MIT recorded a set of Calculus video courses back in 1970s that they have since made publicly available. It is taught by a lecturer named Herbert Gross. His style of lecturing is clear, he states why things are defined the way they are and derives everything from first principles. There is an unusual mix of rigor and focus on building understanding - where everything comes from. It also taught me that math is about reasoning logically and rigorously and we shouldn't always rely on intuition (at least while doing math). Deriving almost all the basic calculus results that were drilled into me from the basic concept of a limit, deltas and epsilons was really refreshing.
Compared to more recent OCW calculus videos, I found this to be better in terms of respecting the learner's intellect, presenting the whole proof rigorously and teaching the student to think a certain way.
I just had a look at the first video in the first course on YouTube [0] and was delighted to see Herb Gross responding in the comments. Clearly this is a man who loves his subject and loves introducing it to others.
Yes, what a gem of a man.He also posted his email address in the comments for a young high school student for any further help. He is 88 years old now, I hope he is in good health.
That's so beautiful that I have to admit my eyes started to tear a little bit. It's lovely to live in a world in which wonderful people are empowered to touch people all over the world in such a powerful and profound way.
Reminds me of the MIT SICP lecture videos from the 80s. The concepts of black box abstraction and the simplicity of using LISP like lego building blocks blew my mind and made me switch from being a UX designer dabbling in Rails to a full blown programmer.
It was still entirely relevant to today even though it was a few decades old as the fundamentals of computer science are still fundamental.
Hearing that intro music still brings a smile to my face.
I just happen to be relearning math right now as I dive deeper into data science and this is perfect timing. Going to watch this series once I get through my math proofs book ("Book of Proof" by Richard Hammack which I recommend to people getting into math https://www.amazon.com/Book-Proof-Richard-Hammack/dp/0989472...).
So a while back, I said I am going to start writing "Colloquial books" about all of the courses I took as I relearn all of the stuff in my and other Engineering Branches (minus the grade-gun pointed at my head). One of the first courses are of course Calculus.
I skimmed through some of these video lectures, I wonder if I even should write books for this. Or simply href.
Thank you for the share. Will be of great help to me come summer.
I learned of this euphemism one night at our family dinner table, when I suggested to our teenage kids that we all take some time after dinner to "Netflix and chill." Needless to say, they were a bit surprised by the suggestion. :)
I watched those lectures in the early stages of my degree and, to confirm what you've stated, they helped me gain a more concrete understanding of the fundamentals of calculus. Highly recommended! Thanks for sharing.
Mathematics requires creativity. Conjectures do not arise from logic and rigor. If that were the case, they would never be wrong, and thus not require proof. If mathematicians don't come up with conjectures, then mathematics doesn't advance.
A formal system's territory is connected together by rigor; but whence comes the formal system itself? That has to be imagined.
You say 1.9MB, the Gutenberg page says 1.8MB, and the download itself is only 1.18MB thanks to gzip. The main difference is that the Gutenberg PDF is a transcribed, cleaned-up version.
As much as this probably makes me sound like an audiophile, I actually prefer the raw scans over what may essentially be a reprint. They show all the blemishes, unofficial additions, and other marks that make the book look more "real" and give it character.
In this instance, the raw scan has a picture of the cover, as well as an interesting note handwritten near the beginning: "Property of Edward M Sumner" with an address. IMHO these sorts of historical artifacts are worth preserving too. I've come across scans with random notes, bookmark fragments, and newspaper clippings included, and it's always fun to ponder how they got there. (Who is this person and how did he get the book? Is he the one who scanned it? Etc.)
This is why I love used books. My copy of Schopenhauer's collected essays has gone through 4 owners since 1914, all 4 of whom have signed and dated the front cover, all of 4 of whom have marked and underlined at different spots (including, now, myself). My copy of Riverside Chaucer went through two students before me. And they all pick up their own unique scents along the way (my girlfriend jokes that I only buy books to smell them).
Just a side note, off topic: I was that king off person who had a shelf in my house with 300 printed books (and some more stored in boxes in the basement).. it was always a pain when I had to move to a new apartment!
After the e-books and pdfs came by we got a lot of flexibility .. now it's easy to have 3000 volumes in the hard disk. You can literally carry a library with you..
I ended up giving away most of them, apart if some classics (e.g. The art of computer programming from Knuth, Operating Systems, from Tanenbaum, and some other classics for image processing..)
