Could anyone comment about Paul's Online Math Notes? I'm working with it.
Calc 2, Calc 3, Linear Algebra, Differential Equations.
I am an adult student. For many reasons, I self-study Science and Maths very lately.
Could anyone comment about Paul's Online Math Notes? I'm working with it.
Calc 2, Calc 3, Linear Algebra, Differential Equations.
I am an adult student. For many reasons, I self-study Science and Maths very lately.
I'm actually a big fan of Paul's Notes. I always recommend them to students as supplemental / alternative material.
I really can't align myself with this sentiment. The phrases "not very in-depth" and "extremely rigorous" really don't have any meaning, and in many cases, I find that people use them when they have no other valid criticism to offer. I think that it's safe to assume that the notes were written with the beginning student in mind, so there is a lot of hand-holding along with motivated examples and explanations. Notation-heavy proofs and expositions are often replaced with a simple picture and description of what is really happening. Fine - it's not always mathematically precise. But it is friendly to the student and allows him to learn. I see no foul in that. (But if one really cares, most of Paul's explanations can easily be converted into a complete proof.)
i find that the phrase "really doesn't have any meaning" really doesn't have any meaning.
yes, they are written for the beginning student in mind (which can be a good thing!). in a way they are like the reader's digest version of a novel. they are good for what they are. they are not an in-depth presentation of the subject, and they are geared to providing a certain type of computational proficiency, rather than a theoretical understanding.
to give an example, some of the constructions in linear algebra (nullspace, direct sum, quotient space, to name a few) make more sense if you've spent any time at all looking at abelian groups, especially given that with many of the examples actually used in exercises for linear algebra, you have integer entries (an abelian group is, by definition, a Z-module, and vector spaces are free F-modules). but you won't find that aspect of linear algebra (the "algebra" part), or at least not much of it, in Paul's notes. you WILL find that sort of thing, however, in most good textbooks on linear algebra.
the reason i did not make this post, as i have written here, instead of post #2, is that i felt it was "too much information" for the reader's original question. i merely wanted to make him aware that although they can be an invaluable asset, that they have their limitations, and that he would be well served to supplment them with other material. i am not, in general, required to justify the comments i make, but your response seems to imply that i said what i did, because i really had nothing else to say. i assure you, this is not the case.
Paul states that he curtailed the complexity of the examples for online reading and sustained enough in-depth on explanation.
I am an adult student. I self-study maths very lately. I was much benefited from Paul's work especially the differential equation part. He made the best serious entrance material for DE.
Graphs are speechlessly nice.
I've tried amazon recommended Schaum's Outline Series and found it is too rigorous for me.
one of the nicer features about Paul's site, is that the problems are worked out (but the solutions are hidden at first, so you can try them yourself), which is one of the main draw-backs to self-study-when you get stuck on a problem, there is no one to help you.
Actually, it means that you should explain yourself more clearly. Thank you for doing so, by the way.
The purpose of my post was to point out that your comments, no matter what your intention was, sounded like a baseless knock on Paul's notes. (Of course, I'm happy to see that it wasn't, even though I still somewhat disagree with your evaluation of the notes.) I'm sure that you know that remarks of that nature occur far too often in academia (and they usually result in unreasonably abstract and unmotivated material for students). So, although thisthe reason i did not make this post, as i have written here, instead of post #2, is that i felt it was "too much information" for the reader's original question. i merely wanted to make him aware that although they can be an invaluable asset, that they have their limitations, and that he would be well served to supplment them with other material. i am not, in general, required to justify the comments i make, but your response seems to imply that i said what i did, because i really had nothing else to say. i assure you, this is not the case.
is true, I wish that you will consider otherwise in the future, especially if somebody is asking for advice. (And to be honest, I'm impressed that a mathematics person is willing to assert this.) I hope that the reasons and attitude behind this are apparent.i am not, in general, required to justify the comments i make
granted, it may have been better to give the original poster more reasons for searching out supplementary texts/web-sites. although you might not think it from my intial post, i have on several sites and forums recommended them. one of the things i think they are especially appropriate for, is adults who may have forgotten a lot of the math they learned earlier, and are contemplating a return to school (they make great "refresher" courses, and give an idea of what will be expected of you to know). if a course you are contemplating enrolling in\auditing\self-study requires linear algebra (for example), everything in Paul's notes will be assumed as "given", and probably used. conversely, if ALL you know about linear algebra is what is contained in the notes, you have enough to start with to "fill in the gaps" later.
i must say that i think his differential equations course, is an excellent introduction to a really difficult subject, and there aren't many on-line resources for it, unlike say, calculus, where there are so many websites it just isn't funny.
In the case of Linear Algebra and your comments I think the significant factor is the target audience. The majority of the prospective audience for a course on linear algebra are not prospective mathematicians but engineers, economists, statisticians ... and for most of these the an abstract approach is counter productive. They need a concrete approach to make the subject comprehensible (even then only to a minority, if experience is anything to go by). Basically you need different linear algebra courses for prospective mathematicians and non-mathematicians.
