Continuous Everywhere but Differentiable Nowhere

So I have started asking my students — Algebra II and Calculus — how they remember things. If they have mnemonics, or if they have funny methods, or what? I don’t know why I didn’t ask this before. It seems so obvious.

How they remember which graph is a sine graph and which is a cosine graph? How they remember certain formulas? How they remember know how to solve absolute value inequalities? 

I’ve gotten some pretty interesting answers, things I wouldn’t have come up with on my own.

For example, when doing absolute value inequalities, you might have . Well, you can actually rewrite that as AND . Similarly, if you have , you can rewrite that as OR . 

How do you know which is an AND statement, and which is an OR statement. Well, one student (who probably got it from a tutor, or another student), said: “You can remember that because the absolute value is less thAND a number, or the absolute value is greatOR than a number.” Love it!

So when you’re doing something particularly new and challenging, remember that students come up with seemingly inscrutible methods all the time. It helps to ask them what they’re thinking. Not only to see if they’re on track, but also because their thoughts might be super valuable to others.

I’m back from my wedding. It was great. One of my favorite moments — well, one of many — was when I got to sit on the porch with my friend T., who taught fourth grade for five years and has just transitioned into a pseudo-administrator position (he’s in charge of staff placements for elementary teachers in this very large school district).

We’ve never talked about teaching since I became a teacher.

It was great talking to him-as-teacher. I dig him-as-teacher. Two things struck with me, which made me realize that we shared a lot of the same values. He said:

“When you give a test and your students do badly, it’s your fault.” In blogs in the educational community, that seems to be an unstated undercurrent to many of the bloggers’s philosophies. There’s a lot to disagree with in a blanket statement like that (e.g. what if you aren’t given the tools, if your curriculum is impractical, etc.). But the sentiment — of accountability, of not immediately jumping to blame the students — is valuable.

“When I was interviewing,” T. said, “I was asked ‘what’s your two-week plan?’ in terms of what was going on in my classroom, and I laughed.” He laughed because he couldn’t fathom knowing what was going to be covered on Thursday if he hadn’t taught on Monday, Tuesday, or Wednesday. He didn’t mean that he was unprepared or directionless; instead, that he doesn’t go off of rote lesson plans that are concerned with teaching material, instead of flexible lesson plans centered around student learning.

(I laughed at that point, because I was like: “T., you’d be so proud of me. I make each of my lesson plans the night before too! Who would have thought that anyone else would find that a virtue?!”)

We disagreed here and there (we had a variation of this conversation/debate). 

But overall, it was nice to see my high school friend as a teacher friend.

Tomorrow I’m taking off to go to a wedding of a close high school friend. It’s a nice thought that some of the students I teach will be friends years down the road, like my friend who is getting married and I are. Years down the road. Okay, yeah, I know, I’m not that old, and I’m sounding positively ancient. But anyway, it’s a nice thought.

Just for a little walk down memory lane, I broke out the old yearbook. There are a lot of people who I barely remember, a ton of people whose names seem utterly foreign to me, and just a few names which I remember well. Partly it’s the hand of time trying to whitewash over my high school memories. However, I also moved to my high school at the beginning of my sophomore year; I didn’t grow up with these people. My histories with them don’t go back to childhood.

Here are two random entries:

Sam, my dear… It’s been an amazing year, and an… interesting… high school career. We are officially cast members of the longest-running Samuel Beckett production in history. You will always be the Godot tree to me. I love Mr. Parent and his countless journeys deep into the Absurd. That English class was the best. Hmm, what else? Oh yeah… WHAM! It was so much fun WHAM-ing it up the past couple of years. Jew power! We are the best there is, the best there was, and the best there ever will be. I wish you the best of luck at MIT and I know you’ll do well. I’m sure I’ll hear about you when you win a Nobel Prize or something. I really hope we can keep in touch next year. Sam, you are an extremely talented person and I’m sure you’ll succeed in whatever you decide to do. Have a great summer and have a great time next year. Talk to ya later.

