Sometimes It's Good to Take a Detour

Probably one of the coolest things about teaching is when a student asks a really good question that lets you detour from your original plan to talk about something really super-awesome.

That happened in my class today.

We were discussing slope and going through a few example problems with the slope formula. I decided to show them one example that resulted in an undefined slope. I gave them the points (7, 3) and (7, 10), then we worked through the problem. We got to a point where we had 7/0 on the board and I asked the students what that meant. The consensus was that the slope was undefined because "we can't divide by zero."

Then, one of my students asked: "Mr. Brenneman, why can't we divide by zero?"

I stopped. I looked at him. I said, "I love that question! Let's put aside what we're doing and talk about this!"

I then launched into a brief explanation of proof by contradiction and asked them to put aside the laws of mathematics for one second. "Let's suppose that you can divide by zero," I said. "Let's consider what 0/0 would be equal to. What do you think?"

Many students chimed in with "0." Others chimed in with "1." I asked each side to back up their reasoning.

"Well, it would be zero because you're dividing zero by another number," one student said.

"I think it would be one, because 2/2 is 1, 4/4 is 1, so 0/0 would be 1," said another.

A few minds were blown when I told them they were both right.

Here's why:

Assuming we can divide by zero, the quotient of 0/0 yields two distinct yet equally valid results.

Suppose we choose a number a from all of the numbers in existence. We say that 0/a = 0 (the zero property of division) and a/a = 1 (a form of the multiplicative inverse property).

In this scenario, division by zero is allowable. (This is an important distinction, because normally the two properties I mentioned above specify that a must be nonzero.) So, 0/0 = 0 by the zero property. But, 0/0 = 1 by the multiplicative inverse property.

Thus, it is reasonable to conclude that 0/0 = 0 and 0/0 = 1.

In other words, 0 = 1.

The discussion can certainly stop here, because we have arrived at a conclusion that is mathematically absurd. Furthermore, this absurdity stems from the initial assumption that we can divide by zero; hence, we must conclude that we cannot divide by zero.

But I knew that ending our discussion at 0 = 1 wouldn't have been nearly quite as fun as proceeding with even more absurdity.

So, I asked the students, "what would 1 + 1 be equal to?"

Many said 2. Some said 1. They were all correct. I showed them why.

1 + 1 certainly equals 2. But, we've already established that 1 = 0, so we can also say that 1 + 1 = 1 + 0 = 1. Or, 1 + 1 = 0 + 0 = 0.

In other words, 0 = 1 = 2.

I extended it one more time by asking the students what 1 + 1 + 1 would equal. Some said 3, some said 2, some said 1. Again, they were all correct. Using similar reasoning as the "1 + 1" case, we concluded that 0 = 1 = 2 = 3.

At that point, the students came to realize that if we kept going, eventually we would conclude that all numbers would be equal to each other.

I told the students one of my favorite mathematically absurd things to say: "If Congress legalized division by zero, we could solve all of our economic problems. We wouldn't have a $15 trillion debt, because if we can divide by zero then 15 trillion would be equal to zero. We wouldn't owe anyone $15 trillion. Problem solved!"

My students seemed to love it. Sometimes it's fun to drop what we're doing and discuss something far more interesting when the opportunity arises.

A Not-At-All Comprehensive Review of Socrative

At the start of the school year, our Assistant Principal introduced me to a student clicker-type program called Socrative. It's free and can be used in your web browser or downloaded as an app to a mobile device (available for iOS and Android devices).

I've been testing this out in my class for the past couple of weeks and have been rather impressed by the results.

There are essentially two modes for using Socrative: you can administer a pre-written quiz to your students with multiple-choice questions and free response questions, or you can administer a quick one-question activity on the fly.

I've been using the pre-built quiz feature for the past few days as a warm-up activity for my students when they get to my classroom. Students log on to their desktop computers (or sign on to Socrative on their smartphones) and complete a question related to the current skill they are working on.

I was asked to demonstrate Socrative to my colleagues at today's staff meeting, so I wrote a sample quiz for them. Here was one of the multiple-choice questions:

You can set the quiz to give instant feedback when an answer is selected. In this case, the answer was obviously "ninjas."

Now, while students are taking the quiz, the teacher can use their end of the software to monitor progress and results in real time:

 Free-response questions can also be built into a Socrative quiz. Here's an example from the quiz from the staff meeting:

Now, obviously I use this in a far more practical manner in the classroom. (That's not to say that questions about ninjas and ice cream aren't important, BECAUSE THEY ARE.) For instance, here is the warm-up question I administered to my students this morning:

Now, here's the really cool part.

When I see that the students have finished, I end the activity. Then, I am presented with the option to e-mail a report to myself.

So this morning when my students finished their warm-up question, I had a report e-mailed to me. A few minutes later, this arrived in my inbox (student names have been removed):

Formative performance data that can inform and drive my classroom instruction to best meet the needs of my students?  All organized and color-coded in an Excel spreadsheet? And this software is free? HOLY CRAP. YES PLEASE.

But wait! There's more!


If you don't have time to write a quiz in Socrative, that's no problem at all. Socrative also allows for a quick one-question option that allows you to assess students on the fly.

On the teacher control panel, you can choose to start a quick multiple choice, true/false, or short answer activity:

You can announce the question orally, or provide it in a written format on paper, dry erase board, online LMS, napkin, ankle tattoo, whatever. Say you wanted to do a true/false question. You select this option, and the students see this on their screen:

Notice that there's no question displayed. As I mentioned, it's up to you to present the question however you want. The point is that you can use Socrative on the fly to formatively assess your students as well. You can also monitor results in real time, though there won't be names attached (so this is also good for taking an anonymous poll). The downside, however, is that you can't e-mail a report to yourself in this mode.

