Alternate ending: draft of a new Chapter 8

I’ve written a rough draft of a very different version of Chapter 8 for Active Calculus (single variable), and I am interested in feedback from users.

TL;DR version: I have come to think that the Taylor series representations of the sine, cosine, and exponential functions are the most important series-related ideas in calculus 2, and the details of convergence tests are not. So I wrote a chapter with its central focus on the former, and very limited discussion of the latter.

Taylor polynomials provide an intuitive way to see why infinite series are important, and their approximations of non-polynomial functions are a powerful application of calculus. Rather than me write on at length about my motivations and the changes, I think the draft speaks for itself, so I encourage anyone interested to check it out: alternate Chapter 8.

While the entire book is found at the above link based at my home page, please know that the definitive current version is always hosted at

Three additional related things:

+ I am teaching calculus 2 this fall and will be using this new Chapter 8 with my own students. In the very near future I will post an invitation to anyone who wants to do a “teach-along” with me this semester.
+ I realize the Chapter 8 draft is rough and needs work: prose editing, mathematical details, more and better exercises, and likely some more images. Feedback is welcome on any of it. I plan to make it much tighter by November 1, which is a couple of weeks before I will share it with my students.
+ I plan to (eventually, not this semester) write one more section and probably also an appendix. The additional section will likely be on using series to solve differential equations, and the appendix would be on some of the details of series convergence that are intentionally omitted in the new Chapter 8.

If you have comments or suggestions on the draft of Chapter 8, please email them to me directly: boelkinm at gvsu dot edu. If you think you will plan to use it with your students, I would welcome knowing that as well, as then I can communicate directly about updates and more with the people who most need to know.

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PreTeXt, Runestone, and the exciting future of open textbooks

In late June, I was fortunate to attend a week-long UTMOST workshop in Ann Arbor that brought together education researchers, PreTeXt developers, Runestone developers, and textbook authors.

The newest part to me was learning more about Runestone. Runestone Academy was founded by Brad Miller in 2011; Brad is an emeritus professor of computer science at Luther College, and his original goal was to have textbooks with interactive code that the reader could experiment with. More than a decade later, he has an amazing platform that is part textbook-hosting service, part open-source textbooks, part tools for writing texts, and part “small LMS”, plus an open-source server to support the texts on the platform.

Brad wrote a great post about the workshop in Ann Arbor and the partnership that he and Rob Beezer have forged between Runestone and PreTeXt: “Going forward PreTeXt will be the primary authoring language for books hosted on Runestone Academy.” In the other direction, I expect that in the near future we will see that Runestone becomes the primary platform on which PreTeXt books are served. Let me say more about what I mean by that.

One of the greatest strengths of PreTeXt is that it allows a book to be interactive. There are currently possibilities for free-response reading questions, interactive Sage cells, and WeBWorK exercises. At present (without Runestone), only the reading responses can be easily tracked in a PreTeXt book where the instructor can see what students did (and even that requires some special dispensation from the PreTeXt developers). Essentially what has been needed is an easy way for students to log in to the book they are using, and then have the results of their work recorded and easily accessed by the instructor. Runestone provides this capability (and more). You might think of it as the textbook living inside of the LMS, with exercises in the text being tracked by Runestone in the way that a WeBWorK server tracks student work on WeBWorK exercises.

This change will also allow authors to move away from WeBWorK. There are new developments regarding WeBWorK-like exercises that can be written in native PreTeXt. These will eventually eliminate the need to be connected to a WeBWorK server, and the work students do on them can be tracked, provided the course is being run through Runestone.

For me personally, I have long thought that the exercises in Active Calculus need to be strengthened, and I can see a path where the combination of native PreTeXt exercises served via Runestone will be a great motivator to get to work on that (big) project. Runestone has a feature that promotes in-class peer instruction that will also merit some serious thinking for how that might be incorporated into AC.

Beyond the exciting developments regarding the partnership between PreTeXt and Runestone, we also heard reports from Vilma Mesa and her team at the University of Michigan about their wide-ranging research into how students and instructors use textbooks. A meta lesson that I took from that is: textbooks can help shape instruction for the better.

Beyond working on the APC, ACS, and ACM books themselves, I will be working over the next year or so to improve the “ecosystem” that surrounds the text by making ancillary materials better organized and easier for new users to access in the hope that these materials will help even more instructors transition to using one or more of the texts.

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Active Calculus is Turning 10

“The days are long but the decades are short.” That’s one of my favorite quotes about the nature of time and life, written in 2015 by a then-30-year-old!

