Mathematical Modeling: The Glass is Half-Full

Where is the horizontal half-way line?

Where is the horizontal half-way line?

By the time they get to me, the students in my calculus class have been given a chance to master a whole lot of math. Typically, though, they haven’t been exposed to many situations where the main challenge of the task is figuring out which types of mathematical tools will best model a problem, and thus, drive the method used to solve the problem.

Take, for example, the problem described here. This is a wonderful, challenging little task that seems so fascinatingly simple and yet becomes quite complicated quickly.

The task: draw a single horizontal line on the cup that represents the half-way line by volume. Almost as soon as I asked the question, you could see the wheels start spinning in the heads of the students.

"Well, it looks sort of like a cone, but it doesn't have a point."

“Well, it looks sort of like a cone, but it doesn’t have a point.”

Some took measurements and prepared to “calculate” it. But did they need a formula? How would they find it? What the heck kind of shape is this thing anyhow? How can we be sure out measurements were accurate?

Making sure measurements are accurate

Making sure measurements are accurate

Some were going to draw it onto paper. But how to they model it? Can they use a 2-D cross section? Which cross section do we use? What do we do with it now that we have it?

Modeled as a two-dimensional shape

Modeled as a two-dimensional shape

Some figured to guess and check. I mean, the half-way is probably going to be somewhere in the middle. There’s not THAT many different values, really. But what do you guess? Do you have to guess more than one thing? How do check your guess to see if it is right?

And all groups had to deal with the question: When you have finished up the work on the page or on the calculator, how do you accurately transfer it back to the cup?

So, let’s get it out into the open: being able to mathematically model a problem that exists in your hands with math that has always existed in a book is not something that comes naturally to most people. In addition to the content, students need to practice modeling the mathematics. They need to learn what the book math looks like when it is in their hands. Jo Boaler (Stanford Math Ed Professor) has a wonderful line about this.

“…students do not only learn knowledge in mathematics classrooms, they learn a set of practices and these come to define their knowledge. If students ever reproduce standard methods they have been shown, then most of them will only learn that particular practice of procedural repetition, which has limited use outside the mathematics classroom” (pg. 126, see bottom for proper citation).

It’s as if the math experiences we are giving most students in class are the kinds of experiences that will do them the least good outside class.

Authentic mathematical modeling requires giving students a task with a simple and clear goal and then letting them decide what kind of math will help to complete the task. The decision is an incredibly important part of the process. They must recognize the variety of available tools, have to choose which ones will work, and of those ones, which one is the best choice.

The #MTBoS is doing a great job of providing tasks of this kind of a nature (Like here, for example… or here, as another example). They are plentiful, well-done, and FREE! Give them a shot and see what happens. We owe it to our students to provide them opportunities to take the math off of the pages of the book and let them see what it looks like when it shows up in their hands.

Reference

Boaler, Jo (2001). “Mathematical Modelling and New Theories of Learning.” Teaching Mathematics and Its Applications. Vol. 20. No. 3, p 121-128.

Circles from Cedar Street

A sloppy bit of construction work makes for an interesting geometry question.

A sloppy bit of construction work makes for an interesting geometry question.

I was driving to the grocery store. This particular trip took me down Cedar Street. I drove past this manhole cover. It caught my eye in such a way that I decided to pull into a nearby parking lot and, when the traffic cleared, tiptoe out to the middle of the five lane road and snap a photo of it.

So, my mind instantly went straight to rotations. (Which is on my mind because transformations are Unit 1 of the Geometry Course that I start teaching in, like, two weeks.)

What if I started out rotations by showing this picture and simply asking the students how much of a rotation would fix the yellow lines.

My goals would be for the students to explore how to investigate an apparent rotation, learn how to visually represent a rotation, and struggle through the task of explaining it out loud to another person.

It would be okay with me if we made it to the convenience of using degrees as a descriptor of how much something is rotated. That would be up to them. I do suspect that after a short time of “about this far” and “about that much” they’ll like to find something to ease the trouble of explaining the transformation.

If you can think of a way to frame this learning opportunity better, please make me a suggestion. I feel like this is a good opportunity that I don’t want to waste.

Penny Circles

This discussion represents the final Makeover Monday Problem of the Year. I must admit, I found this problem to be easily the most engaging for me simply as a curious math learner. (Here’s  the original problem posted by Mr. Meyer.)

Now, I will admit that this is going to be more of a discussion of my experiences exploring this problem and, if you get lucky, I might come up with a recommendation somewhere toward the end.

The problem presents a rather interesting set up: filling up circles with pennies, making predictions, modeling with data, best fit functions. Lots of different entry points and possible rabbit holes to get lost in.

