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Arithmetic
Because evolution endowed humans with a complement of ten fingers, a grouping size of ten seems natural to us, perhaps even ideal. But from the perspective of mathematics, groupings of ten are arbitrary, and can have serious shortcomings. Twelve would be better for divisibility, and eight is smaller and well suited to repeated halving. Grouping by two, as in binary code, has turned out to have its own remarkable advantages.
Paul Lockhart reveals arithmetic not as the rote manipulation of numbers--a practical if mundane branch of knowledge best suited for balancing a checkbook or filling out tax forms--but as a set of ideas that exhibit the fascinating and sometimes surprising behaviors usually reserved for higher branches of mathematics. The essence of arithmetic is the skillful arrangement of numerical information for ease of communication and comparison, an elegant intellectual craft that arises from our desire to count, add to, take away from, divide up, and multiply quantities of important things. Over centuries, humans devised a variety of strategies for representing and using numerical information, from beads and tally marks to adding machines and computers. Lockhart explores the philosophical and aesthetic nature of counting and of different number systems, both Western and non-Western, weighing the pluses and minuses of each.
A passionate, entertaining survey of foundational ideas and methods, Arithmetic invites readers to experience the profound and simple beauty of its subject through the eyes of a modern research mathematician.
Paul Lockhart reveals arithmetic not as the rote manipulation of numbers--a practical if mundane branch of knowledge best suited for balancing a checkbook or filling out tax forms--but as a set of ideas that exhibit the fascinating and sometimes surprising behaviors usually reserved for higher branches of mathematics. The essence of arithmetic is the skillful arrangement of numerical information for ease of communication and comparison, an elegant intellectual craft that arises from our desire to count, add to, take away from, divide up, and multiply quantities of important things. Over centuries, humans devised a variety of strategies for representing and using numerical information, from beads and tally marks to adding machines and computers. Lockhart explores the philosophical and aesthetic nature of counting and of different number systems, both Western and non-Western, weighing the pluses and minuses of each.
A passionate, entertaining survey of foundational ideas and methods, Arithmetic invites readers to experience the profound and simple beauty of its subject through the eyes of a modern research mathematician.
224 pages, Hardcover
First published August 21, 2017
61 people are currently reading
978 people want to read
About the author
Paul Lockhart
4Ìýbooks198ÌýfollowersPaul Lockhart became interested in mathematics when he was 14 (outside the classroom, he points out). He dropped out of college after one semester to devote himself exclusively to math. Based on his own research he was admitted to Columbia, received a PhD, and has taught at major universities, including Brown University and UC Santa Cruz. Since 2000 he has dedicated himself to "subversively" teaching grade-school math at St. Ann's School in Brooklyn, New York.
Librarian Note: There is more than one author in the Å·±¦ÓéÀÖ database with this name.
Librarian Note: There is more than one author in the Å·±¦ÓéÀÖ database with this name.
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Displaying 1 - 29 of 38 reviews
January 31, 2019
Even though arithmetic forms only a small subset of mathematics (when you say something about math, it causes no harm using some of its parlance such as subsets), for many people it is the science of mathematics. And, it is the subject that scares away children from appreciating the art and experiencing the joy of doing math. It is forced learning by rote and mechanical procedures for performing arithmetical operations without stopping a moment to perceive what is going on that repels so many children away from it. Clearly, a new paradigm is needed to teach math in elementary schools, making it a participatory experience for the students. However, the sad fact is that most of the teachers were also churned out by the system of memorizing of multiplication tables and doing long division by carefully choreographed procedures which keeps you on tenterhooks until the final result appears. This book is a welcome change to the genre of popular math. It teaches a teacher of math how to teach it to young boys and girls and to make a generation quite confident of dealing with it. The methods and structures detailed in the book are highly oriented to practical learning, while clearly explaining the theoretical aspects in an easy to follow way. Paul Lockhart teaches math at St. Ann’s School in Brooklyn, New York. He is the quintessential teacher who renounced his prestigious career in the university to teach in a grade school. He is the author of two more books on learning math.