So far, I'm seeing great advantages to using Socrative in my classroom. It's a very handy way for me to quickly collect and organize formative assessment data before, during, and after a lesson. It allows me to more effectively monitor my students' learning and to make appropriate instructional decisions. And, since Socrative can be downloaded as an app to mobile devices, it's also conducive to a BYOT classroom environment.

Probably the one thing I really wish Socrative could do is recognize math type. In the slope question above, I had to settle for typing "1/3" and "5/6" instead of putting them into a less-confusing vertical format. There's also no way to insert charts, graphs, tables, etc. There are ways to get around this, of course. (I can post the full question in another medium that supports math-type and have the students submit their responses via Socrative.) Still, it would be convenient to have these features present. (EDIT 8/22/2013: In the past year since this post was written, Socrative has added the ability to post images. This provides another way around the issue. Sweet!)

Overall, this is a great piece of software and is a very simple way of recording formative assessment data. Works great in a 1:1 technology environment, provides real-time results, and supports data-driven instructional practice. I give it four out of five ninjas.

RATING:

(Trust me, there are four ninjas next to "RATING:" here. You can't see them, because they're ninjas.)

It's School Again! Huzzah! (Part 2 of 2)

The freshmen had their first day of school with us last Tuesday; on Wednesday, the rest of our students returned and I got to see my seniors for the first time since I last had them all as sophomores.

This year, one of my goals for my math class is to get my students writing more and to practice digital citizenship by focusing on communicating with peers in an online environment. To that end, one of our opening activities was for students to respond to a discussion board prompt on our echo course page.

The questions were pretty simple:

Many of the responses were encouraging to read; a lot of students stated they were planning to go to college after high school and that they were excited for graduation. Several students said they were excited to have me as their teacher again because they enjoyed how I teach (which I'm not particularly sure how to feel about since I think I was probably doing many things wrong two years ago).

Some students had their priorities straight:

While other students took a rather avant-garde approach:

Still, I learned a great deal about my students. A few of them said they wanted to go into graphic design; one wants to be a zoologist; one is thinking about cinematography or film; a few are considering getting business degrees; one wants to be a mechanic; one has aspirations of joining the FBI; some are planning to go into the military; and many, many more. There is a lengthy, eclectic list of careers my students want to pursue after high school, which is awesome.

Other students are unsure of what they want to do after they finish high school, which is also okay. I'm hoping that during this year I can connect with these students and help them figure out plans and goals for themselves for a post-high school existence.

At any rate, this discussion board activity served two important purposes. First, as I just detailed above, I learned a lot about my students. I know more about their post-graduation plans and their interests, which will help me a great deal in tailoring our class to incorporate their interests. Second, the activity established a baseline for their ability to communicate and interact with each other in a supervised online environment.

I saw some good things. The students were able to follow directions well for the most part, did an "okay" job of using polite language (save for one student who jokingly said her mom would "beat her ass" if she didn't get a good grade in math), and responded to each other's posts while making an effort to comport professionally.

I also saw some things that need a lot of work. The vast majority of the students re-posted each question and answered them in a list format. I would like to see them get away from re-posting questions and answering in a paragraph form. (Not that answering in a list format is necessarily a bad thing, but I would like to see them practice putting their thoughts together in a coherent, flowing format.) Spelling, grammar, and punctuation remains an issue; I realize that I'm a math teacher, but that doesn't mean I can't give them feedback on these things. (Actually, I minored in English at Michigan State, and am certified to teach the subject in the state of Michigan.) The replies that students wrote to each original post were also, for the most part, superficial. I saw a lot of "I agree with you"-type posts that had little depth and weren't suited to continuing the conversation. Again, not necessarily a bad thing; plus, I wasn't expecting most of the students to be able to do this on the first go. We were simply establishing a baseline to help us identify what to work on. By the end of the year, I'm hoping to see well-crafted, thoughtful responses and replies that result in deeper conversation. (To be fair, this probably requires a deeper topic than what I gave them to start with.)

Outside of the discussion post activity, the majority of the time was spent administering a math benchmark test to establish where the students are in terms of content mastery. This benchmark will help me determine what the students already know, what they still need to master, and thus where we should focus our efforts as far as mastering content is concerned.

Probably the coolest thing of the week that happened was Friday. Many students still needed to finish their benchmark from Wednesday/Thursday, while others were already finished and weren't going to have much else to do. This seemed like a great opportunity to preview the election-themed project that we're going to be launching when we come back from Labor Day weekend.

I decided to get together the students who were already finished with their benchmark in each class for a Critical Friends session. The students had seen the Critical Friends protocol in their sophomore English class and were somewhat familiar with the procedures, so I gave each class a quick refresher before starting.

It went alright in my 1st period class, but the students more frequently got off-task in 2nd period. I realized that I needed to designate a few students to be responsible for steering the "I Like/I Wonder/Next Steps" portion of the session and keeping everyone focused on the task at hand. So, in my 3rd period, I asked the group if anyone was comfortable leading the discussion. Three students immediately spoke up, so I told them they were responsible for keeping everyone on task. I presented the project idea and sat back to let the students discuss it.

I hadn't expected what transpired next.

One of the students I designated to lead the discussion immediately chose a student to read the entry document out loud. The other students all listened intently as she read through the entry doc. After she was done, one of the other student leaders grabbed a dry erase marker to start writing "I Likes," "I Wonders," and "Next Steps" on the board while the other two called on students for their feedback.

