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Art of Problem Solving Math Books Review

What is art of problem solving.

Founded in 1993 by former USA Math Olympiad winner Richard Rusczyk, Art of Problem Solving (AoPS) is a company that produces rigorous math instruction courses and products that can help outstanding math students develop a more thorough understanding of math concepts, as well as help prepare them for success in math competitions. 

From textbooks to online classes to physical learning centers, AoPS offers a variety of educational products and services that can help challenge kids, deepening their knowledge and strengthening their mathematical thinking.

AoPS Math textbooks

Art of Problem Solving has created a series of textbooks for middle and high school math textbooks that are designed to give outstanding math students a deeper and more rigorous curriculum in math. 

Originally designed to help talented math students prepare for competitions, over the years AoPS’s textbook line has expanded to offer full curriculums in middle and high school math courses, and their problem-based and rigorous approach to math has made them very popular with parents across the world as a top enrichment option. 

What Grades and Math Subjects does AoPS Math cover?

Art of Problem Solving textbooks cover middle and high school math, as well as competition prep.  

Generally speaking, the AoPS math textbooks can be broken down into two curricula- introductory and advanced – that roughly correspond to most middle and high school math programs (in terms of overall scope, that is). 

Parents of younger math enthusiasts should note that Art of Problem Solving covers elementary school math (Grades 1-6) in their Beast Academy series, which you can read about in our review .

Introductory Curriculum (Middle School)

Advanced Curriculum (High School) 

When taken as a whole, Art of Problem Solving’s math textbooks cover the topics included in most US Math curricula, as well as touching on a few topics that aren’t usually covered in most public high school programs. 

That said, the point isn’t really to get kids learning college level math or a curriculum beyond high school math, but instead to get students to develop their problem solving skills and develop more creative and flexible mathematical thinking, to get them to recognize and appreciate different approaches to problem solving, as well as getting a better understanding of the why of math, rather than just focusing on how to compute problems. 

As such, AoPS’s curricula tend to go deeper into your typical middle and high school math topics, letting kids examine concepts more rigorously, more thoroughly and with more challenging problems than they would otherwise be able to do in other math courses. 

Art of Problem Solving Contest Prep

In addition to their more academically-focused textbooks, Art of Problem Solving also offers a variety of books designed to further enrich exceptional students or help with preparing for math contests and Olympiads.

These books generally tend to work on developing stronger problem solving skills, going far deeper into various concepts and exploring far more challenging questions and problems, while introducing various approaches for understanding and solving them quickly and effectively.

Geared more for gifted enrichment and contests preparation, each of these books tend to go over a greater variety of concepts and topics, touching on concepts in Geometry, Algebra, Number theory and more, and aren’t really bound to any linear curriculum. 

In addition, the problem sets, geared as they are to helping students prepare for national tournaments and contests, are far more challenging and in-depth than would be expected of even an advanced middle or high school course.

For these reasons we don’t usually think this series is where parents should necessarily start off when working on math at home, but in our experience we do feel they are great supplements to the main textbooks and can be excellent for enrichment purposes and preparing for contests.

How Art of Problem Solving Teaches Math

Aops pedagogical approach.

Art of Problem solving is a big believer in teaching through solving problems. 

The books consequently include a wide variety of problems, many of which kids will have never encountered before.

In fact, some come directly from various math competitions such as:

• The American Mathematics Competitions (AMC)  
• The Harvard-MIT Math Tournament

The general idea is that by getting kids to work through problems themselves, and more importantly discovering how to solve certain problems, kids will develop a deeper understanding of the material. 

As a result, AoPS Math textbooks are quite problem set heavy.

Explanations of each concept are quite short and to the point and are followed by a good deal of exercises for students to try out on their own.

When introducing these textbooks, parents should expect that kids will have to think things through a bit more and work out the answers themselves without a lot of hand holding or spoon feeding, and that there will be a heavier emphasis on logic and proof than other curricula. 

All this really drives home Art of Problem Solving’s place as a resource for outstanding or talented math students who don’t need a lot of time or explanation to grasp the material. 

Consequently, students who are less adept at math may find the instructions a little too short and too quick and may need extra help in order prevent getting frustrated by skill and knowledge gaps as the exercises come rolling in

Lesson structure

Regardless of the book in question, Age of Learning’s lessons tend to follow a particular format.

The books are made up of several chapters, each of which covers a particular topic within the subject and contains several sections. 

Each section is then typically broken down into various related concepts, an overview of the types of problems kids may come across (both common and uncommon) and often the various factors that can affect outcomes.

In Introduction to Algebra, for example, when discussing multivariable linear equations, the chapter is divided up into an introduction, a discussion of substitution, elimination, some word problems, common and uncommon problem sets, different variables and so on.

As kids go through their lessons, they are given lots of examples to try and lessons tend to work through some of them step-by step in a fairly in-depth and rigorous manner to demonstrate concepts. 

Sections typically end with a variety of exercises for that section and, at the end of each chapter, there are review and challenge problems. 

Review problems go over and test what the student has learned with similar problems, while challenge problems go a step further and test mastery of the material with far more challenging questions. 

