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...