I was amazed. I was very proud and excited. I thought to myself, "SOMEONE HAS TO SEE THIS!!"

So I shot a quick Skype message to our assistant principal, who came down a few minutes later as the session got in full swing. We were enraptured by how well the students had taken over the conversation, listing several "I Likes," "I Wonders," and "Next Steps" while conducting themselves in an orderly fashion:

I was very impressed with what the students were able to do on their own. I had actually intended to listen to their conversation and write down all of their feedback myself (as I had done for the first two periods), but they completely took care of that for me! The only thing I'd done was to assign a few students to lead the discussion, and they took it from there! It was really awesome to watch.

I did the same thing with my 4th period class and got similar results. Our principal stopped by my room during that period and was very proud of the students for what they were doing -- she even joined in and gave some Critical Friends feedback herself!

As I said, the students had previous experience with the Critical Friends process in their sophomore English class, so I made sure to track her down and let her know what had transpired in my class. When I told her they not only remembered Critical Friends, but successfully ran a session on their own, she did a happy dance.  I imagine the news must have been incredibly satisfying -- it showed that these students had actually been listening to her two years ago.

So my week ended on a high note. The students gave me some great feedback for our project, and most of them seemed to be interested in the idea.

I would love to expound more on Week 1 (and I did give an update on the Ninja Board), but there's still much I need to do for Week 2! A teacher's work is never done. Until next time!

Ninja Board Update: Week 1

Previously, I talked about this silly idea I had to implement an achievement system themed around ninjas for some reason. The original intention was to see whether or not it would have an affect on student motivation in my class, particularly in the academic respect.

While there have only been three days of school so far and it's waaaaay too early to tell whether or not this will be the case (just as ESPN.com is jumping the gun on projecting my Spartans to go to the Rose Bowl), I made an important realization: the Ninja Board is perhaps going to be far more useful as a tool for developing classroom culture (which, of course, would affect student motivation in turn). This is because I can use it to define and recognize those "awesome moments" in class and capture them for posterity.

Some things went according to plan on the first day. I said absolutely nothing about the Ninja Board. I didn't even point it out. I was secretly planning to award the first ninja point to the first student who asked about the Ninja Board. I figured someone was going to at some point.

The entire first block passed without anyone asking about it.

I was genuinely surprised at first; then I began to think that perhaps nobody would ask about it unless there was more to pique their curiosity than just a blank wall.

I looked for other opportunities to award ninja points to a few students. During my second block class -- which also ended up passing by without anyone noticing the Ninja Board -- one of my students asked me if I wanted to see the folder she was using for my class. I said I would love to, so she reached into her binder and pulled this out:

COOLEST FOLDER EVER.

So I decided to award her a ninja point for being the "first to bring in a ninja item."

I have a particular way of awarding ninja points. When a student does something worthy of a ninja point, I say nothing. I don't announce, "CONGRATULATIONS! YOU WIN A NINJA POINT!" (Doing so would be very un-ninja-like; ninjas don't announce to their victims, "GREETINGS! I AM ABOUT TO ASSASSINATE YOU WITH THIS KATANA! YOU'D BEST ATTEMPT TO FLEE!")

Instead, I jot a note to myself on my iPad: I write down the student's name, how many points they earned, and the reason they earned the points. During my planning time, I make a sign for each student that earned ninja points:

Before I leave school for the day, I tape all the signs to the wall. The students don't find out that they earned ninja points until the next day when they come back and see their names posted.

So after the first day, I picked out three students who earned a ninja point. (And the cool thing is that since I tell the students absolutely nothing about the Ninja Board, I can come up with all kinds of reasons to award ninja points.)

The next day, another block period passed without any questions about the Ninja Board. Finally, however, one student approached me during second block with the question:

"Hey Mr. B, what's the Ninja Board?"

I smiled. I smiled partly because I was happy someone had finally asked me that question after nearly two days of waiting. I smiled partly because she was going to get a ninja point for asking that question. Mostly, though, I smiled because I knew what my answer was going to be:

"That is an excellent question."

And I said nothing else. I think I giggled involuntarily.

A few more students earned ninja points on the second day, and more names were added. On Friday, I have several more inquiries about the Ninja Board, and each time I replied with a non-answer. Slowly but surely, interest in the Ninja Board started picking up.

Even though I'm being incredibly stubborn with my refusal to explain the Ninja Board to my students, I do want them to know what they're earning ninja points for. So, in addition to their names, I also post a list of "unlocked ninja achievements." Here's the complete list from the first week of school:

In all honesty, only about one or two of these "achievements" were pre-planned. The rest are being made up as I go. When I notice my students doing something really awesome, like demonstrating leadership or kindness, that kind of thing deserves ninja points. If I have one of those little student/teacher "moments" where we're building or supporting good rapport, I give ninja points for those, too. To keep my students on their toes (and partly to include those students who are traditionally the "invisible" ones), I also award ninja points for other random things.

It seems to add a certain whimsy to our classroom culture that I particularly enjoy. I'm curious to see how the Ninja Board will continue to play out in week two.

Incidentally, there's much more about the Ninja Board that I haven't revealed yet on this blog -- but that can wait for another time.

It's School Again! Huzzah! (Part 1 of 2)

Happy Labor Day weekend, everyone!

First, a big thank-you to everyone who has checked out this blog over the past couple of days! Please feel free to stick around just in case I write something worth reading someday!

