How to use math textbooks

Question

I'm a higher schooler who was recently gifted a book by my teacher (Schaum's outline of advanced calculus) which is really awesome and I've started working my way through it.

I have run into a problem though. It is so incredibly different from high school textbooks! In high school, most textbooks (I've used) have like max an 80-20 exercise-theory ratio (with that I mean for every 4 pages full of exercises you have 1 page with theory. This book however (and I believe most university math textbooks have) has it extremely differently: first you get a couple of convoluted pages full with theorems/methods/rules (i.e. an information overload) and then you get dozens of exercises. So, I have a couple of questions:

• Did anyone notice the difference too, or am I the only one?

• What would you advise me to use as a supplement to this book?

• Any general tips for a high schooler was has to go through this change alone? (p.s.- I'm extremely motivated to do this! Just skimming the book makes me so excited it's unbelievable. I just need some directions, and I hope I could get them here)

Answer 1

There is indeed a huge difference between high school texts and those used later on. In fact a university text book assumes that the reader is familiar with some prerequisites and is able to do a lot of work for himself, so he/she doesn't need to be guided (and exercised) as much as a high schooler. Furthermore, the extent of topics to be covered is much bigger, hence it is much more condensed.

A general tip would be to go through the proofs and try to fill in all the intermediate steps, thus gaining a lot more insight in the subject.

Hope this was a little bit useful to you.

Answer 2

Yes, I noticed the difference when I started studying Calculus. As a supplement of it, you can use any other Calculus book available. There is a lot of them which you can find for free on the internet and you can also find answers/ask questions about your doubts here.

This theory/exercises ratio may somewhat change when studying advanced topics (and sometimes you'll wish to be back to the "hundreds of repetitive exercises"-routine. My first advice is that you don't let yourself down if you can't understand something the first or second time you read it. It happens to everyone in math.

My second advice: let yourself fall in love with it, if it makes you so motivated. Don't hurry - math is nor a race but rather a marathon.

Answer 3

Undergrad math is less routine and more theoretical compared to high school math. One will need to understand why things work rather than just be able to use formulas to compute. In your first year as an undergrad, you will probably be learning many things you did in high school at a deeper level, such as calculus and vectors. After that, things become even more theoretical especially if you do pure math. What is important is to get used to reading and writing proofs.

There is no need to go too fast now. You can just focus on your school work and read this book in your spare time. Schaum's Outline Series are very good for making the transition to university. They provide some theory but not too much to overwhelm the beginning math student. Just read through it first before looking for others. As to how to use this book, you try to read the text with understanding. Then you do some of the exercises to see if you can apply what you have learnt to solve the problems. Schaum's makes this even easier by providing worked solutions to some problems!

Answer 4

Take a look at the excellent lecture notes by William Chen, probably starting with "First Year Calculus" and looking at the others as needed. The Trillia Group has outstanding texts too.

To learn how to approach problems systematically, take a look at Pólya's classic "How to solve it".

Don't try to read a math book as a novel, a rule of thumb is that it will take you an hour or so to work through each page.

If you need all sorts of advanced exercises to try your hand on, this is the right place ;-) If you need help, ditto. If you formulate your question clearly, and explain what you have tried, where you got stuck or where you have doubts, very often someone more experienced will want to lend a hand. And soon you will be able to answer the questions that have stumped others. Just don't expect to understand everything here, the subject area here is just too widespread.

Answer 5

Yes, indeed, there is a big difference between "high school textbooks, and those that go beyond basic computationally oriented calculus. You might want to see if you can get a pass (perhaps with the help of your teacher) to use a university library, find their section of math books, and just skim the structure and layouts of the books. You could also visit a nearby university bookstore and page through textbooks for intermediate/advanced level undergraduate textbooks.

As you are doing this "on your own," you might want to check out the Art of Problem Solving (AoPS) website to explore resources geared to high-caliber high-school students who, like you, are doing supplementary self-study in more challenging mathematics. There are resources available, helpful suggestions, and a community forum for students to interact. Give yourself plenty of time to explore its vast offerings.

The content of your book (just having previewed Schaum's Outline) seems to touch on topics you'll find in an introductory analysis college course (or an undergraduate course in real analysis). So that would be a good "key term" to use in a search, if you're interested in exploring supplementary text books. Rudin's Principles of Mathematical Analysis (AKA "Baby Rudin") is a classic, but it is very terse and short on examples, and perhaps not a good "bridge" between where you are and what you'd like to learn next.

One reference suggestion that expands on the topics covered in Schaum's Outline of Advanced Calculus is Serge Lang's Undergraduate Analysis.. Of course, it might be good to ask the teacher who gifted you your book for his/her recommendation as to a good supplementary textbook.