Dear Sam-Bam, someday we will change the world, either together or separately. Everyone will know our names because you will invent the formula for world peace and I will be there to document it. I’ll take artsy fartsy pictures of your experience, and I will write poems about your equations. All you have to do is be a genius. Man! I gave you the easy job… But seriously, Sammy. You are one cool cat, and even if you still miss Illinois, I’m so so glad you moved here. Of course I don’t want to get mushy in your yearbook because this isn’t goodbye. We’re going to be friends for a long time. In fact, you’re not going to be able to get rid of me… ha ha! No siree bob, I’m going to show up just when you least expect it. When you’re a world renowned math professor. I’ll enroll in your class and surprise you! (Of course, we might have to work out a “payment plan” so that I can pass) (kidding!). I’m glad that your’e going to MIT rather than Harvard Smarvard. Those people are just stuck up. And I’m not just saying that because they rejected me. (No, I’m really not, because I never applied. I should’ve though. They might have taken me on as a social experiment.) Well, so far its taken me 40+ minutes to write this, so it’s about time to wrap up. Let me leave you with these two thoughts: 1. I love you! 2. “If you’re going to do it, overdo it. That’s how you know you’re alive. Go ahead, take a coma-nap baby; take a Puddle Dive” – ani. You know I had to end with ani.

My high school experience was in no way standard — my group of friends did not easily fit into a single stereotype. We were… unique. We saw ourselves as… unique. Maybe that “we’re so different” attitude is our stereotype, I don’t know. We thought we were just so great. We started an underground zine, and secretly put posters up around the school telling everyone about it’s imminent arrival. We would drag couches and TVs outside of people’s houses and watch movies in nature. We would drive 45 minutes just to have tea at this hip tea lounge because it was a place no one else knew about. Annually, we would all skip school on the same day to go to a giant rummage sale. (Okay, you got me, I had my mother call me in sick.) We would go to the local truck stop (hey, we were in Jersey after all) and drink coffee and order fries and hang out at midnight or one or two in the morning. 

We were also “good” kids. We did lots of community service. Lots. We quizzed each other for American History exams during lunch. We liked talking to (some of our) teachers. We were kids that — barring certain classes here or there — actually liked school. Most of us had jobs (me: restaurant and supermarket). Most of us didn’t drink until late in the game. Most of us didn’t do drugs. 

Maybe secretly we did want to be normal. But we were an eclectic group, and I think we identified as that.

But because of that “I’m so uncategorizable and unique” attitude I copped in high school, I thought “Our yearbook won’t reflect my high school experience at all.” And so I gave each of my friends a page and told them to fill it up with whatever they wanted. Quotes, messages, doodles, pictures, photographs, whatever. Then I made color copies made of each of the pages, bound them, and handed them out. My group of friends had our own yearbook, one that actually reflected who we were in high school and what we did in high school.

On my page, I put a whole bunch of memories strung together (“And the four-boy-trampoline-party in the rain? And my daily doses of “so sorry sams” and people going for the gummy and then making up?… Try not to forget the wrestling matches on the lawn, in my house, at Eva’s house, etc… And then there was the Skynard Concert and Dennis’s Fourth of July Mishap with THE DIP….“). I had a Richard Feynman quotation (“I was born not knowing and have only had a little time to change that here or there“). I had a quotation from My So-Called Life (I won’t embarrass myself by telling you which one). I had a pretentious math formula that I probably copied from my beaten CRC book of tables and formulas. I also stated that Counting Crows was “the best band in the entire world.” 

High school feels like forever ago. And yet, my high school friend getting married? That same friend who let us watch Real World in her basement? The same one with whom I tried to get ice cream from the McDonalds drive through by walking through, after playing ultimate frisbee? The same one who I ate lunch with, sitting in those weird-colored plastic seats in the lunchroom? 

PS. Do I think of any of this when I’m at my high school, teaching? Do I see any of my old self in my students? Do I remember the friendships that get forged, the drama that breaks out — daily!, the heightened emotional response to everything? Honestly, the answer is no. Maybe it’s because my high school was so different than my current high school. Probably it’s because I just have such a bad memory that I remember almost nothing. But for the most part — minus the “when I was in high school, I never would have…” moments — I don’t associate my life as a teacher with anything about my life as a student. At least not consciously. Fascinating, now that I think about it.

I wish this were a metaphor, or something deep, but it’s not. It’s just cute.

