December 4, 2018
Often, when I meet someone for the first time and they learn I’m a math educator, their first reaction is to laugh (or sometimes shudder) and mutter uncomfortably: I hate math. This could have very well been my response as I encountered new friends and colleagues, but I was lucky enough to meet an algebra teacher in the eighth grade who changed my life.
Mr. Thompson taught me that mathematics is about exploration. He taught me that there are a multitude of ways to think about and tackle real-world problems with mathematics and, depending on your approach, you might arrive at completely different and equally valid answers.
Later in my high school years, I tutored younger students and tried to show them that mathematics is not a big, scary mountain that can’t be moved. It’s actually everywhere around us and it connects to every part of our lives: how cities are built, whether or not grocery stores are stocked with food, solving environmental challenges, and how our favorite baseball teams are improved and ultimately win championships.
As I worked with these younger students, I realized two things: I loved teaching, and math had the potential to be a barrier or an opportunity. I knew I wanted to spend my life making sure it was an opportunity for students to see the world in a different way.
Think back to your days in secondary school. Most of us learned mathematics as a straight line: starting at one place with a question and following a predetermined path to arrive at a final answer.
But my experience with mathematics is nothing like this. Mathematics is all about creatively solving problems and exploring different possibilities.
Rather than a straight line, mathematical modeling is more like a never ending circle. Through real-world situations, students determine how to best solve the questions in front of them. For example: how much paint is needed to paint an entire house? How can a school stay within a budget of $6 per student and meet nutritional needs? Or, which baseball team out of the 30 major league teams is the best?
Asking questions like these not only makes mathematics relevant and exciting, but demands that students do more than consider one method for solving a single equation. Modeling makes them leaders in their own learning experience and challenges them to not simply make calculations but to also consider problems from various perspectives.
This focus on critical thinking, problem solving, and developing mathematical skills is integral to preparing students for college and career, and it’s why I’m particularly passionate about mathematical modeling. It’s also why modeling is stressed in rigorous college– and career–ready standards.
Through my work as a teacher and at EdReports.org as a mathematics content specialist, I’ve learned that one key way of ensuring these standards are met, and that students have opportunities to participate in mathematical modeling, is through materials that support teachers’ instruction with problems and resources ready for classroom use.
Modeling makes them leaders in their own learning experience and challenges them to not simply make calculations but to also consider problems from various perspectives.
At EdReports, we believe in the potential of students taking the lead in solving real-world problems. We demonstrate our belief by including an indicator in the high school rubric that examines every series of high school materials we review for mathematical modeling opportunities.
As we assess mathematics programs for these opportunities, we look for tasks that embrace the idea that there could be more than one answer, and that different assumptions and definitions could lead to different results. Mathematics becomes more than just numbers and variables in these materials: it’s about the possibility those numbers represent when combined with student ingenuity, creativity, logic, and analysis.
To better understand why mathematical modeling is potentially transformative for students, it’s important to see it in action. Let’s explore mathematical modeling with the baseball question I posed earlier: which is the best team?
Determining the problem and making assumptions
The first step in modeling often involves answering open-ended questions, focusing and defining subjective words, and exploring the problem through research and brainstorming. To answer the question about which team is the best, students first need to determine what “best”means, and as they define ”best,” they need to consider the assumptions they are making in their definition.
For example: Does best mean the team with the most wins against teams of the same or better record? Does best mean the team with the most hits in clutch situations or go-ahead home runs in the seventh inning or later? Does best mean the team with the strongest pitching and defense? Or does it mean some combination of all these factors? If so, which qualities are we going to preference and by how much?
As students begin to define the problem and make assumptions, reducing the number of factors and simplifying the question will change the variables that make up the calculations and determine the conclusion they reach. Already, you can imagine that two different students could approach this same problem using different assumptions, leading to different mathematical content, and arrive at equally valid outcomes.
Defining variables, making calculations, analyzing and reporting your results
As students begin to determine which team is best, they can combine the variables they've defined with different calculations to arrive at their solutions. While certainly some kinds of questions will lend themselves to specific kinds of mathematics (i.e. algebra equations versus geometric models), the point of mathematical modeling is that students are able to exercise their own creativity, judgments, and knowledge of mathematics as they attempt to reach their conclusions.
What’s especially powerful about modeling is what happens after students get the answers they’re searching for. Then it’s time for them to report out their findings and assess the strengths and weaknesses of their process, analyze whether or not the answer they arrived at makes sense, and consider how they might have done things differently.
The point of mathematical modeling is that students are able to exercise their own creativity, judgments, and knowledge of mathematics as they attempt to reach their conclusions.
This kind of examination and reflection is not as meaningful to students if there’s a fixed, final answer with a limited number of ways to get to the solution. With modeling, not only do students have to find a “best team” mathematically, they have to defend how they arrived at such a choice. Few learning opportunities test and stretch kids in so many different ways.
Through mathematical modeling we can dispel the idea of mathematics as out of reach for most and only accessible to the lucky, talented few. We have the chance to show students what mathematics can mean to their lives but, more than that, we can show them that they are in control of solving and creatively answering the questions they have always wondered about.
Instructional materials will never be a silver bullet in solving all the challenges in our mathematics classrooms, but they are integral to providing teachers with a foundation of resources that make it all the more likely that students will get to experience mathematics the way I did in eighth grade: full of possibility and with every door open.