Calculus Made Easy
calculusmadeeasy.org
calculusmadeeasy.org
You know what blew me away though? Not one textbook I looked in mentioned “why” the dot product is important; that is it’s useful for determining the similarity of two vectors. They all focused on the mechanical details of computing the dot product, but never spelled out the reason it can be useful. I went through a few other resources before I broke down and had a little chat with ChatGPT to discuss the meaning behind it and it makes perfect sense after that.
In comparison to when I was in college, things are much slower paced so I can take the time I need to ensure I have a full grasp of a concept before moving forward. I guess all of this is to say that as I’ve continued forward through more concepts I keep finding that the books I’m reading offer a mechanical view instead of a holistic view of the material. This feels like the biggest issue with most math books I’ve read and it makes me wonder where books that offer more semantic meaning of concepts instead of recipes exist.
Otherwise at the end you get people dissatisfied they cannot get job but they have a degree.
You cannot simply explain to someone complex stuff - best way is to let people grind through to build their own understanding.
Parent poster wrote that "it’s useful for determining the similarity of two vectors" - now I would ask why do I need to determine similarity of two vectors as it does not mean much to me - if I would be grinding through math problems I would most likely find out why, but there is no way I could understand and retain it when someone would just tell me.
To get someone understand something holistically, as in link to their previous knowledge base, requires knowledge of what their knowledge base is. Traditionally this has been done with structuring the teaching with prerequisites etc and hoping it works.
I struggle with this quite a bit when I teach students with heterogeneous background. To be effective, one has to first probe what the students already knows to be able to relate the new stuff to that, and this requires interaction. Hypertext is/would be helpful for self-learning, but it's sadly very underutilized. LLMs may be better. But probably even those can't at least in the current form replace interactive human teaching as they don't really form/retain a model of what the user knows.
Simple:
Start with
a) Suppose you are making a video game..
b) Suppose you are determining ballistic trajectory of your missile system based on model rockets
c) Suppose you are running a fighter robot group..
Or any of the stuff children are supposed to *actually* do and then take these classes with determination to do the actual creative things that they wanna do all life.
There is an aspect of jest in the above comment, but it also contains some likely truth. Children love doing stuff, and these are the things that may enable them.
Edit 2: The teacher probably gave some example from biology or something that you didn't care about and therefore forgot about it.
And that is one of the toughest things to get right. Children are extremely curious, that's how they learn and master absolutely anything including arts, dancing, music, history, skating, catching insects, street smarts etc. It's on us as teachers to not let that curiosity wither into nothingness.
The worst thing was to learn something just because the teacher said so. If I hadn't had the motivation not to fail, I would definitely not have gone this far in my studies and in life.
I think the same goes for mathematics instruction. The thinking is that you will soon take courses that will make specific application of these tools that have broad application.
I am not from the west, but I watched Band of Brothers. From most of the ww2 media there is always a term being used. "Why we fight?"
This is absolutely not true. Many times I've experienced the moment of something complex "clicking" after hearing or reading an appropriate explanation for a phenomenon - finally seeing the right visual or appropriate example or comparison. I find it hard to believe you've never experienced this other than through grinding problem sets.
I believe something "clicked" only because you were grinding or already quite familiar with the topic.
There is no way to simply explain complex topic so someone would just get it or someone would "click" on after reading one book on the topic.
However, I've definitely had the case where person B just explains it better (for me). I still can't completely discount that my brain was primed for it by person A.
Sequential learners prefer to organize information in a linear, orderly fashion. They learn in logically sequenced steps and work with information in an organized and systematic way.
Global learners prefer to organize information more holistically and in a seemingly random manner without seeing connections. They often appear scattered and disorganised in their thinking yet often arrive at a creative or correct end product.
Person B explains X and I don't get it, then 2 hours later person A explains X again and I get it. There's no grinding in the middle. I just needed both perspectives to make sense of X.
The great mathematician Henri Poincare was struggling with a problem on fuchsian functions. He made some progress, and then: "Just at this time, I left Caen, where I was living, to go on a geological exursion under the auspices of the Schools of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it ..." [1]
"Incubation is the work of the subconscious during the waiting time, which may be several years. Illumination, which can happen in a fraction of a second, is the emergence of the creative idea into the conscious. This almost always occurs when the mind is in a state of relaxation, and engaged lightly with ordinary matters. Helholtz's ideas usually came to him when he was walking in hilly country. ... Incidentally, the relaxed activity of shaving can be a fruitful source of minor idea; I used to postpone it, when possible, till after a period of work." [2]
[1] Jacques Hadamard, "The Psychology of Invention in the Mathematical Field" (p. 12--13)
[2] J. E. Littlewood, "Littlewood's Miscellany" (p. 192) [A wonderful book, by the way!]
Actually, I can count the times I’ve applied math from later than 6th or 7th grade on one hand. I’m almost 40 and have been writing code for pay since I was 15. I struggle with this with my own kids and dread their reaching those later classes because I have no compelling answer for “why do I have to learn this boring shit?”
Maybe you prefer to figure out everything yourself, but you have just one lifetime, and having access to guidance while grinding will allow you to learn things faster (and thus more).
I do not argue that guidance is not needed, it also is, but there is no shortcut that lets people skip own hard work by telling someone incantation of words or sentences in special order that will make things click.
Grinding through to build your own understanding when someone can just give you useful meaning and context to connect to your other parts of learning is a core teaching skill, and anyone avoiding that because its "too hard" is doing a deep disservice to their students.
Opposed to the liberal arts were the illiberal or servile arts. These are necessary and good, of course, but necessarily inferior to the liberal arts because their end is not truth or formation; they are instead practical, concerned with effecting some kind of economic end. The point here is not to disparage, but to understand how all of these are related and ranked according to a "for the sake of" relation. A human being doesn't exist to eat, he eats to exist, for instance.
Modern education is very much oriented toward the servile arts, and what passes for the liberal arts today is anything but the classical notion.
