Posts tagged with education

## CALCULUS: the Really Really Short Version

So, you never went to university…or you assiduously avoided all maths whilst at university…or you started but were frightened away by the epsilons and deltas…. But you know the calculus is one of the pinnacles of human thought, and it would be nice to know just a bit of what they’re talking about……

Both thorough and brief intro-to-calculus lectures can be found online. I think I can explain differentiation and integration—the two famous operations of calculus—even more briefly.

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Let’s talk about sequences of numbers. Sequences that make sense next to each other, like your child’s height at different ages

not just an unrelated assemblage of numbers which happen to be beside each other. If you have handy a sequence of numbers that’s relevant to you, that’s great.

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Differentiation and integration are two ways of transforming the sequence to see it differently-but-more-or-less-equivalently.

Consider the sequence 1, 2, 3, 4, 5. If I look at the differences I could rewrite this sequence as `[starting point of 1]`, +1, +1, +1, +1. All I did was look at the difference between each number in the sequence and its neighbour. If I did the same thing to the sequence 1, 4, 9, 16, 25, the differences would be `[starting point of 1]`, +3, +5, +7, +9.

That’s the derivative operation. It’s basically first-differencing, except in real calculus you would have an infinite, continuous thickness of data—as many numbers between 1, 4, and 9 as you want. In R you can use the `diff` operation on a sequence of related data to automate what I did above. For example do

• `seq <- 1:5`
• `diff(seq)`
• `seq2 <- seq*seq`
• `diff(seq2)`

A couple of things you may notice:

• I could have started at a different starting point and talked about a sequence with the same changes, changing from a different initial value. For example 5, 6, 7, 8, 9 does the same +1, +1, +1, +1 but starts at 5.
• I could second-difference the numbers, differencing the first-differences: +3, +5, +7, +9 (the differences in the sequence of square numbers) gets me ++2, ++2, ++2.
• I could third-difference the numbers, differencing the second-differences: +++0, +++0.
• Every time I `diff` I lose one of the observations. This isn’t a problem in the infinitary version although sometimes even infinitely-thick sequences can only be differentiated a few times, for other reasons.

The other famous tool for looking differently at a sequence is to look at cumulative sums: cumsum in R. This is integration. Looking at “total so far” in the sequence.

Consider again the sequence 1, 2, 3, 4, 5. If I added up the “total so far” at each point I would get 1, 3, 6, 10, 15. This is telling me the same information – just in a different way. The fundamental theorem of calculus says that if I `diff( cumsum( 1:5 ))` I will get back to +1, +2, +3, +4, +5. You can verify this without a calculator by subtracting neighbours—looking at differences—amongst 1, 3, 6, 10, 15. (Go ahead, try it; I’ll wait.)

Let’s look back at the square sequence 1, 4, 9, 25, 36. If I cumulatively sum I’d have 1, 5, 15, 40, 76. Pick any sequence of numbers that’s relevant to you and do `cumsum` and `diff` on it as many times as you like.

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Those are the basics.

##### Why are people so interested in this stuff?

Why is it useful? Why did it make such a splash and why is it considered to be in the canon of human progress? Here are a few reasons:

• If the difference in a sequence goes from +, +, +, +, … to −, −, −, −, …, then the numbers climbed a hill and started going back down. In other words the sequence reached a maximum. We like to maximize things, like efficiency, profit,
• A corresponding statement could be made for valley-bottoms. We like to minimise things like cost, waste, usage of valuable materials, etc.
• The `diff` verb takes you from position → velocity → acceleration, so this mathematics relates fundamental stuff in physics.
• The `cumsum` verb takes you from acceleration → velocity → position, which allows you to calculate stuff like work. Therefore you can pre-plan for example what would be the energy cost to do something in a large scale that’s too costly to just try it.
• What’s the difference between income and wealth? Well if you define `net income` to be what you earn less what you spend,