First change was that my circles were not created by radii of increasing inches. For the record, I can’t quite explain why I chose to make that change. Instead, I used penny-widths. Circle one had a radius of one penny-width. Circle two’s radius of two penny-widths. Circle three’s radius… you get the idea.

It looked like this:

I started with 4 circles with radii based on penny width.

I started with 4 circles with radii based on penny width.

Note: I understand the circles look sloppy. I had wonderful, compass-drawn circles ready to go and my camera wasn’t playing nice with the pencil lines. So, I chose to trace them with a Sharpie freehand, which… yeah.

Next, I kept the part of the problem that included filling each circle up with pennies.

It looked like this:

Penny Area 1

Then I filled them with as many pennies as I could

Then I filled them with as many pennies as I could

Here’s where the fun began.

I saw a handful of different interesting patterns that were taking place. If we stay with the theme of the original problem, then we’d be comparing the radius of each circle to the penny capacity, which would be fairly satisfying for me as an instructor watching the students decide how to model, predict, and then deciding whether or not it was worth it to to build circles 5, 10, or 20, or if they can develop a way to be sure of their predictions without constructing it.

But, I also noticed that by switching to the radius measured in penny-widths, the areas began to be covered by concentric penny-circles. So, we could discuss how predicting the number of pennies it would take to create the outside layer of pennies. Or we could use that relationship as a method for developing our explicit formula (should the students decide to do that).

So, if I am designing this activity, I feel like the original learning targets are too focused and specific. I want the students to be able to use inductive reasoning to predict. Simple as that. If they find that using a quadratic model is the most accurate, then cheers to them, but this problem has so much more to offer than a procedural I-do-you-do button pushing a TI-84.

First, I’d start with them building their circles using a compass and some pennies. Build the first four and fill them in. I’m quite certain that the whole experience of this problem changes if I leave the students to explore some photos of the my circles. They need to build their own. Plus, that presents a low entry point. Hard to get intimidated lining up pennies on a paper (although, I have plenty of student intimidated by compasses).

Next, I think that I would be if I simply asked them to tell me how many pennies they would need to build circle five and to “prove” their answer, I believe we would see a fair amount of inductive reasoning arguments that don’t all agree. The students battle it out to either consensus or stale-mate and then we build it and test. It should be said, though, that as they are building the first four and filling them in, I’d be all ears wandering about waiting for the students to make observations that can turn into hooks and discussion entry points.

Then I would drop circle ten on them and repeat the process. This is where the students might try to continue to use iterations if they can figure out a pattern. Some will model with the graphing calculators (in the style the original problem prefers). I’m not sure that I have a preference, although I would certainly be strategic in trying to get as many different methods discussed as possible.

Then, when they have come to an agreement on circle ten, I’d move to circle 20 and offer some kind of a reward if the class can agree on the right answer within an agreed margin of error. Then, we’d build it and test (not sure of the logistics of that, but I think it’s important).

All that having been said, I see an opportunity to engage this in a different manner by lining the circumference of the increasingly-bigger circles with pennies so that the circle was passing though the middle of the pennies and seeing how, as the circles got bigger, how the relationship of the number of pennies in the radius compared to the number of pennies in the circumference. I suspect it could become an interesting study on asymptotic lines.

It just seems like this particular situation has, as Dan put it, “lots to love” and “lots to chew on.”

The Essentials of Project-Based Learning

On Thursday, I will be facilitating a collaborative session for teachers on Project-Based Learning, which, to be fair, is a topic that I have never considered comprehensively. I’ve never been asked to define it. I’ve never been asked to explain it and, quite frankly, until the organizer of EdCamp Mid-Michigan approached me to facilitate the session, I never considered myself a practitioner of it. I just tried to design lessons that were engaging and built authentic, lasting learning.

But, this EdCamp facilitator, who’s name is Tara, has known me for quite a while. We did our undergrad together. We did our Masters work together. She’s been on an interview committee that almost hired me. She’s aware of my shtick. Maybe she wants me to facilitate this through the lens of how I work. If that’s the case, then this takes on an extra degree of difficulty because I still feel like I have a lot of development left to do before people start emulating my approach.

Besides that, my approach isn’t really that complicated. For any activity I consider offering to the students, I simply try to ask a couple of questions.

1. What can I do to maximize engagement? (I’ve seen plenty of ideas that I have had flop simply because the students aren’t drawn in.)

2. What can I do to facilitate collaboration? (The most effective use of class time that I ever see is when a group of students are effectively working together to solve a problem.)

3. What can I do to create a problem that will provide a variety of ways to solve it? (If there is only one real method to solve it, then as soon as one student gets it, the “collaboration” will become that student communicating “the way to solve it” to everyone else. My favorite evidence of this is when I hear a few “I don’t get how they did it, but this is how we did it and the answer came out pretty close to the same.”)

