Do you remember when I said that MP1 (Make Sense of Problems and Persevere in Solving Them) was the “mathematical practice to rule them all”? I may have lied. Or perhaps only spoken a partial truth, like when we tell second-graders that you can’t subtract a bigger number from a smaller number. The 8 Mathematical practices are all important but there are two that really capture the heart of the mathematical discipline: those are MP1 (Make Sense of Problems and Persevere in Solving Them) and MP4 (Model with Mathematics). The ability to reason abstractly and quantitatively (MP2), look for and make use of structure (MP7), and look for and express regularity in repeated reasoning (MP8) are three ways of reasoning that serve those bigger goals of solving problems and using mathematics to understand our world. In order to leverage these ways of thinking, students must construct viable arguments and critique the reasoning of others (MP3), use appropriate tools strategically (MP5), and attend to precision (MP6). That missing hyperlink is a sign of the future. Stay tuned!
Today we’re going to unpack the 4th mathematical practice: what it means, how we teach it, and why it matters. Let’s get started.
Defining Mathematical Modeling
Mathematical modeling is a cyclical process where we use mathematical tools to study, describe, analyze, and reflect on real-world scenarios. This mathematical practice encompasses the ambiguity and complexity that is often involved when mathematics is used in real careers in industry, business, or data science. Authentic modeling tasks should relieve students of the impression that math is always clean, straightforward, and definitive.
Many resources have aimed to describe the mathematical modeling process as it relates to secondary math education. In today’s blog post we offer our own 7-part framework.
Define your question: Analyze a real-world situation and ask a question about it.
Identify essential variables and make assumptions: What factors are at play here? What is important to include? What can we safely ignore to make the model simpler?
Formulate a model: “Mathematize” the situation by formulating a model. This could be an equation, a diagram, a graph, or a collected data set.
Work with the model: Use the model to make a prediction, solve an equation, or learn something new about the scenario.
Interpret results in original context: What does the model suggest about the real world scenario? What have we learned about our original context?
Assess reasonableness of results: Do these results make sense? Are they helpful? What is missing in these results? What needs further study?
Revise or report: Refine the model or summarize key conclusions and the reasoning used to make them.
How is Mathematical Modeling Different from Normal Math Instruction?
Doing math “in the real world” is not a new idea. So how does MP4 differ from the work that is usually done in math classrooms? One important distinction is that modeling is inherently messy whereas classroom mathematics tends to be neat and tidy. We have a tendency to only provide students with stripped-down contextual scenarios where only one variable matters (for example, the cost of the item) and no extraneous information is given. In this scenario, doing a couple calculations gives a definitive answer to the “problem”. In true modeling, this is rarely the case. There is complexity and ambiguity in real life situations that needs to be addressed in the model or at least acknowledged. Modelers might say things like “It’s not a perfect model because…” or “The model points to this conclusion, but it doesn’t account for _____” or “What this model fails to address is…”. Identifying the assumptions made in the model as well as its limitations is a huge aspect of the modeling process that is not often discussed in classrooms.
Another key difference is that true mathematical modeling starts in the real world and then “mathematizes” it, by unpacking the mathematical concepts behind it and then choosing an appropriate mathematical tool or method to analyze the scenario. In the traditional math classroom, we start with the mathematical skill and then see what situations it could be applied to. Whereas authentic modeling says “here’s a real problem we have, what math would be helpful to address it?”, typical dialogue around instruction says “here’s a skill students must practice, what context could this be applied to?”
Now, don’t get me wrong. There is nothing wrong with using a simplified real-world scenario to help students explore a new idea. This is in fact the idea behind using context as a vehicle for learning. We do this all the time in our EFFL lessons, because the purpose is to build conceptual understanding of a topic. Nor is there anything inherently bad about looking for scenarios that exhibit a certain mathematical relationship (like studying half-life problems in a unit on exponentials). But we need to be careful when we assume that mathematical modeling is what we already do all the time in our classrooms, and we need to consider ways we can incorporate the ambiguity and complexity of doing mathematics in the real world.
Finally, modeling includes making a model, but is not limited to making the model. When we ask students to write an equation that represents the relationship between two variables, they are doing a small part of the modeling process, but not the whole thing. Again, that does not mean it is not worthwhile to practice this skill (in fact, we think it’s really important!) but true modeling needs to do something with the model and it always needs to tie back to the original scenario.
How Do We Teach Modeling?
