Most Popular Posts and Favorites

I had a huge month in April, over 25% larger than my last winner, November. My blog has a total of 121,000 page views (since January 1, 2012) and have 178 followers on Twitter. The last probably doesn’t seem terribly impressive, but I literally started with 0 followers. I told no friends or family of my blog, although three or four found me over the months. I had just 7000 pageviews in June 2012, when I created a Twitter account. (First follower: the hyperliteral Paul Bruno, of This Week in Education, who I argue with via twitter but quite enjoy as a writer.)

I have absolutely no idea what this means in relative audience size. What matters to me is that, in a loyal band of regular readers, interspersed between teachers, parents, and Dark Enlightenment folk, I count more than a few policy wonks and reporters—and even a publisher, apparently. I might not have a large crowd following my every tweet, but well over half of my followers do. I started this blog to inform and persuade. So far, so good.

I often check my top posts, reading the growing numbers in awe and wonder, because they, too, confirm that my blogging goals have been and continue to be met. The most popular posts cover pedagogy, policy, some unique data analysis or exposure, and my somewhat scathing opinions about the reform crowd. (I don’t much care for progressives, either, but plenty of people are around to debunk them.)

Since my audience has grown again, I thought I’d remind everyone of my most popular posts, in case someone wanted to check them out. Most of my essays represent at least five or six hours work (I worked on the Philip Dick essay for over a month, the algebra pointlessness one for two weeks), and I think any of the 1000+ view entries are worth a look for a general audience.

Title	Views	Written
Algebra and the Pointlessness of The Whole Damn Thing	4,733	Aug 12
Escaping Poverty	3,664	Nov 12
Teacher Quality Pseudofacts, Part II	3,417	Jan 12
The myth of “they weren’t ever taught….”	2,992	July 12
Homework and grades.	2,576	Feb 12
The Gap in the GRE	2,280	Jan 12
Why Chris Hayes Fails	2,240	June 12
Philip Dick, Preschool and Schrödinger’s Cat	2,102	April 13
The Parental “Diversity” Dilemma	1,907	Nov 2012
An Alternative College Admissions System	1,553	Dec 2012
Why Most of the Low Income “Strivers” are White	1,525	Mar 13
The Dark Enlightenment and Me	1,137	April 13

I left off my “About” page, but both it and “Who am I” right below were nowhere on the horizon last December, so more people are checking out my bio. Neat, if unnerving.

So then we have the 800-900 views, also worth a read for the general audience unless you really have no interest in math pedagogy or curriculum, in which case skip the obvious suspects. But I’m incredibly proud of those curriculum posts; googling modeling linear equations brings up my post in the top two or three as of this writing; likewise a search for binomial multiplication area model brings my post up right near the top.

Title	Views	Written
Who am I?	966	Jan 12
Plague of the Middlebrow Pundits, Revisited: Walter Russell Mead	918	Mar 13
Teaching Polynomials	917	Mar 12
Modeling Linear Equations	907	Jan 12
SAT Prep for the Ultra-Rich, And Everyone Else	871	Aug 12
What causes the achievement gap? The Voldemort View	820	Jan 12
More on Mumford	817	Nov 12
Binomial Multiplication and Factoring Trinomials with The Rectangle	790	Sept 12

And now the less viewed posts that represent my favorites of the rest. I really wish people would read more of these, particularly the Chris Christie post and the Fallacy at the Heart of All Reform. So pick a few to check out. You can also check my year in review for posts I’m fond of.

Policy:

Title	Views	Written
Why Chris Christie picks on teachers	699	Aug 12
Radio silence on Clarence Mumford	660	July 12
Learning Math	605	Aug 12
American Indian Public Charters: What Word Are You Forgetting, People?	602	Apr 13
557
Acquiring Content Knowledge without Hirsch’s Help	555	Jan 13
Jo Boaler’s Railside Study: The Schools, Identified. (Kind of.)	548	Jan 13
Boaler’s Bias (or BS)	521	Oct 12
Picking Your Fights—Or Not	501	Apr 13
Those Who Can, Teach. Those Who Can’t, Wonk.	493	Dec 12
What’s the difference between the SAT and the ACT?	483	June 12
The Fallacy at the Heart of All Reform	454	Sept 12
The difference between tech hiring and teacher hiring	219	June 12

Pedagogy and Curriculum

Probably not too interesting unless you’re a teacher. But I have to say that Modeling Probability is pretty kick ass.

• Algebra Student Distribution–An Example

• Teaching Algebra, or Banging Your Head With a Whiteboard
• The End of Pi
• Mapping Real Life with Coordinate Geometry
• Teaching Congruence, or Are You Happy, Professor Wu?
• Teaching Humanities, Twelfth Night
• Modeling Probability
  
• Geometry: Starting Off

• Modeling Linear Inequalities
  
• Teaching Trig

I realize these probably come off as vanity posts, but for me, they’re a great way to take stock. I have had a genuinely terrific year, between blogging and teaching, and it’s fun to write it all down.

Modeling Exponential Growth/Decay Interspersed with a Reform Rant

Quadratics have become my new nadir, which is cheerier news than it sounds since it means I’ve kicked linear equations into obedient submission. For the first two and a half years of my teaching career, I felt good about quadratics because if nothing else, most kids remembered how to factor, and remembered that factors had something to do with zeros on the graph. Which was a big step up compared to what they retained of linear equations. But then, last year, I cracked linear equations in a big way, which is great except now I just feel bad about quadratics, because as I develop as a teacher I realize the suckers are absurdly complicated and don’t model very easily. The kids learn a lot, but at their level of ability I’d need to do two months to have them internalize quadratics the way most of them internalize linear equations. And I don’t have two months. I just tell myself they still learn a lot. Consequently, I am relieved to see quadratics in the rear view as I move them onto the third of the models that define second year algebra (at least, as I teach it).

Exponential functions are awesome. First, they’re absurdly simple compared to both lines and quadratics. Second, they model actual, honest to god, real life situations. I’m not a big teacher for “Hey, this is something you’ll use again” but automobile depreciation or interest payments are, in fact, something they’ll use again. Third, they provide a memorable and again, useful, reason to review (or learn for the first time) percentage increase and decrease. Finally, they present a situation in which any kid who has even somewhat grasped the course essentials can see hey: Given y, I can’t solve for x. This leads beautifully and meaningfully into logarithms.

So like linear equations, I can kick off the unit with a modeling activity and get the kids moving easily into the math.

I begin with a brief lecture reminding them of the two previous models.

No. Quadratics aren’t repeated multiplication. Exponential functions involve repeated multiplication, as they’ll see in the lesson.

Then I review percentage increase and decrease. I am of two minds about this review. On the plus side, it’s immediately relevant, easy to apply, and gives them a good reason to remember it long term. The downside: the kids never remember what I taught them when they get to the percentage problems. So I explain it up front, knowing that 90% of the kids will forget everything I said just 20 minutes later, when they get to the first percentage exponential increase.

So I explain it, go round the room asking “So, if I want to increase a number by 8%, what do I multiply it by, Jose?” “1 point…..8?” “Watch that leading zero!” “Oh, 1.08.” “Right.” Do that with five or six times, think everyone gets it, and set them to working on models. This is one side of the worksheet, crunched for space so I could “snip” it.

And sure enough, the kids work through the models, making great progress, and stop cold at the third one.

“I can’t do this. How do you increase by a percentage?”

“Excuse me while I beat myself on the head with this whiteboard.”

“What?”

“Nothing. Do you remember me just talking about percentages?”

“Yeah.”

“Do you see it on the board there? All the stuff about turning it from two steps into one step, and why you need to do that?”

“Yeah.”

“DO YOU SEE ANY POSSIBLE CONNECTION BETWEEN THAT CONVERSATION AND THIS PROBLEM?”

“Man, I don’t see why you’re so mean.”

“Read what it says on the board. Right there. In red.”

“Increase x by a%.”

“Yes. Can you read problem 3 and tell me what you think might possibly qualify as x?”

“The population?”

“Yes. And do you see the value that might possibly qualify as a%?”

“Um.” Long pause as the student stares at the problem, and finds the ONLY OTHER VALUE MENTIONED. “Twenty percent?”

“Indeed.”

“Okay.”

I repeat that four or five times to four or five groups and then, miracle of miracles, find a student with a full table of five values for the population problem. There is a god.

“Great.”

“But I don’t know how to find the equation for this one like I did the first two. This one isn’t repeated multiplication. I had to take 20% of 250 and then add it….why are you hitting yourself on the head?”

“We need a function. We need an operation in which we can plug in x—do you have any thoughts on what x might be?”

“How many months?”

“How is it you know that, you smart child, and yet make me go through this torture? Yes. We need an operation that we can plug in the number of months (x) and get the population (y).”

“Right. But this is like three steps.”

“And we need only one.”

“Right.”

“Wouldn’t it be cool if there were a way to increase a number by a given percentage in just one step?”

“How do you do that?”

“LOOK AT THE BOARD!”

“Oh, is that what you were talking about? I was already doing the worksheet.”

And still, the lesson is largely a success. Kids are absolutely freaked out at the cell growth caused just by doubling and yes, I bring up the million dollar mission example, but at the end of the lesson, not as part of it. Most of the kids correctly graph the models, although a few end up with lines that I correct. The flip side of the handout is a blank graph, which they use to take notes on the basic exponential growth model.