then `wealth = cumsum(net income)` and `net income = diff(wealth)`. Another everyday relationship made absolutely crystal clear.
• In higher-dimensional or more-abstract versions of the fundamental theorem of calculus, you find out that, sometimes, complicated questions like the sum of forces a paramecium experiences all along a sequential curved path, can be reduced to merely the start and finish (i.e., the complicatedness may be one dimension less than what you thought).
• Further-abstracted versions also allow you to optimise surfaces (including “surfaces” in phase-space) and therefore build bridges or do rocket-science.
• With the fluidity that comes with being able to `diff` and `cumsum`, you can do statistics on continuous variables like height or angle, rather than just on count variables like number of people satisfying condition X.
• At small enough scales, calculus (specifically Taylor’s theorem) tells you that "most" nonlinear functions can be linearised: i.e., approximated by repeated addition of a constant `+const+const+const+const+const+...`. That’s just about the simplest mathematical operation I can think of. It’s nice to be able to talk at least locally about a complicated phenomenon in such simple terms.
• In the infinitary version, symbolic formulae `diff` and `cumsum` to other symbolic formulae. For example `diff( x² ) = 2x` (look back at the square sequence above if you didn’t notice this the first time). This means instead of having to try (or make your computer try) a lot of stuff to see what’s going to work, you can just-plain-understand something.
• Also because of the symbolic nicety: post-calculus, if you only know how, e.g., `diff( diff( diff( x )))` relates to `x` – but don’t know a formula for `x` itself – you’re not totally up a creek. You can use calculus tools to make relationships between varying `diff` levels of a sequence, just as good as a normal formula – thus expanding the landscape of things you can mathematise and solve.
• In fact `diff( diff( x )) = − x` is the source of this, this

, this,

, and therefore the physical properties of all materials (hardness, conductivity, density, why is the sky blue, etc) – which derive from chemistry which derives from Schrödinger’s Equation, which is solved by the “harmonic” `diff( diff( x )) = − x`.

Calculus isn’t “the end” of mathematics. It’s barely even before or after other mathematical stuff you may be familiar with. For example it doesn’t come “after” trigonometry, although the two do relate to each other if you’re familiar with both. You could apply the “differencing” idea to groups, topology, imaginary numbers, or other things. Calculus is just a tool for looking at the same thing in a different way.

Thank goodness someone with sense and mathematical credentials (W.W.Sawyer) has put the ghastly A Mathematician’s Apology to bed.

That Hardy was a very great mathematician is beyond question…. However, when any person eminent in some field makes statements outside that field, it is legitimate to consider the validity of these statements….

Hardy writes

I hate ‘teaching’….I love lecturing, and have lectured a great deal to extremely able classes. [2.]

Here lecturing means imparting mathematical knowledge to those able to understand it with little or no difficulty; teaching means giving time and effort to make it accessible to those who require assistance…. Good [management] consists in appreciating the merits of a wide variety of individuals and combining them into an effective team. [I]t is precisely this appreciation that Hardy lacks. He makes the extraordinary statement

Most people can do nothing at all well. [3.]

…[H]e regards you as doing well only if you are one of the ten best in the world at this particular activity…. [T]hat very few people do anything well is [then] an [obvious] consequence.

However in life we continually depend on the co-operation of men and women far below this exacting standard….

[E]ven … the … process that links the great mathematicians of one generation to those of the next [depends on them]. There may of course be direct contact, as when Riemann [studied] … under Gauss. But the fact that Gauss was able to reach university at all was due to two teachers, Buttner … and Bartels….[4.]

In science the importance of the expositor is perhaps as great as that of the discoverer. Mendel’s work in genetics remained unknown for many years because there was no one to publicize it and fight for it as Huxley did for Darwin.

He makes this curiously objective division of mankind into minds that are first-class, second class and so on…. There is no part of this that should be accepted as sound advice. If there is something you think worth doing, that you are able to do, that you have the opportunity to do, and that you enjoy doing, wisdom lies in getting on with it, and not giving a second’s thought to what ordinal number attaches to you in some system of intellectual snobbery. As for concern with the self, you are both happiest and most effective when you are so absorbed in what you are doing that for a while you forget the limited being that is actually performing it.

Since [2008], the [US] labor force participation rate (LFPR) has dropped from 66 percent to 63 percent. [Out of 314M people.] Many people have left the labor force because they are discouraged … (U.S. Bureau of Labor Statistics data indicate that a little under 1 million people fall into this category)….

…Knowing the reasons why people have left (or delayed entering) the labor force can help us [guess] how much of the ↓ might … ↑ if the economy ↑ and how much is permanent. (For more on this topic, see herehere, and here.)

The chart … shows the distribution of reasons in the fourth quarter of 2013…. Young people [usually say they] are not in the labor force … because they are in school. Individuals 25 to 50 years old who are not in the labor force mostly [say they] are taking care of their family or house. After age 50, disability or illness becomes the primary reason [given]—until around age 60, when retirement begins to dominate.

Of the 12.6 million increase in individuals not in the labor force, about 2.3 million come from people ages 16 to 24, and of that subset, about 1.9 million can be attributed to an increase in school attendance (see the chart below).

off-topic sidenote: the natural cohort —vs— year adjustments, like “the baby boom has shifted 7 years since 7 years ago” are an economic example of the covariant/contravariant distinction

hi-res

In any formal or semi-formal teaching, there are always two conflicting paradigms being carried out simultaneously.

