Many teachers reading the Standards for Mathematical Practice are confused by what “Model with Mathematics” (Math Practice 4) means. I define mathematical modeling as the process of taking an often real-world context, turning it into something you can manipulate with mathematics, and then returning to the context with the new knowledge. For example if you want to figure out how many stars there are in the universe, mathematical modeling involves creating a representation, or mathematical model, for approximating the number of stars. This is often the hardest part of the process, as once you have your model you can figure out the answer. However, without the model, there are no numbers to calculate.

Fortunately, the California Mathematics Framework Chapters is out and the chapter on mathematical modeling has a very helpful section called “What isn’t mathematical modeling?” Here is the section in its entirety:

- It is not modeling in the sense of, “I do; now you do.”
- It is not modeling in the sense of using manipulatives to represent mathematical concepts (these might be called “using concrete representations” instead.)
- It is not modeling in the sense of a “model” being just a graph, equation, or function. Modeling is a process.
- It is not just starting with a real world situation and solving a math problem; it is returning to the real world situation and using the mathematics to inform our understanding of the world. (I.e. contextualizing and de-contextualizing, see MP.2.)
- It is not beginning with the mathematics and then moving to the real world; it is starting with the real world (concrete) and representing it with mathematics.

It has been my experience that teachers who read “Model with mathematics” for the first time most often think it means the second and third bullets. Students may use manipulatives or make representations during the process of modeling a situation but they are not themselves the models.

Many teachers think that they are providing opportunities for students to model with mathematical when in fact they are doing either the fourth or fifth bullets. The fifth bullet would be a situation where a teacher instructs student on a concept and then has students apply it to a real world problem. While this may be better than nothing, providing students with the real world context first gives students an opportunity to build necessary critical thinking skills as well as develop a desire to learn the skills they need to tackle the problem.

Personally, I would add a sixth bullet: “It is not beginning with a contrived real world situation; it is starting with a situation that is as close to how students would actually encounter it as possible.” For example, consider these two area problems:

- “You are making a garden that is 60 square feet. What dimensions can the garden be?”
- “You are making a garden and have a budget of $100. What dimensions can the garden be?”

Some people say that problem #1 is an example of mathematical modeling. To me, it is not a strong case as it is rare that someone knows the final area of their garden first and then can choose the dimensions. I think problem #2 can eventually lead to problem 1, but it begins with a more realistic context.

What other mathematical modeling misconceptions would you add to this list?

Hi Robert,

You do a terrific job of explaining modeling here. I first heard it explained during a PARCC item review, and loved thinking about modeling in this way, but found the explanation difficult to share in an accessible manner later. You’ve distilled the rather lengthy explanation I heard into one that is very manageable for teachers. Perfect.

Thanks,

Turtle

Thank you very much for the kind words. Also, since writing this post, I’ve come up with a one sentence summary of mathematical modeling…

Mathematical modeling is everything that happens between the problem’s context and doing the calculations.

Perhaps that is also useful.

Wow, I am coming to this part of the discussion late in terms of these posts. You just tweated something out about it though so perhaps you are interested in continuing the discussion. Robert, I wonder if you would stand by this “One Sentence” description still? I feel like it is missing something that you point out adroitly in the post above. Your one sentence here gets at the heart of the process but lacks the return to real context that you include above. Modeling is not complete until you review your results in light of reality. Did your executed model provide a reasonable answer? Are you satisfied with that answer? Can this answer be improved on with further tweaks? etc.

Chris, I agree with you. Verifying that your model makes sense and is reasonable is certainly a part of the process. It’s crazy that it has been over three years since I wrote that sentence. I’d like to believe that I intended on verifying reasonableness was a part of the process, but I can’t remember now. Thanks for the attention to detail!

(I don’t know if you’ll see a comment months after the original post, but I’ll try, anyway! I just discovered your site after reading some about you in the Northwest Mathematics Conference program.)

THANK YOU for this! After majoring in both math & chemistry, I used to be a computational chemist and worked for a software company that did molecular modeling. I started teaching around the time the Common Core standards were being adopted and was very pleased to see modeling with math in the standards, because I think students (and adult professionals) often don’t realize how choosing an appropriate model, with reasonable approximations, will let them use math to solve problems. (Of course there are also a bunch of only-too-true jokes about taking this to extremes by, for example, considering a horse to be a perfect sphere…)

However, I was starting to think maybe *I* was the one who was confused about what this standard meant, after finding out that every teacher I talk to seems to think this standard is about using area models, number line models, manipulatives, etc.