Total Amount = Initial Amount * Rate^time

Initial Amount > 0
Rate > 1

One thing I mull over—the book, and the state test, go through the exponential equation (basically, Initial Amount = 1), along with the transformation model (f(x) = a^x-c +- k. I haven’t focused on this in previous classes, because in my experience the kids don’t even get tranformations of lines and quadratics. But I’m going to give it a try on Monday.

Anyway. Day 2 is exponential decay, but I start by going over percentage decrease. I am nothing if not optimistic.

“So if I take away a third of something, how much is left?”

Pause. Pause some more. Pause still more. I grab three whiteboard pens.

“Rhea, decrease these pens by a third.” Rhea obediently takes one pen.

“Class, how much is left after she decreased the pens by 33%, or a third?”

“TWO!!!”

“Two……?” I wait. No. I sigh, and grab three more pens, getting the one back from Rhea as well.

“Paul, take away a third of these six pens.” Paul takes two pens.

“Class, he’s taken away 33% of the pens. How much is left?”

“FOUR!”

“AUUGGGGHHH!”

It all works out. Seriously. By the end of the exercise, most of the class is shouting back the correct answers as I ask “I take away 30%, how much is left?” 35%? 23%?” and the only mistakes they make are place errors—that is, 100-23 does not, in fact, equal 87.

The second day is always better, because it has slowly permeated their skulls that I’m serious about this percentage nonsense, that it has some relationship to the worksheet. So when they ask questions, it’s more of the “could you run this whole percentage decrease by me again? If they take away a third, I have two thirds left? But what’s two thirds as a decimal?” and trust me, this is a big step up for my blood pressure. Well, a step down. And they do the decay modeling and notes with no small degree of interest:

They have the model graph on the back, too, for exponential decay:

Total Amount = Initial Amount * Rate^time

Yes, it’s the same equation, so what’s different?

Initial Amount > 0
0 < Rate < 1

By day’s end, they have registered the import of the realization that Estefania has 95 cents left after ten days, and they’ve figured out that Jose is right, that his car is worth more than Stan’s after five years, which they managed by using an equation they built themselves, by golly, rather than decrease 25,000 by 5% 5 times.

You notice, of course, that I’ve spent most of this post talking about the percentage issue, something the kids learned were first taught back in middle school, than the exponential growth/decay functions, the actual new material. This should not come as a shock to regular readers.

Back in March, there was much fuss about a study revealing that algebra and geometry classes aren’t rigorous enough.

Of course the classes aren’t rigorous enough. They can’t be. I refer you again to the false god of elementary school test scores and the Wise Words of Barbie.

This twitter debate between reformers Mike Petrilli and Rishawn Biddle is typical of reform debates about “rigor”. Petrilli wants end of course exams to stop us teachers from pretending to teach a subject. Biddle wants more of the same, just shout louder and MANDATE instruction, particularly to those disenfranchised black and Hispanic youth who are being let down by lousy teachers with low expectations.

Both of them assume that the problem is ineffective teaching, that all us math teachers could actually teach percentages and fractions to all seventh graders if we were just smarter and better. Or maybe they just think we take the easy way out, that it’d be really really hard to teach the kids properly, and what the hell, we get paid no matter what and behind close doors it’s easier to just go through the motions. Well, sure.

Petrilli’s proposal, end-of-course exams, would trigger a bloodbath. People really don’t seem to understand how I’d be all in favor of that, if the result were a rethinking of expectations. But of course, what would actually happen is that we’d end the end-of-course exams. That’s what always happens whenever a state or district tries to enforce higher standards (cf Oklahoma and now Texas). And of course, that’s what’s going to happen with Common Core standards, assuming that anyone actually takes them seriously after the testing bloodbath this year. But I’d be all for end-of-course testing if reformers would accept responsibility for the 80% decrease in graduation rates among blacks and Hispanics who would never get past algebra I and understand, finally, that they believe in a myth.

But I digress. And I’m still going to like exponential functions, at least until I crack quadratics. Because you know what? The kids do make progress in understanding percentages, and they learn for the first time not only about exponential functions, but about asymptotes, as I explain Zeno’s Paradox. I don’t use Achilles and the tortoise as an example, but instead talk about how I could throw a stapler right at BTS’s head and know that the stapler would never draw blood because it wouldn’t reach his noggin, so I couldn’t get fired. Or that I could walk to the door and never get there. I do get to the door, of course, and alas, the stapler would eventually crack BTS’s skull. But even though we know that this is true, the tools for proving the paradox false, as opposed to demonstrating it, don’t come around until calculus. They get a kick out of that.

If all that’s not fun enough, I see genuine, honest-to-god intellectual curiosity among most students as they realize that they don’t have the tools to isolate x in the equation 8 = 3^x. That for all these years they’ve been getting along fine with addition/subtraction, multiplication/division, nth power/nth root, but none of those will work here. Which sets us up beautifully for both logs and a proper discussion of inverses, leading into inverse functions. Yes, their skills are still basic, but I can see the glimmering of understanding of the underlying concepts. If the damn state tests would just ask questions about those underlying concepts instead of demanding underlying concepts and advanced operations, I might even be able to get the kids to show that understanding.

And in writing up this essay, I am struck by the obvious solution to the percentage problem on day one: I need a worksheet. They fill it out, and not until they are done with that do I give them the worksheets on growth and decay. Naturally, this solution is again a lowering of expectations, a realization that a clear explanation on a blackboard that they can refer to isn’t enough, that I need to give fifteen to seventeen year olds an activity so the information will sink in and they use the method right away without asking me to explain it all again group by group. But to hell with expectations. It will be much better for my bloodpressure.

Modeling Linear Inequalities

I committed to making a big leap forward in inequalities this year. They’ve always been low priority in my curriculum, nothing more than a subset of equations, even though as a programmer, I can come up with fifteen real-life examples of working with them—much more than I can for linear equations. But once I kicked off linear equations with modeling, I could see some obvious introduction points that would help them fall into place. I committed the time to design some new lessons and build some new handouts. And I completely forgot to take pictures of the boardwork, dammit.

Back to tacos and burritos, but this time I took away the “spend all the money” constraint. Stan could either make the purchase or he couldn’t. I gave them some starting combinations to test, and then they came up with their own.

By this time, they are old hands at modeling and easily came up with more true and false pairs. I then graphed all the points on the board, using blue for “True” values, and red for “False”.

“See that space between the TRUE and FALSE values? You can see pretty clearly there’s a line separating them, like this:

“Take a second. What might that line be?” I’m pleased to see several hands shoot up, but I pick on Karl, slinking away from my glance up front. “Karl?”

“I don’t know.”

“What if it wasn’t an inequality?”

“Well, then it’d be 3x + 5y = 60….oh. It’s the same line?”

“Yeah,” chimed in three other students.

“Yes, exactly. We are now working with linear inequalities, not linear equations. Solutions aren’t defined by a single point, but by entire regions. The line is the same in both equations and inequalities, but in an inequality, the line acts as a border between the TRUE and FALSE values. Everything on one side of the line is TRUE, and everything on the other is FALSE.”

From there, I give them one of the new handouts:

I’m really pleased with this one, as a first pass. The students plot the TRUE and FALSE values in different colors. Then they determine which of the linear inequality borders will correctly separate the regions. In one exercise, they become more familiar with solution regions, while also improving their ability to visualize lines from an equation (especially, god help me, positive and negative slopes).

Next, I give them a notes handout I’ve modified over the years.

as well as board notes a few days later:

I made two HUGE changes this year from previous years, changes I suspect most math teachers will recognize.

First, I abandoned “above and below the line” test for the solution region. This seems so obvious to math teachers, but it’s really only meaningful to about half the students. Worse, the test only works with slope intercept form, and has no validity in standard form. The purple math link above tells the student to solve the equation for y—yeah, because solving inequalities is such a breeze, particularly with negatives. It’s a kluge. I didn’t even mention that test.

Instead, I had them use (0,0) for a test value and had them test the value for all forms. MUCH better. No “hey, remember the first way I taught you? Above and below the line? It doesn’t work in standard form, so here’s another way!” Testing values always works.

(Liam in the comments pointed out that the test value can’t be on the border, and I do cover this. I don’t put it in my notes, though. I used to, but the kids would get very confused at the caveat. So now I wait until they are confident of the (0,0) point test and then introduce an example in which the point is on the border equation. I’m discovering that it sometimes takes me a couple years through before I figure out the best way to create fully complete notes.)

Second, I had them write “true” or “false” by the test point, and then shade the same or opposite side of the line. This gave them a visual and kinesthetic step in the process: One, test the inequality with (0,0). Two, write “true” or “false”. Three, shade the correct side of the line.

I’m sure in later years I’ll further hone this, but this was the first time I felt good about my linear inequalities unit. From there, systems of inequalities was an obvious step:

I did a day of modeling systems, but no, I didn’t go onto linear programming.. Instead, I moved onto practicing graphing without the models, and they did great.

I used one of my favorite worksheets as a quiz. The average grade was a B+. Check out this sample—see the “true” markings? By golly, they listened.

I was so emboldened by my success I went onto absolute values, which I usually only cover for tests. They did all right with that, too.

I’m grading the second unit test, covering linear equations, inequalities, and absolute values. So far, it’s looking very good.

Modeling Linear Equations, Part 3

See Part I and Part II.

The success of my linear modeling unit has completely transformed the way I teach algebra.

