the math of least resistance

Alright, so here we are.  The end of the initiative, which means the continuation of my posting is uncertain.  For one, it’s a time investment and time is something that is in extremely short supply in my life.  For another, I’m by-and-large uninspired.  This is a theme that’s been noticeable since I was in high school.  Give me a prompt and I can produce pretty much anything.  But if I have to prompt myself, I just ruminate and stagnate.  Terrible, I know.

Anyhow, this week wasn’t super exciting for any of my classes.  My algebra 2 kids are working on building intuition about quadratics and my calculus kids are doing a bunch of work with implicit differentiation.  So that leaves my precalc kids, who are developing the unit circle from the ground up.  It’s a bit of a slog and mostly involves conversing with them and drawing lots of pictures together.  So sorta hard to translate here, but…

Last week they came up with radians as a unit of angle measure (using strings and paper plates), so at the beginning of this week I had them derive the formulas for arc length and sector area.  In theory this is geometry, not so much trig, but it’s a good opportunity for them to get their brains working in radian mode and start picking apart a couple of the places that will get them into trouble down the line, conceptually, if we don’t start bumping up against them now.

Warm Up: PREC WU40
At this point they’ve completed a homework where they used the formulas they generated in the previous week for converting between different types of angle measures.  I like this one because most of them were able to think through it without grabbing their conversion notes or using a calculator.  They’ve gotten the hang of 2π being a full circle, so it’s pretty uncomplicated to convert the degrees to revolutions and then the revolutions to radians.  Also, they’re amused by the photo evidence I’ve provided of my skateboarding years.

Correct Homework: PREC HW39
Not an imperative part of the lesson here, but this is where they check all those repetitive conversions they did over the weekend.  Partner-up-and-lemme-know-if-you-have-any-unanswered-questions type of thing.  Clarify that it’s usually best to leave radian measurements with the π still in there and simplify the fraction, rather than a number with weird random decimals (which about half of them did as a result of just typing stuff into the calculator).

Classwork: PREC CW40
Earlier in the previous week they derived the formulas for circumference and area on their own by measuring stuff, cutting up stuff, etc.  I provided them again and gave them the definitions for arc length and sector area, with visuals to demonstrate.  Pose the question and let it sit.  Lots of protests arise.
“But it depends on the angle!”
“I don’t know the angle!”
“I don’t know how to do this!”
Push back.  Bite tongue.  Some ideas start trickling out.
“Oh, wait, I think I know how to do this.”
They come up with a way different than I was thinking, but it works the same, which is always a great moment for me.  My thought was to find the percentage of the given angle to the whole circle, then multiply it by the circumference/area of the whole circle.  They set up a proportion.  The angle out of the whole circle is equal to the arc/sector out of the whole circumference/area.  Then they isolate the unknown (in this case, the angle measure).  They look over their results and the wheels start turning.
“Huh…  That’s weird…”
“The formula for radians is so much simpler than degrees.”
“Why did that happen?”
YES!  Point made, without having to say it.  Flip over page for explanation.  Kids tell me that I’m turning into the science teacher because I put too many comics on things.

Homework: PREC HW40
Here’s what I like about this one.  Some of the kids tell me that they can just find the circumference/area for the whole circle, then divide it into however many parts.  They think they’re being sneaky, by avoiding using the formulas.  I don’t care if they use the formulas or not for this assignment.  Like I said, it’s basically geometry anyway.  I just want to reinforce the connection that sections of circles are proportional to whole circles, and more importantly, start getting them thinking about the difference between angle inputs and numerical outputs.  It worked pretty much perfectly.  Before the next class I had a kid come to me and ask, “Should I calculate parts (c) and (d) using degrees or radians?”  The conversation that came out of that was really valuable.  It doesn’t matter which one you use!  Either is fine (provided you go about it carefully if you are using formulas).  The arc length is the arc length, no matter how you measure the angle.  The area is the area, no matter how you measure the angle.  Spoiler alert: The sine/cosine are the the sine/cosine, no matter how you measure the angle.

I’m glossing over some issues here.  The biggest one is that I only have 4 kids in this class and 1 was absent, 1 was super late, and 1 was with the college counselor for half the class.  So the pacing wasn’t as it should be, which means things have to get shifted and tweaked (e.g. kids can’t check answers together, so instead, I look over individual kids’ homework while they work on the warm up… which happened 3 different times that day as kids filtered in).  I wanted to show the Anti-Pi Rant from Vi Hart, but with the timeliness issues, it just didn’t happen.

