Are people “good” or “bad” at math?


Over at Quartz, writer Allison Schrager says no. I originally spotted her article over at Newser, which summarized it this way: “Think You’re ‘Bad at Math’? You’re Just Being Lazy.

While there are some good points in the article, I feel as though her conclusion is flawed.

Yes, math is difficult – especially when you get up into the higher levels. To learn it requires intellectual rigor, patience, discipline, and hours and hours of repetition. I remember this passage from the first science fiction book I ever read as a child, Heinlein’s Have Space Suit, Will Travel:

“Anybody who thinks that studying Latin by himself is a snap should try it.

I got discouraged and nearly quit-then I got mad and leaned into it. After a while I found that Latin was making Spanish easier and vice versa. When Miss Hernandez, my Spanish teacher, found out I was studying Latin, she began tutoring me. I not only worked my way through Virgil, I learned to speak Spanish like a Mexicano.

Algebra and plane geometry were all the math our school offered; I went ahead on my own with advanced algebra and solid geometry and trigonometry and might have stopped so far as College Boards were concerned-but math is worse than peanuts. Analytical geometry seems pure Greek until you see what they’re driving at-then, if you know algebra, it bursts on you and you race through the rest of the book. Glorious!”

I loved that story when I read it at age 10, and I still love it today, but I’m bitterly disappointed that I never had that breakthrough that the story’s protagonist experienced. I did fair to middling through high school in algebra, geometry, trigonometry and analytical geometry, but I was always doing it like a cook follows a recipe – it never burst on me, and I never saw what they’re driving at. So when I hit college with dreams of being a doctor, and discovered that in order to pursue a chemistry major I would have to take all the concomitant calculus courses, that was all she wrote. It was worse than pure Greek, because that’s a subject I did well in.

Mastering a foreign language is no piece of cake either, and yet for some reason I can master enough of a foreign language to become reasonably fluent in 3 months, whereas learning even the rudiments of calculus has eluded me for over 40 years. The disappointment stems from the fact that I love the world of science, and mathematics is the key that unlocks the door to understanding – a door through which I will only ever be able to peer through a keyhole to where the big boys and girls are playing.

This is music to my ears…

“Per me si va ne la città dolente,
Per me si va ne l’etterno dolore,
Per me si va tra la perduta gente.
Giustizia mosse il mio alto fattore:
Fecemi la divina podestate,
La somma sapienza e ‘l primo amore.
Dinanzi a me non fuor cose create
Se non etterne, e io etterno duro.
Lasciate ogne speranza, voi ch’intrate”.
(from Dante’s Divina Commedia)

whereas this makes my mind shut down completely:


Kepler’s equation for calculating orbits


And all we’re talking about here is limits, let alone the really weird things. To a scientist, this is really simple stuff.

My own experience has demonstrated to me that each person born has gifts and abilities unique to them. As no two snowflakes are a like, as no two stars have identical magnitudes or compositions, so no two people have the same talents. For me, math is terribly difficult, and language is uncannily easy.


As a result, the formula above is the only bit of calculus that I have ever been able to memorize – and I’ll never forget it.  Don’t get me wrong – I can’t evaluate it, even though I’m told by smarter friends that both sides equal ⅓; no, I remember it because it’s a limerick. [1]

Integral zee squared dee zee,
From one to the cube root of three,
Times the cosine
Of three pi over nine,
Equals log of the cube root of e.


Here is a drawing done by my daughter, who has never had an art lesson in her life. This is a gift that springs from within her.

Where I agree with Ms. Schrager is that despite the difficulty, if I worked at it long enough and hard enough, I could master it. But somewhere else there is a brilliant mathematician who trembles in terror at the thought of trying to learn Italian, because he or she knows that it would almost take more effort than it is worth.

In conclusion, I would retitle Ms. Schrager’s article to read: “Yeah, math is hard – but it’s worth the effort.”

The Old Wolf has spoken.

[1] I have hundreds of limericks rattling around in my skull. Unfortunatly,

A limerick packs rhymes anatomical
Into verses quite economical
But the good ones I’ve seen
So seldom are clean
And the clean ones so seldom are comical.

The Sad Death of Cromarty, 7rl

Bobby Hogg, 1920-2012

Today the Daily Mail reported the passing of Bobby Hogg, go ndéanai Día trocaire air [1], the last speaker of Cromarty, or the Scottish Black Isle fishing dialect. Bobby Hogg was 92, and last year his brother Donald left the world at age 86, leaving Bobby alone as the only speaker of the dialect.