From Part II, which I wrote at the beginning of the second semester at my last school:

In Modeling Linear Equations, I described the first weeks of my effort to give my Algebra II students a more (lord save me) organic understanding of linear equations. These students have been through algebra I twice (8th and 9th grade), and then I taught them linear equations for the better part of a month last semester. Yet before this month, none of them could quickly generate a table of values for a linear equation in any form (slope intercept, standard form, or a verbal model). They did know how to read a slope from a graph, for the most part, but weren’t able to find an equation from a table. They didn’t understand how a graph of a line was related to a verbal model—what would the slope be, a starting price or a monthly rate? What sort of situations would have a meaningful x-intercept?

This approach was instantly successful, as I relate. Last year, I taught the entire first semester content again in two months before moving on, and still got in about 60% of the Algebra II standards (pretty normal for a low ability class).

So when I began intermediate algebra in the fall, I decided to start right off with modeling. I just toss up some problems on the board–Well, actually, I start with a stick figure cartoon based on this lesson plan:

I put it on the board, and ask a student who did middling poorly on my assessment test, “So, what could Stan buy?”

Shrug. “I don’t know.”

“Oh, come on. You’re telling me you never had $45 bucks and a spending decision? Assume no sales tax.”

Tentatively. “He could just buy 9 burritos?”

“Yes, he could! See? Told you you could do it. How many tacos could he buy?”

“None.”

At this point, another student figures it out, “So if he doesn’t buy any burritos, he could buy, like,…”

“Fifteen tacos. Why is it 15?”

“Because that’s how much you can buy for $45.”

“Anyone have another possibility? You? Guy in grey?”

Long pause, as guy in grey hopes desperately I’ll move on. I wait him out.

“I don’t know.”

“Really? Not at all? Oh, come on. Pretend it’s you. It’s your money. You bought 3 burritos. How many tacos can you get?”

This is the great part, really, because whoever I call on, and it’s always a kid who doesn’t want to be in the room, his brain starts working.

“He has $30 left, right? So he can buy ten tacos.”

“Hey, now, look at that. You did know. How’d you come up with ten?”

“It costs $15 to get three burritos, and he has $30 left.”

So I start a table, with Taco and Burrito headers, entering the first three values.

“And you know it’s $15 because….”

He’s worried it’s a trick question. “…it’s five dollars for each burrito?”

I force a couple other unwilling suckers to give me the last two integer entries

“Yeah. So see how you’re doing this in your head. You are automatically figuring the total cost of the burritos how?”

“Multiplying the burritos by five dollars.”

“And, girl over there, in pink, how do you know how much money to spend on tacos?”

“It’s $3 a taco, and you see how much left you have of the $45.”

“And again with the math in your head. You are multiplying the number of tacos by 3, and the number of burritos by….”

“Five.”

“Right. So we could write it out and have an actual equation.” And so I write out the equation, first with tacos and burritos, and then substituting x and y.

“This equation describes a line. We call it the standard form: Ax + By = C. Standard form is an extremely useful way to describe lines that model purchasing decisions.”

Then I graph the table and by golly, it’s dots in the shape of a line.

“Okay, who remembers anything about lines and slopes? Is this a positive or a negative slope?”

Silence. Of course. Which is better than someone shouting out “Positive!”

“So, guy over there. Yeah, you.”

“I wasn’t paying attention.”

“I know. Now you are. So tell me what happens to tacos when you buy more burritos.”

Silence. I wait it out.

“Um. I can’t buy as many tacos?”

“Nice. So what does that mean about tacos and burritos?”

At this point, I usually get some raised hands. “Blue jersey?”

“If you buy more tacos, you can’t buy as many burritos, either.”

“So as the number of tacos goes up, the number of burritos…”

“Goes down.”

“So. This dotted line is reflecting the fact that as tacos go up, burritos go down. I ask again: is this slope a positive slope or a negative slope?” and now I get a good spattering of “Negative” responses.

From there, I remind them of how to calculate a slope, which is always great because now, instead of it just being the 8 thousandth time they’ve been given the formula, they see that it has direct relevance to a spending decision they make daily. The slope is the reduction in burritos they can buy for every increased taco. I remind them how to find the equation of a slope from both the line and the table itself.

“So I just showed you guys the standard form of a line, but does anyone remember the equation form you learned back in algebra one?”

By now they’re warming up as they realize that they do remember information from algebra one and earlier, information that they thought had no relevance to their lives but, apparently, does. Someone usually comes up with the slope-intercept form. I put y=mx+b on the board and talk the students through identifying the parameters. Then, using the taco-burrito model, we plug in the slope and y-intercept and the kids see that the buying decision, one they are extremely familiar with, can be described in math equations that they now understand.

So then, I put a bunch of situations on the board and set them to work, for the rest of that day and the next.

I’ve now kicked off three intermediate algebra classes cold with this approach, and in every case the kids start modeling the problems with no hesitation.

Remember, all but maybe ten of the students in each class are kids who scored below basic or lower in Algebra I. Many of them have already failed intermediate algebra (aka Algebra II, no trig) once. And in day one, they are modeling linear equations and genuinely getting it. Even the ones who are unhappy (more on that in a minute) are getting it.

So from this point on, when a kid sees something like 5x + 7y = 35, they are thinking “something costs $5, something costs $7, and they have $35 to spend” which helps them make concrete sense of an abstract expression. Or y = 3x-7 means that Joe has seven fewer than 3 times as many graphic novels as Tio does (and, class, who has fewer graphic novels? Yes, Tio. Trust me, it’s much easier to make the smaller value x.)

Here’s an early student sample, from my current class, done just two days in. This is a boy who traditionally struggles with math—and this is homework, which he did on his own—definitely not his usual approach.

Notice that he’s still having trouble figuring out the equation, which is normal. But three of the four tables are correct (he struggles with perimeter, also common), and two of the four graphs are perfect—even though he hasn’t yet figured out how to use the graph to find the equation.

So he’s doing the part he’s learned in class with purpose and accuracy, clearly demonstrating ability to pull out solutions from a word model and then graph them. Time to improve his skills at building equations from graphs and tables.

After two days of this, I break the skills up into parts, reminding the weakest students how to find the slope from a graph, and then mixing and matching equations with models, like this:

So now, I’m emphasizing stuff they’ve learned before, but never been able to integrate because it’s been too abstract. The strongest kids in the class are moving through it all much faster, and are often into linear inequalities after a couple weeks.

Then I bring in one of my favorite handouts, built the first time I did this all a year ago: ModelingDatawithPoints. Back to word models, but instead of the model describing the math, the model gives them two points. Their task is to find the equation from the points. And glory be, the kids get it every time. I’m not sure who’s happier, them or me.

At some point in the first week, I give them a quiz, in which they have to turn two different models into tables, equations, and graphs (one from points), identify an equation from a line, identify an equation from a table, and graph two points to find the equation. The last question is, “How’s it going?”

This has been consistent through three classes (two this semester, one last). Most of the kids like it a lot and specifically tell me they are learning more. The top kids often say it’s very interesting to think of linear equations in this fashion. And about 10-20% of the students this first week are very, very nervous. They want specific methods and explicit instructions.

The day after the quiz, I address these concerns by pointing out that everyone in the room has been given these procedures countless times, and fewer than 30% of them remember how to apply them. The purpose of my method, I tell them, is to give them countless ways of thinking about linear equations, come up with their own preferred methods, and increase their ability to move from one form to another all at once, rather than focusing in on one method and moving to another, and so on. I also point out that almost all the students who said they didn’t like my method did pretty well on the quiz. The weakest kids almost always like the approach, even with initially weak results.

After a week or more of this, I move onto systems. First, solving them graphically—and I use this as a reason to explitly instruct them on sketching lines quickly, using one of three methods:

Then I move on to models, two at a time. Last semester, my kids struggled with this and I didn’t pick up on it until a month later. This last week, I was alert to the problems they were having creating two separate models within a problem, so I spent an extra day focusing on the methods. The kids approved, and I could see a much better understanding. We’ll see how it goes on the test.

Here’s the boardwork for a systems models.

So I start by having them generate solutions to each model and matching them up, as well as finding the equations. Then they graph the equations and see that the intersection, the graphing solution, is identical to the values that match up in the tables.

Which sets the stage for the two algebraic methods: substitution and combination (aka elimination, addition).

Phew.

Last semester, I taught modeling to my math support class, and they really enjoyed it:

Some sample work–the one on the far right is done by a Hispanic sophomore who speaks no English.

Okay, back at 2000 words. Time to wrap it up. I’ll discuss where I’m taking it next in a second post.

Some tidbits: modeling quadratics is tough to do organically, because there are so few real-life models. The velocity problems are helpful, but since they’re the only type they are a bit too canned. I usually use area questions, but they aren’t nearly as realistic. Exponentials, on the other hand, are easy to model with real-life examples. I’m adding in absolute value modeling this semester for the first time, to see how it goes.

Anyway. This works a treat. If I were going to teach algebra I again (nooooooo!) I would start with this, rather than go through integer operations and fractions for the nineteenth time.

Spring 2013: These students aren’t really prepared, either.

I’m teaching Geometry and Algebra II again, so I gave the same assessment and got these results, with the beginning scores from the previous semester:

I’m teaching two algebra II classes, but their numbers were pretty close to identical—one class had the larger range and a lower mode—so I combined them.

The geometry averages are significantly lower than the fall freshmen only class, which isn’t surprising. Kids who move onto geometry from 8th grade algebra are more likely to be stronger math students, although (key plot point) in many schools, the difference between moving on and staying back in algebra come down to behavior, not math ability. At my last school, kids who didn’t score Proficient or Advanced had to take Algebra in 9th grade. I’d have included Basic kids in the “move-on” list as well. But sophomores who not only can’t factor or graph a line, but struggle with simple substition ought not to be in second year algebra. They should repeat algebra I freshman year, go onto geometry, and then take algebra II in junior year—at which point, they’d still be very weak in algebra, of course, but some would have benefited from that second year of first year.

