By: Maria Hernández, Mathematics Instructor, The North Carolina School of Science and Mathematics, @mathmodeling

In my job as a mathematics educator, I have had the good fortune of working with high school students and teachers in various capacities. Most recently I have been teaching Precalculus and Calculus courses. As the school year winds down, I ask my students to look back over the entire year and reflect on their growth as learners and doers of mathematics. They often write about how much they value their time in class working collaboratively with their classmates, learning from diverse perspectives or novel approaches to problems.

One of my Precalculus students this year wrote, “Group work is extremely helpful. I like that we can reason through problems together and come to a conclusion, especially when we’re trying to figure out a new type of problem. It’s a lot easier when I’m hearing other people’s thoughts because we can build off of each other’s ideas in order to figure out how to solve a problem. I think these different methods of learning are all very effective in helping me learn and remember things we do in class.”

Inevitably, students write about how much they appreciate the opportunity to work on problems that involve some hands-on component or a problem that connects the math they are learning to the “real world”.

Here are a couple of testimonies from my Precalculus and Calculus students:

“I enjoy that optimization, and other modeling problems focused on real world issues. Instead of having an unrealistic problem, I can use an example that relates to something I have actually experienced, which therefore allows me to apply that same method of problem solving in the future.”

“I think problems like the hockey problem are my favorite in the classroom, because it requires us to be ‘math investigators’ where we need to solve the problem ourselves. There is no textbook telling us if we are on the right track. Sometimes homework problems and textbook problems can feel artificial, like there is a certain skill of integration or finding derivatives that it wants us to master. Application problems serve as opportunities for us to use a skill in a way I find quite fun. I love going to the board with other students and trying stuff out.”

Students also include how the struggle they experience when faced with a challenging problem lead them to feel a great sense of accomplishment when they finally solved that challenging problem.

One student wrote, “Math for me was mostly focused on doing what the teacher wanted. Doing the first problem here opened my eyes to real math problems. While trying to solve the problem, I became thoroughly confused and frustrated. Solving it with my group felt great since it felt like we accomplished something.”

The “good” stuff I am referring to in the title of this blog post are those opportunities that I offer my students – the challenging modeling problem or the chance for students to explore their own ideas and share their solutions in their own words.

The valued experiences described in these students’ reflections are what I will strive to hold onto throughout the following school year as I feel pressed for time and am tempted to skip the “good” stuff. When we hit late March or early April, my heart starts racing, and I can feel the anxiety creep in as I realize that we are running out of time. That’s when I am most tempted to abandon student-centered teaching strategies that can offer students the chance to lead the way or be the mathematical authority in the classroom. 

I think to myself, “Won’t it be better if I just revert to the old ways of teaching where I show students a procedure, have them practice the procedure and then assign 10 problems for homework? Just for today or this week, so I can cover more material and catch up.” That’s when I have to take a step back and re-read a previous student’s end-of-year reflection to remind myself that I shouldn’t skip the “good” stuff. I have to push aside the anxiety and focus on what I hope my students will remember about the course down the road or what my colleague, Dan Teague, refers to as the “residue” of our mathematics courses.

So as another school year ends, I appreciate the time to reflect on my practice and remember that including the “good” stuff is not done in lieu of covering mathematical content. My hope is that students may have built a deeper understanding of that content through their engagement in the “good” stuff and that the specific mathematical content might be included in the “residue” for the students in my courses.

In my work with teachers across the country, I have learned that making time for the “good” stuff is particularly challenging when teachers and students are subject to the pressures of  end-of-course tests and final exams scores. 

In my next blog post, I will share lessons learned on how these teachers have found ways to make incremental changes in their practice as they work to infuse math modeling into their courses and incorporate student-centered teaching strategies in their classrooms.

For examples of math modeling problems appropriate for K-16 students and support on how to implement these in your classes, please refer to The Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME) Report.