So, my wife and I spent most of yesterday driving from Chicago to her grandparents' condo in Ohio, where the rest of her family has gathered for the long weekend. On the way, we dropped off our dog, Jake, at a fancy-schmancy doggy bed & breakfast for his first-ever weekend without Mommy and Daddy.

It was a bit hard for us. I mean, LOOK AT HIM.

Anyway, canine parental guilt aside: After driving several hours, heading out for an awesome dinner with the family, playing some games, etc., I found some time to sit down and reflect on the first week of the school year.

SPOILER ALERT: It was awesome.

Tuesday was a freshmen-only half day, designed to get our new students acquainted with our school. The teachers lined up outside the building to welcome each and every freshman as they walked into the building.

It was a total blast to stand outside, welcoming our new students as they trudged past us, one by one, with the sleep not yet completely rubbed out of their eyes. We greeted them with enthusiasm, high fives, and fist bumps. A few responded with the same energy. Many more obliged us begrudgingly. Some looked at us as if we had just picked our noses with an acetylene torch. One or two "anti-establishment" types elected not to partake in such frivolity, to which I replied, "I respect your non-conformity, don't ever lose that." They looked confused.

By 7:30, the freshmen had found their way into our cafeteria for welcomes and introductions. They were seated by their Advisory groups and assigned a senior leader, who would be showing them around the school, helping them find their lockers, etc. Our school's director (which is New Tech lingo for the role of "principal") greeted the students warmly, then had each teacher introduce himself or herself.

As I stood in line waiting for my turn, I whipped out my iPad and loaded the Notability app, quickly scribbling something with my finger. When the time came to introduce myself, I took a few steps out and, silent, stone-faced, held up my iPad for the freshmen to see:

 

After a few long moments (I was purposefully aiming for John Cage-like silence), I spun around and went back into the line without saying a word.

This prompted laughter from most of the freshmen, confused looks from the rest.

(Then I actually introduced myself "normally," discussing who I am, what subject and grade I teach, and that I also run Anime Club during lunch three days a week.)

Throughout the morning, I had the opportunity to meet each freshman group in my classroom. While I'm sure that most of them were already convinced that I was a total weirdo, I threw up a Keynote slide I had whipped together to remove any lingering doubts:

I later tweeted a reaction I got when expounding upon my favorite video games to play:

Only an hour or two into the school year, I had most of the freshmen scratching their heads and wondering what was up with this strange man who was going to be teaching them math in their senior year.

Probably the coolest thing about the interactions I had with the freshmen on day one was that many of them seemed to relax ever-so-slightly from the typical first-day jitters and culture shock.

Several freshmen approached me that morning about joining Anime Club. On the very first day. I'd never had that happen before. A few other freshmen came up to me to declare that I was already their favorite teacher, even though they wouldn't be having me until senior year.

It was a cool feeling to hear them say that, but it was not really the point. The point was to make them feel welcome, and to let them know that I was totally willing to be just as goofy with them as I am with the seniors who are actually taking my math class. The point was to make the feel like they were already part of the family.

Because they are. They are every bit as much a part of our school's family as our first graduating class that started college this fall. They are every bit as much a part of our school's family as our current group of seniors, who I got to see the next day for the start of their school year...

Ninjas: Undeniably Awesome. But Student Motivational Tool?

Tomorrow morning will be my first day with my students. When they walk into the room, this is one of the things they'll see:

 

Yep. A section of blank wall that I have dubbed "The Ninja Board."

What is it?

That's what my students are inevitably going to ask me. And I'm not going to tell them.

But I'll tell you: It's a type of "achievement" system, similar to what one might find in video games. In fact, I'm pretty much ripping this off of the clan rank system from Final Fantasy XII, which I played in my spare time this summer.

The idea is simple: Students do things in class that earn them "ninja points." This can be anything: Completing assignments, demonstrating knowledge at a certain threshold of rigor, developing an interesting project, etc. Lots of things can earn "ninja points." Enough ninja points, along with completing other certain tasks, will allow students to gain a ninja rank ("Level 1 Ninja," "Apprentice Ninja," "Super-Awesome Math Ninja," etc.). Their ranks, in turn, will be displayed alongside their names on the ninja board.

I'm purposely not going to tell my students what they can do to earn ninja points or ranks. I want them to discover that on their own. They'll have no idea what's up until the first student earns ninja points and gets their name put on the board, with a point tally and a rank.

Then everyone will start to get it. And chances are they'll want a piece of the glory, too.

I want them to be curious about what they can do in class to earn ninja points, and then try to figure out on their own what those things are. When a student does do something that earns ninja points, or when a student does gain a rank, the knowledge of how they did so will be published to the ninja board. So they'll slowly learn how to get ninja points and ninja ranks as they go along.

Is it cheesy? Yes.

Is it completely silly? Of course.

But what if my students buy into it?

It's possible some really cool stuff could happen as a result.

I'm hoping to see increased student motivation in different areas of our class. I'm trying many new things this year -- discussion board posts, journal prompts, student blogging, and so forth -- and I would love to see my students get really creative and really deep with these things. The Ninja Board might help facilitate this. Like I said, there are many things that could earn ninja points. Perhaps a particularly thoughtful discussion post; a journal entry where the student talks about a real learning breakthrough they had; or a voluntarily-written blog post on a really cool topic.

Here's what I think might be the real beauty of it: I honestly haven't given much thought to what specifically can earn ninja points. But I'm willing to bet that my students will try out a bunch of different things, or ask me about different ways to earn ninja points. And some of what they try might actually be pretty cool, thus legitimately worthy of ninja points.

