Reflections From #precalcchat: Pre-Calculus Sequencing

I love Twitter chats with other teachers. It's a great way to make connections. It's a great way to get insight, ideas, and resources. It's also a fantastic opportunity to reflect on your own practice and to improve what you're doing in the classroom.

The Global Math Department hosts several weekly Twitter chats for math teachers on a variety of topics. Since I teach Pre-Calculus, I dropped in on the first #precalcchat of the school year last week; thanks to Mimi (I Hope This Old Train Breaks Down...) and Taoufik Nadji for hosting. Couldn't spend much time, but the topic of conversation captured my interest:

I loved that thought. It made me stop and think about how I sequence my Pre-Calculus course and why.

I start with Graphing and Functions first. To me, it's important for students to understand the basics of interpreting graphs of functions and becoming fluent with moving between different representations of a function (graphs, tables, equations). I find this to be a particularly vital theme that I want to drive home with my students, especially those who will be going on to AP Calculus or Calculus I/II in college.

Next, I follow a pretty standard sequence of Quadratics/Polynomials, Rational Functions, Exponential Functions, Logarithmic Functions, Trigonometric Functions, and Analytical Trigonometry. Again, I focus on these topics in particular to prepare my students for success in an AP Calculus course. Other topics such as Analytical Geometry, Series & Sequences, Polar Systems of Coordinates, Conics, etc. come afterward as time allows.

The other chatters all had brilliant things to say, so naturally I felt like I'm probably doing everything wrong (or maybe just some things wrong, and other things not-as-wrong).

When discussing how Pre-Calculus can seem like a re-teaching of Algebra II to students, Tina C (Drawing On Math) mentioned that her school starts with Trigonometry for that exact reason.

This was an interesting idea to quite a few of us: do Trig first semester, slowly build up conceptual understanding of the unit circle, graphing, transformations, identities, etc. Then, move into the other different functions second semester.

The more I think about doing Trig first, the more appealing it seems to me. I've always found that I never seem to have enough time to really properly teach Trig and I need to either rush a few things or cut some other stuff out. I think I probably always had the notion that Trig is "more difficult," and somehow it made sense to put the "harder stuff" at the end of the year. (That's excellent reasoning, isn't it?)

But really though, Trig is a bit of a stand-alone topic. It could go anywhere in the course sequence. There are certainly some underlying concepts that can be applied to other functions: graphing, transformations, moving fluently between representations, and so on. I usually think of these concepts as having to be taught and mastered before doing Trig, as if Trig is the "CHALLENGE MODE" of working with functions in Pre-Calculus.

Who's to say we can't use Trig to teach these concepts instead? Maybe my students would have greater success with Trig if I did it at the beginning of the year, built the concepts slowly with appropriate scaffolding, while still equipping students to be successful in working with other functions. I may have to try it out one of these years. (I already have this year mapped out -- maybe next year?)

Anyway, some great food for thought.

I'm looking forward to more of these chats this school year, and hopefully I'll find time to continue blogging & reflecting on what I take away from them.

Week Zero: Realizing I Might Actually Know Stuff

It's Week Zero. School Year Eve. The last few days of summer before I get to go back into the classroom and spend the next nine months convincing teenagers that math is freaking awesome.

I'm a teacher mentor this year, which still seems crazy to me because I'm only four years into this profession myself. On Monday, I went to an all-day mentor training session to learn about my role and responsibility as a mentor. A lot of the information was about what I had expected: the mentor wears many different hats, has to build a relationship of trust with the mentee, can learn just as much about teaching from the mentee as the mentee does from them, etc. and so on. We talked about how to have positive conversations with our mentees, how to listen and to provide feedback, and best mentoring practices in general.

We also got toys and candy, which was super cool:

One thing that struck me from the mentor training was what distinguishes a good mentor from a not-so-good mentor: the desire to keep getting better as a teacher. Good mentors know that they still have things to learn about teaching, and no matter what the difference in experience is, they can learn a lot from their mentees. (I'm pretty convinced that I'm going to learn more from my mentee than my mentee is going to learn from me.)

