Okay, next is a classic: the **perpendicular bisector**.

All right, another standby: the **angle bisector**.

Let’s try for a fancy one now: the **regular 17-gon**.

Here, to change things up: a **Venn diagram**.

And how about a *real* challenge: the **Cantor ternary set!**

I drew this one at the request of Prof. Jim Propp, who writes the excellent *Mathematical Enchantments* and whose November essay Impaled on a Fencepost explored the kind of off-by-one errors that I make at least 17 times per day. (Or is it 18?)

**Greetings from Heisenberg!**

Tourist bureau of Würzburg, Germany: please feel free to sell these.

**Gotham’s Greatest Nemesis**

Having read (and enjoyed) Glen Weldon’s book on the cultural history of Batman, I know that “Fallacious Policy Analyst” is far from the least credible Batman villain. In fact, compared to actual villains like “Calculator,” “Amygdala,” “Penny Plunderer,” and “Actuary,” I think that my invention is downright menacing.

**“Facts are useless.” “Is that a fact?” **

Saddest/best response on Twitter:

You might be able to google relevant lit, but good luck accessing it on a middle class salary. https://t.co/ZaP4qResqn

— Aaron Thomas-Bolduc (@A_Thomas_Bolduc) December 7, 2017

**Types of Equations**

There ought to be a word for equations that are *never* true, like x + 5 = x.

I guess that word is “false.”

**I “Borrowed” This Joke**

This cartoon was drawn at the behest of Prof. Matt Wykneken:

the word “borrow” – which is typically used in the US, is not so typical in other cultures which avoid whatever is their equivalent of “borrow.” The process, rather, is an “exchange” or “ungrouping.”

**Roy G. Biv**

For the full EM spectrum – radio, microwave, infrared, visible, ultraviolet, x-ray, gamma – we’d need a somewhat clunkier acronym: R Miroy G. Bivux G. Or, if you prefer your nonsense embedded in further nonsense:

Real Men In Real Offices Yell Greedily:

“Beware Very Useless Xylophone Games!”

**The Most Important Baseball Statistic**

Other crucial baseball stats include NOSEs and EYEs.

(The real sabermetricians, of course, tend to look at NPE, or NOSES PLUS EYES, where the league average is 3.000.)

**A Little Piece of a Distant Planet**

My high school math teacher showed us a bit of complex analysis senior year (e.g., a derivation of Euler’s formula from the Taylor series for sine, cosine, and exp). I decided then that I wanted to go to there. Still haven’t; will someday.

**Diversification**

Monoculture: it’s the new biodiversity.

**The Twelve Iterations of Christmas**

I found myself writing a parody of “12 Days of Christmas” as a blog post, then found myself exhausted by the very notion of the song, and wrote this cartoon instead. A definite upgrade.

**F-prime-prime-prime-prime, prime-prime-prime-prime**

The Brits, of course, sing “f-dash-dash-dash-dash-dash-dash-dash-dash” and tend to medley the song with “Jingle Bells.”

*(F-dashing through the snow… in a one-variable function…)*

**Silent 0**

As a former colleague pointed out on Facebook, the singers clearly have a Birmingham accent, affectionately known as “Brummie.” It is reviled by Brits as one of their island’s ugliest, and generally enjoyed by foreigners as above-average for musicality and adorableness. (My colleague and I, respectively a South African and a Yankee, were able to take the outsider’s perspective here.)

**The Cruelty of the Mean**

Not sure how I feel about the colors in the text.

**Memorizing Capitals**

A polite, self-effacing Facebook commenter pointed out that I misspelled “achievement” as “achivement,” which sounds like an award for adding chives to your meal. This is what I get for switching over from hand-written text to typed: typos, but *legible* typos.

**The Purpose of Math Education**

I’m not sure why you’d need functions to understand groups, but the point stands.

]]>Just look at these equations:

After this crowd-pleaser came 2017, a prime year, which engendered this brilliant Tweet from Matt Parker:

That brings us to 2018.

It’s not triangular, like 2016.

It’s not prime, like 2017.

Is it, then, worthless?