In other words, the students themselves will determine what earns ninja points, not me.

So it starts tomorrow, and we're going to see how it goes. It could fall flat on its face. It could be fun for a while and then get old. Or, it could be really super-cool and lead to some unexpected (and pleasant) results.

"Why ninjas?" you might ask.

I'll tell you why: because they're totally sweet.

Related: Ninja Board Update: Week 1

Keepin' It Real

I'm going into my fourth year of teaching math at New Tech High @ Zion-Benton East. I love my job. I've discovered that I'm actually, perhaps, maybe, starting to get halfway decent at it. At least there's a chance that I am. There's perpetual room for improvement, and today I wanted to talk briefly about one area I hope to improve this year.

In my first few years of attempting to teach a PBL/PrBL math class, I've come up with some projects that are pretty good at simulating authentic real-world tasks: Creating a different type of Oreo cookie package for Nabisco; creating a scale drawing depicting how furniture should be laid out in a dorm room; and designing a hole for a miniature golf course.

These projects are certainly useful ways to help students see how math can be used for creating and improving products in a real-world context. On the other side of the coin, they fall short when it comes to real-world results. We didn't actually create a real Oreo package for Nabisco; we didn't actually have a real room with real furniture for our scale drawings; we didn't actually build a real mini-golf course.

They were real-world projects without real-world results.

As Dennis Littky probably would put it, I had the students doing "fake real work" instead of "real real work." Something like that.

I want to change that this year. As I continue on my journey of teaching PBL/PrBL math, I believe one of my next steps is to move my students away from the "fake real work" and into the "real real work." I want students to use math to actually create things; to innovate; to predict; to think critically; to affect their community in a positive way.

How do I do this? I haven't completely figured that out. I think I have a good start with the election-themed project idea I blogged about last time. I'm hoping my students can use their experience with this project to learn more about important issues, about making informed decisions based on available data/information, and about making defensible predictions.

A few of my students will even be voting this year; this might really help them learn about being informed voters.

And, because I want my students to produce real-world results, I need them to have a real-world audience. That's why students will be publishing their findings on our class blog (link coming soon) for the community and the rest of the interwebs to see, as well as sharing them with the Obama and Romney campaigns (fingies crossed that they'll actually take a look).

I think that's a good start in my goal to move away from "fake real work" and giving my students the chance to do "real real work."

But I need more. It probably sounds overly ambitious to the point of absurdity, but I want my students to always be using math to become better citizens and to benefit their community. I think the key to this is "real real work." I would love to have 100% of the school year consist of "real real work." (At this point, I'd be thrilled to even get 25% of the school year that way.)

So that's one of my goals this year. I want my students doing "real real work" that has a positive impact beyond the classroom. I'll certainly be scouring and engaging the blogosphere, Twittersphere, and meatspace for ways to accomplish this.

Project Idea: Math, Social Studies, and 'MURRICA!

I've had a half-baked idea for a project tossing around in my head for the past few weeks that I've been meaning to share. It's nowhere near perfect or ready to go, but I think it has some really cool potential. So, here we go:

It's an idea for a math and social studies project centered around the 2012 election.

(Math and social studies! I know, right?)

The idea is simple: Students work to answer the driving question, "What are the keys to winning the 2012 presidential election?"

Anyone who has been paying attention to the news (or who haven't been living under a rock at any point since 2008) probably have an idea of what the hot-button issues are, or which swing states will be most crucial to securing the presidency. For the math end of this project, however, numbers will tell the story.

As part of the process to answer the driving question, students will examine various sources of polling data. Gallup, for instance, has a daily tracking poll and plenty of polling data broken down by demographics. RealClearPolitics gathers and averages polling data from battleground states. Various electoral maps, such as this one on CNN's website, are available as well. Rasmussen Reports has polling data showing what issues are most important to Americans today. In short, lots of data to examine and interpret.

Students will gather and examine polling data to determine a few key points, including which states the candidates should focus most of their resources on and which issues the candidates should focus on. Their data analysis will be used to justify why they identified particular states and issues as being the most important to focus on.

For the final product in the math portion of this project, students will create a multimedia presentation to deliver their findings and make recommendations to both the campaigns of President Obama and Governor Romney as to how they should focus their campaigns in the final weeks leading up to the election. These presentations are to be posted to our class blog (which I have yet to set up -- I'd better get going on that) and will also be forwarded to both campaigns. (Hopefully, they'll even take time to look at them!)

I've been talking with the social studies teacher on my grade-level team about this project. It sounds like he and his English co-facilitator are planning to run a debate project at the start of the year that this could actually fit into. I think having the students use data to identify what issues are most important to Americans would then lead them to investigate why those issues are important, which would lend itself well to research for a debate. The math can inform their approach to debating various issues.

So that's my half-baked project idea to this point. There's certainly much more that needs to be thought about as I develop this into something workable.

For instance, I talked about students "using data analysis," but haven't gotten very far on how students will actually learn what it is and how to apply the skill. I think I could especially use some help there.

Also, I'm wondering if there's a place for linear modeling in here with the polling data (particularly since the first unit of the year is supposed to be linear equations/inequalities).

Other ideas I've had to far include: utilizing social media to talk directly to people in battleground states and survey them on what issues are important to them; convincing someone from Gallup or another polling agency to Skype with the class and talk about how they conduct their polls; convincing someone from either the Obama or Romney campaigns to Skype with the class about how they use polling data or other statistics to drive decisions about how they conduct their campaigns.

(Also, it would be really cool to come up with a way to make this work with #MYParty12.)