Donald Hogg

Every fortnight, one of the seven to eight thousand languages spoken on the planet passes into history; the National Geographic maintains an intriguing interactive page of linguistic hotspots which illustrates places in the world where languages are the most threatened. Most of these tongues belong to aboriginal or minority populations, languages like Chulym or Tofa, spoken by a hunter-gatherers who also herded reindeer. Specializing in a skill often allowed a language to develop complex meanings with a single word; for example, the Tofa word döngür means “male domesticated reindeer in its third year and first mating season, but not ready for mating.” [2] But as Hogg’s death has pointed out, there are languages and dialects (dissertations have been written about the difference, and linguists love pseudo-intellectual straining at gnats) which are dying a lot closer to home.

I noticed with interest that there is no color on the Geographic’s page along the western coast of Ireland or in Scotland, yet the Gaelic languages have suffered significant losses. Dolly Pentreath who died in 1777 was the last native speaker of Cornish; Ned Maddrell, who passed away in 1974, was the last native speaker of Manx. [3]  Both tongues experienced scholarly revivals, and each language now has second language speakers and a few children who are being raised as native speakers. Scots Gaelic (Gàidhlig), Irish Gaelic (Gaeilge), Welsh (Cymraeg) and Breton (Brezhoneg) are all relatively endangered, although Welsh and Breton are the strongest, and Irish is being valiantly if ineffectively promoted and defended by the government and various groups within Ireland.

Which brings me to the “7rl” up in the title of this post.

In 1970, I had completed my second year of college and was back in New York for a few weeks before heading off to Naples, Italy for a year of work and study abroad. I stopped into a bar to use the phone and ended up talking to the bartender for a few minutes; when he found out I was studying languages, he said to me, “Well, don’t learn Irish.” That was like asking Maru not to jump in a box – I promptly went out and found a copy of Teach Yourself Irish by Myles Dillon and Donncha Ó Cróinín. Unfortunately, this book was printed before Irish spelling reform eliminated most of the silent consonants used in their hellish spelling (you will note that I said “most” – there are still plenty left!) – and without any real guide to pronunciation, I was unable to make any headway with the language, so I put the book on my shelf where it sat gathering dust.

20 years later, however, I stumbled across Linguaphone’s Cúrsa Gaeilge (Irish Course) in the West Valley City library – and it included tapes which became my Rosetta Stone; Irish is a beautiful and intriguing language which I continue to study as time permits. I’ve even attended an Irish Weekend in San Francisco, and would go back every year if resources permitted.

 Irish postal vans used to carry the logo

which stood for “Post agus Telegrafa” (Post and Telegraph); there are still some old manhole covers and other relics around bearing this logo as well. The image below shows clearly that the siglum is not a number 7, but rather a different symbol altogether.

Wikipedia reports that “the Tironian sign resembling the number seven (“7”), represents the conjunction et, and is written only to the x-height; in current Irish language usage, this siglum denotes the conjunction and.” Thus in the Irish language, “7rl” stands for either “agus rudaí eile” or “agus araile,” both of which mean “etcetera” or “and so forth.” [4]

Coming full circle, there are some great (if sparse) resources about the Cromarty dialect out there:

  • Am Baile, the Highland Council’s History and Culture website, published a pamphlet about the Cromarty dialect which includes a lexicon (2.3 MB pdf file)
  • 20 audio clips of Bobby and Donald talking about their dialect can be found here.
  • The Telegraph printed an article in 2007 about Bobby and Donald, including the following phrases:

Talking Cromarty

Thee’re no talkin’ licht You are quite right
Ut aboot a wee suppie for me Can I have a drink too?
Thee nay’te big fiya sclaafert yet me boy You are not too big for a slap, my boy
Pit oot thy fire til I light mine Please be quiet, and allow me to say something

I love this last one; while American English has some colorful dialects buried in remote pockets, our language is pretty bland when it comes to expressions like this.

The death of a language is a tragic thing, because it means the loss of so much culture and history that went along with its speakers. I support the efforts of organizations like Am Baile and Daltai na Gaeilge to encourage the use and revitalization of these beautiful and intriguing tongues.

Tá an sean-fhaolchú labhartha.