The point is that modern education is less interested in leading to understanding, realizing virtuous habits, and leading to freedom, and more interested in churning out workers. Workers don't ask "why" (though we can agree that those who do can, guided by prudence, contribute more economically). Indeed, that is perhaps the key difference between classical science and modern science: the emphasis of the former is truth, while that of the latter is control of nature. Of course, it isn't that you must choose absolutely between understanding and effectiveness, and the classical tradition does not claim either that study precludes work. Study often requires work, for sake of preparing the way for truth. Rather, it is that the end of the modern educational tradition is different from that of classical education, and this end determines the form of the pedagogical methodology. It is a difference in anthropology, of the vision of man.
All men work, but what do they work for? Do they work for work's sake, or perhaps to make money to satiate their base appetites (modern view)? Or do they work in order to be free to pursue higher ends[1]?
Understandably college courses & textbooks meant for CS people will be more focused on computation, while a math major who is taking Linear Algebra will get a more theoretically motivated course. Gilbert Strang is an example of an engineering-focused text while Sheldon Axler or Katznelson & Katznelson is an example of what a math major would experience.
But in general, literature using and/or teaching mathematics does tend to be too algebraic/mechanistic. Languge models can be a very good aide here!
But that fact’s significance and too obvious simplicity, with all its ramifications, only hit me deeply and profoundly when a year later I realized I could use those functions to draw a circle on a screen.
Before that they were abstractions related to other abstractions that I had to memorize to pass a course.
To this day I am frustrated when reading papers about abstract algebraic relations and other such concepts, without even a sentence or two discussing any intuitive way to think about them. Just their symbolic relations.
I appreciate that in the game of math that view becomes natural. But most of us learn math with additional motivations and are interested in any perspective that highlights potential usefulness or connection to the real world. Many of us mentally organize our knowledge teleologically.
Yet even when usefulness is known to exist, it is often neither mentioned or referenced. Or even considered relevant.
Edit: the same goes for not showing a single concrete example of an abstract concept. A kind of communication that would unlock many mathematical papers to a much larger audience of intelligent and relevant readers.
Take parametric curves. I explain that they generalize the concept of a function. Every function can be parametrized in a trivial way. They don’t really understand this concept. They have a hard time parametrizing a function and do so only becuase of a formula.
The fact is most people need to go through the mechanical process of doin g before they can get to a point of understanding. It takes almost the entire semester for me to convince beginning algebra students that the reason that 2x + 3x is 5x is because of the distributive property. And when they do understand it they don’t understand why that is important.
Later on when things click for someone they will often say things like, “Why didn’t they just tell this when we took the course?” Usually we did. You just didn’t have a sophisticated enough understanding of things to grok it at the time you took the course.
And I know picking on your example isn't in the league of a general solution.
But if 2x (which is x + x) is two apples in a box, and 3x (which is x+x+x) is three apples in another box, then you put those two boxes in a bigger box (another +), people already intuitively can see the distributed property of scalar multiplication vs. addition of some unit, they just didn't have a name for it.
Likewise, a 3x4 square of paper next to a 7x4 piece of paper can be easily seen to be a 10x4 piece of paper. Multiplication of numbers over added numbers.
So one way to introduce distribution is to start by showing examples of several places where people already understand the concept of multiplication distributing, and use it every day, but just didn't know it was one concept with a name.
Once people can recognize distribution as an already familiar relationship in everyday life, then the symbols can be visited as the way we write down the already known and useful concept so we can be very clear and general about it.
Anyway, that's just a reaction to one example, which may not mean much.
Sometime later a person will really grok all this and then say, “why didn’t they just tell us this is all just the distributive property?”
A different approach I have thought about, which would really tear the textbooks apart, is introducing every concept in its simplest form as early as possible.
Then when it is eventually expanded on, its familiarity will aid in taking further steps more quickly and intuitively.
For instance, something as simple as adding up the area of a fence of varying heights, or the area of multi-height wall to be painted, being referred to as integrating the area, in early grade arithmetic, creates a conceptual link for down the road.
Systematically going over K-12 materials, just making similar small adjustments to terminology and concepts to be highlighted, would be interesting.
To your point, people do constantly try to tweak things to make subjects easier to understand and more intuitive.
Tossing the word "integral" at younger children won't make that easier or harder.
If you’re talking about research papers, that’s just because they’re written for domain experts and aren’t really for giving you intuition. They’re written in a deliberately terse (one might say elegant) style to convey the research findings in formal mathematical language and nothing much else. If you want to gain an intuitive grasp of things, read a proper textbook in detail or play around with the ideas on paper. Or both!
I guess the reason is that once you’ve acquired the intuition, having the literature cluttered up with the same explanations again and again becomes clunky and increases the volume of material to be sifted through when you’re just looking for a result you need in your research and don’t need all the extra chatter. It’s just cleaner that way. But to an outsider it does look more opaque. It’s a trade off.
I think that really is the best reason for not being more accessible. Along with less work - given a good paper already can take a lot of work to write clearly even for the immediate audience.
But there is tremendous value in reaching a wider audience, for readers, writers, and the very real serendipity of cross pollinating ideas. So an easily skipped concise titled section, that gave a little context or example for the non-inside crowd, would be a nice tradition. Even an appendix - although that might strike the established culture as too quirky.
Some papers manage to do something like that, a colorful example or perspective adding levity as well as clarity. So it is not breaking any barriers. Just not standard or prescribed.
Or maybe it wouldn't have much impact. I tend to find reasons to dive into many different new topics, so it is a prevalent need for one!
I completely agree — especially in the modern era where extra pages cost nothing.
_Make: Geometry: Learn by coding, 3D printing and building_ https://www.goodreads.com/book/show/58059196-make
_Make: Trigonometry: Build your way from triangles to analytic geometry_ https://www.goodreads.com/book/show/123127774-make
_Make: Calculus: Build models to learn, visualize, and explore_ https://www.goodreads.com/book/show/61739368-make
(oddly the Calculus book was published second, so I guess I'll need to re-read it after I finish the trigonometry book)
Hopefully, this will provide me with a sufficient grounding in conic sections that I can solve my next CNC project with a reasonably efficient set of calculations (trying to do it using my rudimentary understanding of triangles from trigonometry had me 4 or 5 triangles deep, barely half-way to the final point I needed, and OpenSCAD badly bogged down performance-wise).
