cial­ly with a door and win­dow involved.

For­tu­nate­ly, the Bul­lit Coun­ty Genealog­i­cal Soci­ety has seen fit to pro­vide an online answer sheet, a dig­i­tal lux­u­ry that would have gob­s­macked their fore­bears.

SPOILER: $8.01. That’s the amount it would’ve cost to kalsomine your room at 1912 prices. (A steal, con­sid­er­ing that a quart of White Wash Pick­ling Water Based Stain will run you $12.37 a quart at a nation­al­ly known hard­ware super­store today.)

Go ahead, take that test.

If you quail at the prospect of far­ing poor­ly against a rur­al 1912 eighth grad­er, just imag­ine how well he or she would do, tele­port­ed to 2016, and forced to con­tend with such mys­ter­ies as cyber bul­ly­ing, gen­der pol­i­tics, and offen­sive egg­plant emo­jis…

via The Paris Review.

Relat­ed Con­tent:

The Open Syl­labus Project Gath­ers 1,000,000 Syl­labi from Uni­ver­si­ties & Reveals the 100 Most Fre­quent­ly-Taught Books

Take the 146-Ques­tion Knowl­edge Test Thomas Edi­son Gave to Prospec­tive Employ­ees (1921)

Take The Near Impos­si­ble Lit­er­a­cy Test Louisiana Used to Sup­press the Black Vote (1964)

Ayun Hal­l­i­day is an author, illus­tra­tor, and Chief Pri­ma­tol­o­gist of the East Vil­lage Inky zine. She lives in fear that her youngest child will pen a mem­oir titled I Was a Home­schooled 8th Grad­er and Oth­er Chillling True Life Tales. Fol­low her @AyunHalliday

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A Short Animated History of Zero (0): How It Started in India, Then Made Its Journey to the West

in Math | April 20th, 2016

Zilch. Nada. Bup­kis. Yes, I’m tak­ing about Zero (0), a num­ber that seems so essen­tial to our sys­tem of num­bers, and yet it has­n’t always enjoyed such a priv­i­leged place. Far from it.

In this short ani­ma­tion, Britain’s ven­er­a­ble Roy­al Insti­tu­tion traces the his­to­ry of zero, a num­ber that emerged in sev­enth cen­tu­ry India, before mak­ing its way to Chi­na and Islam­ic coun­tries, and final­ly pen­e­trat­ing West­ern cul­tures in the 13th cen­tu­ry. Only lat­er did it become the cor­ner­stone of cal­cu­lus and the lan­guage of com­put­ing.

India, we owe you thanks.

If you would like to sign up for Open Culture’s free email newslet­ter, please find it here. It’s a great way to see our new posts, all bun­dled in one email, each day.

If you would like to sup­port the mis­sion of Open Cul­ture, con­sid­er mak­ing a dona­tion to our site. It’s hard to rely 100% on ads, and your con­tri­bu­tions will help us con­tin­ue pro­vid­ing the best free cul­tur­al and edu­ca­tion­al mate­ri­als to learn­ers every­where. You can con­tribute through Pay­Pal, Patre­on, and Ven­mo (@openculture). Thanks!

Relat­ed Con­tent:

Free Online Math Cours­es

Free Math Text­books

The Short­est-Known Paper Pub­lished in a Seri­ous Math Jour­nal: Two Suc­cinct Sen­tences

The Math in Good Will Hunt­ing is Easy: How Do You Like Them Apples?

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