We know it isn't this...
but what is it???
Since a math researcher told me about the work of Richard Lesh, I’ve been reading everything I can get my hands on about modeling in math. One thing I’ve learned is that modeling is a versatile and deceptively simple word! Even the experts use the same word to mean different things in different contexts. (Check out the TCM article, Common Core Confusion about Modeling, for instance!) So first, let’s start with some working definitions. I’m going to focus on two specific types of modeling that I find particularly useful in my classroom: Modeling with Mathematics and Mathematical Modeling.
What is Modeling Mathematics?
Modeling mathematics is using physical or virtual objects to represent math concepts. You’re probably already encouraging students to do this as they make tables, draw pictures, and use manipulatives like Cuisenaire rods, counters, linking cubes, etc. Modeling mathematics is using mental and physical tools to make a mathematical situation clearer. It helps develop math literacy as students fluently travel between multiple representations.
My guess is that you’re more familiar with this type of modeling, so let’s spend the majority of our time clarifying mathematical modeling.
What is Mathematical Modeling?
Mathematical modeling is where the perfection of theoretical math meets the messiness of reality. It’s taking a practical situation, using math to describe and answer questions about it, and, finally, making sure that the answers you come up with make sense back in the context of the practical situation. According to the GAIMME Report, it’s “a process that uses mathematics to represent, analyze, make predictions or otherwise provide insight into real-world phenomena.” It involves constraining a real situation to the aspects you care about so that theoretical math can help solve a problem. It requires making assumptions about some things while neglecting others altogether. In other words, there are value judgments throughout this kind of modeling and there is no one “right” answer, only models that are more or less useful depending on what you need to know. (There’s a great MTMS article, Challenging Students to Make Assumptions, and I love the modeling cycle they use, too!) Assessing, creating, using, and evaluating is mathematical modeling.
Note the emphasized phrases--modeling math starts and ends in the math world whereas mathematical modeling begins and ends in the real world.
What Doesn’t Count as Mathematical Modeling?
Now that you know what it is, what doesn’t count? “Modeling” something for your students isn’t the kind of thing we’re talking about. Likewise, simply making a physical model isn’t either. And while I know you’re all “model” teachers, using the word as a synonym for exemplary also doesn’t count. Finally, don’t be fooled by the marketing of many textbook “real world” or “problem-solving” questions. If the students aren’t translating between the real and math worlds and back again, then those aren’t examples of modeling either. Check out Dan Meyer’s NCTM talk on Fake World Math and Robert Kaplinsky’s article Beware of Fake Math Modeling Problems for more.
There is a powerful chapter in NCTM’s Mathematical Modeling and Modeling Mathematics called “Moving Students from Remembering to Thinking: The Power of Mathematical Modeling.” It gives some examples of authentic modeling tasks and how they promote student thinking and student-invented algorithms instead of student remembering and teacher-invented algorithms. They conclude that “what is learned through mathematical modeling adds a new texture to students’ view of mathematics and of themselves as students of mathematics” and that “the mathematical understanding that students develop when they are ‘thinking’ and not merely ‘remembering’ is more powerful, more flexible, and longer lasting.” I think this positive identity development as mathematicians and as people who can take messy raw scenarios and use math to persevere through meaningful and initially challenging questions is the ultimate reason to teach mathematical modeling in our classrooms. This is what will help our students see math as useful in their everyday lives and as a valuable cross-disciplinary skill, especially in regard to STEM careers.
On a more pragmatic note, modeling is an avenue for us to implement a more integrated curriculum by addressing multiple standards simultaneously. You probably know by now that CCSS devotes an entire standard of mathematical practice to both kinds of modeling. The first half of CCSS SMP 4 refers more to modeling math while the last half better describes mathematical modeling. Furthermore, on pg. 8 in Progressions for CCSS in Mathematics, the authors argue that, like statistics, modeling is a capstone that addresses all eight SMPs and they go on to develop a case for each. I would add that modeling naturally connects with all of NCTM’s standards as a “connected body of mathematical understandings and competencies” (pg. 29 of Principles and Standards for School Mathematics) as well as the Equity, Learning, Assessment, and Technology Principles. In addition, consider the connection between mathematical modeling and NGSS Practices 2, 6, and 7. I especially like this phrase from pg. 6, “Students can be expected to evaluate and refine models through an iterative cycle of comparing their predictions with the real world and then adjusting them to gain insights into the phenomenon being modeled.” That’s exactly what we said about mathematical modeling! Teaching math concepts as naturally connected to other subjects reinforces the relevance of math to students and, practically, we can accomplish this curricular approach by viewing modeling as a vehicle for STEM integration as we promote, borrow, lend, and exchange the standards of traditionally siloed subjects.
We’ve covered a lot of territory here, so let’s recap:
Making tables, pictures, graphs, etc.
Using Cuisenaire rods, counters, etc. to represent
Promotes math literacy through representational fluency
Uses math to describe and answer real world problems
Students do the work of translating between real and math worlds
No one “right” answer
More than problem-solving without context
Teaching students to think, not remember
Makes math relevant and useful to students
Addresses multiple national standards at once
Vehicle for STEM Integration
What’s coming in the next three posts?
Modeling Math Using PhET Sims
The Role of Teacher and Technology: Mathematical Modeling Using Desmos
Examples of Effective Modeling Activities
P.S. - By the way, it actually can be this…
Check out Robert Kaplinsky’s activity on How Much Bigger They Should Make Zoolander’s School. :o)