Anyway, that's it. As I said, I think there's lots of potential here, but I can definitely use as much help as I can get. If even one or two of you out there have thoughts or "I wonders" on this, please share! Otherwise, thanks for reading!

Algebra Isn't the Issue: A Response to "Is Algebra Necessary?"

I have a knack for being fashionably late with chiming in on controversial happenings. Responding to Dr. Andrew Hacker's op-ed piece, "Is Algebra Necessary?" is certainly no exception here.

There have been numerous responses around the blogosphere on this topic already from my fellow math teachers. Dan Willingham posted a particularly well-constructed rebuttal the day after the column was published. The uproar from the math education community comes as no surprise, nor does Dr. Hacker's cheeky response to the outpouring of criticism.

I could certainly dive into the fracas and expound upon the merits of teaching algebra while lamenting the current state of math education under the shadow of No Child Left Behind, but I think a more important issue may be getting lost in the conversation.

In this clip from Monday's episode of CNN's Starting Point with Soledad O'Brien, Dr. Steve Perry of Capital Preparatory Magnet School (Hartford, CT), in discussing Hacker's column, tells O'Brien that algebra "does present a real barrier" for students that come from historically disadvantaged backgrounds.

Perry goes on to refer to algebra as a "gatekeeper," citing a "one-size-fits-all" approach to the academic experience that is detrimental to cultivating success for all students. He asserts that children need experiences that they can be "more connected to" while emphasizing rigor, relationships, and relevance.

Judging by their reactions, O'Brien and co-panelist Margaret Hoover seemed to think Perry was taking Hacker's position that teaching algebra wasn't necessary. Indeed, when one watches this video for the first time, it certainly sounds like Perry agrees with Hacker in many respects.

Hoover seemed particularly incensed, jumping on Perry and pointing out that learning algebra has benefits for developing critical thinking skills that are vital to students later on in life.

That wasn't Perry's point, though. He notes that "it's 2012" and asks the question, "why are we teaching the same things the way we've always taught them?"

The point is this: The problem is not the fact that students are failing algebra. The problem is that we're not doing enough to address why they're failing algebra.

Perry touches on what I think the major underlying issue is with the growing number of students that are struggling with algebra: It's not that algebra is too hard or unnecessary. It's that students from economically disadvantaged backgrounds are not getting the support they need throughout their childhood to be equipped for academic success.

This excerpt from Hacker's editorial reveals a surprising lapse of understanding of the issue on his part:

Algebra is an onerous stumbling block for all kinds of students: disadvantaged and affluent, black and white. In New Mexico, 43 percent of white students fell below “proficient,” along with 39 percent in Tennessee.

This is a rather odd thing to read, coming from the same man who wrote a New York Times #1 bestseller on racial inequality in America. For instance, he only mentions how white students performed on these state standardized tests; though he mentions black students in this passage, he doesn't even bother to mention how they performed, perpetuating an image that black students are incapable of performing as well as white students.  This is an egregious and irresponsible omission.

Equally troubling is the fact that Hacker seems to link being white with being affluent in the same fashion. He makes no distinction between how well low-income students performed on these tests compared to students who are not from low-income households. Yet this seems like an important distinction to make, particularly in the case of Tennessee which has a high population of economically disadvantaged students.

To be fair, comparison data between economic subgroups is not always readily available. The 2011 Tennessee Department of Education Report Card, for instance -- where Hacker got his "39 percent" figure -- provides a disaggregation of test performance data describing participation and results from various subgroups. However, this does not include students from non-low-income households.

That's not too much of a problem, though. We can determine how non-low-income students performed by utilizing basic set theory and a bit of -- gasp! -- algebra. We can then use this information to get a pretty good idea of how many of the "39 percent" of white students that scored below proficiency were also economically disadvantaged.

Taking the time to do some number-crunching, one can determine the following from the data provided by the Tennessee DOE (all figures are from 2011):

• About 443,720 students in total scored below proficiency in math.
• About 318,381 of these students were economically disadvantaged.
• About 262,352 of these students were white; 181,368 students were not.

With these numbers, we can find some overlap between the white subgroup and the economically disadvantaged subgroup:

• Suppose all 181,368 non-white students who scored below proficiency were also economically disadvantaged. If we remove them from the 318,381 economically disadvantaged students that scored below proficiency, there would be 137,013 students left over.
• This means that, at minimum, 137,013 economically disadvantaged students that scored below proficiency were also white.
• In other words, more than half (at least 52.2%) of the 262,352 white students in Tennessee that failed to meet proficiency in math were economically disadvantaged.

This is an extremely conservative estimate, as it assumes every non-white student that didn't meet proficiency also came from an economically disadvantaged background (an unrealistic assumption, if not completely absurd). In other words, the actual number of economically disadvantaged white students in Tennessee that didn't meet proficiency in math is most likely much higher. There is a considerable performance gap between economically disadvantaged students and their peers.

So, intentional or not, Hacker downplays the plight of economically disadvantaged students with his unqualified claim that algebra presents a burdensome obstacle for students regardless of their ethnic or economic background.

This is an incredibly unfortunate oversight, because the truth is that poverty is a major factor in determining a child's preparedness to succeed in school. If Hacker wants to talk about an "onerous stumbling block for all students," he shouldn't be discussing algebra. He should be discussing poverty, which is independent of race (Burney & Beilke, 2008) and perhaps the root cause of many students' failures to complete high school. It is a major issue that warrants our attention and discussion.

