Killing My Own Snakes

When I was hired to teach at Southeastern in May, 1979, the Academic Dean at the time gave me only two pieces of advice: “Make your own way,” and “Kill your own snakes.”-Steven Fettke

One of the most valuable pieces of advice I received, from two different teachers in two different years (student teaching, first year), was that a new teacher had to know what “quiet” is.  If kids wouldn’t shut up, then kick them out until finally, the teacher experiences….silence. Without that baseline, a new teacher has no gauge to assess the ambient classroom noise.

I began teaching as a better than average classroom manager, and somewhat shrugged this wisdom off until I got the advice the second time after five particularly troublesome geometry students wouldn’t shut up during an entire lesson. So the next day, I warned them once and then tossed one then another off to the office. After two were gone, the other three realized I was serious and shut up, after growling a bit about unfairness. Turning back to the board, I suddenly heard…..silence. Utter, attentive, silence. And from that point on, I knew what silence was, and what to expect when I demanded it.

As a mentor, I always advise new teachers to err on the side of excess with disruptive students. If they have an entire class out of control, ask for help. If they have a few students misbehaving, toss them out after a warning. Screw fair. Get silence. Know what it sounds like.

New teachers are often fearful of  sending students out. They worry that administrators will judge them. They’re right to worry. Administrators often notice. At my last job, the volume of my referrals was  a constant source of tension.  In really poorly managed schools, the admins refuse to accept students and send them back. (Note: leave that school.)

This is where mentors come in. Mentors can, and should, give balance to new teachers. My induction mentor’s support and acknowledgement of my unimaginably disruptive students finally forced administrators to take action. If the teacher is weak, by all means help shore up the crumbles. But in the meantime, encourage the teacher to boot students who disrupt teaching time. I get impatient with people who bleat that removing kids from the class is depriving them of education. All students deserve an education. Students who are determined to prevent that can step outside.

In my experience, novice teachers stuck with unusually unruly students will improve their management skills if given the opportunity to remove the disruptors. As time goes on, these teachers will improve their handling of rambunctious students. Part of that improvement involves knowing what silence sounds like.

So new teachers should not try to kill all their snakes, particularly given the likelihood that they’ll have the toughest students.

I assume most teachers kill their own snakes after the first few years. But I’m often amazed at what senior teachers will tolerate. Sample statements, followed by my (usually unspoken) response.

“I’m teaching an Algebra 10-12 class, and the kids start packing up their stuff with fifteen minutes to the bell. Does that ever happen to you? What do you do to prevent that?”

I tell them to unpack their damn books and get back to work. Right now. And if they don’t start moving right away, oh my goodness, pop quiz.

“I’ve been having so much trouble with kids using cell phones constantly in class, not paying attention at all. What do you do?”

I take their damn cellphones away, giving myself extra points if I can swipe it from under their nose without signaling intent. Students who can’t keep off their phones lose them until the end of the day instead of the end of class. And they don’t dare complain, because I can always hand it over to the administrators, whose penalties are far more stringent.

“I have these two kids who constantly talk to each other, but when I try to separate them, they insist on sitting together. It’s so frustrating.”

Why the hell do you give them a choice? Tell them where to sit. In fact, tell everyone where to sit.

“I tell the kids not to bring food to the class, but what do you do when they’ve just bought lunch?”

You take the lunch away and tell them they can enjoy it cold later.

“I’ve tried taking away phones/telling them where to sit/taking their lunch but they refuse to give it over, and I don’t know what to do.”

You call and have them removed from the class.

“What? For something so minor?”

Listen well, little teachlings. Defiance of a teacher is not minor. It’s one of the few snakes that even experienced teachers should hand off to an administrator if they can’t convince the student to comply. Give the kid a chance to walk back. Offer alternatives. Draw a line, though, and if the line gets crossed, have the kid removed for the day.

And of course, logistics get in the way sometimes. More than once, I’ve picked up the phone to call for a supervisor to come take a defiant kid away–and no one answers the damn phone. So I have to call another number. Sometimes no one answers. All that drama and then….man, turning back around to face the class really sucks.

But well over half the time, simply picking up the phone has results, and the defiant one says something like “Well, you want me to give up my lunch AND my drink! No way!” and I say quickly, “No. Just the lunch. I insist on the lunch!” which leads to “Oh, I thought you wanted my drink, too. OK, have my lunch. BUT I KEEP MY DRINK!”

Other times,  the troublesome kid smirks. “Ha, ha, you can’t catch me, copper!” Shrug. Just shrug. And then later, call again, after the smirker has forgotten all about it, and have him pulled from the room, protesting. Don’t gloat. Just go on with the lesson like this is no big deal.