The only thing I miss very much is that I loved the feeling of picking up one at random and expend a couple of hours ... I just can not have the same feeling with e-books.. :(
Must be something with the program you are using. I've just opened the file on my WP10 device with ancient Snapdragon 400 and I don't see any problem with reading, scrolling etc.
LaTeX is not an aesthetic cure-all. The original, at least, has been subjected to the critical eye of a typesetter. There is no disputing taste, but I will anyway. If you're using an e-paper device, btw, you can vectorize the text in scanned pdfs like this by applying "Clearscan" in Acrobat, or the equivalent (after which, it becomes readable).
That was my guess too, but this one actually isn't. Amazon uses "?ref" for reference (innocuous), and "?tag" for affiliates.
For quick reference, a.co and amzn.com are official shorteners and considered safe. amzn.to links however are third-party and often used to conceal affiliate links.
I'm embarassed somewhat to say this, but over the past few weeks I've been taking the courses on Khan Academy on mathematics. I'm nearly 30.
and I'm not talking about brushing up on my linear algebra, that comes later, I'm talking high school level mathematics, stuff that I've largely forgotten or didn't "get" first time round.
I've seen these "machine learning for hackers!" articles who try to dish out a bit of maths saying that's all you need, but I don't think you can escape the fact that sometimes you just need to start from the beginning and work your way up
A great book for brushing up/re-learning pre-calc is "Precalculus Mathematics in a Nutshell"[0] by George F. Simmons (you can find it used cheap). He really boils it down to the essentials. For example here's how he opens his chapter on Trig:
"Most trigonometry textbooks have been written by people who appear to believe that the importance of the subject lies in its applications to surveying and navigation. Even though very few people become surveyors or navigators, the students who study these books are expected to undertake many lengthy calculations about the heights of flagpoles, the widths of rivers and the positions of ships at sea. The truth is that the primary importance of trigonometry lies in a completely different direction - in the mathematical description of vibrations, rotations, and periodic phenomena of all kinds, including light, sound, alternating currents and the orbits of the planets around the sun. What matters most in the subject is not making computations about triangles, but grasping the trigonometric functions as indispensable tools in science, engineering and higher mathematics. These functions and their properties are the sole subject matter of this chapter."
I went to college late, after a few years out in the world. I had not originally intended to go into a technical field, but realized that's where some of my passions were so I went after a CS degree. Not coming from a technical place I had put almost no effort into mathematics during public school and so I found myself very very behind.
I bit the bullet and focused my first few semesters almost entirely on remedial math education, trying to fill in that gap. Because I now had a strong motivation, a little age, and focus, I managed to become a very good math student and even received the occasional award during my undergrad math education.
I've lost much of it, but I now know I can learn it, and many of the topics that had intimidated me at first turned out to be not a big deal once I got it. One day as a hobby I plan on starting over from algebra and work up again for fun.
I do this all the time. There's nothing at all weird or embarrassing about it. 10 years ago I was sure I'd remember and have an intuition for the trig identities, but I'm still looking them up every time now.
Also, even in pretty advanced CS (at least in my field, security and, particularly, cryptography), most of the nuts-and-bolts math is high school math.
Nothing to be embarrassed about. This stuff is very "use it or lose it" so unless you're really using it regularly, it fades from mind very easily. Heck, I've taken College Algebra like 3 times over the years, and I still keep going back to Khan Academy or watching various Youtube videos and the like, for a refresher here and there. And that's just to keep the basic algebra stuff in mind.
I have a master's degree in CS, and I feel the need to do this too. Use it or lose it. Problem is you only use bits and pieces of what you learned here and there, not enough to remember college level math.
My hat is off to you. I did the same thing, mastering something like 85% of the material in the Mathematics area of Khan Academy before finally just enrolling at a university part-time to go deeper. Now that you're older you'll be better able to pause at moments and reflect on the deeper meaning of some of the thing you're learning. Enjoy!!!
I've been doing this as a "wait for queries to run" kind of brain game. I don't think it's embarrassing. I don't think we emphasize "back to basics" in enough areas of life, except when we hit a wall after failing to achieve what we meant to and more often than not B2B solves the problem.
Also attempting to finish CS BS but I have hit a wall in my late 30s unable to pass pre-calc. I shudder at the daunting levels of Calc that come after to the point that I'm debating switching majors just to "get a degree".
