The first thing you have to understand is that asking people to model with mathematics makes them mad. Not in all contexts, though! At a social gathering with a generally amiable and curious group of people, you might try floating a question like:
- I wonder if graduates of more expensive universities tend to earn more in their careers?
- Do you think the time it takes a pendulum to swing back and forth depends on how heavy it is?
- What do you think is the most efficient way to get 2,000 calories a day?
If you catch your audience on an auspicious day, they might be inclined to discuss some different constraints, refine the question and ask a more specific one, look some stuff up, or propose a range of reasonable values. Modeling with mathematics or statistics is a human endeavor and can be fun.
But I’ve noticed that if you ask people to model with a mathematics in a context where they are expecting to learn mathematics, it can incite rage. You quickly run into strong objections, such as:
- You haven’t given me enough information!
- There isn’t a right answer!
- It depends on too many things!
- I am uncomfortable with the ambiguity of this task! (<– direct quote)
Unfortunately, this state of affairs left our high school writing team in a pickle. State and national standards require that high school students engage in some authentic mathematical modeling—even that they engage in the complete modeling cycle (see page 7 of the modeling progression if you need a refresher on what the modeling cycle means). There was no getting around that we would have to present students with these types of tasks that make them—and some of their teachers—uncomfortable. There was also no getting around the fact that many math teachers actively avoid posing questions that require students to make authentic choices. On top of that, a single class period of 45–60 minutes isn’t enough time to engage in the entire modeling cycle, but our curriculum architecture is composed of lessons that come to some kind of resolution within a single class period.
What to do?
Well, in IM’s high school curriculum, we invented a new kind of curriculum component called a modeling prompt. It is “new” in the sense that it’s not a lesson, assessment, nor practice problem. These are prompts that are meant to be launched in class, worked on with a team, and facilitated by the teacher. But it’s expected that students might need to work on them outside of class. An analogy might be a research paper or creative writing piece in another class, where students might have time to work on the paper in class, but it’s sort of going on in the background while they are attending daily classes about punctuation or the causes of World War II or whatever. That solved the “impossible to do modeling in one lesson” conundrum.
When should a teacher assign a modeling prompt? When it makes sense in their schedule! We just want to make sure students have enough math under their belts to approach the prompt, so for each, we let you know after which lesson it’s safe to assign the prompt. (See the “use after” information on these Algebra 1 prompts, for example.)
But how to make it more likely that teachers would choose to assign them, and be successful when they do? A few things.
Each prompt comes with a few different versions that we describe as a lighter lift or a heavier lift. The idea is that when students (and possibly the teacher) are less-experienced modelers, prompts that are a lighter lift will feel more doable and students might be more willing to engage with them. Over time, as everyone becomes more comfortable with the idea of modeling with mathematics and sees several examples of what it looks like to do it, teachers might offer a version of the prompt that is a heavier lift. Or when appropriate, teachers could ask students to choose the version of a prompt that’s an appropriate challenge. For examples of how these different versions of the prompts vary, please check out any of our modeling prompts. You might notice that each version comes with a “lift analysis,” which is a way of communicating how the prompts vary along five different dimensions of modeling with mathematics. If you’re curious about that, more information is available in the teacher guide.
Also, feedback from our pilot teachers indicated that it would be beneficial for students to first see an example of what a response to a modeling prompt might look like. It’s hard to imagine what you’re supposed to do here, if you’ve neither done it before nor seen it done. To that end, we developed Modeling Prompt 1, available in each course, where students are presented with a response to a modeling prompt, and asked to understand and evaluate the response using the same rubric that could be used to evaluate their responses down the road. In addition to a rubric, we also provide some tips on modeling for students and many suggestions for successfully facilitating modeling for teachers.
Finally, there are many activities throughout the lessons in the curriculum labeled with the “Aspects of Mathematical Modeling” tag as an instructional routine. This tag highlights activities where students have an opportunity to flex some of their modeling muscles without engaging in the full-blown modeling cycle.
So, how is this new, unfamiliar thing going over? Hard to say. The pilot teachers that embraced the modeling prompts loved them, and appreciated that there was a bit of an on-ramp to getting their students to engage in some authentic modeling. Many pilot schools didn’t report trying the modeling prompts in the first year due to the demands of implementing a brand new curriculum, which makes sense. We encourage schools using the IM high school curriculum to make a plan for eventually incorporating them, though. If students aren’t engaging in the full modeling cycle, you’re not really addressing the high school standards, after all. So give some a try when you’re ready! Please let us know how it goes, and share any suggestions for making them even better.