Category Archives: pedagogy

Five Things I Learned Remote Teaching Summer School

Attendance is a lot better when grades are involved.

Back in March, a younger, more innocent me argued in favor of excusing students who didn’t show up for high school classes after the shutdown. They didn’t sign up for online school. School is an obligation that society shunts on kids, and it has certain boundaries and (yes, really) choices. So if some students didn’t wake up on time and gave up, or decided that a grocery store hourly wage was a better use of their day, I was all in favor of holding them harmless from that decision. Their choice. In my view, they should not be penalized for that choice. No “F”. No delayed graduation. Take the class off their transcript, reduce the credits for graduation.

Silly me, not to anticipate that some districts and unions would band together and decide that if some kids couldn’t perform, then everyone must pass. Some schools froze grades at point of shutdown, others mandated the even more disastrous credit/ no credit for everyone. In both cases, students could skip school entirely and move on to the next class in the sequence so long as they were passing in March. In the second case, students could work hard and learn or never show up and get the same grade. In my particular school’s case, kids who didn’t show were literally missing an entire semester. Didn’t matter. Even in math. They passed with the same grade as the students who attended zoom sessions, asked questions, learned. And don’t try to feed me any shit about how the hardworking kids earned a moral victory, because without college admissions tests there’s no difference between a kid who didn’t learn anything and a kid who did, if the teachers were forced to give them both a pass.

I wrote half an article about this before I realized I was ranting. Peace.

Anyway. In summer school and, I dearly hope, this fall, teachers can give out grades. I had good attendance all summer.

So from now on when you read those stories about absentee students, remember to email the reporter and ask if the students who didn’t show were guaranteed passing grades and knew it.

Brand new students you’ve never met before? Not a problem. 

One thing everyone seemed certain of last spring was that remote learning only worked because teachers had existing relationships with their students. I worried about that, too. Happy news: that turned out to not be a thing.

I don’t do team building exercises, don’t spend lots of time getting to know my kids. None of this made a difference. I had roughly 25 students in two classes, but all but about five of them were taking both sessions (algebra semester 1 and 2). So about 30-32 students total.  None of them were from my school. I didn’t like remote teaching, but I still routinely received the two plaudits that comprise my success metric: probably 25% of my students said “I like the way you teach so much better; I’m really getting this.” and by direct count eight parents sent me a note saying their kids had mentioned how much they like my teaching. I say this not to brag, but because, well,  I’m a pretty darn good teacher and I get a lot of compliments. And that didn’t change in the move to remote. Students who had no idea who I was still thought I was and only saw me on Zoom a couple hours a day liked me a lot better than their last math teacher.

However, explaining is my go-to skill. So if that’s you, then not knowing your students might not be something to fear. If you are a beloved mentor whose influence is based entirely on in-class conversations and bonding, or lunchtime safe space, good luck.

One challenge left to take on: I sit my students in ability-level groups and they work together productively. I didn’t try this in summer school. Again, I’m not a huge fan of getting-to-know-you activities. I just bunch the kids together and tell them to get to it. I’ll have to be more conscious about this if I try it on Zoom.

Zoom Breakout Rooms

According to David Griswold, Google Meet has some nice features, but the list of limitations he rattled off have convinced me Zoom is my bet. I learned about breakout rooms in my other summer job, teaching test prep (they begged me and hey, I could go on vacation and still teach so why not?).

Breakout rooms solved a huge problem I had during the spring, when I ran “office hours”. I had no training on Zoom, just used what I saw. I used to have different groups sign in at different times, based on what topic they needed to learn, and it was a huge hassle. Breakout rooms are fantastic. You can set them up ad hoc.

Downside #1–to the best of my knowledge, you can’t add rooms after you’ve started, so I always create a couple extra.

Downside #2–if a kid drops off the line and comes back on, you might not see it for a while. If you, the teacher, are in a breakout room, you don’t hear the sound alert for a new entry. So learn to check the icon (it will say “1 unassigned”).

Downside #3–you can’t peek in on the other rooms. Remember when you were a beginning teacher helping one student out while right behind your back mayhem was breaking out? It feels like that. Except it’s not mayhem, it’s just kids not working.

Still, these are manageable problems. Breakout rooms are your friend.

Collect work right away

I quit assigning homework nearly six years ago. With remote learning, I’m no longer wandering around the room monitoring student work, seeing their progress. Last spring, I just asked students to turn in work via Google Classroom.

I wasn’t obsessive about it; students could skip turning in some assignments. But some students never turned in anything. I’d bring them in for special sessions and establish their level of understanding. Which was a lot of work, but remember, the kids didn’t have to show up at all last spring so I was in “sell” mode.

I couldn’t hold those extra sessions in summer school even if I’d wanted to. Students met with me every day for at least an hour. If they had questions, I held office hours earlier, and they’d come to those. Most kids were also turning in the homework, but at least 10 of the 30-some students were turning in little or nothing, despite coming to class every day and answering questions, demonstrating understanding.

I finally realized that they weren’t turning in work for the same reason they didn’t do homework–because once school ended, they were done. They didn’t think about class until the next day. In short, the reason that I stopped assigning homework all those years ago was still a really good reason.

So what I needed to do was consider this work classwork, not homework. Once I’d explained everything, I didn’t dismiss the class. “Do assignment 2, problems 1-8. DO NOT LEAVE ZOOM WITHOUT TURNING IN YOUR WORK. I will give you a zero otherwise.”

That worked. For some reason, the same kids who were untroubled by zeros for homework would religiously turn in classwork to avoid a zero.

By the way, reviewing classwork adds hours to my week, in case you think it’s all daily walks and a few zoom calls.

Google Form Quizzes

In the spring, I used a Classkick hack as a quiz delivery system. Classkick is a great way to administer several different quizzes to students–upload the quizzes into classkick, which allows you to generate a unique code. You can then give the quiz codes to student groups. Classkick’s value-add is the ability for a teacher to share a quiz view with just one student to help them out with questions.

These were just freeform quizzes, suitable only for regurgitation of the basics. That’s all I was able to do in the spring, and I began summer school using that method as well: build my quiz, convert to PDF, upload versions to Classkick.

But Google Classroom offers a Google “Quiz” option, which I learned was just a google form. With a bit of research, I was able to create my“multiple answer” tests:

Exponents:
GoogleFormQuiz1
Algebraic System
GoogleFormQuiz2

Graphed System
GoogleFormQuiz3

I can weight questions, import images, use images in answer choices. It’s very flexible. Not as flexible as paper and pencil tests, alas. I haven’t yet figured out how to allow students to correct answers or if I want to do that. But it’s a start.

None of this is great.

Pacing is incredibly slow. I’m not optimistic about returning to even my notoriously limited curriculum. If you know a teacher who is bragging about covering everything, that teacher has highly motivated and capable students or a lot of lost kids.  I hate being reduced to one mode of instruction. I know kids are only paying partial attention. their lives have been reduced to nearly nothing. This is a horrible way to teach, a worse way to learn, and shame on the people who think covid19 is a reason to shut down schools.

It’s a terrible thing that fearful people are doing to society, to children, to education. And I’m one of the lucky ones.

 


Evaluating vs Solving

Most math teachers start their year with algebra review. I like the idea of “activating prior knowledge“, as it’s known in ed school, but I never want to revisit material as review. It’s so….boring. Similarly, others “reteach” students if they didn’t understand it the first time and again, no, I don’t do that.