Last week, one of my multivariable calc students was late to class. No excuse, he just forgot to leave the break period on time. He was a good 5 minutes late, so me and the other students closed and locked the door, and when we heard him knocking, we starting talking — really loudly — about all this candy we had that we were eating, and throwing around calculus terms. It didn’t make sense, what we were saying. It was just us having a little fun.

After 30 seconds of this, we let him in, and I was like: “Oh, guys, we’re so silly. We should have moved the whiteboard (it’s on wheels in this classroom) in front of the door.”

Of course, Monday comes about, and I open the door to my classroom, and what do I see and hear? I see a whiteboard covering the entrance, and I hear my students — who have all arrived early to do this — talking loudly about candy.

It warmed the cockles of my heart. (Not that I know what a cockle is, nor whether my heart has them or not. But still. You get the point.) [1]

[1] Okay, I had to look this up. This is what I found.

Whatever phrase you want to use to mourn the loss of my start-of-the-year ambition, use it.

I had what I thought were two really good ideas that I wanted to head up in my school this year. 

1. An academic journal, where students could submit research papers they are proud of, for consideration for publication.
2. A professional development group that focused on making a bridge between the math and science curricula in the upper school (high school). So, for example, the math department should teach logarithms before the chemistry students learn about pH. It is especially important that we do this now, since the math curricula in the upper school is being completely redrawn and this is the time we can shift things around easily. [1]

Both were shot down. [2]

What’s the most sad part about this? An invidious seed has been planted in the back of my head. Each time I get an idea of something I want to take on to help make my great school an even better school, to help improve student learning, to get students excited about learning, to get teachers excited about what they’re doing, about anything, I know I’m going to think about these two ideas that never materialized and think twice about pursuing it. 

My more optimistic, excitable self back tomorrow. For the next couple hours, I’m going to be in mourning.

[1] In my school, we are required to join a professional development group which meets half a dozen times a year. These groups are led by faculty and span topics like “Space and Pedagogy” to “The Brain” to “Diversity” to “Critical Friends Group” (don’t ask what that is). Historically these groups have been cross divisional — so we’d have lower school teachers and upper school teachers in the same group — and cross departmental. 

[2] The first was shot down because there are teachers who want to encourage students to submit their good research papers to other journals, and having our own journal would get in the way of this. 

The second was shot down because — even though there was enough interest among math and science teachers, and both math and science department heads were excited enough by it that they wanted to join — the other potential leader of this group and I wanted to restrict the people in it to math and science teachers in the high school. For obvious reasons. It requires a bit of a long-winded explanation about the culture at my school why this would be frowned upon. But I think it boils down to this: if the professional development committee were to approve this, it would be setting a precedent it is hesitant to set. 

Without going on a “oh gosh he’s complaining again” riff, I’ll just say that I find the reasons against both rather specious. But I don’t want to rock any boats, make any waves. I’m going to let them die in peace.

I noticed that in my Algebra II classes and my Calculus classes, students don’t take good notes. Some don’t take notes at all, some take really sporadic notes, some use scrap paper to do the “check yo’self” problems I put up after teaching a concept — to test to see if they get the concept and can do a simple and a somewhat more complicated problem. Then there are the few that take really amazing notes.

My first reaction was: what the heck are you thinkIng?

In both classes, I’m starting off the year not using the book heavily, so their notes are their primary source of material.

But then I had three subsequent thoughts, which tempered how I thought about and approached the situation:

(1) Students learn differently, and they know how they learn best better than I do.

(2) Students might not be used to having a teacher use SmartBoard exclusively, and having the ability to download the class notes each day changes things dramatically. Now the students have the ability to listen, think, and absorb instead of having to listen, think, and absorb all while frantically writing.

(3) No one has probably taught students how to take effective notes in math class, which explains why most of them just write down equations.

Instead of giving my “I’m disappointed in you” talk to students — admittedly, my first reaction — I decided to take a different course.

I decided to deal with my calculus students first. On Thursday, I went about teaching function transformations, and ten minutes into class, I stopped the class mid-sentence and told everyone to exchange their class notes with their desk partner. I asked each person to assign a grade to the notes for the day.

Unsurprisingly, some students had absolutely nothing written down, some had absolutely gorgeous notes, and most had some chicken scratch or just a series of equations written down.