If kids get stuck, there are always hints and solutions that are helpfully included in the back of the book ( no cheating !)

Look and feel 

As you might expect from a problem solving and word problem-heavy methodology, these textbooks contain lots of typical math diagrams and pictures floating about to go along with and illustrate the word problems.

AoPS textbooks also tend to have a lot of floating boxes that highlight important information for kids, including: 

• Pointing out various strategies they can take on given concepts or problems
• Offering extra work
• Giving extra information
• Even offering “bogus” solutions that point out the most common mistakes made by students when solving a problem

Despite its rigor, Art of Problem Solving does its best to keep its material from becoming too dry and boring, which we appreciate. 

The books are written in a very casual tone, which makes it feel as if a math-whiz friend were explaining the material rather than a textbook. 

There are also a good deal of amusing and interesting examples and concept demonstrations sprinkled throughout, sometimes even involving sly pop culture references (some of which may go over kids heads, but parents will appreciate).

Does this approach really work?

Due to its philosophy and the way it teaches, we feel the Art of Problem solving takes more or less a constructivist/Problem Based Learning approach to teaching math where, instead of receiving formal lectures about math, students build up their own knowledge and skill by working through and solving various problems.

This learner-centric approach to teaching math and science actually has been linked to positive outcomes when teaching math and science , fostering greater problem-solving skills, improving self-motivation and encouraging creative and critical thinking skills as they relate to mathematics. 

Past customers have also reported that the series challenges their students pretty thoroughly, increasing the depth of their knowledge on relevant subjects and increasing their speed at solving difficult-math problems, sometimes dramatically. 

It is perhaps unsurprising, then, that the Art of Problem Solving curriculum is often used in honors math classes across the US. 

Some Drawbacks to Art of Problem Solving Textbooks and Curriculum

Can be time consuming.

Due to its focus on doing exercises, exploring concepts and working through problems to gain a better understanding of the subject matter, Art of Problem Solving can take a little more time to work with than some other programs.

This can be particularly true as AoPS tends to use far more challenging questions than kids are used to, some of which are in formats they haven’t seen before. 

While great for learning, this approach isn’t exactly a time saver. It’s not uncommon, for example, for parents to report spending up to 45 min (or more) each day on math (in addition to other homework). 

Can be tricky to jump into from another curriculum

With its particular approach and pedagogy, as well as its more rigorous approach to mathematics and problem solving (including the use of proofs), Art of Problem Solving can be somewhat tricky to get used to if you jump into it from another curriculum. 

Because math is a cumulative process, kids who begin Art of Problem solving without having at least reviewed some of the foundational material in previous books can find themselves lost or slowed down by skill and knowledge gaps they didn’t realize they had. 

Helpfully, the AoPS website does have free, printable diagnostic assessments for each book to help parents determine if their kids are at the right skill level. 

Discovery approach can frustrate some learners

Despite the fact that Age of Problem Solving’s approach has been shown to get results and improve the mathematical thinking and skills of talented math students, sometimes it just isn’t the right approach for the student. 

AoPS often requires students to play around with numbers and concepts and discover missing information themselves. 

Some students, even really talented students, can get frustrated by this approach and may prefer a more straightforward, traditional math course where they can get down to computation and see their results more quickly. 

Who is Art of Problem Solving For?

Overall, we think Art of Problem Solving is a great resource for parents and kids looking for a far more thorough, challenging and enriched math program.

It is an ideal course for students who demonstrate an aptitude for math and are looking to deepen and strengthen their math skills with more challenging grade-level material.

We think AoPS textbooks can be particularly good for students interested for more rigorous preparation for math-heavy STEM subjects in university , where their greater focus on problem solving, proofs and logic skills will be a strong asset, such as with physics, engineering and even computer science,

We also think that Art of Problem solving’s textbooks and methodology can be an excellent base material for students interested in or preparing for math contests and olympiads (AMC 10, AMC 12, MATHCOUNTS and the like), particularly their Contest Math Prep Series, as they promote creative approaches to problem solving and strengthen mathematical thinking that kids can use when faced with new problems.

Who is Art of Problem Solving Not Great For?

That said, Art of Problem Solving textbooks are obviously not for every student. 

These books are not the best curriculum for kids who are struggling with math concepts as AoPS math is primarily aimed at enriching math study. 

AoPS math goes far deeper into the material with far more rigor, exploring various high school and middle school math topics at a more advanced level and with more challenging problem sets, while emphasizing multiple approaches to problem solving and flexibility when approaching new math problems. 

Struggling students, while they often can benefit from learning the why’s behind math, can usually spend their time better by reviewing the fundamentals and practicing basic strategies, as well as by working on more targeted skill development with programs like IXL and Khan Academy .

Similarly, we don’t feel that AoPS textbooks are really the best resource for preparing for the SAT and other timed standardized tests where answering speed and efficiency (and test taking strategies) can be far more effective when it comes to success than gaining a deep understanding of concepts and working through problems.

In these instances, kids are better served through specific standardized prep programs that will work with them on developing their proficiency at solving very particular types of questions. 