The trick is to wrap the review material in something new, something small. It’s wrapping, after all. For example, suppose the kids don’t really get Power Laws 1, 2, and 3 the first time you teach them, even though you went through them in insane detail and taught them both method and meaning. But you give them a quiz, and half the class is like, what means this exponent stuff? so you grit your teeth, yell at them, flunk most of them on that quiz, and go onto another topic for a week or so. Then one morning write ¾ on the board and ask “How would I write this with exponents?” and through the explanation you take them back through all of the power laws.

But I’m not here to write about power laws, although if you want advice on the best way to teach them, even if it takes longer, there’s no better tutorial than Ben Orlin’s Exponential Bait and Switch.

I’m here to explain how I integrate what we usually call “algebra review” into my course, while additionally teaching them some conceptual stuff that, in my experience, helps them throughout the course. Namely, teach them the difference between evaluating and solving functions for specific values.

Evaluate–what is widely recognized as “plugging in”. Given an input, find the output. Evaluate is Follows P E MD AS rules–well, technically P F MD AS, but who can say that? Note–I am pretty sure that “evaluate” is a formal term, but google isn’t helpful on this point.

Solve–well, technically it’s “plugging in for y”, but no one really thinks of it that way. Given an output, find the input(s).  Follows the rules of Johnny Depp’s younger brother, SA MD E P. (I hope I retire before I have to update that cultural reference). And really, it’s SA MD F P, but again, who can say that?

Things that get covered in Evaluate/Solve:

  • Remind everyone once more that addition/subtraction and multiplication/division run left to right, not one before the other. SAMDEP reinforces that, as I put the S first for the mnemonic.
  • “Evaluate”–Evaluating purely arithmetic expressions is middle school math.  At this stage of the game, the task is “evaluate the equation with a given value of x”.
  • “Solve” –Solving is, functionally, working backwards, to undo everything that has been done to the input. Right now, they know how to “undo” arithmetic and a few functions. They’ll be expanding that understanding as the course moves forward.x
  • Hinted at but not made explicit yet: not all equations are written in function format. I believe that, given an equation like  3x + 2y = 12 or x2 + y2=25, the terminology is “given x=4, solve for y” or “given y=3, solve for x”, but I’m not enough of a mathie to be sure. Feel free to clarify in the comments.
  • As I move into functions, this framework is helpful for understanding that evaluating a function must have one and only one answer, whereas solving a function given an output can have more than one input. It’s also useful to start capturing the differences between absolute value and quadratics, which aren’t one to one, and lines and radicals, which are.
  • The “PE” in PEMDAS and SAMDEP stands for exponent, but in fact the laws must be followed for every type of function: square root, absolute value, trigonometry, logs,  and so on. Informally, the “E” means “do the function” or “undo the function”, depending on whether evaluating or solving. So evaluating y=4|x-5| -6  with x=1 means subtract 5 from 1 (the “parenthesis), then take the absolute value (the “exponent”), then multiply by 4 and subtract 6. Solving the same equation would be adding 6, dividing by four, then undoing the absolute value to create two equations, then adding five in each one. (This is more complicated in text than explaining it with calculations on a promethean.)
  • YOU CAN’T DISTRIBUTE OVER ANYTHING EXCEPT MULTIPLICATION. This one is important. Kids will change 2(x-1)2  to  (2x-2)2 to 4x-4  with depressing speed and while many of them will make the last mistake in perpetuity, I’ve found that I can break them of the first, which also helps with 3|x+5| not turning into|3x+15|. For some reason, they never distribute over a square root, but plenty will try to turn 3cos(3x) into cos(9x).

Here’s a bit of the worksheet I  built.

evalsolve

I have found this prepares the groundwork for an indepth introduction to functions, which is my first unit. So when they’ve finished Evaluate and Solve, followed by Simplify ( more on that later), the functions unit:

So by the end of the unit the students can graph f(x) = 2(x-1)2 – 8 , as well as find f(3) and a if f(a)=10, and understand that the x and y intercepts, if they exist, are at f(0) and f(x)=0. They can also do the same for a square root or reciprocal function. Then I do a linear unit and a quadratic unit in depth.

Function notation, particularly f(a)= [value], is much easier for the students to understand once they’ve worked “evaluate” and “solve” with x and y.

evalsolvfuncnot
This also helps the students read graphs for f(4) or f(z)=7.

evalsolvefunction

Back in January, a Swedish guy living in Germany, as he describes himself, read the vast majority of my blog and then summarized his key takeaways and some critiques. His 6 takeaways are a pretty good reading of my blog, but he’s completely dismissive of my teaching and pedagogy, saying I’m mathematically naive and often, due to my ignorance, end up creating more confusion teaching needless information to my students. He explicitly refers to The Evolution of Equals and The Product of Two Lines, but I suspect he’d feel similarly about The Sum of a Parabola and a Line and Teaching With Indirection.

I’m really sure my students aren’t confused. I get pretty decent feedback from real mathematicians. There are legit differences between teachers on this point that approach religious wars, so there’s that.

Besides, these sort of lessons do two things simultaneously. They give weaker kids the opportunity to practice, and the top kids get a dose of the big picture.

Yes, it’s been a while since I’ve written. Trying to fix that.

 


Learning Styles

 

Isaac Asimov’s third robot story, “Reason“, has all the hallmarks of his early work: painful stereotypes, hackneyed dialog. Still, the conflict it explored has always hooked me.

Powell and Donovan, two troubleshooters who fix puzzling problems with experimental robots, are stuck on a remote sun-mining station training a new robot to capture energy from a planet’s nearby sun, run it through an energy converter, and direct it back to the planet. The robot, QT-1, or Cutie, decides that these humans are naturally inferior and must be early models that his superior frame and brain are designed to replace. His world was the station, his god was the Energy Converter, known as the Master, who wanted Cutie to direct beams to the dots. Powell and Donovan try to convince Cutie that the dots are planets, that he is a robot created by humans to do their bidding. Cutie thinks this is absurd and creates his own cult of believers, indoctrinating all the robots on the station with the will of the Master, with  Cutie as the Prophet. Powell and Donovan worry themselves sick with aggravation and fury.

The tale reaches a climax when Donovan spits on the Energy Converter. Cutie is horrified and angry at the sacrilege and refuses to let the two men into the Operations room. Powell and Donovan see a dangerous asteroid storm coming,  a catastrophic event that could cause the energy beam to misdirect and incinerate a third of the planet. Desperate to convince Cutie of his wrongthink, they hit on the idea of building a robot from the box, as it were. They uncrated a spare robot,  disassembled into parts, and spent three hours painstakingly putting the robot together. See? They created the robot! Just like they created Cutie!

Cutie shakes his head. Silly weak humans. Of course, they assembled the parts. But how did the parts get to the station? Only the Master could achieve that. So he turns away and ignores the two men, who stop sleeping and eating in sick anxiety over the incoming storm and the annihilation it will pour down on earth.

When, after the storm, they are finally released into the Operations room, Powell and Donovan rush in to assess the devastation. But no! Cutie protected all the humans on Earth perfectly and kept the energy supply constant. Or, as Cutie describes it,  Cutie “obeys the will of the Master” and keeps the beams directed to the right place on the dots.

Powell and Donovan realize they were worried for nothing. They just have to bring all the robots be indoctrinated in the Will of the Master as told by the Prophet (that is, trained by Cutie)  and the stations will be run beautifully. Cutie waves goodbye to them regretfully, knowing they are bound for “dissolution”, but encourages them to believe they are going to a better place.

