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Calculus Reordered: A History of the Big Ideas
How our understanding of calculus has evolved over more than three centuries, how this has shaped the way it is taught in the classroom, and why calculus pedagogy needs to change
Calculus Reordered takes readers on a remarkable journey through hundreds of years to tell the story of how calculus evolved into the subject we know today. David Bressoud explains why calculus is credited to seventeenth-century figures Isaac Newton and Gottfried Leibniz, and how its current structure is based on developments that arose in the nineteenth century. Bressoud argues that a pedagogy informed by the historical development of calculus represents a sounder way for students to learn this fascinating area of mathematics.
Delving into calculus’s birth in the Hellenistic Eastern Mediterranean―particularly in Syracuse, Sicily and Alexandria, Egypt―as well as India and the Islamic Middle East, Bressoud considers how calculus developed in response to essential questions emerging from engineering and astronomy. He looks at how Newton and Leibniz built their work on a flurry of activity that occurred throughout Europe, and how Italian philosophers such as Galileo Galilei played a particularly important role. In describing calculus’s evolution, Bressoud reveals problems with the standard ordering of its limits, differentiation, integration, and series. He contends that the historical order―integration as accumulation, then differentiation as ratios of change, series as sequences of partial sums, and finally limits as they arise from the algebra of inequalities―makes more sense in the classroom environment.
Exploring the motivations behind calculus’s discovery, Calculus Reordered highlights how this essential tool of mathematics came to be.
Calculus Reordered takes readers on a remarkable journey through hundreds of years to tell the story of how calculus evolved into the subject we know today. David Bressoud explains why calculus is credited to seventeenth-century figures Isaac Newton and Gottfried Leibniz, and how its current structure is based on developments that arose in the nineteenth century. Bressoud argues that a pedagogy informed by the historical development of calculus represents a sounder way for students to learn this fascinating area of mathematics.
Delving into calculus’s birth in the Hellenistic Eastern Mediterranean―particularly in Syracuse, Sicily and Alexandria, Egypt―as well as India and the Islamic Middle East, Bressoud considers how calculus developed in response to essential questions emerging from engineering and astronomy. He looks at how Newton and Leibniz built their work on a flurry of activity that occurred throughout Europe, and how Italian philosophers such as Galileo Galilei played a particularly important role. In describing calculus’s evolution, Bressoud reveals problems with the standard ordering of its limits, differentiation, integration, and series. He contends that the historical order―integration as accumulation, then differentiation as ratios of change, series as sequences of partial sums, and finally limits as they arise from the algebra of inequalities―makes more sense in the classroom environment.
Exploring the motivations behind calculus’s discovery, Calculus Reordered highlights how this essential tool of mathematics came to be.
242 pages, Hardcover
First published July 16, 2019
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Displaying 1 - 9 of 9 reviews
April 11, 2024
The authors draws our attention to a problem in the teaching of calculus: it's taught almost in reverse to how it was invented.
Examples from the book: among the concepts in calculus, integration was the first to be discovered and most intuitive. It has to do with the computation of areas and volumes and is easy to think about as computing by accumulation, i.e. adding together small pieces of a thing to find its size. Limits is the last thing to be understood by mathematicians.
In teaching calculus, however, it was done in reverse: limits comes up first, because formal maths teaching begins with definition.
The author's argument really struck a chord with me as I remember having almost exactly this conversation with my college math professor when I was an undergrad. What really bothered me was how we as students were presently a cleaned-up view of what math is. Hiding from us was the fact that, many of these ideas, and even notations, were confusing to the inventors and discoverers too. It took very smart mathematicians many years, often generations, to figure out what they were dealing with. I felt this knowledge alone would have helped me as a student.
More importantly, in the standard way of learning math, I wasn't introduced to the motivations that drove mathematicians to invent these ideas and tools. As the author points out in the book, often they had real, practical problems that needed solving.
This is a short book. The math examples given are easy to follow and intuitive.
Examples from the book: among the concepts in calculus, integration was the first to be discovered and most intuitive. It has to do with the computation of areas and volumes and is easy to think about as computing by accumulation, i.e. adding together small pieces of a thing to find its size. Limits is the last thing to be understood by mathematicians.
