# Book Recommendations

On one of my whiteboards today, I asked my students to write recommendations for what I should read next. They came up with the following, in no particular order.

Also, to my students that read this – please feel free to add more or email me more.

• This Side of Paradise by Scott Fitzgerald
• Something Wicked This Way Comes by Ray Bradbury
• Go Set a Watchman by Harper Lee
• Handful of Dust by Evelyn Waugh
• Inkheart by Cornelia Funke
• Julius Caesar by Shakespeare
• Lord of the Rings by J.R.R. Tolkein
• War of the Worlds by H.G. Wells
• That Hideous Strength by C.S. Lewis
• The Preist’s Graveyard by Ted Dekker
• The Sanctuary
• The Left Behind Series
• Consolation of Philosopy
• The Oath by Frank E Peretti
• The Alex Rider Series by Anthony Horowitz
• The Ranger’s Apprentice by John Flanegan
• The Brotherbane Chronicles by John Flanegan
• The Magic Tree House
• To Give a Pig a Pancake
• Shutter Island
• The Ishbane Conspiracy by Randy Alcorn
• A Girl of the Limberlost by Jean Stratton Porter
• Number the Stars by Lois Lowry
• Wizards, Aliens, and Starships by Charles Adler
• Sam I Am
• The Last Thing I remember series by Andrew (illegible)
• If We Survive by Andrew (illegible)
• Mysterious Benedict Society by Treton Lee Stewart
• The Roar by Emma Clayton
• Ella Minnow Pea
• The Sweetness at the Bottom of the Pie
• Angela’s Ashes by Frank McCourt
• Atlas Shrugged
• Harry Potter
• East of Eden
• Dante’s Inferno
• Fahrenheit 451
• On the Edge of the Dark Sea of Darkness
• Where the Red Fern Grows
• Proslogion by Anselm
• The Martian Chronicles
• Jane Eyre
• Don Quixote by Miguel de Cervantes
• The Dark Life by Falls
• Eragon
• Hedda Gabler by Ibsen
• Life as We Knew It by Susn Beth Pfeffer
• The Maze Runner by James Dashner
• Thirteen Reasons Why by Jay Asher
• Hate List by Jennifer Brown
• Partials by Dan Wells
• The Great Gatsby by F. Scott Fitzgerald
• The Chronicles of Narnia

So many to choose from…time to hit the library!

So, students read my blog. After last week’s post on blank stares, several of them explained their feelings about me being present to not-answer questions.

Student 1, on Friday: “It’s not that you answer the question. But when look at me, I see in your eyes that you know I can do this. That makes me believe that I can do it.”

Student 2, with help from Student 1, today: “It’s not really you. It’s just helpful to talk it out.”

Me: “So, it’s just the external processing? Could you just write down your thinking?”

“No, it has to be talking. And we can’t talk during a test, but if you’re here, and we’re asking you a question, then we can.”

Those are both things that make sense to me. Confidence, and external processing. Of course, most of their college professors are not going to indulge them on this, so what we need here is a way for them to meet those needs without my involvement.

Still, those are  both good explanations that make sense.

# Blank Stares

I’m giving tests in a lot of classes on Friday. I’m also going to be out, so a sub will actually be giving them. (This means that some of my students will have subs in all of their classes on Friday, leading to speculation that the junior teachers are all going to Astroworld together.) This has happened several times this year. If I’m going to be out, I try to make sure it’s on test day because it’s so much easier to find a sub.

Some of my students have no problem with this. “Oh, you’ll be out? Ok. Who’s the sub? Is it someone I like? Oh, you should get Mrs. X!” Others…not so much. I get looks of pure distress. I’m also starting to get accusations. “Again? You’re doing this on purpose, aren’t you?” (Well, yes, because of the sub thing.) “You just don’t want to answer our questions during the test!”

It’s that last statement that baffles me. I don’t answer questions during tests. It doesn’t matter whether or not I’m there – I’m not going to tell them how to work the problem. If there’s a typo or something, or if the instructions don’t make sense, that’s different, but a good sub can handle that. “Oh, the problem doesn’t work? Ok, well, write her a note telling her why, or how to fix it, or what you think she meant and then work that problem.” Or sometimes a sub will text me with a question. But I don’t answer student questions on how to work the problem. I just stare at them blankly. Sometimes I say, “Ok…”

I do have a lot of them who like to come up and ask me questions during tests. I don’t get it. They know by now that I’m not going to tell them anything.

They claim that it helps. That me just looking at them helps them understand. In class, students will call me over, and say, “Ok, I’m stuck here. I think I should do this next. And then I’ll do that. And then I do this…but now I’m stuck!” I look at them blankly. “Oh, wait, do I do this? I totally get it! Thanks!”

Literally, I stare at them blankly. I mean, of course, there are times that I help, but there are times when they claim that me looking at them helps them. I think that them talking themselves through their thinking got them unstuck, but they don’t agree.

Does anyone else have students who believe this? I’m considering printing out a picture of me staring at them blankly and taping it to the board on Friday so that they can just look up and see it.

