Calculus 2 Simplified

On this page you’ll find a variety of resources designed to support your teaching of calculus.

Exploring a Sequences First Approach to Calculus 2

Calculus 2 Simplified makes it possible to teach the subject by teaching students sequences and series before integration techniques. Why might you want to do this? Here are a few reasons.

• It lowers the prerequisites. Students can contemplate sequences right away: I am 100% confident every one of your students, on Day 1 of your course, can find the next number in the sequence 1, 2, 3, 4, .... By contrast, only those students with sufficient mastery of integration can recall the knowledge needed to jump back into that content on Day 1 of a Calculus 2 course.

• It promotes equity. Because the sequences first approach lowers the prerequisites, it gives students -- and you -- more time to connect and determine what integration knowledge gaps, if any, need addressing before you eventually return to integration. This gives students much more time -- compared to covering integration first -- to review integration and get the support they need to succeed in your course.

• It provides a more natural and engaging arc for the course. The integration first approach results in two weeks' worth of integration technique followed by integration technique (by parts, integrals involving trigonometric functions, trigonometric substitution, by partial fractions, etc.). It's no wonder that this approach to teaching Calculus 2 leaves students with the sense that Calculus 2 is simply a collection of techniques, a grab bag of "do this when you see that." And this sentiment is only further cemented in students' minds by the litany of series convergence tests later covered. 

By contrast, let's consider for a moment the thought experiment of teaching Calculus 2 via the sequences first approach. You first pose a few essential questions: Is it possible to sum an infinite number of numbers together? How might we do that? Could we come to see integration as the result of summing an infinite number of numbers together? (See Chapter 1 of Calculus 2 Simplified -- available for free on the Sample Chapter page -- for how I do this, and for content you could pull from there to use in your own course.) In your first lesson, you lay the foundation for answering these questions by teaching students about sequences. Then, you teach them that in calculus an infinite series is the limit of the sequence of its partial sums (a nice connection of concepts), and that we call that limit (when it exists) the sum of the series. (This would be a great time to point out that yet another new calculus concept -- infinite series -- is being defined in terms of a limit, thereby connecting Calculus 2 to Calculus 1, which used that same approach to introduce its new main concepts.) Now you've got a calculus answer to "How do I add up an infinite number of numbers?" But, you point out, finding a formula for the partial sums is often too difficult in practice. That pivots the work to convergence tests. Later on, you communicate that certain infinite series have special interpretations. You then recall the Riemann sums approach to integration, and note that in the simplest case -- an equipartition of a closed interval of integration -- the value of the associated definite integral is the limit of the (Riemann) partial sums. You point out, therefore, that definite integrals are just infinite series in disguise. (And you might pause here to underscore this new perspective on integration, new particularly from the standpoint of Calculus 1, which didn't cover infinite series.) And you highlight that these special infinite series have special interpretations (net signed area) and applications (later on: volumes, arc length, etc.) that you then explore in future lessons. 

I hope you found this short thought experiment helpful. And I hope you can see how following this sequences first approach unifies the two pillars of Calculus 2 -- sequences and series, and integration and its applications -- and furnishes students with a content arc that seamlessly connects Calculus 2 concepts and elucidates their evolution. 

My own experience teaching the subject this way has been very positive, and reflects these outcomes -- here's a selection of my students' comments: 

• "I learned about how series and integration are integrated together" 

• "This course taught me how to approach mathematics with a creative and innovative mindset" 

• "This course helped evolve my understanding of calculus."

If you're interested in trying the sequences first approach to Calculus 2, feel free to contact me (ofernand@wellesley.edu). I'd be happy to chat about my experiences doing that.

Resources for Teaching Calculus

Various national mathematics organizations have spent decades studying how students learn calculus best. The following links will help you begin to explore the resulting literature that has been produced.

• NCTM’s Principles to Actions outlines eight research-based Mathematics Teaching Practices. I found the six-page Executive Summary particularly helpful.

• The AP Calculus course framework provides calculus-specific Mathematical Practices and maps the content of an AP Calculus AB/BC course to those practices. It also contains detailed learning objectives mapped to both the practices and the big ideas in calculus. Finally, the document overviews various instructional strategies and provides other helpful tips for teaching calculus.

• The MAA’s Committee on the Undergraduate Program in Mathematics published a Curriculum Guide to Majors in the Mathematical Sciences in 2015. The full document sets forth cognitive goals for a mathematical sciences major and discusses graduate study in mathematics and other topics associated with college-level mathematics. The Calculus Sequence document summarizes various aspects of calculus instruction at the college level, including pedagogical approaches, curricular innovations in calculus, and the main takeaways from a large survey of college calculus instructors.

