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How to Think Like a Mathematician: A Companion to Undergraduate Mathematics

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Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. The material has been tested by real students over many years so all the essentials are covered. With over 300 exercises to help you test your progress, you'll soon learn how to think like a mathematician.

274 pages, Paperback

First published February 12, 2009

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Kevin Houston

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5 stars
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91 (34%)
3 stars
42 (15%)
2 stars
3 (1%)
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7 (2%)
Displaying 1 - 22 of 22 reviews
Profile Image for Tareq Elderawi.
6 reviews2 followers
June 11, 2016
*Don't believe everything you read .
*simplicity is the ultimate sophistication .
* If it isn't hurting , then it isn't working .
*Be scaptical of everything and try to prove that the text is wrong , search for hidden assumptions and try to use extreme cases and examples .
*Learn as much by writing as by reading .
*It isn't that they can't see the solution , It is that they can't see the problem .
*Everything is simpler that you think and at the same time more complex that you imagine .
*The highest form of pure thoughts is in mathematics .
*Don't confuse reasons which sounds good with good,sound reasons .
*The first precept was never to accept a thing as true until I knew it as such without a single doubt .
*By proving statements we can build mathematics , one statement on top of another ,
this gives real power to mathematics .
*Little by little does the trick.
*But the fact that some geniuses were laughed at does not imply that all who are laughed at are geniuses .

Profile Image for Ali.
77 reviews38 followers
November 15, 2016
This book contains very useful techniques for anyone who wants to gain a deeper understanding of mathematics, especially chapters on how to read definitions, theorems and proofs. Maybe it requires more acquaintance with mathematics to highly appreciate other advice given, but I found those mentioned the most useful ones. There is also a good treatment of main techniques of proof. Some exercises are a bit challenging for an undergraduate but overall it does not require any especial prerequisite knowledge of mathematics. The only weakness was the part devoted to logic which falls short of exploring subtle points and that is why I decided to drop one star!
Profile Image for Mark Moon.
136 reviews83 followers
June 29, 2022
Lots of good advice here for people setting out to learn undergraduate-level mathematics. The author communicates many things to the reader that I often find myself telling my students. I am likely to recommend this book to many of my students.
Profile Image for Sambasivan.
944 reviews27 followers
May 22, 2016
Written with incredible clarity. This book streamlines your thought process and helps you discipline yourself to think logically as well. Surprisingly good.
Profile Image for Maukan.
54 reviews26 followers
March 6, 2022
Theres always been this question revolved around the field of software engineers from individuals looking in from the outside which is "Do you have to be good at mathematics to be a good software engineer?".

After thinking about this for quite some time, the question is probably too vague to answer without context because what does "being good at math" mean? and What does it mean "to be a good software engineer"? I think a lot of people are intimidated by the field because of the vague connection between the two. I know I was. Let me try to answer this and then I will tell you how this ties in with this book.

First off, there are a number of fields within software development where really no math is required, for instance I know a very talented front end engineer who can build beautifully designed websites without knowing math and hating math. However, there are use cases where being formidable in mathematics ties into software development, particularly in algorithmic style questions or problems, issues where you may be trying to reduce latency, and may have to optimize the application to met certain specifications. In this context the question to be asked is "Do individuals who are more proficient at mathematics excel at software questions when it comes to code optimization as opposed to those who are not sufficient at mathematics? Is there an advantage to be had?" This would be probably clear up some of the confusion between the starting question. However let me begin to tie this into the book, one of the key reasons I enjoyed this book is because there is a lot of similarities when it comes to the type of thinking a mathematician and a software engineer engage in. They have a problem, they use another language to solve this problem, in that process, they often practice induction, reduction, reverse engineering etc to find a solution.

I think one of the reasons why I was never any good at math when I was younger (still not that good right now) is because I was always looking for a formula to plug in but in real mathematics things are not quite so plug in and out. You might have to play with the problem, trying multiple approaches, looking at the results and how they behave and deduce any patterns or rules that you can build on to solve the problem at hand. One of the reasons why I enjoyed this book was because it walked you through how to approach a problem, this type of thinking is something we all have done at one time or another in all fields but when it comes to math. Speaking to people like myself, it becomes more daunting and we become more intimidated by the problem at hand forgetting that we have performed the same type of thinking but just towards other problems. This book helped me make the connection, we all engage in some form of critical thinking (okay maybe not all) but we do not realize that the same type of thinking we engage for other problems is just not extrapolated to mathematics because of how intimidating it can be.

