- #1
aneesh.mulye
- 18
- 0
Earlier today, I just took the last examination paper for my examination in my third year in computer engineering. (One more year left to go.)
Over the past year, now that I no longer have to do mathematics as a subject forced down my throat - it was a compulsory subject until the end of the second year - I've come to appreciate it much more. I've come to the realisation that throughout my mathematical education, there have been serious and structural deficiencies. It is a wish to correct those deficiencies that impels me to ask for advice here. In fact, reading this forum was one of the key motivators for my becoming interested again in mathematics. Thanks are in order.
The next few paragraphs are going to be a bit of a personal mathematical history, so those less patient can skip to the end if they so desire.
In high school, for our first board examination (a standardised test given at the end of the tenth grade) we were forced to do Euclidean geometry. We were mostly expected to memorise proofs by rote, and reproduce them as needed on the examination paper. There was a set of "standard" proofs - quite a large set, by the way, requiring a lot of memorisation - out of which questions were asked. I like the other parts quite well - basic algebra, and so on - but this was enough to make me hate this part of the subject for a long time, and it is only in the last semester that I dealt with the last of the proof-phobia that that phase induced.
Later, in the 11th grade, we were taught the standard pre-calculus sequence - trigonometry, co-ordinate geometry, the needed algebra, and so on. In the 12th came actual differentiation and integration, along with an absolutely elementary introduction to differential equations.
During this phase, the focus was on a set of standard "techniques" one could use to solve the question which would come on the test. The problem was not that we had to learn the techniques. The problem was that we were provided with absolutely no context in which to understand all these things. They were all isolated little islands of information which we had to remember. There was no overall structure into which to fit these things.
Now, when I look back over the curriculum, I realize that the reason we covered the concepts of functions, injectivity, surjectivity, bijectivity, and their relationship with the invertibility of a function before learning about inverse trigonometric functions was because they were needed to understand why ITFs work the way they do. However, when we learned them, we did not relate the two concepts. They were separate. The chapter with the introductory concept of relations and functions was treated as one thing, and the chapter on ITFs as another.
This sorry state of affairs continued throughout my education, up to and including my mathematics courses for my engineering degree. This is in large part to the "high-schoolish" attitude of many of my peers. It isn't really their fault, to be honest - they live with their parents and are not afforded much freedom, so the college considers the parents their clients instead of the students, and attempts to act as an organisation would in loco parentis. Even today, mathematics courses for engineering are taught with the same idea - memorise the techniques, plug in the numbers. It would again be fine if there were understanding and the memorisation was supposed to help it along, but that is not so. Almost all students memorise the techniques, and forget them within a few months of the examination. I do not wish to be harsh, but most would not be able to recall what they are supposed to have learned if you ask them now. Again, I cannot blame them, because this is all that they know. The problem is particularly acute in computer engineering, as most of the mathematics we learn, we do not really need.
If my mathematics education has been so uniformly dull and drab, not to mention off-putting, why am I still interested in mathematics? Well, why does an eagle fly? My innate love the the subject has kept me going through these nightmarish years. I could not believe - and still do not - that the torture I was put through was all there was to mathematics. It was only after finding this forum that I found that I was right and that there was hope.
To come to the crux of the problem:
Right now, I'm in a rather odd position. I know calculus, and have done the standard engineering mathematics sequence, but I'm not satisfied with it. Specifically, I don't think I have mastered anything that I want to.
Therefore, I'm looking to "re-do" my entire mathematical education, beginning with pre-calculus, moving on through calculus and related algebra, and finally doing my engineering mathematics the way it was meant to be done. (The material before that that I need, I have mastered.)
What is it that I want to do over in this fashion?
From high school:
Trigonometry (I was notoriously weak in this, and I want to master it)
Algebra (Binomial Theorem, Sequences and Series, Matrices, Determinants)
Co-ordinate geometry (Lines, Circles, Conics, 3D co-ordinate geometry)
After this is done, I'll move on to revising basic calculus and university algebra. Specifically:
Calculus I & II
Linear Algebra
Fundamentals of Multivariable Calculus
Complex Calculus (AKA Complex Analysis in our courses)
Differential Equations
Vector Calculus and Scalar and Vector Field Theory
Fourier and Laplace Transforms
Partial Differential Equations
Out of these, the only thing I've not studied before as part of the curriculum is the bit about PDEs, and Linear Algebra (our study for our degree was not deep).
The reason I posted this thread is because I want some advice relating to my plan.
First of all, most of what I'll be doing will be revision. However, we've only covered all these things (I refer to everything beyond Linear Algebra in the list) superficially. I want to not only study them, but have mastery over them at the undergraduate level. So what books should I use? I can find good books for university-level material, but I seem to draw a blank for the earlier stuff (basic trigonometry, algebra, co-ordinate geometry, and other high-school level stuff). In high school, we did only what was necessary for calculus. But now, I want an extremely firm foundation - after all, I'm spending the time to re-build my mathematical understanding from the ground-up, I might as well do it right - so I'm aiming for mastery, not merely competence.
Secondly, I don't know what is the right sequence for learning this. The sequence for the high-school level material doesn't matter - I'm familiar enough with the context to learn it in any order. But I'm not sure about the later material. I've put the subjects in the order I hope will be most conducive to learning linearly. I hope someone more experienced can help out here.