Also, we have short classes.  Only 40 minutes and we have no passing period, so classes go bell-to-bell with nothing in between.  Which means the first several minutes are lost to kids getting themselves to me, sitting down, getting stuff out, etc.  Also, at the end of class kids have to record assignments and I have to sign all their planbooks, so that eats a little time.  Let’s call it a net of 35 minutes of available productive time.  Really not very much, if this seems a bit sparse for a lesson.

Lastly, I had the PE teacher in my class that day, doing a peer-to-peer observation.  The kids kept throwing me confused looks, nodding slyly in the direction of the back of the room, trying to figure out what was going on.  As soon as he left, they went bonkers.
“What was that?!”
“Why was he here?”
We demand answers! The ambiance of a class is always slightly (or significantly) different with another person there, so who knows exactly how things would’ve played out if it had just been the usual crowd?

Despite the fact that things didn’t logistically work out the way they were meant to, I think I can say with a reasonable degree of certainty that the objectives weren’t substantially compromised.  Not the most stimulating lesson, but far from the least.  Probably fair to say that this day represented average student engagement and average student learning.  I feel pretty okay with average, at this point.

Comments?  Critiques?  Improvements?  Let me know!

This week’s post is all about the art of the question.  Now with links!  Gettin’ fancy all up in this blog.

A couple things worth discussing here.  The first is that I keep a little box labeled “Stump the Teacher” in my classroom.  Though I originally made the box as a way to let kids anonymously ask questions during sex ed the year I taught health, I decided to keep it after-the-fact because it provides a fun opportunity to engage with the kids in a different way.  Once a week I collect all the little papers in the box and post responses on a whiteboard that’s exclusively dedicated to these questions.

The kids put lot of interesting things in there.  This week’s submissions were, by coincidence, both animal-related.  One asked, “Can spiders be obese?”  The other included all kinds of statements, such as “some animals can survive space naked” and “penguins have teeth,” and I’m supposed to determine if they are true or false.

I’ve noticed 3 distinct uses for the box.

1. Questions tangentially-related to the curriculum that we don’t have time to address in class (e.g. “What happens if we compose a trigonometric function within itself an infinite amount of times?” or “Is there an algorithm for rationalizing the numerator/denominator that lets us skip all the icky algebra steps?”).
2. Genuine attempts to “stump” me with logic puzzles, riddles, or trivia.
3. General silliness (e.g. “Can I have $20?” or “Would you rather fight one horse-sized duck, or 50 duck-sized horses?”).

So there’s that.  But I also thought this would be a good time to reflect on my “How vs. Why” position.  As a teacher, I’m pretty transparent about my refusal to answer most of my students’ “how?” questions.  I always tell them (and remind them over and over again) that I will have a conversation with them if they can rephrase their question in the form of a “why?”

Here’s my theory.  I may be totally wrong about it, but it’s just been my observation over time that “how?” questions tend to be impatient ones.  “How do I…?”  Insert whatever task they’re struggling with.  “Can’t you just show me how?”  No, no I won’t just show you how.  You already know how.  You might not know that you know how, but you do.  It’s not my job to teach you “how.”  It’s my job to set up the circumstances whereby you teach yourself “how” to do something, through observation, analysis, etc.  I am a facilitator of your own learning.

I really try to make them derive as many of the algorithms on their own as possible.  I know, I know…  This is not an original concept, but stick with me.  My precalc kids are just starting trig.  On Thursday we did an activity using strings and paper plates that I’ve professionally titled, “What the Heck is a Radian?”  They observed that it takes a little more than 6 (2π!) strings the length of the radius to go all the way around the outside edge of the plate.  They drew conclusions about the relationship between revolutions, degrees, and radians, and completed a table with some common angles in all 3 measurement forms.  Then, analyzing the pattern from the table, they developed rules for converting between each type of angle measure.  I did not need to give them the formulas for these, they came up with them on their own.  Now, it took a whole class to get there when I could’ve simply said, “To get from radians to degrees, multiply by 180/π” in a fraction (fraction, ha ha) of the time…  But giving them rules they don’t understand the origin of makes me super sad.  It’s no better than answering their “how?” questions.  Don’t try to understand it, just do it.

