# Is there a good “bridge” between high school math and the more advanced topics?

I love math and would like to know more of it. However, whenever I try to pick up a book on what I consider to be "advanced" mathematical topics, I often have a hard time understanding some of the terms and symbols right from the beginning.

I'm a computer programmer and I've used some complicated trigonometry, matrix math, linear algebra, etc in some of my programs. However I feel my understanding of some of these topics isn't very deep, and I don't always completely understand why certain processes work the way they do.

I've picked up some books and articles on calculus, linear algebra, and some other topics I feel I could use some brushing up on, but I often feel lost right out of the gate when I try to read them. They use terms and symbols that I'm not familiar with, and they don't provide any explanation. I feel (perhaps unjustifiably, but whatever) that I could do the math they're talking about if I just understood the explanation and notation, but I don't understand it.

However, when I pick up high-school level books on algebra or trigonometry, everything I read seems very simple and it bores me to death. I don't seem to find the math "vocabulary" I'm missing in them, and I don't find them challenging at all.

How can I get myself to the point where I can move on in my mathematical education when what I perceive to be "high school math" is too easy, but the more "advanced" subjects are incomprehensible? Do you know of a book I should read, a subject I could study, or some other resource I can use to prepare myself for the more advanced topics?

Some of the symbols and terms I don't fully understand include ∑ and ∫ and "coefficient". I admit it's possible that I learned this before and have forgotten it.

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(+1) for a well-thought-out (and formatted!) question. While this is definitely "soft", I'm not as "hard" as others I guess... –  The Chaz 2.0 Aug 3 '11 at 16:25
In order, your last sentence refers to sigma/summation notation, an integral, and ... well, you can Google "coefficient"! –  The Chaz 2.0 Aug 3 '11 at 16:28
Yes! It's called college. Are you currently enrolled / would it be feasible for you to take courses at a nearby college? –  Qiaochu Yuan Aug 3 '11 at 16:53
@Qiachu - No, I am not enrolled, and I don't see college in my immediate future. I've actually been happily employed as a programmer for more than 12 years. I'm just looking to improve my mathematical understanding for my own satisfaction. –  Joshua Carmody Aug 3 '11 at 17:00
@Joshua: hmm. Have you heard of Khan Academy? (khanacademy.org) –  Qiaochu Yuan Aug 3 '11 at 17:06

This is an excellent question.

Regarding your weakness in summation notation, that is one of the first things you need to address. I am a high school mathematics teacher, and I have a lengthy tutorial on summation notation, in PDF format, with exercises, that I give to my students. I have uploaded it to my ipernity account. It is in three pieces, because when I email it, using gmail, I am limited on file attachment size, and so have to email three times, once for each piece.

In order to download the three documents constituting the tutorial from my ipernity account, you would have to be a “pro” member of ipernity yourself. (Becoming an “ordinary” member of ipernity is free, which allows you to blog and send messages to other users, but you can’t upload/download documents.)

If you would like the material on summation notation, but don’t want to go the ipernity route, you can email me at my address given in my profile, and I will be glad to send it to you as email attachements.

The existence or non-existence of the “bridge” you ask about is a debatable point. One point of view, made by one of the other answerers, is that there is no such bridge, that you simply must keep building on what comes before. I take the view that there IS such a bridge, but that it is not an external object, but an internal process. You must become so adept at algebra, and closely related topics such as summation notation, that it is truly second nature for you. Reaching this point in skill constitutes the bridge. You can then move relatively easily into infinitary processes, of which calculus is the customary portal.

An analogy with building a campfire might be helpful. Correctly building a campfire involves three steps (after making sure you’re not building it under a tree!), namely, gather tinder, and light it, gather kindling, and add it to the fire, and then, only when a good blaze is going, add logs. The logs will then easily catch fire, and provide a nice long-lasting fire.

Using this analogy, it is easy to see the two kinds of mistakes that can occur:

1. Being happy with a kindling fire, that is, never adding the logs. The problem is, the fire will not last very long. (This is typically what happens in high school.)

2. Omitting the tinder/kindling steps and just dropping the logs onto the fireplace, and trying to light them with a match. Unless you have a lot of patience and stamina, you will simply give up and have no fire at all. (This is typically what happens in college –it’s the sink-or-swim approach.)

So, navigating that transition between the high school approach and the college approach is pretty much up to you, and will inevitably involve sitting quietly in a room. As Blaise Pascal said, “All the trouble in the world is due to the inability of a man to sit quietly in a room.”