Reasonquote

tl,dr: If learning styles make no difference in outcomes, who the hell cares what teachers believe?


Memorization or Learning?

I originally started to write a post on a memorization technique I’m using for the unit circle, and went looking for representative jeremiads both pro and con. Instead, I found Ben Orlin’s piece When Memorization Gets in the Way of Learning (from five years back):

memoryorlin

…which is the opposite of a standard, boring piece and serves as a good counterpoint to explain some recent shifts in my pedagogy.

It’s a good piece. In many ways, the debate about memorization runs parallel to the zombie problem–students regurgitate facts without understanding. Ben’s against that. Me, too. Ben says that testing requirements create tensions between authentic learning and manageable tests; I have various means of ensuring my students understand the math rather than just hork it up like furballs of unknown origin, so am less concerned on that point.

But I don’t agree with this sentiment as much as I probably did a decade ago: Memorizing a list of prepositions isn’t half as useful as knowing what role a preposition plays in the language. 

Not in math, anyway.

 

A couple years ago, after I’d taught trigonometry two or three times, I suddenly noticed that at the end of the year, my students were very fuzzy on their unit circle knowledge. (It’s no coincidence that Ben’s article and my observations are both focused on trigonometry, a branch of math with a significant fact base.) When working trig equations, they’d factor something like the equation above, use the Zero Product Property, solve for sin(x)…and then stop.

“You’re not done,” I’d point out. You’ve only solved for sin(x). What is the value of x?”

Shrug. No recognition. My tests are cumulative. Many students showed significant recall of concepts. They were using ratios to solve complex applications; they were sketching angles on the coordinate plane–both concepts we hadn’t revisited in months. They could sketch the unit circle from memory and eventually figure out the answer. But they had no automatic memories of the unit circle working backwards and forwards, even though I had emphasized the importance of memorizing it.

Upsetting, particularly at the end of the year. The name of the class is Trigonometry, after all. Solving for sin(x) requires not one tiny bit of trig. It’s all algebra. Trigonometry enters the picture when you ask yourself what angle, in radians or degrees, has a y to r ratio of 1 to 2.

The sine of π/2 is not among [the important things to memorize]. It’s a fact that matters only insofar as it connects to other ideas. To learn it in isolation is like learning the sentence “Hamlet kills Claudius” without the faintest idea of who either gentleman is–or, for what matter, of what “kill” means.

Well, okay, but….if a student in a Trig class can’t work a basic equation without a cheat sheet, what exactly has he learned? He already knew the algebra. Does the same standard hold for SOHCAHTOA, or can I still assume the student has successfully learned something if he needs a memory aid to remember what triangle sides constitute the sine ratio? What else can be on the cheat sheet: the Pythagorean Theorem? The ratios of the special rights?

Ben describes memorization as learning an isolated fact through deliberate effort, either through raw rehearsal or mnemonics, both of which he believes are mere substitutions for authentic learning. He argues for building knowledge through repeated use.

Sure. But that road is a hard one. And as Ben knows much better than I, the more advanced math gets, the more complex and numerous the steps get. Most students won’t even bother. Those who care about their grades but not the learning will take the easier, if meaningless route of raw rehearsal.

So how do you stop students from either checking out or taking the wrong road to zombiedom?

I’ve never told my students that memorization was irrelevant, but rather that I had a pretty small list of essential facts. Like Ben, I think useful memorization comes with repeated use and understanding. But what if repeated use isn’t happening in part because of the pause that occurs when memory should kick in?

So I’ve started to focus in on essential facts and encouraged them to memorize with understanding. Not rote memorization. But some math topics do have a fact base, or even just a long procedural sequence, that represent a significant cognitive load, and what is memorization but a way of relieving that load?

The trick lies in making the memorization mean something. So, for example, when I teach the structure of a parabolas, I first give the kids a chance to understand the structure through brief discovery. Then we go through the steps to graph a parabola in standard form. Then I repeat. And repeat. And repeat. And repeat. So by the time of the first quiz, any student who blanks out, I say “Rate of Change?” and they reflexively look for the b parameter and divide by 2. Most of them have already written the sequence on their page. The memorization of the sequence allows them repeated practice.

But it’s not mindless memorization, either. Ask them what I mean by “Rate of Change”, they’d say “the slope between the y-intercept and the vertex”. They don’t know all the details of the proof, but they understand the basics.

I take the same approach in parent function transformations, after realizing that a third of any class had drawn parent functions for days without ever bothering to associate one graph’s shape with an equation. So I trained them to create “stick figures” of each graph:stickfigures

I drew this freehand in Powerpoint, but it’s about the same degree of sloppiness that I encourage for stick figures. They aren’t meant to be perfect. They’re just memory spurs. Since I began using them a year ago, all my students can produce the stick figures and remind themselves what graph to draw. They know that each of the functions is committed on a line (to various degrees). Most of them understand, (some only vaguely), why a reciprocal function has asymptotes and why square root functions go in only one direction.

So did they learn, or did they memorize?

I haven’t changed my views on conceptual learning. I believe “why” is essential. I’m not power pointing my way through procedures. I am just realizing, with more experience, that many of my students won’t be able to use facts and procedures without being forced to memorize, and it is through that memorization that they become fluid enough to become capable of repeated use.

Like Ben, I think a zombie student with no idea that cosine is a ratio, but knows that cos(0) = 1, has failed to learn math. I just don’t think that student is any worse than one who looks at you blankly and has no answer at all. And addressing the needs of both these students may, in fact, be more memorization. Both types of students are avoiding authentic understanding. It’s our job to help them find it.

So I’ll give an example of that in my next post.


Great Moments in Teaching: The Charge

Friday, two weeks from the end of school, and it’s rally schedule: chop off fifteen minutes from each block for a screaming session in the gym. It’s fourth block, my trig class, and although I try not to have favorites, this semester has been a bit low on students with energy and ability. But even the goof-offs in this class can remember the basics of trig, have put some effort into memorizing the unit circle, reciprocal values, the occasional Pythagorean identity,  know the difference between sine and cosine graphs.  And only two cheaters. The top kids are amazing, enthusiastic, and driven–and there are lots of them, many of whom I just taught Algebra 2.  So a fun class, and really the only one with a genuine personality this semester.

I had given them some extra time to finish up a test from the day before, and it’s now just 35 minutes to rally.

“OK, I want to cover a couple things to set up Monday. Let’s….”

“NOOOOOOO!!!!” the blast of complaints hit me. I turned around and glared.

“Come on! It’s Friday! You can’t make us learn something new!” Tre, who last had a math teacher that wasn’t me in freshman algebra, put on his most ingratiating grin.

“It’s so hot, and my brain hurts. Please, no more math!” Patti slumped dramatically.

“QUIET!” I turned back from drawing a cosine graph to bellow them into submission.

tcgraph1

“I just want to introduce a couple of interesting properties and get you thinking, once again, about…oh, for christ’s sake.”

“WHAT??? What happened?” the students crane their heads forward to see the object of my irritation. I was growling at a student whiteboard sitting on a desk.

“Oh, some student used a fricking sharpie to draw a self-portrait.” and I held up the board so the class could see the penis.

“HAHAHAHAHA!” TJ was cracking up and I whirled at him furiously.

“You know, we use these white boards every day, and if I can’t get the sharpie off, it’s ruined. You think it’s FUNNY that students destroy my stuff?”

TJ was genuinely puzzled. “No. You just called him a dick. Like, without saying so. That was cool.”

“Fine. Ruin the fun of yelling at you. Take one more ounce of joy from my day.” I grinned at him and sprayed cleaner on the board.

