Awesome, glad it helped you! I know what you mean – sometimes we get stuck in the nitty-gritty of integrals and derivatives that we don’t realize that calculus was buried inside the formulas we learned in middle school :).
[…] SF Chronicle Advice to new graduates Calculus: Better explained Possibly related posts: (automatically generated)Weekly Bread for the Russophile- “The Word’s Worth” by Michele BerdyThe setting SunThe Sunday Travel Section: Oh, Those Russians […]
One of the things that I have tried to encourage engineers and mathematicians to do is to tell things to me as though I were 8 years old. You’ve done that here and I am a wee bit wiser for it. Nothing is difficult if the teacher cares enough to make it simple. Congratulations for a [formerly] calculus-shy lawyer.
Hi Richard, glad you found it useful! The funny thing is that many engineers & mathematicians would prefer the 8-year old version too! Many people end up learning the mechanics but not the insight of the operations. Thanks again for the comment.
Kalid, Thanks for such a wonderful article. So far I have never understood maths “the way it needs to be understood”. Your article was enlightening.
I also appreciate your efforts in replying to each of the comments.
My wish is that you write a book on Maths in “Simple & easy to understand” way and i would definitely recommend it. Many Thanks Again.
Hi Vasanth, thanks for the message :). I think math can be understood a variety of ways (intuitively, mechanically, etc.) and you need them all to have a good grasp. Usually, though, we only focus on the mechanical aspects.
On the book, I think it would be a great idea. Currently I’m looking into collecting these pages and organizing them into a series. Maybe after I get a few calculus posts under my belt ;).
I’ve been reading your blog for months now, and I think everything you write is well thought out, informative, and above all, interesting!
Calculus was by far my favorite math subject. I had so many “a-ha!” moments that I felt like the world was different after I learned it.
I think it would be great if in a follow-up article you discuss the relationship between velocity and acceleration. In this modern world there’s so many everyday analogies to be made, and I think determining the rate of change of a rate of change is something that is easily overlooked, yet so elegant once you realize it. It might also be too simple for your blog, but I’d love to see a Kalid explanation for it!
Another related topic I think would be simple yet interesting is events happening in instantaneous vs discrete time, although personally I can’t think of any good examples for that. I just remember how shocked I was that we could determine an object’s velocity at any given instant and totally remove change in time from the equation, yet it is still inherently dependent on time!
Hi Kai, thanks for the message! I’m glad you’re enjoying the articles, I really believe that there are interesting nuggets in any subject – sometimes we just have to dig for them :).
I think the relationship of acceleration to velocity is a good one, I’m thinking about how best to present it. Even in a car, you don’t set your speed – you push down the gas or brake, which accelerates you, which changes your velocity, which changes your distance. So really, the distance you travel is ultimately a “function” of where your foot is on the pedal. I think it’d be an interesting topic – no subject is too simple :).
The use of instantaneous rates is intriguing as well, I’m trying to figure out the best way to approach the limit concept. It’s essentially a machination from the 1800s to deal with “infinitely small changes”/infinitesimals which had been used intuitively before then. Again, a topic that will need a bit of thinking.
Thanks again for the comment!
“
Did you know we don’t learn calculus the way Newton and Leibniz discovered it?
”
I have often thought about this one… Thanks!
Hi, I’d like to first congratulate on that example. And I’d like to point out that that’s the first time i’ve seen that, or anything like that, witch should be mind bogling since i’m a seniour student in a technical college and my knowledge of math and physics is way above that of the average layman.
I have my own example of math being tought moronically. I remember once when our high school physics teacher asked us what was the integral(antiderivative) of 1/VdV(the work being done in an isothermal transformation), and no one had a clue, witch was rather odd given that we we’re pretty good at math, and all of us knew the antiderivative for 1/xdx however the antiderivative of 1/VdV, was a whole together diferent story.
@Sarnath: Yes, I consider it ironic that Newton probably wouldn’t recognize calculus as we teach it today :).
@Paul: Thanks for the message. Yep, sometimes we get so deep into the nitty-gritty that we forget how calculus can help us see relationships between “everyday” equations. And sometimes we get sidetracking when a variable is replaced :).
I “StumbledUpon” this a few days ago, and thoroughly enjoyed it! I had, especially in retrospect, a great calculus teacher in high school. We learned a lot of the intuitive aspects of the subject, but at the time I didn’t realize how unusual and great that was! Unfortunately, it been a long time, and I’ve forgotten a lot. Thanks for publishing this explanation, it makes me want to revisit the subject!
But the real reason I’m posting is because of what Mark said in the comments above. I wasn’t going to say anything, because religion vs science arguments belong in a different forum, and I felt it was very big of Kalid to say, “@Mark: Point taken, and happy for the discussion.” And then he very deftly sidestepped the whole subject and restated his original point without the “offending” reference to scientific theory. It was definitely a very mature way of handling an immature poster, and I’m hesitant to re-open the subject.
However, it really bothers me that we tend to treat these pushy religious types as if their “theories” of the origin of life and the origin of the universe deserve respect, or as if they’re harmless. Now people will always believe crazy things, and that’s fine. But a lot of these people are making an organized, concerted effort to undermine human progress, and with our ecosystem in such a delicate position, we as humans can’t afford to let them.
