Its by the fellow that does Minute Physics on Youtube. Towards the end of his talk, he specifically talks about this topic.
He makes a comparison between mathematics and physics, wherein mathematics you are taught the math without a ton of history. No scaffolding. In physics, you are taught the entire history of physics, i.e. scaffolding, from the ground up.
He also advocates directly teaching modern theory and leaving most of the history out. Make learning physics more like learning math.
As a PhD student in mathematics, I have to say that history is valuable in mathematics because it explains why people are interested in certain problems.
When I was a master level student, I always got annoyed by advanced courses in algebra (things like homological algebra) where it was often unclear why one should be interested in the problems stated there. Prompting the profs to give some short historical overview can be very enlightening.
I do believe you can leave the vast majority of the history out. Advances in notation etc. were made for a reason. But a little bit of history can be quite important for context.
History may be valuable in mathematics, but it is demonstrably skipped. Few know much about it.
When I was in grad school in math, I made the interesting discovery that if p and q are polynomials over a commutative ring, any polynomial that is symmetric in the roots of p and q is actually a polynomial in the original ring in the coefficients of p and q. (The construction works whether or not the ring can be embedded in a field where said roots actually exist.) Using this observation it is trivial, for instance, to write down in fully factored form a polynomial that has sqrt(2) + cube_root(3) as a root.
This construction was news to various mathematicians that I talked to, including a combinatorics prof who studied symmetric polynomials and a number theorist who worked on stuff related to the algebraic integers. Then finally I talked to a very old mathematician with an interest in history. He told me that I had rediscovered an old way to do things. At his encouragement I went to the library, and picked up an algebra book from the 1800s. My construction was taught, and there was a whole chapter full of problems where students were expected to use it to come up with polynomials with specific roots.
Furthermore as I looked into it both of the professors that I mentioned before worked in areas whose history dated back to the observation that I mentioned. It was used in the original proof that the algebraic integers form a ring, and that construction was the original reason that people were interested in symmetric polynomials.
For another demonstration of how little of their own history mathematicians know, ask anyone why the notation for the second derivative is d^2y/dx^2. Then ask them where the f' notation comes from. Then ask them what Cauchy was trying to do that lead to Cauchy sequences. Most will draw a blank on all three.
Don't read on until you're satisfied that you don't know the answers.
In the original infinitesmal notation, d was an operator. It could be defined by d(y) = y(x + dx) - y(x). And you'd calculate a slope as dy/dx (drop any infinitesmal bits). Well when you work out d(dy/dx)/dx it turns out that you get d(d(y))/(dx * dx) which is more compactly written d^2y/dx^2.
The f' notation was introduced by Lagrange in an attempt to get rid of infinitesmals by defining differentiation as a formal algebraic operation on polynomials and power series. This fell apart when Fourier demonstrated that apparently well-behaved power series could be used to construct pathological things like step functions.
Cauchy came up with Cauchy sequences while attempting to define infinitesmals rigorously. His approach fell apart on the seemingly trivial example of how you rigorously prove the chain rule when the derivative of the inner thing is 0. (He was trying to avoid 0/0, but in that special case you get 0/0 all over the place.)
> I have to say that history is valuable in mathematics because it explains why people are interested in certain problems.
Motivation is essential, but there are motivations other than the ones that historically lead to the creation of the field.
In number theory, one of my favorite examples, the motivations that lead people to be interested in it now come from modern cryptography, which flatly didn't exist when the field was founded but provides more interesting and relevant examples than the obsessions of century-old mathematicians.
For example, every math program I've looked at teaches integration of complex functions the way it was arrived at historically, which involves increasingly complex shapes in the plane that eventually lead to the general case - some program spend half a semester going through this construction.
However, if you're familiar with vector calculus, it is an immediate result of Green's theorem in the plane - doesn't even take 5 minutes of work. Engineering programs often go that route.
And if you structured the material in the "right" way, it would be even simpler - Green's theorem in the plane is itself a special case of Stokes' theorem; However, "simpler" is relative - stokes theorem is much more abstract. So it's "simpler" in the sense that you can prove much, much less. It's more complex in the sense that you have to grok all those higher-level abstractions without a mental picture.
Sort of like software engineering - LISP is a better fundamental system. But it is too abstract for most people in the field. And so are APL/J/K, although in a different way.
No, no, no, no. Instinctively I recoil at that idea. I dont quite know how to explain why, but I'll give it a go...
Im good with computers because I was there in the early days when it was pretty simple. Everything since has build upon those simple blocks. So, I can always work it back as it were. It kinda means I can simplify current complexities down to simple fundamentals. So, it leaves me with an ability to be presented something new, or different and very quickly I can understand it.
Does that make any sense? I assume many people here are exactly like that, or recognise what Im trying to say.
Pathetic I know, but if any one can put it better, please do!!!!
Anyway, all I know is that I am so much better off knowing how it all came together than people who don't. I don't understand how any one can claim to understand any subject with out understanding it's journey as it were.