“Ain’t no cleaner taking off sharpie,” Ahmed sympathized.

“Dude, this is Kaboom,” Tre said. “Kaboom’s the bomb.”

“Best cleaner in the known universe.” I spray the board and let it sit. All my kids know I love Kaboom. I tell new teachers about Kaboom, an essential teaching tool. When the kids write F*** in Sharpie, it’s so incredibly satisfying to wipe the obnoxiousness out of existence with one spray. Lesser challenges–gang graffiti, pencil sketches, soda spills, even small patches of gum–all disappeared.

“I hate students, dammit.” I turned back to the board. “I mean, don’t get me wrong. I love you all. But I just hate students. Ruin my stuff, treat it like crap….” I stop, because students breaking my stuff can put me in a foul mood in a hurry.

“It wasn’t us!” Matteo protested.

“Dude, it was you.”

“Screw you, Furio, how do you know?”

“Cuz you’re a dick! That’s your picture!”

I laughed, feeling much better. “Look, back to work. So you know how there’s a line, and then we can square a line, or multiply it by another line, to get a…”

“Parabola,” a reasonable amount of the class chorused, but I could hear talking.

“Shush, whoever’s talking. What happens when we square the cosine function? Take a look at the function and let’s just square what we….BE QUIET BRIAN..see. Cosine starts at…QUIET.” I turn around, wait for quiet. “Cosine starts at what, Furio?”

“1.”

“So 1 squared is..?”

“1”.

td2

I mark (0,1) in a different color, and move to the next hashmark. “Cosine is zero at pi over 2, zero squared is…QUIET.”

Most kids were paying attention, but there was this low level nattering that rose up every time I turned to the board.  But we got through the first one quickly.

“So here’s the square of the cosine function. What do you notice?”

“It’s a cosine graph!” Vicky.

“Sure looks like it. Period? Amplitude?” and we identified all the parameters for a cosine function graph.

tcgraph3

So the square of the cosine function can also be expressed as a regular cosine graph. Amplitude and vertical shift, one half, period one half the usual.”

Ahmed said with faux judiciousness, stroking his chin, “Ah, but how do we know this? It might just look like a cosine graph!”

“Good question. We can see the key points work, but maybe that’s just a coincidence. So pick a value and let’s plug it in. QUIET!”

“How about pi over six?”

tccomp

Carla was impressed. “Wow, when you double the value, it becomes something entirely different.”

“Yes….QUIET!!! I’m always surprised at how the alignments happen. So now let’s go on to the sine function. What do you all think will happ….QUIET!”

tcgraph4

“Jesus Christ, Eduardo and Brian, will the two of you shut.up.? NO! Stop the innocent ‘who me?’ crap. Three times in the past three minutes. I tell you to be quiet, turn to the promethean, turn around and there you are yapping again. Do I need to move you?”

Eduardo (Manuel‘s younger brother) and Benny look abashed, hearing the edge in my voice. I was mad at myself more than anything these two had done. Note to new teachers: don’t push through without attention. Constantly shushing is a sign you don’t own the room..  Don’t push through, stop when you need to. And it wasn’t an accident I’d picked two of the top kids in the class to shut down; it showed everyone else I was serious, if the unusual edge in my voice wasn’t enough.

By now I was furious with myself, and boy, do I get global in a hurry. My rotten students ruin my whiteboards and never shut up. I’m an idiot who decided to teach something complex 30 minutes before the weekend. And there are times when I’ve decided it’s not worth it and call it quits–call a pop quiz, put a problem on the board as an exit ticket, something. But deep breath, act like nothing happened, and push on, vowing to give it one more shot before I bail on an exit ticket activity.

tcgraph5

“Wait.” Joanie, probably my top math student this year, sat up and scowled at the graph dots. “How can that be a cosine, too? That’s weird.”

“What kind of cosine function? What’s different?”

“It’s reflected. So cosine squared is cosine, and sine squared is negative cosine?”

“Looks like it.”

tcgraph6

“But what’s the point of this?” Vicky asked. “Since squaring a sine or cosine function just takes you back to cosine, why do it?”

“Well, math applications will quite often require you to square functions, so it’s good to know how they behave. However, I really just want you to think about exploring functions. Up to now, you’ve been working primarily with transformations or known formats with parameters you can just plug in. But now we’re investigating functions that aren’t familiar with. Notice, too, that we did this all graphically with a minimum of evaluation.”

“So just for fun, what if we add the two functions we just created?”

tcgraph7

“Here they are together. So let’s add the five primary points.”

tcgraph8

TJ puzzled. “They’re all one? Really? That’s weird.”

“Yeah, but you can see it in the graphs,” Juan observed. “They’re equal at one-half, at opposite ends at one.”

I join all the points.

tcgraph9

“So the graph y= cosine squared plus sine squared is always….”

“One!” the class chorused.

And then I threw out casually, oh so casually, “And cosine squared plus sine squared is…”

“One!…”

The pause was the best part. I looked down, and waited as the recognition grew, until by god, the entire room was shouting in approval, clapping and stomping.

It’s one of those things that maybe you had to be there. But in half an hour, at the end of a day, in hot weather, right before a rally and a weekend, I’d not only gotten those kids to apply their knowledge of trig graphs in a new approach, but draw a connection from graphic to algebraic. They hadn’t recognized the familiar equation because their minds were in “graph” mode, and only when I asked about a Pythagorean identity, using almost exactly the same words, did they realize that they already knew what the graph would show. But not until then.

And they thought it was really cool that I’d pulled them around to this recognition.

Literally, a minute of stomping until I waved it down. “All right! Thank you. Remember during the first week, when I told you I’m a stickler for understanding the connection between algebraic and visual representations? Here you go.”

And then, “But what about tangent? What happens when you square that?”

Ten minutes left and I’ve got them asking questions. I realized I haven’t had to shush them once.

And just as the bell rings, we established that tan2(x) + 1 = sec2(x).

The kids rushed out to the rally. Rallies are my one Bad Teacher thing: I don’t go. I checked the whiteboard, Kaboom had wiped out most of the damage. Then I walked to Starbucks just completely charged, reliving the math and the applause. All the yelling, all the grouchiness, wiped away. I’d killed.

I keep telling you: Teaching is a performance art.

 


Four Obvious Objections to Direct Instruction

Recently, I defended teachers from Robert Pondiscio’s accusatory fingerpointing. Why no, sir, twas not teachers at the heart of the foul deeds preventing DI’s takeover of the public schooling system.

I don’t have any great insights into why DI isn’t more popular. But any reasonable person should, without any research, have several immediate objections to accepting the Direct Instruction miracles at face value. Hear the tales about Project Followthrough and spend ten minutes reading about this fabulous curriculum, and a few minutes thought will give rise to the following obstacles.

The weird objection

I’ll have more to say later, hopefully, about the roots of Direct Instruction. But no research is necessary to see the B. F. Skinner echoes.  Direct Instruction looks much more like conditioning than education.  A curriculum sample (I can’t make it bigger, click to enlarge):

NIDIcurriculum

You’re thinking good heavens, those “signals” are just optional, right? Nope. This video , without prompting, tells the viewer that yes, “signals” are required.

Recently Michael Pershan observed that ” while schools are primarily in the business of teaching kids as much as we can, it’s not anyone’s only priority. There are other things that teachers, administrators, parents and kids value besides instructional efficiency.”

Yes. Many of us value public schools that don’t feel like a cult.

