Thanks for the comment. There's this weird notion in Calculus education that we need to start from first principles. Limits were invented a century after Newton died, yet they're taught first. "Oh, students won't understand calculus unless they can build it from first principles. I don't care if Newton worked out gravitation with his understanding, it's not good enough."
My little candle in the darkness (http://betterexplained.com/calculus/lesson-1) is to start with intuitive notions (X-Raying a pattern into parts, Time-Lapsing parts into a whole) and then gradually introducing the terms. Eventually, if people are interested, we can get into the theory (which is like getting into the Peano axioms of arithmetic, if you care to go that deep.)
I think in math there's a tendency to express things in the lowest-level "machine code" we can. We need more comments and pseudocode outlines :).
> Thanks for the comment. There's this weird notion in Calculus education that we need to start from first principles. Limits were invented a century after Newton died, yet they're taught first. "Oh, students won't understand calculus unless they can build it from first principles. I don't care if Newton worked out gravitation with his understanding, it's not good enough."
It depends. In high school, it certainly makes sense to start from an intuitive, rough idea, but in college/uni, I much preferred classes that started from first principles (axiomatic QM, sets → topology → metric spaces → calculus/algebra) etc. rather than the weird classes which define nothing properly and put in a “this is true, but I won’t tell you why :P” every other lecture. Which approach is preferable obviously also depends on your aims, but I found it hard to get a thorough understanding of a topic without a first-principles-based derivation at least at some point.
For a potential math major [i.e. people for whom metric space refers to a measure and not a flat in Europe :)], you definitely want the ground-up understanding. In CS it's similar, where you learn about transistors, logic gates, ALUs, CPUs, machine code, compilers, along with high-level languages.
But, some people just need the HTML "Hello World" to make their webpage. (In the math field, we have students who need calculus primarily to find the min/max of a function, and are wasting time worrying about epsilon-delta definitions of continuity. Limits are interesting, but I'd prefer to ignite curiosity with higher-level topics then dive into the details, instead of forcing someone to learn organic chemistry before being able to drive a car.)
Where I studied, we did have classes on all the low level stuff, but we didn't start there: I took a class taught using a high level language every semester. CS-100 was intro to how computers worked, but right along with it, you had 101 teaching C.
If you are going to end up teaching software engineering material, compiler design and such, you just can't have people that have barely done any programming for a year or two, or all they are going to learn is boxes next to each other, instead of being able to actually write a simple compiler in the compiler class.
Whoops, I probably wasn't clear enough in my reply -- the classes on transistors, logic gates, etc. are available to CS majors but are definitely not a requirement for programming. I completely agree that trying to teach about digital logic, etc. without the general context of programming is backwards.
Drive the car, build your interest, then start taking classes in chemistry, physics, etc. to see how it works.
Same thing with logarithms, which were invented to simplify multiplying two huge numbers. Combined with log tables (think paper LUT for humans) that became a simple matter of looking up two log conversions, adding them, then looking up the inverse of the answer.
Is that what we learn in high school? Nope. So everyone is left wondering what the hell they are good for (or were, before we had calculators) the first time around.
(btw, as far as I can tell you don't have that bit of history on your site yet either - might be an interesting addition?)
Well, I learned that at school. It still didn't answer what the hell logarithms are good for. Everyone was left wondering why we were learning something that can be replaced by calculators.
Seeing some modern application of lagarithms would be great. Even just making a log-log graph at some point would answer every question. But those are not at the official curriculum.
It drives me batty because we learn the properties of logarithms before (if ever) internalizing what they mean. Here's my intuition if it helps someone:
Exponents let you plug in time, and get the amount of growth. e^3 ~ 20, which means "3 periods of 100% continuous growth [100% is implied by e, 3 = 3* 1] will grow us from 1 to 20".
Logarithms let us plug in the growth, and get the time it took to get there.
ln(20) ~ 3 means "It takes 3 units of time [growing at 100%, continuously] to grow from 1 to 20".
Exponents take inputs and find the future state, logs take the future state and work backwards to find the inputs that got us there.
Great tidbit, thanks. I forget if I have a mention that logs were developed before e, but I love it because it's a little mosquito in the ear for the over-rigorous mathematicians who define e up front (as a limit) and then say "the natural logarithm is log base e".
e was only discovered because of logarithms, not the other way around! :)
I find your emphasis of building intuition very important.