Regarding specific books, besides what others have mentioned to you, I would like to recommend “Men of Mathematics” by Eric Temple Bell. Even though it’s been criticizd as not being completely accurate historically, it’s a great read, indeed, I daresay, pretty much an item of “required reading” for any beginner seriously interested in mathematics.

Also, the book “What is Mathematics?”, by Courant and Robbins, is something of a classic. I would suggest that it is pretty much required of any beginner seriously interested in mathematics to have held this book in their hands for at least thirty minutes, leafing through it:)

Also, addressing your concern about “vocabulary”, do you have a copy of a mathematics dictionary? The “Penguin Dictionary of Mathematics”, edited by David Nelson, is the one I recommend to my high school students.

Regarding study technique, there is some excellent advice here on MSE, in the form of an answer to a question. The question was “What are examples of mathematicians who dont [sic] take many notes?”, and the answer that I am referring to, which I upvoted, is that given by Paul Garrett. Here is the link:

What are examples of mathematicians who dont take many notes?

Also, here’s the link to a website you might want to consider:

http://www.mathreference.com/main.html

It “is essentially a self-paced tutorial/archive, written in English/html, that takes the reader through modern mathematics using modern techniques.”

You might want to check out the answers to the question here on MSE “How to effectively study math?” (which is where I pulled the above “self-paced” link from):

How to effectively study math?

How to effectively and efficiently learn mathematics

As a parting thought, remember the story about the two mice who fell into a pitcher of cream. One mouse saw that the situation was hopeless, and so gave up swimming, and drowned. The other mouse could not see any way out either, but did not want to drown, and so kept on swimming furiously. And as it swam, its feet churned the cream, and gradually the cream turned into butter, creating a solid enough surface for the mouse to climb up on and escape from the pitcher.

So, the best of luck in your studies. Press on.

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Thank you very much, @Mike Jones! I was interested in reading the summation tutorials you mentioned, But I wasn't sure I wanted to create a pro ipernity account. Unfortunately, you don't seem to have an e-mail address in your profile...? I was going to post my e-mail address here and ask you to e-mail me, but after reading many of the links provided by others, I think I understand summation notation now, so I guess it's a moot point. :-) Thanks for your help. –  Joshua Carmody Aug 4 '11 at 14:07
Thanks again. There were many helpful answers in this thread, but I found your description of the "bridge" to be especially helpful. On reflection, I can see that there's a lot about math that I "know", but that isn't second nature to me. I've come to realize that's one of my main issues with reading about advanced math subjects. –  Joshua Carmody Aug 18 '11 at 17:31

I worked as a programmer while I was in highschool and starting university. Initially I was a microbiology student and it was unconnected to my side programming jobs.

But shortly after I got interested in mathematics and found my programming background was an excellent "springboard". In that writing a mathematical proof has a very similar mental dynamic to debugging a flawed (but executable) computer program.

There are many ways to get into math. Most of the responses so far have been a relatively linear approach, in that they're talking about the next technical steps you need to see further afield in mathematics. Another approach is to jump over all that and pick an objective. Figure out some aspect of "modern" mathematics you want to learn, and do whatever it takes to get there. When I was an undergraduate, the things along those lines that really turned me on were Whitney's work on manifolds, the Massey Immersion Conjecture, and trying to understand General Relativity. I really liked George Francis's "A Topological Picturebook", and shortly after Rolfsen's "Knots and Links".

This doesn't have to be your path but you can perhaps find something analogous that's more in line with your own tastes. Although the "core" of mathematics is technical and rigorous, the one almost entirely subjective aspect of mathematics is "what do you want to study?" If you can partially answer this question it'll make your next step much easier.

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If the topics that you are looking to learn more about include linear algebra and (possibly) calculus, it might help to take a look at the Linear Algebra and Single/Multivariable Calculus video lectures on Open Courseware (http://ocw.mit.edu/courses/audio-video-courses/#mathematics).

Textbooks aren't always the most easy things to read on your own, because they're usually a supplement to classroom lectures (at least when you are learning linear algebra and calculus).

In any case, those lectures are actual classes that students usually take right after high-school mathematics, so the professors will likely be explaining things with the mindset that they are speaking to people with your type of background/understanding of math.