The age objection

From the meta-analysis that’s given rise to all the recent stories:

The strong pattern of results presented in this article, appearing across all subject matters, student populations, settings, and age levels, should, at the least, imply a need for serious examination and reconsideration of these recommendations.

It’s behind a paywall, but I can’t help but be skeptical. I’ve never heard of Direct Instruction implementations at high school.  High school is leagues harder than elementary school and middle school. How would DI work?

Teacher script: “Hamlet Act One Scene One Word One What Word?”
[tap]
Class: “Elsinore!”

Or math:

Teacher script: “Y=mx + b is the slope intercept form. Word m What Word?”
[tap]
Class: “Slope!”
Teacher: “Word b What Word?”
[tap]
Class: “Intercept!”

How many subjects have been broken down to that level? How many books have they scripted for instruction? Or is the high school curriculum like this US History sample, a few questions every paragraph?

I don’t know. I’d guess the researchers don’t know, either.

If DI’s curriculum isn’t entirely defined for high school students in all subjects, then how can the claim be made that DI works for all age levels?  How can we be sure that the gains made in elementary school aren’t subject to the dreaded fadeout? What if DI is simply a good method of teaching basic skills but won’t address the gaps that arise in high school?

Maybe answers–good answers, even–exist, maybe DI works for fifteen to eighteen year olds, maybe Romeo and Juliet can be broken down into tap-worthy chunks. Or maybe those writing paeans about Project Followthrough have no success stories about older kids to tell.

The money objection

There’s a new meta-analysis [that] documents a half-century of “strong positive results” for a curriculum regardless of school, setting, grade, student poverty status, race, and ethnicity, and across subjects and grades.–Robert Pondiscio(emphasis mine)

If it works for all income levels, why aren’t rich kids using it?

I mean, surely, this incredible curriculum is what they use at Grace Church School or Circle of Children to teach these exclusively and mostly white little preschoolers how to read. Distar is the gold standard at  exclusive Manhattan elementary schools. All the teachers are going word one, what word? (tap) and all the little hedge fund progeny obediently repeat the word, or Word.

Except, of course, that’s not the case at all. Check all the websites and you’ll see they brag about their inquiry learning and discovery-based curriculum.

 

Zig Engelmann has written that he focused his attention on the “neediest” children, but that his curriculum helps all students achieve at the highest level. In which case, Zig, go sell your curriculum to the most exclusive private schools. Public schools spend much time arguing that poor children deserve the same education rich children’s parents pay for.

The race objection

I almost left this section out, because it is necessarily more detailed and less flip than the others. At the same time, I don’t see how anyone can hear about DI the miracle and not ask about race, so here goes.

About thirty years ago, Lisa Delpit wrote a stupendous essay, The Silenced Dialogue that just obliterated the progressive approach to education, effectively arguing that underprivileged black children needed to be directly taught and instructed, unlike the children of their well-meaning progressive white teachers.  As I looked up her article to cite  her comments about the “language of power” I realized that Delpit actually discussed this using the context of Direct Instruction (Distar is the primary Engelmann brand):

DelpitonDistar

Note that Delpit, who so accurately skewers progressives for withholding the kind of information that black children need, then rejects the notion of “separating” students by their needs.

She wants it both ways. She wants to acknowledge that some kids need this kind of explicit, structured curriculum while denying the inevitable conclusion that other kids don’t.

DI claims that all kids, regardless of race, see strong improvements.  But take a look at the videos, like this one from Thales Academy, and notice all the students reciting together. They all learn at exactly the same pace?

 

Really?

So I’m going to spoil alert this one. A quick google reveals that Direct Instruction doesn’t allow a student to progress until he or she has mastered the level, and yes, there is ability grouping.

History suggests that the students who move forward quickly will be disproportionately white and Asian, while the students who take much longer to reach mastery will be disproportionately black and Hispanic.

In fact, public schools are strongly discouraged from grouping by ability, and by discouraged I mean sued into oblivion. So how can Direct Instruction achieve its great results without grouping? And if DI helps all races equally, then won’t the existing achievement gap hold constant?

It’s quite possible that DI is an excellent curriculum for at risk kids, particularly those with weak skills or a preference for concrete tasks. It’s not credible that DI instituted in a diverse school won’t either lead to very bored students who don’t need that instruction or the same achievement and ability gaps we see in our current schools.

As I said, these are the relatively straightforward objections that, I think, make a hash out of Robert Pondiscio’s claim that teachers, those foul demons of public instruction, were the source of all DI discontent.  Next up, I’m going to look at some of the actual data behind the claims.

 


Great Moments in Teaching: From Dead Animals to Disney

ESL this year hasn’t been particularly enjoyable, unlike last year, which troubled me ideologically but was a joy to teach. I am primarily challenged by a hard truth: my students simply aren’t interested in learning English. In fairness, they’ve had a tough year, the details of which I won’t share. When I arrived, they weren’t grateful, but rather annoyed that they had a teacher who expected them to speak English rather than watch movies.

Most are eager to learn, having been out of regular school for a year or more. They’re just not  eager to learn English, and they particularly don’t want to speak English. I’ve been having trouble getting any conversation going; my questions are met with either utter silence or a request, in Spanish, that someone give them a one word answer to get me off their backs.

I can focus on any content, anything that sparks their interest while reading or at least hearing English.  I taught them ratios and fractions. We constructed some robots. They enjoy grammar, primarily because they just like completing worksheets instead of talking.  I showed them Zootopia, a clever little movie, and tied it into “prey” and “predators”, which then expanded into “producers”, “consumers”, and “decomposers”, then into “herbivores”, “carnivores” and “omnivores”. This went over pretty well, so I found an ESL science book and reinforced all that with pictures and text.

I’m a teacher tailor-made for covering a wide range of topics, and I’ve improved their compliance and cooperation. But they are still a sullen lot, with no cohesion and they aren’t that crazy about me, which is a hard ego hit for someone who’s quite used to being “favorite teacher”.

So I needed a day like last Friday.

Notably, Reyes was absent. “Behavior problems” and “ESL students” don’t see a lot of overlap; unhappy ESL students act out by passive inaction, in my experience. But Reyes, a junior from Mexico, became a huge behavior problem once the others started showing even minimal compliance and improvement.  He chases girls around the room. He pulls his hood over his head when he’s trying to ignore me. He constantly speaks Spanish, interrupting me and making crude comments  that cause the other Spanish speakers to giggle.  He refuses to speak English, even simply to ask to go to the bathroom. He’s not a bad kid, really, but nonetheless a disruptive force in the room was gone, and that mattered a lot.

We’d left the day before on “food web” and “food chain” and I brought the image of a spider web up again, intent on explaining in some way  that the original meaning of “web” has transformed, to start to get across the notion of metaphor. Then  I googled “web” without spider and bring up one of the results.

You get this sound, in ESL classes–at least you do in mine. It’s a genuine “Aha” of comprehension and connection. It’s a great sound.

“See? We use ‘web’ to describe the connection because it’s many connections to many other connections. It’s not one way up or down. Now look at ‘chain’” and I googled the word and tabbed to images.

Again with the “aha”.

“See the difference? In a chain, every link is directly connected to only two. See this one? In English, we often use the word ‘chain’ to mean one up and one….”

“Down!” they chorused.

“So when we talk about food web, we are talking about many to many.  See the many connections? All these animals exist in a web, with different relationships. Now look at a food chain. See the clear cycle, or circle?”

So far, so good. Then I lost them: “First, we’re going to focus on food chain, which is a basic way of seeing who is eating, and who is being eaten.”

