Last weekend I attended the Math for America Utah Fall Conference. The topic of the conference was Mathematical Modeling and the presenter was Kara Imm, Co-Director of Math in the City. My mind is still spinning a little from some of the ideas I received, but I wanted to share one or two in particular that have already had an impact on my teaching practice.
Less is More
Dan Meyer has been talking recently about how "You can always add. You can't subtract." Usually a mediocre math task or even a good math task can be made much better by removing some of the information. For example, in this task called Styrofoam Cups by Andrew Stadel, rather than presenting students will all of the information they need to solve the problem (how many cups will stack to reach the height of the door), Andrew Stadel asks students "What information would be useful to know here and how would you get it?" Since we're not in Andrew's classroom and can't actually measure the height of the door and the height of the cup, he provides us with links to pictures of that information which can be given to students when they request it. By leaving that information off to begin with, students are given the opportunity to think about the problem and why that information would be useful.
I find that sometimes when students are given a problem that gives them every piece of information they need, they have a hard time then knowing how to process the information. In contrast, when I allow students to think about and request information, they have a much better idea of what to do with it because they've thought about why the information would be important and how it relates to the situation.
Processing the Information
As students ask me for information, I ask them "Why do you think that would be important to know?" After that student gives their reasoning, I like to ask a few other students if there is any other reason they would want to know that information. I don't tell them how to use the information they request. They tell me!
Choosing the Model
Moving Along the Modeling Continuum
Each of these ideas about mathematical modeling (Collecting and Selecting Information, Processing the Information, and Selecting the Model) can be represented along continuum. Some tasks are high in one area and low in another. Sometimes all a task needs is a small tweak to make it fall a little higher on the modeling continuum.
A Final Thought
In the four classes that have done the Styrofoam Cups task, my students were able to determine the information that they needed rather quickly. However, all of those classes were honors classes. This has me wondering if I perhaps ought to make some changes before I run this with my regular ed classes next week. One thing that really got me thinking was this blog post by Joe Schwartz from Exit10A. Joe talks about working on his own to solve this swingwraps task, also written by Andrew Stadel. What fascinated me most about Joe's thinking was this image that he shared:
To help him think about how many times a swing would wrap around a pole, he grabbed a poster tube and a chain, and wrapped the chain around the tube. What a beautifully simple model! I would have never thought of doing that!
At our MfA conference, Kara Imm suggested providing students with a Tool Table. For the task we did at the conference, the tool table held paper (lined and graph), rulers, string, markers, tape, scissors. There were also pieces of tape on the wall to use as 'measuring stations'. We had each been instructed to bring a graphing calculator to the conference. Kara later told us that she wished this hadn't happened, because it limited the types of models that we constructed. She mentioned that she sometimes puts a limited number of graphing calculators on the tool table - but not enough for all students/groups to have one. Students are given full access to all the tools at the tool table, but are not told how to use them.
My students have tool baskets at their tables, in which I vary the tools that are available depending on the activity. When I run the Styrofoam Cups lesson with my regular ed math classes in a week or two, I'm going to include some cups of varying sizes. If students struggle to identify the information that they need to solve the problem, I will give them some time to experiment with stacking the cups in their tool baskets, if they choose to do so. Just another testament to the power of online collaboration and the #MTBoS! Thanks Mr. Schwartz! I would have never thought of that without you!
About the author: MaryAnn Moore (@missnarymm) teaches 8th grade math in Davis School District. She coordinates the UCTM teacher blog and is also a regular contributor to the UCTM teacher blog. Please email MaryAnn at email@example.com if you are interested in contributing to the UCTM blog.
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