This may not work for you, but I honestly didn't understand calculus until I worked through Knuth's Concrete Mathematics. There's a portion of it detailing rules on summations, which were (I realized at the time) the discrete equivalent of integration (summation of functions over integers versus integration which is summation of continuous functions). With my (stronger) CS than math background, it just "clicked" for me. You could check out the book from your university's library and see if this material helps you. I can't put my finger on which specifically now (too many years later) but various calculus concepts just fell into place as I worked through those portions of the book.
I'm doing something similar. As an undergraduate I studied biological sciences in the U.K. and in the U.K. they don't think biologists need to know any college-level Math(s). Or at least they didn't when I did it. Just a stupid Maths for Biologists class with one of those ridiculous "learn Statistical Tests as recipes" syllabuses. (This was at Cambridge) I really regret not taking the first year intro to applied math courses that those studying chemistry and physics took.
I dropped out of college and never went higher than pre-calculus, and I went back to learn all of this stuff with Khan Academy in my 30s, as well. It's an amazing resource.
I'm almost 34 and when I watch Calculus One videos on coursera, I sometimes get stuck when the guy easily expands (a + b)² or folds it back. We used to to this at math class at school and then probably in university, but I haven't been doing anything like this since then (talking about importance of math for programmers!).
Very much this. My son is taking his first course in HS Algebra. Fortunately, he generally gets it, but every now and then asks me for help in understanding a concept since his textbook is awful. I've found myself having to Google topics that I know I mastered decades ago. It does come back, but sometimes it takes a bit of thinking.
I did something similar. I returned to college in my late-20s to get my degree, and started at trig and worked my way back up. It wasn't strictly necessary, but I'm glad I did it.
Good luck with your studies, and never feel embarrassed to better yourself!
Summer of 1980, going into my senior year in high school, I mentioned I'd be taking Calculus next year to a co-worker a couple years older than I. He said he had the best book in the world on Calculus, and he loaned me his copy of Silvanus P Thompson's Calculus made easy. I thoroughly enjoyed that book, benefited from its intuitive explanations, and forever appreciated his recommendation.
If I may similarly influence anyone here, for themselves or someone they know, to read Calculus made easy to supplement their calculus coursework, I will be happy to have paid the favor forward in some small way.
By the way, Kalid Azad may be our modern day Silvanus P Thompson. And he has better tools[1], which he wields masterfully, than just pen and paper. Recommended too.
I vividly remember cramming for a test my freshman year of college, not having things click, and the final Aha! when a semester of pain disappeared with a few visualizations. The contrast between how most classes presented the material and what actually worked for me was jarring, and I had to share what helped. I hope other people share what works for them, in any format they can.
Betterexplained is right up there with the Khan Academy and Wikipedia. Between those three there aren't many subjects you can't improve your knowledge on from your livingroom.
Would you happen to know of any websites, video series, books, etc that would be in the same vein as these 2 sources but geared towards very young students (elementary/middle school math)? ...so that they can understand it conceptually from a young start.
EDIT: I'm working through Khan Academy with one of my kids...would like to find other resources as well.
Hmm, how about a technique rather than a resource...
Look for opportunities to expand what they're doing in school. For example, after my kids had covered place values in school, I taught them binary. Their knowledge of base 10 is so much more robust when they learn base 2 as well.
Point is, I bet you already know plenty to be that ideal resource you seek for them. Good for you for wanting to start early.
Thanks for that affirmation. I actually taught them binary counting too over a weekend a few months back...I was amazed how quickly they were able to pick up on it (3rd and 5th graders). With limited time/energy, I was hoping to find some great resources to add to what I can provide them individually.
Keep up with teaching your kids. I took calculus in 8th grade and having it as a base understanding has been invaluable in surviving college engineering.
Concur - Kalid approaches things through first principles and teaches the 'how' very well. The ideas and approaches can be translated into almost anything else. I hope he keeps on keeping on.
I have an embarrassing amount of Calculus books. My dad taught the subject in a high school and community college; I suppose I have a soft spot for it. "Calculus Made Easy" is a good book though I do think there are better ones these days. Some of the lexicon has changed and there are topics covered in a modern Calculus textbook that aren't covered in the original book (that I personally think are worthwhile spending time on). The updated version with Martin Gardner does have blurbs where necessary to point it out. The Kline book is a MUCH larger read, but is what I would recommend if you want a reasonably priced Calculus book that's easy to grok. Otherwise, I think it's hard to go wrong w/ the Stewart books. Work through the problems as they do in the book, you will come away w/ what you need. Finally, if you want a whirlwind tour, Calculus for Dummies by Mark Ryan is great.