It's quite recent, but seems very good to me, but my math education is about non-existent, so anything would be an improvement.
Most textbooks motivate it by the angle between the vectors or as projections (e.g., for hyperplanes). Numerics-focused ones will further emphasize how great it it is that you can compute this information so efficiently, parallelizable etc.. Later on it will be about Hilbert space theory or Riemannian geometry and how having a scalar product available gives you lots of structure.
> This feels like the biggest issue with most math books I’ve read and it makes me wonder where books that offer more semantic meaning of concepts instead of recipes exist.
All of the good ones do both. They first give the motivation and intuition and then make matters precise (because intuition can be wrong).
With that being said, I do remember my math and physics teachers in high school spend lots of time talking about the why and intuitions and let the books state the how.
It wasn't until years later that I found that it was all about "the area under the curve" and why that would be useful. At no point in those high school classes did we ever work a practical example. I was pissed off all over again when I found out how useful that stuff could be, and how much I'd missed out on.
I'm sure most teachers mean well, and I'm sure most of them try. But by god there are some truly awful twats out there who should never set foot in a classroom again.
He had to confess he didn't know why or how either of them works, he just uses them :-)
What an awful person. Chances are she was getting defensive and covering for her own lack of understanding. If I were a parent, I would confront her about that, not least of all her contempt for students and for learning, but toward parents.
Teachers don't know everything, and when they don't know, they should be able to admit that without hesitation or defensiveness. This sets a good example in general, of humility, instead of inculcating the notion that life is about having all the answers, or rather, pretending to have all the answers. All this does is set up people to become imposters. Of course, if you're teaching calculus, you should have at least a basic grasp of the material, and if you don't, you should say so, so that you've not put in a position where you have to teach it.
> I'm sure most teachers mean well, and I'm sure most of them try.
I think it is generally accepted that primary education isn't exactly packed with the best candidates, both from the point of view of pedagogical ability as well as mastery of the material.
https://betterexplained.com/articles/a-calculus-analogy-inte...
Perhaps a bit oddly I didn't have any problem with the pure math prof. also not offering any use cases, although of course it also does!
The difference in pedagogy seems to be which of these perspectives is treated as fundamental. Math education tends to treat the mathematical operations as fundamental. Physics treats the concept as fundamental and regards the operation as an implementation detail. It is very similar to how in software development you (for the most part) treat an API's interface as more fundamental than its implementation.
Unfortunately even physics books don't go over the intuition for underlying math very well, to their detriment. They seem to just assume everyone already perfectly understands multivariable calculus and linear algebra. I think it's because by the time you've gotten through a physics PhD you have to be completely fluent in those and the authors forget what it was like to find them confusing.
Colleges tend to have two tracks for physics, one that's closer to high school physics, which is as you described. A collection of algebraic equations that you have to either remember or, if your professor was kind, given a crib sheet of.
The other is the "Engineering" or "Calculus" based physics track where, as you can imagine, you're taking Calc 1 and Physics 1 at the same time.
I have seen some, kinder, programs where you take Calc 1 in your first semester and start the Physics classes in your second semester.
I guess if you want to learn math, only a math textbook will actually care.
It's not really that it measures similarity. Physics isn't interested in that. It's that it tells you lengths and angles, which you need in all sorts of calculations. In more advanced settings, a dot product is generally taken as the definition of lengths and angles in more abstract spaces (e.g. the angle between two functions).
In machine learning applications, you want a definition of similarity, and one that you could use is that the angle between them is small, so that's where that notion comes in. A more traditional measure of similarity would be the length of the difference (i.e. the distance), which is also calculated using a dot product.
In physics, the dot product is used to losslessly project a vector onto an orthonormal basis, and the angle measures how much of the vector's magnitude is distributed to each bases vector.
The angle can be defined in terms of the dot product, because you don't need the angle (as in a uniform measure of rotation) in order to compute important physical results.
The angle can be defined in terms of dot product because |a|=sqrt(a•a) can be shown to be a norm, and because a•b/|a||b| can be shown to always be between -1 and 1, and because those things agree with length and cosine of the angle for Euclidean spaces. It's not that you don't need the angle. It's that the dot product gives a good definition of angle in settings where it's otherwise not clear what it would be (e.g. what's the angle between two polynomials `x` and `x^2`)
In programming terms, there are interfaces for things like length and angle (properties that those things should satisfy). If you implement the dot product interface, you get implementations of those other ones automatically. The "autogenerated" implementations agree with the ones we'd normally use in Euclidean geometry.
One of the best books on Electronics according to HN crowd is The Art of Electronics, and it is filled with pages over pages of how-to of designing circuits of more than 1000 pages. But if you want to know why a Colpitts oscillator is the best for your design, all the best for that.
Even the textbooks produced by professors from the best engineering schools (e.g MIT, Stanford, etc) are not spared of this issue. One of my former lecturers (not MIT) for linear algebra and numerical analysis courses claimed that he worked and consulted for NASA, but how I wished that he had cover some of the motivations of doing a dot product. For the ChatGPT responses of the reasons of doing dot product for two vectors see ChatGPT 4 prompt below. I think once ChatGPT/Gemini/etc can operate properly on math equations at least as good they can for NLP, I think we are going to have a wonderful treat even for the understanding the higly complex concepts for examples thermodynamics, electromagnetics and general relativity.
Another classic example is try to find inside your favorite DSP book on why analytic signal in complex number generated by Hilbert Transform is necessary for communication and there is a very high probability that that's not a single explanation for that but there are many equations and graphs related to analytic signal. If we understand why or the raison d'etre, perhaps we can further ask why stop at analytic signal in complex number form, can we use quaternion number and what is the benefit of having more elaborate quaternion number form rather than complex form. If you are trying to do that and check this fact with most of DSP textbooks on this particular important issue then you will probably end up dissapointed even if you have bookshelf that contains majority of DSP books that are published since the invention of FFT.
I have also included the ChatGPT prompt for the dot product, and the responses are delightful to say the least.
>User prompt: Why you want to perform dot product of two vectors
>ChatGPT 4:
The dot product of two vectors, also known as the scalar product, is a fundamental operation in mathematics and physics with various applications, including:
Computing angles between vectors: The dot product can be used to find the cosine of the angle between two vectors. This is particularly useful in geometry and 3D modeling to determine orientations and angles.
