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 first draft of 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?