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Open Courseware definitely is a good option for picking up knowledge. I happen to know a programmer who went from being a pre Ph.D. in Physics to using OCW to self-teach himself low-level programming mainly for mainframes and algorithms. –  theonlylos Aug 3 '11 at 19:29

Given that you come from a computer programming background, the number of basic mathematical symbols is rather low and it's not really necessary to learn them from a textbook.

Aside from basic arithmetic, you should know

All other notations should be properly introduced and defined in the appropriate subject area. For example "∫" means integral, which is found in calculus.

The great thing about math is that once you learn the fundamentals, everything is built upon that foundation. It's not about "high school math" vs "more advanced" math. The more "advanced" topics should follow directly and intuitively from the "less advanced" topics. So instead of viewing it as a bridge, see it as more of a prerequisite.

To learn area W of mathematics, you should have a knowledge of X, Y and Z first. It's not like you know the basic A,B,C and to know advanced X,Y,Z you need to have a single all-important bridge M. There is no such general M that will do that.

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### Learn to pronounce symbols

Learning to pronounce mathematical symbols can be a challenge if you are completely self-taught.

### Understand dependencies between domains of mathematics

It can be frustrating when you want to dive into an interesting mathematics book, but you are finding the concepts and notation impenetrable. My advice is to take anything in the preface of a book about prerequisites seriously. For example, mathematics books that I read often state that a general understanding of calculus and linear algebra is required.

• If you are trying to get a sense of dependencies between topics, you can (a) check out university mathematics curricula; (b) read the preface of mathematics books; (c) ask the experts on this site.
• In very general terms, you might want to have a look at some 1st year university calculus and linear algebra material. If that seems too advanced, then check out some pre-calculus or final-year high school material.
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You may start by reviewing high school math from a more advanced viewpoint, such as in Lang's Basic Mathematics. See also Good book for high school algebra.

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There have been plenty of excellent books mentioned already, so I'll give you some more general advice. I've had a lot of experience teaching myself mathematics, from the high school to graduate level, so hopefully I can say something helpful to you.

1) Don't get bogged down in notation. If you using a good text, what they are doing should be explained in multiple ways. Try to read through a section or proof and understand what they are trying to do, even if you don't remember all the notation yet. Once you have a sense of the ideas they are trying to get across, the notation will be much easier to pick up.

2) The situation you have described, where everything you have already learned seems ridiculously trivial, and everything you haven't impossibly difficult, is not uncommon. Don't be discouraged by this, math takes time and effort to learn. If you are motivated, you'll be able to work through anything.

3) Don't rely exclusively on texts. Over and over, I've been impressed by how much easier it is to learn something from a (good) teacher than from a book, however well written it is. Take classes, go to office hours, and think about things on your own time, and you'll make up for your weaknesses in no time. A reasonable substitute is to watch lectures on the internet. Khan academy and MIT open courseware are great resources that have plenty of helpful lectures at your level.

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+1 for #2 especially. Glad to hear that! –  Joshua Carmody Aug 4 '11 at 14:39

I liked An Introduction to Mathematical Reasoning by Peter J Eccles. Moreover it has solutions to all problems.

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Definitely using OCW to begin studying something like Calculus will certainly help you out a lot. Typically if you're able to master basic calculus, almost all of higher mathematics will be much more accessible to you. If you've gotten most of high school math out of the way and are confident in your algebra abilities, calculus really won't be that difficult as only the first line of the vast majority of basic calculus problems is actual calculus and the remainder is algebraic manipulation.

Barring that, or working on it concurrently, you might consider reading Robert Ash's excellent text "A Primer of Abstract Mathematics" which will give you a lot of the introductory material on basic notations, definitions, logic, proof structure, and flow that the vast majority of advanced mathematics textbooks leave out and generally take for granted. Don't let the jargon stop you from succeeding as sometimes the difference between the words "into" and "onto" can make a huge difference.

If OCW doesn't seem as easy, take a look at what your local colleges and community colleges have available. In the past 5 years I've been able to take evening courses in areas as diverse as complex analysis, group theory, fields, galois theory, differential topology, manifolds, and algebraic topology. When I started, I wasn't that much different than you...

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If you want to get an expository account of the breadth of mathematics (with in many places lots of depth, too) and places to go to pursue individual topics more, you can not do better than the book edited by Timothy Gowers: The Princeton Companion to Mathematics, Princeton U. Press, 2008. (Many of the items are written by Gowers himself.)

