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Comment by nickff | Hacker News Reader

>"So we teach some of every subject to every kid. That way no matter which path they end up following, they are as prepared for it as we can make them."

We tend to waste a lot of time teaching subjects which they're unlikely to use, and fail to teach them about the ones that they would really benefit from. A basic understanding of criminal and civil law, along with accounting and statistics would be extremely useful to almost everyone as individuals and as citizens. Music, history, and calculus are useful to some people, but not nearly as many.

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I've never liked how people say that statistics is useful but calculus is not. I do not believe that you can actually understand statistics without understanding at least some calculus. So much of statistics is about areas under curves!

The problem with this is that the first classes in calculus are usually focused on continuous functions, which don't really exist in statistical datasets. The math has a lot in common, but most people don't really see or use that to their advantage, as evidenced by the literature on "transfer of learning".

Have you actually studied calculus based probability/stayistics though? Your comment seems characteristic of my own former thinking from when I had only taken an algebra based intro stats course (AP statistics) and hadn't yet learn it the calculus based way a few years later.

There is a lot of cool stuff you miss out on in the basic stats course because of having to dumb it down to avoid the calculus. Some I remember off hand:

- proof of the central limit theorem, which gives the shocking result that if you sum several uniform distributions you get rapidly more precise approximations of the normal distribution, which looks similar to exp(-x^2) if I recall. This central result is the foundation of all statistical sampling. This is why in real life if you see something follow a normal distribution you can guess it is probably caused by a moderate to large number of somewhat independent factors, and vice versa. This is genuinely useful, but if you don't know it you won't miss it - poisson distribution which relates the mean time between events to the probability of failures. Obviously very applicable to a lot of real life tbings

Also, pretty much every fancy formula you learn in Stats 100/ AP stats that looks weird but is very useful, can be derived and proved using calculus. Without calculus you just have to take it on faith, and may not have as intuitive an understanding of why it's true and what the significance of the terms is.

The same is the case in basic physics. No, V does not = IR, nor does F = ma. That's the simplification they tell us so they can explain a simplified version to us. In fact, the correct equations have derivatives in them and thus are differential equations.

Look, nobody needs calculus but nobody needs to read either. After all you could hire someone to read everything out loud to you. All knowledge is like this.

I hate to pull rank, but I've seen this enough to realize almost everyone saying "statistics but not calculus" are simply ignorant to what calculus (or analysis) entails. It is true that the classes as taught focus on continuous (smooth actually) functions, but data needs not to be continuous for you to use calculus, otherwise the field would be useless in real life applications and wouldn't even be a part of high school education like number theory isn't.

If there is any issue, there is an issue in how it is taught: I feel like there is too much focus on symbolic manipulation. The algebra essentially prepares you to take a physics course, and that's it really. The underlying concepts however do lead you to things like optimization and approximation (which is fundamentally what calculus is anyway) and that needs to be communicated to students somehow.

I actually do a lot of discrete math (numerical analysis) and statistical analysis for work, and completely agree that there is too much emphasis on symbolic manipulation in school-math. That said, I’m trying to address the reality of what exists in high schools, and the current reality is that a little extra discrete statistics, and a lot less continuous calculus seems like a good trade-off to me.

Alright, sorry to assume less of you. It is a sentiment across the thread but you do know that it's analysis and that calculus is just analysis. Really, calculus of continuous smooth functions is a special subset of calculus.

The thing I remember vaguely is when I was taught calculus first, we "took limits" by hand, including derivatives, numerically, and then we did the formulae and spent the rest of the time doing nonsense like difficult trigonometric integrals and integration by parts. The thing is as you go onto proofy classes including real analysis and such, you go back to the original concept and learn that that was the important bit and actually useful piece after all, as most of life's data cannot be well modeled by analytic solutions you can write down.

I think this is what I contend the problem is. Unfortunately, I don't have much contact with people who actually teach students high school calculus, but almost every mathematician and physicist I know (apart from the theorists may be) agrees with me, that at the end of the day, there is a lot of value to the concepts underlying calculus because they are general and help both naive models of data in your head and eventually statistical and numerical (read computational) models that vastly more people use, while the trigonometric substitutions are much less useful, and are really only useful if you're going to go on to being a theoretical physicist (or at least get a degree in physics where you'll need to do derivations).

The problem is kids just don't know.

I spent my entire university degree convinced that I was going to go into the video game industry. It took only a few months to realize that it's not what I wanted for a career, and I've spent the next 20 years loving my industry but doing anything but gaming.

I was an arrogant teenager that thought I knew what I was doing. I disrespected the arts, music, history, and focused exclusively on stuff like Math and Calculus.

Now I don't feel like a well-rounded adult, and I wish I spent more time when I was younger on music and humanities.

Your experience is what I think of whenever someone discounts a liberal arts education. It seems like the perfect second degree!

> Now I don't feel like a well-rounded adult, and I wish I spent more time when I was younger on music and humanities.

Easy remedy. Learn an instrument. Find someone local to take weekly lessons, and practice several hours a week. If you never did this before I think you will be overjoyed by how well you play with consistent practice.

I think the kids are also misled to a great degree. They basically told us if we'd want to go to uni and go for a STEM degree we should absolutely, totally do the focused math courses etc.