Arithmetic is the skillful arrangement of numerical information for ease of communication and comparison. Lockhart makes it categorically clear that being good at it does not make you particularly smart or mathematically inclined. It is like any other craft you can get good at if you wanted to, but it is no big deal either way, assures the author. The book gives an excellent introduction to the two main schemes of arranging numerals � the marked-value and place-value systems. It starts right from the activity of counting where you take stock of something by assigning a numeral to denote something in the real world. Actual examples from the Egyptian, Roman, Chinese and Indian systems are given. It is the place-value system developed in ancient India that revolutionized the way the world did its calculations. Invention of the symbol and concept of zero as a place holder to denote a null value was a novel concept, without which the explosion in information exchange wouldn’t have come into being. The Indian numerical system reached Europe in the thirteenth century through Arab traders. Leonardo of Pisa � better known as Fibonacci � showcased the new ideas in his book Liber Abaci (Book of the Abacus). Strangely, such a fool proof system was slow to catch on with the general public in Europe. Such is the aversion of people to change! Even as late as the eighteenth century, well-educated adults found it confusing and overly technical. Eventually, convenience and increasing availability of inexpensive paper won out over the traditional Roman numerals.
The book is an attempt right from the first page to the last to kindle the spirit of appreciation for mathematical beauty among the readers. In case anyone still ‘feared� numbers, Lockhart encourages them with sage advice that mathematical operations are only good strategies for encoding and manipulating numerical information, and we can use them in any way we see fit. Instead of thinking in terms of systems and rules, we should think of it more as options and tools at our disposal (p.73). The entire gamut of basic mathematical operations like addition, subtraction, multiplication and division are covered in detail and provide new insight into the heart of the problem. Even those who are well-versed in math would learn one or two new ideas from this impressive book. Lockhart also attempts to detach numbers and its representations from its homologues in the real world. It is not always possible to find a related process happening around us. Math or arithmetic is abstract, making reification � the process of searching for counterparts in the real world � extremely difficult and sometimes impossible. Whether or not it is physically possible, arithmetic imagine or invent some sort of logically coherent structures and forms a part of mathematical reality. This disclaimer is issued in the context of negative numbers which can’t be compared readily to any real thing. But it provides an excellent tool for making calculations to arrive at the result that has real significance. Even though Lockhart doesn’t mention it in this book, complex numbers � which are square roots of negative numbers � are another realm of arithmetic which has no physical meaning, but is immensely useful in calculating and comparing real world data.
There is nothing to be said against the book, except its small type size and the absence of categorization into distinct chapters. The entire volume appears to be divided into several topics of varying lengths, more akin to an encyclopedia rather than a normal work of learning. Even though the focus is on aspiring teachers, the author has cleverly included several practice questions without making it appear as an exercise. The chapter on mechanical calculating machines could’ve been eliminated, as it proves a redundant chapter that doesn’t contribute anything to the general thread of argument. This book is an excellent choice for anyone who has set his heart on becoming a teacher of elementary schools, which will surely help them become an arithmetician of wit and intelligence.
The book is highly recommended.
Arithmetic is the skillful arrangement of numerical information for ease of communication and comparison. Lockhart makes it categorically clear that being good at it does not make you particularly smart or mathematically inclined. It is like any other craft you can get good at if you wanted to, but it is no big deal either way, assures the author. The book gives an excellent introduction to the two main schemes of arranging numerals � the marked-value and place-value systems. It starts right from the activity of counting where you take stock of something by assigning a numeral to denote something in the real world. Actual examples from the Egyptian, Roman, Chinese and Indian systems are given. It is the place-value system developed in ancient India that revolutionized the way the world did its calculations. Invention of the symbol and concept of zero as a place holder to denote a null value was a novel concept, without which the explosion in information exchange wouldn’t have come into being. The Indian numerical system reached Europe in the thirteenth century through Arab traders. Leonardo of Pisa � better known as Fibonacci � showcased the new ideas in his book Liber Abaci (Book of the Abacus). Strangely, such a fool proof system was slow to catch on with the general public in Europe. Such is the aversion of people to change! Even as late as the eighteenth century, well-educated adults found it confusing and overly technical. Eventually, convenience and increasing availability of inexpensive paper won out over the traditional Roman numerals.
