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I'm a rising sophomore in high school. So far, I've taken Algebra One, Two, and Geometry in school. I want to learn higher math such as precalculus/trigonometry, calculus, linear algebra, and more, so I can go into topics such as cryptography, advanced computer science, and possibly take the AMC and other olympiad tests (I'm not too interested in that).

The only problem, though, is that my abilities in problem solving and other stuff in math aren't that good. I do pretty well in my classes (high As) but that doesn't mean anything. The U.S. system doesn't seem too good in actually teaching math.

For example, I can do whatever is on my homework or tests. But, if I'm given a more difficult problem than usual concerning a topic I learned (say logarithms or something), I can't solve it.

I feel like this is going to be a hindrance to me learning higher math, doing well in more difficult subjects like calculus and linear algebra, doing well on olympiad tests, and going into math-heavy fields like computer science and cryptography.

So, how can I change all of this and improve my skills? Are there any books that teach problem-solving, mathematical thinking, and higher math (or something like precalculus)? Again, I want to better these skills so I can do well not only in math, but other fields.

Any help is really appreciated.

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    $\begingroup$ Several things will help you learn higher math. You are enthusiastic about the subject. You invest time to really understand the material (forget about 'plugging into formulas). You have confidence in you abilities (don't say "I can't solve it" - try "By investing more time I will learn valuable things"). $\endgroup$ – CopyPasteIt Jun 9 '17 at 0:05
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    $\begingroup$ Probably someone says that this is wishful-thinking but in my experience if you want something you generally found it (at least to some degree where you find comfortable). In this case in particular, learning mathematics, you already showed that you are good in many aspects of it. I read somewhere that some very good mathematicians were bad at calculus when young, or at least not very well. Indeed, at today, I do many mistakes in simple calculus (horrible mistakes) and I dont think I cant learn what I want to learn. $\endgroup$ – Masacroso Jun 9 '17 at 0:26
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    $\begingroup$ Huh. I never thought of it that way. I've almost always heard that great mathematicians displayed talent from a young age, like Gauss or Terrence Tao and his gold medal in the IMO at 13. I just wanted to learn higher math and math skills so I can apply those in other fields (like computer science). $\endgroup$ – Ansh.23 Jun 9 '17 at 0:50
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    $\begingroup$ Ansh.23 I would be careful with the use of "talent." Tao didn't win the gold medal on his own, he had world class training and people behind him giving motivation. I believe talent played some role, but Tao's case does more to prove the value of early training and (this is key) being taught by people who actually know what it is like to do math than prove the innate talent hypothesis. You should look at The Road to Excellence by K. Anders Ericcson(a collection of Psychology papers on deliberate practice and elite performance) if you're interested in learning from an academic perspective. $\endgroup$ – Retired account Jun 9 '17 at 3:01
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    $\begingroup$ @Ansh.23 If it seems good it probably is. This stuff hasn't really changed recently. But if you want something more interactive you could also go with Keith Devlin's intro to Math thinking course on Coursera(I actually learned proofs from this, it's a very good course). The most important thing, regardless of which source you use, is to (obviously) solve problems, but also to discuss them with others. It is very easy in writing proofs as a beginner to think your proof is good when in reality it has large logical gaps or is extremely over complicated. $\endgroup$ – Retired account Jun 9 '17 at 5:03

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I'm going to take a different approach. Yes, you should buy the Polya books, I also recommend looking at learning how to learn on Edx for an interesting take on learning techniques. But do something else as well: watch the Khan academy videos on trig, then the first couple of MIT OCW calc videos. Then take Robert Ghrists Calculus course on Coursera and take the A.P. Calculus exam (I actually did this in one year, and it wasn't very hard-not because I'm so smart, I know from experience that I am at best mediocre in a real math class. It's just A.P. Calculus doesn't take a ton of real math skill). Then, for the final step, see if you can take classes at a local Univ. in real math. Their is no way to learn math like learning from actual mathematicians, this will get you college credit, and it will look good applying to college.

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    $\begingroup$ He's right. AP Calculus really doesn't require much thinking to get a 5/5. They are quite basic and reiterated problems, and much what you'd find in any textbook. $\endgroup$ – Saketh Malyala Jun 9 '17 at 2:12
  • $\begingroup$ Can the down voter explain themselves? @Bob1123 was this you? $\endgroup$ – Aperson123 Jun 9 '17 at 22:34
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I highly recommend George Polya's Induction and Analogy in Mathematics. The link is to a free version on the web, but if you find it engaging you will want a hard copy. Also How to Solve It, by the same author, although I don't find it as compelling.