 

So you might be reading all this saying, wow, Ed’s a tyrant. Which is hysterical, because I’m one of the loosest teachers you’ll ever run into. Remember, I don’t assign homework. My kids sit in groups. I have a non-existent detention rate, the lowest in the school. I rarely give an F grade.  To my considerable pride, I’ve gotten the coolest of the Student Nominations three years running (best story teller, most unpredictable, most dramatic).  My classes are noisy and boisterous affairs. In many ways, my classroom environment is a progressive’s dream, the kind of place that Ed Boland dreamed of having before he realized he hated students.

I have five rules, handwritten seven years ago on still bright yellow poster paper. Students should avoid:

1. arguing with the ref (me)
2. eating, drinking, or grooming
3. setting objects airborne
4. travelling without consent
5. incessant yammering

But bottom line, do what I tell you.  My lines are very clearly marked, albeit occasionally negotiable. Just pay close attention to when I say “when”. As  I tell my kids every year at syllabus time: in order for “all this”–school, teaching, classroom environment–to work, I have to be in charge. Students have to obey my direct orders.

I realize that many teachers feel that schools already exert a great deal of control over student lives. They feel that rules about eating, phones, and seating are an unfair imposition. These same teachers often feel that “consequences” must be “deserved”, that their restrictions on those who have made bad choices, are somehow more reasonable.

Shrug. I’m not saying there’s only one way. Other teachers can make their own choices. Me, I avoid morality plays. I don’t talk about what students deserve or earn, simply about what helps me teach and others learn.  I handle even cheating as a pragmatic issue, not a value judgment.

From students’ perspective, their least  favorite of my management techniques is  my yelling, specifically  calling out or putting a student on blast.  They prefer teachers who rebuke quietly and in private. But they also agree that when you aren’t being the one called out, it’s fun to watch me rant.

As I invariably mention when going through the syllabus, the only action a student can take to earn a permanent black mark is deliberate cruelty to another student. I will punish that and I’m much better at being mean.

Note that I prohibit being mean to other students.  Nowhere in my rules is it verboten to be mean to me, the teacher.

At least once a year, I (usually inadvertently) get a student furious, and the exchange goes something like this:

Student: “F*** YOU!!!!”

Me, unfussed and occasionally confused: “Sit down.”

Student: “NO!!! You F******* *****! F*** YOU!! F*** OFF”

Me: “Sit down.”

Student, walking to the door: “NO WAY. EAT SH**. I’m OUT! YOU #*@#W%@#W%!”

Me: “DO NOT WALK OUT THAT DOOR!”

Student: “WHY NOT?”

Me: “BECAUSE UP TO NOW, YOU HAVEN’T DONE ANYTHING WRONG!”

This usually stops the student for a minute or so, giving me a chance to calm things down. In every case, after a brief talk with a fascinated class watching on, the student sits back down and everyone gets back to work. Show’s over.

Which is not to say I let students take nasty potshots at me. Like I said, I’m much better at being mean than your average adolescent. But I don’t demand respectful behavior, and don’t get upset at rudeness.  This will not come as a shock to people who know me online.

Look. Teaching is very much an expression of personality.  Mine is a teacher-centered classroom. But nowhere is it written that teacher-centered classrooms must be ruthlessly controlled environments of churchlike stillness.  My classroom is, like me, loud and often disorderly, friendly, sarcastic. It sometimes changes on a dime. But its purpose is always there, driving things along, moving everyone forward.

New teachers: does your classroom environment reflect your personality, your values? Experienced teachers: are you setting rules that matter? Are you sure?

 

Modeling Rational Expressions

As part of our Teacher Federalism agreement, I now include the reciprocal function as one of my parent functions in Algebra 2. But time constraints don’t allow me to really dig into the function–plus, the kids are on overload by the end of the term, what with exponential functions, logarithms, and inverses. I don’t really have time to switch gears. Besides, they’ll be exploring rational expressions in depth during pre-calc.

But then I noticed, during Trig, that my students really weren’t completely understanding that four of the major trig functions are rational expressions and how they differed from sine and cosine.  Meanwhile, I’m always doing a bit of algebra review…and so I decided to kick off my trig class with a rational expressions unit. A brief one, that enabled a review of quadratics and rational expression operations (aka adding and subtracting fractions with variables).

Starting Activity

Task 1 is a straightforward linear function, so almost every kid who has made it to trig, no matter how weak, is able to quickly build the function.

