[Karen Arth] Mathematical Modeling

This is the second session I attended from Karen Arth at this conference and I’ll re-iterate again that I appreciate her communication skills. She also has a lot of good ideas, even though I disagreed with one of her premises in this particular session.

She started by asking for our definitions of “modeling” and I thought the audience’s suggestions were astonishingly perceptive. Two well-turned phrases:

  1. “Start with life, do the math, check back with life.”
  2. “Approximating the world.”

“So what I’m hearing from you guys is ‘application’,” Arth said, and I made a note to ask her a question about that later.

She gave the following Algebra II problem, which she called “traditional”:

Write the equation of a parabola going through the points (0,0), (20,50), and (40,0).

a) Use the form y = a(x-h)^2 + k.
b) Solve for y when x = 25.

Then she gave us the following revised problem, which she implied was the same task only with modeling:

McDougal’s Restaurant has a play area for children under and around their giant arch (in the shape of a parabola with negative orientation). They plan to set up a new activity that allows children to bungee jump from the arch. The manager, upon hearing of your team’s expertise, hires you to calculate the maximum stretch of the rope that will keep the kids safe. The arch is 50 feet high and 40 feet wide at the base. The jumping location will be 5 horizontal feet away from the axis of symmetry of the arch.

a) Write an equation to model the shape of the arch.
b) What’s the maximum length to which the cord could stretch to keep McDougal’s safe from lawsuits?

She had us create a graphic organizer for our work that included sections for “Notes,” “Labeled Picture,” “Table,” “Estimates,” “Assumptions,” “Calculations,” and “Recommendations.”

“If I look at your picture, if I look at your table, if I look at your notes, I want to be able to tell the whole story,” she said.

We worked on the task in groups and presented it. Then I asked my question. “Earlier you equated modeling with applications. Is there ever an applied task that doesn’t involve modeling?”

She told us about several SBAC performance tasks in the modeling strand. Bruce Grip asked if my question had been answered. I said I was still unclear on what features of those performance tasks made them modeling. Is it enough to have “real world” objects in your problem?

Our own definitions of “modeling” are interesting, of course, but the CCSS and other documents have offered their own, very specific definition, which includes, among other skills:

  • Identifying essential variables.
  • Formulating a model that uses those essential variables.

Arth had us talk about our model’s assumptions, which is a core component of modeling, but her task also gave us the essential variables (the height of the arch, the width of the base) and it gave us the model also (“the arch is in the shape of a parabola with negative orientation”). This seemed to make the task something other than modeling.

I said all of that and she replied that, given her student population, she had to offer them some of that information because the task might be too foreign and unmanageable otherwise. I’m sure she knows her students’ capabilities better than I do. Whatever our students’ capabilities right now, though, I hope we’re all moving in the same direction, towards the same definition of modeling.

2 thoughts on “[Karen Arth] Mathematical Modeling

  1. kmorrowleong says:

    I did not attend Arth’s session, but I did attend #204 with Kathy Morris and Brigitte Lahme. After working individually and then in groups on the McDonald’s problem, we created posters. normal stuff. Then the interesting piece – Kathy asked each person to take two post-its, read each poster, and then write questions about the ASSUMPTIONS that the group made as they were generating a judgment as to whether it was reasonable that 8% of American ate at McDonald’s each day. Brigitte, as a mathematician, was invited after to comment on aspects of modeling, which included:
    1) Seeking new information
    2) Addressing the reasonableness of assumptions
    3) Defining relevant language and also parameters (ie Americans, people (unique or distinct individuals?), meal, etc.)
    4) What are the upper and lower bounds of a reasonable answer?
    Apart from encountering one of the rudest young teachers I’ve ever met, it was a satisfying session and discussion that is helping me formulate what we mean by modeling.
    I was still left with my original question, however: what is the same and what is different between this form of modeling and the kind of modeling that is done by younger students with concrete manipulatives and pictures?

  2. Fawn Nguyen says:

    I attended Bruce Grip’s modeling session at UCLA and found it very helpful. The McDougal’s Restaurant problem does not fit what my definition of what modeling is. It still seems like a word (textbook) problem with all the variables given. And you pointed this out.

    “I hope we’re all moving in the same direction, towards the same definition of modeling.” Yep.

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