Finally, AoPS textbooks are also not the best solution for kids looking to explore college level math as, despite its more challenging nature, AoPS math goes deeper into middle school and high school math topics (algebra, geometry, number theory, single variable calculus), rather than beyond it.

Price: How much do AoPS Textbooks Cost? 

The price of AoPS math textbooks really depends on the particular book and subject you’re interested in. 

Generally speaking, though, each book costs between $45 and $70, which is roughly the same as the average middle or high school textbook .

The length of each book varies, however, from just under 300 pages of instructional material in some cases to well over 700 in others.

Unlike many other middle and high school textbooks, however, these are designed to serve as a complete curriculum for each topic as every book contains instructional material as well as hundreds of practice problems, hints, and a step-by-step solution guide that itself is usually a couple hundred pages long as well. 

Bottom Line:

If you have a talented middle or high school math student and you’re looking for ways to nurture their excellence, Art of Problem Solving’s math textbooks might be right for you.

Although certainly not for everyone, with their challenging curriculum and in-depth exploration of math concepts, AoPS can foster better problem solving skills, stronger analytical ability and improved creative and critical math thinking, all of which can help students take their math skills to the next level.

About the Author

David Belenky is a freelance writer, former science and math tutor and a tech enthusiast. When he’s not writing about educational tech, he likes to chill out with his family and dog at home.  

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The Math Evangelist Who Preaches Problem-Solving

September 13, 2022

Richard Rusczyk, 50, at the Art of Problem Solving campus in San Diego.

Philip Cheung for Quanta Magazine

Introduction

When Richard Rusczyk became interested in math competitions as a middle schooler in the early 1980s, the contest problems looked nothing like the ones in his math classes. He couldn’t find any book to guide him — there were only the problems themselves.

In some of the more advanced competitions he participated in as he moved on to high school, he couldn’t solve a single problem. Gradually, though, he figured out how to “kind of connect the dots, and back out what was actually going on,” he said. He learned a lot of math, but also something he considers even more important: the art of problem-solving.

Later, as an undergraduate at Princeton University, he saw classmates struggling in math classes despite having gotten perfect scores in high school. Their earlier classroom experiences had taught them to memorize a grab bag of tricks, he said. “When you get to college, that doesn’t work anymore.”

So Rusczyk and a competition-loving classmate, Sandor Lehoczky, set out to write the book their 13-year-old selves would have devoured. The resulting two-volume series, The Art of Problem Solving , opens by addressing readers: “Unless you have been much more fortunate than we were, this book is unlike anything you have used before.” From the start, the books sold 2,000 copies per year — “enough to cover rent,” Rusczyk said. Word of mouth grew, and over the 30 years since, well over 100,000 math enthusiasts have bought copies.

Today, Rusczyk’s company, Art of Problem Solving (AoPS), offers not just a large array of textbooks but also online and in-person math classes for “ambitious problem solvers” that serve nearly 25,000 students each year. These courses include both contest prep classes and subject-matter courses, but they have a common goal of fostering a problem-solving mentality. The company is currently expanding its elementary school materials, called Beast Academy, into a full curriculum, with the goal of bringing the problem-solving mindset to more than just self-selected math lovers.

This mindset “should be baked into the curriculum,” Rusczyk said. “It shouldn’t be the thing you do on every third Friday.”

Quanta spoke with Rusczyk about how to turn math learners into problem solvers. (In the interest of full disclosure, our interviewer’s child has taken AoPS classes, and her sister taught AoPS summer camps online in the first year of the pandemic.) The interview has been condensed and edited for clarity.

Your Beast Academy textbooks are comics, and you introduce concepts through story. The characters are talking about their math homework on the school bus, or they’re in woodworking class, or they’re on a field trip. What made you choose that approach?

You can’t lecture a third grader. You need to have a back-and-forth. The comic book structure we use has little kid monsters in conversation with each other, parents, teachers, the different characters in the universe.

Beast Academy’s illustrated math guides.

So you can model exploration, you can model overcoming challenges, you can model being OK with being wrong. You can create the environment for the child emotionally and intellectually. Every year we have parents sending in pictures of their kids dressing up as various characters for Halloween. They are putting themselves in these spaces.

We spent months trying to figure out: What is our delivery mechanism? We had 150 pages of worksheets, and we’re like, “No, this doesn’t work.” And then in one five-minute stretch, someone said comic books, and someone else said monsters. And we got a fantastic artist and started building out the books.

The lessons you’re trying to teach seem to go far beyond any specific math content, or even specific problem-solving techniques.

One of the main things we’re trying to get across is just the mindset of openness and willingness to engage with things we don’t understand at first. This is something kids are naturally inclined to do. But then something happens during elementary school, particularly in math classes, and we train that out of them.

We’re trying to encourage kids not to lose this curiosity or get into a mindset where the goal is to do everything perfectly. Because we have machines for that now. When we set kids up to compete with computers, we’re setting them up for failure, because anything a computer can do, it’s going to do better.

Within Beast Academy, the kids have different strengths. There’s one that’s wacky and does outlandish things that are sometimes not right, but sometimes really insightful. There are characters that are very precise and organized. And there’s a character who emerges over time as just plain brilliant. These are all different aspects of approaching different types of problems.

