Category Archives: teaching

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.


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
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.

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.

Meanwhile….midterms again

I originally thought I could blog daily. Ha, ha. I managed close to that in January 2012, the first month of the blog, but it made me unhappy. Since then, I’ve usually managed an entry every five days or so. Sometimes I’d go 8-9 days after a really successful post, hiding away and watching reruns of Quantum Leap or The Mentalist, worried that I’d written my last good thing.

But this gap is more like the two-week plus hiatus I had in April of last year, when I spent most of a month working on a piece (under my real name). When I’d finally finished, I couldn’t even think of a thing to write about for a week or more. And so it was with my Philip K. Dick and preschool piece, which I’ve been thinking about since late December and working on for over a month. Kicked my ass, a bit. Happily, it was extremely successful—already #8 on my greatest hits list. My Twitter followers know that the piece had one fan that truly made it all worthwhile, and it kills me not to namedrop, but I won’t.

And then, midterms. My school covers a year in a semester, so we just finished the “semester” grades for the second “year”. We teach 4 block periods a day, so the kids can take 8 classes a year. Not great for test scores, since half the kids haven’t looked at the material in four months, and the other half haven’t entirely finished the courses. But it works in other ways.

One thing I noticed in my first “year” at this school: I wasn’t testing enough. In a normal schedule, I test or quiz kids every 8-10 school days, with the occasional longer gap. But in a block schedule covering a year in a semester, that’s the equivalent of testing kids every three-four weeks. Nothing terrible, but I decided I should up the pace slightly.

Except I’m teaching four classes and three preps—one of them for the first time. So I upped my quiz/test activity, thus increasing both my test creation and grading load, at the same time I was working a heavier teaching load, both in time and in preps. I work straight through from 8 to 3, in front of an audience of 35, with a brief 10 minute break at 9:30 and a half hour lunch at 12:35.

You know how teachers complain about grading, the tyranny of tests? I never have. I never even knew what they were talking about. I work fast, read fast, think fast, and I’m an insomniac. So the occasional two hour grading stint never struck me as particularly onerous. I’d generally schedule the grading session on a weekend, but if I had to grade things on a Tuesday or Wednesday every so often, I could manage easily. I pride myself on a rapid test turnaround, returning quizzes and tests within 3-4 days, a week at worst.

And suddenly, I was giving tests or quizzes to 140 students in three different subjects every week. Every time I turned around, I had a stack of tests to grade. At a modest estimation, I’ve spent 6 hours a week grading since February.

Then midterms. My geometry midterm is fifty multiple choice questions (multiple choice), my intermediate algebra 33. I have two sections of algebra, the test was four pages long. I graded it a page at a time on Sunday. Figure it took me 30 seconds a page to grade. 30*4*70 is 8400 seconds, or over two hours just to correct them. Then I had to calculate the curve and write the scores on each test, which takes another half hour or so, so three hours. And most teachers will tell you that’s a fast job.

Then my geometry test, which was five pages. I did two of them on Monday night, then just couldn’t take it any more. So Tuesday, I passed the tests out to the kids (geographically arranging them so no kid had his own test or a near neighbor’s), gave them red or blue pens, and we went through the whole test in 10 minutes. The kids loved the activity, were scrupulously exact, and asked why we didn’t grade tests this way all the time. I told them I usually look over the tests closely to see what kind of mistakes the kids make, what areas need revisiting, and so on. But I wasn’t going to be managing that little activity on this test.

Then Pre-Calc, the test that I spent five hours building the night before I gave it, and four hours grading on Wednesday night/Thursday morning (grades due 1:30 pm on Thursday).

The next day, Friday, I gave my intermediate algebra kids a quiz, so I have seventy quizzes to grade this weekend. I also have to build a pre-calc test (polynomials) and a geometry quiz (similar triangles).

I am not whining. Were I teaching a normal schedule this work would be entirely manageable. The longer day itself exhausted me for all of February, but by the first week of March I’d acclimated to the extra strain. I am getting well-compensated for the extra work, if you assume my normal salary is adequate, and I do. I get a 33% bump for the extra class.

But I finally know what teachers are talking about when they complain about the workload. It took an extra class and a concentrated effort on my part to assess my kids more regularly to get me to the point that many teachers reach on a normal schedule. Given that I’ve always had an inordinate ability to work long hours and do a lot of work without noticing (I worked full-time through all three of my degrees), I figure that I’m odd and the rest of teachers aren’t lazy.