So, I like their “why?” questions and I try pretty hard to make them interact with me in that way.  “Why?” questions are patient, and they are rooted in a desire to understand rather than to simply get things over with.  When a student asks “why?” he or she is genuinely trying to make sense out of something.  For example, “Why does the government get the same revenue from a lower tax rate as from a higher tax rate?”  That came up in algebra 2 while studying the Laffer Curve (and watching this clip from Ferris Bueller’s Day Off) in an effort to build some intuition about quadratics.  I drew a couple points on the graph and looking at them, the students said, “Wait, what?  Why???”  Then we had a conversation.

One last thought here.  I’ve been reflecting a lot about how many of my questions to them are in the form of “how?”  “How did you get that?”  “How do you know?”  Etc.  I wonder if there’s a way for me to prompt them to explain their thinking, using more “why?” questions, without sounding accusatory.  “Why did you do that?” sounds a bit… harsh.  I’d like to incorporate more “why?” questions from me to them, but I’m having trouble thinking of good ones.  If you’ve got some ideas or any go-to ones that you like, feel free to share them with me.

Look at me!  On time this week!  Barely, but still.  #nailedit

Alright, this week’s challenge is to blog about one of my favorite things.  Of course the “no d’uh” answer is the kids.  However, last week I didn’t really talk about teaching or math at all, so in an effort to direct this back towards the overall teachy-mathy theme at hand, I thought this would be a good time to share my grading policy for tests and quizzes.

I’ve actually been reflecting a lot on that this week, because the kids just took their midterms.  We run on a different schedule and exams happen the last week of the 2nd quarter (the last week of the 4th quarter for final exams), so I don’t get to grade these the same way I would their normal assessments during the rest of the year.  Which is, I must say, kind of a bummer, because I’ve really come to embrace the way I usually grade them.

A lot of teachers do “test corrections.”  Kids get the opportunity after their exams to make corrections and improve their scores.  I do something similar, but with a little twist.  It’s an idea I picked up from my first student teaching mentor.  He called it “rough grading,” so that’s what I call it too.

Kids take their tests.  At the end of the period, I collect them.  That afternoon/evening, I go through all of them and mark their correct work with a little “+” symbol.  If there’s anything incorrect about their work, it just gets left blank.  The next day, they get the beginning of class to look over their rough graded tests/quizzes and make adjustments, fix things, etc.  It gives them the advantage of not losing points for silly mistakes (assuming they can find them), of course.  But they also have the opportunity to at least see the quiz, then go home and study up that night so they don’t completely bomb the whole thing if they’re not adequately prepared.

One of the things I really like about it is that if a kid is just in a bad mental space on that particular day (reminder: I work with kids who have behavioral and emotional difficulties), his/her grade can really improve, simply because of being in a better place the next day.  Another thing is that there’s still a lot of personal responsibility the kids have to take.  They don’t get to do corrections open-book or with a partner; they still have to finish their assessments in testing conditions, on their own, and it happens immediately following the original attempt.  Lastly, some of the best learning I’ve seen from kids takes place when they’re looking for errors in work, rather than trying to solve something from scratch.  They often flounder when I try to give them “find the mistake” problems, but when they’re looking for a mistake in their own work, they’ve seemed much more invested and really proud of themselves when they finally do find it.

Which brings me back to midterms this week.  I really wish the kids had the opportunity to have them rough graded.  I mean, I could, but it wouldn’t go towards their Q2 grade.  And, realistically, they’re not going to have this luxury in college.  But when you see a kid quit halfway through who you know could finish most of the exam, it’s pretty disappointing.  On the bright side, it makes me realize how much I love the rough grading system, and I honestly don’t think I would’ve had the idea on my own.  So all my students have Mr. Denny to thank for that.

Here’s the effect of the rough grade in action.  Kid arrived for the quiz and before even beginning he wrote this sticky note and stuck it on his own forehead.  I took it from him and the next day, after he got to finish (and did so strongly), I added another sticky note to it.  We put it on the wall to remind him why he shouldn’t quit before he’s given things his best effort.  He added the next two sticky notes himself after subsequent quiz.

The first week and I’m already behind schedule!  This is one of the things about teaching that I found has made my life hard.  Like, really hard.  There is always more work to be done.  And that work always feels like the priority, so everything else takes a backseat.  But better late than never, I suppose.  Onward & upward.

