Introduction to Mathematical Thinking

by Keith Devlin

In the twenty-first century, everyone can benefit from being able to think mathematically. This is not the same as “doing math.” The latter usually involves the application of formulas, procedures, and symbolic manipulations; mathematical thinking is a powerful way of thinking about things in the world -- logically, analytically, quantitatively, and with precision. It is not a natural way of thinking, but it can be learned. Mathematicians, scientists, and engineers need to “do math,” and it takes many years of college-level education to learn all that is required. Mathematical thinking is valuable to everyone, and can be mastered in about six weeks by anyone who has completed high school mathematics. Mathematical thinking does not have to be about mathematics at all, but parts of mathematics provide the ideal target domain to learn how to think that way, and that is the approach taken by this short but valuable book. The book is written primarily for first and second year students of science, technology, engineering, and mathematics (STEM) at colleges and universities, and for high school students intending to study a STEM subject at university. Many students encounter difficulty going from high school math to college-level mathematics. Even if they did well at math in school, most are knocked off course for a while by the shift in emphasis, from the K-12 focus on mastering procedures to the “mathematical thinking” characteristic of much university mathematics. Though the majority survive the transition, many do not. To help them make the shift, colleges and universities often have a “transition course.” This book could serve as a textbook or a supplementary source for such a course. Because of the widespread applicability of mathematical thinking, however, the book has been kept short and written in an engaging style, to make it accessible to anyone who seeks to extend and improve their analytic thinking skills. Going beyond a basic grasp of analytic thinking that everyone can benefit from, the STEM student who truly masters mathematical thinking will find that college-level mathematics goes from being confusing, frustrating, and at times seemingly impossible, to making sense and being hard but doable. Dr. Keith Devlin is a professional mathematician at Stanford University and the author of 31 previous books and over 80 research papers. His books have earned him many awards, including the Pythagoras Prize, the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. He is known to millions of NPR listeners as “the Math Guy” on Weekend Edition with Scott Simon. He writes a popular monthly blog “Devlin’s Angle” for the Mathematical Association of America, another blog under the name “profkeithdevlin”, and also blogs on various topics for the Huffington Post.

Genres: Mathematics, Science, Nonfiction, Stem, Logic, Textbooks, Programming

102 pages, Paperback

First published January 1, 2012

About the author

Dr. Keith Devlin is a co-founder and Executive Director of the university's H-STAR institute, a Consulting Professor in the Department of Mathematics, a co-founder of the Stanford Media X research network, and a Senior Researcher at CSLI. He is a World Economic Forum Fellow and a Fellow of the American Association for the Advancement of Science. His current research is focused on the use of different media to teach and communicate mathematics to diverse audiences. He also works on the design of information/reasoning systems for intelligence analysis. Other research interests include: theory of information, models of reasoning, applications of mathematical techniques in the study of communication, and mathematical cognition. He has written 26 books and over 80 published research articles. Recipient of the Pythagoras Prize, the Peano Prize, the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. He is "the Math Guy" on National Public Radio.

Reviews

[Matthew Dowd]

Didn't complete all of the exercises since many of them are nearly the same as the MMOC Devlin teaches through Coursera, and some of them took longer than I had time for (I'll be revisiting, this was a pretty cursory read). Renders poorly on the kindle but very instructive nonetheless.

[Brian · Rating 5 out of 5]

I've been reading this book while taking Dr. Devlin's Coursera class of the same name. I have greatly enjoyed both the textbook and the online course. If you have any interest in mathematics, I recommend both the course and this textbook.

[Olga Grytsenko · Rating 4 out of 5]

Nice read :)

[Dimitar Dimitrov · Rating 3 out of 5]

Good attempt, but lacking considering the intended audience

The first few chapters where the author defines the linguistic forms axioms, theorems etc. take and the discussions of connectors and quantifies are decent. The explanation of implication is over complicated due to avoidance of truth tables and the standard definition (even though the author already introduced both truth tables and negation) in the attempt to build intuition. There are some good parts around symbolic manipulation, but the proof sections are lacking in explanation of the mental models and oscillate between too simple and “figure it out without any answers/hints though you are learning this”. In fact there is a lot of “figure it out yourself” without actually teaching you a new way of thinking. Not the best work of this author for sure. I would recommend reading through the first few chapters and ignoring all the sections on proofs. One can then pick up the supremely awesome “Proofs: A long-form mathematics textbook” by Jay Cummings and really do the work while savoring the fun of the subject.

