The key step with math is to do the exercises. Videos can get you interested, and help explain what you read, but that's it. In the end, you need a textbook, judged according to the quality of its exercises and then a maybe a study partner to make it fun and maintain commitment.
Anybody have insight into how to actualize these
nuggets into some semblance of a self-learning course?
Buy Calculus by Micheal Spivak. Solve at least one problem every day. Make it ritual and a daily requirement. Watch MIT lectures for corresponding chapter you are on.
To learn this, don't trouble over the path and reason at present. Buy the book and start. Right now.
Strang's Calculus book is available for free. Is the Spivak book a much better resource? I'm not familiar with either one, but I've seen both recommended before.
Haven't read his calculus book, but Strang's linear algebra book is the best math book I've ever read. It's actually readable! Certain chapters are available on line. Based on it I would definitely try his Calculus book if I needed one.
Spivak's book is more of an analysis text. If you are interested in mathematics, go with Spivak. If you're more interested in engineering or physics, you may be happier with Strang.
I have been intermittently trying to teach my 6 year old niece calculus graphically. She can visually tell what the slope of the curve is and filling in the area under the curve is, well child's play. :-)
I believe there was some work done using small talk to teach children physics where they were able to do quite complex things without having to understand calculus... can't quite find the link right now.
Try looking at the constructionist articles, specially Papert's works, as there are a lot of them online[1].
Basically, the idea is that you can introduce deep mathematical concepts very early by adopting a more intuitive media. Also, take a look at McLuhan[2] to understand how the media shapes the message if you really want to go deep into it.
While in university I took linear algebra and didn't understand much of it, especially on a deeper level. Then I stumbled upon Gilbert Strang's linear algebra lectures and watched them... After watching his explanations I got all of it and actually understood things at a much higher level. It was a sweet revelation and today I find linear algebra beautiful. I highly recommend watching his linear algebra lectures: http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-...
Edit: I also find linear algebra to be useful and much more important for programming/CS than calculus, especially for implementing various ranking algorithms (e.g. Google's PageRank algorithm is mostly rooted in linear algebra).
Thanks for reminding me of those lecture videos. I have a linear algebra test tomorrow, Wednesday, and these videos will certainly help me study for it.
I've to second that, it was a bit slow at times but overall pretty decent and far better than my university class.
Is there anything similar for Analysis I and/or II?
While that seems like a bad idea now, there's a very necessary reason for Newton doing it that way.
Newton presented Calculus exclusively in the more complex language of Euclidean geometry in order for it to be taken seriously by the intelligentsia, the royalty, and clergy of the era, as well as to make a bold statement about the historical magnitude of his discovery. In that era, Euclid and Newton were the book ends bounding all that was known of mathematics and natural philosophy(proto-science). So of course he had to write it the same way way Euclid would. Euclid's manner of logic and geometrical reasoning was lauded and unchallenged by both the church and the king, so there would be no way to refute a proof built upon the same language.
I think there are two levels of refutations which Newton was guarding against. First, it's the typical peculiar trait of mathematicians to achieve perfectionism in a proof. The margin for error is zero-it's either all right or it's all wrong. Everything you are working on is either the most important thing in the world, or utterly worthless folly. So it's a point of immense personal pride to present a work that is complete, irrefutable, and entirely self-consistent.
Second, considering the political & religious climate of the time, had Newton shown calculus without Euclid, the religious implications could have easily blown up into a scandal because his physics starts the ball rolling in formally challenging
several major foundations of Christian theology. If you consider that Newton and all professors of the era at Cambridge had to be approved as official clergy in the Church of England, so his job and his future career success would depend highly on not outraging his superiors. Many men who spark revolutions by challenging ancient power structures end up tragic martyrs. But Newton navigated that problem with equal mastery. He ended up the Master of the Mint, was elected to parliament, grew immensely wealthy, had the ear of the Queen, and became so famous that visiting foreign celebrities would mark him as the first person to visit upon arriving-even before the Queen!
For many decades after the publication of the Principia, Newton in fact concealed the derivation of the calculus using mathematical terminology we would now easily recognize as being similar to our own. But he did write it all down, he just never published it. There's an interesting story about this manuscript. Newton's friend, the secretary of the Royal Society kept this paper in safekeeping and private. Leibniz once visited England before he was famous and influential, and Leibniz knew Newton's secretary and visited him in London. The secretary allowed Leibniz to read Newton's private manuscript, describing the derivation of calculus, which no other people had ever seen. Not too much later, after Leibniz returned to the continent, he published his own book demonstrating calculus. Newton recognized the plagiarism, Leibniz denied it, and they maintained an intense personal feud for the next 40 years.
