A Gentle Introduction To Learning Calculus(betterexplained.com)
betterexplained.com
A Gentle Introduction To Learning Calculus
http://betterexplained.com/articles/a-gentle-introduction-to-learning-calculus/
24 comments
Thanks :). I don't know what it is, but for a field concerned with abstraction people miss the forest for the trees.
I try to catalog insights right as they happen, and try to capture the before-and-after: what, exactly, was the turning point that made it click? Unfortunately that makes articles slow coming, but I think they can be more genuine than explaining something after you're already an expert. Most of us can't remember what it was like trying to learn to read, for example -- it's too hard to relate to that problem.
I try to catalog insights right as they happen, and try to capture the before-and-after: what, exactly, was the turning point that made it click? Unfortunately that makes articles slow coming, but I think they can be more genuine than explaining something after you're already an expert. Most of us can't remember what it was like trying to learn to read, for example -- it's too hard to relate to that problem.
The author is obviously under the delusion that an introduction to calculus should be gentle and soothing, like slipping into a warm bubble bath. It should not. It should be swift and shocking, like jumping into a river in early Spring.
Mathematics requires an exact and expert hand. They are EMTs. This of your Calc 1 mid-term: "Quick, you have two minutes, resuscitate this equation!"
Mathematics requires an exact and expert hand. They are EMTs. This of your Calc 1 mid-term: "Quick, you have two minutes, resuscitate this equation!"
You would have eliminated Newton with this kind of thinking. The author's point was math is supposed to be exciting, mind-blowing, and gosh darn it, even fun. And the way they teach it at school is neither of these three. Extremely smart people love things that are exciting, mind-blowing, and fun, and what a horrible atrocity to scare these people away from math!
Rarely do math and science instructors teach material the way it is discovered. Also, math and science courses are typically taught the way history or English is taught--you have some topics, you have things to memorize, you're given questions that are very similar to those you've seen in school or on your homework. I can imagine that only top schools teach this material in terms of tools and knowledge, versus learning and writing down the right equations on a test.
That's exactly it. We show math as this complete and perfect system, whereas it was discovered, often times, through trial-and-error. Retracing the steps can help show why certain decisions were made.
For example, why does C have a separate short and long data types, but javascript just has "var"? Because when C was made, memory was at a premium. When javascript was invented, everything could be a 4-byte value.
For example, why does C have a separate short and long data types, but javascript just has "var"? Because when C was made, memory was at a premium. When javascript was invented, everything could be a 4-byte value.
Actually, the reason was more that C was designed for systems programming, and that sort of code often has to interface with hardware devices through memory-mapped registers. To do this, you need to be able to write code that uses variables of the correct width. This is why C has all the various sized variables in signed and unsigned variants.
Good point -- C was designed to interface with hardware ("high level assembly") whereas javascript was designed to interface with web pages & browsers. Either way, the history helps understand the design decisions.
Heh. Unfortunately, a large fraction of the students tend to die of hypothermia :).
The cold water is harsh, but not necessarily a "bad thing" as it weeds out the weaker ones.
The point is, "Do we want to spoon feed sciences?" Looking at what we, western society, boil into pablum to get people to pretend to comprehend, in my opinion, causes more problems than helps. It gives a false sense of understanding, like the solar system model of the atom.
Having said that I concede that there is no reason to purposefully obfuscate sciences and higher level math.
The point is, "Do we want to spoon feed sciences?" Looking at what we, western society, boil into pablum to get people to pretend to comprehend, in my opinion, causes more problems than helps. It gives a false sense of understanding, like the solar system model of the atom.
Having said that I concede that there is no reason to purposefully obfuscate sciences and higher level math.
Yeah, that's a good point. I definitely believe in having rigor when needed.
Concepts like calculus, unfortunately, wait until high school or college, so it's being thrown in the river without ever having swum. Sure, some people survive, but most people end up having an aversion to math/science which leads to innumeracy and a host of other problems.
Kids could be taught the basic ideas (like the circle -> triangle) just as a way to get thinking. Later, they can cover them more rigorously, but they've had experience doing integrals on their own.
Even the solar-system model is "ok" to start with, and later you explain about probability distributions etc. You don't want to jump into quantum mechanics without covering Netwonian Physics (which, technically speaking, is "incorrect but useful").
Concepts like calculus, unfortunately, wait until high school or college, so it's being thrown in the river without ever having swum. Sure, some people survive, but most people end up having an aversion to math/science which leads to innumeracy and a host of other problems.
Kids could be taught the basic ideas (like the circle -> triangle) just as a way to get thinking. Later, they can cover them more rigorously, but they've had experience doing integrals on their own.
Even the solar-system model is "ok" to start with, and later you explain about probability distributions etc. You don't want to jump into quantum mechanics without covering Netwonian Physics (which, technically speaking, is "incorrect but useful").
Some of the discussion is reminiscent of the "A Mathematician's Lament" essay:
http://www.maa.org/devlin/devlin_03_08.html
http://www.maa.org/devlin/devlin_03_08.html
this is copying the message and style of A Mathematician's Lament
I was definitely enjoyed that essay and want to spread its message; it's why I linked to it and quoted from it :).
However, I've been writing about these concepts in this way since 2001, way before I ever heard of the essay (here's my earlier website from school, which was the the precursor to this blog: http://www.cs.princeton.edu/~kazad/resources.htm).
However, I've been writing about these concepts in this way since 2001, way before I ever heard of the essay (here's my earlier website from school, which was the the precursor to this blog: http://www.cs.princeton.edu/~kazad/resources.htm).
oh, thought you were just another me too blog.
good on you then. :)
good on you then. :)
Hey, no prob -- I dislike me-too posts as much as the next guy :)
[deleted]
I stopped reading here: How does an equation grow and shrink? Accumulate over time?
Equations, don't grow or shrink. Maybe he was confusing equations with functions.
Equations, don't grow or shrink. Maybe he was confusing equations with functions.
You may be one of the people this article was explicitly not written for.
For some level of rigor, functions don't grow or shrink either.
Don't hate - participate!
[deleted]
I think the author's main point-- that most teaching of mathematics tends to undervalue the "aha!" types of insights that connect various pieces-- is a good one, and it looks like he has some concrete ideas about how to go about it.
I, for one, look forward to seeing the author follow through with additional concrete examples.