Meanwhile, I’m beat.

Random notes on the classes:

Geometry: my most difficult class, academically speaking, since it’s a 10-12 class. Six kids are probably bored, because I can’t give them enough to do—the bottom half of the class is not unmanageable, but I can’t leave them unattended for too long without havoc ensuing. But they’re learning despite themselves, and the algebra progress is extraordinary. The kids right out of the top 6 but above the halfway mark are appropriately challenged, I think, and doing well. But I rarely feel good about the class.

On the other hand, last week, right at the time I was feeling really low, I gave the kids a worksheet (looked something like this, with a wider range of difficulty). I told them their task was twofold: 1) use what they knew about isosceles and equilateral triangles and 2) most importantly, build on the diagram, using what they knew, to solve for x. The second part, I said, was incredibly important. They couldn’t just look at the diagram and create an equation. They had to take the given information and work forward. The equation might be two or three steps away. I was not hopeful. And glory be, the kids surprised me by doing a bangup job—the weakest kids finished at least half the handout without begging for help every second. So I must have instructed them well! Need to remember what I said. Midterm performance: exceptional. They averaged 81% on the midterm (with a 15 point curve), and even the weaker kids did a great job on a long test.

Intermediate Algebra class: Having spent too long last “year” on linear equations, I used the same amount of time to cover linear equations, inequalities, and absolute values. We’re now wrapping up quadratics. Now that I’ve kicked linear equations and inequalities into submission, I’m struck by how damn complicated quadratics are. They don’t easily model. They have multiple forms and multiple methods of solving, all of which are complicated. But these kids, too, are doing well. Midterm: average 73% with almost no curve.

Precalc: longer post coming, eventually. Fun class. Midterm average was 69%. I have about 5 kids who simply do not understand the material, and I’m a tougher grader at the precalc level.

I was going to write this post first, but I’d seen one too many wails about American Indian Public Charter High, so I got back into it with a bang.

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?”


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….”


“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).


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.

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.



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: exprob.

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.
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.


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.


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.

Push the Right Buttons

Since my school only has four block periods, teachers often sub for each other during their prep periods. Cheaper than hiring an outside sub, and with an experienced classroom manager running things, there’s an outside chance that the class won’t be a complete waste of time, which is normally the case for all but honors and AP courses when teacher’s not around.

Today I subbed for an economics teacher during fourth block, last class of the day. She told me to go to the library, that she’d put a note on her classroom door telling the students where to go. Ten minutes after the late bell, no students. I wander over to her classroom, where about half the students are milling around trying to get the nerve to leave. No note on the door. (I suspect someone pulled it down.)

“Are you Ms. L’s Econ class?

“Class was cancelled.”

“Yeah. What is this, college? Go to the library and work on your business project.”

“Who are you?”

“I’m the person telling you to go to the library and work on your business project. Come on, move. You’re late.”

(Note to new teachers, particularly you little twenty-somethings bleating about relating to students: it goes much smoother if you expect obedience. Better yet if you demand compliance. And like John Travolta says in Get Shorty, what you’re not doing is feeling about it one way or the other.)

They all trudge to the library and get to work on their project with little fuss; most of the remaining third trickle in over the next 25 minutes.

Last in, half an hour late, are two African American young men who clearly took the opportunity to pick up a second lunch at Wendys, and tried to get out of a tardy by telling me that they’d been waiting by the door “the whole time!” I laugh at them and tell them to get to work.

“We don’t know what to do.”

“Here.” I produce the assignment handout.

“Oh, yeah, the sports project. But there aren’t any computers open, they’re all full.”

“You two?” I speak to two kids on the end row of computers. “You’re not in this class?” They try to ignore me. “Are you in this class?” They still ignore me. “Let me find your name on the class list.”

“Okay, we’re not in this class. But we have homework.”

“Good for you. Do it later. Bye.” They are not happy, but they leave. (The computers are booked for the class, in case you just think I’m mean.)

“Look, guys! Two computers! Put the food away, and get to work.”

It’s a trip watching these two. They try. I watch them page through the assignment, puzzling over it, telling each other man, this is f**** up, she doesn’t tell us what to do. “It’s just a list of things we need to have done! But what do we do, man?”

After five minutes, I see them both paging through search lists, getting “Site blocked” messages. I wander over.