Wait, what was my point? Oh, yeah–this geometry class class is 10-12, so the students took one or more years of high school algebra. Some of them will have just goofed around and flunked algebra despite perfectly adequate to good skills, but a good number will also be genuinely weak at math.

On the other hand, a number of them really enjoyed my first activity: visualizing intersecting planes, graphing 3-D points. I got far more samples from this class. I’ll put those in another post, also the precalc assessment.

I don’t know if my readers (I have an audience! whoo!) understand my intent in publishing these assessment results. In no way am I complaining about my students.

My point in a huge nutshell: how can math teachers be assessed on “value-added” when the testing instrument will not measure what the students needed to learn? Last semester, my students made tremendous gains in first year algebra knowledge. They also learned geometry and second year algebra, but over half my students in both classes will test Below Basic or Far Below Basic–just as they did the year before. My evaluation will faithfully record that my students made no progress—that they tested BB or FBB the year before, and test the same (or worse) now. I will get no credit for the huge gains they made in pre-algebra and algebra competency, because educational policy doesn’t recognize the existence of kids taking second year algebra despite being barely functional in pre-algebra.

The reformers’ response:

1) These kids just had bad teachers who didn’t teach them anything, and in the Brave New World of Reform, these bad teachers won’t be able to ruin students’ lives;

2) These bad teachers just shuffled students who hadn’t learned onto the next class, and in the Brave New World of Reform, kids who can’t do the work won’t pass the class.

My response:

1) Well, truthfully, I think this response is moronic. But more politely, this answer requires willful belief in a delusional myth.

2) Fail 50-60% of kids who are forced to take math classes against their will? Seriously? This answer requires a willful refusal to think things through. Most high schools require a student to take and pass three years of math for graduation. Fail a kid just once, and the margin for error disappears. Fail twice and the kid can’t graduate. And in many states, the sequence must start with algebra—pre-algebra at best. So we are supposed to teach all students, regardless of ability, three years of increasingly abstract math and fail them if they don’t achieve basic proficiency. If, god save us, the country was ever stupid enough to go down this reformer path, the resulting bloodbath would end the policy in a year. We’re not talking the occasional malcontent, but over half of a graduating class in some schools—overwhelmingly, this policy impacts black and Hispanic students. But it’s okay. We’re just doing it for their own good, right? Await the disparate impact lawsuits—or, more likely, federal investigation and oversight.

Reformers faithfully hold out this hope: bad teachers are creating lazy students who could do the work but just don’t want to. Oh, yeah, and if we catch them in elementary school, they’ll be fine in high school.

It is to weep.

Hey, under 1000 words!

A new year begins—midyear

As I mentioned, my school runs a block semester schedule—we cover a year in a semester, four classes each semester. So Monday starts the new year.

I will be teaching Geometry and Algebra II again, although the geometry class will be 10-12 instead of the freshmen of my first semester. One Geometry, two Algebra IIs and, for the first time, pre-calculus.

Note that I am teaching four classes, which means no prep period, and 33% more pay. I am pumped. Okay, a little bit because of the pay, but mostly for two reasons.

Reason 1: Admins don’t give a teacher extra work unless they are happy with the teacher. I have now had two observation writeups so glowing that I keep checking back to see if the name is mine. I know I’m a good teacher. I’m just not used to the principal agreeing with me. More importantly, the administrators seem to like me for the right reasons. Both the principal, who did my observations, and the AVP of the master schedule (the one who reawakened my algebra terrors finally came to see me teach. We had a whole conversation.) both mentioned the quality of my explanations which, many will be surprised to learn, is rarely considered important by either administrators or ed schools. I don’t know why. They also like the fact that I pass most of my students, and keep low ability kids engaged, which is a skill of mine that all previous administrators have used for their own purposes, but never acknowledged as rare or useful. Anyway. I don’t know if they’ll keep me, but it’s very nice, if unusual, to be appreciated.

Reason 2: In my state, a math credential has two levels. One qualifies the teacher to teach through Algebra II, but a third test is required to teach Algebra II/Trig and beyond. I have passed all three tests, passed them first of all the subjects I’m qualified to teach, passed them before I went to ed school by a whole year. And yet, in my first three years of teaching, administrators have on several occasions given advanced math classes to other teachers, teachers who had not yet passed the third test, despite several attempts. Their decisions, done purposefully and in full awareness of the facts, then necessitated a letter home to the students’ parents, per NCLB, that their kids’ teacher was not qualified to teach the class. This is what is known as a penalty. And they took this penalty instead of giving me classes that I was actually qualified to teach. In one case, the administrator didn’t like the other teacher any better than me, had told him to his face he was a “useless brick”, and yet still gave him three Algebra II/Trig classes, giving me Algebra II, taking the penalty, instead of giving him A2 and me A2/Trig. She did not mean this as a personal affront, and I did not take it as such. Such madness as this is pretty normal, and it’s why teachers laugh hysterically when eduformers yammer on about giving principals complete control over hiring and firing. Anyway. As a result of these previous administrator decisions, I have never taught an advanced math class. Not once. Ever. I have no idea how to teach pre-calc. I have no idea how to talk to students who are taking a math class for some other reason than “I need it to graduate”. I have even less idea how to teach an entire class of people who–please, please, PLEASE god—know a positive slope from a negative one. I can’t wait.

I have a friend who is a professor at an elite public university, in a field that requires a lot of math. Back when I was first tutoring and learning math on the job, and got hired to teach a student pre-calc, I asked him “What topics are in pre-calc?” He sniffed, snootily, and said “Precalc isn’t a subject. It’s an administrative category.” I must have learned a lot of math in the intervening years, because I get the joke now.

I’ve got a book, so I’ll figure it out. But if any pre-calc teachers have broad topics to organize around, I’d love to hear about them.

In addition to teaching a full-schedule, no prep period, I start my yearly ACT class on Monday, and in a month I begin my AP US History review classes, two of them. I dropped my English enrichment class, though, so for the first time in seven years, my Saturday mornings are free. I love late winter/spring. But with all this extra money I may just take the summer off for the first time ever.

Algebra 1 Growth in Geometry and Algebra II

Last September, I wrote about my classes and the pre-algebra/Algebra 1 assessment results.

My school covers a year of instruction in a semester, so we just finished the first “year” of courses. I start with new students and four preps on Monday. Last week, I gave them the same assessment to see if they’d improved.

Unfortunately, the hard drive on my school computer got wiped in a re-imaging. This shouldn’t have been a problem, because I shouldn’t have had any data on the hard drive, except I never got put on the network. Happily, I use Dropbox for all my curriculum development, so an entire year’s worth of intellectual property wasn’t obliterated. I only lost the original assessment results, which I had accidentally stored on the school hard drive. I should have entered the scores in the school grading system (with a 0 weight, since they don’t count towards the grade) but only did that for geometry, the only class I can directly compare results with.

My algebra II class, though, was incredibly stable. I only lost three students, one of whom got a perfect score—which the only new addition to the class also got, so balance maintained. The other two students who left got around 10-15 wrong, so were squarely in the average at the time. I feel pretty comfortable that the original scores didn’t change substantially. My geometry class did have some major additions and removals, but since I had their scores I could recalculate.

	Mean	Median	Mode	Range
Original	just above 10	9.5	7	22
Recalculated	just below 10 (9.8)	8	7	22

I didn’t have the Math Support scores, and enough students didn’t take the second test that comparisons would be pointless.

One confession: Two Algebra II students, the weakest two in the class, who did no work, scored 23 and 24 wrong, which was 11 more than the next lowest score. Their scores added an entire point to the average wrong, increased the range by 14 points, and you know, I just said bye and stopped them from distorting the results the other 32 kids. (I don’t remember exactly, but the original A2 tests had five or six 20+ wrong scores.)

So here’s the original September graph and the new graph of January:

The geometry class was bimodal: 0 and 10. Excel refused to acknowledge this and I wasn’t sure how to force it. The 10s, as a group, were pretty consistent—only one of them improved by more than a point. The perfect scores ranged from 8 wrong to 2 wrong on the first test.

In short, they learned a lot of first year algebra, and that’s because I spent quite a bit of time teaching them first year algebra. In Algebra II, I did it with data modeling, which was a much more sophisticated approach than what they’d had before, but it was still first year algebra. In geometry, I minimize certain standards (proofs, circles, solid shapes) in favor of applied geometry problems with lots of algebra.

And for all that improvement, a still distressing number of students answered x² + 12 when asked what the product of (x+3) and (x+4) was, including two students who got an A in the class. I beat this into their heads, and STILL some of them forget that.

Some folks are going to draw exactly the wrong impression. “See?” these misguided souls will say, nodding wisely. “Our kids just aren’t being taught properly in early grades. Better standards, better teachers, this problems’s fixed! Until then, this poor teacher has to make up the slack.” In short, these poor fools still believe in the myth that they’ve never been taught.

When in fact, they were taught. Including by me—and I don’t mean the “hey, by the way, don’t forget the middle term in binomial multiplication”, but “you are clubbing orphan seals and making baby Jesus cry when you forget the middle term” while banging myself on the head with a whiteboard. And some of them just forgot anyway.

