Relearning Math: Comparing Resources for the Best Learning Experience

[Hsopitalist]

One last thing. I gave myself a year to do algebra, precalculus and trig. Starting with those two EdX courses made a HUGE difference there. Now I'm on to calculus I'm doing at least ten problems per day and the steady exposure really helps.

And I use a trig text by Lial. I've found it quite useful in explaining the really basic stuff.

Good luck and get to work! Keep us posted on your progress

 

[econreader]

I noticed the page count of Gelfand's books as well. I figured they were so short because they weren't filled with a bunch of pictures, and popups and whatever else algebra books printed in the 2010s contain.

Well, trigonometry is the only one which is a bit like a textbook. It is still kind of short (but I guess the length is normal for a school textbook). Of course, Gelfand being one of the top mathematicians, one can still learn a lot from anything he writes. Same with Polya. I just wouldn't expect these books to replace read & drill from more conventional sources. I don't know much about precalculus books since I learned it while in secondary school, but you may want to check one of the books by Allendoerfer and Oakley which is often recommended here. I read their book as a refresher some time ago and I can confirm that it is awesome. Also, it's very cheap for a second hand copy.

Yes, I read the same book by Eccles as what you ordered. I was skeptical at first but it is actually quite good unlike some "introduction to proofs" books which are often superficial. I supplemented it with Norman Biggs' Discrete mathematics for a couple of topics, but since you have one by Rosen, you should be able to use it instead.

The books by Euclid are mind-blowing but will require a very serious commitment - so I personally put them aside till I have free time to study them. Same with Euler - it's fun to read a genius (and actually quite easy to follow this one), but he goes into topics which are not so immediately pressing to learn, so I'd keep it aside and turn to it for fun reading and use the time to focus on the shortest reading path to calculus.

 

[MidgetDwarf]

I just checked out the courses from ASU. It looks like they have precalculus and college algebra/problem solving ones open for enrolment right now.

I think I just got so many resources now that I might wait until I get through the more elementary stuff and use the precalculus course they offer to supplement Santos' precalculus book that I downloaded.

As for what your brother said, I already feel it. This year that just passed, I had 2 programming courses. Although I did very well, it took me a lot more effort to get through some of the more challenging assignments than my classmates who just graduated high school and took calculus and advanced functions in grade 12. That's part of what motivated me to dig deeper into math and make sure I fill in the gaps by 3rd year.
I actually placed an order for a few other supplementary texts that might give me more problems to work through. I chose not to go with the AoPS content as no one really brought them up, but compared to my first list, I've also added the following:

Euler - Element's of Algebra
Euclid - 13 books Vol I-III, translated by Thomas Heath (the Dover editions)
Jacobs - Elementary Algebra
Jacobs - Geometry
Cohen - Precalculus with Unit Circle Trigonometry

My approach has kind of changed as well due to how much the list of material increased since making this post. I plan on using Lang and Gelfand as a main source of what I should be learning, and dig for those topics in the supplemental books for extra problems. Of course if the initial explanations by Lang and Gelfand get confusing, it'll be nice to have multiple perspectives as well.

I noticed the page count of Gelfand's books as well. I figured they were so short because they weren't filled with a bunch of pictures, and popups and whatever else algebra books printed in the 2010s contain. Again, if they aren't detailed enough, I plan to supplement them with Jacob's Elementary Algebra. I will say that I'm glad you claim they are aimed at high school students though. As of right now I consider my algebra skills to be at a grade 9 level at the most. To paint a clearer picture, I had to have a friend teach me FOIL while we were learning mathematical induction last semester.

I regret not looking deeper into Polya's book before purchasing it. Until you mentioned it, I didn't realize it was just a bunch of definitions. It looks like there's a page or two of problems at the end but they look rather lacklustre. Luckily it was one of the cheapest books in the list, so I hope I'll get at least some use out of it. Between the book by Eccles, the free discrete math book I downloaded by Oscar Levin (thanks MidgetDwarf), and the copy of Rosen's discrete math book I currently own, Polya will undoubtably be on the back burner for quite some time.

The book by Eccles that you mention helped you learn abstract mathematics, do you mean the one that I ordered?

Lastly, I definitely will be keeping the book by Moise. First book or not, I found a pdf online and did a quick skim. It looks like it'll be an incredible resource down the road.

Glad you liked Moise. I find his exposition very clear, informative, and sometimes funny. However, his Elementary Geometry From An Advance Standpoint, can be sometimes dry. Its a good book tho, but many people confuse the audience which the book is aimed for, ie., the elementary part. This kind of reminds me of Komogorov: Introduction To Real Analysis. This book is not really meant for a first course in analysis, however, it has some weird chapters that are aimed for intro courses., ie., the intro to sets.

 

[MidgetDwarf]

I would hold off on buying any more sources. You have enough to last at least 2 years. The Gelfand book on trig is very neat, although it lacks problems/topics. But everything contained within it is lucid. There is a neat proof of the Pythagorean theorem, that is not so trivial. It uses some advanced Theorems about geometry. However, he dumbs it down that even a kid can understand. Although, the proof given in Gelfand is more intuitive than mathematical rigor. You have a pre-calculus book. Combine that with Gelfand, and you are good at intro trig. One can dive deeper, but I think its good enough.

This same proof was required in my Modern Geometry class, of course, the more rigorous proof. But the teacher explained it so confusing (we used his lecture notes) during a previous lecture. Needless to say, I was the only person in my class to understand and write a correct proof, since I seen the intuitive explanation in Gelfand 5 years prior.