When I arrived for my CompSci they basically said: "We'll leave what you already learned in math behind around christmas (so after half of the first semester), no matter what kind of math you learned before".

I can't 100% grade both judgments, but I did not take the advanced math thing, but if they hadn't said these (apparently) completely wrong thing, I 100% would've taken French at school. (Which is another problem I'll not go further into, some fixed tracks of what path you need to choose in which grade)

Good luck convincing anyone here that they might be acting a little arrogantly.

> Music, history, and calculus are useful to some people, but not nearly as many.

I couldn't imagine not introducing my kids to History, Music, the Classics and so on. I value them far higher than my experience with Computing, Finance, Law, what have you. What a pointless life to only have interest into things that are productive.

You're welcome to introduce your kids to all kinds of things, but more people end up in prison than with record contracts.

Aspects of music can benefit people who do not have record contracts.

I would rather analyze deep finance than listen to music. I just don’t enjoy music, at all.

Spreadsheets and algorithms on the other hand I find highly entertaining. I love many board games for this same reason: it’s an opportunity to build novel algorithms in strange domains to achieve a specific purpose.

And most can see that boardgames are more similar to “productive things” you find disdainful than music.

Introducing kids to hobbies they might enjoy is good. But the fact that we teach them upper-class hobbies, and only upper-class hobbies, in school, suggests we're not doing it solely for their benefit.

I find teaching history essential. It helps understand the present, why things are as they are, and avoid repeating mistakes.

I had many different teachers with different approaches. Ones it was all about memorizing events and dates. Of course that is trash. But others is was about understanding why it went the way it went, why not other way. What were the key events that triggered another events, under which circumstances... alone the critical thinking that went into that, is every minute worth it.

History is absolutely fascinating (one of my favorite subjects), but not very useful. History has too high of causal density to allow drawing any clear lessons. We’ve studied WWI and WWII extensively, and nobody’s really sure what ‘caused’ either. Was it the shooting of an archduke that caused both? What about the arms buildups? The alliances? Just flukes? It’s not clear, and it never will be.

History is a beautiful subject and a great hobby, but almost completely useless. On the other hand, every student needs to understand finance and the law, both on the individual level, and in order to be a thoughtful voter.

That part of history, like the two WW, of course. The sample size is too small. It is not like: "I will look what happened in the past, and apply that today." That is most likely impossible, just because of context. But trying to understand what happened, really helps as a gymnastic to understand world dynamics. For example, good history teaching should have helped Europe to not be so dependent of Russian gas... I think the people making the decision there were not paying attention in history classes, or had bad teachers/professors.

But if you for example analyze the rise and fall of cities, empires and civilizations, there are some things you can learn (e.g. overuse of resources). Also the the economical crisis, bubbles and inflation teaches a lot of things.

Studying totalitarian regimes in the past, can help to detect the first signs of alarm.

It is not an exact science, yes. But knowing the past helps to understand the present, and helps to not repeat mistakes. I really think it does.

History in Austria is 2 years about how Hitler was the bad guy and Austria had zero fault in WW2, was essentially forced into everything.

We all knew that's not the truth. The teacher knew, but that is what the text book said so we learned it.

My point is, while history is important it shouldn't be a marked lesson where you just have to remind right and wrong facts.

It’s always been my impression that school history is more about indoctrination than any broadly applicable lesson. I took as much as I could in high school, and loved my teachers, but I’m not sure I learned anything I can apply outside the classroom.

Would be interesting to know when or if that's still the current case.

In Germany I found the focus on WW2 a tad much (but that was '93 to '02). Sure, it's important, but I think they could've gone at least one year out of nine where it wouldn't come up, and instead ignoring a huge amount of epochs outright, or with a laser focus on central europe.

How do you define "introduce"? When I went to school I'm pretty sure I had to take the music subject for around 10-11 years. First we had to learn the flute (no singing), then our third class teacher was shocked that no one could hit a not when singing. A little later it was a mix of good and bad, but mostly getting grades for singing, without ever properly being taught anything (not talking about the first 2 grades here, also ever onward, in a class of 30) and then some mix of tests on musical history.

The outcome? I still enjoy music despite this torture of lessons, but I never properly learned to play an instrument, and was mostly dissuaded instead of encouraged.

I feel that music and calculus are very different to history. I believe that history should be a fundamental course taught all the way through, we can't understand where we're going if we don't understand where we came from.

I would say that these subjects are more likely to turn into vocations than the teaching of how law and economy works.

I see it like when I learned about programming, I was frustrated to learn about language theory, complexity, graphs, etc. I wanted to learn langages, frameworks, specifics for being ready to work right at the end of my degree but it would have made me more fragile and less versatile to future changes. Although law and economy are less likely to change as fast as the latest cool tech stack so this example is not the best.

We are subjected to the law everyday and we all need to know about money to support our addictions to food and shelter.

Most of us are subjected to computers everyday too.

If everyone had at least a brief understanding of game theory, maybe people would cooperate better (not just at work, but in society at large)?

That’s an interesting idea, but I’m not sure people who have leaned game theory in an academic setting actually apply it to their lives. This would be evidence of ‘transfer of learning’, which is alarmingly uncommon. If the students did manage to benefit from learning game theory, I’d support it being added to the curriculum.

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