There is another post in response to this comment containing four full paragraphs explaining why Newton wasn't succinct, but the post is now marked as dead. If anyone familiar with the Leibniz vs Newton history wants to turn showdead on in their HN settings and read the post, I'd appreciate commentary on its accuracy and origins (and, perhaps, an explanation of why it was killed).
I'll taks a stab at guessing the killing: parent^2 was being facetious and a lengty serious answer was not considered appropriate.
I am mostly guessing though, and I don't think it should have been killed.
Based on my reading of Jim Gleick's biography Isaac Newton and my background as a theoretical physicist, I believe your comment is essentially accurate. In particular, Newton's desire to connect calculus with Euclidean geometry is certainly correct. As you note, the Principia eschews calculus in favor of geometric arguments: Newton used calculus for his private calculations and then translated the results into more conventional geometry to meet his audience's expectations. I'm not sure about the religious aspect, but Newton was privately a heretic (he believed in a unitary god—an awkward belief for a professor at Trinity College), so he definitely knew how to placate the religious authorities. Finally, the Leibniz anecdote is new to me, but it sounds plausible.
In any case, I don't think your comment should have been killed.
wow, GS is the man! he still looks remarkably similar to the way he did almost 10 years ago when i took his class.
EDIT (to make this post not as content-free): Prof. Strang keeps the hope alive that some distinguished faculty in top research universities still place an emphasis on great undergraduate teaching
No idea actually, but it was something he mentioned the first day of class. He introduced it as "K, the second difference matrix, and my favorite". That was good enough for us.
I haven't watched the videos, but be wary of anything that claims to simplify math into some awesomely brief time frame. In my experience you come away with a conceptual understanding, but no ability to apply it. Convolution, for example. 99% of pages on convolution spend a long time describing what it represents, and using nifty animations to show you, but you still come away unable to solve all but the most basic problems.
Anyone happen to know a good resource that takes a single problem to show how Geometry, Algebra, and Calculus can each be used to solve it? I'm hoping for something that can quickly demonstrate how each builds on the other to get better and faster results.
Prof. Gilbert Strang is a great teacher — I am amazed at how well he explains complex concepts in a simple way.
Does anybody know of a similar resource on probability, especially the Bayesian approach? All I could find were lectures of significantly worse quality than prof. Strang's teachings.
I struggled with high school calculus. I just couldn't wrap my head around the concept. My teacher kept making noises about "rate of change" but it made no sense. Luckily for me, I was taking physics at the same time, and we ran an experiment to calculate acceleration due to gravity.
So we ran the experiment with a weight and a ticker tape and a little hole punch tool and we got these data sets measuring the distance between each consecutive hole on the tape. Plotting distance against time on a graph, we produced a curve somewhat reminiscent of a y = x^2 function.
Then, given d2, d1, t2 and t1, we were able to calculate a set of velocities between each point. Plotting velocity against time on a graph, we produced a sloped line somewhat reminiscent of a y = 2x function.
And then, of course, given v2, v1, t2 and t1, we were able to calculate a set of acceleration rates between each point. Plotting acceleration against time on a graph, we produced a horizontal line somewhat reminiscent of a y = 2 function.
Then it hit me. Looking at the three graphs, in a flash I suddenly understood exactly what "rate of change" meant. I understood why d(x^2) = 2x, and why d(2x) = 2. Calculus made perfect sense, and I plowed through all the exercises that had plagued me since the start of the year.
So when I clicked on the first video in this OCW set [1] and watched Professor Strang put distance/speed and height/slope side by side as his two canonical examples, a big smile spread across my face.
38 comments
[ 2.8 ms ] story [ 71.5 ms ] threadAnybody have insight into how to actualize these nuggets into some semblance of a self-learning course?
To learn this, don't trouble over the path and reason at present. Buy the book and start. Right now.
http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098...
Buy it. To learn this- buy it and start. Right now.
http://ocw.mit.edu/resources/res-18-001-calculus-online-text...
Basically, the idea is that you can introduce deep mathematical concepts very early by adopting a more intuitive media. Also, take a look at McLuhan[2] to understand how the media shapes the message if you really want to go deep into it.