Projecting vectors: The dot product is used in the projection of one vector onto another. This is crucial in physics for resolving a vector into components parallel and perpendicular to another vector, and in computer graphics for rendering and shading calculations.
Checking orthogonality: Two vectors are orthogonal (i.e., the angle between them is 90 degrees) if their dot product is zero. This property is vital in many areas including computer graphics, signal processing, and linear algebra.
Work calculation in physics: In physics, the work done by a force (a vector) moving an object through a displacement (another vector) is calculated using the dot product. This is important for understanding how much energy is being used or transferred in a system.
Machine learning and data analysis: The dot product is used in algorithms such as those for finding similarities between data points, calculating vector norms, or in the training processes like those of neural networks where weights and input signals are combined.
Each of these applications demonstrates why the dot product is a key tool in both theoretical and applied disciplines, linking geometry, algebra, and practical computation.
Things like:
> Then (dx)^2 will mean a little bit of a little bit of x;
is one of the pillars for one of my last struggles that has taken tens of my hours just to have a basic understanding of stochastic calculus and why this actually matters in this specific case.
When I see things like this makes me think that humanity is progressing as new generations having access to this information will make them learn faster. Thanks :)
Also, it should be easy to ask people attending school now if they would prefer books to internet. And to me if the justification is that youth doesn’t know what they are talking about is like denying progress.
We often find ourselves sifting through materials that always assume you have (or can figure out) the intuition and make it hard to find something that digs a little deeper and allows something to stick. This happens in math, programming, and you name it. We kind of assume the intuition is already developed most the time. Writing depends on our audience. If you're a programmer like me imagine someone trying to explain a modern and complex algorithm to you while simultaneously explaining every small part of how a program executes all the way down to machine code. That would be a horrific way to try to learn just the algorithm itself and it would be drudgery to try and fight through finishing an article or book that contained that much information, and because of that it would be a waste of the author's time writing such a thing. The simple answer is usually is to either stick with simpler examples and try to build intuition like this book does (though that doesn't mean there is no complexity in the example still due to digging into the example from a fresh perspective), or to just write a book full of rules and examples through problem sets (think like a textbook on math).
I swear this is related, but my favorite course in college was Discrete Mathematics which is sometimes titled something like Mathematics of Computer Science. The interesting thing is it was the first time in Math and at school that someone has explained to me the meaning of things that build a foundation in Logic in mathematics like "for all", "there exists", logical operators, and the negation of any of those things and what they mean. It was enlightening to say the least. It gave me an intuition behind the language of proofs. It allowed me to write my own proofs and to be able to read other proofs. Which proofs are everywhere in mathematics and understanding a proof is exactly like building an intuition of the underlying math. I couldn't believe after taking that one course that I actually understand textbooks way more often, and could understand that most the learning didn't happen by going example to example and solution to solution, but by understanding the underlying rule. Math always seemed, so much more ambiguous to me before then like the rules didn't clearly define every edge case and outcome, but they almost always certainly do.
The difference quantinent ended up as the Official Blessed Formulation of differential calculus, but it's very rarely used in practice, even though that's how calculus is used. And in practice calculus is still done using ad-hoc infinitesimal notations, but they are some weird thing with rules of their own which very few actually know (at least I don't).
Nonstandard calculus allows using infinitesimals in algebra with more or less the usual rules. Not sure if it's not more popular due to some fundamental technical or philosophical problems, or if it's just conservatism.
Stochastic calculus is quite bizarre indeed. Never understood e.g. the "proper" formulation of continuous time Kalman filters. Just limiting the timestep to zero seems to make sense and produces the right result with some massaging, but I've understood it's not really formally correct.
The hard part isn't the highest level concepts, which are actually fairly easy to grasp and somewhat intuitive.
The hard part is all the foundational knowledge required to solve actual math problems with Calculus.
The most difficult parts of Calculus (for me at least) are:
1. Having a very thorough grasp of the groundwork / assumed knowledge. Good enough that you can correctly solve an unexpected problem, from completing the square to long division of polynomials to an equation involving differentials.
2. Understanding and correctly applying the notation and graphing techniques, from Leibniz notation to sketching curves.
This is why large books and courses exist covering only introductory Calculus, not even beginning to scrape the surface of more advanced math.
Math isn't like programming. In programming you can often solve a problem using a library, framework, language facility, etc. without entirely understanding why it works all the way down to the binary level.
In math you can't often solve a more advanced problem such as Calculus problem without understanding the more foundational math such as algebra, fractions, etc.
If "information hiding" / layers of abstraction was possible in math, I would have completed my university entrance course months ago, but here I am still struggling.
Sure, we could have Algebra made easy and also Trigonometry made easy, Fractions made easy, Functions made easy, etc. etc.
I just find it personally irritating that all this foundational knowledge is brushed aside when it's really core to someone's actual competence dealing with actual math problems.
Maybe it's just assumed that people went to a good high school or had a private math tutor and already learned the foundations very well, but I think at least that assumption would be coming from a place of privilege.
It's similar to telling someone to take a Bootcamp in React and that will be enough for them to succeed as a software engineer. But to solve the kind of problems they are going to face in reality they will eventually have to learn at least some foundational Javascript and maybe a little about algorithms and data structures.
Certainly not everyone is in the same place in their learning journey as you. Material on calc, at a university level, is typically going to focus on calc. Yes it is assumed that you have learned the fundamentals before taking that course.
I was in a similar situation as you. If you really want to learn it there's no substitute for skipping over the fundamentals. I did that and did fairly well but it's all long forgotten. Never use the stuff :)
So many people tell me this that it's become cliche at this point.
I find it demotivating, but unfortunately I have to press through, as there is literally no other way I'm going to gain entry to my university's bachelors program.
A part of me wonders if this kind of fundamental knowledge could be actually useful, similar to being able to cook your own food instead of takeaway.
Kind of like how "first principles" thinking can apparently lead to new discoveries because you're not just mimicking / re-using the same structures that were already built.
Good luck you’ll get there.