Personal anecdote related to Pythagorean theorem/euclidean distance.
When in ninth grade, I asked my math teacher to explain why why the the standard deviation is like it is and why the square root and not something else. He could not explain that to me satisfactorily.
I spend few hours looking at the formula myself and plotting and drawing different datasets until I made the following association: if dataset is like N dimensional vector and its values x1, x2, .. xN are coordinates, then I'm looking at euclidean distance from mean plus adjusting for the number of dimensions. I think this single moment of cryptic formula making sense changed my attitude towards statistics and math. Dataset ~ vector and data-point ~ vector-coordinate association helps to understand statistics using geometrical intuition.
That's pretty amazing you came up with that in 9th grade. You may know this already but for anyone who doesn't, this is a result of covariance being an inner product [0]. Remember that the inner product of a vector with itself is the length of that vector squared. So if you had another data set, y1, ..., yN, the correlation ( = covariance / (std.dev(X)*std.dev(Y)) ) is the inner product divided by each length which, if you remember, is the angle between your vectors (datasets after removing the means.)
Great example, I like that. I started realizing that many instructors don't necessarily have an intuition for the topic they're teaching, and it's often up to the individual student to seek it out. I'm hoping to do more stats concepts down the line, this is a nice analogy for why standard deviation is defined as it is.
Kalid - First of all, your site kicks ass! Keep doing what you're doing. You probably can't even imagine the 2nd and 3rd order impacts of the benefits you provide. Someone gets one insight from your site, that makes something they learn in school more comprehensible, and then it explains it to someone else, who then can learn more quickly....
Bithive123 - I found that practicing Project Euler has helped my understanding a lot. Part of it was that it made math a joy again, and I tied it to learning new programming languages.
Thanks so much. My meta goal is to help people gain confidence in what it means to learn (in general, not just math) and ideally encourage people to share their insights with others. No reason for us to all bump our shins on the same coffee table in a dark room, someone turn on the lights on your way out!
Unfortunately, most of the mathematical text miss the historical context which they are developed in. I personally enjoy a text in any subject, where the author develops the content with historical perspective.
Any learning course should include one textbook which develops the content from an historical perspective.
I agree. A few years after doing articles, I started seeing the power of historical context.
For example, the Fourier Transform was originally _rejected_ as implausible when presented to the mathematicians of the time. They needed a decade of debate to verify its truth, and yet we teach it to students in a week and expect them to internalize it without issue. We need to acknowledge the difficulty/counter-intuitive nature of the idea up front.
Kalid, I just read your article on Prime numbers (not-quite randomly selected from the homepage). I just wanted to say how well written I found it - even beyond the technical side of things, your writing style, tone, and humility ("if I didn't mess it up") are fabulous.
Thank you Jacob, I really appreciate it. I try to imagine that I'm writing to a younger version of myself, who hasn't seen the material yet. It helps me remove potential anxiety about not knowing everything, or trying to impress -- there's no reason to deceive yourself about what you do or don't know.
Since we've mentioned the article, I can't help asking about a particular tit-bit that caught my attention --what do primes have to do with quantum mechanics?
Primes appear to be zeroes of the Reimann Zeta Function. Nature loves minimizing energy, so there might be some reason the Zeta Function models how atoms behave. Then, primes are the most "stable orbits" or have a similarly useful property.
Thanks! I am a mathematician, so am fine with the discussion of the 𝜁 function. I get the impression that de Sautoy is overselling the connexion a little bit in his zeal to make an analogy, but (1) I don't know the area, and so shouldn't really comment, and (2) it's interesting even if it's oversold.
Thanks for the comment. There's this weird notion in Calculus education that we need to start from first principles. Limits were invented a century after Newton died, yet they're taught first. "Oh, students won't understand calculus unless they can build it from first principles. I don't care if Newton worked out gravitation with his understanding, it's not good enough."
My little candle in the darkness (http://betterexplained.com/calculus/lesson-1) is to start with intuitive notions (X-Raying a pattern into parts, Time-Lapsing parts into a whole) and then gradually introducing the terms. Eventually, if people are interested, we can get into the theory (which is like getting into the Peano axioms of arithmetic, if you care to go that deep.)
I think in math there's a tendency to express things in the lowest-level "machine code" we can. We need more comments and pseudocode outlines :).