Time spent learning Calculus is worthwhile; and if nothing else, understand the fundamental theorem. Overwhelmingly impressive.
For what it's worth, this is available as a much leaner pdf [1] on Project Gutenberg now, including the TeX source that some kind person used to update it.
The prologue is 100% dead on and what I have always thought in terms of the easy part of calculus needs to be taught early. I would have never been able to vocalize what he said in the short half page of text.
We teach math backwards. We have a population that can barely do 4th grade arithmetic and it is socially okay.
Children and people believe that decimal points are accurate and that fractions are abstract when the exact opposite is true. 1/3 of a pizza is real and 0.333333 is a fake number.
We also teach movement and change as a word problem i.e. a train leaves Chicago at 25 MPH and another train leaves Flint, MI at 35 MPH when will the trains meet? That answer is an estimation of an unattainable constancy but people believe it is logical conclusion.
Pre-Calculus (needs to be repackaged with the idea of the one consistent in our world is change) should be taught before Algebra and Geometry. Make math into something where people can truly understand abstract and concrete. People actually think calculus is a hyper abstract algebra when in fact it is putting math into real world solutions.
The number one problem is calculus is 100% dependent on the teacher. A great teacher will make this work and a lower skilled teacher can absolutely kill almost all learning.
I assigned this in an Honors Calc II class I taught at a state university.
The main text was Stewart (decided at the department level), but I was teaching the Honors section which provided a good opportunity for me to ask for something extra. I had my students read this book alongside Stewart, and write weekly short essays comparing the two approaches. Many of the students turned in some quite good writing.
A lot of the elements of what we'd call "modernity" were either introduced or became widespread in the course of the first World War so I'm not trying to just be a jerk here; I think it really is a useful dividing line.
I'm not sure I agree to it being anything stronger than a 'catalyst' period, but regardless; as elsewhere, by 'modern' I meant 'more recent than it is'.
It's less than a decade from the cutoff date after which books stopped entering the public domain, so by Project Gutenberg standards, it is pretty modern.
Okay, fantastic, arguably a poor choice of word then. I think it's pretty obvious what I mean, and whether or not 1914 qualifies as 'modern' doesn't affect it in the slightest...
Not a poor choice of words at all, modern must be understood in context and you used it correctly. It would be impossible to ever use the word without a debate on definition otherwise. "How do I build a modern web app? .. Here, I found some relevant resources on vacuum tubes!"
exactly. this was driven home for me when I read babbit by sinclair lewis [1]. It's all about "keeping up with the jones's" consumerism in America. It's amazing that it was written 95 years ago.
I guess what I'm saying is, for me, '... made easy' sounds just as of our time as '... for dummies'. I'm in the UK, maybe it's a more recent phenomenon here? (I naïvely imagine cross-atlantic publication was rarer in the early C20?)
Good lord. I somehow took (and passed!) a year of high school calculus plus a semester in college, and I never had a good sense for the word "Integral" in a math context as anything but arbitrary jargon. And I've spent a not-tiny amount of time with BetterExplained's calculus. 30 seconds with this book and it's obvious. Now I'm making connections with the French (of which I have barely any, but any port in a storm) and it's solid in my mind.
After scanning for a few minutes, I was amazed at how readable the book is given its age and my experience with books of similar age.
The book also does appear to go out of its way to keep language simple. For example:
We call the ratio dx/dy "the differential coefficient of y with respect to x." It is a solemn scientific name for this very simple thing. But we are not going to be frightened by solemn names, when the things themselves are so easy. Instead of being frieghtened we will simply pronounce a brief curse on the stupidity of giving long crack-jaw names; and, having relieved our minds, will go on to the simple thing itself, namely the ration dx/dy.
This is an incredible book. I'm a visual thinker, and while this book lacks all the glossy pages of illustration found in a modern Calculus textbook, the writing style helps develop that visual intuition. In the same vein of concise Calculus books, Serge Lang's Short Calculus is also great (if you need a refresher, or if you're just starting out).
This is, actually, a really good book. Don't let the cheesy title throw you off. I learned more in the first few chapters than I did after a semester of calc classes. Highly recommended.