Video : Richard Rusczyk, founder of Art of Problem Solving, discusses how to bring out the joy, creativity and beauty in math.

Photo by Philip Cheung for Quanta Magazine; Video by Emily Buder/Quanta Magazine and Noah Hutton and Jesse Aragon for Quanta Magazine

Your materials for older students don’t incorporate a storytelling framework. But one striking thing about them is how each new chapter or class session begins not by introducing concepts, but with a collection of problems. What made you choose that format?

This was how I learned math. It was a pretty powerful way to learn.

When I started experiencing high school math Olympiads, it was two years of getting zero right on every single test. That was really frustrating. But it was safe, because it was a math contest, and who really cares? It wasn’t the first-year math class in college, staring at four problems and thinking, “I am not going to be able to do this, I am not going to be a scientist, I’m not going to be an engineer.”

That’s the experience our educational system gives to a lot of students. They think they’re not good enough, because the first time they’ve had this experience is when they get to college. They’re good enough, they just haven’t been prepared.

So we show the problems first. If a student discovers math for themselves, it becomes their math, instead of just something that was told to them. They’re not always going to get there, and that’s fine. Or sometimes they’re going to do it very differently than we did. That’s great too.

Your classes tend to attract kids who are already excited about math, and that in turn attracts teachers with strong math backgrounds. It’s one thing to make a system that works well for such enthusiastic and experienced participants, and another to make something that will work in classrooms everywhere. What challenges do you anticipate in scaling up your Beast Academy materials to a full curriculum?

We are approaching it first as a learning experience for us. We have a strong perspective on a certain type of student, and a strong conviction about some of the approaches we think should be taught to students. As to how to best deliver those resources to teachers and students in different environments, that’s something we’re more than humble about.

I’ll step back further and say I believe a lot of the troubles in education right now are technology companies going to schools and saying, “This is how you should do things.” It has to be a partnership between the content providers and the most important delivery mechanism these kids will ever have, which is the teacher in the room and the other kids.

Two or three years ago, we started working with schools using Beast Academy as a supplement, and that’s been pretty successful. But to reach more students and have a deeper impact on them, you really want to be the entire experience.

Rusczyk with his staff at the San Diego office.

When you say that Beast Academy has been successful as a classroom supplement, how do you measure that?

We just had a study completed in a school district in Minnesota. It was a little over 1,000 students in three groups: a “gifted” group, that passed some test; “Rising Scholars” students, who I think are defined as kids from diverse communities that didn’t pass this test but were close; and other students. They looked at the students’ performance on the Minnesota [standardized] test, and how that varied with the number of lessons they did on Beast Academy online. And they found a very strong relationship — the students who did more than, like, 150 or 200 lessons grew by a much larger margin than the kids who did 15 lessons, or no lessons. One really interesting thing is, the effect size was largest in the Rising Scholars group.

Who chose how many lessons kids did — the teachers, or the kids themselves?

It was during the pandemic, so my guess is a little of both. The outliers are almost certainly kids choosing it themselves. Whether this is revealing that the material teaches the kids or the material unlocks the kids, I’m not sure it matters, right? You have to give them material that’s going to make them want to do it. Getting the student to a place where they are interested in struggling with whatever you’re showing them, that for a lot of kids is the whole game.

There’s a lot of debate in educational circles about whether kids at both the high and the low end of performance are best served by being put on separate tracks or the same track. It sounds like you feel pretty strongly about giving extra challenge to kids who are ready for it.

We want to give students the materials that are most suited to help them realize their potential. If you give students material that is not speaking to them, you’re not giving them the opportunity to realize that potential.

When you remove advanced programs, you remove them for all students. So there’s going to be some kid who’s brilliant, but she will never know. And that’s a missed opportunity for her and for us, because these are the highest-leverage people in terms of making medical and technological advances.

Rusczyk at Torrey Pines State Natural Reserve in San Diego.

Creating those experiences also helps the students find their people. Part of what we do with Art of Problem Solving is our online community. For some students, it’s the only place where they feel safe expressing a love of math and science, because it is not part of the culture of their schools.

When I went to math competitions for the first time, the thing that resonated with me was, not only were there other kids who liked the same geeky stuff I did, there were adults who were excited about me being good at math, and they weren’t my parents, they weren’t my teachers. They weren’t required by profession or relation to be happy that I could do math. I had never seen that before.

Math competitions can be great for kids who are naturally competitive, but that’s not all kids. What can we offer the other kids?

It’s one of the great failings of the math community that the primary way you can explore deep interest in math is through competitions. When I was a student, contests were the only game in town.

This has gotten less true in the last 10 to 15 years, which is great. Now there are summer camps that are not contest-focused, and math circles that came out of the Eastern European tradition where professors work with the top students in their city.

I started one of these math circles at UCSD here in San Diego before I started Art of Problem Solving. And we had Efim Zelmanov, a Fields medalist, come give a talk. This was joyous, beautiful math — he was just so magnetic and happy to be there. So I thanked him for coming, and his answer was, “Well, I’m here to do this because this is what people did for me growing up.” And I’m sitting here thinking, I have exactly the opposite answer. We’re building these things because we didn’t have this sort of stuff.

