Vi Hart: Every school should replace calculus with recreational math

  > What many -- perhaps even most -- kids really want is a
  > course where they do only simple mechanical work, and
  > get a boost to their GPA.

  > ... not all teachers care. 

And largely all of your points are exactly right, except for your misconception about what kind of math we want kids to be doing.

Going back to that point, the IMO is like playing at Carnegie Hall. What we want is like messing about with instruments to see what they do, and getting hooked on the curious things that are possible.

I'm not offering solutions here, I'm just helping define the problem, and explore directions.

For example, it's quite important for me to know when I might need to compute an integral, and what I'd do with it. But when it comes to banging out the symbolic manipulations, remembering tables of what integrates to what, recalling integration strategies for common kinds of integrals, etc., Mathematica is more skilled than I am, so I defer to its expertise, and haven't tried to keep any of that stuff in my head since high school.

Appealing to symmetry or applying Green's theorem (e.g.) will save time over "mechanical processes". Also avert mistakes and be more comprehensible.

{if you don't know what I mean: what's ∫sin x · cos^55 x · abs|x|^131 · x^444 · exp(-x^2) ?}

Moreover only conceptual understanding allows the development or use of certain things, like SVD, wavelets, eigendecomposition, game theory, theory of distributions, topology, single-crossing curves...

I'm hesitant to endorse any paradigm that aims to make math more intuitive, since at the end of the day that's a lie.

Is it a lie? It sounds like maybe you just didn't experience the intuition personally.

math ... will contradict your intuition

To me the point is to re-ground intuitions in true mathematical facts. For example I intuitively think of the reals as transcendentals ∪ algebraics or as the completion of the rationals, since I learned counter-intuitive things about the reals that disfavour the "possibly nonterminating decimal" view.

paradigm ... pace

Great distinction. Grand changes are often proposed when small changes might do.

But actually inventing that time saving machinery is different from applying it.

But let me at least thank you for this: "The primary aim of math class up through calculus is to teach kids to perform the calculations needed to do physics, which is needed to do engineering, which is generally perceived (correctly I think) to be greatly beneficial to society."

This is the best summary I have yet seen about why math is taught the way it is (in high-school, not in university) -- and by extension why many people, incl. myself, feel or felt uncomfortable with math up to high-school.

But let's be realistic about this. "Interesting" is an opinion, and the number of students who don't already find it interesting who would if we took the time to explain it would be fairly small.

    why do you need assessment?

1. Are very desirable (lucrative, secure, etc.).

2. Require very high mathematical proficiency.

3. Have drastic consequences if performed poorly.

It's self-evident why assessment is an important function of the education system.

The question becomes: is it more important to correctly identify potential engineers and scientists and to maximize their proficiency at their careers, or more important that Joe the Plumber can balance his checkbook when he goes straight into the workforce after his high school diploma?

I doubt that we can optimize both of these cases simultaneously.

  > If you don't get the fundamentals down,
  > you can't handle the applications.

  > The real question is, why do you need assessment?

FWIW - I advocate a radical shift in what's done, and am working subversively on a ten year plan to make changes. Just becuase I can identify some problems, don't assume I believe they are real, or insurmountable.

Puzzles arise from the real world, and they arise from abstract structures. Regardless, they are puzzles to solve, and the techniques of pure math and applied math can be used.

Thing is - not all students like puzzles, not all students like curiosities, some students actively want stabilty and assurance.

Some people aren't cut out to be entrepreneurs.

I'm just trying to be charitable to people who don't want music, drama, driver's ed, health class, ........ cut out.

Why should recreational mathematics replace calculus, either? Ideally math would be taught well and written so that people could engage with it throughout life. (But how many adults do you see reading those yellow spiney Kluwer books at coffee shops?)

slippery slope

Thought Vi Hart was talking about high school. Maybe I was wrong. I agree that calculus is the right kind of brain-stretching for college.

Really it depends on what you want to do...but Python is widely used in science, finance, math, and the web (thru django). So that covers all of what I'd want. And it's a high-level (read: easy), widely used (read: supported) language.

If I were just starting out I'd use zed shaw's book. As it was I found the django tutorial good; and Norman Matloff's tutorial on scientific python.

I'd learn programming in linux. Install Ubuntu if you don't already use linux. You can dual boot and leave the windows side for video games.

HTH

And once you've struggled to solve the problems that can't be brute forced away, you start wondering if maybe a little linear algebra wouldn't do you good...

Primary sources are way above the level of most college mathematics students, much less high school or younger students. Almost anything written in the last 50 years requires very advanced mathematical ideas, while anything written before that uses outdated notation which is very difficult to understand. Moreover, the primary sources for common ideas are often spread out over huge periods of refinements - going from a small idea to a general notion over the course of years, perhaps involving dozens of papers by multiple authors. One notable exception to this: Euler's papers. These might be readable by some students, yet when I see them presented in college courses, they are often presented by a secondary source for clarification.

The other thing I don't agree with is your use of the word "interpret." I don't think that it is common to find a mathematician who believes that mathematics is open to interpretation.

Lastly, original research in mathematics is hard. It's not the interpretation of previous ideas (although it is occasionally relating concepts previously thought to be unrelated - some consider this the most important type of mathematical result, but it is much harder to do than you might think), but the formulation of new ideas. Even to know if an idea is new requires a great deal of mathematical training.

Perhaps I'm wrong about this, but I think that for young mathematics students your ideas are unfeasible. Remember that the original sources for a lot of basic mathematics are hundreds of years old. Even Euler's famous writings on geometry (the exception that I said might be workable) were largely a rewriting of previous ideas into a coherent whole - basically a textbook - and could not really be considered an original source.

Edit: my apologies, I misunderstood. I did not realize that we were talking about teaching history, in which case I defer to someone of greater experience.

The Elements of Euclid is a model of clear, concise, beautiful mathematics which is easily accessible. Archimedes as well. Follow that up with Apollonius, Ptolemy, some Descartes, and then Newton, and you have a junior high and high school curriculum in mathematics that would give students a real advantage over the rote "here is what Disembodied Authority says you should know" learning.

Note -- I am not advocating the study of mathematical history in junior and high school, but rather the study of mathematics historically. It gives access to mathematics as a branch of the humanities, less focused on the answers, and much more focus on the questions and how some really smart people have addressed those questions in the past, which gives a good guide on how to address new problems that will come up in the future.

When it comes to history specifically, as the OP asked, I would still agree with going with some of the great works of history. Historiography can be left for college, but give the kids access to the letters and diaries and personal accounts of people who were at the scene of history, as well as the great works of history that have been written (Thucydides, Herodotus, all the way to Toynbee, Gibbon, and, hell, even Spengler). Drop the mundane and milquetoast textbooks.

I have had some experience with reading primary sources though. I had an excellent history teacher in high school who had us read The Autobiography of Benjamin Franklin to prepare us for American History. Now, Benjamin Franklin is a fascinating person, and he did plenty of extremely interesting things, but I had a lot of trouble getting through the style it was written in. I love learning about the things he did, but the language and style of his autobiography just did not work well for me.

Primary sources are an option, but it doesn't work for everyone.

There's a lot more that could be done, and it's not clear it imparts only accurate information, but for me at least, it's where I learned that "fording a river" was a thing that existed; that Chimney Rock is a landmark; that there was a period in history when wagon trains were regularly heading out west along this lengthy route; etc.

Of course, this approach does rather depend on having a lot of reasonably interesting historical resources within easy reach.

The experiment has been done - the vast majority of the kids don't watch the videos.