I was reminded of this the next day (Tuesday) when I attended the first-day morning session of new teacher orientation. I sat with my mentee throughout the morning as we introduced ourselves and learned various things about the teacher-mentor program. We had time to talk about the upcoming school year and I was able to answer some questions about curriculum and how we do things at our school.

The experience made me think back to my first Week Zero in our district, when I went through new teacher orientation. I remember feeling excited and nervous about my first year of teaching. I also remember thinking that I was probably going to make a lot of mistakes, I was going to have to learn from them, and there was so so much about teaching that I didn't know yet.

I had the same excited, nervous feeling this week. I still feel like there is so so much about teaching that I don't know. But, in the act of answering my mentee's questions, I was struck by another thought: I actually, maybe, perhaps, do know stuff about teaching now. I had never really thought about it until someone else was asking me. When I was answering my mentee's questions, I really had a lot to say. I had a place of experience to speak from. Holy crap, I have experience. And it might even be useful to someone else.

That might be my important realization from this week: There are many things about teaching I still don't fully know. But I'm also starting to understand how much I do know about teaching. Maybe I'll actually be a decent mentor.

Anyway, back to work! Students come back next week!

Never Be (Fully) Satisfied

The past few days, I've been reflecting on how much time I spent this summer working on writing and tweaking curriculum for the new school year. It's not exactly a new activity for me -- I pretty much write and tweak curriculum every summer -- but I think I probably got more done this summer than I've ever managed to.

I actually fleshed out two different curriculum maps with topics & aligned standards (first attempt at aligning Common Core, so probably lots of mistakes). I'd never made curriculum maps with a great level of detail before, and I'm pretty sure I'm going to be very thankful I did so this summer.

I also spent a lot of time this summer working on incorporating more Problem-Based Learning (PrBL) tasks & lessons into the curriculum (one such idea I had is detailed here; feedback is more than welcome!). I teach at a New Tech Network school, so a rigorous PrBL curriculum is my goal. I've spent hours and hours looking for ideas, researching, thinking, scribbling in my notebook (particularly for those middle-of-the-night ideas), typing pages of details, and probably making my wife very annoyed that I was spending so much time working. I hope the result is that my students do some really awesome, really meaningful learning this year.

Another goal of mine is to learn more about Common Core (I admittedly am still a novice), so when my principal e-mailed the staff earlier this summer about attending a Common Core workshop in September, I was all like "MEMEMEMEMEME!!" So, I'm excited to go, learn some more about Common Core, and hopefully take away valuable knowledge that I can incorporate into my professional practice.

And the idea of improving my professional practice is something I've been thinking about over the past few days.

I've found myself thinking a lot about all the things teachers do to try and improve their teaching. I see many teachers who I follow on Twitter talk about all the conferences they attend and share what they've learned. I have several friends who are enrolled in masters programs, learning more about educational technology, developing curriculum, or otherwise broadening their skill sets as educators. I've thought about the things I've done each summer since I started teaching: working on curriculum, participating in the professional community, working on my own masters, and constantly thinking (and often worrying) about how I can be a better teacher.

And as I thought about all of this, I realized something: I'm not sure I ever want to be satisfied with the kind of teacher that I am.

I'm sure not satisfied with my teaching right now. Frankly, I'm not that great at it. (Sure, I'm funny, handsome, irresistibly charming, and very humble; but from a pedagogical standpoint, those traits can only carry me so far.)

But I don't think I want to ever be fully satisfied with my teaching, not even after I've been teaching for thirty (forty? fifty?) years. Sure, I want to feel happy about my teaching, which I think is a different thing. But not satisfied.

I think it's probably easier to feel this way now, since I'm only going into my fifth year. I know that I have a lot more to learn about teaching. Any fool can see that. There are roughly eleventy billion areas where I can to improve my teaching. I have rather lofty goals for myself this year. I might not meet them all this year, but that just means I'll regroup next summer and try again the following year. And the following year. And the following year. And so on.