I don’t know how my kids will do on their state tests, but it’s safe to say that the geometry and second year algebra I exposed them to was considerably less than it would have been had their assessment scores at the beginning of class been the ones they got at the end of class. And because no one wants to acknowledge the huge deficit half or more of each class has in advanced high school math, high schools won’t be able to teach the kids the skills they need in the classes they need—namely, prealgebra for a year, “first year” algebra for two years, and then maybe some geometry and second year algebra. If they do okay on the earlier stuff.

Instead, high schools are forced to pretend that transcripts reflect reality, that all kids in geometry classes are capable of passing a pre-algebra test, much less an algebra one test. Meanwhile, reformers won’t know that I improved my kids’ basic algebra skills whilst still teaching them a lot of geometry/algebra II, because the tests they’ll insist on judging me with will assume a) that the kids had that earlier material mastered or b) that I could just catch them up quickly because after all, the only problem was the kids’ earlier teachers had never taught them.

Modeling Probability

This is a lecture class, but I put all the instructions in the handout as well, mostly so I can remember the outlines of the activity.

In the lecture, I explain that video games run on probabilities, that balancing probabilities is an essential element for strategy video games. Any games that give the user a running series of choices has to balance outcomes and create choices with tradeoffs. Otherwise, the user learns that the “good” choices are and the game gets boring. (I have no idea if this is true, but it sounds reasonable.)

Obviously, if the students were really at a software company, these scenarios would be automated, but hey, it’s a math class.

Materials: Scintilla Handout (reproduced here), and 9 Oracle advice cards (3 gryphons, 3 dragons, 3 excaliburs)

Here’s the main handout.

Crossing the Scintilla

You’re an intern at a major software development company! You’ve been assigned to work with the team developing Sorcery & Shadows, a new game scheduled for fall release.

S&S is “retro”, harking back to the 70s and 80s fantasy games—less violence, more strategy. At various points, the player’s choice of avatar must consult the three Oracles for permission and guidance. The Oracles determine the character’s actions, but the response varies based on the avatar chosen:

The software team is working on the following scenario:

The player (Vlad, Dulcinea, or Chaos) is trying to cross the Scintilla River. The Oracles must be consulted. The Oracles will each advise one of the following options:

1. The gryphon, which swims across.

2. The dragon, which flies across.
3. Excalibur, the sword, which magically transports the carrier.

Gryphon

Dragon                                         Excalibur

The player will follow the Oracles’ advice if the right number of them agree. The Oracles are pleased when the player follows their advice and give the player five silver coins.

If the player can’t follow the Oracles’ advice, then the player must pay 1 silver coin to cross on the ferry.

The software team has already decided that the Oracles’ responses are randomly generated, and they need to determine the probability that each character gets across the river. They’ve asked you to work this out.

Before you start, have a brief group discussion and make your predictions.

Is it equally probable that Vlad, Dulcinea, and Chao will be able to follow the Oracles’ advice?
1. If not, who do you think is the most likely to be able to follow the advice?
2. Least likely?

I wander around to listen in on the predictions. The two most common predictions are Vlad and Dulcinea, which is interesting.

I always do two trials as a class before I set them on their own, stressing that the actual advice—Gryphon, Dragon, Excalibur—is immaterial. They are tracking which character is able to accept the Oracle’s advice. Back to the handout—this is on the flip side.

Experimental Probability
Experimental Probability—Performing an event repeatedly and measuring the results as a ratio of a particular occurrence to the total number of trials: .

Once a pattern has been established, we can rely on this data as an empirical probability.

You are going to perform 30 simulated trials of the event “Asking the Oracles for Advice”.

Each group has three sets of three cards: a gryphon, a dragon, and Excalibur. Three group members will “play Oracle”, by randomly and simultaneously throwing down one of the three cards. (IT MUST BE RANDOM!). The other group member tracks the outcome of each trial, using the table below.

When you have completed 30 trials, compute the experimental probability of each character’s likelihood of following the Oracles’ advice. Remember to keep careful track of how many trials you run, as that’s going to be your denominator.

Send someone from your group up to the front whiteboard and report your results. Report the totals, not the experimental probability percentages. We’re going to calculate the results for the class.

The kids love the trials. They are religiously random, and really get a kick as they see a clear pattern emerge in the results.

After all the results are on the board, I tote them up and calculate the experimental probability for the class.

Then I transition to theoretical probability and discuss the difference between experimental probability—what actually happens in a series of trials—and theoretical, what is expected to happen. I ask for examples of trials that would have no theoretical probability: medical trials, new treatments, new procedures. I point out that researchers run thousands of trials because they want to have a reliable experimental model in order to begin to build theoretical probability.

In other cases—coin tosses, lottery tickets, and asking the Oracles’ advice—the theoretical probability is easily modeled. And that’s what we do next. Back to the handout, page 3.

Who Crosses the River?–Theoretical Probability

In Part B, you ran trials and calculated the experimental probability of each character’s being able to follow the Oracles’ advice. Now you’re going to determine the theoretical probability for each character and compare them.

Theoretical probability can be calculated when the number of possible outcomes is fixed.

In this case, we can define the following:

Sample Space:
The set of all possible outcomes for a trial.
Event:
Particular outcome(s) of the sample space. An event may contain other events, each with its own sample space.
Target event:
The desired outcome to be tested.

For example, the sample space for one instance of “asking for an Oracle’s advice” is: gryphon, Excalibur, dragon.

But in this case, we are asking for all the Oracles’ advice. So how do we find all the combinations possible from three Oracle requests?

There are three useful tools to help you model theoretical probabilities for a multiple-event scenario.

• A tree diagram can help you determine all the possible outcomes for a “compound” (multiple event) outcome, particularly complex events such as this one. Trees model a new “branch” for each component of an event.

• An area model is simple and easy, but limited to two or three events.
• A counting diagram is useful for ordering and calculating the number of outcomes.

See the flip side of this page for detailed descriptions of area models and probability trees.

Probability Tree

Working with your team, create a probability tree for all possible combinations of Oracular responses.

Using different colored pens for each character, trace the different outcomes: all three match (Vlad), two match (Dulcinea), no matches (Chaos).

Count up the possible crossings for each character, and the total possible outcomes.

How many different outcomes are possible? __________________

How many outcomes allow Vlad to cross? ________

Dulcinea? _______________

Chaos? _______________

Compare these numbers to your own experimental results, as well as the class totals. How do they compare?

I use a CPM handout (page 14), which isn’t all that great but gives visual examples of both models on one page. All I really want is the visual, which I haven’t gotten around to building for myself yet.

Building the probability tree diagram for “asking the Oracles” is a great activity that really brings home the difference between theoretical and experimental probability. The kids can see why Dulcinea gets the necessary agreement more often, and why Chaos wins more frequently than Vlad.

In earlier years I would have just had the students create the probability tree on paper, in their groups. But earlier in the school year, I came up with a fun way to work on bigger, multi-step problems. I have a lot of whiteboards. So in their groups, the kids go to a whiteboard section, and start working on the assigned problem(s). They have more room, I can see the work and be sure everyone’s got the correct answers. It shoves the math right under the nose of the weaker students, who might otherwise (ahem) sit quietly hoping I don’t notice they aren’t working or paying attention. And it mixes things up, which is always useful. I used this about three times during the year (our semester is a year); here’s a picture of them working on parabolas:

(Note: smudged the faces, took out the color, and asked the kids permission to use the photo.)

As I was driving to school the morning of this class, I suddenly realized that whiteboard work would be perfect for the probability trees. If I had them do it on paper, at least half of the kids would be looking on as someone else did all the work anyway, so I might as well be sure they were actually looking on, instead of tuning out. I usually use this method as review rather than for new concepts, but in this case it was the right call. Each of the groups were all involved in their own tree, and none of them were simply copying some other group’s work—some surreptitious checks to see if they were on the right track, sure. But that’s a bonus.

There were several gorgeous versions done in red, blue, green, and purple that I forgot to photograph before I erased them, but this one is nicely functional. Except the misspelling.

So we take a few minutes to compare the theoretical outcomes with the experimental, and since we have close to 200 trials (8 groups, 25 trials on average), they match up beautifully.

I tell them that science and research engage in experimental probability, but in math, we focus entirely on the theoretical by modeling the possible outcomes. I always start with the most flexible and visual, but the least useful, model. I then outline the other two models I want them to use, both of which I think have limitations, but are much more helpful: the area model and the “counting diagram”.

Counting Diagram

I always snicker at a formal name for a very simple concept. But it’s extremely useful as an organizer.

One blank line for every event. So asking the Oracles, there are three events.

______      ______      _______

How many outcomes are possible for each event? The events are independent of each other.

___3__        __3___        ___3___ = 27

Multiply across. That’s the total number of outcomes that can result in asking the Oracles.

Now, model each character. In these cases, the number of desired outcomes for subsequent events is conditional. (I point out that the Oracles’ response is still random and independent, but the desired response is conditional, and that’s what we’re counting).

Vlad is the easiest. The first event can be any of the three responses. But the second and third events must match the first, so there’s only one acceptable outcome for each.

___3__       __1___       ___1___ = 3

Dulcinea is more complicated, but the trees are very helpful in getting the kids to see that the first two events can have any outcome. The third event must match one of those first two.

___3__       __3___       ___2___ = 18

Finally, Chaos. Why is Chaos twice as likely as Vlad to get the correct Oracle response? The counting diagram helps students see that as each event occurs, he loses the possibility of that outcome—and yet, this gives him more outcomes than Vlad.

___3__        __2___         ___1___ = 6

The diagram can also calculate probabilities of multiple events, but it’s primarily useful for counting.

