My friends in a top tier high school had no intuition about numbers because they used calculators for everything, including basic addition and subtraction.
Some of them had customer service jobs where they became utterly confused if the change was 99 cents and the customer gave them an additional penny.
Everyone crying "calculator" about ChatGPT has me convinced that ChatGPT is bad for your education. Learning to do mental math as a child sucked, but now that I can do it, my brain is so much more free to think about stuff that matters. There's no intervening step of "let me pull out a calculator to see what that is," I just know the answer. The thoughts can just flow freely.
This is true of all memorized facts. It enables thinking at a higher level. “Why should I learn multiplication when I have a calculator on my phone” ignores this.
The more you have memorized the more nimble your thinking is. If you have a large vocabulary you can effortlessly express yourself with precision while others are thumbing through a thesaurus (or these days asking an AI to “rewrite”).
If you know the history of something you can have more interesting perspective and conversations about it.
There is almost no situation where the person with a lot of memorized knowledge is at a disadvantage to the person who needs to look everything up or rely on tools to do the work.
True. I have friends that have refused to learn algorithms and instead insisted that they only needed to master $FRAMEWORK. Then they got stumped by any problems that can not be solved by $LIBRARY, spending days on it with no result.
Yes, it takes time, but learning is exponential, and overtime, the pace will increase greatly.
That's true for every skill that comes fluently. We've only got limited time, which skills really matter?
I'd say yes to basic arithmetic; but I can't really use my own experience as a software developer who started off in video games to justify why a normal person needs to understand trigonometry and triangle formulas, any more than I can justify why they need to study Shakespeare and Alfred Tennyson over e.g. Terry Pratchett and Leonard Cohen — "I find it intellectually stimulating" is perhaps necessary, but certainly not sufficient, given there's more to learn than we can fit in a lifetime.
Because they give you flexibility. One does not need to master everything, if you have a wide and stable foundation, your options become more numerous later in life. And expertise is a pyramid, so the more diverse your basic skills are, the farther you can reach.
> I can't really use my own experience as a software developer who started off in video games to justify why a normal person needs to understand trigonometry and triangle formulas
We live in a 3d world, every human has to reason about geometry many times in their lives and trigonometry is the most useful and easiest of geometry math. Same reason we learn about volumes etc, everyone learns cos, sin, tan, its just 3 things it doesn't take a lot to learn that.
And yet, outside my hobbies of "being a massive nerd", I've not needed the maths of trigonometry for at least a decade.
Outside hobbies and having worked as a game developer, I can't recall a single time I've needed them.
I suspect most learn them for a test them forget them
forever.
Now, statistics, percentages, and compound interest, those are things people would do better to know, as those are the tools of advertising and money and are unavoidable in modern life.
> statistics, percentages, and compound interest, those are things people would do better to know
They do teach those in middle school before they teach trigonometry, so that is already covered.
> And yet, outside my hobbies of "being a massive nerd", I've not needed the maths of trigonometry for at least a decade.
Lots of people craft things or work with schematics in some way, not just highly educated people, to them having the basics around angles is often very useful. People like mechanics, carpenters etc, that sort of math is useful to solve so many basic everyday problems in more handy jobs.
Literacy is important. Understanding Shakespeare and/or Tennyson is as much a part of that as is Pratchett. You will read Pratchett on your own, though, and you probably won't go reaching for the Shakespeare, so you might as well get some forced exposure. Understanding rhetoric is a very underappreciated skill, but an important one.
Reading isn't studying — that most won't read Shakespeare means studying the deeper meanings of the ingredients in the witches' brew in Macbeth is pointless and forced exposure is as much a waste of time as Roman numerals or Latin — do it on your own time as a hobby, don't make it mandatory.
Conversely, understanding the deeper meanings behind the stuff people do read, perhaps Pratchett, is valuable — the use of speciesism between Dwarfs and Trolls used as a substitute for racism, etc.
It's not like rhetoric stopped with old classics; therefore it is better to learn how it works in newspapers or viral memes, as if you keep to Tenyson you're just going to pattern match it to 1800s writing style.
On rhetoric: you learn rhetoric by writing more than reading. That's why they make you write about books: it is practice for writing with evidence since the evidence to back up your ideas is contained only in the pages of the book (children are idiots and the book is a replacement for knowledge). While you're doing this exercise, you might as well get exposure to a variety of different books to learn about the culture and history of the world you live in.
Modern writers including Pratchett make lots of references to this canon. Some people also learn that they enjoy Shakespeare in school, and they would never read a play like that otherwise.