Resources for Becoming a More Effective Instructor

It is a sad fact of our educational system that most college instructors are not adequately trained in graduate school to teach. (They are, by and large, trained to produced research in a particular discipline.) Teachers at the K-12 level, on the other hand, are provided with some training in effective teaching. Nonetheless, as we instructors know, effective teaching is a lifelong endeavor–a mere few years of studying how to teach better will not make you an excellent instructor forever (student populations change over time, student needs change over time, and new research on teaching and learning is continuously being produced).

The following links are intended to furnish some exposure to the latest evidence-based instructional strategies, in addition to providing links to useful tutorials and guides on effective teaching moves.

Syllabus Resources

• Syllabus Design: Check out this article from Change magazine on content- versus learning-focused syllabi, why the distinction matters, and its implications for student learning. Complement that with this collection of useful advice and “how to…” guides for developing, writing, and updating syllabi from Yale’s Center for Teaching and Learning.

• Assessing Your Syllabus: Here is a very helpful (and validated, research-based) rubric for determining how learner-centered one’s own syllabus is. Complement this with example syllabi scored using the same rubric.

Inclusive Teaching

• Toolkits: These links take you to comprehensive resources aimed at helping you create a welcoming and inclusive classroom atmosphere in which every students feels they belong and has a voice: Inclusive Pedagogy Toolkit (Georgetown University); Inclusive Teaching Toolkit (Center for Instructional Innovation, Western Washington University); Guide to Inclusive Teaching (Center for Teaching and Learning, Columbia University).

• Specific Tips for Creating Inclusive Classrooms: Creating Inclusive College Classrooms (Center for Research on Learning and Teaching, University of Michigan); Creating Inclusive Classrooms (The University of Arizona); 7 Ways to Create an Inclusive Classroom (University of California, Berkeley).

• Research-based principles for boosting learning for women and students of color: Check out my chapter on this topic, which appeared in the edited volume Constructivist Education in an Age of Accountability.

Student Development Theory

No two students are the same. Among their differences is their level of development along each of the various learning domains. The following links provide exposure to various student development theories.

• Applying Theory to Advising Practice (NACADA). Contains a useful summary of student development theories and their usage in academic advising of students.

• Theories and Models of Student Development (Illinois State University).

General Teaching Tips

• Teaching Tips (Center for Advancing Teaching and Learning Through Research, Northeastern University). Over 90 Teaching Tips that “briefly present a strategy that can be incorporated into your class to improve student learning, along with the research that supports it.” 

• Interactive Techniques (Academy for Teaching and Learning Excellence, University of South Florida). A list of over 240 techniques for engaging and assessing students in the classroom. 

• K. Patricia Cross Academy. This site contains dozens of short video tutorials on effective instructional techniques (e.g, the jigsaw method) along with associated handouts for each technique.

Comprehensive Teaching Support

• Preparing to Teach (Center for Advancing Teaching and Learning Through Research, Northeastern University). A list of several resources particularly useful for new faculty.

• Faculty Resources (Center for Teaching and Learning, Yale University). A collection of strategies, tools, and resources on everything from learning theory to concrete instructional strategies.

• Teaching Strategies (Center for Advancing Teaching and Learning Through Research, Northeastern University). Discussions of several teaching strategies (e.g., group work, problem-based learning) and how to implement them in the classroom.

• Resource Library (Office of Faculty Development and Teaching Excellence, University of Florida). A repository of resources “developed to support your teaching and optimize student learning.” 

Useful Books on Teaching

The following books contain many, many useful insights and frameworks you can draw on to start exploring new instructional strategies.

• Small Teaching. An excellent book that summarizes the takeaways from the science of learning and distills them into actionable components you can add in to your course that won’t take too much time.

• What the Best College Teachers Do. Based on a study of effective college teachers, this book outlines several principles for effective teaching. It correlates these with the research base for them and what we know about learning to produce a valuable resource for college (and, I would add, high school) instructors.

• Make It Stick. This classic book overviews the science of learning. It’s a must read for anyone interested in revamping their classes to promote deep learning.

• How Learning Works. Featuring “seven research-based principles for smart teaching,” this book sets forth a useful framework that you can use to tie together the various structures in your course (e.g., its grading system) in ways that promote learning for all students.

• The Spark of Learning. Many (most?) classrooms tend to focus on the intellectual development and don’t touch on–or actively discourage–students’ emotional development. This book makes the case for thinking more carefully about your students’ emotions, overviews the evidence that doing so boosts learning and motivation, and provides practical advice for bringing emotion back into your classroom.

• Grading for Equity. Grades have become the central object in the classroom, as evidenced by the frequency of questions like “will this be on the test?” and “what percentage of the grade is this assignment worth?” Such inquiries point to the power of grades to detract from the point of education: to promote learning. This book takes a deep dive into grades and grading systems, their history, and how we might grade differently to lower anxiety and promote learning.