It has hopefully become clear at this point that we need better strategies for teaching students authentic mathematical modeling, but we also need to be realistic. Students are not going to be doing only modeling tasks all the time. They are not going to choose the mathematics they work on everyday based on a scenario they find interesting and want to explore. Can we find a middle ground? I think so!
Perhaps the biggest thing we can do is help our students adopt a modeling mindset. This means establishing a classroom environment and choosing tasks that reinforce the idea that mathematics is beautiful, messy, helpful, cyclical, open-ended, relevant, and requires creativity. Additionally, there is not always one right answer and there are almost always multiple methods for arriving at a given solution. If you’ve ever watched people playing strategic games or comparing prices at the grocery store, you know that this is true!
Furthermore, the teacher’s role is to be a facilitator, supporting students in posing and exploring problems, offering suggestions and asking questions as needed, and making connections to the instructional goal. The teacher is a resource in learning, not simply one who provides the answer.
Helping students transition to a classroom environment where they are active producers and not passive consumers requires work. Teachers can help in this transition by starting with entry-level tasks and gradually increasing the complexity of the modeling tasks. The hope is that over time students take on more and more of the responsibilities of the modeling process in contexts that are increasingly authentic and relevant to the students.
Ideas for incorporating MP4 into your classroom:
Have class discussions around:
Identifying the aspects of the scenario that are most important
Defining the essential variables
Determining assumptions that need to be made to simplify the process
Reflecting on the reasonableness of a model
Regularly ask students to consider the limitations of a model
Provide contexts and ask students to generate questions that could be answered about that context
Help students narrow down a question that is reasonable to explore and model given time, knowledge, and resource constraints
Use “almost modeling” tasks to help students build skills
Slightly less authentic contexts
Eliminating some of the variables from the get-go (simplified contexts)
Whimsical problems (contextual, but not real-world)
Have students practice some of the subskills required for authentic modeling
Representing a scenario with an equation or system of equations
Choosing a linear, quadratic, or exponential model to represent a scenario, and give reasons for the choice
Representing a scenario in a visual way (with a graph, table of values, flow chart, etc.)
Finding Resources that Support Modeling
Many of the EFFL lessons in the Math Medic curriculum can be used as is or adapted slightly to feature the aspects of authentic modeling. To get you started, here are some of our favorite modeling tasks from each course:
Algebra 1:
A Dicey Game (Exponential modeling)
Crossing the River (Using math to understand and solve a problem)
How Much Revenue Can a Waterpark Make? (Quadratic modeling)
Geometry
The Joy of Jenga (Volume and surface area)
How Far Apart Are the Exits? (Arc length)
Algebra 2
How Long Until the Zombies Take Over? (Exponential modeling)
Racquetball Bounce (Exponential modeling)
Clifford’s Cliff Diving (Working with a model)
Precalculus
How Often Should You Take DayQuil? (Exponential modeling)
It's Getting Hot Out Here! (Trigonometric modeling)
AP Precalculus
Do Females Live Longer than Males? (Constructing a function model)
Eating Out vs. Eating at Home (Regression models)
AP Calculus
Canalysis (Optimization)
Intro Statistics
Can You Predict Shape? (Modeling with different types of distributions)
AP Statistics
Barbie Bungee (Regression)
Is Climate Change Real? (Confidence interval for a mean difference)
Looking for more? Search for words like “model” “represent”, “interpret”, or, my personal favorite, “ambiguity”, in the search bar of the teacher portal.
Why Should We Care about Mathematical Modeling?
To boil it all down, mathematical modeling is about making sense of our world using mathematical tools. Even outside of math-teacher-land, we do this all. the. time. This is because we make hundreds (thousands?) of decisions every day, and weighing all the possible options tends to involve some kind of analysis. For the sake of this blog post, I started paying attention to all the times during my day when I was making a decision about my current scenario using math. Whether it was deciding whether to roll again in a dice game based on risk and reward, or determining whether it makes sense to continue my Panera Sip Club subscription, or deciding which size shampoo bottle I should buy, I was often faced with having to make a decision, and I needed some kind of math to help me analyze and weigh the options. While I might not have consciously gone through all seven steps of the modeling process, there were certainly many of them at play. What am I trying to figure out? What variables matter to me? Does comparing cost per ounce make sense here? Will this analysis hold six months from now? What am I not considering?
Math is a powerful tool for helping make sense of our world. Whether we use it in informal or more formal ways, let’s equip our students with the understanding and skills they need to wield it.