Ten years ago in early August 2012, I wrote this post that presented the first public version of part of Active Calculus.

I had just returned from Mathfest 2012 in Madison, WI, where I met Rob Beezer for the first time. A lot has happened in the decade since:

  • August 2012, chapters 1-4 of Active Calculus Single Variable are made public (PDF)
  • January 2013, chapters 5-8 of Active Calculus Single Variable are made public (PDF)
  • August 2014, first print version of Active Calculus is made available via print-on-demand services
  • August 2015, Active Calculus Multivariable is first made public (PDF)
  • April 2016, I attended the first PreTeXt workshop (then called Mathbook XML)
  • August 2017, Active Calculus Single Variable is made available in HTML format; source files post on GitHub
  • January 2018, a new online home for the books:
  • January 2019, Active Prelude to Calculus is made public (HTML and PDF)
  • January 2020, the Google group “active-calculus-users” is launched
  • March 2020 to May 2022 …

During most of the pandemic from March 2020 to May 2022, my teaching and administrative duties at GVSU expanded and took priority. Throughout that 26 month period, I basically ignored the plans I had for Active Prelude and Active Calculus. And I missed working on them.

This June, I was fortunate to attend the UTMOST workshop in Ann Arbor that brought together education researchers, PreTeXt developers, and Runestone developers. I’ll write more about that in a separate post soon, but for now I will simply say that I found the workshop invigorating and it made me excited to return my textbooks to a place of high priority in my professional life.

In the next couple of weeks, I expect to post frequently here, on at least the following topics:

  • a reflection on the June UTMOST workshop and some of the future goals for APC and ACS and ACM it has inspired
  • news about a new version of one of the chapters of ACS that I’m working on and expect to make public during the fall semester
  • my hopes for a fall “calc 2 teachalong”
  • a short adoption survey to help me get a better sense of where APC, ACS, and/or ACM are being used
  • an updated post with available ancillary materials

As always, thanks for reading, and thanks for your interest in and support of APC, ACS, and ACM.

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A (partial) mastery-grading approach for Active Prelude to Calculus

For the Fall 2020 Pandemic Semester, I taught two sections of GVSU’s 5-credit MTH 124: Functions and Models, our calculus-prep course. This is the course for which I wrote Active Prelude to Calculus, and this was my first time getting to teach the course from my own text.

I taught the course in an unusual pandemic-induced hybrid format that is likely familiar to readers. My classroom could hold up to 20 socially-distanced students wearing facemasks (plus me, also masked), but I had 30 students registered in each section of the course. Some students needed to self-isolate or quarantine periodically, and other students needed the option to attend online at all times. This led to a format where some students would attend class in person, while the remainder were present in a synchronous Zoom meeting that I hosted. (The logistics of that were surprisingly successful and not as big a deal as I feared in August; that should be the subject of a future post, as it is not my main purpose here.) All of this is to say: the pandemic forced some structural changes, and the changes tied to assessment ended up being really good for my students and me.

Working with three other colleagues also teaching the course, we decided to rethink our assessment plans since we had to navigate the issue of some-students-in-person, some-students-online, all simultaneously. We centered the course around 12 Core Learning Targets and weekly Checkpoints (quizzes) that assessed student understanding of those learning targets in a mastery-based system. Here is the basic gist of it:

  • In the last ~30 minutes of our final meeting of each week, students would download that week’s Checkpoint from Blackboard in PDF.
  • After working to respond to as many of the questions as they could, they would use a scanning app on their phone to submit a single PDF of their work.
  • Each learning target typically had a cluster of 2-4 short questions that students needed to answer nearly perfectly in order to earn a mark of “M” (mastered) for credit on that learning target; if they didn’t demonstrate mastery, the mark was “NY” (not yet), and they could re-attempt that target on a future Checkpoint.
  • Checkpoints were open-book, open-note.  The only constraints on students were:  no collaboration with other human beings, and no using resources beyond the text and their self-generated notes.

Checkpoints constituted 36% of students’ semester grades — effectively 3% for each learning target.  (You can see my full syllabus if it’s helpful.)  Each of the 12 learning targets appeared on 3 separate Checkpoints: for instance, Checkpoint #5 had learning targets 3, 4, and 5, and then Checkpoint #6 had learning targets 4, 5, and 6, and so on.  Once a student had mastered a learning target, they could ignore that question on subsequent Checkpoints.  And, if a student hadn’t mastered a learning target after three attempts, they had two options:  they could (for a limited number of targets) have an oral assessment with me by Zoom to demonstrate mastery, or, during final exam week, take a Custom Checkpoint with a 4th attempt on up to 3 learning targets (I wrote 25 Custom Checkpoints during final exam week ;-)).