“I enjoy your articles, but weakly weaving religion into an article on math is unnecessary and, frankly, I didn’t think it was your style.” The first word or concept on the page having anything to do with religion is in Mark’s previous paragraph, when he brings up “Creation.” Apparently, Mark is saying that the theory of evolution is a religion. The simple fact that he believes this shows how ignorant we can be if we hold false beliefs. Ignorance is often very dangerous. Especially widespread ignorance. This also shows how rude and pushy these religious types can be, while making it seem as if we are slighting them, by (in this case) not making allowances for somebody’s random, wacky religious beliefs in a discussion about math.
“Your implication appears to be that evolution is THE theory . . .” Just as the theory of gravity is THE theory we have to explain the observable fact that gravity exists, the theory of evolution by natural selection is THE theory that explains the observable fact that evolution exists. There are no others. Go ahead, try and name one. But remember, in order to qualify as a theory, it must explain the available evidence, and it must make predictions which are testable. In other words, it must be refutable. Otherwise, it is not a theory.
“It came off a bit preachy to me.” This is so ridiculous I just had to include it. Sorry. It would make me laugh if it didn’t make me lose so much hope for our future.
I just think that the most important way to combat the kind of ignorance that leads to the election of incompetent public officials is to combat ignorance whenever we encounter it in our daily lives. Sort of a “think globally, act locally” plan.
Now this post really is preachy! Sorry, Kalid. Now I’ve had my say, I’ll leave it alone. I Promise.
I have to disagree with you on your Shakespeare example. Sure, it gets at the idea “be yourself,” but there’s a reason Shakespeare didn’t just say “be yourself.” The Shakespeare quote is beautiful, and to figure out why, we can diagram the sentence, figure out the meter, look at word choice – in short, figure out why it is beautiful. All of those components contribute to the way the sentence functions, so it’s important to look at them. There’s a reason we don’t read simplified, abridged, plain-language versions of Hamlet in English class!
This is great!
I flunked math all through high school and ended up doing basic math and algebra in a community college. I found a great teacher there who could turn the lessons around 180º and explain it so I could finally understand it. I found out I was a visual learner, but when I got to calculus class, it all fell apart again. I could understand the concepts - I couldn’t put together the equations.
Funny thing is, I love numbers and thanks to Mechanical Universe, I like physics. I found a book called Physics Without Calculus and truly enjoyed it because I could “see” the problems. As soon as calculus was added, the pictures “disappeared” again.
Hi,
it’s probably not your favorite topic, but since you mention evolution, couldn’t you write an article about it? So many people walk around and think they can argue against it, while the effects are so plain obvious.
excellent job on this one
@Jeff: Thanks for the comment, glad you enjoyed the post! Yeah, mixing “religion & science” can be a touchy subject. To me, science is about knowledge and religion is about ethics, and you shouldn’t use one to determine the other. I usually don’t address it in posts since it’s unlikely for either person to change their opinion based on a few paragraphs on a website. In this particular example, I’m not as interested in anyone’s reasons why something is so, as much as the effects (sugar/fat serve a higher purpose of encouraging survival). But great points :).
@Gabe: Good point. I think the key is being able to appreciate the intricate structure and the meaning (not just one or the other). Unfortunately, math education tends to focus on the former.
@Kelly: Thanks, glad you enjoyed it! Yes, calculus can be visual and intuitive, but often it’s buried underneath a pile of equations. The equations are useful, but can be really dense without any kind of intuitive grasp (I had a similar experience with vector calculus – it didn’t really start making sense until I visualized it). Appreciate the book suggestion, I’ll have to add this to the reading list.
@darwin: It would be an interesting article, though I’m currently not that well versed in the biological particulars :). But yep, it would be cool as it’s a very powerful and far-reaching observation.
@wlid: Thanks, glad you liked it.
To start, forgive my english, its my third language.
Ive always enjoyed math classes, I guess that Ive been lucky enough to have good teachers although they did use the conventional teaching system. The things is that the only way I could study math and remember formulas was to make sense of them. So, while others tried to memorize what formula and when to aply it I was trying to understand why that formula and why use it there. After a while I realized that studying math was quite enjoyable and easy if you went and understood what were you doing with the numbers. That was back in 9th grade.
Its been more or less 7 years since that revelation was made and Im still studying math related degree, economics. Coming across this article has been interesting since it describes exactly what I felt back in that day, and even though its elemental math for me, its reminded me of the beauty that math has and I had forgotten with subjects like Econometrics and advanced statistics, cause if you dont go to class its very hard to internalize and understand huge formulas. LOL. But anyways.
Very well written but most of all, I admire someone whos making this effort just for the sake of math and the bad name it has among the young, and I hope that many of them come across this to learn to see math with different eyes.
Luck and thanks!
Manu
Spain
Hi Manu, thanks for the message! Yes, I’ve always enjoyed knowing the “why” not just the “how” – unfortunately, for many students it’s not obvious that this understanding is important unless they stumble upon it themselves or their teacher encourages this line of thinking.
Glad you’ve enjoyed the article :).
I can’t imagine how I did pass all my math subjects back in college. My professor presented the subject just the way you presented your circle-triangle area presentation.
This is awesome. I love mathematics!