# Visual U-Substitution

Last week, I wrote about u-substitution and ended by wondering what it would look like to give students a visual representation of what it does to the graph. I came up with this.

The red function is $f(x) = \frac{4x}{\sqrt{2x^2-1}}$, and I’ve shaded in the area represented by $\int_2^3 f(x) dx$.

The blue function is $g(x) = \frac{1}{\sqrt{x}}$, and I’ve shaded in the area represented by $\int_7^{17} g(x) dx$.

When you work through u-substitution on the first integral and change your bounds, you come up with the second integral and the new bounds. If you estimate the shaded area, you’ll see that they’re both just under 12 blocks.

When we do u-substitution, we’re not just doing some algebraic trick. (Well, yes, ok, it is an algebraic trick, but it’s not just an algebraic trick!) We’re doing a transformation that gives us a new function that’s easier to integrate (when we change x to u), then undoing the transformation by changing it back into the old one (when we change u back to x). If we’re doing a definite integral, then we’re finding a new function that’s easier to integrate, but doing the new definite integral over some bounds that give us the same area.

I don’t necessarily expect my students to walk away with a deep understanding of transformation of the x-axis from this. I’m still working out the details of it myself – most of the transformations I’ve learned about (other than the ubiquitous function transformations) were of the complex plane, and that was a while ago and fuzzy to begin with. But I like that this reinforces that we’re finding area, and that we’re changing the integral in such a way that the area doesn’t change.

Also, my students often struggle with why we have to change the bounds. I think this visual might help instead of me babbling about transformations of the x-axis and finding our new u-coordinates.

I didn’t think of this until after school on of our last day of u-substitution, so I showed it to my students the next day, after going over their homework. That was fine. I think it may have helped someone solidify their understanding. At the very least, it’ll give them something to chew on. I didn’t really have any questions for them, though – I did all the talking. And we weren’t really in the moment when they were wondering about it any more.

But next year, I want to show it to them when we start u-substitution with definite integrals. And I’ll start by showing them the red graph and the shaded area. We’ll work through the u-substitution together and come up with the blue graph, then I’ll have them find the bounds that give the same area – maybe with sliders. I need some functions that give friendlier areas, instead of “just under 12 blocks.” Then ask how the new bounds relate to the old bounds, and they can find the connection. This way, they’re doing more thinking, and we start with the idea of new bounds but same area, instead of coming back to it at the end.

Clearly, this is turning into a Desmos Activity.

If I ever build it, I’ll post a link here.

# Students in the Comparison Test

Background: I have two students in Calculus II.* A boy, M, and a girl, G, who have been really good friends for a very long time. They’re very comfortable with each other.

Today we hit the Direct Comparison Test.** (And the Limit Comparison Test, too.) The quick-and-dirty summary of the DCT says that if you’re trying to determine if a series converges or not, compare it to a different series. If your original series is smaller than your new series and the new series converges, then so does the original. If your original series is bigger than your new series and the new series diverges, then so does the original.

The standard statement of the theorem says it something like this: If $\{a_n\}$ and $\{b_n\}$ are sequences and $a_n \leq b_n$ for all n, then …. . But I have a goal this year of making less complicated and more memorable statements. So…the smaller series became G and a the bigger series became M. Because in my classroom, M is bigger than G.

Is it anything special? Eh, probably not, but it made all three of us smile.

Of course, I’m writing this so that I’ll remember it for next year, and the two students taking this class next year are about the same size, so I’m not sure what I’ll do then.

*Calculus is AP Calc AB. Calc II is roughly BC, but it’s for students who have already taken AB, so it’s everything in BC that’s not in AB. Which is a ton of integration methods, polar, vectors, and series. We covered the first three last semester. I have no idea what we’re going to do between series and AP review, because there’s going to be a ton of time. Graph theory? 3d graphing? Combinatorics?

**After going over this today, I’m wondering: why do we need the direct comparison test if we have the limit comparison test? The limit comparison test lets you get away with murder and still come up with an answer! Is it because we use the direct comparison test in the proof for the limit comparison test? Or is there a series where the direct comparison test works when the limit comparison test fails?

# Music

When I was in college, I was part of the marching band. (I played violin. Yes, we had violins in our marching band. No, that’s not a traditional band instrument. Yes, it was strange – but the marching cellos were stranger.) I went to every home football game and an away game or two every year, as well as a lot of basketball games

One of the things that sports arenas try to avoid is dead air. If something isn’t going on in the game, they want music, or the announcer, or an ad to be going. It shouldn’t be quiet. The one exception to this is when a player is injured – then it’s quiet, out of respect. Every year, the new drum majors learned that if they didn’t get the band to start playing quickly enough, the press box would put on some canned music. (We were not fans of this.)

Math class should not be boring.

Therefore, I play music in class.

This started last spring. We were in the midst of AP review, which involved students doing a lot of work and me doing almost no talking to the group as a whole. It got kind of boring. Also, class met an hour before the rest of the school that year, so the kids were pretty sleepy. One day, I opened up Pandora, put on a very safe station (usually soundtracks), and let it play in the background.

