Active Calculus 2.0
Matt Boelkins, Grand Valley State University
David Austin, Grand Valley State University
Steve Schlicker, Grand Valley State University
Pub Date: 2017
ISBN 13: 978-1-9742068-4-1
Publisher: Grand Valley State University
Conditions of Use
This text contains all of the core ideas that I would include in Calculus I & II. It is not trying to be a comprehensive tome, which is for the best, read more
This text contains all of the core ideas that I would include in Calculus I & II. It is not trying to be a comprehensive tome, which is for the best, especially because it allows for a text that is readable for learners of Calculus, which is the stated purpose.
Most of Calculus is in the canon at this point, so there is little controversy about the concepts. However, many of the root concepts, such as limits, are extremely subtle and are not usually discussed accurately until Real Analysis courses. I still like to talk with students about these subtleties, but I think there is almost no chance that they would be able to do that work by reading a written text (and I like to have them reinvent it anyway), so it's appropriate for limits and other similar-level concepts to be defined more heuristically, as they are in this text. Like most mathematics texts, there is a Platonic bias
As above, the Calculus canon seems quite stable at this time, so I expect that this text will have great longevity. This text has been revised successfully, demonstrating that such work is sufficiently straightforward when desired. Movements like integration-first, series-first, project-based, integrated STE(A)M, or community- and culturally-relevant approaches to Calculus would require large changes for this text, but that is true of almost every Calculus textbook. Moreover, I think it will be a very long time before there is not a sector of Calculus courses that would still want a text like this. Conversely, this text is much better aligned with active learning pedagogies than texts like Stewart's, so calls to incorporate active, constructivist teaching at all levels will make those other texts obsolete soon while making this text even more appealing to a wide audience.
The stated purpose of this text is as a readable resource for Calculus students, and it succeeds in being accessible while also maintaining concision and appropriate depth. Readers are asked to connect work across the text, which is hard for many Calculus students, but this work is not possible without it. An instructor will certainly have to teach students to read effectively (both in general and for mathematics specifically), but this is true of all university-level course work. This text uses physical motion as a context regularly, though there are other contexts present. This text would be a better fit for a STEM population than a Business Calc population.
This text is extremely consistent. The concepts build on each other and are connected, making use of the consistency. The guiding goals and values of the text are also consistently employed; for example, the motivating questions and preview activities are used throughout and are central to the structure of the text.
This text uses subsections well; they are appropriate sizes and well titled. It would be easy to add a section for a special topic of interest to a course; most of the sections are important to later work, but it seems that the few topics that could be skipped are stand-alone sections, making that skip straightforward. It would also be easy to use a single section or some sections from this text as a module to replace other course materials in a course if, for example, an instructor preferred this development of a particular topic.
The flow of the concepts is logical and clear. Each section is consistently and appropriately structured with motivating questions, preview activities, development, examples, activities, summaries, and exercises.
The online interface for this book is REALLY strong. The table of contents is useful and fluid. The expandable content works well, especially the WeBWorK exercises, which are user-friendly and focused. The use of subsections keeps individual pages manageable. It might be nice to have PDF or printed versions for reading, in-class usage, and note-taking, but this text lives online.
This text is well edited. In some ways, Calculus is about learning to be precise with vague and intuitively defined terms such as approaching and smooth, and this text handles language professionally without a hint of pedantry.
This text is abstract and symbolic in a way that is normative and expected; many students and instructors will find it neutral and hence appropriate. In a larger context, the assumption that mathematics and its artifacts are culture-free is a deeply problematic issue with which we are not engaging. I am not aware of better models for university-level topics. This text is at least as relevant as texts for comparable courses.