Task 2, of course, is the introduction of division. In function terms, we’re dividing a line by a line, while I will eventually make clear. But practically, the big hop occurs when students realize that cost per hoodie is not constant. Usually students do this incorrectly, graphing either a constant line of 20 or confusedly dividing total cost by 20. So I’ll meander by and ask:

“According to your table for L(h), we spent $520 and got one hoodie. What was the cost of that hoodie.”

“Twenty dollars.”

Silence. I wait. And inevitably, a student will gasp, “No! One hoodie costs $520!!”

It usually takes about 45 minutes for the kids to work through both tasks, including graphing the unfamiliar rational expression. Then I call them back up front for explanation and notes.

After putting the two equations on the board (linear and rational), I point out that our cost per hoodie equation is basically a line divided by a line. I point out the two asymptotes , vertical and horizontal. Why do they exist? Most students, by trig, know that you can’t divide by zero, but why doesn’t the vertical asymptote intersect y=20?

This usually prompts interesting discussions. I usually have a couple students to correct when they build the graph, as they make it linear. So now I redraw it, making clear that the drop is sudden and sharp, followed by a leveling. Why is it leveling?

Usually, a student will suggest the correct answer. If not, I ask, idly, “Can anyone tell me why the cost per hoodie isn’t $20? After all that’s the price.”

“Because you have to pay the $500, too.” and this almost always leads to a big “aha” as the students realize that the $500 is “spread out”, as many students call it, among the hoodies. The more hoodies purchased, the higher the total cost–but less of the $500 carried on each one.

At some point, I observe that certain forms of equations are much easier for modeling than for graphing. For example, when modeling linear functions, we use standard form and slope intercept form all the time–many real-life (or close to real life!) applications fall naturally into these formats. John has twice as much money as Jane. Tacos are $3, burritos are $5, Sam has $45.  But you’d never deliberately model an application in point-slope form. You might use it, given two points, to find the equation. But it’d be an operation, not a model.

So take a look at TL(h) and what does it look like? Usually, there’s a pause until I remind them that we could have negative hoodies, and we graph that in. Then the kids recognize the reciprocal function.

“If  we take a look at the graph and think of it as a transformation of a parent function, what’s the vertical shift?”

Silence.

I draw the parent reciprocal function . “Remember this? Where are the original asymptotes?” and eventually the kids remember y=0 and x=0.

“Right, so the original parent function, the horizontal asymptote is y=0. Where is it in this function? y=20. So what’s the vertical shift?”

Now they get it, and I hear “20” from all corners.

“Right. Is there a horizontal shift?”

“No.”

“So we know that h=0, k=20….what’s a=? What’s the vertical stretch?”

Someone always remembers that it’s the vertical distance between (1,1) and the actual output value for x=1, which is….

“Right. a=500. So TL(h) could also be written as 20+500(1⁄_h). Notice that if we split the numerator into two terms and simplify, we get the same thing. But we’d never model it that way. Much more intuitive to create the linear equation for total cost and divide it by the line.”

In other words, I point out, the hoodie activity is actually the same function that they learned about last year, but instead of just graphing or solving transformed functions, they’re modeling with it.

The second part of this activity is about 20 minutes, and comes at the end of my 90-minute block. In between, I do a lecture on  the meaning of rational expressions, vertical and horizontal asymptotes, usually bringing up something like this:

But while it works in the context of the lesson, it just pulls the focus of this post so I’ll write about that some other time. Suffice it here to say that yes, I discuss what the defining criteria of rational expressions are, what asymptotes are, and so on. One of the main reasons I teach this now is so the kids will understand both as they bump into them.

Part  three comes quickly because the students see the pattern, and that alone is enough to please a lot of them.  Suddenly, they’ve added an entire model type to their repertoire.

Part 4 is where so much gets tied together. Most students get all the way to part 3 without anything more than nudges. At that point, I usually bring it up front.

Using Desmos, we graph the same system. At this point, I’m obscuring the solution value.

This sets off discussion about the shift, how one starts out cheaper but stabilizes at a more expensive base cost. And then, look, the lines intersect? What do intersections mean, again? SOLUTION!

And in this first equation, the solution is quite simple because the equations have the same denominator. (note: I’m using x instead of h because I quickly copied these from Desmos):

So it’s a quick matter to solve the system, but again, good reminder. At 40 hoodies, they are $32.50 apiece.

Right around here, I point out that it would be convenient if one equation could show us the information we needed.  How could we show the difference between the two functions?

Fortunately, a number of the kids have lived through my algebra 2 class, and call out “subtract!”. I briefly explain that functions have operations and can be combined, for the rest. So we can subtract one equation from the other. Since ultimately the Hawk’s function will be more, we perform TW(h) – LW(h) and graph it as D(h).