[Philipp · Rating 3 out of 5]

This would work very well if you're about to start a university degree that supposed you did some "higher" mathematics in high school and you need an introduction, or a refresher - you'll learn what all of those Greek symbols do (university-level notation), you'll learn about the kind of precise language you need, how to structure proofs, types of proofs, etc.

The language is the best part. It's never particularly dry or boring, Devlin knows how to structure what he wants to convey. However, most of the content I already had back in my "Mathematik Leistungskurs" [in Germany's gymnasium you can kind of "major" in a few classes so you'll have more classes and more depth] so I've had all this before - why did I read it then? Curiosity about the title, I guess. I thought there would be more "how to think", less "how to write". But you've got to start somewhere, I guess.

[Alison · Rating 3 out of 5]

This was basically a retread of the Coursera course, and while I still really really love the preface to this book with its overview of the difference between school mathematics and actual Mathematics, the details of how to come to an attitude of puzzling over math and logic problems didn't really make for a readable book.

[Sebastian · Rating 4 out of 5]

This is a book you shouldn't only read but live with it for however long it takes you grasp the content. This includes thinking deeply about almost every sentence, solve or at least try to solve every exercise, even if it costs you many hours of your life. If I follow my own advice I am far from done with the book but intend to start again right away.

[Marina · Rating 3 out of 5]

#logicalthinking
#reasoning
#mathematical thinking
#analytical skills
#math exercises
#21st century skills

[Wannia · Rating 4 out of 5]

i picked this up almost a year ago to prep for my first university courses on mathematical proof and didn't have the need to read it all the way through. but it helped introduce me to the mind-bending and attention-demanding world of pure mathematics, and for that i'll forever be grateful. as such, i decided to finish it many months later just to see how much i'd learned since then. and now i can officially mark it as read :)

ok now for the actual review lol: it is a great read if you seriously want to begin understanding maths beyond calculus, but to fully grasp a lot of the proofs will require further reading from elsewhere (which, to be fair, devlin himself does point out).

[Mayur Sinha · Rating 4 out of 5]

I read this book while taking the Coursera class of Dr. Devlin. Mathematics is not about solving abstract problems with no meaning in real life, it's about the higher-order form of thinking where you combine multiple streams of knowledge to solve a real-world problem. No one can teach you how to think mathematically, one has to learn on his own. The author recommends knowing at least algebra, logic, set theory, and real analysis to be able to be proficient in mathematical thinking but I think he should add Calculus as well. I recommended this book for high school students at least.

[Patrick Coakley · Rating 4 out of 5]

Great way to jump into mathematical concepts that many people didn't learn in school. I used it as a way to catch up on understanding some of the basics of computer science-oriented math and it helped tremendously. Includes exercises but no answers, which really makes it sort of frustrating to do them because it can feel like a waste of time and there are other meatier books with more problem sets.

[Krzysztof Wydrzyński · Rating 3 out of 5]

This book is good introduction to maths for people who are not familiar with it on university level. When I was reading it I had to skip a lot of parts which I already knew, because people after even one semester of STEM studies have already done that. Besides that, book is very resourceful and exercises are very similar to academic once.

[William Darian · Rating 5 out of 5]

A great book for introductory math course. Beside the minor errors in some math notation, the explanation was adequately expressive and the progression was just right for an undergraduate study pursuing bachelor in mathematics. The book is as dense as it could be before omitting important details.

[Christopher Byram · Rating 3 out of 5]

I first discovered this book when it was referenced in an online mathematics & logic course I am enrolled on, offered by Stanford University, entitled 'Introduction to Mathematical Thinking'. The author of this book is the course instructor, and this book is the course textbook.

[Warren Mcpherson · Rating 3 out of 5]

This book has a couple of very good insights and is fairly short.
I think it is useful for getting reacquainted with reading mathematical material.

[Alex Gomez Villa · Rating 3 out of 5]

Although the concepts and explations are pretty good I sincerely think that the book is very short. Anyway it is a must read to any engineering freshman

[sarah chang · Rating 5 out of 5]

I'm grateful for Prof. Keith Devlin. Saved my life from math anxiety inflicted by my high school math Olympics.