The book is an attempt right from the first page to the last to kindle the spirit of appreciation for mathematical beauty among the readers. In case anyone still ‘feared� numbers, Lockhart encourages them with sage advice that mathematical operations are only good strategies for encoding and manipulating numerical information, and we can use them in any way we see fit. Instead of thinking in terms of systems and rules, we should think of it more as options and tools at our disposal (p.73). The entire gamut of basic mathematical operations like addition, subtraction, multiplication and division are covered in detail and provide new insight into the heart of the problem. Even those who are well-versed in math would learn one or two new ideas from this impressive book. Lockhart also attempts to detach numbers and its representations from its homologues in the real world. It is not always possible to find a related process happening around us. Math or arithmetic is abstract, making reification � the process of searching for counterparts in the real world � extremely difficult and sometimes impossible. Whether or not it is physically possible, arithmetic imagine or invent some sort of logically coherent structures and forms a part of mathematical reality. This disclaimer is issued in the context of negative numbers which can’t be compared readily to any real thing. But it provides an excellent tool for making calculations to arrive at the result that has real significance. Even though Lockhart doesn’t mention it in this book, complex numbers � which are square roots of negative numbers � are another realm of arithmetic which has no physical meaning, but is immensely useful in calculating and comparing real world data.
There is nothing to be said against the book, except its small type size and the absence of categorization into distinct chapters. The entire volume appears to be divided into several topics of varying lengths, more akin to an encyclopedia rather than a normal work of learning. Even though the focus is on aspiring teachers, the author has cleverly included several practice questions without making it appear as an exercise. The chapter on mechanical calculating machines could’ve been eliminated, as it proves a redundant chapter that doesn’t contribute anything to the general thread of argument. This book is an excellent choice for anyone who has set his heart on becoming a teacher of elementary schools, which will surely help them become an arithmetician of wit and intelligence.
The book is highly recommended.
December 6, 2017
This is not a book for mathematicians. If you’ve never pondered the wonders of different number bases, then you may find this very interesting. It is well-written and there is a wry sense of humor throughout, but don’t look for any earth-shattering revelations if you were already a math fan before reading.
December 27, 2017
Loving exploration of numbers and basic concepts. Starts with symbols and origins, tally marks and grouping, then moves through other operations. Chapters on mechanical aids (abacus, soroban, odometers and calculators) are particularly interesting. Nice illustrations!
Would be an excellent reader for latter elementary school; focuses more on concepts of arithmetic than rote practice. Italicized "problems" are scattered through the text, providing more examples of the concept at hand. Later chapters gloss over commutative and associative aspects, eventually using them to examine negative numbers. This felt a bit rushed, and didn't go far enough into the inverse of multiplication (as one example).
Would be an excellent reader for latter elementary school; focuses more on concepts of arithmetic than rote practice. Italicized "problems" are scattered through the text, providing more examples of the concept at hand. Later chapters gloss over commutative and associative aspects, eventually using them to examine negative numbers. This felt a bit rushed, and didn't go far enough into the inverse of multiplication (as one example).
August 4, 2024
Excellent! It should not have taken me so long to read this, but I enjoyed the small bites here and there. I took the time to do many of the problems to get the full experience. Lockhart has the right idea: teaching math as play. Definitely recommending this for anyone wanting to see some beauty in even the simplest math.
July 9, 2021
This book might seem super basic for some, but it gave me a new perspective to look at arithmetic.
It is mix bag of history, evolution and art of arithmetic. It breaks down the most basic, taken-for-granted concepts, and pushes you to look at all those concepts with new lens.
It is definitely something worth reading.