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The Companion, while remarkable, is, I think, at a slightly higher level than appropriate for the OP, given that the OP is not familiar with sum notation. –  Qiaochu Yuan Aug 3 '11 at 16:54

Sometimes you just have to jump in straight into the deep end. However that being said because you are not familiar with some of the notation used in mathematics, I could recommend Mathematics - The Core Course for A-Level.

I learned a lot from this book, and examples are easy to follow. For example, if you don't understand summation notation you get to practice it in the book. For example, "write $1 + 2 + \ldots n$ in summation notation, and conversely write out what $\sum_{i=1}^n \frac{i(i+1)}{2}$ means."

After a while when things become more familiar you will feel more confident!

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(Too long to comment on those responded to my suggestion about Gower's book.)

It seems to me that the person who posed this question may belong to a large group of people who don't have a good sense of what mathematics is about nor the broad range of its applicability (making possible cell phones, HDTV, cheaper delivery of municipal services, etc.). This is in part because a lot of good books about mathematics and courses where it is taught emphasize a very symbol driven hierarchical approach - you can't possibly understand what algebraic geometry, say, is about until you have studied this long list of prior subjects. There is in my opinion a big shortage of "big picture" looks at Mathematics. While there are many books that have selections of charming topics in mathematics most of these still have a limited view. Gower's Princeton Companion is noteworthy for giving a comprehensive, surprisingly symbol free and somewhat unhierarchical account of a large part of the mathematical landscape. I think this a very rewarding book for people at all levels of mathematical sophistication.

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OP here. I appreciate your suggestion of The Princeton Companion to Mathematics, and will look into it to see if it can help me. I'm surprised though that you'd think a seasoned computer programmer who has used some advanced math in the systems he creates wouldn't understand the role of mathematics in "cell phones, HDTV, cheaper delivery of municipal services, etc". After all many of the things I create serve purposes such as these. (Just to be clear - I'm not offended by your statement. I just wonder if you fully understood what I'm looking for here). –  Joshua Carmody Aug 4 '11 at 17:47
@Joshua: I am not sure what you are looking for actually. So perhaps you know about Hamming codes (error correction ideas needed for cell phones), Hamming codes (data compression codes), graph coloring problems for frequency assignments for cell phones) and the mathematics behind GPS systems. However, many programmers and even people with doctorate degrees in mathematics don't. (What I know of the mathematics of these subjects I had to learn on my own. They were not part of mathematics courses I took.) Try reading parts of the Princeton Companion and report back! –  Joseph Malkevitch Aug 4 '11 at 18:07
Ah, is that what you meant? I didn't mean to imply that I knew the specific math involved in all of those things. Just that I was aware of the importance of math in our society and what it can do. There are plenty of high school students who study math and groan "why do I have to learn this? I'll never use it in real life!". I am not one of those people. :-) What I am looking for is some kind of "stepping stone" primer or review to teach me enough of what I didn't learn in high school that I feel competent to study things like statistics, calculus, and linear algebra. –  Joshua Carmody Aug 4 '11 at 18:16
Incidentally, when I went to look at the Princeton Companion, I saw Amazon had another book by Timothy Gowers called "A Very Short Introduction to Mathematics". It looked interesting and the electronic version was only \$6, so I decided to grab that before ordering the Companion when I have more money. I'm reading 'a short introduction' on my cell phone right now. –  Joshua Carmody Aug 4 '11 at 18:18
Gower's very short introduction is nice, too. By the way, there is a typo above. I meant Huffman codes for data compression. Given your goals, though some of the others who have commented here might think this is not a good choice, I would consider (given your programming background) reading a book with charm about discrete mathematics. (For example: L. Lovasz, J. Pelikan, and K. Vesztergombi, Discrete Mathematics: Elementary and Beyond.) This will help with some traditional notation and ideas about proving of theorems, and then you can take on a traditional book in stat., calculus, etc. –  Joseph Malkevitch Aug 4 '11 at 18:36

I recommend looking at the following:

Edna Kramer's "The Nature and Growth of Modern Mathematics"

http://www.amazon.com/dp/0691023727

MAA's New Mathematical Library series

http://tinyurl.com/3l6gxfo

"Gelfand School Outreach Program" books and English translations of the Russian "Popular Lectures in Mathematics Series"

http://mathforum.org/kb/message.jspa?messageID=6640965

http://www.eldar.org/~problemi/singmast/russian.html

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