For just plain fun, look at Hugo Steinhaus, Mathematical Snapshots. Dover, so very inexpensive.

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  • $\begingroup$ Thanks for the recommendation. I've heard of Polya and How to Solve it, but not this. Do you have any recommendations for books which teach math skills (like problem-solving) and a particular subject (like precalculus or calculus)? $\endgroup$ – Ansh.23 Jun 9 '17 at 0:00
  • $\begingroup$ You've plenty of time to learn particular subjects. Now just have fun. This answer elaborates: math.stackexchange.com/questions/1714966/… . If you really want problem solving practice find a way to get involved in competitions. $\endgroup$ – Ethan Bolker Jun 9 '17 at 0:09
  • $\begingroup$ Thanks for the link to the other answer. I liked the number theory book you suggested, and I think I'll get it from a library or something. I mainly wanted problem solving practice so I can do better in other math-related stuff like programming and cryptography. Do you have any suggestions for studying for olympiads/competitions (other than the Polya suggestion)? $\endgroup$ – Ansh.23 Jun 9 '17 at 1:07
  • $\begingroup$ Can you comment on why you prefer Induction and Analogy in Mathematics to How to Solve It? $\endgroup$ – littleO Jun 9 '17 at 2:39
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    $\begingroup$ @littleO Induction and Analogy has somewhat more advanced mathematics (though not very advanced) and somewhat less introspection about strategies (though a lot, of course). $\endgroup$ – Ethan Bolker Jun 9 '17 at 12:06
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For a plain precalculus textbook that's good, but not extremely challenging, you can use Basic Mathematics by Serge Lang.

Good (short) books that will improve both your problem-solving ability and your ability to appreciate proofs at the high-school level include:

  • Algebra by Gelfand and Shen
  • The Method of Coordinates by Gelfand, Glagoleva and Kirillov
  • Functions and Graphs by Gelfand, Glagoleva and Shnol
  • Invitation to Number Theory by Oystein Ore
  • Introduction to Inequalities by Beckenbach and Bellman
  • The Mathematics of Choice by Niven
  • Numbers: Rational and Irrational by Niven

Please also have a look at the excellent bibliography in the Mathematical Olympiad Handbook by Gardiner, which is viewable on Google Books. See here: https://books.google.com/books?id=zyFLrAEVgv8C&lpg=PA41&pg=PA41&redir_esc=y#v=onepage&q&f=false

These books are all great preparation for rigorous calculus and linear algebra later on.

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    $\begingroup$ This is the first time I've seen someone recommend Lang's book unironically. It's always struck me as in that odd place between basics and rigor, and I guess these situations really do fit the bill. I just had to chuckle. Anyway, +1 for the books by Niven! $\endgroup$ – Alfred Yerger Jun 9 '17 at 13:26
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    $\begingroup$ @user49640 Have you ever used the Art of Problem Solving book series? I'm debating whether I should use those or the Gelfand books and the other books you recommended. Thanks for the list and link, they're both very helpful. $\endgroup$ – Ansh.23 Jun 9 '17 at 14:53
  • $\begingroup$ Why didn't you suggest "Linear Algebra" from Gelfand? $\endgroup$ – Dac0 Jun 9 '17 at 15:18
  • $\begingroup$ @Dac0 I see linear algebra at that level as being something one would study around the same time one studies calculus, and if possible, after one has studied some basic vector geometry. $\endgroup$ – user49640 Jun 9 '17 at 18:02
  • $\begingroup$ @Ansh.23 I haven't actually seen the Art of Problem Solving books, so I can't comment on their quality. What I know is that they were mostly written by people who did well in math competitions when they were young, as opposed to actual mathematicians. All the books I'm recommending were written by accomplished mathematicians. If you're not sure, visit a library to compare the books. $\endgroup$ – user49640 Jun 9 '17 at 18:09
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I often use https://brilliant.org/ as a site with a lot of questions for every level, with really great solutions. The wikis expect a lot of focus from you if you want to learn more, but go really in depth. Pretty much every math topic is covered. If you look at topics you think you already know, you will often find questions that you will be unable to solve because they offer new perspectives.

