Monday, July 30, 2012

Seeing Less is Seeing More

I'm a couple of years late to the party on this one, but the other day I watched Dan Meyer's 2010 TED talk and had a rather salient "AHA!" moment. There are many takeaways from this talk, but here's the one that really stuck with me: Seeing less is seeing more.

To get an idea of what I mean, watch the video from 4:33 to 6:28 as Dan talks about the "structural layers" of a ski lift problem. As presented in the textbook, the problem lists out the specific steps that students need to take to solve the problem. What Dan does is take away the steps and the mathematical structure, leaving only a visual and a simple question: Which section is the steepest?

Stripping away the layers and leaving only two things -- the question and the scenario -- leaves ample room for discussion, debate, and ideas. (For my purposes here, question refers to the problem the student is asked to solve, in its simplest form. Scenario refers to the situation being modeled in the problem, in its simplest form.)

I admit, in my first three years of teaching, I've given my students so very many math problems that have self-contained instructions for how to solve them. It's not something that lends itself well to deeper learning, and that's something I'm trying to work on this upcoming school year.

I'm trying to look at math problems in this new light, which is to strip away the layers and leave just the question and the scenario. As Dan Meyer demonstrated in the ski slope problem, you certainly can create more room for mathematical discourse and problem-solving.

But... what if you go even further, and remove the question, leaving only the scenario?

Seeing Less: Baseball Diamond Racing Problem

Full disclosure: I am a lifelong Chicago Cubs fan. From time to time, I like to write math problems about baseball. Here's one such problem I wrote for a Pythagorean theorem unit a few years ago:

The problem itself is not really a bad problem; it does require students to recognize that the segments connecting 1st, 2nd, and 3rd base form the sides of a right triangle. (In fact, an isosceles right triangle.) Once this is realized, the student calculates the hypotenuse using the Pythagorean theorem (or by multiplying the leg length by the square root of 2), then finds the difference between the sum of the two legs and the hypotenuse.

What the problem doesn't do is leave terribly much room for discussion. The information needed to solve the problem is given. The exact path that Jeff and Peter each take during the race is described. The problem also takes great pains to mention that the baseball diamond is in the shape of a square and that Jeff makes a 90-degree turn at 2nd base, strongly hinting at the existence of a right triangle. Once students realize that there is a right triangle in the diagram (which is really the only major thing in the problem that there is to "realize"), the rest is calculation.

Now, let's strip away everything until we have only the question and the scenario:

While perhaps I may have taken away too much, there is plenty to talk about with this problem now. Students will have to decide what they need to find out in order to answer the problem, and ask questions accordingly. Of course, with how much information I took away, a few might be asking: "What does this diagram even mean?"

Which is a very good question.

Seeing Even Less: A Scenario Without a Question

Recall the question I posed earlier: What if you remove the question, leaving only the scenario? Let's do that now:

Now we just have a diagram of a baseball diamond, with emphasis on the distance from 1st to 3rd. We don't have a footrace anymore.

So what?

Ever since I wrote the original version of this problem, I couldn't help thinking that there had to be more math, deeper math involved with the baseball diamond scenario. There had to be more than just a Pythagorean theorem footrace problem with this. But, I couldn't see it.

I couldn't see it because my mind was stuck on the original problem and blocked my way to other possibilities.

Seeing More: The Third Baseman Problem

Yesterday, I was at Wrigley Field for the Cubs vs. Cardinals game. I was watching the players take batting practice before the game. In one moment, when I watched a ground ball dribble toward third base, watched the third baseman scoop it up and throw it to first, a question popped into my head:  

"How hard does the third baseman have to throw the ball to get the runner out at first?"

That question fits perfectly with this scenario:

Now we have an entirely different problem, using exactly the same scenario as the racing problem, that involves a heck of a lot more math.

There is so much conversation that can go on here! What information do students need to solve the problem? What skills are required to find an answer? There are plenty of factors at play in this situation:

  • We need to know how fast the batter is running to first.
  • Consequently, we need to realize that the runner accelerates to his top running speed (thus putting quadratics into play).
  • We need to know at what point in time the third baseman fields the ball; in other words, how far away from 1st base is the runner at that point?
  • We need to know if any natural factors (such as wind) need to be accounted for in figuring out our answer.
  • We need to know where the third baseman is when he fields the ball.
  • We need to know when the third baseman throws the ball; he doesn't throw it at the same instant he fields it!
  • We need a way to figure out how far away from 1st base the third baseman is when he throws the ball.
  • We need to know how we're expressing "how hard" the third baseman throws the ball.

This problem is more mathematically rich and complex than the racing problem and would almost certainly result in different groups of students coming up with different yet justifiable responses. It's open-ended, rife with ambiguity; messy, just how real-world math tends to be.

I probably would not have come up with it had I not found myself in a moment where I was only observing the scenario -- a baseball diamond with an emphasis on the space between 1st and 3rd -- in the absence of a question.

The point is this: Math is freaking everywhere. If you're a math teacher, you know this all too well. The problem is that sometimes there can be so much math in a scenario, that we have a really hard time seeing it until we strip away everything except the scenario itself.