Right around now, I bring up lottery tickets, and we go through the hugeness of the numbers in a diagram. And here, I mention a key difference between theoretical and experimental probability, aka Why All Math Teachers Tell You Not To Buy Lottery Tickets.

Theoretical probability says it’s utterly pointless to buy lottery tickets. But every time the lottery runs, someone achieves the functional equivalent of getting struck by lightening while finding a four-leaf clover while getting abducted by aliens. Someone wins. Reality occurs. Gambling exists because of experimental probability. So my students won’t get the Lottery Tickets lecture from me. Go ahead and buy. Cross your fingers when your plane takes off, even though the car ride to the airport was the riskier trip. But if you blow your entire salary on poker, find a 12-step program.

Area model

My students are already

For this, I have a handout, because the area model makes the two most important probability operations beautifully clear: when to multiply probabilities, and when to add them.

Limitations—only two sample spaces, alas. You couldn’t model “asking the Oracles” with the rectangle.

So there’s the three basic models that we use for the rest of the unit. I often return to the “intern at the software shop” scenario, which gives me endless possibilities. Here’s a couple more.

This one is usually homework for the first day—what is the probability of getting various payouts? I don’t mean expected value—we go through that the next day, with the original Scintilla scenario and this one.

And here’s one that goes nicely with an area model, which can really help students visualize conditional probability.

Thanks to binomial expansion, probability and elementary combinatorics are sandwiched into second year algebra and it’s hard to go into the subject in depth. AP Stats is pretty joyless. Example 99,521,325 on the list of Why We Need to Offer a Broader Range of Math Classes.

Teaching Congruence, or Are You Happy, Professor Wu?

I first ran into the writings of Professor Hung-Hsi Wu in ed school, and never forgot him. How often do you see a math professor hyperventilating about elementary teachers as the abused children of math education:

For elementary teachers, there is at present a feeling that they have been so damaged by their K–12 experience…that we owe it to them to treat them with kid gloves…. Those that I have encountered are generally eager to learn and are willing to work hard. The kid-glove treatment would seem to be hardly necessary. …There is another school of thought arguing that for elementary teachers, one should teach them not only the mathematics of their classrooms, but at the same time also how children think about the mathematics. Again, I can only speak from my own experience. The teachers I observed usually had so much difficulty just coming to terms with the mathematics itself that any additional burden about children’s thinking would have crushed them.

More:

There are ample reasons to believe that at present most teachers are operating at the outer edge of their mathematical knowledge. Now when one finds oneself in that situation, one is prone to being tense and inflexible, and is consequently not likely to create a friendly atmosphere for learning. There should be a study to look into how much of the so-called math phobia in this country can be traced to this fact (especially in elementary schools). The other simple reason is that no matter how elementary the topic, some students would bring up deep or at least non-elementary related questions.2 If the teacher fails to answer such questions too often, the students’ confidence in the teacher is eroded and, again, a non-productive learning atmosphere would result.

Abuse victims grow up to be abusers, you know?

He trains elementary school teachers in these special recovery workshops for abuse victims:

The main difficulty with the Geometry Institute, and the relative lack of success thereof, was the teachers’ unfamiliarity with anything geometric. With but mild exaggeration, some teachers literally trembled at the sight of ruler and compass or when they were handed a geometric solid. As mentioned in the preceding paragraph, we were prepared for teachers’ being ill-at-ease with geometric reasoning and lack of geometric intuition, but not for the degree to which both were true. School education in geometry is in deep trouble.

I know many who read Professor Wu and take his descriptions at face value, using him as evidence of teacher stupidity. Anyone who believes a math professor—a mathematician, that is—can be objective about elementary school teachers’ math knowledge has never met any mathematicians. Read the above link in which he describes the bare minimum of what fifth grade teachers need to know in order to prepare their students, and ask yourself if elementary school math teachers have ever known that much. How on earth did we all get to the moon and create the internet?

Still, his fulminations about geometry caught my attention over the summer, when I read this:

As to the subject of school geometry, the problem is that if universities do not teach it, or do not teach it well, then the only exposure to school geometry that geometry teachers ever have will be their own high school experience in geometry. The latter of course has been scandalously unsatisfactory for a long time, to the point where many school geometry courses cease to prove any theorems.

Well, hang on. As I’ve mentioned before, I don’t teach proofs and rarely prove theorems. I also typically dump transformations, most of construction, and solids. But I’m teaching non-honors geometry in a Title I school, where geometry is but a brief respite before the kids are dumped back into Algebra Hell. As a tutor, I’m very familiar with geometry as it is taught in the high-performance schools in the area, which are some of the highest performing schools in the country, and they are getting proofs a-plenty. Moreover, geometry hasn’t been “scandalously unsatisfactory” since we began forcing everyone into college prep math (an absurd notion in and of itself). What’s scandalously unsatisfactory is the idiocy of trying to teach proofs to low ability kids. But I digress. The point is, I dispute his notion that geometry is taught badly at all schools. A highly modified version of geometry is taught at some schools, for a very good reason, and given the kids involved there’s little evidence that it’s doing any damage, and regular “old school” geometry is routinely being taught to the top students.

But I was curious, nonetheless, as to what dire notions Prof Wu had about geometry as it is taught in schools, and so I googled. Teaching Geometry According to the Common Core Standards is the first article I read, but this interview with Rick Hess spells out the key point with the fewest words (would that I valued the same behavior in myself):

As another example, when state standards ask that the concept of congruence be taught in middle school, they do not realize that what students will end up getting is that congruence means same size and same shape. As a mathematical definition, the latter is completely unacceptable.

I sat up straight at this. Remember, I’m not a mathematician (which of course means that Prof Wu wouldn’t let me near fifth graders, much less high schoolers), and had never given much thought to congruence and similarity until I began teaching public school, as opposed to reviewing similarity for college admissions tests (congruence is not a tested subject). At that point, however, the book definitions seemed a tad circular. Check out the CPM section on similarity. It uses dilations, which (as you will see) is the right start, but at no point does the text explain the link between similarity and dilation. It’s all “cut out the figures, talk with your group, what’s the same, what’s different” crap. And when I taught geometry two years later using Holt, the official definition of congruence, straight from the book, was “Two polygons are congruent if and only if their corresponding sides and angles are congruent”. Really? Polygons are congruent if they are congruent?

I actually apologized to my class last year when we got to that point, because I hadn’t noticed this bizarre definition until the day before I taught congruence. I said yeah, a tad circular, and I hereby promise to investigate but for now, let’s go with it. (Sorry, Professor Wu, but most of them are never going to use congruence again.)

So here’s Wu saying that yes, this is a problem? How does he want it to be taught?

Holy Crap. Rigid motions? Isometries? (not isometrics. I make that mistake all the time) The sections I ignore? They have a purpose? I should have majored in math. No, not really.

This was a huge revelation, and incredibly easy to put into action. Most students got two “transparency triangles” and a white board. Some students used the graph paper with the transparency triangles. Three students (strong students) used the white boards with different colored pens and no manipulative. I wanted to see which methods worked best and if any problems came up with a particular method.

Day 1: Introduce translations and reflections, moving to increasingly complex reflection equations. Emphasize that reflections occur over a line; evaluate the change in coordinate points with pre-image and image, and then start calculating new coordinate values without using the manipulatives first.

Day 2: Rotations (by far the most difficult, in my mind). We focused on rotations of 90 degrees, and on reviewing the definition of perpendicular slopes, since that’s how the students found the new point—find the slope from the point of rotation (usually the origin) to the vertex to be rotated, convert to the perpendicular slope. Stressed that rotations were around a point, in contrast to reflections.

I foolishly didn’t take pictures in class that day—or if I did, I can’t find them. Here’s roughly what it would have looked like for a student using graph paper and the manipulatives, except they used colored pencils for the different slope connections. This is an example rotating a triangle 90 degrees clockwise.

First step (click on image) was to identify the slope from the point of rotation to each vertex. Then they identified the perpendicular slope for each of the rotation points—reinforcing perpendicular slope relationships being a big ol’ secondary point of the lesson—and sketched that line in as well. The students used different colors for each vertex, so they could easily see the before and after for each point, and recognize the 90 degree nature of the turn.

Then, with the points sketched, they did the actual rotation. Put the triangle on the original point, hold the manipulative at the point of rotation, and turn. Voila.

And then put the first manipulative in the original position to see what the rotation before and after looks like.

I’ve always had a difficult time teaching rotations, but the manipulative really helped.

I end the day pointing out that transformations preserve both degree and distance. They can see this because they are using identical manipulatives, but I have them calculate some side lengths and slopes to confirm.

Day 3: Congruence
And now, congruence. Instead of a circular definition, I have a clean syllogism:

If Polygon A is congruent to Polygon B, then A can be mapped onto B using a series of transformations. If the figures can be mapped into the same space, then their corresponding angles and sides are congruent, because the mapping preserves degree and distance. Therefore, congruent polygons’ corresponding angles and sides are also congruent.

From there, I go onto congruence shortcuts and proofs, blah blah blah. But it started much more cleanly. I taught transformations, reviewed perpendicular lines and other coordinate geometry formulas, and linked it all to congruence in a meaningful way.

A few weeks later, it was onto similar polygons. Again, instead of just saying “Similar polygons have congruent angles and proportional sides”, I can link it to dilation.