Since you bring up food. As a former professional baker it would also take me some time to make croissants professionally at the level I used to. At least for me personally, if I don't use it I lose it. But I can certainly pick up faster than someone seeing it for the first time if I needed to.
Along the way you'll pick up some intuition that you can use elsewhere that's hard to quantify. Outside of the loans I don't regret taking any of the maths required for my CS degree.
Personally, I found the calculus lifesaver by Adrian Baker to be helpful in my studies as someone that was missing some fundamentals
Yep, this is the number one reason people think they aren’t suited for math. Everything is built on everything else, and if you missed anything you’re screwed. It takes a while to realise you are screwed, you can get by on rote for a surprising distance.
Ultimately, “there is no royal road”, but a good tutor will help you find those gaps and build out the missing bricks.
That's exactly what happened to me!
This is why I'm learning about differentiation yet struggling to factor simple fractions with a surd.
It's similar to the "expert beginner" problem described by Erick Dietrich (https://daedtech.com/how-developers-stop-learning-rise-of-th...).
I didn’t enjoy math as a child, and I used to be a lot more bitter about this when I first started to grasp what mathematics actually was. As a child, mathematics seemed like a small amount of “learn and understand a new abstract concept” (which I was pretty good at) bogged down with a huge amount of “okay now you have to solve a a bunch of problems based on that concept over and over again before we’ll trust you with another concept”. Eventually I figured out that mathematics itself really is the concepts, and that the concepts eventually build up to a level of complexity where it was increasingly challenging and fun to grasp them.
Maybe the reason it’s taught this way is because the vast majority of people aren’t mathematicians and aren’t really attracted to mathematics out of an abstract intellectual appreciation for the beauty of mathematical concepts; they just want to solve problems. And this is perfectly reasonable. But if I had it to do over again, I probably would have put more effort into mathematics and study more of it, at much higher levels, if I knew it would eventually get a lot more interesting.
And eventually things do start to branch out a bit. The standard K-12 curriculum up through calculus mostly builds up like a single tower where everything is built on everything else, but there are parts of mathematics where you can just sort of go in a different direction for awhile.
This is true to a good degree, but maybe a bit less so than you believe. Trigonometry is a topic that only clicked for me after finishing my uni calculus curriculum, I didn't get a great grade, but got by with a technique similar to how we handle complex numbers: Instead of giving up after being unable to solve an eg. weird chain of sin, arccos etc. functions, just declare it to be u(x) and do the calculus bits around it. In the last step substitute the actual function back in and you have an incomplete, yet technically correct solution.
These techniques definitely won't work in a tough online multiple-choice test (of the kind I'm getting) where they deliberately sprinkle in subtle quirks to deceive you, which would require very disciplined algebra, fractions, powers, etc. to identify.
It is common for mathematicians to rely on theorems as black boxes(ex: classification of surfaces) even without knowing the proof. Secondly, people can even write research papers without knowing how to work with some object covered in the paper, by working with collaborators who are experts on a different topic.
It would be helpful to isolate the essence of calculus itself from the symbolic techniques, for ex to actually calculate integrals(especially magical seeming substitutions and nontrivial factorizations) as many of these symbolic techniques will appear in different topics even outside calclus.
Math classes build up, and at some point unfortunately they do have to start assuming that your previous classes were solid. Calculus is where algebra and trigonometry gets some of that treatment. It is extremely common for a calculus class to reveal some shaky algebra foundations though, so I’d hope your school has some help there…
My guess is that most people hit a wall of abstraction at certain point.
I don't think it's a limit to their abstraction, I think it's that they didn't work properly on the fundamentals, so they had a superficial understanding of the abstractions.
To give a fitness analogy it's like trying to do heavy barbell presses before you can even do 10 pushups in a row.
My experience with programming is that once you get really really good with fundamentals you suddenly leap ahead and pick up new languages, paradigms, etc. incredibly fast.
Maybe this partly explains the 10x phenomenon - it's because they worked very hard on the fundamentals.
this is one big hurdle in learning math backwards. You discover new missing pieces at every corner. Each missing piece leading to another missing piece.
Learning math from the basics to advanced (as recommended by most) is very frustrating at how slowly you actually develop the math muscle.
At a deeper level, conceptual grasp does not make you good at math, its not enough. You may fool yourself into thinking you "get it" till you try to solve a few exercises. You need to repeat the lower levels enough to make it into muscle memory (which some people refer to as math intuition or groundwork) before embarking onto higher levels that build on it.
So working your way bottom up is slow and frustrating, top down is slow and frustrating. What do you do?
Just keep at it. One key observation for me was that at some point the misery and rabbit hole nature diminishes, quite rapidly. The groundwork of solving all those exercises repeatedly pays off and the next set becomes a little easier. Getting to calculus after spending ridiculous amount of time on algebra is the only way I have known to work.
And this is true for learning progamming too. knowing the concept of loops is essential but, you still can't write efficient code to sort an array. You need to get the syntax and write enough loops and then progress to exercising writing specific sorting algorithms repeatedly to get them into muscle memory.
But there is an inflection point beyond which the same concepts repeat but in different variations and they take progressively lesser time to get a grasp on.
thats just how I've learned math and programming. Also why a large percentage of people just give up hope and accept they just don't have the math gene. Meh.
Yes this is exactly what happening to me.
E.g. I got up to Week 2 of the course and suddenly made the big (to me) discovery that sqrt(a/b) = sqrt(a)/sqrt(b).
It seems trivial I know when you see it written like that, but the problem is to recognise and apply that principle in the context of a broader problem such as factoring.
> Just keep at it.
Thanks, this gives me confidence that I'm not wasting my time haha
I am beginning to get better at it, to the point that I can often work out why I got a question wrong on my own without referring to the answer.
https://hn.algolia.com/?query=mathacademy&sort=byDate&type=c...
The really important part for me was to rip these small but critical parts out and form somewhat like mental workout routine that I kept repeating multiple times per week. By week 5/6 I could solve the same/similar/related problems which weeks ago took me several minutes with ease and I had more brain power left to think about higher level and related concepts and techniques that formed more connections, making the experience a lot more fruitful, productive and faster. Without that mindless, disciplined mental routine to get the basic and critical stuff in muscle memory, I do not believe I could have made it through.