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I'll give you my advice since until now it has not been given: "Think a lot on the easy stuff". Meditate the easy definitions, work examples and exercise only to understand what's going on really well.

Try to deduce implications of "easy statement" at whatever level they are. Generalizing is easy once you understood what's going on. If you're smart you can understand a lot of geometry from the geometry of surfaces. If you're smarter you can understand a lot of geometry from the geometry of lines.

If you're in the first year an easy statement can be the definition of a group, the property of associativity, the properties of integers or elementar linear algebra stuff. If you're in a master course it can be the definition of manifold, group actions, modules. If you're first year doctorate it can newly be the definition of a group, the property of associativity and newly linear algebra :D :D :D .

I mean that the solutions to a lot of problems appear when you see the same old thing that everyone knew from a totally different perspective that gives you a hint to a road that noone had seen before. This doesn't come from knowing a lot of fancy technical stuff rather than from knowing really well what's going on.

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Don't worry at all if you can't solve harder problems (in your topic) now, I've used to be just like you.

The main thing that helped me was to browse through answers to harder problems I knew I couldn't do; after following through proofs I've tried to solve similar problems, if I couldn't solve it in around ~$20$ mins I would read through the solution and I would follow that procedure until I was able to solve such problems on my own.

Such knowledge needs time to better, I was very week at $1$st grade at $2$nd grade I was way better but still sometimes I chocked on problems and had to read through solution. At $3$rd and $4$th grade I was able to exactly pinpoint what technique I could apply to the problem.

I suggest trying out easier olympiad problems they usually have elegant solutions and are usually made to test problem solving skills of the brightest minds. I suggest Art of Problem Solving site for resources.

I would suggest perfecting the math topics you learned by now than when you start learning higher math topics it will be much easier.

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I have nothing to add in terms of books and such.
However, for critical thinking and problem solving part, one of the key things that helped me is to learn to identify what i don't know in a given problem. This doesn't necessarily correspond to the wanted variable in a problem. It is more like a sense of direction.
You look at a problem, and most of the time you have a sense of direction, a rather flexible set of approaches that come to your mind, for the problem at hand. Some parts of the problem makes you loose that sense of direction, like "now what?!", or "i lost track of what is happening here" moments, being able to identify those parts beforehand and train yourself accordingly, is a valuable skill. I personally believe it comes with proofs as they are already mentioned, i would also add formal logic, which would teach you how anything with a set of axioms and rules work.

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You know what you know and don't know what you don't know. Just learn what you don't know so you know.

Everything is easy once you've learnt it.

The number of hours you put in is everything.

You will put enough hours if you like the subject. Otherwise it will be an horror.

You've got to grow your mental model of maths. It grows with every fact you add to it. The best ones have ridiculously huge mental models of their fields. How do you want to solve something without having a mental model of it? You've got no tool then.

If you feel someone's better than you, it's because they've put more hours into it than you, and maybe of better quality (hours). You know, they might be from a family with mathematician traditions lasting centuries, they might have been being taught by the best mathematicians every day since the age of 3.

It's great though that you're wondering and asking such questions. That puts you in a better position than 99% of society. It's good, but you're probably gonna want to compete with the 1%.

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I believe it's similar to becoming good at Golf. Yes, you want to have good instructors and learn efficiently but by far the largest factor is years and years of practice and dedication to your craft.

I've thought about this a lot when speaking with someone who doesn't appear to have strong critical thinking skills, or unable to easily go abstract with concepts.

The interesting part - When I get the sense they are as least as genetically smart as I am I think to myself, how are you not able to quickly slice and dice logic, generalize, etc. things that seem pretty simple to me?

My best guess is there is simply a large difference between us in the time spent practicing these skills. How do you explain that even the most gifted athletes in the world usually can't switch sports, like how Michael Jordan was not very good at baseball?

When you come across an interesting problem you can't solve, find someone to help you slowly pick it apart end to end until you understand it fundamentally. Dedicate yourself to this systematic practice and think of the amount of time it takes to become a pro at anything.