But when I've been teaching for a few decades, I don't know how easily I'll still see all of that. I don't know if I'll still be this enthusiastic about improving my craft or if I'll be like, "meh, I've been teaching for thirty (forty? fifty?) years, I'm awesome enough." I don't like that idea. I really hope instead that I'll always want to be a better teacher than I was the year before. Even if it's just a teensy bit better. My students deserve that much, I think.

I talked about this with my wife the other evening. She understood where I was coming from, and noted that this is true about many professions. I mentioned that I was (and am) nervous about meeting my new students on the first day. She said one of her past supervisors once told her that's normal; "that means you care." And my feeling nervous doesn't really stem from being scared about meeting a new group of people, but more from really really wanting to be a better teacher this year than I was last year. I don't want to let these kids down.

Summer is great. It's a time when teachers can work on improving themselves and do what they can to make the next school year better than the last one. I spent a lot of time this summer working on that. I always want to be doing that. I want to be happy with who I am as a teacher. But I also think that I want to never be satisfied. Maybe mostly satisfied. But not fully satisfied.

For the Interns and the First-Years: 2013 Edition

It's almost time for another new school year to begin! Some of you have already started. Some of you start Monday. Some of you still have a few weeks left and are itching to go.

Last year, I wrote a post with some advice for new teachers and students entering their internship, and received many positive responses. I figured it might not be a bad idea to re-post it this year, with a few alterations based on comments I've received and my own newly-learned experiences.

Other teachers, please feel free to chime in with your own advice for first-years and interns in the comment box below! I would especially love to hear from second-year teachers with what advice they would give to a new teacher.

Without further ado, "For the Interns and the First-Years: 2013 Edition." Eleven pieces of advice that I have for interns and new teachers (that I never followed myself because I was more of an idiot then than I am now):


1. This year is not going to kill you.

This is the thing you need to know, first and foremost: you are going to survive this year. I mean, it's not really a life-or-death situation, but it sure as hell will feel like it sometimes.

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

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

You're not going to do everything right. New teachers never do. Veteran teachers never do. What matters is that we try to improve, all the time.

We try new things. We refine old ideas. We reflect. We seek feedback from others. It's a process that continues long after you've left college. Some of us have been teaching for years and we're still just getting some things about this craft of ours figured out.

Your most important skill as an educator isn't your ability to teach; your most important skill as an educator is your ability to learn.

Learn what you can from this year. From yourself, from your colleagues, from your students. It's wonderful. It's possibly one of the most challenging and exhausting years you'll ever have, but it's worth it. All you can really do this year is your best, and grow from there. 


2. Use your support systems.

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

I called my mom so many times during my internship.

I called, texted, and IMed (this was in the AIM days, mind you) other math ed students in my cohort. (They called, texted, and IMed me, too.)

I vented to my professor (who I remain friends with to this day). I vented to other teachers.

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

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

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

Even reaching out to total strangers on the Internet (such as the mathtwitterblogosphere for instance) is an excellent way to lean on other teachers for support. Seriously, get a Twitter account. Find fellow teachers who are active in various online professional learning communities. Participate in weekly Twitter chats. Start a blog. You'd be amazed by the support and positivity you get from teachers you've never even met.

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


3. Leave school at school.

There are many times that teachers do have to take work home. Whether it's grading or spending extra time planning, we often end up working more than we probably should. That's just part of the job. But we do it.

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

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

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

It drove me crazy and kept me awake many nights.

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

You can't always leave school at school; but when you can, learn how to do it effectively. It might take time (and experience), but it will come. 


4. Be realistic when you do take work home.

A couple of teachers gave me a great rule of thumb when it comes to taking work home: "only take what you can carry in your hands." Realistically -- unless you're feeling unusually energetic or having some kind of manic episode (kidding, kidding) -- you're not going to get terribly much done when you get home.

You're probably either going to be too tired to get much done, too busy with other after-school obligations, or just plain unmotivated. These are feelings that we refer to colloquially as "normal."