Students who come from economically disadvantaged households have parents who not only have low incomes, but often a lower level of education than parents from other households. Both of these are indicators of how likely a student is to be successful in school (Davis-Kean, 2005). Such students are less likely to value education and to have the necessary resources at home to prepare them to succeed in their academic pursuits.

Many economically disadvantaged students live in concentrated urban settings that do not always attract high-quality teachers, further diminishing their chances of academic success (Burney & Beilke, 2008).

On top of this, poverty is often viewed as being an "individual problem," associated with laziness, apathy, amorality, lawlessness, poor parenting and a lack of education (Bullock, 2006). This stigma is an incredible barrier for economically disadvantaged students, particularly when their teachers accept this stigma as reality.

There is truth in what Dr. Perry said about algebra being a barrier for students from historically disadvantaged groups. None of the factors described above bode well for a student's ability to succeed in their K-12 education, let alone in algebra.

Blaming algebra for the failure of these students to graduate from high school or finish an undergraduate degree is like blaming the 20th mile for a one-legged runner's failure to finish a marathon. We shouldn't be addressing whether or not the 20th mile is too hard, we should be addressing the fact that the runner is missing a leg.

So before we question whether or not algebra is necessary, we should be questioning whether or not we, as a society, are doing everything we can to equip all of our students to be successful in their K-12 education. All students need equitable access to the support and resources necessary to successfully complete their education. Facing this challenge must be a priority if we really want our students to realize their potential.

In the meantime, we must also heed Dr. Perry's call to emphasize rigor, relationships, and relevance in our classrooms. We are going to continue getting students that are ill-prepared for educational success, and we are going to need to be creative to support their needs. This requires getting to know our students: what their interests are and how they learn. Doing so equips us to provide such students with opportunities for meaningful, authentic learning experiences that can capture their attention, connect new knowledge to old, and help them see the value in what they're learning.

For the record, I do think teaching algebra is necessary; but that's not the issue here.

Bullock, H. (2006). Justifying inequality: A social psychological analysis of beliefs about poverty and the poor (National Poverty Center Working Paper Series #06-08). Ann Arbor, MI: University of Michigan. Retrieved August 4, 2012, from www.npc.umich.edu/publications/workingpaper06/paper08/working_paper06-08.pdf

Burney, V.H. & Beilke, J.R. (2008). The constraints of poverty on high achievement. Journal for the Education of the Gifted, 31(3), 295-321.

Davis-Kean, P.E. (2005). The influence of parent education and family income on child achievement: The indirect role of parental expectations and the home environment. Journal of Family Psychology, 19(2), 294-304.

For the Interns and the First-Years

It's August! A new school year is just around the corner! Huzzah!

I'm very excited for the school year. It will be my fourth year teaching math at New Tech High @ Zion-Benton East. I have great students and awesome colleagues who have become dear friends of mine. I also have some new ideas that I'm excited to try out in the classroom!

For many teachers, 2012-13 will be the first year of their careers. For many pre-service teachers, this year will see them taking on an internship in preparation for entering the profession.

I attended Michigan State University for my undergrad. Michigan State's teacher education program requires a year-long internship, as opposed to the standard ten weeks at many other colleges and universities. In many ways, my internship was like my first year of teaching.

If you're about to start your internship or your first year of teaching this year, you need to know: this is going to be a hard year. An exhausting year. For me, it was the most stressful year of my life.

"Thanks for the rosy outlook, Jeff," you might be thinking to yourself.

Look, I would be doing you a disservice by sugar-coating it and saying everything is going to go smoothly. I also think you're perceptive enough to know that things won't go smoothly. But there are some things I think you should hear/read now, before the school year begins.

This isn't a definitive "survival guide," and most of what I'm about to say really has little to do with actual teaching. These are just some lessons I picked up from my internship year. I hope you can take something away from this.

So, to the interns and the first-years, this is for you:


1. This year is not going to kill you.

That's probably far more dramatic than I needed to put it. But, you are going to survive this year.

You're going to work very hard, you're going to have some awesome moments along the way, and some not-so-awesome moments as well.

This is a year to discover who you are as a professional and as an adult.

You're not going to do everything right. New teachers never do. Veteran teachers never do.

There is so much to know about teaching; there's not a single teacher prep program out there that can teach everything there is to know about teaching before entering the profession, because nobody lives that long.

Just do your best.


2. Use your support systems.

There are going to be times when you feel very alone and overwhelmed. These are the times that you need to reach out to people.

I called my mom so many times during my internship.

I called, texted, and IMed other math ed students in my cohort. (They called, texted, and IMed me, too.)

I vented to my professor. I vented to other teachers.

I vented to my girlfriend (who somehow thought it was still a good idea to marry me later on).

If I hadn't had all of these people around to listen to me when I needed to talk to someone (or ask a panicked question about what the hell a "unit" was and how to plan it), I'd probably be locked up in a padded room right now.

Let your friends and family know that you might need to rely heavily on their emotional support this year. You're going to need people to listen to you and to advise you. Know that you have these people.

Whatever you do, however you do it, just use the hell out of your support network.

(Addendum: As noted in the comments section below, blogging is also a great way to get support and feedback, too!)


3. Leave school at school.

There are many times that teachers do have to take work home, particularly assignments to grade. That's just part of the job.

There are other times, though, when you can leave work, go home, and not have to worry about tomorrow until tomorrow. This is when you need to learn to leave school at school.

During my internship, I would stay at school until 4:00, sometimes 5:00, before packing up and going home. I spent this time grading, planning new lessons, creating assignments, and so on. I'd be reasonably prepared for the next day and decide it was time to head back to my apartment.