It is mix bag of history, evolution and art of arithmetic. It breaks down the most basic, taken-for-granted concepts, and pushes you to look at all those concepts with new lens.
It is definitely something worth reading.
January 4, 2022
I really enjoyed the portions dealing with the history of mathematics. It's always bothered me that the abbreviation for pound is lb. Now I know why.
After hitting my head against a wall enough times, counting with different bases usually makes sense, and most of his counting tips were interesting. The book warbled weirdly between seeming too easy and making my head hurt.
Anyway, here we are a month later, and I actually might use those factorial counting tips. Not a bad investment.
After hitting my head against a wall enough times, counting with different bases usually makes sense, and most of his counting tips were interesting. The book warbled weirdly between seeming too easy and making my head hurt.
Anyway, here we are a month later, and I actually might use those factorial counting tips. Not a bad investment.
October 1, 2024
We really should be centering our early math education on this book. This was incredibly interesting, witty, relatable and just made certain math concepts make sense. Really I can’t say enough about how good this book is
May 5, 2022
don't think i'm smart enough for it, sorry
November 14, 2021
Some interesting thoughts, but I was bored by the end and barely skimmed the last two chapters. I’m not sure who Lockhart’s target audience was: people who already like math? who don’t? young people or adults? I like math and numbers, but didn’t love the book, so who knows.
January 10, 2025
I don't know what I was really expecting when I bought this book, but I was a little disappointed. It felt very pop sciency in the sense that I left with a smorgasbord of fun facts to tell other people when I'm out of ways to start a conversation as opposed to having actually learned any new techniques I can apply. I liked the discussions about marked value versus place value systems, the creation of additional subgroups by the Chinese, and how the Hindu system functioned as its own abacus. I disliked how half the book is literally just overexplained example problems of some of the most basic math I have seen in my entire life. I think my main issue was that when I bought this book I was expecting a more in depth look at the math surrounding arithmetic itself rather than a blowhard rehashing the same shit I learned in first grade. I found his opinion that the invention of zero was not groundbreaking to be plain dumb. Also when he said that people who decide to write a number down in the next column in order to avoid having to carry digits by memory might as well just use a tabula or count coins I was pretty taken aback. How can you develop a superiority complex about such simple math? Overall, I would not recommend this book to anyone who is interested in math. However, if you are one of the three diehard math history buffs then look no further, this book is for you.
May 30, 2024
The author, a mathematician, explains the concepts of arithmetic (numbering, addition, subtraction, multiplication, division, fraction, negative numbers, permutation, and combination) in a way that focuses on understanding of ideas rather than the algorithm for problem solving. I really appreciated how he challenged his readers to think of numbers outside the decimal system and re-present them in three's, four's, and six's. He explained the origin of pounds as "lb" and the British money system, which was something I never fully understood but always wanted to know. I also appreciate his presentation of negative numbers as a an imaginary "negation/canceling/anti-" of actual numbers and clarifying that these concepts do not exist in the real world lol The chapter on permutation and combination is fun too, like a series of brainteasers.
March 22, 2018
Numerical representation has changed across human civilizations but is unified by an ultimate goal of counting numbers of different things. The book provides a witty description of different numerical systems from some of the history's greatest civilizations. In this sense, the book connects something as seemingly trivial as counting to the forces that drove the evolution of arithmetic.
The second half of the book dived into the meaning behind basic arithmetic like division and multiplication and covered the invention of machines to perform these functions.
This was a quick read, the authors light hearted writing makes the book approachable but I was hoping for more depth/philosophy/history rather than the sometimes overly detailed mathematical descriptions.
The second half of the book dived into the meaning behind basic arithmetic like division and multiplication and covered the invention of machines to perform these functions.
This was a quick read, the authors light hearted writing makes the book approachable but I was hoping for more depth/philosophy/history rather than the sometimes overly detailed mathematical descriptions.