Reading it as a kid I wasn't sure I trusted the author - Silvanus P. Thompson FRS ('FRS' - which I knew to be something rather prestigious) when he wrote in the prologue "Being myself a remarkably stupid fellow, I've had to unteach myself the difficulties . . . What one fool can do, another can."
As someone who struggled through calculus, this book would have made a huge difference for me. Just reading through the first few pages brought a smile to my face that someone could so plainly explain these critical concepts in such a familiar way. Why didn't my professors do that?
Because to them it's obvious. Most maths teachers are so far ahead of the students they forgot they once were students themselves.
I've had 3 different ones in high school and the difference was incredible. All the way from 'only the best students learn anything' to 'everybody earns at least a passing grade'.
Maths and physics were the classes where the quality difference between the teachers stood out the most.
And to them it's wrong! Much is said in this book which is difficult (but not impossible) to rigourously justify. It took centuries for calculus to be placed on a rigourous mathematical foundation; this foundation (called "real analysis", largely developed in the 19th century) is quite different from the intuitive ideas presented in this book. The presentation here (in particular the idea that dx² is negligible) can be made rigourous through "non-standard analysis", but this 20th century development is less-known (in fact, unknown to many mathematicians) and perhaps even more difficult than real analysis.
Rigour is not necessary to understanding, and can even be fatal to understanding, but it's how mathematicians work, and generally mathematics is taught by mathematicians.
> this foundation (called "real analysis", largely developed in the 19th century) is quite different from the intuitive ideas presented in this book. ... (in particular the idea that dx² is negligible)
Thank you for posting this! Such things always bothered me in high school, seemed like approximations that ought to bite you in the behind at least in some corner cases. Another example from TLA:
> dy = 2cos(θ + 1/2 dθ) · sin 1/2 dθ
> But if we regard dθ as indefinitely small, then in the limit we may
neglect 1/2 dθ by comparison with θ, and may also take sin 1/2 dθ as being the same as 1/2 dθ. The equation then becomes:
> dy = 2cosθ × 1/2 dθ
This again seems like very sloppy and careless kind of approximation that ought to bite you in the back - but knowing there are just (supposed-to-be) intuitive non-rigorous methods, and that these have actual rigorous backing, somehow soothes me.
But this is true of every level of mathematics. See humorous proofs that 2=3 amounting to manipulations like 0a = 0b, cut the zeros, a=b. This is no fault of "nonrigorous elementary algebra", it's a matter of remembering all notations are abbreviations and you have to know how to manipulate them.
You can get away with dy and dx if you just think "I'm not writing integral signs because they're annoying" and remember the rules of working with integrals. This is how stochastic differential equations work -- they're not even differential equations because Brownian motion is not differentiable, they're just notation.
"dx^2 is negligible" can be alternatively (rigorously) obtained in standard analysis/calculus in the following sequence:
0) Intermediate value theorem (continuous function with image between b and c must be such that there is x such that f(x)=d for d between b and c. This can be first proved where d=0 -- it's the method of bisection by nested intervals, really -- and then generalized)
0.5) The actual definition of derivatives known to everyone.
1) Rolle's theorem, i.e. the Intermediate Value Theorem on first derivatives (if something goes up and down then it must have a critical pont).
2) Mean Value Theorem (Rolle's theorem but rotated)
3) Taylor's theorem.
> and generally mathematics is taught by mathematicians
So that leaves the door wide-open to a whole army of mathematicians to who it is unknown that are teachers.
And those are the ones for who all of the above does not apply and who still treat their 'entry level problems' as obvious to all their students even when the evidence strongly suggests that to the students it is not at all obvious.
And I'm writing this as a high school kid who was pretty good in math but totally lost interest due to such teachers (that, and computers were far more interesting).
Totally agree with you. I did advanced research in computer science without fully grasping integrals - only if I had this book!
But you're right, why can't things be like this? Then I think, there is a certain art to having the ability to explain something so easily.. props to reddit's elif subreddit (https://www.reddit.com/r/explainlikeimfive/)
> ... a 1998 update by Martin Gardner is available from St. Martin's Press which provides an introduction; three preliminary chapters explaining functions, limits, and derivatives; an appendix of recreational calculus problems; and notes for modern readers. Gardner changes "fifth form boys" to the more American sounding (and gender neutral) "high school students," updates many now obsolescent mathematical notations or terms, and uses American decimal dollars and cents in currency examples.