4. What can I do to make sure that the solution(s) are approaching effective learning outcomes? (While it can be interesting to occasionally have a group find a way to solve (or estimate a solution to) a problem effectively in ways that circumvent my desired learning outcomes, if I am continually putting together problems that don’t push the curriculum the community is trusting me to teach, then my students will be missing something.)

My favorite examples of this process working well are The Lake Superior Problem, The Pencil Sharpener Problem, The Wedding Cake Problem, and the Speedometer Problem. In each case, engagement and collaboration were high. Multiple solutions were present and most of the solution techniques required the students to make sense of mathematical procedures that could be correct or could be incorrect. “Why this formula is better than that other one.” Or “neither of our answers are perfect, this this one is better because this process has less of a margin for error.” Stuff like that. These conversations provide great opportunities to move beyond the memorization of a formula and to push toward sense-making of how that formula is used effectively.

While I’m not sure if this is “by-the-book” project-based learning, I have seen improvements in my students’ learning through the lesson design process I described above. Does anyone have any advice for me? Questions for me to think about? Corrections? Obvious holes in my logic? I appreciate this community because of your willingness to share, so feel free to do so.

Learning from Playing Around

These past two weeks have been an awesome time of learning for the students we’ve been working with, but I’ve also done a bit of learning myself.

I’d like to have my students love math and science and naturally be interested in it. But they’re kids. They would prefer to play. People get the most out of that which they put the most in to. If given the chance, students will put a ton into playing around.

These past two weeks I’ve been working with upper elementary-aged students. I normally teach high school students. I’m not sure if the age difference changes anything. The stuff they want to play with might be different, but not the desire to play.

And after 7 years of teaching math, there’s something appealing about a situation where students will be voluntary and enthusiastic participants.

Play Science 1

I have just spent two weeks watching students play with two activities. The first was an activity called “Table Timers” where they were challenged to design and construct an apparatus on a table top that reliably moved a marble down an inclined table in ten seconds. Second, The Helium Balloon Problem challenges students to keep a helium balloon rising, but to have it travel as slowly as possible upward. Not every group worked well and not every group achieved these goals, but the engagement level has been high. I suspect this is because they we allowed to play.

Here’s what caught my attention the most: In the midst of their play, the students demonstrated some authentic problem-solving techniques. They had to identify the major challenges to their goal, which they often did. They had to brainstorm possible ways to overcome the challenges, which usually took the form of raking through a tub of blocks or looking through the supply table. They discerned which seemed like the most realistic and then test. Following a test, they discussed what happened, why, and revised. And the students were often quite excited when they got the right answer (knowing themselves that it was right and not relying on me to tell them).

That’s a pretty good learning model. That’s something that I have a hard time getting my students to do with book work.

Play Science 2

So, the sharing, the idea-making, the consensus-building, the authentic assessment are all good things. Obviously I am not simply advocating letting the students play around all day. But perhaps by using play, we can improve engagement and the students seem to more naturally fall into a more authentic problem-solving mindset. When I consider helping them draw out the learning, some thoughts come to mind.

First, it seems like during the whole process of exploration, design, construction, testing, revising, and demonstrating, there needs to be an abundance of contents-specific vocabulary. The marble didn’t “bonk into that block.” That block “applied a force” to the marble. Students don’t “figure out how big the shape is.” The “find the area” or “circumference” or “volume.”

Second, students don’t seem naturally inclined to take data or to keep records. In the past two weeks, it seems that students are avid experimenters and do a pretty good job of verbally analyzing the problems if the plan didn’t work. Practically NONE of them documented anything on paper. No sketches, no data, no records of updates. This is an important part of the problem-solving process that would have to be established as a norm.

Third, the activities have to be tiered. Video games are great at this. The entry point tends to be quite low. The first couple of levels are pretty manageable and then the intensity and difficulty pick up. People get locked into video games through that model and people get unlocked quite quickly once the game has been beaten. Both Table Timers and The Helium Balloon Problem worked with this model. Then entry point was low for both activities and it was easy enough to begin to approach the goal, but perfecting the design and executing the plan took much more care. Then, once they hit ten seconds, we’d challenge them to add five seconds to their timer.

Fourth, I think that the groups need to be expected to summarize and present their work to each other and to field questions from the class. Class norms should allow for questioning of each other’s work and students can learn a lot about their own design, but also about the content when they know that they are going to have questions coming from their peers. Also, it would seem like this would encourage more thoughtful designs, too. Besides this, idea sharing gives the students an opportunity to look at other designs, integrate specific vocabulary into more regular use, and get the students comfortable with collaborating.

 

Play Science 3

I don’t think that playing around is the answer to everything, but I know that in my own experiences, it seems to be the forgotten learning model and if I’ve learned anything these past two weeks, it’s that an environment that produces enthusiastic student participation shouldn’t be ignored.