I was quite surprised to hear a big groan from Allie. “I HATE English!!!”

Taio agreed. “Both eating! Why eaten sometimes, sometimes eat?”

Ah. “So when is it eat? When is it being eaten?”

Allie threw up her hands. “They are both the same thing!”

“No, they’re just the same verb root. But…. Huh. Let me think.”

“See? English is stupid!”

“No, no, I get that! And you’re right. English can be insane. But I’m not teaching you verbs right now. I just want to figure out how to make you see the difference. Oh, wait.”

And I quickly googled up “rabbit eating carrot“.

“The rabbit is eating the carrot. The carrot is being eaten by the rabbit.”

Pause, but I could see they were thinking. So I googled up “fox eating rabbit”.

“The fox is eating the rabbit. The rabbit is being eaten by the fox. So if you are eating, you are the one getting food.”

“If you are eaten, you are the food?”

“Exactly!”

Elian stood up and came to the front by the projector. “Who eats fox?”

“Great question. I don’t know? Who would kill and eat foxes?”

“Birds?” Allie again.

“Hey, that’s an idea.” I google “eagles eating foxes“.

“So then someone eats eagles?” Taio asked.

“Maybe. But some predators aren’t eaten. Like humans. We kill other predators, though, because of competition. So we kill foxes because foxes will eat our chickens and rabbits. Or we kill eagles because we like their feathers.” Elian nodded, and leaned against a desk, still up front.

“Let’s try another chain.” I google “mouse eating“.

“Elian, is the mouse eating or being eaten?”

“Eating!”

“Yes! So Taio, what is happening to the blackberry?”

“The blackberry is…eaten?”

“Allie?”

“The blackberry is eaten by the mouse?”

“You got it! So who eats mice?”

“SNAKES!” I had all seven kids playing along as I google snake eating mouse.

“The snake…” I prompted.

“the snake is eating the mouse!” even my non-English speakers, like Chao, was moving his lips, at least.

“THE MOUSE IS EATEN THE SNAKE!” announced Hooriyah, my lone Afghan student.

“No. Eaten BY,” from Elian.

“Yes. The BY is very important. Otherwise, in English, it sounds like you are saying ‘eating’.”

“That’s why I don’t like English. Eaten and eating sound the same!” Allie nodded.

“So remember the ‘by’. That will help.”

“Do snakes eat deer?” Taio asked.

I can’t begin to explain how pumped I was. We’d now kept steady conversation for close to ten minutes, where everyone was chiming in without prompting. So I googled “snake eating” and we paged down looking.

“THERE!” Taoi pointed.

“I have a question,” Allie announced. “What do you call that word that snakes do to….” she paused. Kept pausing and then shrugged. “I don’t know the word.”

“Crushed? Constricted? Squeezed?”

Allie had come up to join Elian, standing by the Promethean, looking at the images for one specific thing. “No. The other way. Before.”

“Poison? Some snakes bite their prey and the poison kills or at least paralyzes–makes the animal not able to move.”

“No, not that. It’s….” and here Allie gave up  in frustration, looking at me, trying to “think” the word at me.

Up to now, I’ve been doing a good job, but it was all ad hoc teaching, taking what comes.  But I don’t think all teachers grasp the essential moments of their job. This was an essential moment and I made it a great one.1

Nothing is more important to me in that minute than identifying Allie’s word. Writing this a week later,  I have a vivid memory of standing next to the projector, looking intently at Allie, oblivious to everything else, trying to grab the word out of her brain. And best of all, I could see that she knew this. She knew I was absolutely intent on figuring out her word, that I wanted this, that I wanted to be useful because hell, she’s stuck in this class learning a language she hates, can’t the teacher give her information she actually wants? For once?

My second great moment arrived, but I’m not sure it’s a pedagogical moment or just that of a very good and quick thinker. Because instead of trying to prompt more information from her, I started thinking about snakes. What are the ur-Snake things? I’d gotten constriction, gotten poison, what other snake categories are there?

Cobra?” Allie stared intently at the google results, but shook her head. “No, it’s…” she paused again, giving up.

“What do you call that?” Elian pointed.

“That’s a hood. Cobras have a really distinctive look. That’s why I thought maybe Allie was thinking of them.”

More ur-Snake. What else? I stare at the cobra images, and suddenly, miraculously, I think of Indian snake charmers.

“HYPNOTIZE!” I practically shouted.

“YES! WITH THE EYES!” Allie was overjoyed. “It makes the animals….something.”

“Obedient. Calm.”

“What’s hypnotize?” Hooriyah.

Third great moment, back to teaching. How to show kids what Allie is thinking of, and the meaning of “hypnotize”? I switch over to youtube.

“This is a famous Disney movie. Has anyone seen it?”

“Yes!” Allie was over the moon with excitement. “This is what I was thinking of!”

So as the scene progressed, I showed the students the broadly caricatured meaning of hypnotize.

When this was over, Allie rested content, sitting back down.

“How do snakes hypnotize?” Taio asked, saving me the trouble of raising the issue.

“I don’t think snakes actually do. I think people just think it is true.”

Allie nodded. “My neighbor has a snake. He says they don’t hypnotize.”

So I googled again, and we found a few highly verbal sites that seemed to deny it, but I didn’t dwell on this much.

Final pretty great moment in teaching: I brought it back to food chains!!

“So. Remember where this all started? Eating and…..”

“Being Eaten!”

“Let’s go through some food chains that you might see in a farm.” I wrote on the board.

corn->mouse->owl

“Owl?” asked Hooriyah, and I googled “owl eating mouse”.

“So now we know three bird predators: owl, hawk, eagle.”

Another food chain: wheat->caterpillar->black bird

“What’s wheat?” Taio again. “I don’t know wheat.”

“Every country has a primary grain. In South America, the big grain is corn. Maize.” Elian nodded. “In China, in most of Asia, it’s rice. In Europe and in America, also the Middle East, wheat is big.”

Allie, who has Brazilian parents but was born in Germany, nodded. “Yes. Bread is made from wheat.”

“And the Germans do amazing bread.”

“Bread!” Suddenly Taio is galvanized. “We have bao bread!”

I know a lot of Chinese food, but this one was new, so I googled.

“Oh, like in pork buns! I didn’t know that.”

“Dumplings. I hate dumplings,” Maria, Salvadoran, my best English speaker, had been missing from most of the class and had just arrived.

“No, this isn’t dumplings.” I corrected her. “Dumplings are like shu mei. It’s food wrapped in a pastry.” Chao sat up and chattered excitedly to Taio, who answered in English.

“Yes, that’s dumpling.”

I grinned at Elian, my only repeating student. “This feels like last year,” and he smiled in recognition. Last year, we’d talked about food in class all the time, going around the room talking about various foods just for fun–what they eat in Afghanistan for breakfast, what they eat in Vietnam for dessert, why Westerners make the best desserts (that was my claim, anyway, although my students roundly disputed this assertion).

We finished up with explanations of caterpillars and cocoons, and discussing the difference between blackbirds and crows–“One is just a black bird, the other is a blackbird.”

The bell rang off for once on an animated conversation.

I started this article a week ago, and was originally going to finish it with the hope that my class had turned the corner. My perpetual lagtime in writing allows me to say that it is better. Last week was a distinct improvement on every day that came before the great moments. More conversation, less lag time, and a much improved sense of camaraderie, even Reyes is speaking with a bit less prompting.