The prologue had me hooked... I have never read anything like this in a text book. The self-deprecating humor immediately disarms you if you're the type that would go into something like this intimidated. I am definitely reading this.
I've been looking for something similar to this but for Algebra, Trig, etc. before relearning Calculus and came across this: Mathematics For The Pracicle Man by Howe (1918?) http://www.aproged.pt/biblioteca/mathematicsforthehowe.pdf
It's targeted to Engineering students and it's not lengthy, won't take me two years to learn. Hopefully it'll be of same quality as Thompson's book?
I love calculus, it's such an amazing collection of fascinating, elegant (and useful!) concepts that give you a transformative insight on mathematics in general. It's very frustrating to see it taught so poorly so often though.
One problem that teaching calculus has is that it's very dependent on having a solid foundation in other mathematics such as advanced algebra, trig, etc. In today's school systems that encourage gaming the system as students and teaching to the test as educators it's rare for most students to actually understand or be competent with material they've allegedly studied. When they hit something that starts off where they left off and builds upwards, if they have any weaknesses in that foundation it will show immediately and slow them down immensely.
Add on to that all the other problems of typical calculus instruction such as a desire to make it hard as a matter of protecting calculus education as a status symbol for "smart" people, the ability to ratchet up the difficulty arbitrarily through requiring memorization of a potentially infinite set of "trivia" (every trig. identity, every method of differentiation/integration, and so on), while generally not concentrating on the abstract concepts or the fundamentals.
There is a 1998 update to the book with "modernized" english (I think it is clearer while preserving a dated style) and some additional chapters. ISBN-10: 0312185480
It's funny that a book from 1914 is formatted in a way that it is much easier to read on my mobile phone than pretty much anything I can download from google books.
IIRC this the book Richard Feynman said he checked out of the library and learned calculus from. Also, Feynman made comments similar to those in the Prologue.
IIRC, Feynman mentioned Woods in the context of differentiation under the integral sign, whereas J.E. Thompson was in the context of learning Calculus as a kid.
In Italy there is a "similar" (in the sense that it manages to explain calculus in a friendly and easy manner) book, that has been re-edited and re-published since - I believe - 1929 or so:
As someone who dropped out of Cal II, the first few paragraphs of this book gave me more understanding than I ever had in college... Either I was very lazy, or my teachers couldn't express these simple ideas clearly, or both.
Fun fact: The author Silvanus Thompson was an early pioneer of what is now known as transcranial magnetic stimulation. You can see him posing with his head in a device for testing neurophysiological effects of alternating magnetic fields in figure 4 of this paper: http://rsnr.royalsocietypublishing.org/content/61/1/5.figure...
This book is staggering. I'm on page 1. I've read a couple of dozen lines. This has explained more about calculus then a decade and a half of seeing related concepts.
My problem with advanced math is not so much the understanding of principles but the application of these to solve new problems creatively.
I was able to master partial differential equations and pass exams but was never able to apply what was learned to solve new problems which I found very frustrating and was what ultimately led me to not pursue a career in the field.
Terrible advice! I tell all of my students to find as many analogies as they can. You can never lose by increasing the number of ways in which you understand something. Analogies, some may argue, are at the heart of mathematics.
> Mathematics is the art of giving the same name to different things
Poincare.
> The art of doing mathematics consists in finding that special case which contains all the germs of generality
Hilbert.
> The vast majority of us imagine ourselves as like literature people or math people. But the truth is that the massive processor known as the human brain is neither a literature organ or a math organ. It is both and more.
John Green.
> Sometimes I think that creativity is a matter of seeing, or stumbling over, unobvious similarities between things - like composing a fresh metaphor, but on a more complex scale.
David Mitchell.
There is an ounce of truth in what you say -- metaphors can be abused to draw false conclusions. This does not mean one should cower from using them.
Everyone has their own philosophy of mathematics. To me, the core idea is the idea of representations, such that those representations allow you to extract patterns (aka similarity or commonality) from various cases.