It seems like Beast Academy, the imaginary school in the comic books, is the kind of place you would have dreamed of attending as a kid. You’ve said that some kids dress up as their favorite Beast Academy monster for Halloween, but what about you? Is there a monster you especially identify with?

Bits and pieces of various characters. But I might have identified most with Fiona [the math team coach]. In her day, she was pretty strong. But her interest is in sharing beautiful, interesting things with students, and helping them become stronger than she was.

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Getting better at proofs

So, I don't like proofs.

To me building a proof feels like constructing a steel trap out of arguments to make true what you're trying to assert.

Oftentimes the proof in the book is something that I get if I study, but hard to come up with on my own. In other words I can't make steel traps, but I feel fine buying them from others.

How does one acquire the ability to create steel traps with fluency and ease? Are there any particular reference books that you found helped you really get how to construct a proof fluently? Or is it just practice?

• soft-question

• 29 $\begingroup$ "How does one acquire the ability to create [proofs] with fluency and ease?" The same way one gets to Carnegie Hall: through practice, practice, practice, and more practice. $\endgroup$ –  Arturo Magidin Commented Oct 25, 2010 at 0:23
• 8 $\begingroup$ Community wiki? $\endgroup$ –  Arturo Magidin Commented Oct 25, 2010 at 0:34
• 4 $\begingroup$ Try explaining the proof to someone who only has knowledge of the givens. Ask "why?" again and again. Why, why, why. Why do we know this is true? Because this other thing is true. But why is that true? Keep on asking why until you get to a point where you don't need to ask why anymore (your givens). $\endgroup$ –  Justin L. Commented Oct 25, 2010 at 1:52
• 1 $\begingroup$ @J.M.: Maybe you could remove the [proof-theory] tag? I don't see any connection here. $\endgroup$ –  AD - Stop Putin - Commented Oct 28, 2010 at 19:30
• 3 $\begingroup$ @bobobobo: I'd also like to add that I think that one should be able to verify your solutions to the problems your working on, especially if you're working independently. Many problem books do not include solutions to the problems posed within the book itself, but some do. I did a little cybersearch for you, and I found the book "Mathematical Olympiad Challenges" ( amazon.com/Mathematical-Olympiad-Challenges-Titu-Andreescu/dp/… ) , which, upon closer examination, I perhaps intend to buy myself. Perhaps it could help you, too. $\endgroup$ –  Max Muller Commented Dec 19, 2010 at 20:21

5 Answers 5

I'd like to second one part of Qiaochu Yuan's answer: the recommendation to read Polya's book. Unlike many other books I've seen (albeit none of the others recommended above), it actually does contain guidance on how to construct a proof "out of nothing".

And that's one problem with the "practise, practise, practise" mantra. Practise what? Where are the lists of similar-but-not-quite-identical things to prove to practise on? I can find lists of integrals to do and lists of matrices to solve, but it's hard coming up with lists of things to prove.

Of course, practise is correct. But just as with anything else in mathematics, there's guidelines to help get you started.

The first thing to realise is that reading others proofs is not guaranteed to give you any insight as to how the proof was developed. A proof is meant to convince someone of a result, so a proof points to the theorem (or whatever) and knowing how the proof was constructed does not (or at least, should not ) lend any extra weight to our confidence in the theorem. Proofs can be written in this way, and when teaching we should make sure to present some proofs in this way, but to do it every time would be tedious.

So, what are the guidelines for constructing a proof? You'll probably get different answers from different mathematicians so these should be construed as being my opinion and not a(n attempt at a) definitive answer.

My recommendation is that you take the statement that you want to prove and apply the following steps to it as often as you can:

• Expand out unfamiliar terms.
• Replacing generic statements by statements about generic objects.
• Including implicit information.

Once you've done all that, the hope is that the proof will be much clearer.

Here's an example.

Original statement:

The composition of linear transformations is again linear.

Replace generic statements:

If $S$ and $T$ are two composable linear transformations then their composition, $S T$, is again linear.

It is important to be precise here. The word "composable" could have been left out, as the statement only makes sense if $S$ and $T$ are composable, but until you are completely familiar with this kind of process, it is better to be overly precise than otherwise. In this case, leaving in the word "composable" reminds us that there is a restriction on the domains and codomains which will be useful later. (However, one has to draw the line somewhere: even the word "composable" is not quite enough since it leaves open the question as to whether it is $S T$ or $T S$!)

Include implicit information:

If $S \colon V \to W$ and $T \colon U \to V$ are linear transformations then $S T \colon U \to W$ is again linear.

Here's where remembering that $S$ and $T$ are composable in the previous step helps keep things clear. As $S$ and $T$ are composable, we only need $3$ vector spaces. Then, since we explicitly have the vector spaces the fact that $S$ and $T$ are composable is plain, though some may prefer to keep that fact in the statement. Also, some may like to have the fact that $U$, $V$, and $W$ are vector spaces explicitly stated.