In teaching calculus, however, it was done in reverse: limits comes up first, because formal maths teaching begins with definition.
The author's argument really struck a chord with me as I remember having almost exactly this conversation with my college math professor when I was an undergrad. What really bothered me was how we as students were presently a cleaned-up view of what math is. Hiding from us was the fact that, many of these ideas, and even notations, were confusing to the inventors and discoverers too. It took very smart mathematicians many years, often generations, to figure out what they were dealing with. I felt this knowledge alone would have helped me as a student.
More importantly, in the standard way of learning math, I wasn't introduced to the motivations that drove mathematicians to invent these ideas and tools. As the author points out in the book, often they had real, practical problems that needed solving.
This is a short book. The math examples given are easy to follow and intuitive.
July 22, 2024
A refreshing retelling of calculus that gives an appreciation and deeper understanding through historical context of the concepts that when taught in traditional classrooms felt rushed and unintuitive.
Will definitely be coming back to this book for years to come, and recommending this to anyone in the college calculus slog.
Will definitely be coming back to this book for years to come, and recommending this to anyone in the college calculus slog.
January 26, 2020
The author is clearly extremely knowledgeable and an accomplished mathematician, sharing mathematical and historical insights spanning 2500 years. The book does an excellent job discussing mathematical innovations with a global perspective (highlighting key examples from India and China) while bringing a strong rationale for why the 1600s and beyond typically emphasize developments arising in Europe.
Highlights:
Having completed my undergrad in math many years ago, and now teaching AP Calculus for 6 years, I enjoyed the mission of the book which is to reimagine a Calculus course taught in historical order that emphasizes contextual understanding. This is a novel thought and potentually influential text in how to enhance math education for future high school students. I myself will definitely be leveraging many of these ideas into my own classroom.
The chapter on Analysis was a joy. I had fond memories of my Intro to Real Analysis course upon reading this chapter, and thought the author did a wonderful job in building it up as a tangentially related field to Calculus, yet an entirely different beast!
Additionally, the chapter on series is fantastic. Anytime I am prompted to follow through the thoughts of Euler and Archimedes, I get goosebumps. The legends of their character, historical contexts, and mastery of mathematics is a necessary inclusion for truly understanding math as a body of knowledge. The author brings them to life in stellar fashion.
Why 4 stars? My quarrels with the book are listed below:
Due to the vastness of the body of knowledge that is Calculus and its historical contexts spanning 2500 years, the author feels forced to be overly brief in moments that truly need more thorough explanations. Often, key mathematical steps are skipped over. I found myself having to stop reading and take out a pencil and scratch paper in order to fill in the gaps and enhance my own understanding of what was happening.
The author sometimes chooses to use verbal language instead of the beautifully concise mathematical notation we have at out disposal in the 21st century. To be fair, this may have been an artistic choice to highlight the lack of modern math notation until Euler, however the net result is overly verbose paragraphs with diminished clarity. To anyone who has read accumulation problems knows, what one integral represents would take a paragraph to verbalize. The author very often chooses the paragraph option instead of the concise and clear symbolic language option.
Highlights:
Having completed my undergrad in math many years ago, and now teaching AP Calculus for 6 years, I enjoyed the mission of the book which is to reimagine a Calculus course taught in historical order that emphasizes contextual understanding. This is a novel thought and potentually influential text in how to enhance math education for future high school students. I myself will definitely be leveraging many of these ideas into my own classroom.
The chapter on Analysis was a joy. I had fond memories of my Intro to Real Analysis course upon reading this chapter, and thought the author did a wonderful job in building it up as a tangentially related field to Calculus, yet an entirely different beast!
Additionally, the chapter on series is fantastic. Anytime I am prompted to follow through the thoughts of Euler and Archimedes, I get goosebumps. The legends of their character, historical contexts, and mastery of mathematics is a necessary inclusion for truly understanding math as a body of knowledge. The author brings them to life in stellar fashion.
Why 4 stars? My quarrels with the book are listed below:
Due to the vastness of the body of knowledge that is Calculus and its historical contexts spanning 2500 years, the author feels forced to be overly brief in moments that truly need more thorough explanations. Often, key mathematical steps are skipped over. I found myself having to stop reading and take out a pencil and scratch paper in order to fill in the gaps and enhance my own understanding of what was happening.
