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.
In my last post, I mentioned that I read a ton of math teaching blogs and am constantly collecting ideas from them. Great math tasks are just one of the many things I glean from reading math blogs. Since I'm attending a conference this weekend and have been asked to bring one of my favorite tasks, I thought I'd take this opportunity to tell you about it too.
Noah's Ark by Fawn Nguyen
(click on the link above to check out Fawn's original post about this task and download a Word Doc of the task)
Mr. Noah wants his Ark to sail along on an even keel. The ark is divided down the middle, and on each deck the animals on the left exactly balance those on the right — all but the third deck. Can you figure out how many seals are needed in place of the question mark so that they (and the bear) will exactly balance the six zebras?
This task is pure gold. It's problem solving; it's balancing equations; it's systems of equations. And there are no numbers. A sixth grader could do this task (it was written by a 6th grade teacher), but it is engaging enough for a calculus student. Here are a few snippets of dialogue from my students as they were working on this task last year.
"So one bear equals three zebras."
"How many seals does the elephant equal?"
"Can you have half a kangaroo?"
"Get a smaller kangaroo!"
Student: So there are 6 zebras on that side, so then we take zebra out.
Me: Wait, you can't just take a zebra off one side. Your ark isn't balanced any more.
Student: Well, we're not really taking it off. We're just saying 'Sit down, Zebra. Wait for us to catch up.'
This all sounds completely crazy, but believe me, it's very fun. Last year, I gave the task right before Thanksgiving. Many of my students figured it out in class, but for those who didn't, I informed them that it was NOT homework and they didn't have to do it over the break. The following Monday, a few of those students complained to me that I had ruined their holiday. They spent hours working on that task because they just had to figure it out.
Actually, Noah's Ark sort of took over my own family Thanksgiving last year too. My oldest nephew, Max, was in eighth grade at the time, so I printed out a copy of Noah's Ark and handed it to him before dinner. Intrigued by Max's intense frustration/determination, one by one, each of the adults in my family found a copy of the puzzle and started working on it. In case you think this sort of thing is a regular occurrence in my family, believe me when I tell you that I'm the mathy oddball in my family. We don't sit down and do math together. Ever. It was one of the most strangely awesome holidays I've ever spent with my family.
About the author: MaryAnn Moore 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 firstname.lastname@example.org if you are interested in contributing to the UCTM blog.