Good luck.
> all of it really
The link is not a pamphlet (unless you read only the linked HTML page). It is an entire book, published in 1910 by Silvanus P. Thompson, and sufficiently well-regarded that it was re-edited in 1998 by Martin Gardner, and (independently) lovingly re-typeset in TeX by volunteers (and also turned into this website). Clearly it serves a need, and is not merely a “trite” pamphlet.
(The edition by Gardner is actually recommended against by some, who see in it a clash of two strong personalities, individually delightful.)
(As someone who's surprisingly bad at math and trying to undertake a Calculus pre-req to get into university!)
Calculus Made Easy is an amazing book btw, by far the best introduction and much better than the way I was taught at school as it actually builds your intuition.
Which is fair, but if you believe that you shouldn’t have insulted the work itself by dismissing the value of its content and calling it a trite pamphlet.
It comes down to Leibnitz Vs Newton, and the world has standardised on the notation of one (I forget which). However the notation is a destination when learning it all, and the foundational ideas behind calculus were best explained taking ideas from both of them.
That's what this book does. It takes you through with every simple jumps in logic allowing you to discover calculus yourself and you therefore have the foundations to reason about it yourself. You don't just have to learn the final answers by rote.
For most people—who won't solve complex math problems daily at work—the takeaway from learning math is not their mechanical ability at solving math problems. The takeaway is their understanding of math concepts and ideas, which will shape their thinking skills in general.
The later reading suggests a more intuitive (to me) definition: a limit f(x)→q as x→p exists if, for every open interval Y containing q, an open interval X containing p exists such that f(x)∊Y for all x≠p in X (and then if f(p)=q, f is also continuous at p).
Another nice property of the above definition: replace "interval" with "ball" or "neighborhood" for analogous definitions for functions between metric and topological spaces, respectively.
Agree, but for some others, there are real world consequences, e.g. whether they get accepted into a university or whether they can read and properly understand an academic paper.
https://press.uchicago.edu/ucp/books/book/chicago/C/bo548572...
Genetic
But I didn't really "know" calculus until I read a book not too dissimilar from this one, "A Course of Pure Mathematics" from 1908 (!), which constructs calculus up from number theory (I think the fundamental theorem of calculus comes halfway through the book?). From that point it's impossible to forget.
As for why it's not taught this way today, I'd blame our testing regime and large classrooms, which incentivizes temporary memorization of key formulae and knowing where to mindlessly apply them, over deep/lasting/semantic understanding. I'd also blame the fact that we have a different maths teacher every year, so students come in with heterogenous understandings of the pre-requisites for the next year's material, so the first part of a section is spent reviewing + consolidating.
It takes maybe 10-20% more time to get a rich understanding of the subject that lasts a lifetime. But we value compression and instantly-measurable results more than actual learning. :/
Encountering material like this makes you really happy, but it's kind of bittersweet because it makes you realize that the modal state of modern pedagogy is pretty abysmal.
My end goal is to be able to follow Andrej Karpathy's "Neural Networks: Zero to Hero"[1] without any big problems So starting basically from "zero" in order to learn the prerequisites before learning what you actually want to learn on your own can feel daunting at times. But I think taking shortcuts will result in frustration. So, here I am taking algebra courses on YT with 38 years.
[0] https://youtube.com/@ProfessorLeonard?si=0kiGvmbZv4b9Sgf9
[1] https://youtube.com/playlist?list=PLAqhIrjkxbuWI23v9cThsA9Gv...
The link isn’t to the front page with his name.
"How can this be possible?!" I hear you ask.
Well, the book was written in 1910, 50 years before category theory appeared.
But do not worry! There is a book that uses categories to develop ordinary differential and integral calculus!
What could be easier than that? I don't know! But if I find it, I'll let you know!
Enjoy!
https://books.google.com/books?id=gaE5EAAAQBAJ&newbks=1&newb...
IMO, the author is right about what's wrong with the most "proper" books, but overcompensates, and thus makes it unnecessarily complicated to understand, maybe even more so, than some "proper" textbooks. It is too wordy, too informal and pretty hard to read and follow. I don't need any Dean Swift poetry and references to "Queen Elizabeth's days", I just want to know what is calculus, why do we need it, and how do I actually do it, goddammit… That's if we follow the seemingly easier "engineering" approach. And we need it to be a bit more formal still if we follow even slightly more mathematical approach, which I think is actually necessary if you want even a bit of real understanding, because math is this thing where it's very easy to fool yourself into thinking you understand something, when you actually don't, and then get all flabbergasted and helpless against paradoxical "fake proofs" or being asked to prove anything yourself. And to tell valid derivation from invalid, yes, you do need some formal definitions.
And in fact, the (formal) basics really aren't difficult to understand at all. Everyone can readily agree that x² grows faster than x, and being introduced into concept of limits to see why (dx)² can be considered negligible compared to dx. You don't need to be comparing weeks to minutes for that. If anything, the latter only throws you off. And I feel like it takes way less patience to read a handful of formal definitions than pages of these "old British"-stylized ramblings.
Edit: oh, it isn't "stylized". Still, doesn't change the main point, that there are far better more recent resources to learn calculus.
Daniel Kleppner and Norman Ramsey, "Quick Calculus" - We used this for math background when I was taking AP Physics in high school. The physics textbook was Resnick and Halliday, which uses calc throughout. It is not particularly intuitive, but the "programmed learning" approach makes it very easy to pick up enough basic computational ability to do applications. (The phrase "programmed learning" is so old that maybe I should clarify that it has nothing to do with computers.)
H. M. Schey, "Div, Grad, Curl, and All That" - This might be helpful for people taking vector calculus, or using it in an electromagnetism course in physics. In fact, a lot of his examples are from E-M. Very good at providing intuition without too many technicalities.
