Multiple Solutions (A follow-up to "When Is the Right Answer the Right Answer?")

A couple of weeks ago, I wrote this post about how I wanted my students to determine equations of lines, given certain information. The broader point, I think, was realizing that my students had more than one option for determining answers to the problems they were working on, and being okay with that. (Why wouldn't I be?)

I had another "when is the right answer the right answer?" moment in class yesterday that I thought was really super-cool.

Two students were working together on the same problem. They came up with what they thought were different answers, so they were wondering who was correct. Their work is shown below:

So both students used point-slope form for their equations, and came up with two answers that looked different. This peculiarity made them wonder who was right and who was wrong. (Which, in turn, makes me realize that I still have a lot of work to do with teaching them about making sense versus being right.) They called me over to ask me who had the correct equation.

I must have been really busy at that moment and not really thinking, because I looked at their answers and said, "actually, you're both right." Not that I was wrong in saying so; but I regret that I didn't recognize the teachable moment that had presented itself. This would have been a great opportunity to ask each of them what they thought about their equations, how they came up with them, why they thought their answers made sense, why the other person got something different, and whether or not it made a difference which point they used for point-slope form. Still, it was a really cool moment: two students have a spirited debate over who had the "right" equation, when really they were both right. It was my favorite moment of class from yesterday.

Fortunately, the same thing happened today, on the same problem, with the same work as shown above, between a different pair of students. Grateful for a second chance, I was able to stop and facilitate an awesome math discussion between the two of them.

One student was adamant that the "first" point, (-4, 3), had to be plugged in for point-slope form instead of the "second" point, "because they're X₁ and Y₁," she reasoned. She said this because she had labeled the coordinates as such when using the slope formula to determine the slope:

And point-slope form was written on the board as Y - Y₁ = (X - X₁). So I could see where she was coming from.

I asked her, "so, how would you label these points if the order was swapped?" In other words, what if the problem listed the points "(6, 1) and (-4, 3)" instead of the order they were given? She responded that she would have labeled (6, 1) as (X₁, Y₁) and (-4, 3) as (X₂, Y₂).

My next question was, "So would that change things? Would you get a different slope, for instance?" The student initially thought that yes, she would get a different slope. The other student, who was working with her, said that the slope should be the same. I had both of them determine the slope of the line with the different designations for the coordinates; naturally, the slopes turned out to be the same as in their original work.

I asked, "how did changing the order of the points affect the slope?" The student replied that the order of the points didn't change the slope at all. "Cool," I said. "So what about the two different equations you guys came up with? What difference does choosing one point over the other [when plugging a point into point-slope form] make?" The first student still wasn't quite convinced that it didn't matter what point she chose; her partner said it didn't matter what point was chosen for the point-slope form of the equation.

We decided to have each of them solve their equations for y, so they'd both be in slope-intercept form. When they did so, they came up with the same equation, and the first student was finally convinced that it didn't matter which of the two points she chose. Both students were convinced that they'd both determined correct equations for the line described in the problem. "Why doesn't it matter which point you choose?" I asked. The first student wasn't quite sure. The second student guessed, "because both points are on the same line?" I replied, "that sounds like it makes sense."

I love when students find different (yet equally valid) solutions to problems like this. It makes for some great discussion. I need to keep myself aware that it's more important to ask my students to make sense of their work instead of telling them that they're right; I missed out on having a great conversation with two students yesterday, but I'm glad I had another chance at it today.

"When Am I Ever Going to Use This?" ...Sometimes I Don't Know the Answer

For some reason, the question of "when am I ever going to use this in real life?" seems to pop up at a disproportionately higher rate in math class than in any other subject. I'm willing to bet that's the case.

Do I have any scholarly research or statistics to back up this claim? No.

But I did go to Google and type in the phrase, "when am I ever," to see what popped up:

See? Algebra and Calculus! Obvious proof that this question is asked more in math class than in any other class! And if you aren't convinced by this, then... uh... um... well, then you probably think critically about your info sources and have good judgment.

At any rate, the new school year is around the corner. I've enjoyed the time away from the classroom and have spent many hours mapping curriculum, trying to keep up with the happenings in the MTBoS and preparing some new tasks to try out this year.