Day One: Review of Proportionality, then onto dilation

The kids did straight dilations as well as transformations and dilations in combination. I started with straight dilations, because I wanted the students to confirm the elements of similarity. The kids generally remember that parallel lines have the same slope, but I thought it would also be useful to see the transversal relationships with the parallel lines. We could prove, algebraically, that the lines of the dilated triangle were parallel to the original, and we could then extend those lines to prove that the corresponding angles on each triangle were congruent. Here’s an example (again, one I just sketched up) that shows how the kids determined the angles were congruent.

The kids colored the corresponding angles—there are three in each case (one of the green ones in my image is an error, you can see I xed it out, just too much hassle to draw again).

So again, the point was to algebraically and visually confirm the parallel relationship, and then follow the dual sets of parallel lines and transversals to confirm that the angles are congruent.

I had them do a combination transformation/dilation, confirming that order didn’t matter, and identifying which of the isometries had the parallel relationship.

Day 2: Review of Dilation, then onto Similarity.

Done.

Linking isometries to congruence and similarity was so much better, and whenever I tell math teachers about it they go oooh, ahhh and think about trying it themselves. And yet, I can’t point to why it’s so obviously superior. I can’t swear that my students learned congruence or similarity more thoroughly—in fact, I think they learned it as well but not any better than my students last year.

But it’s just more….cohesive, maybe? Not only am I finally linking in rigid transformations, which I never gave more than a quick review at test time (two of the three are intuitive, rotations are tough), to the rest of geometry, and creating an organic reason to review the relevance of perpendicular and parallel lines/transversals, but I am also linking both of these concepts to congruence and similarity, rather than just giving that annoying circular definition. While congruence doesn’t have much relevance past geometry, similarity runs through the next three years of math in a big way. So anything that makes the introduction more meaningful is probably a good thing. Moreover, transformations are easily grasped by even weak students, and their interest kept them going through the review of perpendicular and parallel lines.

None of this required complicated worksheets. I taught congruence using notes and Kuta worksheets; for similarity we used the book (Holt). I taught the transformations with boardwork–I really, really could have used a document camera, but that just came a couple weeks ago. Still, the kids got it well.

The really important thing, though, is that I have to feel mildly guilty about mocking Professor Wu. Next time I see him at Math Survivors Anonymous, I’ll grovel.

Kicking Off Triangles: What Method is This?

How do math teachers kick off a new unit? When possible, I like to do something with manipulatives, or some sort of activity that introduces interesting questions. Last year, I came up with a triangle activity that I’d originally conceived of for congruent triangles, and then realized it wouldn’t work. But something at the heart of the idea struck me as fascinating, and the single day activity was extremely successful. This year, I used it to kick off triangles.

As you read this (if you read this) ask yourself: is this a constructivist lesson, in which kids discover their own meaning, and the teacher is the “guide on the side”? Or an instructivist lesson, with the teacher as “sage on stage”, telling the students the facts?

I think describing key aspects of the lesson through my interactions with the class will help clarify the lesson. Or not. Maybe it’s just a goofy delusion.

Prep: I have several questions written on the white board:

1. What constraints exist in triangle construction?

2. Can a triangle be made out of any three lengths?
3. How many triangles can be made out of any specific three lengths?
4. How can we classify (group) triangles?
5. How many degrees in a triangle?

I also have some vocab words written out: constraints, properties, and room for more. I created seven of the bags, the Triangle Activity sheet, the Classifications graphic organizer. Planning this took a while until I realized I could leave the logic lesson (see below) for homework; then it all fell into place.

So first, I hand out the Triangle Activity Sheet and a bag for each group of kids (my kids sit in groups of 4, roughly arranged by ability, with strongest kids in the back):

So today we’re kicking off our triangle unit. I could lecture and give you an introduction, but I thought it might be fun to give you some specific memories about triangles to introduce the shape and help you understand that it’s a little more than ‘just take any three sides and put them together.’ So start off by emptying out the bag of everything EXCEPT the three identical paper triangles, okay?”

The kids obediently dump out the contents. They see paper fasteners (you know, those little gold things?), and strips of paper in various lengths and colors. The strips all have holes on each end, put there by a paper punch.

“Okay, first, I want you to know that this activity is going to create some really cheesy looking triangles, but it’s precisely because they are so cheesy that I like this exercise, because it proves an important point about triangles. So no giggling or mocking my strips of paper, okay? I spent hours making them.”

Naturally, a wiseass in the back of the room says “Oh, man, these are so flimsy and cheap! Who made this crap?” all to get the Killer Stare from his teacher.

“But before you start making triangles, I want you to NOT make triangles. Take a second to read the instructions in Part I.”

(Brief pause. But I do not make the rookie’s mistake, dear reader, of assuming that the kids are all obediently RTFM just because I provide them time and orders to do so.)

“Okay. Eddie, what do the instructions say?”

“Huh?”

“Alice?

“Um, what part again?”

“Craig, wanna go for a trifecta?”

“You just said it. We have to find strip combinations that DON’T make triangles. But we can’t bend them.” (Craig, btw, is the wiseass. He has redeeming qualities.)

“Exactly. See the holes on the end?” Wait for yeses, don’t get enough. “SEE THE HOLES ON THE END?”

“YES!”

Line those up. Some of the combinations will not extend sufficiently. Remember that you can change the angle measures, and try different lengths. Now. Turn over the instructions.” I listen and look for the half sheets to turn over. “Does everyone see the smaller of the two tables? When you find a combination of strips that can NOT make a triangle, you put the three lengths in this table. Got it? No, that wasn’t loud enough. One more time: You will put the strip lengths that do NOT form a triangle in this little table. All of you should keep your notes updated, but I’m going to collect one handout from each group. Ready? Louder?” YES! “Okay, go.”

I wander around the room, periodically reminding students that they can alter the angle measure, and within 5 minutes all student groups, even the weaker ones, have at least three combinations. I call on each group to provide an example, and we soon have 8 or 9 combinations on the board.

“Excellent! Now, I’m going to say a combination of strips that’s not here on the board, and I want to close your eyes and visualize those strips. Don’t say anything, just visualize. Close your eyes!”

“Okay, visualize the strips 2, 3, and 9. Don’t answer this question, just think: can you make a triangle with strips of 2 inches, 3 inches, and 9 inches? Think for a minute, don’t say anything. Okay, now: Put your thumb UP if you think you can, thumb DOWN if you think you can’t.” The bulk of the students have their thumbs down (this is a very strong class), and even many of the struggling students are visualizing correctly and have thumbs down. A few kids, no thumbs.

“Ike, no thumb? Do you think you can make a triangle?” (Ike is in the front of the room and didn’t think to look around and get a thumbcount. He is chagrined.)

“Um, sure.” I hold up my hand to forestall the kids who want to correct him.

“Try it.” Ike tries to arrange the strips.

“No. You can’t.”

I add 2, 3, 9 on to the board. “Okay, Ike, how about 2, 3, and 3?”

“Yeah,” Ike says instantly.

“Yes, you can. That time you could see it, right?”

“Yeah. Two strips have to be longer than the other one or you can’t make a triangle. “

“Very good! Anyone else see that?” A bunch of hands go up. “We’re going to formally learn this later, but Ike has articulated half of what’s called the Triangle Inequality Theorem. No need to write it down right now.” (I’m writing it on the board.) “In a triangle, the sum of two sides MUST be greater than the third side. There’s more to the theorem than that, but we’ll stop on this point, because I want you all to consider something. Before right now, how many of you thought you could just take any three lengths and make a triangle?” Most of the hands in the room go up, and most importantly, I see the top kids going “Hmm.” They realize that yes, indeed, they had thought that, which meant that this little exercise with no clear agenda had, in fact, taught them something. “Here’s what’s even stranger: before today, if I had asked you to make a triangle with a toothpick, a pencil, and a pogo stick, what would you have told me?”

“Wow,” said Mary. “You can’t.”

“You can’t. And you all would have instantly realized that, had you been given specifics, right?” Many nods, I own the class right now. “So this brings up something important about triangles: they have constraints. They have constraints that you have never considered, and yet if you had considered a triangle with specific questions, you would have instantly realized some of the constraints. Alan?”

“What’s a constraint?” I’m walking over to the vocabulary section while several students explain that it’s a restriction, or “something you can’t do”, and I write their definition on the board.

“So now, we’re going to build triangles. See Part two of the handout? Create triangles that meet the criteria. There are more than one combination of lengths for most, but not all, of the triangles listed. I recommend getting them all identified and lined up BEFORE you use the paper fasteners. And remember, if a hole breaks I’ve got more of the support doohickeys.”

So off they went, creating triangles with enthusiasm and precision. After 15 minutes, they all had completed tables that looked something like this:

Then I passed out this simple graphic organizer. Amazingly, many triangle classification graphic organizers (yes, there are a variety out there on the internet) miss the opportunity to visually emphasize the simplicity of triangle classification. For example, this one completely mucks it up, and this one buries the lede. This one is perfect, but it gives out all the information up front, when I wanted the kids to pull it together in a class discussion. I had already created mine which looks like a simpler version of the last, but really, how many ways are there to show it visually?

Now, some of you are saying, “WTF???? You need a GRAPHIC ORGANIZER to tell kids how triangles are classified?” Well, yes. I’m not a big fan of taking “math notes” for all but the very top kids. If a kid wants to take notes, fine. But
I hand out graphic organizers for most of the information I want kids to keep, so I can tell them what handouts to review for the test. If I want them to take down a page of notes (for example, coordinate geometry), I tell them specifically to copy down my board notes as is—and then, a few weeks later, I ask about those notes and check to see if kids have them (compliance rate this year close to 90%).