Dan Meyer's way of stripping away the layers of a problem until only a question and a scenario are left is a fantastic means of getting our students hooked into having patient, thoughtful conversations about math and problem-solving. I found his talk to be inspirational. Going further and removing the question, I think, can be a way to help math teachers look more deeply at a situation and uncover even more math that they weren't seeing before. The more math we can see in a scenario, the more complex the questions we can ask our students, and in turn the deeper their learning. But in order to do this, I think we sometimes have to make ourselves forget the question and just look at the scenario from a fresh perspective.

Is this true for every math problem? I doubt it. But seeing less really can be seeing more.

Wednesday, July 25, 2012

Twenty-Six-Point-DOOM: The Marathon Man Task

Okay, sometimes I enjoy a bit of over-the-top drama when I title stuff. Something you should probably know about me going forward.

This summer, I have been training for the Prairie State Marathon on October 6th of this year. It's my first-ever marathon, and I'm totally psyched for it. In fact, I'm so psyched about running that I've already registered for the F^3 Lake Half Marathon -- which takes place along Lake Michigan in freaking January -- and am planning to run the Wisconsin Marathon next May. Needless to say, I'm addicted to running beyond all measure of common sense (which I'm not certain is actually measurable).

My fanaticism for endurance running aside, I was feeling particularly inspired after reading this post by Nat Banting (@NatBanting) about a problem he's developing to determine how to minimize the amount of water wasted by his sprinkler. I found myself wondering what real-life situations I could use to similarly create an authentic PrBL experience for my students.

The Scenario

As I was out for a 7-mile run this morning, I realized that such an experience might lie in this:

This is a Nathan Trail Mix 4 hydration belt, which I take with me on my long-distance training runs. Each bottle has a capacity of 10 ounces, which means I can take 40 ounces of liquid with me. Most of the time, I consume Gatorade while running. With longer distances, however, I also take packets of GU energy gel with me for supplementing my glycogen stores.

(Incidentally, Chocolate Outrage is my favorite flavor. 
Also, "GU" is pronounced "goo.")
Energy gel needs to be consumed with water to dilute it enough for the body to absorb it quickly, or else cramping and sometimes vomiting (eww) can occur. (Also, taking it with Gatorade causes the gel to become thick and sticky like molasses, which doesn't sit well either.) Thus, I have to take both Gatorade and water with me on my long distance runs:

My longest training run to this point has been 16.5 miles; thus far, I have been able to ration my Gatorade and water appropriately to get me through each run. In addition, there will be hydration stations spread throughout the course stocked with water and sports drink. However, as race day approaches, I've been frequently asking myself this question: Will I be able to take enough Gatorade and water with me to get me through 26.2 miles before I run out of both?

The Task

This leads to the driving question I would put to my students for this task: What is the best plan to keep Mr. B hydrated and energized during the marathon?

I am not yet sure what the entry event will look like, but my intention is for it to include only a few pieces of information:
  • The task is to create a "consumption schedule" that tells Mr. B when to drink liquid or eat a gel packet.
  • A marathon is 26.2 miles long.
  • Mr. B has a hydration belt for carrying water, Gatorade, and energy gel.
  • Energy gel must be taken with water, and must not be taken with Gatorade.
  • There are also hydration stations located throughout the course that carry water and sports drink.

Potential Student Need-To-Knows

My hope is that the limited information from the entry event leads to several student-generated need-to-knows. Here are a few that I was able to come up with on my own:
  • How much liquid does each bottle hold?
  • How much Gatorade and how much water should be taken?
  • How often should Mr. B consume Gatorade?
  • How many hydration stations are there on the course?
  • Where are the hydration stations?
  • How many gels will Mr. B consume?
  • How fast does Mr. B run?

Some of these need-to-knows can be answered rather quickly. As I mentioned above, each bottle on my hydration belt has a capacity of 10 ounces, for a total capacity of 40 ounces. If the students ask, I can simply tell them this one.

If students inquire about the number/locations of hydration stations on the marathon course, I will be able to furnish them with this map from the race web site. The map marks all of the hydration stations throughout the course.

The frequency and amount of Gatorade consumption is where I'm sure many groups will diverge in their solution paths. Some runners consume a few ounces of Gatorade every few miles or so. My personal preference is to sip about an ounce or two or Gatorade at every mile marker, though for the purpose of this task I'm not married to that notion. Chugging an entire 10-ounce bottle at any point, however, would be inadvisable, as it would probably result in me throwing up (eww).

In any case, the Gatorade consumption can be modeled with, say, a linear inequality. Is the total amount of Gatorade consumed going to be equal or less than the amount of Gatorade available to me during the race? Each group's plan will need to address this question, and there are a number of ways to answer it.

Now, I haven't forgotten about those last two need-to-knows:
  • How many gels will Mr. B consume?
  • How fast does Mr. B run?

These two questions are very closed tied to each other; plus, the gel consumption will also dictate the water consumption.