As I was reflecting on my first four years of teaching and looking ahead to year five, I kept thinking about what I'm going to do when, inevitably, the question is asked:

"When am I ever going to use this kind of math in real life?"

This question nearly always evokes some kind of emotional reaction from me. One of two types, in fact:

(1) UNADULTERATED, ABSOLUTE JOY, because I have an answer to the question that is totally satisfactory, underscores the relevance of the current mathematical topic, lets me talk about math (which is super-cool because I love talking about math) and helps the kid to see just how motherfreaking awesome math is,

or

(2) MURDEROUS RAGE that someone, who's half my age, who hasn't even learned as much math as I've forgotten, would have the impudence to ask me that question. Not just to ask me that question, but to ask me that question when I have no idea whatsoever how to answer in a way that isn't complete bullcrap.

Okay, I don't actually get mad at students for asking me that question. Or any question. Not ever. I like having curious students. And a job.

I do try my best to be prepared to answer the question of "when am I going to use this?" for any mathematical topic that comes up in my classroom. But sometimes I do feel annoyed when I don't really have a good answer. Not annoyed at the student (not much, anyway), but more at myself for not being prepared with a brilliant, insightful response. After all, I'm the math teacher, right? I should know, in great detail, when the hell someone would ever use an inverse tangent function when they get out into the real world. I should be able to spell out the exact situation in which one would need to know everything there is to know about the latus rectum. (Tee-hee.)

But I don't always know the answer. When that happens -- depending on my mood/how busy I am/what's happening in class/how many cups of coffee I've consumed in the past five minutes -- I tend to go with one of the following responses (I don't recommend using any of these):

• "That's a great question." *flees without saying another word*
• "I can't tell you that; it would ruin the surprise!"
• "'Real life?' Math is real life, son."
• "You know, I find that the best answers in life are the ones we find for ourselves."
• "What're you talking about? I use it all the time!" ("But you're a math teacher," the student replies. "Yep, that's why I use it all the time!" I reply back.)
• "Uh, come see me after class and I'll be happy to talk to you more about your question." (Nobody has ever taken me up on this.)

I feel terrible when I don't have an answer for that question right away. There are times when it seems like it would be easiest just to say, "you know, there's a pretty good chance that you're not actually going to use this; but hey, gotta know it to pass, right?"

I mean, I've actually uttered those words to a student once or twice. I'm not proud of it. At the time, I felt like I was just being straight with the kid(s) who asked. I guess I figured that kids appreciate honesty and have a pretty good nose for B.S. But when I think about it, I realize that what I really did was cheat those students out of a great learning opportunity. I cheated myself as well.

I'm a math teacher, yes, but I'm also a math learner. A lifelong math learner. I shouldn't be ashamed or annoyed when I don't know exactly where or when stuff like hyperbolas or the mean value theorem are used in real life. Instead, I should be seeing an opportunity to learn something new. I should be excited that I've discovered something new to learn about a topic I absolutely love. I should be, like, absolutely jacked that there's stuff about math that I don't know but can find out about for myself. I mean, that's what I'd want my students to do, right?

So this year, I'm going to start using this response (or something similar, still a work in progress):

• "You know what? I'm not quite sure. I know there's a use for [insert mathematical concept here], but I'm still trying to figure that out. I'll try to do some research on it after class. Maybe you could also look it up, and let me know what you find. That would be really helpful."

I'm a math teacher; I shouldn't be shrinking away from that question when a student comes asking. I should be full-on body tackling that thing like it's a quarterback's blind side.

I do think we math teachers try to know where the stuff we teach can be used in the world beyond school; but we won't always know. When we don't know, we need to find out. We need to include our students in the process of finding out, because they asked the question in the first place.

We can't always know, but we can always care. We can always care enough to try and find out. We can always care enough to try and do better.

So this year, I'll try and do better.

Should I Even Bother Reviewing For Final Exams?

Happy New Year, everyone!

It's been a while since I last released one of my incoherent ramblings into the wild jungles of cyberspace, and I have much to talk about, so expect to see a few more posts in the coming days. (And if you don't see said posts materialize, please nag me until I get them done.)