June 13, 2020
This nice little book is another of those enlightning works by Paul Lockhart that are written for the general public but are well deserving a readership as wide as possible. Although, to my taste, his previous book "Measurement" is far more interesting, this one is also a very well written and precious introduction to central parts of elementary Mathematics: the rational numbers, the elementar arithmetic operations, and the art of and counting. The slow construction and explanation of the algorithms for the four arithmetic operations is particularly interesting. Both this book and Measurement should be attentively read by those thinking to become Elementary School teachers, and also by those already in the profession.
November 16, 2017
Lockhart has a great writing style and voice, and this book offers many insights into how the math we learned in school works, as well as great thought experiments that can change the way you think about math, numbers, and counting. I highly recommend this book for teachers who are going to need to explain basic arithmetic to students or be flexible enough to see if a student is understanding the material in a different way than the teacher is used to explaining it.
It's been long enough since I did math that a few sections were technical enough I lost interest, but that's less important than the perspective shifts this book can offer.
It's been long enough since I did math that a few sections were technical enough I lost interest, but that's less important than the perspective shifts this book can offer.
January 2, 2018
I really loved the early sections and exercises on different counting systems. There was something exciting about seeing the pros and cons of each and learning to translate between them without going through base 10 Arabic numerals. I stayed interested through the basic arithmetic operations, but really lost interest as he went off into combinatorics and probability� I no longer felt I was learning or stretching my brain.
June 23, 2020
Great voice, encouraging the fun of mathematical thinking and fuller understanding of the premises behind arithmetic. Encourages the reader to stretch beyond base ten and other conventions in which she's embedded. The opening chapters on theoretical and historical counting systems and the final chapter on counting problems that can be described by patterns and thus equations are the most enjoyable. Enjoyed reading it with my eleven year old, but it's great for adults, too.
January 1, 2022
This is a great book for a number theory discussion. I know most teachers don't have the option to create such a class, but in a world where math was not kill-by-drill, this would be a textbook. I plan on giving it to high schoolers at my Classical Christian school. Very enjoyable. Great for discussion. Well written and fun. Leaves things out in just the right amount to lead to classroom conversations and interaction. You might need to buy some abacuses.
July 25, 2018
Imagine a world where humans have first encountered the need to count things and compare amounts of things. This is the way Paul Lockhart takes the reader through a history of math that is both enlightening and engaging. For anyone who thinks they are not a "math person" this book will have you rethinking that assessment.
April 5, 2022
Amazing. I really detested math, and still somewhat do 😅, but reading this has given me another prespective of mathematics. This also gives a lot of insight on how we came to use numbers the way we do, and the different methods of calculating them and ease our minds of stresful work with the beautiful art of math.
May 1, 2025
Lockhart didn't fail to dispense his lucid intuitions on this read. I got to pick up new tricks and techniques and understandings on a subject as old as Arithmetic with its transformations throughout the ages, cultures and histories. I wish I read this in Primary School; my passion for mathematics would have been inflamed greatly then. Still, it's never too late to begin new quests.
May 17, 2018
Okay, but not as cute as the author thinks it is. It was fun to read for awhile, but in the end, it just was really slow moving. It reminded me of taking math in school, where I could see where they were going, and just wanted them to hurry up and get there.
March 18, 2020
The early sections about grouping systems, that number systems are solution to the perception problem are especially interesting, by which I meant bringing up new perspectives. I tend to rush through and didn't find them as enjoyable and insightful as the early part.
June 30, 2024
Witty, informative, and enjoyable! This is a great refresher on all sorts of the fundamentals of counting and where numbers and number systems come from. It even reminded me how to use permutations and combinations!
December 2, 2017
Book talks about very basic math but in a carefully crafted English language. Just a nice book.
January 11, 2018
The author explains Arithmetic and how it came to existence and how it evolved in a very easy to understand way.
February 15, 2018
A good book that gives a bit of a historical perspective to the evolution of counting and clearly indicated the shortcomings of the earlier counting systems
August 30, 2018
This book has nice conversational tone.
August 26, 2019
Never read a book written with more passion for the subject!
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