The level of mathematical competence of "professional" engineers in the UK is a matter of some worry to me. I have given talks on signal processing to engineers where most of the available time has been wasted on explaining dB's - the standard logarithmic representation of "things" (relative signal levels, gains, ...) in engineering - and don't get me onto the statistical competence of safety engineers ...
CB
that's a valid point, many engineers look at math as just a tool for solving problems in the areas they are really interested in (sizing a steel beam, designing a dampening signal circuit, detemining whether a certain economic interaction represents a net gain or loss).
the sad thing is, abstraction, in itself, isn't really to make things harder, but to make things easier, by giving a less specific criterion for applicability. things that could be settled by Jane Q. Professional once and for all, for many cases, cannot be imparted at once, but instead she must realize this internally after noticing the repetitive nature of a thousand calculations. groups aren't harder than vectors, they're easier (for one thing, there's a lot fewer rules to remember), but it's hard to explain them to peole with the mind-set of "it's not real unless i can hold it in my hands" (never mind the fact that the relatively advanced abstraction of number was accepted unquestionably at the outset of schooling).
and so, many linear algebra courses (including Paul's Notes) focus on the numerical aspects, because adding and multiplying numbers (often integers at that) pose no threat of them having to "learn something weird that i won't understand".
here is a personal anecdote: i moderate another forum, for an online game. one of my fellow moderators was studying group theory, as part of his chemistry degree in the UK. except, he really wasn't...what he was taught to do, is recognize molecular symmetry, by checking for a list of characters. and he was finding it terribly hard, because he had NO matrix experience, he had no idea what a "general linear group" might be, so he was forced to assimilate a lot of information, with no understanding of what it meant. oh...these squiggly symbols match...they're the same....these don't....let's try another. in the future, if he needs the information molecular symmetry encodes, he'll have to look it up in a book...he won't KNOW.
and this happens all the time in engineering. people don't know what the results mean, in terms of the consequences of their calculations, and they make mistakes. and sometimes, people get hurt, or even die because of this (building collapse, anyone?). i understand that there is a lot "out there" to assimilate in any field. i understand, people are pressed for time, and time is money, and the pressure is on to get there first.
and yet, i am dismayed by how eager people are to learn "cookbook math", and abdicate the one thing that makes humans so special: we can think. and abstraction is at the very core of that process of thinking....otherwise "you never step in the same river twice" and everything is ad-hoc. i don't know about you, but i'd rather not fly in an airplane that was designed as fast as humanly possible, by someone learning the minimum they needed, just before they needed it.
i don't say this as an academic, wanting more pedagogal purity in the ivory tower, but as someone with 20 years work experience in the construction industry, who has seen first-hand the terrible costs of ignorance.
if "forward thinking" corporations and governments want people who think and work like machines, they should just hire machines, they make fewer mistakes, anyway. if they want brilliance and innovation, they should train for that. i've said it before, and i'll say it again: mathematics is not about numbers and calculation it's about ideas (some of which involve numbers and calculation). and the ideas are important, and they should be getting to students sooner, rather than later. there's a beautiful article on cut the knot, about an experimental geometry class where they didn't actually do that much geometry. but they did think about things, and many of the students said it was the best learning experience they had in their entire lives.
we are selling the next generation short, but it's not Paul's fault. he's filling a need, and i'd have to say his target audience is spot-on. who is to blame? the parents (and the teachers) and the society at large who say: it's ok to hate math, it's a dull, burdensome subject which you don't really need anyway. and it's true, technically, you don't need to be able to think logically, or to have clear definitions of terms, or to be able to reason and conceptualize clearly to survive...but...come on now, have we (as a species) no self-respect?
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I don’t want to derail this thread or be overly nit-picky, but I feel that this part of your rant requires some clarification.
Buildings are actually built, at least in Canada, with a probability of failure in any given year of 10 to the minus 6 (nuclear facilities are 10 to the minus 7). So the fact that buildings collapse does not necessarily indicate faulty engineering; it is quite possible to follow code to the letter, and have a building collapse. And as to your second point, that engineers (or at least civil engineers) are accustom to following a procedure without understanding runs quite opposite to how civil engineers are trained. In practice, most structural engineers who end up in design positions use computer modelling and the like to design structures. But in school, at least in Canada, they learn to design via hand but not only that, they learn the theory behind those hand calculations. The entire education of a structural engineer is dedicated to teaching them how to spot an odd result, and knowing why such a result is odd; you cannot possibly do that without an intimate understanding of what is going on.
There are shitty structural engineers, who don’t care to understand but in my experience that is the minority. It’s no different than any other field.
well, i don't know how the licensing of engineers is done in Canada, but in the US, someone who got a degree in mechanical engineering (maybe they focused on hydraulics, or manufacturing) can easily start sealing drawings as a structural engineer (with the exception of a few states, who specifically require training in structural engineering. Illinois does, and so does California, i think). i worked for a structural engineering firm for 3 years, and i had opportunity to consult with many of them.
to be fair, the vast majority of them ARE competent, but there are more idiots out there certifying structures then there ought to be. in Texas, where i live, one of the largest commericial engineering firms in the state stopped doing structural work entirely for several years, because of several large pending lawsuits. and sure, there are problems that occur because of the sub-contractors being ignorant, even when the engineering is spot-on.