I was recently playing “Simon Says” with one of my friend's young kids. They found the game very entertaining, their brows wrinkled in concentration to make sure they didn’t accidentally do something that Simon didn’t say. I enjoyed playing with them too, but it also got me thinking about the ways in which school can seem like one big game of Simon Says for some of our students. Just do what the teacher says and follow the example problem! Watch out for the trick questions!
I don’t want mimickers-- students that can replicate worked examples -- I want students to have a deep understanding of how the big ideas of Calculus are related. I want students who can make sense of problems, communicate mathematically with one another, and think flexibly about how they can apply a known concept to a new type of problem. By putting activities in front of our students where they have to figure something out without being told how, we are positioning students as capable doers and knowers of mathematics and promoting lifelong learning. This is why the Experience First, Formalize Later (EFFL) model works well even in (and I might add, especially in) a class like AP Calculus.
I like to consider AP Calc instruction as consisting of five buckets.
EFFL lessons fuel sense making and build conceptual understanding of key Calculus concepts. We introduce every new topic with this approach.
Homework helps build procedural fluency and allows students time to process and practice new ideas.
Warm-ups can be used to clarify misconceptions, review ideas from previous days or units, and can be used to highlight particular problem types.
Assessments expose students to the rigor of the AP Exam by including multiple choice and free response questions, with and without the use of calculators.
Released free response questions give students opportunities to reflect on scoring guidelines and what counts as justification; later in the year, they offer mixed review of previous units.
EFFL lessons coupled with meaningful practice and intentional exam review are the primary ways we prepare students for the AP Calc Exam. But even more important than each of these individual buckets is when we use them. The goal is to gradually move students towards complex, and rigorous problem solving.
How Do We Build Rigor in an EFFL Classroom?
You have probably noticed that the activity on the front page of the EFFL lesson (the “Experience” portion) often has a very low floor and builds up gradually. The numbers generally work out nicely, and the functions themselves aren’t overly complicated. This is on purpose! When introducing a new topic, we want students to make sense of the Calculus concept before considering more difficult cases that require a lot of Algebra or Precalculus. This is to keep students engaged, thinking conceptually, and to avoid students quitting because they’re too frustrated. We begin by focusing on the why rather than the how.
The Check Your Understanding (CYU) questions build from there, offering more traditional Calculus questions and incorporating more algebra skills. By being intentional about when and where we add rigor, we allow students to build confidence in their skills and pinpoint where their specific struggles are. While it can be tempting to expose and prepare students for every type of problem they could encounter early on, we run the risk of losing student engagement and seeing struggling students fall further behind. If the problems are too hard too soon, students face too many obstacles to getting themselves unstuck. If we instead start by building the concept so that all students have access, and then find students struggling with a more challenging problem in the CYU, we can pinpoint what specific aspect of that problem is causing them trouble. Most likely, there are some gaps in a student's understanding of, for example, the logarithmic function, and not in their understanding of how the first derivative gives information about a function's behavior. I always try to be explicit with my students that much of what makes Calculus difficult is not the Calculus itself, but applying all the knowledge from previous courses.
Can We Ever Use Explicit Instruction in an EFFL Classroom?
A common misconception is that the EFFL model allows no room for direct instruction, and thus can’t fully prepare students for everything on the AP Exam. The question is not whether we should use direct instruction, but when. Instead of preteaching students all the formulas, theorems, and definitions and then having them try problems on their own, we have students start in groups to explore and make sense of big ideas before the teacher layers on the formal vocabulary, notation, and formulas. While students may not discover the four things needed to earn full points on a “interpret a derivative in context” question, they can certainly be asked to make sense of instantaneous rates of changes within a context before being officially taught how to do so. The “Formalizing Later” stage allows the teacher to make explicit note of new vocabulary and the necessary elements of a complete response.
Conclusion
Peter Liljedahl explains that as teachers, we have the tendency to design our lessons in ways that will allow students to answer the hardest question of that type they will face (we’re AP teachers, we know which questions he’s talking about). But when we consider students’ learning progressions, we realize they only need enough to get started on the first task. This, in turn, will provide the new learning that they can take to the next stage. The information students need to get started on a new topic, even in AP Calculus, is surprisingly little. We have to trust the process!