Upon getting home, my brain wouldn't shut off. I kept thinking about everything: wondering if I wasn't doing enough or if I was doing too much with my lesson plans; wondering if I was going to do a great job the next day, or totally bomb; wondering what other ways I could explain material to the students, or what other activities I could possibly have them do; and so on.

It drove me crazy and kept me awake many nights.

One way I found to cope with my inability to stop thinking about school 24/7 was to keep a notepad with me and write down ideas as they came. I also wrote incredibly detailed lesson plans, sometimes pacing things down to the minute. This let my brain have a chance to get everything out and wind down. Gradually, I was able to get to a point where I could leave school and flip off the "teacher switch."

You can't always leave school at school; but when you can, learn how to do it effectively.


4. Exercise.

I wish I'd done more of this during my internship.

As I mentioned in another blog post, I'm currently training for a marathon. I've actually been running regularly every week since this past February, and plan to run more races through next spring. I've lost weight, lowered my blood pressure, and lowered my cholesterol.

Exercising regularly can help you be more energetic and feel more positive during your pre-service experience or your first year of teaching. It also serves as a way to keep you in an established routine, which can be tremendously helpful in organizing the rest of your time.

How do you best like to exercise? Running? Biking? Swimming? Rollerblading? Basketball? Tennis? Tae Kwon Do? Find out your preferred method of staying active, and set aside time to do it at least 3-4 days a week. Some teachers prefer to work out before school, some prefer to work out after.

Find out what works best for you, do it, and stay active. (Even better, see if any of your colleagues or anyone from your cohort will exercise with you!)


5. Eat healthy (or eat, period).

I lived by myself during my internship, and I was God-awful at keeping my place stocked with food. I'd skip breakfast. I ordered out a lot. As I like to tell people, I was sustained by Pokey Stix and Insomnia Cookies during my internship year.

You'll definitely be busy this year. You may think you won't always have time to cook, let alone eat healthy. It's even more difficult if you happen to be living alone. But, there are ways to make it happen.

My advice: Plan out all of your meals for a week (or even two) ahead of time, put together a grocery list, and shop for everything at once. Do this on the weekend. For healthy recipes or recipes that don't take very long to put together, I highly recommend EatingWell.

Teaching makes you hungry. Eat three meals a day. Have some snacks handy, too. Just be sure to eat healthy and prepare your own meals whenever you can. (You'll even spend less money!)


6. GO. TO. BED.

You need to sleep. Seriously. Go bed at the same time, every night. Get at least 7 hours.

You won't be able to function at your best if you're up past midnight worrying about lesson plans or grading papers. Yes, the grading needs to get done, but you have to balance that with your health.

Pick a bedtime and stick to it consistently, with little or no exception.

You might actually save time by doing this. I got myself into a very messy pattern of staying up very late working on plans or grading, getting up at 5 AM (sometimes as early as 4 AM) the next morning, going to school all day, coming home and immediately napping for three hours on the couch out of sheer exhaustion. Sometimes I wouldn't wake back up until 8 or 9 PM, and I'd still have work to do. If I'd stayed awake, I'd probably have gotten all of my work done and then had the rest of the evening to myself.

Get into a regular bedtime routine and avoid naps if at all possible. Use daylight to work, nighttime to rest. 


7. Friday night is a "FUN ONLY" zone.

This may seem obvious now, but many of you may actually need to be reminded of this at some point during the year.

Just as important as learning how to turn off the "teacher switch" is taking some time during the week to have fun. For my money, Friday night is the holy sanctuary of "me time."

You may feel overwhelmed with work, even on Fridays. There may be a lot on your plate during the weekend. Give yourself permission to take a break. If you don't stop to enjoy the weekend, you're going to burn out in a hurry.

Designate Friday night as a "fun only" time. Go to a movie, go have drinks with friends, go on a date, whatever you want. Just make sure you're taking time every Friday to do something fun.

Just don't get too out of control on Friday nights, because...


8. Saturday morning is a great time to get some work done.

Yes, I said to leave school at school whenever possible. But as one week ends, you'll have an entirely new week to plan for.

By all means, sleep in on Saturday. You've earned it.

But don't wait until Sunday evening to start planning for the week ahead. You'll save yourself a lot of needless stress and worry by taking a couple of hours on Saturday mornings to get some work done.

I was masterful at the art of procrastination; having all of that work hanging over my head each weekend not only detracted from my ability to enjoy my free time, it had a cumulative effect throughout the year. I got to the point in March and April of my internship year when I was having a hard time falling asleep on Saturday nights, let alone Sunday nights.

You might not want to do it, but chances are you'll be able to enjoy your weekend a little more if you're proactive on Saturday mornings.


9. Year 2 will be better.

Ask just about anyone who is in the teaching profession.  When you've gone around the block once, things start to click. You feel more confident about your teaching during your second year because you have a sense of "I've done this before." You have something you didn't have in your first year: experience.

Your experience will inform your practice as you take time to reflect and make adjustments. You'll have a better idea of what works and what doesn't. You'll find ways to manage your classroom and engage your students that are better, more efficient.

(By the way, you can do yourself a great service during your first year by making notes of what works well and what needs changed. Take time to reflect on your practice as often as you can. Future you will thank past you for it.)

If the idea that your second year of teaching will be better than your first gets you through, then by all means hold onto it. I can tell you that the third year is even better than the second year. I imagine the fourth year will be better still than the third.

It gets better.

If teaching is your passion, that's all the truth you need this year.

Good luck. Be awesome.
    

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.

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!