“Please tell me you aren’t googling porn sites in open view of a teacher and librarian.”

“Naw, I’m just finding shoe prices.”

“Shoe shopping?”

“For the team.”

“Oh, so you’re pricing shoes for your business. How much are you spending on personnel?”

“We don’t know how to do that. She doesn’t say how.”

“What are you staffing?”

“We have to staff a professional high school basketball team for $100,000.”

I look at them expectantly.


“Nothing comes to mind?”

“She didn’t tell us what to do.”

“But you’re pricing shoes.”


I wait, to see if enlightenment will dawn. But it is still well before midnight.

“Okay. Who will be wearing the shoes?”

“The players. Oh.”

“Indeed. So maybe start with people instead of shoes. How many players on a professional basketball team?”

“Twelve. And we’ll need a manager, and at least three coaches….”

“and what about a guy to manage endorsements at the team level?”

And now they are cooking with gas, telling me what they need to staff the team, what prices seem reasonable for a “basketball farm team, these are high school players, they’ll be chill with maybe $500 a season”. I walk away, and ten minutes later they come back to me asking if I know how they can put this “in a file or we’ll lose this paper with our notes.” I bring up Excel, they carefully enter all their data.

A bit later, one of them comes over to me and says “I took this class on Excel when I was a freshman and you could, like, add things up?” So I look at what they have, and suggest that they create one column for count and one for price, so they could then change values and automatically change the sums and they caught on and by god if they didn’t have a damn spreadsheet with personnel variables and starting costs entered, ready for tweaking. They were brainstorming equipment and facility costs by the end of class.

Do not imagine, dear reader, that these two boys will come on time to class tomorrow, ready to jump in and pick up where they left off. More likely, they will cut class one day, get pulled out for some activity on another, and by the time they get back to it will have forgotten the file name and where they emailed it. Still, for 45 minutes, these kids did productive work, thinking about what they’d need to create a small business. Count it in the win column, once I pushed the right buttons.

There’s a larger point in this story somewhere, about the degree of scaffolding low ability, low incentive kids need to do any sort of project based work. The teacher had put together a good lesson, too: achievable, relevant, and interesting. But many of the kids would need far more support—leading questions and a project broken down into key milestones, Excel templates for business plans and budgets, and so on. And as always, I am boggled by the gap between the idiots calling for project based learning which is, to their thinking, essential to modern education, and the actual students who simply don’t have the ability or motivation to meaningfully engage in the learning required for the projects as they are currently envisioned.

But it’s late and I’m pleased with the win, so I’ll stop here.

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?”



“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?”


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.


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

“They were in different circles?”


“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?”


“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?”


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


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


“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?”


“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?”


(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.

100 Posts

I started this blog on January 1 with two primary goals. As I mentioned in my initial post, I’d gone a whole year without writing anything for publication (under my real name, which is not Ed). I wanted to initiate more and respond less, and I wanted my writing here to spur me to write more under my real name. As a second goal, I hoped to reflect the full spectrum of my views on education and teaching—and nothing more. I wanted to write both about teaching and educational policy. I teach a great many subjects, and the joys of teaching composition, literature, American history, and text prep ideally needed some of my writing attention, while I expected to write primarily about the challenges of teaching math (yes, I am saying there are relatively few joys in teaching math, but that doesn’t mean I’d give it up.) When the subject turned to educational policy, I expected to focus on the degree to which Voldemortean avoidance prevents us from sane, realistic objectives, but I also intended to discuss the very real problems I saw with both eduform and progressive math philosophies.

Thus far, the blog has exceeded my goals. I had one very successful piece go out under my own name, and to the extent I haven’t written more it’s been because of time constraints. I have plenty of ideas, which was not the case a year ago, when I felt hamstrung. I also think my posts fairly reflect all my teaching interests.

What I didn’t expect, and has been deeply satisfying, is the degree of attention many of my posts have had. Here are the top six posts:

  1. Algebra, and the Pointlessness of the Whole Damn Thing
  2. The myth of “They weren’t ever taught…”
  3. Teacher Quality Pseudofacts, Part II
  4. Why Chris Hayes Fails
  5. The Gap in the GRE
  6. Homework and Grades

I wrote all but one of these hoping they’d get a big audience—Homework and Grades is the exception; while it got a nice bump when it first came out with a link from Joanne Jacobs, most of the activity has been from consistent attention over time. People refer to it a great deal, for some reason. (Actually, half of my big pieces got link love from JJ, and a host of smaller ones as well, which means a lot because she’s the best pure education blogger out there. Other bloggers who contributed a lot of readers to the above pieces: Steve Sailer and Gene Expression.)

