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Comment by tommiegannert | Hacker News Reader

I.e. mathematical history.

I think one power of calculus (and everything else) is that you don't need the history. The same way I don't need the history of the hammer to use a nail. What drives my interest is the examples. At the end, the book author talks about students not paying attention until the teacher starts talking about examples. To me, that suggests you should first contextualize and create a need for the tool, then describe the tool. Not to start introducing names of dead people, no matter how great they were. (At least where I grew up, this will always devolve into the curriculum requiring you to know those names and dates, draining focus from what actually matters.)

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I disagree. The biggest thing that can help you learn something complex is by introducing it as it was discovered, and making someone feel like they could have come up with it on their own.

Things were usually discovered in a confusing way that kind of made sense at the time, but had false starts and conceptual limitations driven by people still being in an older mindset when they were trying to discover it. A good teacher sands off those edges, leading to an oversimplified history, but not starting from the firm conceptual footing we only got later on still makes things overly difficult to grasp. Good teaching builds on concepts, even if history took a messier route.

>> Things were usually discovered in a confusing way that kind of made sense at the time, but had false starts and conceptual limitations driven by people still being in an older mindset when they were trying to discover it

To know this gives us some perspective on our own biases/limitations and not take things as gospel.

> some perspective on our own biases/limitations

This is sometimes prescribed in such form as: "Fraction X of your reading should be of authors writing at least 100 years ago."

Similar: In defense of the reading of old books by C.S. Lewis ...

https://www.youtube.com/watch?v=jFHQDIE7CSw

"All contemporary writers [of any period] share, to some extent, the contemporary outlook."

That's not the biggest thing that can help you learn something complex. Practice is the biggest thing. And there are many variations of what motivates people to practice.

I don’t see why the history of discovery of something should be a particularly good way to rediscover it.

As a concrete example, if someone asks “how do we know the Earth is round” or “how do we know the planets orbit the sun”, one could go through ways to figure this out with math and minimal technology. But one could also point out that we have spaceships and probes and have actual pictures! You can see the Earth from a satellite or the ISS or the moon, and you can see the planets from space!

I agree that a process of rediscovery can be a useful teaching technique, but that doesn’t mean that rediscovering something the way it was first known to have been discovered is the right approach.

Another concrete example of the conflict is classical vs quantum physics which is nearly exclusively taught as the historical story of the quantum revolution or whatever you call it around 1900.

It would be trivial to skip the historical story and just "yo here's a FET transistor, a solar cell, and a vacuum tube, and the math just works, so get used to it" but nobody is ever taught that way.

Would be interesting for someone to attempt to write a non-historical physics textbook.

On the other hand, in IT, this style of learning is considered normal. "This is how the kool kids do things today. The past does not matter and probably didn't exist anyway. The end."

> Would be interesting for someone to attempt to write a non-historical physics textbook.

I recall some prominent physicist saying they've never read Einstein's original papers on relativity; and they don't have any real desire to, since we understand it much better now, and have better examples, notations, etc.

Most physicists have never read them, fwiw, as interesting as they may be for their historical role. Most classic papers go unread, and it’s not a bad thing pedagogically. How things get sorted out is usually a huge mess and following the real history can be an enormous distraction (and can confuse and misdirect students).

While that is true, I would highly encourage everyone to read the original EPR (Einstein Podolski Rosen) it is so amazingly clear in its writing and can be understood even by laymen.

A downside to the modern approach is that “plug the equations into a computer and let it churn for a bit” plays a big role I think. This is seems a bit annoying from a teaching point of view

* “here’s what the computer came up with” doesn’t really build as much intuition as doing the equations out by hand

* there’s enough material in “how to plug it into a computer and make it run quickly” for a whole degree

>On the other hand, in IT, this style of learning is considered normal. "This is how the kool kids do things today. The past does not matter and probably didn't exist anyway. The end."

And yet nand2tetris is one of the most common suggestions for people who actually want to figure out how computers work. Because you actually are posed with problems that real world engineers used to struggle with, and are allowed to play with the problem before finding out the answer. It builds a much deeper understanding than "it works like this because it is"

But nand2tetris doesn't have a historical approach if i'm not mistaken.