Well, I myself am neither triangular nor prime. But if the roles were reversed, I like to think 2018 would do its best to uncover my special qualities and catalogue them in a blog post. So I went to do “research” (my codeword for “Google searches”).

What secret mathematical properties and pleasures will our new year contain?

To 2018’s credit, there are a few special days to mark on your calendars:

That’s just around the corner! And there’s another one coming:

And then in February, the first of our factor days:

The second will follow in March:

And the third in June:

With another special day in August:

And the final factor day in September:

Exciting as these days are, they pale in urgency alongside the fact that 2018 is the year in which the film Iron Sky takes place. So if there’s a shadow civilization of Nazis living on the dark side of the moon, expect them to surface sometime in the next 12 months.

In another dark turn, 2018 marks the 500^{th} anniversary of the great Dancing Plague, in which four hundred citizens of Strasbourg danced for days without rest, some to their deaths. The cause remains unknown.

(Seriously. I’m not making that one up.)

Aside from that… I’m afraid 2018 is a rather bland number. Well, not bland; let’s go with “understated.” I can’t call it outright “boring” because of the classic proof that there is no uninteresting positive integer:

Still, I have to confess that 2018 is below-average for mathematical swagger. The best I can offer is this little trifecta:

Or, if you prefer strange conversions to and from binary:

I’m afraid I’m not serving well as 2018’s advocate, since this is all rather arbitrary and numerological. We’d get similar answers from an astrologer or a fever dream.

The harsh mathematical truth of 2018 is that it is “semiprime,” i.e., a product of two primes—in this case, 2 x 1009.

That’s not the most exciting property. Other semiprime numbers include 6, 9, 10, 14, 15, 21, 22, 25, 26… and indeed, more than a quarter of all years that have happened so far.

Is that the best we can say for the forthcoming year?

Luckily, no. 2018 has one last trick up its sleeves.

Although semiprime years are quite common, this is the first since 2005. That 13-year drought is rather impressive; it’s the longest since Shakespeare’s death.

With any luck, that’s interesting enough to last us until the Moon Nazis show up.

]]>And what goes for journalists, goes double for stick-figure cartooning math teachers. Thus, as one who loves truth even at its ugliest, I choose to divulge a fact sure to rattle your faith in humanity itself:

The game of dreidel is built on a lie.

Dreidel, of course, is a beloved Chanukah game. (Happy Chanukah, everybody!) First, each player places a chocolate coin in the center. Then, you take turns spinning a four-sided top (the dreidel), obeying the commands that appear on its ides:

The top functions like a die, with an equal chance of landing on each side—at least, in theory.

The reality is far more sinister.

Fearless and groundbreaking research by Robert and Eva Nemiroff reveals that on the typical dreidel, **not all sides are equally likely**.

I quote here from their startling abstract:

all three dreidels tested—a cheap plastic dreidel, an old wooden dreidel, and a dreidel that came embossed with a picture of Santa Claus—were not fair… it is conjectured that hundreds of pounds of chocolate have been distributed during Chanukah under false pretenses.

It I worth asking: Why?

No, not “why does a Jewish toy come embossed with a picture of Santa Claus,” although this too is a vexing matter. I mean: Why is the dreidel unfair?

Is it shoddy craftsmanship?

A manufacturer’s deviousness?

Anti-Chanukah sabotage?

The likeliest answer: none of these. It seems that, across the board, spinning is a poor randomization process. A classic study by three Stanford researchers called Dynamical Bias in the Coin Toss found that spinning coins on a table was less effective for randomization than flipping them through the air.

One can imagine why. The long duration of a spin, from rapid beginning to wobbly end, allows time enough to amplify a tiny difference in weight distribution. The heavier side falls down. The lighter side lands up. Invisible deviations in density become visible disparities in chocolate allocation.

What’s the solution?

One drastic measure: change randomizers. Use a tetrahedral die, or two coins (with HH, HT, TH, and TT as the four outcomes). But this would remove the dreidel from dreidel. Unacceptable. When a patient comes with chest pains, you don’t yank out her heart.

Instead, I have a different solution: each turn, you spin the dreidel *three times*, and interpret the outcome according to this table:

Each row follows the same pattern. It consists of four permutations: one without nun, one without shin, one without he, and one without gimmel. Because order does not affect the probability of a permutation, each row is therefore equally likely.