Thats what this books reminded me of which is why I rated it so high. Do I think this book will make you a better mathematician? Possibly, there are approaches and constructs to view mathematics problems in ways that are relatable to other problems but the main point is seeing this not as a math problem but just as a problem. I think this book delivers in that instance and although it could just be me, that point is profound if you can extrapolate it to other areas.
Profile Image for alex stanley.
15 reviews2 followers
March 4, 2022
was VERY easy to understand and should hopefully help me pass my proofs class :)
Profile Image for Fotis Koutoulakis.
75 reviews8 followers
November 16, 2021
This book is simply outstanding.

It's a book for transitioning to higher-level mathematics (more proof/concepts based than procedural based, i.e. encouraging deep understanding and engagement with the material rather than a superficial application of a calculating procedure to solve a problem).

In it you will find it all:

* Advice on how to read mathematics (with specific advice on how to engage with definitions, theorems, proofs, etc).
* Advice on writing mathematics.
* Advice on approaching higher level problems (i.e. proof tactics such as proof by contradiction, direct proof, proof by cases, etc)
* A small logic compendium (from propositional to a bit of first order logic (quantifiers), with a couple of chapters spent helping to instil a deeper understanding about implication statements (what they mean, what they do not mean, the inverse, converse and contrapositive, etc).
* General thought process/approach advice.

It is hard to state just how good this book is. I'm just going to say that it revived my love for mathematics after it had been butchered by incompetent tutors at school and at uni.
Profile Image for Dan.
124 reviews
February 16, 2023
The mathematical education I got in high school and college was excellent. But it was heavy on calculation and light on proof. And as I’ve gotten older I’ve forgotten most of it.

I’ve tried a few courses over the last decade to try to strengthen my understanding of proofs, but always stalled out. I never stalled out reading this book - it proceeded at just the right pace, and spent a lot of time at the beginning covering basic logic. The exercises were challenging but not impossible. Some were even fun. I really got a lot out of this book, and I can finally explain why the square root of 2 is irrational and that there are an infinite number of prime numbers.
Profile Image for Aleksandra Singer.
27 reviews5 followers
September 12, 2017
I found the book too basic, but it does give good study tools to people beginning to learn mathematics. However, I was often annoyed by the pompous tone and snide digs at other disciplines like history/sociology that showed ignorance about how they work. There was a lot of 'this matters in math' but you could just look up a date in history to know an event happened. That's a strawman and not how high level history works at all. That said, the math ideas are very clearly written and accurate and I could see this being very beneficial to first year undergrads.
Profile Image for Anand.
12 reviews
October 26, 2019
Should be required reading for everyone, not just people studying for a maths degree. While it's intended for an aspiring mathematician, the thought processes outlined in this book give practical advice to just about anyone. Knowing the ways that a mathematician deducts their arguments and their thought processes is a valuable life skill, and this book shows the way.

It provides good study advice too if you're just trying to do well in math, if that convinces you further.
Profile Image for Châu Nguyễn.
10 reviews
June 22, 2020
I'm going to university for mathematics in September and found this book in the university's suggested reading list. The author did a superb job in delivering his ideas! His friendly tone of voice, neat blocks of information, and tips on reading academic books efficiently certainly helped me to NOT fall asleep as I usually do.

In conclusion, this work is a neat collection of tips and foundational pure maths notes. I think it would be helpful for all mathematics students.
Profile Image for Sam.
46 reviews28 followers
October 30, 2020
Lmao the book reads like a Lecturer who got tired of his student's shit😂.

But more honestly it's a guide filled with things I wish I knew earlier, habits I wish I adopted earlier, and conventions that will make your professor's job way easier.
Profile Image for Philip.
221 reviews6 followers
December 13, 2018
Was hat mich dieses Buch gelehrt?
Besser noch: Was hat dieses Buch getan?

Also zunächst einmal hat es mit einigen grundlegenden Begriffen und Symbolen aufgeräumt. Langsam aber stetig wird man näher an die komplizierten Kernstücke der Mathematik heran geführt. Und dabei bekommt man wieder und wieder Methoden vorgestellt, wie man mit diesen Teilen arbeiten kann, wie man damit denken kann, und wie man denken sollte, um die Mathematik zu ergründen und für sich zu erobern.

Ich finde, es ist in der Tat ein grandioses Buch, um Neulingen zu helfen.

