The research article referenced earlier “Foundational Reasoning Abilities that Promote Coherence in Students’ Function Understanding” by Oehrtman, Carlson, and Thompson recommended that pre-calculus instructors spend more time helping students develop a deeper understanding of function outside of the traditional symbolic representation. Through our development and implementation of the activities we have seen that this is really needed for students both in the high school classroom and in the college setting. We have addressed this initially through tasks that could be brought into the classroom. However we wonder if this is sufficient. Are these tasks seen just of “special activities” that students do for a day? Do these activities really help them develop a deeper understanding? Or is what is really needed is a more comprehensive approach to functions when they are first introduced? As pre-calculus and calculus instructors we would like to investigate these questions further.
One of our initial thoughts was that once we gave the pre-calculus readiness test in class and we identified gaps in student knowledge we could empower them to use on line resources to address these gaps. We wonder now how effective this can be. While students may be able to use on line recourses to focus on improving procedural skills, there seem to be few resources available to address the need to strengthen their ability to see functions from multiple perspectives. We wonder if this can be skills that students can do outside of class time as it seems to us these are addressed most effectively collaboratively while in class. All mathematics instructors struggle with the issue of what do you emphasize during class time. This is true not just as the high school level but also for every college instructor. When students enter our classrooms with a symbolic background in functions we are now convinced this may not be sufficient to allow to succeed in calculus. However this requires us as instructors to provide enough time in the classroom to address these issues. We feel that students need time and space to productively struggle with these ideas during class. We personally feel that we need to make a commitment to including this as a major focus of our courses.
One big advantage of these activities is how it helps us as instructors identify areas of student misunderstandings. Many times as mathematics instructors we are surprised that students struggle with ideas we have covered in depth or perhaps even assessed earlier. Perhaps at times we felt distressed by this and wondered how they could still struggle with an idea we had emphasized. Now we “expect to be surprised” and see this as an opportunity for our students to deepen their understanding. We also see it as an opportunity for us to learn more about how our students understand.