[Michael A · Rating 4 out of 5]

Helped a lot

I never took this course,and had a lot of lightbulb moments reading this before each lecture. That’s all I have.

[Johanna · Rating 4 out of 5]

Very witty for a mathematic book. I'm currently taking Dr. Devlin's MOOC and it was as fun as reading the book and doing the exercise. I thought this course was going to be an easy one because this is an introduction, but the energy required to think in the course is enormous.

[Minh Nhật · Rating 3 out of 5]

cuốn này là background reading theo course introduction to mathematical thinking của stanford, 1/2 đầu nói khá hay về toán, với một người lần đầu tiên tìm hiểu sâu sâu tí về toán thì thấy nhiều thứ rất thú vị.

1/2 sau tập trung vào phân tích ngôn ngữ hơi chán vì mình đã từng theo 1 lớp khác chuyên sâu và thú vị hơn là Language, Proof and Logic

The dramatic growth in mathematics led in the 1980s to the emergence of a new definition of mathematics as the science of patterns. According to this description, the mathematician identifies and analyzes abstract patterns—numerical patterns, patterns of shape, patterns of motion, patterns of behavior, voting patterns in a population, patterns of repeating chance events, and so on. Those patterns can be either real or imagined, visual or mental, static or dynamic, qualitative or quantitative, utilitarian or recreational. They can arise from the world around us, from the pursuit of science, or from the inner workings of the human mind. Different kinds of patterns give rise to different branches of mathematics. For example:

- Arithmetic and number theory study the patterns of number and counting.
- Geometry studies the patterns of shape.
- Calculus allows us to handle patterns of motion.
- Logic studies patterns of reasoning.
- Probability theory deals with patterns of chance.
- Topology studies patterns of closeness and position.
- Fractal geometry studies the self-similarity found in the natural world

[Omar · Rating 4 out of 5]

This was nice - Prof. Devlin throws in some humor, history, and down-to-earth language into this "introduction" to mathematical thinking. I'd say it certainly would work better if used in a class (e.g. a course complementary text) because although for the already initiated the proofs that are given are "basic" -- for those who are truly starting their journey into mathematical thinking it will still be tough and would benefit from the advice of a teacher (for some of the practice problems, for example).

Overall, excellent, and Devlin does not stray much from his usual approach in teaching (compared to some of his public university lectures, videos, etc).

[Thai Son · Rating 3 out of 5]

The introductory essay on math is great, I got a great deal of insights out of that. The rest was nice, but I was too bored by the time I got through the major half of the book, having learned proofs and other things beforehand. I certainly am not with the camp who thinks a proofs class should be required before taking any math-major course class (e.g. Abstract Algebra, Real Analysis, etc.) . But that is just me, coming from a rich pre-university background, having exposed myself to some interesting math. It probably won't work for people from the US public schools system.

[Corin · Rating 4 out of 5]

The first chapter of this short book is a terrific explanation of the value of math and why it matters. As a proponent of the rethinking of education in general and the needs of a changing economy in particular, I found that Devlin really understood what too many people do not. The rest of the book is good, too, but that whole first section was what really piqued my interest!

[Violet Marcelo · Rating 3 out of 5]

Most problems at the latter end of the book were too technical for readers who have no background in higher maths. And this is supposed to be 'Intro to mathematical thinking'. The outline is good to follow though, so you can take it from there.

[Alex · Rating 4 out of 5]

One of my (multi) NY resolutions is to study the math pretty much from scratch and this is a good first book - the material is pretty simple though even if you took math in college it still going to be a workout.

[Gnuehc Ecnerwal · Rating 3 out of 5]

It tends to reaffirm the status quo by suggesting ways to be better equipped to handle it. Not what I expected from Devlin, who had been a proponent for reform.

[AZ ZA]

In the present moment I'm thinking: "may be I should learn about mathematics as a hobby".

[Rohaan]

aasdasdasdasdasdddddddddddddddd

[Izhan Noorzi · Rating 4 out of 5]

It's a good book especially for non-mathematician to learn the essence of mathematics.
The author described in depth how the framework of mathematical thinking deal with every day related matters.

[Raymond Lewis · Rating 4 out of 5]

Deceivingly short.