Und wie ist das bei Quereinsteigern und Autodidakten wie mir? Die vielleicht so ähnlich wie ich kein Gramm Bildung aus der Schule mitgenommen haben, sich ihren Lebensalltag durch alltagtstaugliche Logik erkämpften und für die Mathematik ein Horrorzirkus aus Auswendiglernen von schier unbegreiflichen Formeln und Tabellen war?
Also für mich war dieses Buch ein Segen. Beruhigend und ermutigend zu gleich, denn es gibt nun nichts mehr an der Mathematik, dass mich ängstigen muss. Sie lässt sich ja genauso erschließen, wie sich alle anderen großen Unbekannten im Leben erschließen lassen, wenn man denn nur will. Man muss zur Mathematik nicht geboren sein, wie ich oft den Eindruck hatte. Man braucht nur Zeit und Beharrlichkeit.
Somit schließe ich hier voller Freude diese Einführungsliteratur und freue mich schon auf meinen nächsten Schritt im Studium der Mathematik. Es wird mir in der Anfangszeit ein treuer Berater sein, zudem ich immer wieder greife. Denn auch wenn ich alle Methoden schon aus dem Hobby-Studium an Philosophie und Code-Quality kenne, so hilft die Spezialisierung der Methoden auf die Welt der Mathematik, um die großen Unbekannten in mundgerechte Happen einzuteilen.

Besten Dank an die Fernuniversität Hagen für diese Empfehlung,
und ein besonderes Lob an Houston für diese bereichendere Lektüre!
Profile Image for Avesta.
238 reviews19 followers
July 14, 2022
It can be an incredibly useful guide for getting into the frame of mind for mathematical sciences and physics - but I personally found the majority of it to be fairly trivial as a university physics student in the UK - and much of the content covered in this is also covered in A-Level further mathematics. This does not mean, however, that the majority of others won't learn a lot from it - it's incredibly informative and well written.


I Study Skills For Mathematicians

1 Sets and functions
2 Reading mathematics
3 Writing mathematics I
4 Writing mathematics II
5 How to solve problems

II How To Think Logically

6 Making a statement
7 Implications
8 Finer points concerning implications
9 Converse and equivalence
10 Quantifiers – For all and There exists
11 Complexity and negation of quantifiers
12 Examples and counterexamples
13 Summary of logic

III Definitions, Theorems and Proofs

14 Definitions, theorems and proofs
15 How to read a definition
16 How to read a theorem
17 Proof
18 How to read a proof
19 A study of Pythagoras’ Theorem
IV Techniques of Proof
20 Techniques of proof I: Direct method
21 Some common mistakes
22 Techniques of proof II : Proof by cases
23 Techniques of proof III : Contradiction
24 Techniques of proof IV: Induction
25 More sophisticated induction techniques
26 Techniques of proof V : Contrapositive method

V Mathematics That All Good Mathematicians Need

27 Divisors
28 The Euclidean Algorithm
29 Modular arithmetic
30 Injective, surjective, bijective – and a bit about infinity
31 Equivalence relations

VI Closing Remarks

32 Putting it all together
33 Generalization and specialization
34 True understanding
35 The biggest secret


A Greek alphabet
B Commonly used symbols and notation
C How to prove that . . .
12 reviews
May 5, 2019
Good book for the confused

The book is good enough for college students to get hang of how to think in mathematics, especially proofs & statements. Ample examples & no assumption by the author that only experienced math students will read it.
Profile Image for Ben G.
113 reviews2 followers
Currently reading
November 11, 2018
I bought this so I could understand the language of mathematics (e.g. theorems, proofs, etc.). I'm not a mathematician by trade, but use mathematical techniques (i.e. constructing ODE models of biological systems) in my everyday work. Thus, an absolute necessity to understand more of maths beyond the crash course I did in biomathematics at university.

Beyond chapter 1 which gives a thorough overview of sets theory and brief introductions to maps and functions. Chapter's 2 and 3 are concerned with reading and writing mathematics, however the advice is equally useful to anyone studying for school / college level qualifications in other subjects also.
1 review
August 7, 2016
Definitely worth reading even for people who don't study maths, but I find it hard to comprehend, not the mathematical bit but it's a bit dull for me and sometimes it feels like the author spent pages and pages to say about one thing, which should be a lie to explain or clarify in one chapter. Also there are many excises without solution, if you are reading it on your own, how can you know your answer is right?
Profile Image for Isaac.
41 reviews
January 12, 2016
A lot of good material explained well. Wish I had something like this when I was an undergraduate.
Displaying 1 - 22 of 22 reviews

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