Three policy pieces that I was personally pleased with, audience or not: Why Chris Christie Picks on Teachers, On the CTU Strike, and The Fallacy at the Heart of All Reform.

Three teaching pieces that are regularly linked to or used as references by teachers: Modeling Linear Equations, Teaching Algebra, or Banging Your Head with a Whiteboard, and Teaching Polynomials.

Total views at the time of this post: 38,000

I’d also like to shout out to my commenters and Twitter readers. Thanks for your great feedback.

So on to the next 100. I’d say I’ll try to keep them brief, but that’s a big lie.

Mapping Real Life with Coordinate Geometry

Yesterday, I wanted to close off the coordinate geometry section (distance, midpoint) before I moved into logic. Rather than put a few random problems on the board, I came up with a map description.

Han is a driver for Harley’s Restaurant Supplies, making his Monday morning route.

  1. He went due north for three miles, dropping off supplies for diNardo’s.
  2. He then cut northeast along Steep Street to Patel’s Naan and Curry shop. He could have gone due east for 8 miles along Grimley, and then two miles due north along Freeman, but he wanted the shortest route.
  3. He then drove southwest along Morespark, back past Harley’s, all the way down to Bob’s Burgers. Harley’s is exactly halfway between Patel’s and Bob’s.
  4. Next stop, 17 miles due east along AutoBahn Boulevard to Andy’s Noodle Shop.
  5. Then it was northwest along Bracken Drive to Tomas’s Taqueria, which was just 6 miles due north of Harley’s.
  6. Back to Harley’s for his lunch of his noodles, naan, tacos, and burgers before he started out for the afternoon.

A. Create a map of Han’s route, including street, restaurant names, and coordinates. Suggest using (0,0) for Harley’s.
B. How far did Han travel?

All the students had their own whiteboards, so they could sketch and erase as needed. Step A, the sketch, went really well. As I expected, step 2 gave students the most difficulty, but a third of the class understood it without assistance, and the rest had drawn the two descriptions as two different locations.

First student finished with the sketch on whiteboard:

(Yeah, no street names. He put them in after I took the picture).

As students began moving from the sketch to calculating the distance, I brought it back up front. What was the difference between finding the distance from Bob’s to Andy’s (due east) and Andy’s to Tomas’s (northwest)?

This is the final product—I forgot to take a picture mid-lesson. But see how some trips are starred, and some have a plus. The class identified the stars, which required further calculation to get the difference. I stress the “slope triangle” in all aspects of coordinate geometry (slope, midpoint, distance), and you can see my light colored sketches of the three relevant triangles.

Later, the class identified the missing distances, and then we added it all up. Final instructions: transfer all of this to a quality sketch in your notes. Use color to identify the triangles.

When I teach the Big Three of Coordinate Geometry (slope, midpoint, distance), I emphasize the triangle because for so many students, the formulas are just one more reason to get negatives and subtraction all hosed up. Sketch in the triangles, and you’ve got a backup. Does this answer make sense? Yes, it’s fine if they use the formulas. I will forgive them. Provided they don’t muck up the math. And remember, knowing the formulas is essential. I want them to recognize the format of each formula, even if they never use them.

This took about 45 minutes? Wrap up and transition, maybe 55 minutes.

A few days ago, an a**l obsessive overly rigid teacher called me lazy for not having weeks of lesson plans written in advance. I am usually pretty nice to commenters (which is, like, so not me) but while I don’t object to teachers who plan, I vehemently object to teachers who confuse planning with teaching, and this guy is a prime example of the moralizing putz who never got over his potty training and wants everyone else to suffer his pain.

But here’s the thing: I built this lesson in the fifteen minutes before the day started. I do not think my ability to do so is an essential aspect of good teaching. But it’s a part of teaching I really enjoy, the combination of a) my understanding of my kids’ immediate need and b) my strength at creating interesting lessons on the fly. Forcing me to put together a schedule weeks in advance would either make a liar of me or take away that essential piece of my teaching. I’d become a liar, of course. But why go through the farce?