[1] http://www.papert.org/works.html [2] http://en.wikipedia.org/wiki/Marshall_McLuhan
http://www.amazon.com/Mindstorms-Children-Computers-Powerful...
Edit: I also find linear algebra to be useful and much more important for programming/CS than calculus, especially for implementing various ranking algorithms (e.g. Google's PageRank algorithm is mostly rooted in linear algebra).
Newton presented Calculus exclusively in the more complex language of Euclidean geometry in order for it to be taken seriously by the intelligentsia, the royalty, and clergy of the era, as well as to make a bold statement about the historical magnitude of his discovery. In that era, Euclid and Newton were the book ends bounding all that was known of mathematics and natural philosophy(proto-science). So of course he had to write it the same way way Euclid would. Euclid's manner of logic and geometrical reasoning was lauded and unchallenged by both the church and the king, so there would be no way to refute a proof built upon the same language.
I think there are two levels of refutations which Newton was guarding against. First, it's the typical peculiar trait of mathematicians to achieve perfectionism in a proof. The margin for error is zero-it's either all right or it's all wrong. Everything you are working on is either the most important thing in the world, or utterly worthless folly. So it's a point of immense personal pride to present a work that is complete, irrefutable, and entirely self-consistent.
Second, considering the political & religious climate of the time, had Newton shown calculus without Euclid, the religious implications could have easily blown up into a scandal because his physics starts the ball rolling in formally challenging several major foundations of Christian theology. If you consider that Newton and all professors of the era at Cambridge had to be approved as official clergy in the Church of England, so his job and his future career success would depend highly on not outraging his superiors. Many men who spark revolutions by challenging ancient power structures end up tragic martyrs. But Newton navigated that problem with equal mastery. He ended up the Master of the Mint, was elected to parliament, grew immensely wealthy, had the ear of the Queen, and became so famous that visiting foreign celebrities would mark him as the first person to visit upon arriving-even before the Queen!
For many decades after the publication of the Principia, Newton in fact concealed the derivation of the calculus using mathematical terminology we would now easily recognize as being similar to our own. But he did write it all down, he just never published it. There's an interesting story about this manuscript. Newton's friend, the secretary of the Royal Society kept this paper in safekeeping and private. Leibniz once visited England before he was famous and influential, and Leibniz knew Newton's secretary and visited him in London. The secretary allowed Leibniz to read Newton's private manuscript, describing the derivation of calculus, which no other people had ever seen. Not too much later, after Leibniz returned to the continent, he published his own book demonstrating calculus. Newton recognized the plagiarism, Leibniz denied it, and they maintained an intense personal feud for the next 40 years.
In any case, I don't think your comment should have been killed.
I can't take credit for the comment. It was posted by HN user korch. I only wanted to know if it was correct.
EDIT (to make this post not as content-free): Prof. Strang keeps the hope alive that some distinguished faculty in top research universities still place an emphasis on great undergraduate teaching
Incidentally, what makes that his favorite matrix?
The Indian Institutes of Technology have full video for a lot of their engineering curriculum on youtube.
Calculus Made Easy, 2nd Ed (1914)
http://www.gutenberg.org/ebooks/33283
One of the best explained calculus texts I've read.
Does anybody know of a similar resource on probability, especially the Bayesian approach? All I could find were lectures of significantly worse quality than prof. Strang's teachings.
So we ran the experiment with a weight and a ticker tape and a little hole punch tool and we got these data sets measuring the distance between each consecutive hole on the tape. Plotting distance against time on a graph, we produced a curve somewhat reminiscent of a y = x^2 function.
Then, given d2, d1, t2 and t1, we were able to calculate a set of velocities between each point. Plotting velocity against time on a graph, we produced a sloped line somewhat reminiscent of a y = 2x function.
And then, of course, given v2, v1, t2 and t1, we were able to calculate a set of acceleration rates between each point. Plotting acceleration against time on a graph, we produced a horizontal line somewhat reminiscent of a y = 2 function.
Then it hit me. Looking at the three graphs, in a flash I suddenly understood exactly what "rate of change" meant. I understood why d(x^2) = 2x, and why d(2x) = 2. Calculus made perfect sense, and I plowed through all the exercises that had plagued me since the start of the year.
So when I clicked on the first video in this OCW set [1] and watched Professor Strang put distance/speed and height/slope side by side as his two canonical examples, a big smile spread across my face.
[1] http://ocw.mit.edu/resources/res-18-005-highlights-of-calcul...