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How to Start the Year in AP Calculus

Updated: Aug 15

I love the beginning of the school year! The first few days of school set the tone for the year and help acclimate students to the ways we will interact with each other and with new learning. If the hallway buzz is true, then AP Calculus is one of the hardest high school courses, and students come in wondering if they have what it takes. My goals in the first week of school are to build the kind of classroom community that will support deep learning and allow students the space to struggle, take risks, and respond resiliently to the emotional and cognitive struggles that will inevitably come their way. For the students who have always been successful in previous math classes and who have little tolerance for imperfection, it becomes especially vital to instigate a new kind of culture around learning, problem solving, and academic growth.


The activities I choose for the first week in AP Calculus are centered around the following overarching goals:


  • Establish a culture of care and build trust: We know from neuroscience that feeling safe in an environment is essential for learning and risk taking. Throughout the school year we will ask our students to share ideas in their rough-draft form, to present ideas to the class, to give and accept feedback from peers, and to leave their comfort zones to wrestle with challenging content. All of these have some level of social and emotional risk associated with them and we can not expect our students to engage in these ways if they do not first feel safe, cared for, validated, and a sense of belonging.


  • Shift mindsets from knowing to learning: Many students come to us expecting math class to consist of receiving information in the form of a lecture, doing practice problems, and then memorizing as much as humanly possible the night before the test. Calculus students have usually been successful in this model but we know that the AP Exam requires students to think deeply, flexibly, and apply conceptual understanding. We need to establish this class as a thinking class where students engage in the messy, non-linear, idiosyncratic process of problem solving. For more on this, I would recommend Peter Liljedahl’s fabulous book Building Thinking Classrooms in Mathematics (Corwin, 2021). Not only will students need to think differently but they will need to work differently. Discussion among group members will be the primary tool for introducing and processing new information in this course.


First Week of School

On the first day of school, we have students sit in assigned seats in groups of four. Not knowing where to sit or having to choose a seat without knowing anyone in the class is a weighty and anxiety-inducing task for some of our students. Even high schoolers deal with nerves on the first day of school, so we want to eliminate as many potential threats as possible to make students feel safe and excited for the school year.


I generally start with a quick (5-10 minutes) get-to-know-you activity. I share a little about myself to establish trust, then quickly turn to having students introduce themselves to their group members. Try to be as explicit as possible with what information you want them to share, and avoid any questions that might be triggering or too personal. The goal here is not deep connection, but safety and rapport. Here are some of my favorite ice breaker questions.


Next, we jump into a mathematical task. These are tasks that require deep, non-routine thinking and encourage collaboration. Here are some of my go-to activities:


  • NRICH Short Problems--Interesting problems of various lengths and challenge levels; I generally choose the ones related to rates and averages to start the year in AP Calc

  • Bike Shop

  • To Run or Not to Run? (I extend the question so that students have to represent their thinking using diagrams, colors, words, tables, equations, graphs, etc.)

  • Don’t Be Late

  • Revolutions

  • Well Read

  • Jo Boaler’s Galileo Investigation (from her Exploring Calculus resources)

  • Open Middle Math--this is a fantastic puzzle structure that I like to introduce early in the year, before we do Open Middle puzzles with actual Calculus content

  • The X-Y Game--this was taught to me by my colleague, Barb Montgomery, and it is a fun way to help students see the importance of collaboration. You can modify the number of rounds if you choose. The reflection at the end is important!


Days 2-3 continue in a similar manner, with a short community-building activity and then jumping into a task. We use tasks to teach about group norms and class norms. While we do have to make time for some school-wide initiatives like PBIS and pre-testing, we try to fit these around the other tasks we’re already doing.


When should we talk about the syllabus?

We generally don’t spend more than 10 minutes talking about the syllabus (and not before day 3!). Many of the items on the syllabus can be shared on a need-to-know basis as we get closer to the first test, start assigning homework, etc.. Students are being inundated with grading policies and rules in all their classes at this time of the year, so memory of these conversations tends to be low, and many things are not immediately applicable.


By day 4 or 5, we start with our first Experience First, Formalize Later (EFFL) lesson:

A Wonder-fuel Intro to Calculus!


Is spending time on non-curricular tasks worth it?

As AP teachers, we know that the standards are many and the minutes are few. While it’s tempting to dig into content as soon as possible, we are convinced that spending this time up front to establish class and group norms and to set the stage for the deep thinking we will be doing all year is absolutely worth it. In our experience, students are much more willing to engage in our EFFL lessons, share their thinking, and get to work quickly, after having these first week of school experiences. We have to go slow to go fast!


Additionally, you may have noticed we don't spend the first couple weeks reviewing ideas from Precalculus. Instead, we build students' Algebra skills throughout the year with Calculus content. The tasks mentioned above for the first week of school will also help students review key ideas about rates, averages, fractions, and proportional reasoning. All of this saves time and allows us to move on more quickly to developing deep understanding of Calculus.

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