Expand out definitions:

If $S \colon V \to W$ and $T \colon U \to V$ are such that $S(v_1 + \lambda v_2) = S(v_1) + \lambda S(v_2)$ and $T(u_1) + \mu T(u_2)$ for all $v_1, v_2 \in V$, $u_1, u_2 \in U$, and $\lambda, \mu \in \mathbb{R}$, then $S T(x_1 + \eta x_2) = S T(x_1) + \eta S T(x_2)$ for all $x_1, x_2 \in U$ and $\eta \in \mathbb{R}$.

Note that I have been careful not to repeat myself with the newly introduced symbols. It would be technically alright to reuse $u_1$ and $u_2$ in place of $x_1$ and $x_2$ since these are local declarations (restricted by the phrases "for all ..."). However, humans are not good at differentiating between local and global declarations so it is best not to reuse symbols unless the scope is very clear.

If $S \colon V \to W$ and $T \colon U \to V$ are such that $S(v_1 + \lambda v_2) = S(v_1) + \lambda S(v_2)$ and $T(u_1) + \mu T(u_2)$ for all $v_1, v_2 \in V$, $u_1, u_2 \in U$, and $\lambda, \mu \in \mathbb{R}$, then whenever $x_1, x_2 \in U$ and $\eta \in \mathbb{R}$, $S T(x_1 + \eta x_2) = S T(x_1) + \eta S T(x_2)$.

Up to now, the rephrasing has not taken into account the fact that there is a conclusion and a hypothesis . This rephrasing modifies a part of the conclusion to turn it from a generic statement "$P(p)$ is true for all $p \in Q$" to a statement about a generic object "whenever $p \in Q$ then $P(p)$ is true". We do not do this for the similar statements in the hypothesis. This is because these two pieces are treated differently in the proof.

Replace generic statements, and reorganise to bring choices to the fore:

Let $S \colon V \to W$ and $T \colon U \to V$ be such that $S(v_1 + \lambda v_2) = S(v_1) + \lambda S(v_2)$ and $T(u_1) + \mu T(u_2)$ for all $v_1, v_2 \in V$, $u_1, u_2 \in U$, and $\lambda, \mu \in \mathbb{R}$. Let $x_1, x_2 \in U$ and $\eta \in \mathbb{R}$. Then $S T(x_1 + \eta x_2) = S T(x_1) + \eta S T(x_2)$.

In this form, the distinction between hypothesis and conclusion is all the clearer. Parts of the hypothesis use the word "Let", parts of the conclusion use the word "Then".

With this formulation, the proof essentially writes itself. With all it's gory details:

Let $S \colon V \to W$ and $T \colon U \to V$ be such that $S(v_1 + \lambda v_2) = S(v_1) + \lambda S(v_2)$ and $T(u_1) + \mu T(u_2)$ for all $v_1, v_2 \in V$, $u_1, u_2 \in U$, and $\lambda, \mu \in \mathbb{R}$. Let $x_1, x_2 \in U$ and $\eta \in \mathbb{R}$.[^quick] Then:

$$ S T(x_1 + \eta x_2) = S \big( T(x_1) + \eta T(x_2)\big) $$

using the hypothesis on $T$ as $x_1, x_2 \in U$ and $\eta \in \mathbb{R}$. So:

$$ S T(x_1 + \eta x_2) = S T(x_1) + \eta S T(x_2) $$

using the hypothesis on $S$ as $T(x_1), T(x_2) \in V$ and $\eta \in \mathbb{R}$. Hence the conclusion is true.

Notes: 1. This could be condensed, but the important thing here is how to find it, not what the final form should be. 2. Notice that I wrote "as $x_1, x_2 \in U$" rather than "with $u_1 = x_1$ and $u_2 = x_2$". This is partly style, and partly because in the statement of linearity, $u_1$ and $u_2$ are placeholders into which we put $x_1$ and $x_2$. So saying $u_1 = x_1$ is semantically incorrect as it equates a virtual vector with an actual vector. This is a very minor point, though.

Finally, I would like to disagree with one part of Qiaochu's answer. I actually like the imagery of a steel trap. A proof is a bit like a trap: we want to capture the theorem in a trap so that it can't wriggle out. We construct the proof so that there is no possibility of escape. Eventually, yes, we want the proof to be beautiful but when it's first constructed we just want it to do the job. Only once the theorem is caught can we spend a little time decorating the cage to make it look pretty and set it off to its best advantage. So build the trap because theorems can be dangerous! An escaped theorem can do untold damage, rampaging across the countryside, laying waste like an unchecked viking.

(Okay, not quite finally. The step-by-step proof above was taking from a page I wrote for my students on the nature of proof. The original can be found here .)