The course was not a full instantiation of mastery grading, but the vast majority of the graded work allowed for multiple attempts:  36% for Checkpoints, 20% for weekly WeBWorK sets (10 exercises each, with up to 8 attempts per exercises, but students could ask for more attempts if they were unsuccessful after 7), 15% for weekly Writing Assignments (graded on a points scale, but with some options for revision & resubmission), and 10% for Daily Prep Assignments that were marked on the basis of effort and completeness.  These together constituted 81% of the course; there was an additional 4% for a pair of metacognition assignments, plus 15% for a relatively traditional final (though it, too, was administered online and was open-note, open-book).

There is lots of evidence that mastery grading is good for learning.  I absolutely found this to be the case for my students, in my experience and theirs.  The combination of frequent, lower-stakes assessment; high standards for mastery; emphasis on conceptual understanding; and the more encouraging message of “not yet!” led to one of the most successful teaching experiences I’ve had.   

I know that I was fortunate to have a really good group of students this fall (generally hard-working with positive attitudes and great patience for navigating the circumstances of COVID) and that certainly influences the results, but the group didn’t seem that different from those I’d taught in two sections of the course in Fall 2017.  In fall 2017, the DFW rate was 33%; in fall 2020, the DFW rate was 21%.  And in fall 2020, a very small percentage of students earned marks in the C-range; the vast majority of my students demonstrated very strong understanding in the core ideas of the course and ended up earning semester grades of B or better.  

Finally, my students self-reported that they really liked this system:  that it lowered the pressure for them, that it allowed them to focus more on learning and understanding, and that it was not demoralizing to get “NY” on their paper.  From my perspective as instructor, it made my grading more straightforward and meaningful, and there was abundant evidence that my students adhered to the expectations for academic honesty.  The combination of multiple attempts and open-book, open-note assessments resulted in all or nearly all students doing the work themselves.  Frequently, students would write on their own paper before submitting: “not yet” or “I need to work on this more.”  

Along with a link to this blog entry, I will post on the AC-users list some more information for interested instructors.


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Screencasts for Active Prelude to Calculus

My GVSU colleague Mandy Calvillo has developed a fantastic resource for students and instructors using Active Prelude: a YouTube library of 122 (!) short videos that offer additional perspective and explanation on key ideas in the text. The videos are numbered and labeled in alignment with the sections of the text. This is a wonderful complement to the libraries for chapters 1-4 and 5-8 of Active Calculus (single variable), and something I know will benefit students and instructors alike.

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Back to school updates for Active Prelude, Active Single, and Active Multi

Here is my just-in-time post for back to school, Fall 2020 Edition.


The two main things to know about the Active Calculus textbooks (Active Prelude to Calculus (APC), Active Calculus Single Variable (ACS), and Active Calculus Multivariable (ACM)) are that (a) the main landing site ( is always the place to find the current versions of the texts themselves (in HTML, in PDF, and in print), while (b) the best location to receive updates and communicate with other instructors using the texts is If you are an instructor who is not yet a member of the active-calculus-users group, just click the link and ask to be added.

Updates for F20:

+ the HTML versions of APC and ACS have both been slightly updated to fix a small number of typographical errors. Due to the PDF being more complicated to produce, I’ve not made the changes to the PDF (and thus not the print edition). If you care about the list of small differences that now exist between HTML and PDF/Print, see these documents: APC errata, ACS errata.  You may need to reload individual pages or clear your cache in order to get the updates if you’ve been already using the texts.

+ thanks to Mitch Keller and Nick Long, there’s now a chapter on vector calculus (line integrals, divergence, curl, Green’s Theorem, Stokes’ Theorem, etc.) as part of ACM:  instead of the usual ACM link, use the following one instead to access this new content (along with all of the previous content)

This is a fantastic development and I’m deeply grateful to Mitch and Nick for writing this.  Cheers for OER, too.

All else in APC, ACS, and ACM remains essentially the same from a year ago.


Historically, I’ve had people email me directly for ancillary materials, largely so that I have better information on who is using the texts. Going forward, I’m going to make these available to anyone who’s a member of the AC-users group.