I had students whisper the grade they assigned to the notebooks to the owner of the notebooks. I made some jokes, they made some jokes, and we diffused the atmosphere, which was tense. Initially a lot of people felt like they were caught with their pants down. They thought it was a pop quiz and the grade was important. I told them I wouldn’t ever hear the grade.

We used this to launch into a discussion of notetaking. I prompted three questions:

1. How do you take notes?

2. How does my using Smartboard everyday and uploading it change how you take notes?

3. Why do we take notes?

Different people had different strategies they shared when answering 1 and 2. The four things I emphasized/brought out in the discussion:

1. Different people learn differently and hence take notes differently, and until I see you start slipping, I’m not going to get on your case. But learning how to take good notes in math classes is an important skill so you might want to get on it now.

2. Keeping your notes organized (by date!) and neat can help you a lot.

3. You don’t need to take notes on everything. But when we’re first learning a concept, you want to really put a lot of attention into how you’re taking notes.

4. WORDS! WORDS! WORDS! A math notebook needs WORDS to explain each step, each tricky point, each concept. If you try to study from a series of equations without any words, you’re going to forgot why we were studying those equations or doing each step or technique. You need to understand the concepts we’re learning and words are key.

My two calculus classes meet either first or last period most days (we have a weird daily alternating schedule, but for some reason, my calculus classes always end up first period when everyone is dog tired and still waking up or last period when everyone is beat from the day and ready to go home.

I didn’t really let students answer “Why do we take notes?” even though I put it out there. They were going to come up with obvious. I wanted to say the slightly less obvious.

The number one reason: TAKING NOTES KEEPS YOU AWAKE.

The number two reason: TAKING NOTES KEEPS YOU ACTIVELY ENGAGED WITH THE LESSON.

I presented these with some humor, too. We talked about these ideas, and then moved back to the lesson. From start to finish, this aside took only 10 minutes. I noticed, at least for that day, students were being a lot of conscientious about what they were writing down and how they were writing it.

(I am probably going to have this “stop what you’re doing and exchange your notes with your partner” moment every so often.)

I’m going to deal with my Algebra II students’s notes next week, and slightly differently. They are younger, and me preaching to them won’t be as effective. So I asked the help of one of my students from last year, who took the most beautiful notes, wrote the most beautiful homework (with words and answered word problems with complete sentences!), and was always engaged in class. I’m having him come to my class and talk with them about his strategies for succeeding in my Algebra II class last year. How he studied from the book, how he took notes, how he did his homework, etc.

He met with the learning specialist last Friday to just hash out his ideas and get them in order, but these are his words. I didn’t tell him what to say, how to say it, anything. He has 5-10 minutes of time to say whatever he wants.

I don’t know how it’ll work. But I’m hoping that it’ll be an early wake up call to my class.

With that, I’m out.

UPDATE:

So my former Algebra II student said this is what worked for him:

1. Read the section BEFORE going to class, so it at least looks familiar
2. Take notes on the bolded terms in the book (e.g. “leading term”) — the math terminology
3. Try every problem to the furthest, even if it seems like it’s going nowhere. Because it might go someplace good, and if not, it’s fun to see how far you can get.
4. Write the entire solution to problems done in class, even if you know the process. Writing things down helps.
5. Ask questions, but wait until the teacher (me!) finishes his thought; the question may be answered if you give the teacher a chance.
6. Make a formula page, with all the formula we learn.
7. Study 2 days before the quiz, not the night before.

Two new Mersenne Primes have been in the news recently. (Mersenne Primes are prime numbers of the form .) Finally, finally, after their primality (primeness?) was independently verified, they were revealed to the rest of the world:

and .

There was a lot of speculation about the number of digits that these numbers would have. Not least for the fact that the first person to find a Mersenne prime with more than 10,000,000 digits would win $100,000. And indeed, the newly discovered primes have 11,185,272 and 12,978,189 digits respectively.

To put that in perspective, The Math Less Traveled shows that if you write out the number of atoms in the universe, that number would have a paltry 80 digits.

Of course, I think: how can I use this in my own classes? We don’t really talk about primes in Algebra II, Calculus, or MV Calculus. However, we do talk about logarithms in Algebra II.