The first one is a capitalist paradigm, where the teacher functions as a boss who assigns work, for which the students earn credit.

(Source: www2.hawaii.edu)

The spirit of mathematics is not captured by spending 3 hours solving 20 look-alike homework problems. Mathematics is thinking, comparing, analyzing, inventing, and understanding.

The main point is not quantity or speed—the main point is quality of thought.
Geometry and the Imagination with Bill Thurston, John Conway, Peter Doyle, and Jane Gilman

(Source: geom.uiuc.edu)

Over a year ago, I wrote a letter to the editor of the Journal of Computational Sciences, urging the retraction of Bollen, Mao, and Zeng’s paper, “Twitter Mood Predicts the Stock Market.” Since JoCS is an Elsevier journal, one does not simply email the editor.

Rather, one has to register with the Elsevier author system, … submit `LaTeX` source code of a letter, along with supporting documents, author bio, .… I distilled the main arguments into two:

1. first, that the Granger causality tests presented in BMZ’s paper are … datamining, and present no evidence for a connection between Twitter and the Dow Jones Index;
2. and that the quoted predictive accuracy of the forecast model is so high, it would … [contradict] the experiences of … [traders] … and so this forecast accuracy is likely to be erroneously reported.
I included references to BMZ’s failed attempts to commercialize their patented techniques with Derwent.

Following the strictest protocol, the editor of JoCS duly sent this letter to reviewers . After roughly seven months, …

The reviewers’ comments were more than fair. If my arguments were unclear, I was more than happy to reword them and provide additional evidence to get my point across. So I edited my letter to the editor, and re-sent it. …

…within two months or so (the equivalent of overnight in journal-time), the editor sent me a rejection notice with … review, quoted below. This review—this review is sensational. As one afflicted with Hamlet Syndrome, I admire Reviewer #4’s conviction. As someone too often in search of the right phrase to dismiss a crap idea, I take delight in Reviewer #4’s acid pen: I have never seen a reviewer so viciously shit-can a paper before. Reviewer #4 tore my letter to pieces, then burned the pieces. Then poured lye on the ashes. Then salted the earth where the lye sizzled. Then burnt down the surrounding forest, etc.

Discussing the first year of life.

• In-born aptitudes include
• theory of other peoples’ minds (looking at where they’re looking, show surprise when they know something they “shouldn’t”) (this gives some idea of what autistic people go through—they lack a skill that one-year-olds have!)
• animate versus inanimate things—causality, essentially

Infants do reason. They just have less knowledge than adults.

Brain hemispheres act the same at birth but begin to specialise over time. (So could there be an alien environment where human babies would specialise their brains along different lines?)

There’s no one thing that makes the human brain superior to other animals’ brains (at least not that we’ve found). It’s thought to be the interaction of various factors—as well as our long developmental period—that make us able to build rocket ships, paint potato eaters, invent radio and discuss these things on it.

(Source: BBC)

Believing that people believed the Earth was flat is a good example of a modern myth about ancient scientific belief. Educated people have known it was spherical (and also how big it was) since the time of Eratosthenes.

Geoffrey Carr

(And, via tolerablescamp.tumblr.com, here’s Carl Sagan making the same point quite powerfully:)

Sagan: "Eratosthenes’ only tools were sticks, eyes, feet, and brains."

(Source: edge.org)

Playing around with polaroid screens at Google.

If you shine light through

• a light polariser `A`
• another light polariser `B` that’s perpendicular to `A` (i.e., `A ⊥B` or `A×B=0`)
• i.e., `AB` represents “shine the light through `A` then through `B` which` ⊥A`
• then no light comes through (it’s black.)

If you shine light through

• a polaroid `A`
• another polaroid `B ⊥ A`
• a third polaroid `C` that’s halfway between `A` and `B` (either halfway)

then no light comes through (it’s black.)

So far, formulaically, we have:

• `AB=0`
• `BA=0`
• whence it follows
• `ABC=(AB)C=(0)C=0`
• `CBA=C(0)=0`
• `BAC=0`
• `CBA=0`

But! This is surprising to watch and surprising to see the formula.

• If you shine through `A` then `C` then `B`, it’s kind of light!
• `ACB≠0`
• furthermore
• `BCA≠0`

Woo-hoo, Noncommutativity!

Earlier in the talk Ron Garret does a two-slit experient with two mechanical-pencil leads and a laser pointer. Wave-particle duality with an at-home science kit.