Btw, I hope you already know that people like Einstein didn't sit around working 40 hours a week and just come up with these amazing insights. He was obsessed, consumed, maniacal in his work. Not terribly different from some pro athletes. HBO Real Sports has a new episode about Larry Bird. Raw talent not withstanding, there are simply not a lot of people you can't compete with when you put in that kind of time.

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    $\begingroup$ The necessity of extremely hard work for years on end is a common theme in the accounts of scientists such as Weinberg and Feynman. $\endgroup$ – Steven Thomas Hatton Jun 10 '17 at 19:33
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You should also look for a good Applied Mathematics course. This really helped me go from just rules and formulas to understanding how to use mathematics to solve real world problems. Also, math is a contact sport. You just have to sit down and work on problems.

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The most essential aspect of critical thinking is being honest with yourself.

Read Discourse on the Method of Rightly Conducting one’s Reason and Seeking Truth in the Sciences, by Rene Descartes.

Write your math work out as if you were explaining it to someone else.

Establish patterns for writing out expansions, etc. For example, if you have

$(a+b)(c+d)$, always do it as

$(a+b)(c+d)$

$=a(c+d)+b(c+d)$

$=ac+ad+bc+bd$.

Even though you know you could just as well write

$(a+b)(c+d)$

$=(a+b)c+(a+b)d$

$=ac+bc+ad+bd$.

Try to preserve the order of expressions as much as possible. For example, when I produced the second example above, I copied and pasted and then edited. The original correct result was

$(a+b)(c+d)$

$=(a+b)c+(a+b)d$

$=ac+ad+bc+bd$.

But, in my opinion, that is bad form.

Of course, there may be good reasons for deviating from those rules from time to time. For one; the exercise of doing something the alternative way may be instructive. But having set patterns avoids a lot of mental clutter.

Read with a pencil and paper (or whiteboard, or computer, etc.), and write out the theorems and proofs in your own words and symbols. Be sure you can justify every step to yourself. But don't over do it. Sometimes it makes more sense to just read through material trying to get the gist of what is being presented. Then go back and try to get a fuller understanding.

Learn to use LyX. It's free, and it will help you post to math.stackexchange.com. https://www.lyx.org/

Consider getting a student version of Mathematica.

I dropped out of high school in the tenth grade, and was never very good at participating in academic environments. My advice comes from not doing things that way for a long time. Edit to add:

Read the paragraph prior to the discussion of Decartes' contribution to the area of critical thinking. What I mean regarding being honest with yourself is that you should guard against believing in what Francis Bacon calls "idols". A Brief History of the Idea of Critical Thinking

For example, if someone gives you sound advice on how to improve your problem solving techniques and improve your critical thinking, but it gets voted down. You are obligated to decide for yourself who is in error.

As an example of how I approach learning math, see my Mathematica notebook recording my study of C.H. Edwards, Jr.'s Advanced Calculus of Several Variables. They are a work in progress. My notation is non-standard. And the notes are terse and cryptic. (But I expect a seasoned mathematician could follow them.)

Notice also that I have noted where Edwards made an assertion for which I do not provide proof. That typically means I'm flagging that for later consideration. Also note that there are some errors in Edwards treatment which I discovered by attempting to rewrite his discussion in the way I understand it.

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  • $\begingroup$ That's the same Descartes who invented analytical geometry. $\endgroup$ – Steven Thomas Hatton Jun 10 '17 at 8:47
  • $\begingroup$ To be clear. I did get a GED for which I did no focused preparation. I just like to learn, so I knew enough inadvertently. I also have a degree in computer science. But that was a very long process. I even took extra math classes, in which I got decent grades. I just knew I was faking it. I wasn't really learning. $\endgroup$ – Steven Thomas Hatton Jun 10 '17 at 19:14
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Agreed on the US school system... the best way to fix it is to forget everything they're trying to teach you! Okay maybe don't forget everything, but in my experience US math classes often try to teach you how to solve problems the wrong way, which is why many students end up hating math. Math isn't about following a series of steps towards a defined problem, computers do that. It's about creatively finding those steps.

My best advice is to do your best to forget about memorizing methods, and focus on understanding why those methods work. I'll always remember my Calculus 1 final exam where I derived the equation for arc length in the margins because I forgot it. My professor thought I wanted extra credit.

It's all well and good to be able to spit out that the area of a triangle is (1/2)bh, but it is much, much more helpful to know that a triangle is just half of a rectangle.

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