What is the purpose of a Calculus textbook? In the authors' words: "It is our opinion that in the 21st century—an age where the internet permits seamless and immediate transmission of information—no one should be required to purchase a calculus text to read, to use for a class, or to find a coherent collection of problems to solve." This text is clearly a success for these goals. I think, for many people, a textbook is a resource like an encyclopedia in which the concepts of a discipline are collected. Viewing a textbook in this way makes us focus on the ways that the textbook transmits information to a reader or structures their thinking. This text meets those needs and goes beyond by offerings easy on-ramps for educators to help students engage the concepts of Calculus actively. However, I structure my courses using inquiry-based learning, and the presence of a pre-set authoritative summary of the concepts can be at odds with my goals. How does this text fit in this context? Well, the sections start with questions, which I think it EXCELLENT. The very first is "How do we measure velocity?", which is great. This question is a section title; however it is the only section title that is a question - the rest are topics. In the second section, the text asks "What is the mathematical notion of limit and what role do limits play in the study of functions?". In contrast to the first question, this question makes a lot of sense from the perspective of the expert and even the student who has learned a little Calculus. In my assessment, most of the motivating questions in this text are of this type; they frame Calculus as an existing body of knowledge that is to be learned rather than something that the reader is participating in building. This is not a critique of this text, since this isn't what it's trying to accomplish, but it does mean that I don't see how to use this text as the core element of an inquiry-based course. If you are looking for a text you can ask students to read before class, such as a flipped/blended or project-based context, this text is a strong choice. If you are looking for a text students can read after class discussions and when reviewing for high-stakes assessments, this is an excellent option because it does a great job of making the core questions and goals that structure Calculus visible to the reader with enough experience to engage those questions from within the topic. In particular, if I had to have a text for a course but didn't intend to ask students to read it regularly (such as in my inquiry-based courses), this would be a top option because it focuses on conceptual questions, asks the reader to read actively, and is free! And if you are looking for a text from which you can pull active preview, exploration, and extension tasks to make your course active, this is a great choice.
This textbook is intended for a two semester Single Variable Calculus sequence. I was mostly pleased with the textbook, although it lacks sections read more
This textbook is intended for a two semester Single Variable Calculus sequence. I was mostly pleased with the textbook, although it lacks sections on the Mean value Theorem and parametrization/polar coordinates. However, the book is presented in both online and pdf formats. The online version is somewhat interactive, so one simply has to click on the appropriate section and the web browser is directed to the respective section of the text. On the other hand, the book includes a section on the Qualitative Behavior of Solutions of Differential Equations, as well as a section on Euler's Method. Overall, the book could be used for a standard 2-semester Calculus sequence, however, the instructor will have to introduce additional material and proofs.
The concepts are presented clearly and accurately. I did not find any errors on concepts, notation, or grammar.
As a Calculus textbook, this book will remain relevant for generations to come. It mostly follows in the typical layout of a traditional Calculus textbook. It's online format, though, has opened interesting venues since the book contains links to Java applets using GeoGebra in the author's website. This ability makes it outperform paper or PDF texts. Additionally, the book can be downloaded and customized (if the instructor is experienced in web publishing).
The material is presented clearly. Each sections begins with "Motivating Questions" as a means to introduce the material. In general, I found this text to be less "wordy" than a traditional Calculus textbook. However, I found the lack of proofs to be a considerable weakness in the presentation. Calculus without proofs transforms from an exercise in logic to a belief system.
The structure throughout the textbook is very consistent. All sections are organized in the same format and the notation is very clear and consistent. The exercises are appropriate, although the problems at the end of the text are few.
The essential ideas of Calculus cannot be segmented, and this book succeeds in presenting the material very cohesively. However, the material is grouped in a traditional format, so the sections in any one chapter are relevant to each other.
Every section in the book has the same structure and organization. The sections begin with a "Motivating Question" followed by conceptual explanations. In may of sections there are "Preview Activities" and WeBWorK exercises at the end of the section. Each section is properly referenced to relevant chapters, sections, and graphs throughout the text, so a simple click on a link will open a window with the information pertinent to the material being presented. I found this to be very valuable.