In keeping with this (actually last) week’s assignment, I thought this would be a good place to recount our girls’ basketball game on Friday.  Even though being an assistant coach means more responsibilities and more drain on my already limited time, it provides such a valuable opportunity to connect with the kids outside of the classroom.  Plus, the two real coaches are men, so having a regular female presence is probably good all-around.  Who else is going to go into the locker room and tell them to hurry up when they’re goofing off instead of getting dressed?

The first thing to know is that the school they played is sort of our unofficial rival.  The second thing is that they’re pretty good.  The third thing is that we only have 5 girls this year.  Total.  That means no subs.  Ever.  And if someone gets hurt or has too many fouls, then we have to automatically forfeit*.  Going into the game we were all understandably nervous, but with our 3-0 record I think our hopes were pretty high that we’d manage to be okay.

Then the other team shows up.  With 12 girls.  The coach tells us, “This is only half our team!”  We tell her back, “Half?!  It looks like an army!”  Our hearts all sink a little bit.

But once the game gets underway, things are going surprising well.  Our team holds their ground in the first quarter and neither team scores.  Then in the second quarter, one of our girls scores and the doors just open up.  Three more successful shots for us, only one for the other team.  The game is 8-2 at the half and we were feeling pretty confident.

Then the reality of having no subs against a team that has 7 subs sets in.  Our girls start getting tired.  They start losing ground.  We put up a couple more, but the other team makes two as well, plus a free throw after a shooting foul.  It’s 12-7 at the end of the third, the lead just slightly narrowed but with things moving in the wrong direction.

The other team makes another basket and halfway through the fourth quarter things stand at 12-9.  Our girls’ nerves start mounting, and with them, a noticeable increase in tension and irritability.  One of our girls has a few physical altercations.  She gets elbowed and pushed over, has her glasses knocked off shortly after, and ends up in a collision with another girl that leaves them both on the ground at the sidelines.  One of our coaches says aloud, “She might not make it through this game.”  We’re all holding our breath.

We make another shot, but then something terrible happens.  The other team has a beautiful 3 pointer.  Respectfully, it’s really just inspired.  What can you do?

14-12 with 8 seconds left on the clock and no time outs for our team.  We have possession and the other team calls a time out.  All we have to do is hold the ball for 8 seconds.  That’s all!  Unfortunately, there is some sort of issue going on with the people running the scoreboard.  They’re not paying attention and the ref is calling out to them, asking if they’re ready to start running the clock.  It’s very distracting.  Meanwhile, the other ref signals our girl on the sidelines to resume play.  This is where things go wrong.  Very wrong.

The girl who should be receiving the ball is also distracted by the other ref yelling at the scoreboard people.  The girl on the sidelines passes her the ball, but she’s looking the other way and it just bounces off her hand.  The other team snatches it up and immediately calls another time out.

So now it’s 14-12 with 5 seconds left on the clock and we’ve lost possession.  They’ve got the ball at about half court.  Our team is pretty confused and understandably, very nervous.  The other team sets up to resume play and, of course, they’re giving it to their best player.  The one who made the 3 pointer.  She can dribble, she’s fast, and she can shoot better than anyone else on the court.  Things are not looking good.

They pass the ball in to her and the clock starts.  She takes it towards the basket, weaving right past all of our defense.  She puts up the shot at the buzzer and…  It somehow hits the back of the rim and miraculously bounces out and away!  It’s over!  We won!!!

There was a lot of jumping, shrieking, high-fiving, and laughing.  In the privacy of the locker room, there was also a fair amount of gloating about how the boys (who are not having a very good season) are gonna be so mad that the girls won another game.  Two years ago there was no girls team.  There was a PE class of all girls that took two “field trips” to play a couple scrimmages.  The boys teased them a fair amount about how they were… uh, not good.  How the tables have turned.

What these girls are doing is amazing.  4-0!  I’m so impressed and proud of how far they’ve come.  Better than that are their attitudes.  They’re so positive and they have so much fun.  It’s a blessing to be around that kind of energy.  No matter how many hours I spent this week writing review packets and midterm exams, the opportunity to share this experience with these girls is something I’m immensely grateful for.

*This may or may not be true.  It’s what the other two coaches told our girls, but maybe that was just to scare them.  Perhaps they can play down a woman, I’m not actually sure.