What makes Pratchett more "relevant" than Shakespeare? I would posit to you that the same order of magnitude of people (or more) read and watch Shakespeare plays by choice as read Pratchett books. Let alone modern adaptations like West Side Story. I get that you prefer the one, but not everybody does.
English classes also have you read lots of other more recent and more modern works. It's not exclusively Shakespeare. I remember, for example, Death of a Salesman, 1984, and Slaughterhouse-Five being part of my high school's English curriculum.
Lord of the Rings is apparently in several English curricula now.
Comedies of all kinds (from Aristophanes to Pratchett) usually take a backseat to tragedies, bildungsromane, and other more "serious" genres in English classes because they tend to be much less timeless and much more current in terms of their themes and references.
> That modern writers reference old works is true, but so too did Shakespeare, and we don't reference his cultural ancestors to understand his works.
Schools do generally have you read ancient writers, often including Greek comedies and tragedies, and some have you read medieval British literature and those references do, in fact, come up. It's possible that you did not fully enjoy Shakespeare if you did not get that context.
> That fine, but it's not the job of school. If it was, we'd also have mandatory watching of foreign language horror films.
Believe it or not, if you take advanced French classes, you're going to watch a lot of French films and listen to a lot of French music for school! Since most students in English-speaking schools spend a lot more time studying English than French, they read a lot more English literature than French.
> What makes Pratchett more "relevant" than Shakespeare?
The language, the content, and the characters.
We don't live in a world where someone could fail to hear from their ship for a few weeks, thinking themselves unable to pay a weird debt, and have their loan shark's daughter dress up as lawyers and waltz into the trial to save the day. And we definitely don't encounter people who speak in Shakespearean English (even Victorian English is pushing it).
We do live in one where people try to commit insurance fraud as soon as they hear about it, a world where Magrats' have excessive candles (even if the magic isn't real), and a world of Vimes' Boots.
> Let alone modern adaptations like West Side Story
Or the Lion King. The adaptations are fine, the success make the connections to the world in which they are created.
> English classes also have you read lots of other more recent and more modern works. It's not exclusively Shakespeare.
I didn't say otherwise; I'm saying focus on those other things. I think the most modern thing we had was Ethan Frome.
> Believe it or not, if you take advanced French classes, you're going to watch a lot of French films and listen to a lot of French music for school! Since most students in English-speaking schools spend a lot more time studying English than French, they read a lot more English literature than French.
Again, that's fine for those learning French (and etc. for each other second language) to a higher level — given my experience with teaching myself a second language, and my self-tests over the years, there's a huge quality gap between what's a good grade in school and what's enough to get by with in practice — but that's not most people, and as there's exactly 18 years in the first 18 years of your life, my point is: what should be mandatory in schools? You've got limited time, what is the "you must" rather than "here's something you may like"?
except that the typical case is ... the charge is $1.01, I give you a $5 and a penny. A penny from the client to the house on top of a charge of $0.99 by the house, does xxxxxxxxx ... (edit) as pointed out below, a penny plus a $0.99 charge means that the cashier can return a whole number of bills, avoiding any coins..
The change, not the charge. If the change is 99 cents, the customer gives an extra penny, the change is now 1 dollar, avoiding a handful of coins.
It seems to have become the norm for young cashiers to be unable to understand. And if you try to explain, they'll insist "I can't change it now I've rung it through". Some seem to think the system keeps an exact record of the quantity of each individual coin (or they just don't even know where to begin to think about it).
If you have basic comfort in arithmetic, this is not a calculation that involves giving a damn. Being confused about why someone would give you an extra penny and having a discussion about it with a stranger burns 100x more calories than knowing it. If basic arithmetic involves taking a deep breath and closing your eyes for half a minute, or looking around the room for a calculator, that's a different cost/benefit analysis.
It's like the difference between a language you are fluent in and a language you are tentative in. If you're fluent, you have to make an effort not to listen to somebody's loud conversation, or not to pay attention to a billboard. They intrude into your consciousness. There's never a situation when I don't do simple arithmetic when exposed to it. I don't have to consciously figure out what 4 times 9 is. Subjectively, the number just pops into my head when I see the question.
edit: If you can't do this with explanations of identities or related rates, etc., it's hard or impossible to follow any quantitative or especially probabilistic argument. Even the simplest ones. I think this results in people for whom arithmetic is difficult faking it by trying to memorize the words used during quantitative arguments without having any real understanding. Just sort of memorizing a lot of slogans and repeating them during any argument that shares similar words. I think discomfort with arithmetic ruins people politically (as citizens), so I really do think calculators are a problem.
Some of them had customer service jobs where they became utterly confused if the change was 99 cents and the customer gave them an additional penny.