Note first that it’s still a rational expression, although not the same type we’re working with. See how nice and clean the break even point is displayed!

So what we want to do is move this break even point further to the right. Luvs is ahead and has no reason to bargain. Obviously, we need to talk to Hawk’s Hoodies.

The administrators want to buy the better hoodie, but 40 isn’t enough to have a fundraiser–they want over 100. How can we get a better deal?

 

Suggestions? The ideas come fast.

First up is always “Hawk’s should sell the hoodies for cheaper.”

“Right. Hawk’s could lower its asymptote and slow the rate of increase in total cost. What would be a good price?”

We try $23/hoodie:

$22 is even better, putting the break even point at 100 hoodies.

“But here’s the thing–Hawk’s has real pride in their hoodies. They know they’re charging more, but their hoodies are worth it. That’s why we want those hoodies to begin with! They’re softer, better colors, hold up to wear and tear, whatever. So cutting their price by 12% sets a precedent. There’s a whole bunch of marketing research showing that customers don’t value luxury items if they sense the vendor will cut prices at a later date. So while we might like the price of the hoodies, Hawks could be hurting its brand if it cuts the retail price per hoodie. Take this as a given, for the moment. Is there any other way we could cut the price per hoodie that still maintains the same retail price?”

This always leads to good feedback: give a school discount, cut the price of the logo conversion, and so on.

At some point, I break in (unless the solution I’m looking for has been mentioned):

“All of you are coming up with great suggestions that involve reducing the value of the numerator. How about the denominator?”

Puzzled looks.

“If I have a division problem stated as a fraction, reducing the numerator (the dividend, if you must) will reduce the result, or the quotient. But is that the only way I can reduce the quotient?”

Pause. New teachers, let the pause hang. If it still gets no response, say “What else can I change?” because that will lead to someone saying…

“The denominator. But why would reduce the denominator?”

“Why indeed. 6 divided by 3 is 2. If I reduce the numerator to 3, my answer is 1. But….”

“Oh, I get it! Increase the denominator?”

“What would that do? Or put it this way: what would Hawks have to do to increase the denominator?”

And eventually, everyone figures out that Hawks could throw in some hoodies for free, which would also let them maintain their higher prices while still getting the sale.

“So go figure out the equation if Hawks includes 10 hoodies for free.”

Someone will always realize that this means we could get negative hoodies. So I tell them to test some negative values and remind them to think about what this might do to the asymptotes.

When they’re done, we put the whole thing on Desmos, showing that the vertical asymptote has changed, but not the horizontal.

“See, this way, Hawks is decreasing the time it takes for our purchase to get to the lower prices, getting us to just a little over $25 per hoodie with far fewer purchased, because we’re getting $10 for free.”

Now, take a look at the new equation to find the breakeven point:

“So how many of you remember being assigned these ridiculous equations with variables and fractions and thinking oh my god, none of us will ever use this? Who would ever have to add or multiple or subtract fractions? And yet, here we are. This one has them set equal to each other, but as we said above, function D(h) is the difference between the two :

…look at that! Your math homework in real life!!!

So we discuss what d(h) is doing. I point out that “solving the system” of TL(h)  and TW(h) is nothing more than “finding the zeros” for D(h).

From a curriculum standpoint, I transition pretty quickly from rational expressions to a review of binomial multiplication and factoring. So the D(h) subtraction equation gives me a great opportunity to review the procedures before I set them on their way. I work the problem–which requires the quadratic formula at the last step, ironically, but still gives us a chance to review the steps to determine whether or not a quadratic can factor. Then I show again how Desmos takes the equation and shows us how far we’ve “moved to the right”:

Then they all work out the comparison between a $3 reduction in price and ten hoodies thrown in for free. I take a moment to point out that math drives business analysis. Today, we have technology to do the work for us, but the best analysts have an understanding of the rational expressions driving the graphs.

If I had time, I’d do this in Algebra 2, but from a time perspective, I have a choice between introducing exponential equations and logs or go deep on rational expressions. That’s a nobrainer. They need to at least be introduced to logs, and there’s no opportunity in trig to bring that topic up. Rational expressions, on the other hand, forge a connection that makes sense when we get to the graphs.

And yes, it’s made a difference. I’ve been using this activity for two years, and have seen a noticeable improvement in their understanding of the four rational expression graphs. Remember, I’m not just teaching my kids, so even those who got a full dose of the rationals with other teachers are showing increased understanding. I would like to do this and more in Precalc, and will report back.

Sorry I’ve been so long without writing. We had a ridiculous heat wave and I responded by sticking to Twitter and playing Fallout Shelter, which is kind of cool.