It takes about 8 minutes to go through the organizer; I look sternly at the top kids and stop them from blurting out the top-level categorizations, giving the mid-levels time to think and suggest. Then we complete the organizer working together; I tell the kids to put an example in each.

I hand out a triangle classification worksheet from KUTA (I used pages 2 and 3), and the kids work through it busily. The top kids rip through it the quickest; it’s not that hard. But this section here gave them opportunity to think:

Two of the triangle classifications aren’t possible, so when the top kids finished this, I made them write out why two classifications weren’t possible. This gave the less adept kids time to finish.

“So Jasmine, did you find a triangle type that wasn’t possible?” Jasmine stays silent, desperately hoping that someone else will shout out the answer, but I have poleaxed the usual suspects with a stare, and wait her out.

“Right….obtuse?”

“Very good! Why isn’t that possible?” Another long wait—an eternity to Jasmine, no doubt.

“They were in different circles?”

“And…“

“You..said that a triangle could be in only one circle?”

“I did! Clark, can you add to that?”

“A triangle can only be in one angle group?”

“Exactly. A triangle can only be classified by one angle type. Actually, if you put some information together, you may be able to figure out why a triangle can’t have both a right angle and an obtuse angle, but don’t worry about that now—we’ll work on that more tomorrow. Ellie, how about the other impossible triangle?”

“Right equilateral?”

“Good! Now that one is a bit tougher. Why isn’t it possible? Kevin?”

“Well, wouldn’t all the angles have to be equal, if all the sides are? And if they were all equal, the total degrees would be 270.”

“Very nice! We haven’t established all of the facts you used, but your reasoning is good. There’s another way, too. Maya?”

“The hypotenuse of a right triangle is the longest side, right?”

“Nice. Candy, can a triangle with three equal sides have a longest side?”

“No.”

“Does a right triangle have a longest side, class?” “YES!” “So can a right triangle have three EQUAL sides?” “NO!”

(okay, I know some of the weaker students don’t quite grok this, but I’m aiming for the top students now, so I’ll pick up the pieces later.)

“Okay. Very good. The big idea, again: every triangle fits into an angle classification and a side classification. All of them. Turn over your handout.”

“This is your homework. I’m going to go through it during our advisory, so don’t sweat it now. But this organizer reveals something critical about triangle classifications that I want you to think about tonight. Also, more logic. Don’t groan!”

[Groan. They do not like logic.]

“Okay, now put the graphic organizer away. Did everyone make a parallelogram? Everyone take the parallelogram they built, and any one triangle. Apart from the number of sides, what differences do you notice?”

It takes the kids a while to figure it out, even as the parallelograms collapse the minute they pick them up. I let them mull it for a while, and they do come up with some creative offerings. I hint at it by convincing them to move them around.

But I finally tell them to hold up each figure.

“Hey! Look at that!”

“That’s not a real difference! They’re just made differently!”

“Made differently? You made them! What did you do differently?”

“Nothing!”

“Okay! They were both made of index card stock and paper fasteners. So what’s keeping the triangle up?”

Silence.

“So take a look at the board, question 3. How many different triangles can be made from three specific lengths?”

Silence.

“Ooookay, how many different parallelograms can be made from two pairs of strips?”

“Lots,” says Angie promptly.

“Show me.” Angie takes her parallelogram and moves it from a rectangle to a steeply slanted parallelogram.

“Perfect! Now, which has the biggest area?” Silence. “Angie, stand up and show everyone your parallelogram in the super slanted position. Okay, now push it up a little bit. Class, what just happened to the area of the parallelogram?”

“Bigger,” says Ron.

“Angie, push it up more.”

“Oh, I get it,” Karinna says. “It always gets bigger if you make the….um. I don’t know how to explain it.”

“Think in terms of height. Angie, put it back to super slant.”

“Oh!” Ike says, “If it’s taller, it has a bigger area!”

“And where does that end? Angie, keep moving it..moving it….Craig?”

“When it’s a rectangle. Because after that, it’s going to start going down again.”

“Nice. So you see, guys, a parallelogram has an infinite number of areas, although the largest area is going to be when the sides are perpendicular to each other. No need to write that down; I am just making a point really about triangles. Back to our question. How many different triangles can be made from three specific lengths?”

Pause, then “Just one?” asks Ron.

“Everyone made a 5,5,9 triangle, right? Hold it up. Look around, everyone. See any variety? Or do they all look the same? “

“But why?” asks Effie.

“It all goes back to Euclid. How many non-collinear points in a plane?”

I kid you not, the class gasped as they realized the connection. “THREE!”

“That’s right. Any three points define a unique plane. Only one possible plane. So the triangle has structural integrity and rigidity. The parallelogram does not. If I tell you the three lengths of a triangle, it’s mathematically possible to determine the area. Not so with a parallelogram. And there you have the reason I love this little exercise. You made these shapes. You can see the triangle hold up, the parallelogram collapse. You know there’s no trick. It’s just the triangle. So if you’re going to build a bridge, or a skyscraper, what shape is going to ensure your structure won’t fall down?”

“Triangles!”

“But look around this room. See any triangles?

God love ‘em, they really do look around the room.

“How about squares, rectangles and parallelograms in this room?”

“They’re everywhere,” says Zeus.

“Isn’t that weird? I don’t know if it’s human or just cultural, but we have a thing visually for rectangles. When was the last time you saw a triangular table of figures, or even a triangular dining room table? I was thinking of making the table on the back of this handout triangular, but it would distort the information! But even given our fondness for rectangles on the outside, we know to build and support our rectangles with triangles. And this goes back to the earliest times. Look at Roman architecture, and you can see triangles everywhere. When you leave the class, look outside at the classroom wall and the overhanging roof. You’ll notice struts holding it up, and what will the shape be?”

Bert: “But can’t you just add something to the rectangle to make it stay the same?”

“What, you mean like this?” I hold up two parallelograms that keep their shape:

“What’d I do?“

“Made triangles!” the class chorused.

“Indeed! Okay, for my last trick, take out the three identical triangles in the paper bag. See that each triangle has a different angle colored in? The Angle Addition Postulate says that if we align these angles so that each of them share a side with one of the others, the sum of their angles is equal to the sum of the larger angle that they form together.“

(I forgot to take a picture of this, but it’s a well known demonstration and I just stole an image off the web. Normally, you have one triangle and tear it into three parts, but I want to keep these bags as kits for future use.)

I wander round the room and check; all groups have their triangles arranged so that they can answer the key question.

“Okay. Your triangles are of all different shapes and sizes. But when you align their angles together, what do they form?”

“A straight line!”

“And how many degrees are in a straight line?”

“180!”

(I am getting very loud full-class responses here, not just the top kids in the back.)

“So the angles in a triangle add up to….“

“180 degrees!”

I wrap it up by going through the questions I posted on the white board, ensuring general understanding of the key concepts. And then.

“Okay, before we go through the homework, I want you to realize something. All this work you did today wasn’t proof of the facts you learned. They were demonstrations. You have demonstrated visually that a triangle has 180 degrees, that a triangle has rigidity, that the sum of two sides of a triangle must be greater than the third. And I like demonstrations! They are not to be sneered at. But don’t confuse a demonstration with proof. We’ll be proving some of these facts during the unit; in other cases, the proofs are more complicated than I want to go into. But don’t ever confuse a picture with a proof. A proof is an argument built with logic, facts, and definitions. But a picture or an experience is much easier to remember.“

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

So, what do you think? Was this a “discovery” lesson, the classic progressive ideal that traditionalists sneer at? Was it constructivist or instructivist? Teacher-centered or student-centered?

Instructivist, teacher-centered…..and discovery-based.

Most progressive, constructivist “discovery” lessons are insufficiently sculpted. Check out the pictures of this lesson, which is clearly covering much of the same material that I did. But the kids are spending a lot of time creating triangles with a protractor and ruler. I don’t want them spending time on that right now, so I give them the strips. Moreover, the kids are spending a lot of time putting together a presentation, which they are then going to, well, present—and everyone is going to hear the same basic information 8 or 9 times, as each group presents their “findings”—which, in the happy talk, differs for each group but in reality is the same thing over and over again. Tons of wasted time, and if you ever see a classic group-work class, at least a third of the kids aren’t doing a thing (cf Boaler’s bias).

While I wouldn’t do such an open-ended lesson, my point isn’t to argue for my lesson’s merits. I want people to notice the difference between a classic constructivist, discovery-oriented lecture and one in which kids put things together or complete discrete tasks that lead to immediate, unambiguous findings that we then translate into facts. I’m still the sage at center stage, but the kids aren’t just listening. They have sense memories and are participating in long class discussion interspersed with tasks that they can all do.

A classic instructivist approach would be a lecture, in which the teacher gives notes and the students listen and take notes. I could, of course, have covered this material in about 15 minutes. The top kids would remember it. The bottom kids would not. Plus: boring.

I am not a fan of straight lecture every day—well, actually, I rarely but ever do a straight lecture. Even when I’m talking, I’m engaging in a class discussion to move things along. But for the teacheres who do lecture daily, why not vary the routine? Find some good activities that demonstrate essential principles, with some handouts. It takes a little work the first time, and you often have to modify the activity a few times until you find the right balance of tasks and fact delivery. But the end result is almost always an enjoyable activity that gets the same information across.

But the larger point is this: many people sneer at constructivist teaching. I am not a fan, either. But so long as we are teaching kids who don’t want to learn math, we need to accept that the lecture is just zooming right over the heads of 75% of most classes. I’d rather reach a larger audience.