Before last Friday, I’d been telling myself regularly that tough classes are good for me. They keep me humble, keep me looking for answers, for methods, for strategies to help my students want to learn.

Besides, I’d tell myself grimly, tough classes make the triumphs all the sweeter.

I love being right.

*********************************************************************************

1Again, the great moment is mine. I’m standing there going oh, my god, this is a great moment in teaching, in my life. For me! The kids, hey, if they liked it, that’s good.

 

 

 


Why Not Direct Instruction?

Robert Pondiscio calls it the Rodney Dangerfield of curriculum as he berates the teaching community for disrespecting and neglecting  Zig Engelmann’s Direct Instruction program. Despite showing clear evidence of positive educational outcomes, Direct Instruction has been at best ignored, at worst actively rooted out for over forty years.

And whose fault is that?

..Direct Instruction, however effective, goes against the grain of generations of teachers trained and flattered into the certain belief that they alone know what’s best for their students.

Emphasis mine own, because oh, my goodness.

Trained and flattered.

Trained and flattered?

Trained?

Flattered?

Teachers?

I’ll leave you all to snorfle.

I do not dispute that many teachers think DI is creepy and horrible.  Here’s a fairly recent implementation [tap] that might [tap] help [tap] explain why [tap] teachers shudder. Word one, what word? Oorah!

But now, a question for serious people who want serious answers that don’t require the pretense that teachers are trained and flattered and capable of shutting down educational developments they dislike: why isn’t Direct Instruction more popular?

I’ve read Zig Engelmann’s book, Teaching Needy Kids in Our Backwards System,  and he doesn’t blame teachers. He thinks teachers are backwards and not terribly bright, but argues that most teachers introduced to his curriculum love it.

No, Engelmann puts the blame elsewhere.

 

For example, Direct Instruction unambiguously won Project Follow through. Originally, the program director had intended to identify winners and losers, to prevent schools from picking weak curriculum. But ultimately, the results were released without any such designation. Such a decision is well beyond any teacher’s paygrade.

According to Engelmann, the Ford Foundation was behind the effort to minimize his product’s clear victory. The foundation awarded a grant to a research project to evaluate the results.

The main purpose of the critique was to prevent the Follow Through evaluation results from influencing education policy. The panel’s report asserted that it was  inappropriate to ask, “Which model works best?” Rather, it should consider such other questions as “What makes the models work?” or “How can one make the models work better?”

Engelmann believes that Ford Foundation wanted to feel less foolish about funding all sorts of failed curriculum. I have no idea whether that’s true. But certainly Project Follow Through did not declare winners and losers, and thus from the beginning DI was not given credit for an unambiguously superior result.

Teachers didn’t turn Ford Foundation against DI.

But Engelmann and Becker were expecting decisionmakers to appreciate their success even if Project Follow Through didn’t designate them the victor. Becker wrote up their results for Harvard Educational Review, expecting tremendous response and got a few responses bitching about the study’s design.

I mean, cmon. Teachers don’t read research. That wasn’t us.

Engelmann and Becker fought for recognition all the way up the federal government food chain,  including politicians, and got no results. Shocking, I know.

Zig reserves his harshest criticism for district superintendents, describing a number of times when his program was just ripped out of schools despite sterling results. Parents, teachers, principals complained. One principal was fired for refusing to discontinue the program.

Throughout his memoir, Engelmann seems extremely perplexed, as well as angered, by his program’s failure, and to his credit is still determined to pound down the doors and win acceptance. His partner, Wesley Becker, was less copacetic. After years of rejection by his university and policymakers, Becker left education entirely and drank himself to death in less than a decade.   A few disapproving elementary school teachers aren’t going to induce that degree of existential despair.

Teachers didn’t kneecap Direct Instruction curriculum because it imposed an “intolerable burden” upon them, as Pondiscio dramatically proclaims. No. Decisionmakers killed DI programs. Time and again, management at the federal, state, and local level refuse to implement or worse, destroyed existing successful programs.

Blaming teachers and educators for what are manifestly management decisions is not only contradicted by all the available evidence, but failing to engage with a genuine mystery.

Why have so many districts refused to use Direct Instruction? Why has it been the target of so much enmity by power players in the educational field?

Those are questions that deserve investigation.

 

I did some more digging and have some data to talk about. I also want to discuss Engelmann’s book, since he often contradicts the claims made about his program.

But I’ll leave that for another day, because every so often I like to prove I can get under 1000 words.

 


The Structure of Parabolas

A year ago, I first envisioned and then taught the parabola as the sum of a parabola and a line.The standard form parabola, ax2 + bx + c, is the result of a line with slope b and y-intercept c added to a parabola with a vertex at the origin, with vertical stretch a.   This insight came after my realization that a parabola is the product of two lines (although I wrote this up later than the first).

I didn’t teach algebra 2 last semester, so I’ve only now been able to try my new approach. I taught functions as described in the second link. So the students know the vertex form of the parabola. Normally, I would then move to the product of two lines, binomial multiplication, and then teach the standard form, moving back to factoring.

But I’ve been mulling this for a few months, and decided to try teaching standard form second. So first, as part of parent functions, cover vertex form. Then linear equations. As part of linear equations, I teach them how to add and subtract functions.  As an exercise, I show them that they can add and subtract parabolas and lines, too.

So after the linear equations unit, I gave them a handout:parabolastructure

I don’t do much introduction here, except to tell them that the lighter graphs are a simple parabola and a line. The darker graph is the sum of the parabola and the line. What they are to do is explore the impact of the line’s slope, the b, on the vertex of the parabola, both the x and y values. We’d do that by evaluating the rate of change (the “slope” between two points of a non-linear equation) and looking for relationships.

Now, I don’t hold much truck with kids making their own discoveries. I want them to discover a clear pattern. But this activity also gives the kids practice at finding slopes, equations of lines, and vertex forms of a parabola. That’s why I felt free to toss this activity together. Even if it didn’t work to introduce standard form, it’d be a good review.

But it did work.  Five or six students finished quickly,  found the patterns I wanted, and I sent them off to the next activity. But most finished  the seven parabolas in about 40 minutes or so and we answered the questions together.

Questions:

  1. Using your data, what is the relationship between the slope of the line added (b) and the slope (rate of change) from the y-intercept to the vertex?
    Answer: the slope (b) is twice the slope from the y-intercept to the vertex.
    b2= rate of change
  2. What is the relationship between the slope of the line added (b) and the x-value of the vertex?
    Answer: the x value of the vertex is the slope of the line divided by negative 2.
    b⁄-2= x value of the vertex
  3. What is the relationship between the y-intercept of the line and the y-intercept of the parabola?
    Answer: they are the same.

Note: I made it very clear that we were dealing only with a=1, no stretch.

The activity was very useful–even some strong kids screwed up slope calculations because they counted graph hash marks rather than looking at the numbers. Some of the graphs went by 2s.

So then, they got a second handout: parabolastructure2

Here, they will find the slope (rate of change!) from the y-intercept to the vertex and double it. That’s the slope of the line added to the parabola (b!). The y-intercept of the line is the same as the parabola.

The first example, on the left, has a -2 rate of change from vertex to y-intercept. Since a=1, that means b=-4. The y-intercept is 8. The equation in standard form is therefore
x2 -4x + 8. In vertex form, it’s (x-2)2 +4.

Tomorrow, we’ll finish up this handout and go onto the next step: no graph, just a standard form equation. So given y=x2 -8x + 1, you know that the rate of change is -4, and the x-value of the vertex is 4. Draw a vertical line at x=4, then sketch a line with a slope of -4 beginning at (0,1).

