"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.

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.

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.

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.