 

[econreader]

This kind of reminds me of Komogorov: Introduction To Real Analysis. This book is not really meant for a first course in analysis, however, it has some weird chapters that are aimed for intro courses., ie., the intro to sets.

I haven't read it myself but I know that the Russian original is based on material taught in the third part of an analysis course at the top national university. No idea why the English edition was titled "Introduction".

 

[norxport]

One last thing. I gave myself a year to do algebra, precalculus and trig. Starting with those two EdX courses made a HUGE difference there. Now I'm on to calculus I'm doing at least ten problems per day and the steady exposure really helps.

And I use a trig text by Lial. I've found it quite useful in explaining the really basic stuff.

Good luck and get to work! Keep us posted on your progress

That's basically what I'm doing right now. Going to try and get the bulk of the foundation laid out over the summer, and then try to keep it fresh in my brain during the school year.

I'd love to keep you guys posted on my progress. It'll be a good way to motivate myself to stay committed!

Well, trigonometry is the only one which is a bit like a textbook. It is still kind of short (but I guess the length is normal for a school textbook). Of course, Gelfand being one of the top mathematicians, one can still learn a lot from anything he writes. Same with Polya. I just wouldn't expect these books to replace read & drill from more conventional sources. I don't know much about precalculus books since I learned it while in secondary school, but you may want to check one of the books by Allendoerfer and Oakley which is often recommended here. I read their book as a refresher some time ago and I can confirm that it is awesome. Also, it's very cheap for a second hand copy.

Yes, I read the same book by Eccles as what you ordered. I was skeptical at first but it is actually quite good unlike some "introduction to proofs" books which are often superficial. I supplemented it with Norman Biggs' Discrete mathematics for a couple of topics, but since you have one by Rosen, you should be able to use it instead.

The books by Euclid are mind-blowing but will require a very serious commitment - so I personally put them aside till I have free time to study them. Same with Euler - it's fun to read a genius (and actually quite easy to follow this one), but he goes into topics which are not so immediately pressing to learn, so I'd keep it aside and turn to it for fun reading and use the time to focus on the shortest reading path to calculus.

I've seen the Allendoerfer and Oakley books go for very cheap second hand. But as MidgetDwarf said, I'm going to try to put a pause on any more purchases. If I can't get to a point where I feel like I'm ready for calculus with my current resources, I may have other things to worry about, haha.

I imagine the book by Eccles and Hammack's free proof book will be more than enough to get a better grasp on proofs. I should've mentioned that in first semester we did a couple weeks on proof by contrapositive, contradiction and mathematical induction. The thing is, between the rush of being back in school for the first time in a number of years, and not having had done math in nearly a decade, a lot of it was learned at a very shallow level. I plan to dive much deeper this time around.

The books by Euclid and Euler were part of the additional materials I purchased after I initially made this post. I don't expect to make them part of the core part of this curriculum. But I do think I'll get a lot out of them after finishing some of the material I mentioned in the first post. I plan on waiting to go through Euclid's books until I get a copy of Hartshorne to accompany it with. This probably won't be until next summer.

I would hold off on buying any more sources. You have enough to last at least 2 years. The Gelfand book on trig is very neat, although it lacks problems/topics. But everything contained within it is lucid. There is a neat proof of the Pythagorean theorem, that is not so trivial. It uses some advanced Theorems about geometry. However, he dumbs it down that even a kid can understand. Although, the proof given in Gelfand is more intuitive than mathematical rigor. You have a pre-calculus book. Combine that with Gelfand, and you are good at intro trig. One can dive deeper, but I think its good enough.

This same proof was required in my Modern Geometry class, of course, the more rigorous proof. But the teacher explained it so confusing (we used his lecture notes) during a previous lecture. Needless to say, I was the only person in my class to understand and write a correct proof, since I seen the intuitive explanation in Gelfand 5 years prior.

Couldn't agree more. I got a bit carried away with this new found excitement.

You got me very excited for the trig book by Gelfand. I am curious about one thing though. Since there aren't any problems to practice with in it, do you think Cohen's Precalculus with Unit Circle Trigonometry will be enough practice to hammer home those concepts? My ignorance might show a bit here as I'm unaware if the trigonometry that Gelfand covers and the things covered in Cohen's book are different from each other.

Lastly, that's the thing with teaching. You can know the material inside out, forwards backwards. But that doesn't make you a great teacher. I imagine there's a reason why out of the endless list of publications out there, I see the same few authors get recommended over and over.

 

[MidgetDwarf]

Yes. Teaching is an art. I have a very intelligent teacher (numerous publications). However, his lecture just goes over my head. He does try to explain it so that students understand.

But sometimes, while taking a bath or walking. Things click. I can replay some of his lectures by memory years later. So I don’t think he’s that bad. How do you like the books so far?

 

[norxport]

Yes. Teaching is an art. I have a very intelligent teacher (numerous publications). However, his lecture just goes over my head. He does try to explain it so that students understand.

But sometimes, while taking a bath or walking. Things click. I can replay some of his lectures by memory years later. So I don’t think he’s that bad.How do you like the books so far?

Haha, I certainly didn't mean your teacher was bad. Sometimes being introduced to a new topic by a teacher who's extremely intelligent can be confusing. It's tough for them to put themselves in the students' shoes.

The books are great! So far I've only received the Gelfand books, Lang's Basic Mathematics, Euclid's books and Velleman's How to Prove It.

I'm working through Lang right now and it's exactly what I needed. With the rest of the shipments that are yet to come, I've got a very busy summer ahead of me!

 

[jimrasmussen50441]

I realize I'm late to this party but I would say my reaction to getting Velleman was ... meh. Daniel Solow's "How To Read And Do Proofs" is immensely superior in my opinion.