Eventually, it spread to my other classes. It was the end of the year, and I was willing to try anything to keep them focused on this year instead of college in the fall. Music helped. This year, after a week or two of quiet, I missed it and started up again.

Now, I don’t play music all the time. Not while I’m talking – no need to tire my voice out anymore. Not while they’re taking a quiz or a test. But if I’ve given them problems to practice or explore on their own, then I’m probably playing music. This does a number of good things to my classroom culture.

1.) Students ask for help more.

There’s something about the silence of a quiet classroom that’s intimidating. If they ask for help, everyone will know. So, they have to really have given up in order to risk everyone knowing that they’re asking for help. When music’s going, someone across the room can’t hear them, so it’s safe to admit that they don’t know something.

2.) They collaborate more.

I think that they’re afraid that if they talk to a neighbor, I’ll get on them for talking. I’ve no idea why – I want them collaborating! – but in addition to giving them cover from their peers, it gives them cover from me. I can actually still hear most of what’s said – they music isn’t that loud – but they feel safer. If I don’t turn the music on, they don’t start talking.

3.) Class is way less boring.

A good party has atmosphere, which involves music. Math class should be a party, so it should have music. Dead air = boring.

4.) This plays well into the shots game.

Yesterday, I talked about the mini-basketball hoop in my room. Students earn the right to take a shot and earn house points for doing certain things. Correctly identifying the movie that the song is from is one of those things. Telling me the composer for the piece of music is another. (I play classical sometimes, too, although rarely.) This is one of the few times that not everyone has an opportunity at earning a shot; normally, if you finish the problems, you can shoot. Here, only the first person gets a shot. So, yes, my class is a running game of name-that-tune. I like that there’s an element of competition that doesn’t favor stronger math students.

5.) It starts conversations.

“Oh, I know this! I know this! It’s, uhm, it’s….Star Wars!”

“Oh, I’ve never seen it.”

“What? You’ve never seen Star Wars?! What kind of movies do you watch, then?”

And we’re off…

What I love most about teaching is spending time with students. Don’t get me wrong, the math is great, but I can do math in an office by myself. It’s students that make teaching special. And this is one more thing that we can talk about.

Now, if I taught a different group of kids, would I do this? I’m not sure. My kids are mature enough and responsible enough that they can refocus when I hit pause. I know that they can handle it. If they ever prove me wrong, or if they just have a really excitable day, then I can always turn the music off.

The best compliment I’ve ever received on my teaching was when a student said, “This class is like my therapy.” It wasn’t the math that prompted the comment either – it was the atmosphere. And a big part of that is the music.

# Shots

Last year, I spent a couple of days shadowing students. Each day, I got a sub and followed a student around all day, going to their classes, doing their assignments, eating lunch with them. I even did the reading ahead of time so that I could participate in the discussion in English. I learned a lot. For instance, I thought I knew a good bit of theology, but I failed the quiz they took that day in worldview.

It was however, a very long day of sitting.

There’s a lot of research about how kids need to move, and it’s mostly geared at little kids. “Give them four recesses a day!” are headlines that I see on Twitter and the like. I teach juniors and seniors. They don’t get recess. But I know that after a few hours of sitting in in-service, I’m bored and tired and my brain shuts off. I want my students to move. Even a little bit.

All of that leads me to my favorite classroom game: shots.

No, no, no! Not alcohol! They’re underage! Although I routinely wonder when an administrator is going to walk in right as a student says, “Can we do this for shots?” or chants “Shots! Shots! Shots!” These are daily sayings in my classroom. *sigh*

Anyways, I have a mini basketball hoop on the back of my door, and some spots marked on the wall: 1 point, 2 points, 3 points. If a student makes a shot from behind that spot, they earn their team that number of points and go and mark it on the poster board that I’ve had hanging in the room since August.

My school has every high school student divided up into a house (think Harry Potter, minus the boarding school aspect), so we play where they’re earning teams for their houses. You could do it a lot of ways, though – just create teams out of thin air. Period 1 vs Period 5. Guys vs girls. Blondes vs brunettes. Whatever.

There are a variety of actions that earn a kid a shot, and I occasionally add more randomly. The standard ones are turning in papers and completing a certain number of problems. The goal is to get them moving – if you turn your own paper in, you can take a shot; if you hand your paper to a friend to turn in, you don’t. I don’t really care about whether or not they turn their own papers in, but I care a lot about whether or not they sit the entire hour that they’re in the room.

Most of the kids love it. I have a few who will do anything for shots. They first thing they ask me when they walk in is will there be any shots today. They are excited about doing their examples because then they can take shots. There are times when the entire class will pause their work to watch someone take a shot.

A few don’t want to participate, and that’s fine. They’re almost adults, and they have the right to decide what they’re going to do with themselves in my class, to certain limits. But most play. And more importantly to me, most get out of their desks without complaining several times an hour. Is it the same as recess? No, of course not. But it’s better than sitting still, and it makes class a whole lot more fun.

*This post was going to include pictures, but I came down with a cold and forgot to take them today. Oops.

This post is part of the MTBoS Blogging Initiative. For more posts like this, go here.