Gel needs to be consumed at regular intervals throughout the race in order for me to maintain my energy stores; in fact, most packages of gel carry the advice of consuming one packet every 45 minutes. With that in mind, it becomes incredibly important to know how quickly I can be expected to finish the race. Without a sense of how fast I run, students will be unable to determine how much gel and water I will need to consume.

I happen to have recorded the times of each of my long-distance training runs from the past few months. Mostly this is due to the fact that I am a shameless braggart:

Since I've recorded all of my times, however, it means that instead of giving my students my own estimate of how fast I run, I can provide them with a table of data and have them estimate how fast I can run. Linear regression models, anyone?

Where To Go From Here?

I'm only about 8 hours removed from when the idea for this task first formed in my head, so naturally it's nowhere near perfect. There are many pieces of the task that I was admittedly rather vague in articulating. But, I do think I'm onto something cool here.

One thing I am not sure about is how to have my students present their solution. Live presentation? Blog post? Physical document? Scribbles on the back of a napkin? All of these options and more?

Any thoughts? This is the first PrBL idea I've really come up with on my own, so I gladly welcome feedback!

Saturday, July 21, 2012

Deeper Learning: Passion + Conversation = Want-To-Knows

This past week, I participated in #PBLChat on Twitter for the first time. I'm still pretty new at this whole "chatting on Twitter" thing (I hadn't even known about TweetChat until halfway through), but the experience was awesome. I "met" a lot of great teachers in the network and had a chance to build my PLN, which I think will be invaluable as we continue our conversations in the coming months.

With the theme of NTAC 2012 being "Dive Into Deeper Learning," the question posed to the chat was, naturally, "What is 'deeper learning?'" As I read response after response, two over-arching themes grabbed my attention: passion and conversation.

It really all begins with passion. Many in the chat agreed that deeper learning requires an initial deep desire to learn. I remember when I was in high school, I used to write out the proof of the quadratic formula in my notebook whenever I was bored, because I preferred math more than any other subject. I liked doing it, so in turn I gave it more attention. It's what we like and what we want to do that we seem to become best at. Passion is probably why I can quote every line of "Anchorman," but I couldn't tell you the laws of thermodynamics without looking them up. I was never that interested in science, but dude, I can go on for days about whether Brick actually loves lamp or is just looking at the lamp and saying he loves it.

Where passion fuels deeper learning, conversation helps us make sense of our passion. In the chat, this aspect of deeper learning was commonly articulated as "talking about process" or "being able to explain or teach the concept to someone else." These are fine examples; I think, more generally, the conversation aspect of deeper learning means to engage in an exchange of interpretations or viewpoints in order to refine one's own knowledge. In a way, conversation is confrontational. It forces the learner to articulate what they believe they know about a problem or a topic, and can even lead them to realize or admit what they do not yet know. This is vital to deeper learning; the very act of identifying what we do not know gives us direction for our pursuits.

To a PBL teacher, this probably sounds suspiciously like I'm talking about "need-to-knows." In the context of deeper learning, it might be more appropriate to call them "want-to-knows," since deeper learning is driven by passion. Maybe I'm overgeneralizing, but I think there's an important difference between the two. The idea of "need-to-knows" is definitely important as an organizational and learning tool in PBL, to be sure. "Needing" to know something can sometimes, I think, imply that the learner has to know the thing for the sake of completing the project. "Wanting" to know something, on the other hand, comes from sheer curiosity and is motivated by genuine interest rather than academic requirements.

For example, one (not-really-that-great) project I ran in one of my geometry classes had students design a new type of mini-Oreo package based on different types of 3-D shapes they were learning about (spheres, prisms, cylinders, pyramids, cones). So, in order to complete the project, students "needed" to find out what these shapes were and how to calculate their volume -- not a very exciting or motivating prospect for anyone who isn't already a math geek. One student, however, "wanted to know" about other, more complex 3-D shapes. There was a desire within him to research and find out what other shapes existed. So, I told him to have at it. A few days later, he came back to me with a package prototype in the shape of a conical frustum. I was amazed! I had never even heard of this shape before; had the student not presented me with his "want-to-know," I might still have no idea what a frustum is.

This is only a small example of a student pursuing a "want-to-know," but I think part of our responsibility as educators is to give our students room for such pursuits of any magnitude. We must continue to provide our students with opportunities to find their passions and make sense of them. One of the greatest things we can do for our students is to equip them to chase down their "want-to-knows."

Wednesday, July 18, 2012

Hello, World!

A.D. 2012 - And so, a new blog is born.

I'm still tinkering with the layout, fonts, tabs, etc., but the entire point of this blog is content and conversation. The bells and whistles will come; I'd rather just get down to business.

While I'm not going to pigeonhole myself into saying "this is a math blog" or "this is an education blog" or "this is a PBL blog" or "this is a super-cool ninja blog," I can say that one of my purposes here is to continually reflect on my teaching practice. I think reflection is an important aspect of improving my teaching practice, and I don't really do it enough. So, here I am.

I don't know what I have to offer, but I'll share what I can. There are three parties I want to benefit from this blog: myself, my students, and my colleagues. If I can learn from other teachers, and other teachers can learn from me, then all of our students will benefit, right?

Anyway, let's get this started.