I think the best uses for books like these are: (a) You want to get the "big ideas" without worrying too much about technical facility (maybe because you won't ever need it - e.g. you're just learning for your own interest); (b) To provide a fair amount of the math you'll need in (say) a physics course; (c) As prep before taking a standard calc course, or maybe as a refresher. Books like Thompson, Kleppner and Ramsey, or Schey will help with the big picture and to boost confidence but it's best to read them before taking the "real" calc course - since once the "real" course is running you'll be too busy with the work in the course to read something else on the side. But do whatever works for you. Calculus really is wonderful - have fun!
the most important idea in calculus is often glossed over, weakly presented or omitted entirely: the continuity of the reals. i feel that once this is fully understood, most of the ideas in calculus become intuitive.
Just out of curiosity, does anyone know of similar sites for linear algebra, discrete mathematics, statistics, etc?
[1] https://math.mit.edu/~gs/linearalgebra/ila6/indexila6.html
[2] https://www.pearson.com/en-us/subject-catalog/p/linear-algeb...
Here is an additional link to the Spring 2023 course materials that follow along the 6th ed. Of his textbook [0].
But my perspective changed. "Memorizing to free up brain space" is very real.
In my opinion, college calculus is poorly named and instead should be named "Pre-Calc" and the real calculus course is then physics.
Most of those courses are not calculus, it's just advanced algebra, and we do those trying for success in them a disservice with the dishonesty.
This is probably due to the education I got. Every maths or science exam in my country hands copies of this booklet out for reference: https://www.examinations.ie/misc-doc/BI-EX-7266997.pdf
Either way, thanks! I'm saving this. It's a wonderful reference.
While I enjoy Silvanus book, I don't think calculus can be made easy without preparing the students to accept the handwaving and trickery involved in the jump from an equation to its 'area' or 'velocity'. Compared to the solemn majesty of euclidic trigonometry or relatively straightforward step-by-step solving of quadratic equations the techniques foundational to calculus are rather devious (as are those that bring in complex numbers).
In high school the combination with physics made it harder for some students, they had the impression that learning math amounted to learning about nature, rather than a language for expressing fictions about a view of nature. In turn the approximative nature of the problem solving and calculus didn't fit very well.
Yes this hits the nail on the head
The explanations overall look simple, maybe too long sometimes, but that's a killer prelude
First, credibility: Ah, I never took the first year of college calculus -- to make faster progress in college math, got a good book and taught myself. After an oral exam at a black board, was admitted to the second year. Majored in math. Took (so called) advanced calculus from Rudin's Principles .... Took applied advanced calculus from a famous MIT book from an MIT Ph.D. Taught calculus at Indiana University. Studied Fleming's Functions of Several Variables, right, through the inverse and implicit function theorems, Stokes formula, exterior algebra, etc. Published some advanced math, essentially advanced calculus. Once, with some calculus, at FedEx pleased the most serious investors on the Board, had them return their airline tickets back to Texas, stay after all, and saved the company.
Okay, the 90 seconds:
Consider a car, its speedometer and odometer.
Calculus has two parts.
For the first part, you read the data from the odometer and reconstruct the speedometer readings. Doing this, you take changes in (increments of, differences in) the odometer readings, say, every second. This is called differentiation.
For the second part, you read the data from the speedometer and reconstruct the odometer readings. Doing this, you add the speedometer readings, say, every second. This is called integration.
Starting with the odometer readings and differentiating to get the speedometer readings and then integrating the speedometer readings will give back the odometer readings -- this is the "fundamental theorem of calculus".
~90 seconds.
For more, instead of the 1 second steps could use 0.1 seconds, .... 0.0001 seconds, etc. With really small steps, making them smaller will make no or nearly no difference. So, the reconstructions will have converged, reached a limit.
Reaching this limit is mostly what was novel when Newton, Leibnitz, etc. invented calculus.
It is fair to say that the first big application was to Newton's law F = ma where have some object -- baseball, airplane, rocket -- with mass m and are applying to the object force F. Then a is the acceleration of the object. Integrate the acceleration and get the velocity v. Integrate v and get distance d. So, can find where the rocket is after, say, 10 seconds. Other early applications were to planetary motion.
There are applications to areas, volumes, classical mechanics. fluid flow, mechanical engineering, electricity and magnetism, quantum mechanics, relativity, electrical and electronic engineering, e.g., Fourier theory.
Physics and engineering are big users. And there are applications in economics, e.g., work of Arrow, Hurwicz, and Uzawa on the Kuhn-Tucker conditions.
By the early 20th century, calculus was refined, e.g., presented with careful assumptions, definitions, theorems, and proofs, e.g., B. Riemann and, soon, H. Lebesgue. By then there was the idea of the highly irregular Brownian motion and the observation that differentiation wouldn't work there -- Brownian motion was differentiable nowhere!
Calculus? A pillar of science, technology, and, thus, civilization.
How I fixed it:
1. I learned to become bored and be okay with it. I sometimes lie on the ground for 15 minutes to an hour and do nothing (and that's different than meditation!). I feel incredibly bored, bored to tears actually. I have noticed when I'm in this state, I'm in a much better position to do math or anything else that I slightly find boring but also really interesting.
2. I went to the root of my problem. I'm Dutch. I failed a course in high school called wiskunde B (math B). It teaches calculus, vectors and trig. I'm currently doing that [1]. It's been a few months, but it's going well.
Results and observations:
I think I'm almost at the level to do actual college level math. It helps that I studied CS, because I know a ton of math channels on YouTube and they are now immensely helpful. The thing is the course I'm following isn't explaining the theory well. It is amazing in making me practice, so fortunately I just need to go to good YouTube channels such as 3Blue1Brown or Khan Academy for an adequate explanation on theory.
Another observation is that it has made me a better programmer. My debugging style has always been "turn on the debugger". Now my debugging style is: think deeply about code execution first. Math makes me comfortable with "code execution" because with math you have no choice!
__Looking for a math tutor (email in profile)__
I'm looking for a math tutor. I'd love to ask questions during the week that I don't need an immediate answer to. I think it'll take about 1 hour per week for you. I have noticed I'll need one after the wiskunde B course because I have asked about 100 questions to them in total over the duration of 4 months.
Edit:
To add. I think for me what made calculus so hard was not having a strong understanding of the fundamentals. The gist of calculus of slope and area is easy. Manipulating the equations perfectly 100% of the time. That used to be really hard and it is getting easier the more I practice.
I just happen to watch a YouTube video where a math professor mentions the same thing [2].