If you're anything like me, taking home everything will be self-defeating. You'll be staring at that huge pile of paperwork and just get to a point where you're like, "screw this."

I mean, there will be nights where a marathon grading session can't be avoided. It happens. But most nights, be realistic. Take home only what you can carry in your hands, and get that much done.

 
5. Exercise.

I wish I'd done more of this during my internship. Actually, I wish I'd done more of this my 20's entirely.

I love to run, and I'm currently training for the Chicago Marathon (because I'm out of my mind). I started doing long distance races a couple of years ago and have been doing a lot of running. I've lost weight, lowered my blood pressure, and lowered my cholesterol.

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

Also, it's pretty freaking cool when your colleagues start to notice you've been losing weight. The compliments can be very uplifting, especially when you're in the doldrums of late winter/early spring.

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

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


6. Eat healthy (or eat, period).

I lived by myself during my internship, and I was God-awful at keeping my place stocked with food. I'd skip breakfast. I ordered out a lot. As I like to tell people, I was sustained by Pokey Stix and Insomnia Cookies during my internship year. Thank goodness my wife started making me eat healthier when we moved in together, or I'd probably weigh eleventy billion pounds by now. Y'know, give or take.

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

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

Bringing leftovers from last night's dinner for lunch is another great way to go with meal preparation. 

If you're really lucky (which I am) and have access to ample refrigerator and cupboard space at your school (which I do), then keep your own supply of breakfast and lunch items around. I keep milk and cereal for breakfast, and stuff to make sandwiches or salads for lunch. This way, I don't have to mess with making meals at home; I can just do it when I get to school.

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


7. GO. TO. BED.

You need to sleep. Seriously. Go bed at the same time, every night. Get at least 7-8 hours.

You won't be able to function at your best if you're up past midnight worrying about lesson plans or grading papers. Yes, the grading needs to get done, but you have to balance that with your health. (This is another good reason to obey the wisdom of #4, above.)

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

You might actually save time by doing this. I got myself into a very messy pattern of staying up very late working on plans or grading, getting up at 5 AM (sometimes as early as 4 AM) the next morning, going to school all day, coming home and immediately napping for three hours on the couch out of sheer exhaustion. Sometimes I wouldn't wake back up until 8 or 9 PM, and I'd still have work to do (because I was stupid and brought everything home and didn't know how to turn off the "teacher mode" switch, all while not exercising or eating healthy and thus violating pretty much every piece of advice I'm giving you). If I'd stayed awake, I'd probably have gotten all of my work done and then had the rest of the evening to myself.

Get into a regular bedtime routine and avoid naps if at all possible. Use daylight to work, nighttime to rest. (I mean, not literal daylight, because the days are really short in the winter, but... well, you know what I'm getting at.)


8. But before you go to bed, be sure to unwind first.

Going to bed at a regular bedtime isn't quite as effective if you're working frantically on grading or planning lessons up until the last minute. You're probably more likely to have fits of insomnia if you do this.

Case in point: Just the other night, I was up working on a lesson for the upcoming school year. I worked until it was pretty late (hey, it's summer, the rules are different) and then went straight to bed.

That was a mistake; I lay awake half the night thinking about the lesson I was planning on, because it was the last thing I was doing immediately before going to bed.

Be sure to take at least an hour or so before bedtime to unwind and relax. Read a book. Play some video games. Netflix a TV show. A few months ago, my wife and I started watching Doctor Who, and we'll probably be continuing that as an evening ritual when the school year starts. It gives you time to relax before bed, and it gives your brain a chance to separate itself from work mode and sleep mode.

Go to bed at a regular time, but don't go to bed without distancing yourself from your work first. 


9. Choose one night as your "FUN ONLY" zone.

Just as important as learning how to turn off the "teacher switch" is taking some time during the week to have fun. You need one night during the week to be your inviolable holy sanctuary of "me time."

For me, this night was Friday. Most people pick Friday as this day, but some people prefer to use Friday to work ahead and enjoy the rest of the weekend (see #10, below). You may feel overwhelmed with work, even when the weekend is just kicking off. Give yourself permission to take a break. If you don't stop to enjoy some free time, you're going to burn out in a hurry.