Check it out.

How do you think we know how many digits are in and ? It’s a simple application of logarithms.

We know that that a number is written in the form , it has digits (if is an integer). Think about it: has 2 digits; has 4 digits; has 6 digits.

If isn’t an integer, it’s just a hop skip and jump away to saying that the number of digits is the next higher integer. So if we have , we have 4 digits.

Where do logarithms come into play?

Well, has a certain number of digits. Since that 1 probably won’t affect anything since it’s such a huge number, we will ignore it. How many digits does have? Let’s use what we just discovered:

Then solving for , we get . Hence, we know there are 11,185,272 digits.

And a good question for the really ambitious student is to ask: can we be sure we can ignore that 1? (Answer: yes.)

(You can find the number of digits for in the same process.)

You know you’re a bit rusty from the summer when a teacher asks you to — without using L’Hopital — to prove that: 

And you are like, oh, that’s easy.

And then — after two false starts — it takes you 3 minutes to figure out.

Just remember: you are a calculus teacher. 

I’m going to say it again: you are a calculus teacher. 

Maybe if I say it enough times, it will be true.

(more…)

Every so often, I get a reminder of how completely different this independent school world is to the rest of the universe of schools out there. I guess after my first year in this microcosm, the shock and infinitude of differences have become so naturalized that I fail to recognize the weirdness, except when something jars me out of this strange reality.

Then you’ll usually hear me mutter “back when I was in school…”

And Sarah and Jackie’s comments to my last post did exactly that. I spoke about letting students out of my classes a few (not many) minutes early, if they finished and checked over a quiz. How is that even possible?

Let me paint a scene for you.

A school where there are no hall passes, no late passes, no detentions, no bathroom passes [1]. Students, when they have a free period or two, can sign out and leave the building. To get lunch, to get coffee, to enjoy some fresh New York air. Sometimes students sign out three our four times a day. We send our official attendance to the main office only once daily — after homeroom — and then teachers are responsible for keeping track of their classes attendance. Students are trusted that they’ll be where they need to be.

Not that there aren’t the occasional breaches of trust. A substitute comes and a student sneaks out of class. A student skips out on a gradewide meeting. Students who aren’t allowed to sign out — because of being late to school too many times or being put on academic probation — sometimes do sneak out. (Not on my watch, mind you.)

But they are occasional, and definitely the exception.

Right now, as I type, I’m sitting with my laptop at the sign out table in the front entrance. Students come by to say hi as they walk to Chipotle for lunch. I often get to have really nice conversations with teachers who walk by.

It’s a different world from what I grew up in, where our bathroom passes were toilet seat covers spraypainted in neon colors, where we had official pink hall passes, and where there were detentions for being late to class too many times. I guess that’s my “back when I was in school…” moment.

There are amazing benefits to working at my New York City independent school. And, as you would suspect even though I don’t write about them here, there are problems too. I don’t see my school through rose-tinted glasses. Seriously, I don’t. Still, I have a lot of admiration for this community that has been cultivated over the past hundred and some years. This school does something right… a lot of somethings right.

I gave my first official quiz today in my Algebra II class. Since so many of my students have 50% extra time accommodations, I designed a 30-35 minute quiz and let the class take the entire 50 minutes. Usually there are a number of students who finish early.

Before students turn in their quiz, I tend to say “are you absolutely, absolutely sure you want to turn this in? Once it’s in my hands, I won’t hand it back, and it tends to be exactly 1 second after students hand in a quiz that they realized they made a mistake and want to check over the test again.” Except for those with some sense of usually false bravado, they wisely go back and check over their work.

But I’ve come up with a good way to keep students occupied once they’ve finished their test. I put up a problem — somewhat based on what we’ve been doing but *just* different enough that students will be forced to make a new conceptual leap on their own. If they show me the right answer, with correct work, I let them leave class a few minutes early. A luxury, for sure. The solution to the problem becomes their key out of my class.

Today in Algebra II, we had a quiz that covered — among other topics — inequalities. Students learned simple linear inequalities and how to solve them. So, for the challenge problem, I put:

Find and write in interval notation where: . 

Two of them got it. Most of the rest of the class wanted to get it. I liked them thinking about problems that are just beyond what we’ve done.