Via this system, the underlying probabilities of the dreidel itself are rendered irrelevant. Even a grossly asymmetric dreidel can be used to play a fair and balanced game.

Now, is this hyper-complicated? Yes.

Liable to confuse and alienate children? Almost certainly.

Totally unnecessary, given that nobody cares whether the four sides of the dreidel come up with equal likelihood? Perhaps.

But mathematics has never been about “understandable” or “desired.” It has always been about insinuating itself, over all manner of protests, into nostalgic memories and cherished holiday sentiments. And I refuse to let that tradition die.

]]>It blooms only under rare and perfect conditions, when you’ve given the seedling absolutely everything it needs.

There’s no perfect recipe. What gets my 6^{th}– and 8^{th}-graders’ thoughts blooming might flop with my 7^{th}-graders. This work is wonderfully and maddeningly specific. Each seedling presents its own unique and irreducible case. The best you can do is kneel down in the soil and try to help it along.

Even so, I find a few recurring themes: three crude reasons why deep thinking fails to bloom, and the hardy but colorless perennial of “rote learning” surfaces instead.

**As students, we seek the cognitively easier path.**

Earlier this year, I had a typical conversation. A 7^{th}-grader wasn’t sure why 3√2 + 4√2 should equal 7√2.

“Well,” I said, “what’s 3x + 4x?”

“7x.”

“Why?”

“Because you add the 3 and the 4, and keep the x the same.”

“That’s an accurate description of what you’re doing,” I said, “but let’s try to figure out why it’s true. What do ‘3x’ and ‘4x’ mean?”

They mean, of course, “three groups of x” and “four groups of x.” That totals seven groups of x, no matter how large or small *x* happens to be. That’s elemental, but not elementary. It demands that you (1) think about specific cases; (2) look past their superficial differences to the underlying similarity; (3) articulate a general principle; and (4) translate your discovery into algebraic notation.

Or… you can ignore all that, and just learn a rule for moving symbols around.

“Oh, I get it!” he said before long. “You just add the 3 and the 4, and leave the √2 the same.”

**As a teacher, I seek the administratively easier path.**

Back to my 7√2 student: why had I, his teacher, put him in such a pickle to begin with? Why was I asking this student to extend a rule that he didn’t even understand?

Well… because that’s what came next in the syllabus.

Sure, I could have found a better personal activity for him. Or I could have found a richer task for the whole class, so that he could explore this idea while his classmates explored others. Or I could have stayed with him—collating numerical examples in a table, producing visual and verbal models, testing his symbol-based rule on cases where it would fail—until he developed an actual understanding of why 3x + 4x = 7x.

So why didn’t I?

Because lessons are short, teaching is hard, and I had a classroom full of 7^{th}-graders to manage.

Symbol-pushing isn’t just easier for students. It’s easier for me. It takes less planning before class, less improvisation during class, and less mop-up with struggling students afterwards.

To help a room of students think deeply—that’s no easy task. To help them learn superficial facts and mechanical rules? Well, that’s a heck of a lot easier.

**As assessors, we seek clear-cut standards by which to rank students.**

It was there in the room, as I spoke with my 7√2 student. I’m not sure I can pinpoint where—hovering by the ceiling? lurking under my desk?—but it was there:

The system’s need to assess.

Schools play a lot of roles in society, and one of them is the crude business of sorting. In June, the school’s students all take a year-group test. Top scorers win prizes and are called on-stage in front of the whole school. Low scorers feel the gut-punch of failure, no matter how meaningful or meaningless the test is.

Three years after that, the same students sit the IGCSE exams. Top scorers have a better shot at prestigious university degrees. Low scorers have a worse one.

This stuff matters.

The tests are written to be “objective” and “fair,” which means they ask for scripted performances of technical skills rather than for flexible improvisation. On such tests, deep thinking can be more an impediment than an aid.

So what is there to be done? How do you help healthy flowers grow in a climate that can feel so ill-suited to them?

Well, that’s called teaching, and I’m still learning how to do it.