(As a side note, as of this writing, my blog has about 25,000 views accumulated since my first post in July. 20,000 of those views are attributed to a post I wrote in August about the ninja board. Apparently Google likes ninjas.)

My school resumed classes today, and 1st semester final exams are coming up in a week and a half. That means the time has come to start reviewing for finals.

I've been wondering about this lately, the idea of spending a week and a half of class time reviewing for final exams.  I'm not completely sure I ever do it the right way. Actually, I often wonder if there even is a right way.  

Does it even do any good to review for final exams?

Every semester, I take the last week and a half or so before final exams to review with my students everything that we learned over the prior 16-17 weeks, tell them what kinds of questions to expect on the final exam and how many, give them time to work on review packets/assignments/flaming obstacle courses, etc. and so forth.

I've tried various ways of helping my students to take stock of what they learned (or were supposed to learn) over the semester. We've done the "review for finals process" as a project (with a rubric and everything) where students had to develop and publish their own study guides. We've done the classic "Jeopardy!"-style review game. We've done notecards that students were allowed to use on the final. We've done review assignments with the final exam questions literally lifted from the exam itself, with the numbers changed.

And what bothers me is this: Not once, that I can recall, in the four years I've spent teaching so far, have I been able to discern whether or not these methods of reviewing have done any good to any of my students.

What appears to happen is that the students who more or less have been "getting it" (or have been perpetually on the cusp of "getting it") all along are best equipped to understand and solve the problems set before them on the review assignments. Students who have been struggling all semester -- for whatever reason -- also struggle to find success on review assignments. It strikes me as a situation where the students who benefit the most from reviewing for finals are also the ones who need it the least, and the ones who benefit the least are the ones who need it the most.

I don't know why this appears to happen. (Or, if I'm really being honest, if it actually is what happens.) Maybe I haven't been making enough of an effort to find out. Maybe it's some bizarre phenomenon that can't be explained, like Honey Boo Boo. Maybe I suck at teaching. (Okay, maybe not.)

I was discussing this matter with my lovely wife the other night, and she asked me, "well, how do you know whether or not it's helping your students?" I thought about it, couldn't come up with a great answer, got childishly frustrated then stammered something like, "it's just based on what I've observed in class, I don't know how to explain it!" Then I pouted and decided to go do something else, because I'm so mature.

The bottom line is, I've never really been confident in my approach to reviewing for finals. I haven't made it easy for myself to tell whether or not my approach has a positive (or negative) effect. Maybe that's what makes me wonder if reviewing does any actual good.

Perhaps in the naivety of being a young teacher, I've been thinking of it the wrong way.  I think the best way to describe how I've approached reviewing for finals is that I've seen it as an eleventh-hour scaffolding activity, intended to give students one last hope at having a mathematical epiphany, a lifeboat that will float them safely through the perilous, shark-infested tides of the final exam.

It never seems to really work that way. No lifeboats. Sharks with happy tummies.

Maybe I should be looking at reviewing for the final exam as part of the cumulative assessment itself. Reviewing should really be more of a time for reflection and fine-tuning, not making a last-ditch effort for comprehending something for the first time. That's not to say there won't be a few students that do get that benefit from reviewing, but that shouldn't be the point. The point should be to look back at all of the work we've done all semester, take stock of what we've learned and what we still have questions about, address areas that still need addressed, and perhaps even celebrate.

My angst aside, here's what I'm trying this time around. The other day, I remembered something I read on David Coffey's blog about giving students the answers to the problems and having them explain how to get that answer. In my case, I'm going to provide students with a set of problems that are similar to what's on the final exam, give them all of the answers, and require them to explain how to get each answer. This way, they focus on how to solve the problem as opposed to focusing on getting the right answer.

I don't really know if this will be any better or any worse than what I've tried in the past. But, I think it will at least alleviate some of the anxiety and second-guessing that comes with reviewing for final exams. We have a week and a half, which should be plenty of time to address any questions or concerns that arise as students work through their review assignments, particularly since I am putting the focus on articulating their mathematical thought processes.

Will it do any good? Your guess is as good as mine.

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.

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.

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.

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

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

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

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

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

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

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

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