“How do we know the world is round” is an interesting example to use. Eratosthenes got an estimate for the radius of the Earth that was pretty close using, I dunno, sticks and his eyeball or whatever ancient Greeks had.

The history is useful there, I think. First off, it is motivating — that’s a pretty big question, really shows what math can accomplish without much hardware. Second, it provides some sort of useful constraints if you are trying to work out the computation yourself. In a textbook the problems are always a bit artificial, some info will be given, some hidden. “Anything an Ancient Greek would have” is a bit more real-world, you can see where the constraints come from.

Erastosthenes did it by comparing the angle of the shadow cast by a stick on the summer solstice at noon in two cities, Alexandria and Syene (now Aswan), which are about 800 km apart. What was special about Syene was that it is so close to the Tropic of Cancer line that the sun would hit the bottoms of wells on the solstice, meaning that he only needed one stick, in Alexandria, and the distance between the cities, to do his estimate. His error was about 2%.

He's also famous for the sieve of Erastosthenes for finding prime numbers and figuring out the earth's axial tilt.

Darn, at some point I started imagining he had a protocol for synchronizing clocks over a distance. assuming the difference was in longitude. (E.g. Symmetric NTP over carrier pidgeon.)

Longitude was a much tougher problem to solve.

Yea, recalling the book of that name might have contributed to my drift of recollection. Solvable earlier on land (but not sea) on terms of clock synchronization.

You can get a vaguely credible estimate by standing on a (small!) hill and measuring the distance to the horizon. Works better if you have a body of water handy, because bodies of water are nice and flat.

I think it can depend on the particular discovery whether we should teach "here's how the first person figured it out" or skip right to "here's how a later person found a much better, easier to understand approach".

I've been interested for awhile in a distinction between math and philosophy here. Just descriptively speaking, we don't tend to teach math from original source (at least, not well-established math like you'd see in high school and early college). We don't teach calculus out of Newton/Leibniz/Cauchy/etc, we don't teach Fourier analysis out of Fourier, etc. Maybe geometry students will work from the Elements, but I certainly didn't.

But in philosophy, students do read everyone from Plato to Rawls in the original. There are (lots of) supplementary texts, of course, but they're companions.

I think that shows an interesting divide that the way we actually think about philosophy is not just a catalogue of arguments and positions one could take, but that the famous works are things worth reading unto themselves. There are a few math papers like that ("God Created the Integers" tries to anthologize them) but they are the exception rather than the rule.

But maybe there's another explanation?

Two contributers I can think of

1. Being able to tease apart language from ideas can't be 'taught' you have to do it

2. People disagree much more in philosophy than math

I agree, one book I recommend that does this for computing is "Code: The Hidden Language of Computer Hardware and Software"

The way calculus was discovered was via highly geometric methods that are beyond the comprehension of people today. So the historical development of calculus needs to be modernized into modern notation and concepts, which at that point basically becomes examples.

While calculus is actually easy to develop and motivate, the way we "discover" it today is nowhere close to how it was originally discovered.

In my opinion, calculus is most easily motivated via physical examples. Here is a curve, it represents some information, and what are some things we'd like to know about this curve and thus the information it represents? What are some things we can do with that information? Such motivation can be more abstract and not rely on physical examples, but I think the physical examples help tie it to the so-called real world and somewhat mirrors, in spirit, how and why calculus was originally developed.

The comparison between a hammer and math doesn't make sense, math is much more than a tool. Personally, I'm able to learn best when I understand the context and the "why" questions that people had which led them to the answer. I know other people learn differently so to each its own. Also, I agree with you in that the traditionally-taught history is awful because it lacks relevance. It's now about "how is this useful for me" or even "why should I care" but rather "why did they decide to act that way?"

I think the dual role of means (tool) and end in itself is part of the reason mathematics is so difficult to motivate. There is this constant tension within the pedagogy between the desires to teach the beauty and the practicality of mathematics.