I make a distinction here between know how to use something and understand it. If all one wants to do is "use", then understanding isn't absolutely necessary. You know, plenty of people drive cars with out understanding anything about how they work.
No one is saying that knowledge of the history isn't useful. But it's not the most efficient way to learn the concepts. For example, it would be crazy to start learning basic programming by first studying the physics behind computers.
Learn the history behind your subject, but separate it from the pedagogy. Otherwise you'll have a much harder time learning knowledge that you can actually apply.
> For example, it would be crazy to start learning basic programming by first studying the physics behind computers.
Technically, a truly historical development in terms of teaching programming would involve plugboards and specialized hardware design freshman year, punch cards sophomore year, teletypes junior year, and then 'intelligent' terminals senior year. We simply wouldn't have time to teach anything similar to modern networking, which only really came about in the 1980s, let alone GUI design or Web programming.
> I was there in the early days when it was pretty simple.
I think you have an interesting idea of 'simple': A lot of software becomes orders of magnitude simpler to design and reason about once you have enough RAM to organize it in the obvious way.
For example, for many parsing tasks, a recursive-descent parser is the obvious way, but you can only design your software that way once you have a language that can directly express recursion and a computer that can grow a big enough call stack (on the stack or in the heap) to allow the parser to juggle nontrivial sentences. Otherwise, you're left the laborious, non-obvious, labor-intensive, but ultimately pointless task of turning your design into a program your tools can implement.
> So, it leaves me with an ability to be presented something new, or different and very quickly I can understand it.
Honestly, I think that says a lot more about your intelligence than your history. That's actually a pretty fair definition of intelligence, in fact.
How about this: Teaching networking now is teaching TCP/IP and everything that TCP/IP rests upon, such as Ethernet and Wi-Fi. Theory naturally flows from practice, and that is the practice.
The OSI Model? Gone. Out. Forget it. Nobody actually implements all seven layers; at most, we have four.
NCP? What? If you know what NCP stands for, congratulations, you know something entirely useless. "Jeopardy!" would love you.
So why teach the stuff we know is useless? It actively discourages students. It makes them think they're just wasting their time, primarily because they are.
The history of computer networking is a valid topic. It should be taught in its own course, it deserves its own course. It does not deserve to be shoved into a course on how networking actually works.
In chemistry class (where we learned particle physics in high school) I really hated having to unlearn all the crap we were quizzed on in the last section to learn some new crap that I'd be tested on and then expected NOT to use later because it was obsolete! /rant
Anyway, if you really want to teach the history of a subject like math, you should probably do it after teaching the modern understanding of the subject. In math class our teacher explained some of the controversy over the invention of calculus, and the origin of the different notations. It gave us some appreciation for the unintuitive and tricky nature of the subject, which looks very simple now.
... which lead to the less-intuitive epsilon-delta proof framework replacing the much more intuitive infinitesimal framework calculus was originally founded on, until the 1960s when infinitesimals were reformulated as part of nonstandard analysis.
(Unless the controversy you're talking about is the utterly uninteresting one about Leibniz vs Newton.)
Another example where a historical development would lead students through a completely pointless diversion (epsilon-delta proofs) simply because Robinson was born in 1918 instead of 1618.
I was trying to play devil's advocate but I honestly couldn't think of a good reason to teach Calculus using epsilon-delta vs. Nonstandard Analysis. The way Calculus is taught today is already unrigorous until you get to Analysis, so there's no strong reason to teach students using epsilon-delta first.
Right, because students struggling with "for all P there exists Q" won't have any problem with higher-order logic?
It's always seemed to me a more sensible road to both simpler and more rigorous calculus courses would somehow involve reducing the scope of functions under consideration, since most of the exercises involve analytical functions anyway. Then, when students are ready for analysis, it can be more about "how to reduce nasty cases to problems you know how to solve, and how to recognize the truly pathological specimens where you can't" rather than "everything you thought you knew is wrong."
You don't need higher order logic to present a nonstandard analysis approach at the same level as of rigour as a standard limits-based calculus course. Keisler wrote a great infinitesimal-based Calc book (http://www.math.wisc.edu/~keisler/calc.html).
In terms of pedagogy I've found that there's a huge leap that students make between Math focused on computations and anything involving proofs. The reason it's difficult to make Calculus rigorous is not because you have to address a ton of cases-it's because understanding proofs is really hard. This is why students who take Algebra and then Analysis or vice versa tend to do much better in their second course-because they're already used to proofs. So I don't really think it's possible to make a first-year Calculus course more rigorous by sticking to analytic functions, because you still have to get over the proof barrier.
http://www.youtube.com/watch?v=_Cv5ldhxpLA&feature=g-his...
Its by the fellow that does Minute Physics on Youtube. Towards the end of his talk, he specifically talks about this topic.
He makes a comparison between mathematics and physics, wherein mathematics you are taught the math without a ton of history. No scaffolding. In physics, you are taught the entire history of physics, i.e. scaffolding, from the ground up.
He also advocates directly teaching modern theory and leaving most of the history out. Make learning physics more like learning math.