The author sometimes chooses to use verbal language instead of the beautifully concise mathematical notation we have at out disposal in the 21st century. To be fair, this may have been an artistic choice to highlight the lack of modern math notation until Euler, however the net result is overly verbose paragraphs with diminished clarity. To anyone who has read accumulation problems knows, what one integral represents would take a paragraph to verbalize. The author very often chooses the paragraph option instead of the concise and clear symbolic language option.
March 23, 2024
I'll admit that I skipped most of the math parts in this one. I could generally get the gist of it but rarely followed through the proofs. That being said, I still enjoyed it, seeing all the connections between famous mathematicians was incredibly rewarding. I felt though that a sort of visual representation of all of these connections would have helped tremendously, especially one that would sort of build up as you go through the book.
It did make me think of the education that I got in high school and even university, where the focus often was on learning symbolic manipulations and not on truly understanding the fundamentals. Also, very often, the real-world applications of the math we were learning weren't even mentioned and it felt like we were learning it for the sake of learning it. This book does a good job of reminding us that these concepts very often appeared as a consequence of real-life problems that needed solving.
It did make me think of the education that I got in high school and even university, where the focus often was on learning symbolic manipulations and not on truly understanding the fundamentals. Also, very often, the real-world applications of the math we were learning weren't even mentioned and it felt like we were learning it for the sake of learning it. This book does a good job of reminding us that these concepts very often appeared as a consequence of real-life problems that needed solving.
This entire review has been hidden because of spoilers.
May 24, 2023
I came all the way to finishing the book after being introduced to it in one of the Calculus History Explained video on utube. While I started reading it a few months back, I quickly lost interest due to other commitments but decided to stick along regardless as I wanted to fulfill this promise that I've made to myself.
I took calculus BC in high school so the mathy content still triggers some of my memories (not all parts rings bells though...) so really instead of a maths book I treated Calculus Reordered as more of a history book. With that mindset immediately I felt like the stories and the descriptions made sense to me. And I appreciate the author for giving me valuable insight on how there are actually more than just Newton and Leibniz in the course of developing modern calculus.
4.25/5 (rounded to 4)
I took calculus BC in high school so the mathy content still triggers some of my memories (not all parts rings bells though...) so really instead of a maths book I treated Calculus Reordered as more of a history book. With that mindset immediately I felt like the stories and the descriptions made sense to me. And I appreciate the author for giving me valuable insight on how there are actually more than just Newton and Leibniz in the course of developing modern calculus.
4.25/5 (rounded to 4)
August 5, 2025
Many of us learn math in a very matter-of-fact fashion, with formulas handed down from on high divorced from any context beyond a vague increase in difficulty. This book accomplishes something very impressive with a subject that many find dull, reminding us that mathematics is a science with major discoveries being made across history. It is so refreshing to read about math as a mystery with a storied past, and to put names and faces with familiar formulas. This book puts the spotlight on the development of calculus, beginning with the thesis that most students learn calculus concepts in the reverse order that they were discovered. By “reordering” these concepts, the author makes it easier to fundamentally understand them.
August 22, 2021
Bressoud has yet again done a wonderful job presenting the development of calculus and analysis *in context,* an extremely valuable pedagogical contribution. This book shows that the logical presentation of calculus in most courses is backwards from how the subject was discovered. For students to truly grasp the power of calculus, the discovery-based approach of walking through the organic steps of its conception, not its retrospective logical organization, provides a much more powerful lens into the beauty and subtleties of the subject. Thank you to Bressoud for providing yet another such book, it will influence my teaching!
August 21, 2025
A satisfying summary of the history of calculus, motivating the big ideas in their original context -- long enough to cover the ground, but short enough to be approachable. One wonders how much of the academic landscape would have to change to allow us to adopt, on a large scale, the pedagogy for which Bressoud advocates.
December 25, 2023
As a calc geek, I LOVEDDDD this book!!!! The author explains concepts in very digestible ways (:
Displaying 1 - 9 of 9 reviews