The interface works well and the full online version is available on github. The only problem I found was my computer not being able to open an Java Applet in the author's website in one instance. On a different day, I was able to open the same applet without a problem.
I did not find a single grammatical error. While I may have missed a run-on sentence, nothing distracted my attention to recognize as an obvious grammatical flaw.
Calculus has been relevant for 400 years and will continue to be relevant for generations to come. No issues here.
I liked the presentation in general. This book has a lot of potential. Using WeBWork exercises is great because the students can practice concepts on their own textbook with immediate feedback. As mentioned earlier, lack of mathematical proofs is a serious omission for a well-rounded Calculus textbook, which this one can be. The hyper links and applets make this book an interactive tool that can be continuously improved to become a fully interactive Calculus website.
The text was fairly comprehensive. The first portion of the book, which is dedicated to differential calculus, was very thorough. However, the read more
The text was fairly comprehensive. The first portion of the book, which is dedicated to differential calculus, was very thorough. However, the sections on integral calculus was lacking in some of the integration techniques and methods commonly taught in a Calc II course (such as trig sub, higher order partial fractions, etc.) and the Differential equations and Series sections were nice introductions.
Overall, the accuracy was decent in the book. I found some of the notations to be confusing. In the sequences and series sections, s_n was used to represent terms of the sequence, while later a_n was used for the terms of a series, which are just a sum of the terms of a sequence. Then the partial sum was S_n, which could easily be confused with s_n. It's also not consistent with other math texts I've seen that discuss this topic, which could also be confusing. There is also a slight question mark over the topic of continuity as far as what constitutes being continuous over it's domain.
The text could be easily rearranged or updated if necessary. The examples were relevant to the topics at hand and there weren't any examples that were outdated.
The text was quite clear and should be accessible to most students. The only trouble I had was with some of the notation, as mentioned above.
The book was very internally consistent with it's terminology.
The text could be rearranged without much disruption to the reader, however can only be rearranged to a certain extent. (As with most math texts, certain topics have to go together.)
I really enjoyed the structure of this text. I liked that each section started with some motivational questions to start conversations, then progressed to a discussion of the topic and hitting the high points I appreciated that there were places for students to help build up the material themselves alone with the instructor. I also enjoyed that there was a summary at the end of each section.
I had the e-book version of the text, and the only issue I had was that the table of contents page wouldn't take me to an individual section, it would only take me back to the cover page.
There were only a handful of errors that I saw in the wording of some of the questions. For example, in chapter one, there is a graph about calories burned and the question asks if the number of calories increased or decreased, instead of the number of calories used/ burned increased or decreased. Still, over 500 pages and a handful of grammatical errors is impressive!
There were no culturally insensitive examples that I found. These topics don't generally involve examples that could potentially be racially or culturally offensive, but those that were included were appropriate.
On the whole, I felt this was a great Calculus I (differential calculus) and introductory Calculus II (integral calculus) / Differential Equations text. I feel it would be a great addition to a Calculus I course, but would caution against using it in an upper level course or in a more in depth program (such a math or engineering majors.)
The book covers all major topics of differential and integral calculus. However, the emphasis is on "big-picture" understanding of the topics and read more
The book covers all major topics of differential and integral calculus. However, the emphasis is on "big-picture" understanding of the topics and has relatively few (in comparison to other texts) formally stated theorems and even fewer proofs. This could be seen as an asset, if the goal is introducing students to the overall ideas of calculus.
The text is accurate.
This is as up-to-date of a calculus textbook as I've seen. The examples and activities are interesting and draw from a number of disciplines (physical and social sciences, etc.) which gives the reader a sense that calculus has many applications. While specific, the examples will not feel dated in a decade or two and could be replaced with new examples or new data as the need arises.