I can't help but indulge a bit and now ask whether or not we can't extract a common pattern from the two representations of mathematical cognition offered by yourself and GP. :-)
As an opposing view, I offer you the following analogy, by mathematician Charles C. Pugh, which he draws between conceptions of metaphor as understood in natural versus mathematical language:
From "Real Mathematical Analysis" 1st edition, p. 9:
Metaphor and Analogy
In high school English, you are taught that a metaphor is a figure of speech in which one idea or word is substituted for another to suggest a likeness or similarity. This can occur very simply as in "The ship plows the sea." Or it can be less direct, as in "his lawyers dropped the ball." What gives a metaphor its power and pleasure are the secondary suggestions of similarity. Not only did the lawyers make a mistake, but it was their own fault, and, like an athlete who has dropped a ball, they could not follow through with their next legal action. A secondary implication is that their enterprise was just a game.
Often a metaphor associates something abstract to something concrete, as "Life is a journey." The preservation of inference from the concrete to the abstract in this metaphor suggests that like a journey, life has a beginning and an end, it progresses in one direction, it may have stops and detours, ups and downs, etc. The beauty of a metaphor is that hidden in a simple sentence like "Life is a journey" lurk a great many parallels, waiting to be uncovered by the thoughtful mind.
Metaphorical thinking pervades mathematics to a remarkable degree. It is often reflected in the language mathematics choose to define new concepts. In his construction of the system of real numbers, Dedekind could have referred to A|B as a "type-two, order preserving equivalence class", or worse, whereas "cut" is the right metaphor. It corresponds closely to one's physical intuition about the real line. See Figure 3. In his book, Where Mathematics Comes From, George Lakoff gives a comprehensive view of metaphor in mathematics.
An analogy is a shallow form of metaphor. It just asserts that two things are similar. Although simple, analogies can be a great help in accepting abstract concepts. When you travel from home to school, at first you are closer to home, and then you are closer to school. Somewhere there is a halfway stage in your journey. You know this, long before you study mathematics. So when a curve connects two points in a metric space (Chapter 2), you should expect that as a point "travels along the curve," somewhere it will be equidistant between the curve's endpoints. Reasoning by analogy is also referred to as "intuitive reasoning."
Moral: Try to translate what you know of the real world to guess what is true in mathematics.
- Baddeley, Alan, Michael W. Eysenck, and Michael C. Anderson. Memory. NY: Psychology Press, 2009.
- Cat, Jordi. "On Understanding: Maxwell on the Methods of Illustration and Scientific Metaphor" Studies In History and Philosophy of Science Part B32, no. 3 (2001): 395-441
- Derman, Emanuel. Models of Behaving Badly. New York, NY: Free Press, 2011
- Foer, J. Moonwalking with Einstein. NY: Penguin, 2011
- Lutzen, Jesper. Mechanistic Images in Geometric Form. NY: Oxford University Press, 2005
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I think that is a very "mathematician" thing to say; my own experience of doing mathematics is largely in accord. A little experience of teaching mathematics shows that this doesn't work for most people.
IMHO the best way to teach any concept is to start with a problem or question that can eventually be solved using the concept. Start with the question in general terms but don't provide the answer, let people think about it, make it engaging conversation and slowly and steadily build the concept that leads to the solution.
It's not rare that supports the people teaching the class. Nothing like pushing your own book to students that just have to buy it. In some cases the books don't even get used.
This is a good book. But on page 5 you have an error: it should say trillionth instead of billionth. I find myself liking your approach, though. You should change the font and insert some more diagrams.
Oh, what's that... you can't just republish instantly because electronic computers haven't been invented yet? Well, just iterate and do things that don't scale :)
The word is still used by countries that use the 'long scale', which has the nice property that a billion = (1 million)^2 and a quadrillion = (1 million)^4 etc. If you can count in greek this also means that an n-illion times an m-illion equals an (n+m)-illion.
Apparently [1], both ways of counting originate from France. This 'superior' way came first and was adapted by the British, then our current way of naming was created and adopted by the US. Wiki says this one then became the standard because of USA's dominance in the financial world.
Compared to more recent OCW calculus videos, I found this to be better in terms of respecting the learner's intellect, presenting the whole proof rigorously and teaching the student to think a certain way.
Calculus Revisited: Single Variable Calculus | MIT OpenCourseWare - https://ocw.mit.edu/resources/res-18-006-calculus-revisited-...
Complex Variables, Differential Equations, and Linear Algebra - https://ocw.mit.edu/resources/res-18-008-calculus-revisited-...
Calculus Revisited: Multivariable Calculus | MIT OpenCourseWare - https://ocw.mit.edu/resources/res-18-007-calculus-revisited-...