Designate one night as a "fun only" time. Go to a movie, go on a date, go have drinks with friends (note: you should probably only do this at the end of the week), whatever you want. Just make sure you're taking time every week to do something fun.

But don't let yourself get too out of control before the weekend, because... 


10. "Early in the weekend" is a great time to get some work done.

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

By all means, go out on Friday night if you want to. Sleep in on Saturday if that's what you need to recharge. You've earned it.

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

When I say "early in the weekend," I'm being a bit broad. When I posted about this last year, I was specifically talking about Saturday morning. However, a few teachers mentioned to me that they prefer to get extra work done on Friday nights and then enjoy the rest of the weekend. One teacher commented that a colleague would put in extra time on Thursday nights, so she could enjoy Friday night, Saturday, and Sunday. So really, if you're the forward-planning type, you can get started on "next week" before "this week" is over.

I never followed this advice during my internship. (Really, did I follow any of this advice? Damn you, past Jeff.)

I was masterful at the art of procrastination; but having all of that work hanging over my head each weekend not only detracted from my ability to enjoy my free time, it also had a cumulative effect throughout the year. I got to the point in March and April of my internship year when I was having a hard time falling asleep on Saturday nights, let alone Sunday nights. I would actually be curled into a fetal position around noon on Saturday. That's no way to spend your weekend.

You might not want to do it, but chances are you'll be able to enjoy your weekend a little more if you're proactive during your weekend time (or if you're really ambitious, on Thursday night).


11. Year 2 will be better.

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

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

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

If the idea that your second year of teaching will be better than your first gets you through, then by all means hold onto it. I can tell you that the third year is even better than the second year. I just finished my fourth year, and it was by far and away my best year ever. I'm super-excited for my fifth year; in fact, I don't think I've ever been as excited to start a new school year as I am now.

It gets better.

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

Good luck. Be awesome.

-----

Once again, veteran teachers, if you would like to add anything, please comment below!

Second-year teachers, if you have any advice, I'd love to hear from you too. :-)

First-year teachers and/or interns, join the conversation!

Mathspotting (Because Math Hides In Plain Sight Like a Ninja)

One of the coolest things about doing math for a living is having a higher sensitivity to its presence in the world during day-to-day activities. For me, this seems to be particularly true right before the school year when my brain is constantly in planning mode. So I'm, like, on HIGH MATH ALERT.

I went to the beach with my dog yesterday morning, and noticed several sets of tire tracks in the sand. There are many different types of tire patterns, of course, but this particular set caught my eye (so I took a photo and tweeted it):

Of course, I'm not completely sure that these are exactly trig patterns, but... but... close enough, right? RIGHT?

In any case, it had me wondering about the application (if there actually is any) of periodic functions in designing certain types of tire treads. I don't really know anything about how tires are designed, so take it for what it's worth. But it's cool to think about; I mean, if I could legitimately tell a student that trigonometry is what keeps them from hydroplaning in a downpour, that would be awesome. I just don't know if that's actually true or not. *shrug*

In the evening, my wife and I were walking our dog and made a quick stop at the grocery store. As I was waiting outside with the dog, I found myself staring at this sign in front of me and wondering mathy things:

I actually look at signs like this and think about symmetry problems all the time. Like, probably an unhealthy amount. If you see me staring at a sign, chances are I'm probably thinking about symmetry. I try not to do it while driving.

Anyway, when I posted this on Twitter, one of the comments I got was: "What kind of symmetry? Even or odd?" Which is exactly the kind of question I was hoping to see. If I posed this problem to my students (and I may very well do that), I would love for this issue to arise. The "N"s certainly have odd symmetry, and I never did specify any particular type of symmetry. So, we'd have to include the "N"s in our answer, yes?

So those are just a couple of math nuggets that I spotted yesterday. Maybe I should post more mathy pictures on Twitter and start hashtagging them with #mathspotting or something. Feel free to join in!

"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.