This is a really well written book. The style is clear and easy to understand. The tone is conversational and uses appropriate vocabulary and description of the mathematics - it doesn't "dumb down" the topics, but also doesn't rely too heavily on "mathy" words when an explanation using regular language would suffice. However, there is a lot of text between and explaining the very few examples and it is hard to say, having never taught with this text, whether or not students would enjoy reading it as much as I did. The authors choose not to state many formal theorems or show detailed proofs. The emphasis is on overall understanding of the topics, not the underlying theory that makes the mathematics possible. This approach is successful for this textbook. If, however, you prefer a formal step-by-step build-up of theory and theorems, this is not the textbook for you. Also, the flow of this book and paucity of detailed examples and homework problems would make it difficult to use as a reference book.
Very consistent in layout and use or terms/vocabulary.
The chapters and sections of the book are of the appropriate length for breaking over class periods throughout the term.
The organization of each section is nicely done: "Motivating Questions", text with examples, activities for students, and the a "Summary" at the end followed by a homework set.
The interface is fine.
No errors found.
I have yet to see a culturally relevant calculus textbook! This particular text is in no way insensitive or offensive.
To take full advantage of the unique approach of this textbook would need a lot of instructor planning/preparation before class. The text has very few examples and the instructor would need to provide many of his or her own examples during lecture. This is especially true for the more algebra-intensive topics. In class, students would need a lot of involvement from the instructor. It would be very important for the instructor to make sure that the students were completing and drawing the correct conclusions from the Activities within the text before moving on. With the right preparation and implementation by the instructor, this "lab"-style approach could be very successful and enjoyable for the students. There are only a few homework problems listed for each section. Additional problems from elsewhere (either another textbook or something like WebWork) would be helpful.
PLEASE BEGIN BY READING THE "OTHER COMMENTS" SECTION AT THE BOTTOM FIRST. It seems to cover all of what we need for the first two quarters of read more
PLEASE BEGIN BY READING THE "OTHER COMMENTS" SECTION AT THE BOTTOM FIRST. It seems to cover all of what we need for the first two quarters of calculus except surface integrals, which we could add or move to the third term.
i found no errors
It is the most up-to-date book on Introductory Calculus that I have seen so far.
This is a book designed to teach. As such, it will not be a good resource for student who have already studied calculus. That would be a very different book.
not appropriate question for this subject.
Excellenet. See "other comments" for more details.
I found no grammatical errors.
While that would be a great idea, no one has yet attempted to write a calculus textbook which was "inclusive". The closest thing was an environmental calculus book, but that included only covered the applied calculus material.
It is hard to get a good sense of how well a book will work before one has taught a class using it. Nonetheless, the approach articulated in the preface follows the the best of what is known about student learning as it relates to calculus. The approach would be challenging for graduate teaching assistants to accomplish, but possible with sufficient support and worth the effort toward the improvement of student learning. I would do a "Dan Meyer" ( https://www.ted.com/talks/dan_meyer_math_curriculum_makeover?language=en ) on the activities and the initial questions. However, the formatting of questions and then activities seems a sound one. For example, I would not foreshadow the answer to the questions by using terminology too soon. For example, I would change the question "How does the notion of limit allow us to move from average velocity to instantaneous velocity?" to "How do we manipulate average velocity to compute instantaneous velocity?" Example 2: Instead of "What is sigma notation and how does this enable us to write Riemann sums in an abbreviated form?", say "How can we write Riemann sums in an abbreviated form?" I should have more examples, after testing this book in a class. Our challenge is that this book would cover only 2 quarters, not the 3 quarters that we teach. We would be required to use a more traditional book (presumably open source) for the third term. Likely do-able, but challenging.
This book is thorough and up-to-date in all areas of a single-variable differential and integral calculus course. I have been using it in my courses read more
This book is thorough and up-to-date in all areas of a single-variable differential and integral calculus course. I have been using it in my courses for over a year now, and I haven't found it to be lacking any topic, theorem, or technique. It is current in its reduced emphasis on algebraic technique and greater attention to the underlying concepts and engineering-based applications. For example, integration techniques have been reduced in coverage and in emphasis in most calculus textbooks and this book is no exception. Substitution and Integration by Parts are featured, Partial Fractions gets a nod, and then students are introduced to the idea of a CAS (Computer Algebra System). This is in keeping with the reduced treatment of "by hand" integration techniques in most modern calculus textbooks.