• 6 $\begingroup$ Great answer, it is a sorrow that I can't vote up but once. $\endgroup$ –  user1869 Commented Oct 28, 2010 at 20:43
• 2 $\begingroup$ +1. I would like to add that the books which are not Polya's in my answer are about half filled with lists of things to prove, together with discussion of some general principles useful for solving Olympiad problems. The focus might seem narrow but Engel was a revelation for me in my high school days. I would also like to add that this seems like a great set of steps for solving routine exercises in an undergraduate class, but it won't necessarily help for solving a problem which genuinely requires thought. $\endgroup$ –  Qiaochu Yuan Commented Oct 28, 2010 at 23:12
• 6 $\begingroup$ @Qiaochu: I think that the "genuinely requires thought" remark needs a whole blog post to respond to it! (So if you post about it on your blog, tell me and I'll respond there where the space is less confining.). In short, I'm advising learning to walk before hiking across the Sahara; and it's amazing how many problems that appear to "genuinely require thought" actually don't once the above steps have been carried out. $\endgroup$ –  Andrew Stacey Commented Oct 29, 2010 at 7:20
• $\begingroup$ +1. +2 If I could. The link to your paper is broken. Where can I find your papers? I really enjoyed your writing style. $\endgroup$ –  Enzo Ferber Commented Aug 7, 2016 at 23:35
• 1 $\begingroup$ @JonasEschmann In the above then $\mathbb{R}$ stands for the reals. The original version of that technique was written in a context where "vector space" meant "vector space over $\mathbb{R}$" and since the exact choice of allowable coefficient field doesn't alter the point of what I wrote then it didn't occur to me to make that explicit. Yes, the statement isn't as general as it could be but it still suffices to illustrate the technique. $\endgroup$ –  Andrew Stacey Commented Sep 15, 2022 at 23:13

You get better at proofs the same way you get better at basketball or carpentry: lots and lots of practice. (In particular, like in basketball and carpentry, you can only get so far by reading books.) Of course, there's good practice and bad practice. For general experience writing and coming up with proofs I think the same kind of material that people use to prepare for Olympiad-style mathematics is very helpful. To that end, here are a few references you might find helpful:

• Problem-Solving Strategies , Engel (somewhat high-level)
• How to Solve It , Polya
• The Art and Craft of Problem Solving , Zeitz

For writing proofs specific to a course (e.g. real analysis) you'll generally find that the same basic ideas are being used over and over again, and once you learn to recognize when these ideas will be useful your life will be much easier. But I think this is a hard skill to teach.

It doesn't help that many proofs in textbooks are written in a style that makes it nearly impossible to see how someone could have come up with the proof from first principles. This is an unfortunate tendency, and you should find a different textbook if this is too much of an issue. (Alternately, you should see if there are better proofs online, for example on someone's blog. Tim Gowers and Terence Tao are fond of writing up conceptual proofs of things, and more generally their blogs are a great source of insight into how mathematicians think.) Once you build up a little problem-solving skill, another way to fix this is to reprove things yourself. It helps if you can't remember what the proof in the textbook is.

But I'm serious about the practice. $10000$ hours and all that.

Edit: I also find this "steel trap" analogy very depressing. While it might seem that way in some textbooks, a properly presented proof should feel more like a poem.

Edit #2: I'd also like to mention that the general outline of Polya's advice is on Wikipedia .

It seems you are asking two questions. The first is how to get better at doing proofs, and the previous answers are better than I can do. The second is why to do them, which has not been addressed. From your questions on this site, you seem more an engineer than a mathematician. This is not a negative-I am educated as a physicist and practicing as an engineer(ing manager), but I enjoy proof-type math. It just is a different view.

In response to your question "How can I characterize the type of solution vector that comes out of a matrix?" Greg Graviton not only gave an answer of what matrices met your criterion, he proved that matrices that did not met his answer did not work. This is a major improvement as you know there aren't others out there.

Proofs are designed to make sure you covered all the loose ends. Ideally, each statement in a proof would be derived from the previous statements through an accepted rule of logic. But aside from the automated theorem provers, this is unachievable because it would make the proofs too long. So you only need to display enough steps to convince the audience it can be done. How big a leap is acceptable varies with the audience.

As physicists/engineers our numbers and functions are better behaved than the ones the mathematicians worry about. The mathematical world is not uniformly continuous, but ours is. As such, we interchange limits, integrals, derivatives, and sums freely.

As Arturo and Qiaochu have pointed out you get better at proving things by practicing a lot, solving exercises and by seeing how other people have proved things. There's a big part in learning how to prove statements by repeating certain strategies that may have worked for particular types of problems.

For instance, a particular argument that is used countless times in analysis is the "technique" of adding and subtracting a quantity in a way that lets you group things in a useful way. And there are lots and lots of tricks such as this one that you will learn during the way by seeing them being used several times (and also by using them yourself).

But there's certainly something about proving things that I must say. I don't know how common this is for people studying at "top universities" around the world where only really talented and smart people are admitted, but in my country it is relatively easy to get accepted in a mathematics program.

As a consequence, I've seen lots of people fail several times in the first mathematics course in my university for mathematics students (where actually students are introduced to proof techniques and basic logic and set theory, basically what in some American universities is referred to as a transition course from the usual calculus courses to proof oriented courses).