[1] Irish for “May God have mercy on him”

[2] Astute readers will say, “Oh yeah – just like Eskimos have 23 (or 42, or 50, or 100) words for snow.” One such published list of Inuit words for snow follows:

Aiugavirnirq – very hard, compressed and frozen snow
Apijaq – snow covered by bad weather
Apigiannagaut – the first snowfall of Autumn
Apimajuq – snow-covered
Apisimajuq – snow-covered but not snowed-in
Apujjaq – snowed-in
Aput – snow
Aputiqarniq – snowfall on the ground
Aqillutaq – new snow
Auviq – snow block
Katakaqtanaq – hardcrust snow that gives way underfoot
Kavisilaq – snow roughened by snow or frost
Kiniqtaq – compact, damp snow
Mannguq – melting snow
Masak – wet, falling snow
Matsaaq – half-melted snow
Mauja – soft, deep snow footsteps sink into
Natiruvaaq – drifting snow
Pirsirlug – blowing snow
Pukajaak – sugary snow
Putak – crystalline snow that breaks into grains
Qaggitaq – snow ditch to trap caribou
Qaliriiktaq – snow layer of poor quality for an igloo
Qaniktaq – new snow on ground
Qannialaaq – light, falling snow
Qiasuqqaq – thawed snow that re-froze with an icy surface
Qimugjuk – snow drift
Qiqumaaq – snow with a frozen surface after a spring thaw
Qirsuqaktuq – crusted snow
Qukaarnartuq – light snow
Sitilluqaq – hard snow

Well, it turns out that the truth is both far more simple and far more complex. There’s a difference between packing semantic density into a single discrete lexical item, and using multiple suffixes to produce new meanings from a single root. At Language Log, Geoffrey K. Pullum, author of The Great Eskimo Vocabulary Hoax, said:

“If you wanted to say “They were wandering around gathering up lots of stuff that looked like snowflakes” (or fish, or coffee), you could do that with one word, very roughly as follows. You would take the “snowflake” root qani- (or the “fish” root or whatever); add a visual similarity postbase to get a stem meaning “looking like ____”; add a quantity postbase to get a stem meaning “stuff looking like ____”; add an augmentative postbase to get a stem meaning “lots of stuff looking like ____”; add another postbase to get a stem meaning “gathering lots of stuff looking like ____”; add yet another postbase to get a stem meaning “peripatetically gathering up lots of stuff looking like ____”; and then inflect the whole thing as a verb in the 3rd-person plural subject 3rd-person singular object past tense form; and you’re done. Astounding. One word to express a whole sentence. But even if you choose qani- as your root, what you get could hardly be called a word for snow. It’s a verb with an understood subject pronoun.”

The entire page is worth reading if you’re interested in such things.

Another example of aggressive word formation comes from the Turkish language. It is said that the single word Avrupalılaştırılamayabilenlerdenmısınız is the equivalent of an entire sentence: “Are you one of those who is not easily able to be Europeanized?” This, however, is misleading because Turkish agglutinates (i.e. crams whole bunches of stuff together); it’s not really fair to call the monstrosity above a “word.” Here’s a breakdown of how the thing is put together:

Avrupa: Europe
Avrupa-lı: European
Avrupa-lı-laş-mak: become European (mak is the infinitive ending)
Avrupa-lı-laş-tır mak: to make European
Avrupa-lı-laş-tır ı l mak: (reflexive) to be made European (with the linking consonant “l”)
Avrupa-lı-laş-tir ıl abil mek: to be capable of being Europeanized (the infinitive ending mak changes to mek because of vowel harmony)
Avrupa-lı-laş-tır ıl ama mak: not to be capable of being Europeanized
Avrupa-lı-laş-tır ıl ama y abil mek: this time the  abil is probability: that there is a probability that one may  not be capable of being Europeanized
Avrupa-lı-laş-tır ıl ama y abil en: the one that may not be capable of being Europeanized
Avrupa-lı-laş-tır ıl ama y abil en ler: the one that may not be capable of being Europeanized (-ler, -lar is the plural suffix)
Avrupa-lı-laş-tır ıl ama y abil en ler den: of or from the ones who may not be capable of being Europeanized
mı?   question tag (officially, this should be written separately, but it’s very common usage not to do so)
mısınız?  are you (Second person plural, also used for formal second person singular)

[3] At one point as I was following my passion for all things Celtic, I stumbled across a Manx Language resource page and discovered to my delight that the spoken samples by Ned Maddrell and John Kaighin were close enough to Irish to be understandable. I regret that I don’t have time to dig deeper into this language.

[4] Which reflects the nature of many of my posts here… pretty much a free-association experience. Sorry.