There are no issues with the book's mathematical accuracy. Another kind of accuracy, though, is how well an individual activity "hits its mark" in taking the student through an illuminating example of a topic. Generally, I think the text succeeds here, but there are some edits I might suggest. For example, in Activity 1.15, after working through this activity in class with students, I altered the graph a bit to create more variation so the resulting discussion about displacement, velocity, and acceleration would be a bit more fleshed out. Teaching with this new version of this activity has had better results in terms of student understanding.
The content is not only up-to-date, but I think very forward-looking in its approach to the subject matter. The book has an almost conversational tone that I find very appealing. However, to remain relevant going forward, I would like to see the "book" revised to take advantage of its medium. It's presented and used (especially by students, who are perhaps more open to using electronic resources than their older, more traditional instructors) as an *online* resource. To remain relevant, I hope that future editions will take advantage of the power of the computing devices on which this book is often read, and feature more video, applets, maybe some Desmos-type graphs with movable parts. When students want to learn how to do something, they are searching YouTube, not looking for a page of text that describes how they might do a thing. They want to try it on, see it in action, engage with it. We should encourage and provide more opportunities for students to do that.
My students did not much care for the text. I am teaching this course using a flipped model, so there is reading and also instructional videos that students are asked to do outside of class. Not surprisingly, most students prefer the videos. Some student comments (from an anonymous end-of-course survey) about the assigned reading in the text: --I didn't find the textbook explanations very user-friendly, as they were much more difficult to comprehend than the videos. I don't know if there are textbooks with clearer explanations? About mid-way through the course, I also didn't find the reading to be necessary for most modules, as the videos and class explanations were clearer teachings of the same book concepts. --The book tends to be confusing, as student that learns from examples, I find this book to be hard to understand. --I liked that the textbook wasn't expensive, but I don't think the examples given were very helpful. I think they were a bit distracting from the point of the section at times. --more videos less reading I am not surprised that students prefer videos, but I don't think this is the fault of the text, but rather that they would prefer video explanations over ANY text. Nor am I surprised that they wanted more "example problems" from the text. Students have been taught that math is mostly manipulating expressions and equations. This book takes a very different approach. One student expressed his discomfort: --...also there aren't very good examples and explanations in the text. For instance; A section has about a paragraph and then the preview activity.....there's no explanation or good examples of problems. We are kind of just thrown into a pit of fire. ...which is exactly the point of the text, that you learn this content by interacting with it. The text is interspersed with Activities (as the book's title implies). I used most of the activities (either as-is or modified a bit) as group work in my Calc I and II classes. Students resoundingly preferred this "active" approach to learning calculus to the traditional lecture-based approach, and I think the quality of these activities was a big factor in students' satisfaction.
The book is consistent in terminology and framework, absolutely. There is a bit of variation in the consistency of the relative difficulty of the activities, however. For example, in the section on Implicit Differentiation--a topic that students often find challenging--Activity 2.20 features an expression that is algebraically quite complicated for students. I used this once in class. Students just laughed out loud, most refused to try it! I removed it from the set of activities I use. At other times, there are questions that seem to confuse students because they are "too easy", like (d) in Preview Activity 1.3: "Write a meaningful sentence that explains how the average rate of change of the function on a given interval and the slope of a related line are connected." Students ask "Do they just want me to say that they are the same? Is that all?"
The text is easy to pull apart and put back together. It is suitably modular.
The text gets full points for organization/structure/flow. I would like to suggest perhaps an alternate version of the text where it is organized more like a workbook, with more room left between the questions/problems where students might write their responses. I don't like asking students to copy down the text of a problem when they are working; their resistance to doing so is firm and vocal! But a bunch of answers on a piece of paper with no context is not good work product, nor very helpful as a study device. A version of this text that invited that kind of "active" participation from the reader would be a marked improvement, I think.