This may seem forced, but students really have no idea how b influences the position of the vertex. I’m hoping this will start them off understanding the format of the standard form. If not, well, there’s the whole value of practicing slope and vertex form I can fall back on. But so far, it’s working really well.

By late tomorrow or Monday, we’ll be formalizing these rules and determining how an increase or decrease in a changes these relationships. So I hope to have them easily graphing parabolas in standard form by Monday. Yes, I’ll show them they can just plug x to find y.

Then we’ll talk about factored form, and go to binomial multiplication.

I’ll try to report back.


The Evolution of Equals

High school math teachers spend a lot of time explaining to kids that all the things we told you before ain’t necessarily so. Turns out, for example, you can subtract a big number from a smaller one.  Fractions might be “improper” if the numerator is larger than the denominator, but they’re completely rational so long as both are integers. You can take a square root of a negative number.  And so on.

Other times, though, we have to deal with ambiguities that mathematicians yell at us about later. Which really isn’t fair. For example, consider the definition of variable and then tell me how to explain y=mx+b. Or function notation–if f(x) = 3x + 7,  and f(3) = 16, then what is f(a)? Answer: f(a) = 3a+7. What’s g(x)? Answer: A whole different function. So then you introduce “indeterminate”–just barely–and it takes a whole blog post to explain function notation.

Some math teachers don’t bother to explain this in class, much less in blogs. Books rarely deal with these confusing distinctions. But me, I soldier on. Solder? Which?

Did you ever think to wonder who invented the equal sign? I’m here to wonder for you:

Robert Recorde, a Welsh mathematician, created the equal sign while writing the wonderfully named Whetstone of Witte. He needed a shortcut.

“However, for easy manipulation of equations, I will present a few examples in order that the extraction of roots may be more readily done. And to avoid the tedious repetition of these words “is equal to”, I will substitute, as I often do when working, a pair of parallels or twin lines of the same length, thus: = , because no two things can be more equal.”

First of his examples was:  or 14x+15=71.

Over time, we shortened his shortcut.

Every so often, you read of a mathematician hyperventilating that our elementary school children are being fed a false concept of “equals”. Worksheets like this one, the complaint goes, are warping the children’s minds:

I’m not terribly fussed. Yes, this worksheet from EngageNY is better. Yes, ideally, worksheets shouldn’t inadvertently give kids the idea that an equals signs means “do this operation and provide a number”. But it’s not a huge deal. Overteaching the issue in elementary school would be a bad idea.

Hung Hsi Wu, a Berkeley math professor who has spent a decade or more worrying about elementary school teachers and their math abilities, first got me thinking about the equals sign: wuquotenu2

I don’t think this is a fit topic for elementary school teachers, much less students. Simply advising them to use multiple formats is sufficient. But reading and thinking about the equals sign has given me a way to….evolve, if you will…my students’ conception of the equals sign.  And my own.

Reminder: I’m not a mathematician. I barely faked my way through one college math course over thirty years ago. But I’ve found that a few explanations have gone a long way to giving my students a clearer idea of the varied tasks the equals sign has. Later on, real mathematicians can polish it up.

Define Current Understanding

First, I help them mentally define the concept of “equals” as they currently understand it. At some point early on in my algebra 2 class, I ask them what “equals” means, and let them have at it for a while. I’ll get offerings like “are the same” and “have the same value”, sometimes “congruent”.

After they chew on the offerings and realize their somewhat circular nature, I write:

8=5+2+1

8=7

and ask them if these equations are both valid uses of the equal signs.

This always catches their interest. Is it against the law to write a false equation using an equals sign? Is it like dividing by 0?

Ah, I usually respond, so one of these is false? Indeed. Doesn’t that mean that equations with an equals sign aren’t always true? So what makes the second one false?

I push and prod until I get someone to say mention counting or distance or something physical.

At this point, I give them the definition that they can already mentally see:

Two values are equal if they occupy the same point on a number line.

So if I write 8=4*2, it’s a true equation if  8 and 4*2 are found at the same point on the number line. If they aren’t, then it’s a false equation, at least in the context of standard arithmetic.

So if the students think “equals” means “do something”, this helps them out of that mold.

Equals Sign in Algebraic Equations

Then I’ll write something like this:

4x-7=2(2x+5)

Then we solve it down to:

0=17

By algebra 2, most students are familiar with this format. “No solution!”

I ask how they know there’s no solution, and wait for them all to get past “because someone told me”. Eventually, someone will point out that zero doesn’t in fact, equal 17.

So, I point out, we start with an equation that looks reasonable, but after applying the properties of equality, otherwise known as “doing the same thing to both sides”, we learn that the algebra leads to a false equation. In fact, I point out, we can even see it earlier in the process when we get to this point:

4x = 4x+17

This can’t possibly be true, even if it’s  not as instantly obvious as 0=17.

So I give them the new, expanded definition. Algebraic equations aren’t statements of fact. They are questions.

4x-7=2(2x+5) is not a statement of fact, but rather a question.

What value(s) of x will make this equation true?

And the answer could be:

  • x= specific value(s)
  • no value of x makes this true
  • all values of x makes this true.

We can also define our question in such a way that we constrain the set of numbers from which we find an answer. That’s why, I tell them, they’ll be learning to say “no real solutions” when solving parabolas, rather than “no solution”. All parabolas have solutions, but not all have real solutions.

This sets me up very nicely for a dive back into linear systems, quadratics with complex solutions, and so on. The students are now primed to understand that an equation is not a statement of fact, that solutions aren’t a given, and that they can translate different outcomes into a verbal description of the solution.

Equals Sign in Identity Proofs

An identity equation is one that is true for all values of x. In trigonometry, students are asked to prove many trigonometric identities,, and often find the constraints confusing. You can’t prove identities using the properties of equality. So in these classes,  I go through the previous material and then focus in on this next evolution.

Prove: tan2(x) + 1 = sec2(x)

(Or, if you’re not familiar with trig, an easier proof is:

Prove: (x-y)2 = x2-2xy+y2

Here, again, the “equals” sign and the statement represent a question, not a statement of fact. But the question is different. In algebraic equations, we hypothesize that the expressions are equal and proceed to identify a specific value of x unless we determine there isn’t one. In that pursuit,  we can use the properties of equality–again, known as “doing the same thing to both sides”.

But  in this case, the question is: are these expressions equal for all values of x?

Different question.

We can’t assume equality when working a proof. That means we can’t “do the same thing to both sides” to establish equality. Which means they can’t add, subtract, square, or do other arithmetic operations. They can combine fractions, expand binomials, use equivalent expressions, multiply by 1 in various forms. The goal is to transform one side and prove that  both sides of the equation occupy the same point on a number line regardless of the value of x.

So students have a framework. These proofs aren’t systems. They can’t assume equality. They can only (as we say) “change one side”, not “do the same thing to both sides”.

I’ve been doing this for a couple years explicitly, and I do see it broadening my students’ conceptual understanding. First off, there’s the simple fact that I hold the room. I can tell when kids are interested. Done properly, you’re pushing at a symbol they took for granted and never bothered to think about. And they’ll be willing to think about it.

Then, I have seen some algebra 2 students say to each other, “remember, this is just a question. The answer may not be a number,” which is more than enough complexity for your average 16 year old.

Just the other day, in my trig class, a student said “oh, yeah, this is the equals sign where you just do things to one side.” I’ll grant you this isn’t necessarily academic language, but the awareness is more than enough for this teacher.