I would be grateful for any feedback.
Over the past year, now that I no longer have to do mathematics as a subject forced down my throat - it was a compulsory subject until the end of the second year - I've come to appreciate it much more. I've come to the realisation that throughout my mathematical education, there have been serious and structural deficiencies. It is a wish to correct those deficiencies that impels me to ask for advice here. In fact, reading this forum was one of the key motivators for my becoming interested again in mathematics. Thanks are in order.
The next few paragraphs are going to be a bit of a personal mathematical history, so those less patient can skip to the end if they so desire.
In high school, for our first board examination (a standardised test given at the end of the tenth grade) we were forced to do Euclidean geometry. We were mostly expected to memorise proofs by rote, and reproduce them as needed on the examination paper. There was a set of "standard" proofs - quite a large set, by the way, requiring a lot of memorisation - out of which questions were asked. I like the other parts quite well - basic algebra, and so on - but this was enough to make me hate this part of the subject for a long time, and it is only in the last semester that I dealt with the last of the proof-phobia that that phase induced.
Later, in the 11th grade, we were taught the standard pre-calculus sequence - trigonometry, co-ordinate geometry, the needed algebra, and so on. In the 12th came actual differentiation and integration, along with an absolutely elementary introduction to differential equations.
During this phase, the focus was on a set of standard "techniques" one could use to solve the question which would come on the test. The problem was not that we had to learn the techniques. The problem was that we were provided with absolutely no context in which to understand all these things. They were all isolated little islands of information which we had to remember. There was no overall structure into which to fit these things.
Now, when I look back over the curriculum, I realize that the reason we covered the concepts of functions, injectivity, surjectivity, bijectivity, and their relationship with the invertibility of a function before learning about inverse trigonometric functions was because they were needed to understand why ITFs work the way they do. However, when we learned them, we did not relate the two concepts. They were separate. The chapter with the introductory concept of relations and functions was treated as one thing, and the chapter on ITFs as another.
This sorry state of affairs continued throughout my education, up to and including my mathematics courses for my engineering degree. This is in large part to the "high-schoolish" attitude of many of my peers. It isn't really their fault, to be honest - they live with their parents and are not afforded much freedom, so the college considers the parents their clients instead of the students, and attempts to act as an organisation would in loco parentis. Even today, mathematics courses for engineering are taught with the same idea - memorise the techniques, plug in the numbers. It would again be fine if there were understanding and the memorisation was supposed to help it along, but that is not so. Almost all students memorise the techniques, and forget them within a few months of the examination. I do not wish to be harsh, but most would not be able to recall what they are supposed to have learned if you ask them now. Again, I cannot blame them, because this is all that they know. The problem is particularly acute in computer engineering, as most of the mathematics we learn, we do not really need.
If my mathematics education has been so uniformly dull and drab, not to mention off-putting, why am I still interested in mathematics? Well, why does an eagle fly? My innate love the the subject has kept me going through these nightmarish years. I could not believe - and still do not - that the torture I was put through was all there was to mathematics. It was only after finding this forum that I found that I was right and that there was hope.
To come to the crux of the problem:
Right now, I'm in a rather odd position. I know calculus, and have done the standard engineering mathematics sequence, but I'm not satisfied with it. Specifically, I don't think I have mastered anything that I want to.
Therefore, I'm looking to "re-do" my entire mathematical education, beginning with pre-calculus, moving on through calculus and related algebra, and finally doing my engineering mathematics the way it was meant to be done. (The material before that that I need, I have mastered.)
What is it that I want to do over in this fashion?
From high school:
Trigonometry (I was notoriously weak in this, and I want to master it)
Algebra (Binomial Theorem, Sequences and Series, Matrices, Determinants)
Co-ordinate geometry (Lines, Circles, Conics, 3D co-ordinate geometry)
After this is done, I'll move on to revising basic calculus and university algebra. Specifically:
Calculus I & II
Linear Algebra
Fundamentals of Multivariable Calculus
Complex Calculus (AKA Complex Analysis in our courses)
Differential Equations
Vector Calculus and Scalar and Vector Field Theory
Fourier and Laplace Transforms
Partial Differential Equations
Out of these, the only thing I've not studied before as part of the curriculum is the bit about PDEs, and Linear Algebra (our study for our degree was not deep).
The reason I posted this thread is because I want some advice relating to my plan.
First of all, most of what I'll be doing will be revision. However, we've only covered all these things (I refer to everything beyond Linear Algebra in the list) superficially. I want to not only study them, but have mastery over them at the undergraduate level. So what books should I use? I can find good books for university-level material, but I seem to draw a blank for the earlier stuff (basic trigonometry, algebra, co-ordinate geometry, and other high-school level stuff). In high school, we did only what was necessary for calculus. But now, I want an extremely firm foundation - after all, I'm spending the time to re-build my mathematical understanding from the ground-up, I might as well do it right - so I'm aiming for mastery, not merely competence.
Secondly, I don't know what is the right sequence for learning this. The sequence for the high-school level material doesn't matter - I'm familiar enough with the context to learn it in any order. But I'm not sure about the later material. I've put the subjects in the order I hope will be most conducive to learning linearly. I hope someone more experienced can help out here.
I would be grateful for any feedback.