The interface is fine; I've encountered no issues.
I think the book is not only grammatically correct, but very well-written. Not always the case with math textbooks!
I have encountered nothing even remotely insensitive or offensive in this text.
This book helped me to understand how I might teach calculus in a more learner-centered way, and for that I sing its praises! I recognize, though, that the "active" approach is a bit different from what most students are used to/expect, and they will need instructor support to make the most of this book and what it has to offer.
I thought that the book was thorough in the subjects that were listed, including limits, derivatives, integrals, differential equations, and read more
I thought that the book was thorough in the subjects that were listed, including limits, derivatives, integrals, differential equations, and sequences and series. I would have liked a few chapters on multi-variable calculus, but that wish should not degrade the comprehensiveness of the book. The book is hyperlinked throughout, so if on the PDF you look up a terming the index, clicking on the link will bring you right to the page that the term is introduced.
The book builds upon 400 years of calculus understanding, so most of the book is accurate and unbiased in terms of the content.
The text will not be obsolete for a long period of time. The topics covered, and the problems presented are relevant. I conjecture that if an application problem is ever out-of-date, it could be easily replaced.
This book is written contrary to many mathematics textbooks in a fresh, active, and accessible manner. The layout of each section of the text has a summary of what will be discussed, preview activities to get the reader situated, activities throughout the prose, and a summary of what was discussed prior to exercises. It seemed as though the activities and the mathematics had purpose and understanding built in, which I cannot say the same for some other textbooks. I was excited to move to the next section when reading.
The consistency of the textbook is fine. Every section has the same layout, and problems at the end of the section are probing no matter which section is discussed.
I think that it is slightly difficult to be modular with a mathematics textbook. With that being said, I thought that the authors had a different approach than other textbooks in terms of what they wanted to introduce first. For example, I have always learned to prove the fundamental theorem, I would need the interplay between derivatives and integrals. The authors prefer to conjecture the fundamental theorem from observations of velocity and position, and in the next chapter approach the proof.
I have already commented on the flow in the modularity section. I think that many parts flow in this textbook, but there were some parts that I had trouble with initially.
I found little to no grammatical errors in this textbook.
I did not see any portion of this text that referred to any ethnicity or race, so technically it is inclusive of all races and ethnicities.
Table of Contents
Chapter 1: Understanding the Derivative
Chapter 2: Computing Derivatives
Chapter 3: Using Derivatives
Chapter 4: The Definite Integral
Chapter 5: Finding Antiderivatives and Evaluating Integrals
Chapter 6: Using Definite Integrals
Chapter 7: Differential Equations
Chapter 8: Sequences and Series
About the Book
Active Calculus is different from most existing calculus texts in at least the following ways: the text is freely readable online in HTML format and is also available for in PDF; in the electronic format, graphics are in full color and there are live links to java applets; version 2.0 now contains WeBWorK exercises in each chapter, which are fully interactive in the HTML format and included in print in the PDF; the text is open source, and interested users can gain access to the original source files on GitHub; the style of the text requires students to be active learners — there are very few worked examples in the text, with there instead being 3-4 activities per section that engage students in connecting ideas, solving problems, and developing understanding of key calculus concepts; each section begins with motivating questions, a brief introduction, and a preview activity, all of which are designed to be read and completed prior to class; following the WeBWorK exercises in each section, there are several challenging problems that require students to connect key ideas and write to communicate their understanding.
About the Contributors
Matt Boelkins, Professor, Department of Mathematics, Grand Valley State University. PhD in College Teaching of Mathematics, Syracuse University.
David Austin, Professor, Department of Mathematics, Grand Valley State University.
Steve Schlicker, Professor, Department of Mathematics, Grand Valley State University. PhD, Northwestern University, specializing in Algebraic K-Theory and the Cohomology of Groups.