I would like to see many more examples of tasks for this practice. I think traditionally math problems are set up with the modeling already pretty explicitly laid out for the student: for example, a diagram of the wheelchair ramp is presented as a triangle with one side and the angles known, and the student is asked to find the length of the ramp. In real life, you’d have to decide what conditions were important, decide what mathematical model you would use, figure out the information you needed, make measurements, and apply your model.

Thanks Julie. I agree that this is one of the SMP that has many misinterpretations. If you are looking for more examples of tasks for this practice, check out my “Lessons” page or anything from Dan Meyer’s lessons or Andrew Stadel’s lessons at Estimation180.com.

Hi Robert. Thank you for the clarification of mathematical modeling. I wonder what your thoughts are on the difference between mathematical modeling and statistical modeling?

Kim

Hi Kim. My best understanding is that statistical modeling is a type of mathematical modeling like a square is a type of rectangle. I think it is just using mathematical modeling and applying it to a statistical question. I generally just call that mathematical modeling as well, but I guess it might have a more specific name. Thoughts?

I agree with the definition describing modeling as going from real world context to math presentation to real world again. I would add that, that “something you can manipulate with mathematics” is a quantitative model. The quantitative model that fits a real world situation is unique, but it can have multiple representations.

Interesting. So, if a student changed 46 + 99 to (45 + 1) + 99 and then 45 + (1 + 99) and 45 + 100, would that be mathematical modeling to you?

I like the use of the sixth bullet. Even though it isn’t required, it seems to push the problem in a direction that engages kids. Without that bullet the problem seems naked and doesn’t provide the critical thinking within children. A proper question like the one stated in the garden problem, should generate more questions from the student. The use of their language shows they are thinking abstractly and their modeling and use of mathematics will come inherently once they are satisfied with the information they feel is needed to solve their dilemma.

Thanks Jaelani. I’ve been thinking a lot about the role of student questioning and data acquisition in mathematical modeling. I will give what you wrote more thought.

Hi Robert

Thanks for floating this question of what constitutes mathematical modeling. I’m wondering how Cathy Fosnot & Maarten Dolk’s thoughts on mathematical models fits into your idea of mathematical modeling, if at all. In their Young Mathematicians @ Work series of books, they talk about mathematical models undergoing a transition from a model of a student’s thinking to a model for student’s thinking.

Does your idea fit anywhere in this paradigm? To me, it seems that what you’re saying about models and manipulatives is that it’s not enough for a model to simply be a representation of how someone did some thinking about a problem.

Rather, the process of modeling occurs when the model is utilized as a tool for thinking. Thoughts?

Hmm. I’m not familiar with their work and am not really in a place to comment.

I have a new framework coming out soon in a blog post series called “Spies and Analysts”. That might add to the conversation.

Sorry I couldn’t be more helpful.

I hate to even bring up the standardized testing we are administering into this extremely helpful discussion but in light of the importance placed on these assessments today, how do you see mathematical modeling being tested in these situations? We specifically use the Smarter Balanced Assessment.

I think that the reality is mathematical modeling is very challenging to authentically assess. Smarter Balanced (which we also use in California) uses performance tasks, but those only loosely resemble true mathematical modeling.

In general, I believe that if we help our students become better overall problem solvers, success with standardized tests will follow.

Nice to see you at NCSM. Thank you for sharing your thinking regarding Modeling in Mathematics. I loved the words “modeling is a process.” In my head, it is iterative. It is required when there are multiple feasible/possible solutions. Think FedEx or UPS trying to decide the best routing for delivery including breaks for the driver. I think the words (modeling and model) get muddy when we confuse “modeling (a process) and a verb” with “model (a thing) and a noun.”

Thanks Linda. You might also enjoy this post (https://robertkaplinsky.com/mathematical-modeling-need-better-spies-analysts/) and the related examples (https://robertkaplinsky.com/tag/math-modeling-can/).

If you ask in #1 that you want to put a fence around the garden with minimum fencing what should be the dimensions of the garden? could you know ahead? what would happen if the garden was 64 sq. ft?

I’m not sure I understand what you are saying here John. Can you clarify what you mean?