From what I’ve seen, the beauty side tends to come from mathematics teachers and professors who have developed a passion for the subject and genuinely want to share that passion with their students. The practicality side, on the other hand, seems to be driven by industry, administrators, lawmakers, parents, and others involved in the curriculum-setting process.

Ultimately, both sides struggle mightily for students’ attention and a typical result is a collective sigh from the class. If were to make any progress I think we need to recognize that student motivation comes from within. Some students are genuinely attracted to the beauty of mathematics while others are interested in different subjects (science and engineering) for which mathematics is only a prerequisite. Perhaps one approach for dealing with this disparity might be to offer students a choice of projects with differing mathematical styles (proof vs calculation).

Another thing that makes calculus hard to teach today (and not unrelated to the points being made) is that while a couple generations ago, every calculus student also took calc-based physics, that is no longer true today. With the way we compartmentalize teaching today, it's hard not to teach entry-level math abstractly, and that's only going to work for a small subset of students.

Learning names for the sake of learning names is never helpful. However the history of mathematics can help to contextualise the material and its motivation. For example, group theory has myriad modern applications. It can be (and usually is) presented in the standard mathematically-polished form, beginning with the axioms and perhaps using matrix groups or simple geometric symmetries as examples. For many learners this is a perfectly sensible way of approaching it and they wouldn't be interested to know more, yet I found it rather clinical and dry.

As I'm sure you know, group theory arose as a novel yet natural way of investigating the conditions under which polynomials can have closed-form solutions, and when Galois began to sniff this out it allowed him to get to grips with the impossibility of a closed-form solution to the general quintic polynomial: a problem which had beguiled mathematicians for centuries, solved by a theory which crystallised a more structural way of approaching abstract mathematics. Even though Galois theory is a relatively niche topic within the broad context of academia, for me it draws out the true character of group theory in a way that matrix groups don't, and neither do contrived examples formed from the natural numbers under various quotients. And it clearly solves a problem that required a new way of thinking.

There's certainly a balance to be struck here. If you fixate on historical origins then you aren't engaging with the reasons why the topic is still relevant today, so you risk missing the point. It also slows down the pace at which you can absorb the tools needed for applications. But if you only engage with the modern approach you risk building a disconnected, lifeless archipelago of knowledge, unable to see the beautiful links that unify so much of mathematics.

I totally second this. For someone who has a tortuous relationship with mathematics, I always lovedthe beauty of som concepts but to a certain point, got lost mainly due to knowledge base not really understood. Getting to know some historical facts would have help to be more critical to myself ( rather than just copy and paste formula).

Years later I surprised myself spending time on new maths problems thanks to YouTube maths contents(3brown1blue, Michael Penn, mindyourdecisions,etc...) Which make me reboot my knowledge. The challenge here is how is it inseminated during your scholarship.

From the what I remember, complex number theory were introduced to me as a rule that need to be acknowledged ( with no incentive to look further).

Welsh lab vid below made me reconsider my own shortcomings.

https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...

While I agree with your sentiment, studying abstract algebra at the undergraduate level we never got more than a broad sketch of a proof of the general insoluability of the quintic. It more seemed an opportunity for the professor to tell the life story of Galois -- mathematical genius and political firebrand killed in a duel over a woman at age 20.

That being said, it is difficult to find a motivating example for much of elementary abstract algebra -- which is doubly hard, because for many students abstract algebra is a new level of abstraction, math without numbers and it takes something to make that leap.

As others have argued, one can learn mathematics without its history, but it is very helpful to motivate many of its abstractions. A good example of this are some of the textbooks of David Bressoud. In a "Radical approach to Analysis", concepts are introduced in historical order. The book starts with the heat equation and its solution by Fourier series which was rejected at the time, in constrast to most books that start with the real numbers and limits.

I think something quite different than you, but that isn't important. The important thing is that different people are learn differently and are motivated in different ways. Curriculums should take this into account and accommodate different learning styles. Unfortunately, they usually don't.

Look up Lockhart's lament, if you haven't heard of it

> he same way I don't need the history of the hammer to use a nail

This has to be the world's worst analogy ever used.

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