Active Calculus 2.0

Reviewed by Steve Leonhardi, Professor of Mathematics and Statistics, Winona State University on 8/2/18

Comprehensiveness rating: 3

Reviewer’s note: Please read my “Other Comments” in section #11 first, where I’ve written most of my review, and then return here to read my condensed, individual section comments.

Comprehensiveness: The text covers most areas and ideas that I would expect, although there are some “missing” sections and additional exercises that I would like to see included, as described in the “Comments” section.

Content Accuracy rating: 5

The text is very well-written and includes appropriate diagrams.

Relevance/Longevity rating: 4

The subject matter is relatively timeless, but I would like to see more contemporary examples using recent real-world data.

Clarity rating: 5

The exposition is straightforward and comprehensible, at a level appropriate to my students.

Consistency rating: 5

The exposition and notation are internally consistent.

Modularity rating: 4

The modularity is generally good, except for the presentation of Antiderivatives being sprinkled throughout Chapter 4 rather than presented as a separate section.

Organization/Structure/Flow rating: 4

The organization is generally good, except for the presentation of Antiderivatives being sprinkled throughout Chapter 4 rather than presented as a separate section.

Interface rating: 4

The interface is generally good. I do wish that each “activity” could be linked as pop-up pdf page for easy printing when desired.

Grammatical Errors rating: 5

I found only a very few typographical errors when using the text for a full semester.

Cultural Relevance rating: 3

The text is not offensive or insensitive in any way, but includes very little mention of the history of the subject or biographical information. I understand the desire to “streamline,” but I would prefer greater inclusion of this context.

As implied by the textbook title, the design of Active Calculus 2.0 is based on the view that an active, inquiry-based approach is the best way to help students learn Calculus. The experience of the instructor and the students using this textbook will depend largely upon (1) the extent to which they agree with this inquiry approach, and (2) their interpretation of how “active learning” is most effectively implemented.

I used this textbook in a first semester Calculus course in the Spring 2018 semester. Students were asked to read the assigned section before class and attempt to complete the Preview Activities for each section via WeBWork exercises. In class, students worked in pairs through two to four “activities” from the text. (I required that students purchase a printed $10 “course pack” of the activity worksheets from our campus bookstore, available as a pdf file from the authors upon request, so that they could easily turn in their work at the end of each class period.) I circulated throughout the room checking students’ progress and answering questions, and then typically spent the last 10 to 15 minutes of each class summarizing possible approaches and results (with as much input as possible from the students), then collected their worksheets and gave credit for effort and participation (about 10% of their final grade). As the semester progressed, I adjusted this approach by starting most classes with a brief (10 minute) introduction to the material or sometimes completing the preview activity in lecture format, rather than simply starting the class with them working on worksheets. I also learned to break the class into smaller “chunks” of activity, so that by the end of the semester we were generally alternating between 5-10 minutes mini-lectures or recaps directed by me, interspersed with having students work on activities for only 10-15 minutes at a time rather than longer stretches.

The reason I go into such detail describing my pedagogical approach in this textbook review is to clarify that this textbook’s effectiveness and suitability will be highly dependent upon the instructor’s preferred pedagogical approach. If your own approach to teaching matches the active-learning approach that I have described, then you will find this text to be a valuable resource, especially compared to OER alternatives for Calculus. If you and your students prefer the lecture-discussion model, this textbook is not a good choice, at least not as your primary text.

Overall, using this textbook and this approach was a very positive experience for me and my students. I used a variety of resources associated with this text, generously provided by the primary author Matt Boelkins in response to my email requests. As previously mentioned, he sent me the pdf file of the activity worksheets only. (I would recommend using this separate pdf file, as I could not figure out a way to easily print the activities only from either the pdf or live versions of the online text.) Matt also shared with me sets of WeBWork exercises that align well with the Active Calculus sections. Instructors at Carroll College (MT) have developed a free “Chapter 0-Preliminaries” of Precalculus review as a supplement to this text. Instructors at Grand Valley State University (MI) have posted a series of instructional “screencasts” (videos) on YouTube that are matched to the sections of this text. The authors also suggest using the online Calculus applets developed by Marc Renault at Shippensburg College (PA). My students found these all these additional resources quite useful.

My comments so far have focused on the suitability of this text with an active-learning approach. I would next like to shift now to my second initially-posed question: How is “active learning” most effectively implemented?

My own experience with active learning has led me to believe that active learning using worksheets in groups is most effective when the following ingredients are present: engaging questions, student-led discovery, and supportive scaffolding.

Active Calculus succeeds for the most part in providing all three of these ingredients, to different degrees in different sections and different activities. If I use this text again, I would select more carefully which activities I reuse unaltered, which activities I use with some edits, and which activities I skip entirely or replace with my own personal worksheets.

Despite the active learning spirit embodied throughout the text, its contents are remarkably similar to a traditional text. I wish that the text included more real-world data and examples, in the spirit of a text such as the Hughes-Hallett Calculus. My students find such problems and examples more engaging and compelling than such questions as “How do we find the slope of the tangent line?” and “How do compute the area beneath a curve?”

I concur with the review comments of Professor M. Paul Latiolais (Portland State University) that some of the “motivating questions” at the beginning of sections could be reworded to better facilitate student discovery of the relevant concepts as opposed to just asking students “How do we define [fill in the standard Calculus term]?” and then proceeding to tell them in the following exposition. I also think that some of the activities could be improved by providing more scaffolding. Most of my students had difficulty correctly completing the Preview Activities (inWeBWork) before each class, to the point that I decided that scoring 50% on a Preview Activity would count as 100%, and any score above 50% would count as extra credit.

For example, Preview Activity 3.1.1 on critical numbers and extreme values was an especially effective activity because it breaks the concepts up into small enough questions and concepts that allow the students to proceed step by step to feel like they are discovering the notions of critical values and Fermat’s Theorem for Extrema on their own. Preview Activity 3.5.1 on related rates is also effective because it poses an interesting question and lays out a series of steps for the student to develop understanding of a procedure to answer the question.

On the other hand, some of the Preview Activities (1.8.1, 2.4.1, 2.7.1, to name a few) felt simply like homework problems and did not generate much student interest or feelings of discovery.

Several other features of the text could be improved, in my opinion. The text is missing several sections that I would expect to see in any Calculus text, even if included only as an optional section. The definition of “limit” is stated correctly but somewhat informally, in the sense that the epsilon-delta clarification of “sufficiently close” is never introduced. I agree that a strong case can be made for delaying and underemphasizing the formal definition and notation until students are better able to appreciate and understand it, but I would prefer to see the formal definition presented and explored intuitively. More surprisingly, the Mean Value Theorem is not included. Again, I am sympathetic to the view that most first semester Calculus students are not ready to appreciate rigorous proof or lengthy formal derivations, but I choose to introduce these notions to my students in intuitive visual contexts, so that they can gradually absorb their meaning and significance over several semesters. Of course, these “missing” sections are easily filled in using personal notes and/or other, more traditional texts (such as the OpenStax Calculus).

Also, I would prefer that the text contained both more examples in the exposition, and more exercises at the end of each section. Active Calculus does contain most of the standard examples found in any Calculus text, and any instructor using WeBWork has access to as many additional exercises as they could ever care to assign. I believe that the authors were intentionally trying to keep the text streamlined to force students to discover