Concurrent with this blog post, I’ve now posted most ancillary resources on the AC-users Google group. Here is what will be listed there, with direct links to them:

+ APC ancillaries: PDF workbook, print workbook on Amazon, WeBWorK .def files, link to Edfinity course

+ ACS ancillaries: PDF workbooks, print workbooks on Amazon, WeBWorK .def files, link to Edfinity course, Daily Prep Assignments, Geogebra labs (for chs 1-4), Screencasts on YouTube

+ ACM ancillaries: WeBWorK .def files, link to Edfinity course
(note: there is not yet a PDF Activities Workbook for ACM, in part due to the challenge of moveable 3D graphics)


I’m teaching from APC this fall. I’ll be working on a bunch of different things, and I hope to be able to find time to make some of those efforts public as I go. I expect that my GV colleagues and I will be creating video content, developing daily prep assignments, and adding to our WeBWorK offering for the course. Anything I have time to share I will endeavor to post on the AC-users Google group.

A wish for us all:

As the fall semester gets underway everywhere amidst the ongoing challenges of COVID-19 and considerable remote teaching and learning, I just want to say: I wish us all well. May we stay healthy and safe, may we care for our students as human beings first and as learners of mathematics second, and may we all look forward to a return to easier circumstances in which to do this work.

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Active Calculus Activities in Google Sheets

Erica Miller of VCU had a brilliant idea:  set up AC activities in Google sheets for remote teaching so that her students can work in breakout groups on a shared document.  Dave Kung learned about this in an MAA-sponsored session on teaching remotely and had the idea to crowdsource building the sheets; he posted on the Active Calculus Users Group and lots of people have pitched in.

Currently there are templates for Activities 2.7.2 – 4.4.4 for the first four chapters.   Make sure you start with “Read Me First” in the first tab, and follow the instructions to either (a) contribute missing activities, or (b) use the activities with your students yourself.  Activities for Chapters 5-8 in this format are just getting started; more people are needed to contribute to making the tab for each activity.

The basic idea is that each tab in these two shared spreadsheets will hold a template for student responses to the activity, and then anyone who wants to use it can copy the tab to a new worksheet, make n tabs for n groups in their own course, and then have their students work synchronously in breakout groups in Zoom, BB Collaborate, or however.

Erica, thanks for having and sharing such a great idea.  Dave, thanks for getting the community to work to pitch in on this.  I hope others will join and help round out the collection.


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reminder: Active Calculus YouTube playlists

My GVSU colleagues have developed extensive video content to support all 8 chapters of Active Calculus.  Each screencast is enumerated with the section number that aligns with the text.  At GV, our Calculus 1 is MTH 201, which corresponds to chapters 1-4, while Calculus 2 is MTH 202, chapters 5-8, and thus you’ll see the “MTH 201” and “MTH 202” playlists:

Calc 1 – Chapters 1-4
Calc 2 – Chapters 5-8

I expect these videos will prove to be an especially valuable resource in spring 2020 as nearly all US colleges have moved online for the near future due to the coronavirus pandemic.

Take care, everyone — of your students and of yourselves.

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Desmos versions of Active Prelude Preview Activities

My colleague Marcia Frobish at GVSU has instantiated each of the preview activities in Active Prelude to Calculus as a Desmos activity.

If you are a Desmos user, you should check these out.

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Google groups for Active Calculus

I’m launching two groups for information about the Active Calculus texts!forum/active-calculus-announce, where there will be infrequent posts announcing new editions and features, and!forum/active-calculus-users, which is a forum for instructors who use (or are considering using) the text.

At the latter, I hope there will be a lively exchange of ideas that center on the user experience, feedback and suggestions, and requests for new content or features.  My goal is to develop a user-driven forum that can further improve all three textbooks.

I have long maintained an email list of people who have contacted me about one of the texts.  I’m using that list to directly add everyone to “announce”, and invite everyone to “users”.  (Google only permits adds/invites to 100 people per 24 hours (and only 10 at a time!), and I have close to 400 people on my list, so by the end of the week I expect to have all of these out, at which time I’ll start posting some items for discussion on the -users list.  If you are on my email list, haven’t yet gotten an invitation/notification, and don’t want to wait, follow the links above:  anyone can join the “announce” list, and anyone can request to join the “users” list.  I intend to reserve the list for people who are HS or college instructors.

Finally, I’d like to thank Mike Shulman of the University of San Diego for prompting the formation of these groups.  I look forward to the discussion that follows.

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