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Introduction to Mathematical Thinking, by Keith Devlin
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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.
- Sales Rank: #243875 in eBooks
- Published on: 2015-02-23
- Released on: 2015-02-23
- Format: Kindle eBook
About the Author
Dr. Keith Devlin is a mathematician at Stanford University in California, where he is Executive Director of the university's H-STAR institute. 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 31 books and over 80 published research articles. His books have won a number of prizes, including the Pythagoras Prize, the Peano Prize, the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. In 2003, he was recognized by the California State Assembly for his "innovative work and longtime service in the field of mathematics and its relation to logic and linguistics." He is "the Math Guy" on National Public Radio.
Most helpful customer reviews
183 of 198 people found the following review helpful.
Changed the way I thought of Math
By DanP
This goes out as a "thank you" to Professor Devlin, but should prove informative to people who can relate to my situation,
All my life, I absolutely hated math. I mean the deep pains in my cortex, blood boiling derision of the concepts kind of hatred. I remember being 8 years old and absolutely loathing the thought of studying times tables. Then, variables were added - great, numbers and letters now! No one took the time to explain why we were studying such seemingly needless information. I understand now that I am quite the pragmatist, needing a defined practical application or goal for almost anything I do. Hence the reason I hated mindless calculations so much.
My disdain for the field grew to a point in my teenage years that I could no longer even look at the subject without my well-established biases taking over, driving me away. It affected my SAT score, with a perfect score in verbal comprehension being blemished by an embarrassingly subpar result in math. My educational experience was inhibited for almost two decades, simply because I saw no use for math past counting how many apples I wanted to buy at the grocery. After law school, I became interested in finance. From fairly light interactions with financial valuation methods grew a shameful realization that my hatred for math had caught up with me, finally affecting my daily life. I was a quantitative infant in the worst way, entirely handicapped in a vast and important arena.
Then I found Coursera, and through its curriculum I came across this course. Though I have not finished it, as it has just started, the book has opened my eyes. Math is a language, just like the ones we speak and master. Its algorithms are logical thought, its concepts proven by deductive and inductive reasoning. All you need is a good handle on core concepts and, if you are like myself, a tangible application (even if its somewhat metaphorical or anodyne), and you will be able to appreciate what Professor Devlin has done with this course.
If you are willing and able to learn past the age of 20, hated math your entire existence but are finding useful applications for it as you age, then this book and the corresponding course are entirely for you.
Thank you, Professor Devlin, for doing what no teacher was able to for decades - finally get through to a smart kid who just lacked the appropriate impetus to apply himself.
60 of 66 people found the following review helpful.
Thinking like a mathmatician
By Tim Crawford
Keith Devlin's book "Introduction to mathematical thinking." is the textbook, the unnecessary and ridiculously inexpensive ($10) textbook, for his class on Coursera.com. The class obviously designed to be an introduction to mathematical thinking, a transition from the problem solving math of secondary school to college level mathematics where simply finding an answer is not the final goal.
I wanted to take a look at the free, non-credit classes that Coursera offers and this looked like a good one to try. It has been 30 years since I last took college calculus, and I have not looked at a math book since then. I knew I could do the work, I wanted to see just how a free, non-credit class, with 50,000 students worked.
Both the class and the book are excellent. Devlin begins by showing us that imprecision is often acceptable in spoken English. "One American dies every hour from heart disease" is his favorite example. Literally it says that there is one single American who dies, and apparently recovers, from heart disease every hour. We all understand the true meaning because in English we have background knowledge which allows us to make sense from nonsense based on the context. Mathematics requires precision because with it we will be dealing with concepts with which we do not have the background to guide our understanding.
Dr. Devlin focused on developing logical thinking and managed to arrange the lessons and exercises such that the mathematical logic required quickly evolves from simple "and" "or" statements into doing formal proofs, no small feat for a class only seven weeks long. The book was not really necessary for the class, the video lectures were very close to the text and the problem sets could be printed out and worked on offline but I do feel that having the book helped me. I find it impossible to highlight a point in a video lecture.
I was concerned about how a class this large could be taught, in all the math classes I have taken the learning takes place not during the lecture but answering questions that come up after attempting to work problems related to the lecture. Advanced students were recruited to act as Teaching Assistants to keep an eye on the discussion threads and answer questions when they could. Threads that had heavy traffic were brought to professor Devlin's attention and he, most often, just confirmed what the TA's had said. Learning directly from the book is possible but would mean losing what for me was the most helpful part of the class, the forums.
I learned a lot taking the class, principally that you can forget a lot in thirty years. I was over a week behind in the work and struggling not to fall farther behind when it came time for the final exam. Needless to say I will not be getting a certificate of completion. I also learned several other things, Coursera classes really are college level, someone only a few years out of high school would have had to work to finish the class and would have been prepared for higher mathematics classes and would be better able to think logically.
It was expected that out of the over 50,000 students that started the class as few as 5,000 would successfully finish it but as Dr. Devlin pointed out, teaching 5,000 students in traditional classes of 25 students would represent one professor's entire career. This one class introductory level class freed up one career to teach higher level classes.
45 of 54 people found the following review helpful.
Good topic, but not well explained
By Quantstyle
This topic is very important and I appreciate Dr. Devlin taking the time to focus on what can be considered the glue that holds the other pieces together. However, the examples are few (most of the time just one), and not articulated well. When he explains the bi-conditional , he explains how he proved that A => B (A implies B), but pulls out a completely unexplained argument for why B => A, and concludes that's how you prove it. And because it's in the middle of the book, you can't go on without understanding it thoroughly.
I am probably missing something simple, but again, the examples are few in numbers (most of the time one), and the logic is not well articulated.
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