[1] The math course I've followed (400 euro - includes exam): https://kdvi.uva.nl/nl/onderwijs/wiskundecursussen/e-winterc...
It's in Dutch, but they have an English version in the language settings.
[2] https://www.youtube.com/watch?v=M7febmLhS6E&ab_channel=BigTh...
My tool to get me through slogs is streaks, so I commited to doing a lesson (or at least part of a lesson) every day and I'm at 198 days so far.
I wrote this at 100 days in case it's helpful: http://gmays.com/math I'm not sure if I'll have time to write an update for 200 days, but maybe at the 1 year mark.
There are psych experiments to back this up. Also Seinfeld's central thesis is that humans are essentially always just frantically trying to avoid becoming bored.
Seinfeld seems to be right :P
I used to tutor math (10 years ago), but would be happy to provide some asynchronous help if you like. Time permitting of course. Let me know how I can help.
Math background: Have completed and done well in all math courses I’ve taken, thru graduate level Chaos Theory (last math class I took).
I can't seem to contact you. My email is in my profile.
Can you share links to some of those?
https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PL0-GT3co4r...
and Khan Academy
I just use YouTube a lot and ChatGPT a lot too. ChatGPT is a pretty okay tutor at high school / early college level math
Sure sounds like https://encyclopediaofbuddhism.org/wiki/Dzogchen
We can approximate the slope (derivative) of any function y = f(x), at (x,y) as the slope of a straight line from (x,y) to another nearby point (x',y') where y' = f(x').
The slope of this line is (y'-y)/(x'-x).
Since we need x' close to x to make this accurate, we use x' = x + dx where dx is small, and we can also represent y' as y' = y + dy.
So, then our slope is (y'-y)/(x'-x) = dy/dx = (f(x+dx)-f(x))/dx
And that's all there is to it.
e.g.
For f(x) = x^2
dy/dx = ((x+dx)^2-x^2)/dx = 2x + dx
We get the true value of the slope at (x,y) in the limit of dx -> 0, giving us the derivative of f(x) = x^2 as f'(x) = 2x.
For anyone reading this, I highly recommend attending the best college you can. The reason why is: the caliber of professor. I was lucky to receive (IMHO) a world-class engineering education at a university in Illinois. It would have been even better at an Ivy League school. But I have heard various accounts from students at other state schools and community colleges, and there's a distinct lack of nuance in their coursework.
For example, my statistics and probability professor explained everything by counting apples and bananas on his fingers. It sounds hilarious, but that's all probability is - counting the number of certain outcome(s) out of all possible outcomes. He broke down even the most complex problems into a series of basic counting operations. A less dedicated professor might have just regurgitated the course material verbatim.
Where this really matters is, I don't believe in complexity. Learning to learn is about breaking down problems into smaller pieces that are at most the maximum size that can fit in a human brain, and then working each problem until reaching the final answer. Once you can do that, you can tackle any challenge, like running hurdles, with full confidence that you will succeed in time and with enough resources.
Which is why I'm so hard on fields which I don't feel are mature yet, due to their byzantine terminology and barriers to entry. Some low hanging fruit might be: AI's obsessions with tensors, CUDA and the entire concept of a GPU (instead of just using symmetric multiprocessing), quantum physics, organic chemistry, biology, medicine, genetics, law, accounting (the CPA test for example), imperative programming (vs functional programming and fundamentals like Lisp and spreadsheets) and design/repair of internal combustion engines.
Now, paying for college is a whole different story. It stopped being affordable in the late 2000s, by design to enrich private interests and create a subjugated class that needs a job to pay off student loans, much like the scam of attaching health insurance to employment. The only feasible answer there is systems thinking. Which looks like working the problem backwards by enumerating the core problems and then deriving a general solution which allows everyone to attend for free, funded by taxes paid by automated robots and billionaires. I mean, that's how Star Trek works, why settle for less. Getting rid of billionaires is just a bonus.
>> A very simple example will serve as illustration.
>> Let us think of xx as a quantity that can grow by a small amount so as to become x+dxx+dx, where dxdx is the small increment added by growth. The square of this is x2+2x⋅dx+(dx)2x2+2x·dx+(dx)2. The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x2x2.
You need something like smooth infinitesimal analysis [0] to make this rigorous, but it's much better than anything involving limits.
I've come around to infinitesimals, but mostly through exposure to the large hyper-reals. (for context for someone who doesn't know, the idea is to define a number, k which is greater than all real numbers. If you take 1/k, you have a very small number and you can fit an infinite number of 1/k's between 0 and the "next" real number. This concept is what sold me on infinitesimals.)
Yes it's equivalent to limits, but limits are a very cumbersome machinery, specially if you use the epsilon delta definition (there exists .. such that all ..).
But note that I just linked you a PDF that does fully 100% rigorous calculus using only infinitesimals with no limits. Yhey aren't disregarding small terms willy nilly (like it was done in the early history of calculus)
The only catch about SIA is that it requires you to use intuitionistic logic rather than classical logic in your mathematical arguments (which I admit is a barrier, but it also buys you some things). And what it offers is much simpler proofs that support intuitive reasoning.
There is also this book, "A Primer of Infinitesimal Analysis" [0], which develops a big chunk of calculus and classical mechanics using only infinitesimals, and is fully rigorous.
[0] https://www.cambridge.org/br/universitypress/subjects/mathem...
I made the mistake of taking algebra-based physics, then calculus, and only after the calculus course did I realize how much harder I made my life by not starting with calculus (and learning it as the mathematical language of physics).
i.e something like, Distance, Velocity and Acceleration with respect to Time.
Velocity is rate of change of Distance in respect to time (ds/dt) Acceleration is rate of change of velocity in respect to time (dv/dt)
You can derive the equations of motion v^2 = u^2 + 2as etc.
Things like the Bernoulli equation from Fluid Dynamics and a lot of other engineering principles can be derived this way.
>Let us think of x as a quantity that can grow by a small amount so as to become x+dx, where dx is the small increment added by growth. The square of this is x²+2x⋅dx+(dx)². The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x².
... and you need to know what a shilling and a farthing are to understand some of the examples.
-eastern european