The main reason for most of them failing such a course, from what I've seen, is that some of them just don't seem to get used to the idea of having to prove something rigorously (as a mathematician will expect). I remember very well when one my classmates argued with the professor because he was asked in the test to prove an equality of sets, and my classmate drew a Venn diagram which showed the sets in question. The professor told him repeatedly that as an aid for intuition the drawing was perfectly fine, but that the drawing didn't constitute a proof by itself.

In the end (as you'll imagine) my classmate didn't win the argument (or extra points for that matter) and he ended up going home frustrated and finally he gave up on the course.

My point is that for some reason some people are just better than others at proving things, and I'm not saying that they're just smarter, it's only that the thought processes involved just seem to come more naturally to some persons.

That being said, there are some books that specifically address the topic of introducing students to the task of proving things. What they usually do is to begin by explaining some basic logic and them they build up some easy facts from set theory, binary relations and functions and in the way they introduce some proof techniques, such as "proof by contradiction", "proof by contrapositive", etc.

For example, one book that helped me a lot when I was starting is Proofs and Fundamentals: A First Course in Abstract Mathematics , but there are a lot of other books that I'm pretty sure work really well when making your first steps in proving mathematical statements, such as How To Read and Do Proofs by Daniel Solow or How To Prove It by Daniel Velleman.

• $\begingroup$ I think you should separate proof creation from proof writing . The former is primarily about convincing someone (yourself, the reader) that the claim is true on logical grounds. From that standpoint, why can't a proof be visual or have a visual component? For your classmate, a diagram might be more convincing than any written argument. Of course, our geometric intuition can mislead us, but so can our logical intuition (e.g. forgetting a case in case analysis). $\endgroup$ –  augurar Commented Apr 8, 2014 at 22:51
• $\begingroup$ (ctd.) I wouldn't conclude from the anecdote that your classmate necessarily lacked the ability to prove things, only the ability to put his ideas in writing. $\endgroup$ –  augurar Commented Apr 8, 2014 at 22:53

I recommend the book by Lay, "Analysis with introduction to proofs". The first chapter focuses on logic and proofs and personally I found it rather helpful. At least it helped me. Of course to master the art of proofs you have to keep practicing, but knowledge of some basics such that instead of proving the statement "A implies B you can prove contrapositive "not B implies not A", and how it differs from "Proof by contradiction", and the fact that negation of "A implies B" is "A and not B", and how to deal with existential and universal quantifiers, and so forth and so forth. Those are useful techniques one should master.

Incidentally, as an introduction to analysis, the aforementioned book is quite mediocre. I had a first hand experience with that book, since it was a textbook when I took the "low level introduction to analysis (wasn't sure if I was ready for Rudin's Principles of Analysis). So, this is only the chapter on logic and proof which is worth reading. Of course, purchasing the whole book just for a single chapter is funny, but may be you can find the used one or get one in the library.

But that chapter on proofs did help me. When subsequently I took a course on analysis and we used Baby Rudin as a textbook, I was in a very good shape and in fact did quite well.

And, of course, another suggestion: learn by example. See how other people write the proofs. Rudin's style, for instance, is very terse, you literally have to dig through his proofs, but after a while you start getting used to it, and still after a while you start writing similar proofs.

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Introduction to Algebra 2nd Edition

• ISBN-10 1934124141
• ISBN-13 978-1934124147
• Edition 2nd
• Publisher AoPS Incorporated
• Publication date March 30, 2007
• Part of series Art of Problem Solving
• Language English
• Dimensions 8.75 x 1.5 x 11.25 inches
• Print length 656 pages
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• Publisher ‏ : ‎ AoPS Incorporated; 2nd edition (March 30, 2007)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 656 pages
• ISBN-10 ‏ : ‎ 1934124141
• ISBN-13 ‏ : ‎ 978-1934124147
• Item Weight ‏ : ‎ 3.26 pounds
• Dimensions ‏ : ‎ 8.75 x 1.5 x 11.25 inches
• #221 in Algebra & Trigonometry
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• #776 in Decision-Making & Problem Solving

About the author

Richard rusczyk.

Richard Rusczyk founded Art of Problem Solving (AoPS) in 2003 to create interactive educational opportunities for avid math students. Richard is one of the co-authors of the Art of Problem Solving classic textbooks, author of Art of Problem Solving's Introduction to Algebra, Introduction to Geometry, and Precalculus textbooks, co-author of Art of Problem Solving's Intermediate Algebra and Prealgebra, one of the co-creators of the Mandelbrot Competition, and a past Director of the USA Mathematical Talent Search. He was a participant in National MATHCOUNTS, a three-time participant in the Math Olympiad Summer Program, and a USA Mathematical Olympiad winner (1989). He graduated from Princeton University in 1993, and worked as a bond trader for D.E. Shaw & Company for four years. AoPS marks Richard's return to his vocation: educating motivated students.

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Books with math and logic puzzles for graduate level math people?

I really like math and logic puzzles. Especially there are plenty of nice books by Raymond Smullyan. But, I want to hear your opinion about some nice puzzle books, that after you have read and tried the puzzle you actually feel like your problem solving skills have increased and that you have upped a few IQ points. This is a bit vague definition for a book, but shorty, I want to hear puzzle books that made you feel something. Please note that I'm graduate level university student.

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