One suggestion: This might be a good time to dig into getting familiar with proofs and proof strategies, as you will encounter them in both the expository reading, and in exercises. In particular, my first most recommendation is that you read and work through How to Prove It: A Structured Approach. by Velleman. It will be worth the read, and is preparation for just about any mathematics beyond straight calculus and such, and more theoretically oriented math. Indeed, many undergraduate programs in math offer a "bridge" course in "proof-methods" (e.g. courses with names like "Introduction to higher-level math", "Proofs and Proof-Methods", etc.) to help "bridge" the gap between more computationally-oriented math, and higher-level math. And they do so precisely for the reasons you are encountering now: it requires a huge "cognitive" shift to move from into more advanced math.

Stick with it, though! Move at a pace which challenges you, but does not overwhelm. It is well-worth the journey on which you are embarking! Your enthusiasm and motivation will be key assets to a successful journey.

I wish you well!

Answer 6

While I am not familiar with the book by Schaum, I think you are on to something. There is definitely a difference between textbooks aimed at high school students and textbooks aimed at college students. Furthermore, this isn't just true of mathematics, but of other subjects as well. I would go as far as to say the whole learning process is generally a little different at the college level than at the high school level.

Part of this difference is made up by a change in how the information that you are expected to know at the end of a course is presented.

In high school new information is presented to the student verbally, through the medium of one's teacher. Ideally the teacher also serves as a motivational aid, but the textbook is relegated to a marginal role as a supplementary aid and reference. While textbook exercises are very important, they usually do not present new concepts, but merely help a student internalise already familiar ones. Perhaps high school textbooks are generally light on theory because it is assumed that the teacher will present new concepts during class discussion, and that many students will not bother to read the text.

In college new information is transmitted mostly via the written form, through the medium of one's textbooks (and, especially later on, papers and journals). The professor thus becomes relegated to the marginal role of a supplementary aid and motivator. This becomes increasingly the case as one progresses to the higher levels of college education. In fact, some, like the eminent philosopher David Hume, completely dismissed the usefulness of professors as knowledge repositories: "There is nothing to be learned from a Professor, which is not to be met with in Books." I would not go as far myself, but very often the professor's most important role is to put the right books in one's hand, and to motivate one to read them, as your mathematics teacher seems to have done. Plus, someone has to write the books.

What is certain is that you will be doing a lot more reading at the college level. This is true for the subject of mathematics and for other subjects as well.

A few years ago I came across a book that can help with this transition in acquiring information. It is called How to Read A Book, by Mortimer J. Adler and Charles Van Doren. Adler was a writer and philosopher who was the main force behind Encylopedia Britannica's 60-volume Great Books of the Western World series, many of which are mathematics textbooks, or at least mathematically themed (e.g. Euclid's Elements, Apollonius of Perga's On Conic Sections, Nicomachus' Introduction to Arithmetic, Descartes' Geometry, Newton's Principia Mathematica, Whitehead's Introduction to Mathematics), which suggests that he wasn't unfamiliar with the subject. The book has been in print since 1960 and the latest edition is from 1972, but its subject matter is timeless. Books have been around for a long time, and will no doubt continue to exist in some shape or form in the foreseeable future.

At first it might seem a little silly to read a book about reading, but the subject is approached quite formally, and the reader is given a series of rules and guidelines that are to be observed when reading analytically, or reading certain types of subject matter, such as mathematics. Many of these rules were completely new and extremely helpful to me, despite considering myself an avid reader prior to tackling this book. If you read this book carefully, and one can do this especially well if one recursively applies the concepts presented within the book to the book itself, you will emerge a much more analytical reader. This will greatly help you handle both mathematical text books at the college level, as well as books dealing with other subjects.

How to Read a Book deals with all kinds of reading matter, but is especially useful in handling expository books, or, to use your terms, books that contain large amounts of "theory". There is a specific section on how to read mathematical books. The critical reading section, however, which makes up the majority of the book, applies to all expository books, of which mathematical books are merely a subset.

I wish you great success on your quest of mastering Schaum's book, as well as any others that may come after.

Answer 7

One of my suggestions that I don't think has been mentioned before is to try and read through several theorems, take a break, and then come back and see if you can reproduce the proofs yourself. Try to imagine you're explaining a theorem to a relatively smart middle school student. Attempt to fill in any gaps in a textbook's proof (sometimes they can be awfully short) or elucidate concepts by introducing drawings, figures, and other props.

Doing textbook problems is also a good idea, but if there's 100 of them per chapter, it may be to your advantage to review the proofs for a theorem instead of grinding your way through repetitive questions. Long after a course ends, you may not encounter "homework-style" questions again, but you may have to use the theorems. As an example, I proved the Central Limit Theorem in my probability class, which really helped my intuitive understanding of how it works, and I am using the CLT in my current classes.

And since you're posting here, if you ever run into difficulty in understanding a proof, use this site as a reference.

Answer 8

I would look into courses on line like MIT's OCW or courseara. That was you can get a lecture to figure out what is important. I read lots of math textbooks too but it helps me to have a guide to sort out what in a textbook is worth your focus. I have the advanced calculus book you mentioned and I found it helpful through my graduate level courses, so it's not the friendliest book to start off your higher studies with. Good luck!