But I’ve got some ideas.

To overcome the first obstacle—students’ preference for easier thinking—I’ve got to give them motive and opportunity. I’ve got to help them see the steeper path as an exciting adventure, not a pointless side quest.

To overcome the second obstacle—my own preference for easier administration—I’ve got to play it smart and conserve my energy. I’ve got to avoid the mire of self-created busywork, and lay the groundwork (with routines and class culture) to make open-ended tasks go smoothly.

And to overcome the third obstacle—the system’s preference for clear-cut assessment—I’ve got to fight a multi-front war. I’ve got to seek better, richer, more varied assessments. I’ve got to help students see their results not as irreversible judgments but as guiding feedback. And—hardest of all, in a system that puts all kinds of pressures on teachers and students alike—I’ve got to know when to hold ‘em, and know when to fold ‘em.

Now, the goal isn’t for students to think deep thoughts every minute of every day. That’s as unsustainable as a full night of top-intensity dancing. You need cool-down songs, slow dances, chances to catch your breath.

But I know this much: I want my students dancing as hard as they can.

]]>A pal on Twitter read this as “elven witnesses.” I wish. Eyewitness testimony is unreliable, but elf-witness testimony is foolproof.

**Those Who Do Not Learn Recursion…**

Also, those who do not learn their cliches about history are doomed to repeat them.

**A Researcher’s Politics**

I’m not saying *all* researchers are like this. I’m just saying that the Darwinian process of funding selects for some strange behaviors.

**Hyperbolus, the Emperor of Hyperbole**

The real Hyperbolus got ostracized from Athens. He was literally the last person to suffer that punishment – no hyperbole.

**The Squabble of the Inverses**

*Plot twist*: Both of the depicted functions are f(x) = 1/x.

**Undecidable**

My new home in Northampton has one of the world’s finest ice cream shops: Herrell’s. Their flavor list is gargantuan and horrifying. I could stand there for hours.

**My Podcast Listening Habits, in a Single Graph**

At this point the river is getting full, too.

**A Glass of Red Puns**

I’m truly sorry.

**A Dangerous Epsilon**

The limits only apply to positive epsilons, after all. A negative epsilon is a lawless rebel.

**Changing Minds**

You are right not to believe this theory, because I just made it up. *Math with Bad Drawings*: another empty calorie in the wasteland of your informational diet.

**Invariants of Doughnut Pricing**

The classification of finite donuts has been a major project of topologists/pastry enthusiasts for decades.

**The “Amazon Delivery Drone” Problem**

“Salesmen sold things like encyclopedias.”

“What’s that?”

“A print-out of Wikipedia.”

“Whoa, they print it out every day?”

“No, just once, and then it sits on your shelf for 20 years.”

“I see…”

**Beer Review**

An essential part of the academic process, from what I gather.

**Estimating Variance from a Sample**

To be fair, this is just a heuristic – it doesn’t explain why n – 1 is exactly the right adjustment, as opposed to other options – but it’s more satisfying than the blank stare (or page of algebra) that most textbooks will give you.

*These cartoons appeared throughout the month of November on Facebook and Twitter. Check there for more!*

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And the occasional search term will tap into a matter of real depth, like this one: **“my students are failing my math class.”**

It’s bleak. It’s discouraging. And if you’ve taught math, it’s an experience you know.

We’ve all endured days when it felt the whole class was falling short. You demonstrate how to bake brownies; the kids burn theirs into coal. You model a skateboard move; they gasp for air, having somehow placed the elbow pads over their nose and mouth. You say, “Don’t divide by zero.” They light their hair on fire and then, while you sprint for the extinguisher, they divide by zero.

On such days, my psychic antibodies kick in. “This isn’t my fault,” I say. “Their prior teachers didn’t teach them anything. Or maybe they’re lazy. Or wait—why didn’t I think of it before?—it’s all the administration’s fault.” School administration, federal administration, National Aeronautics and Space Administration; it doesn’t really matter, as long as the blame weighs upon shoulders other than mine.

Of course, “blame” has no good place in the classroom. The question “How did we get here?” matters only insofar as it informs the real question: “What do we do next?”

When a whole class fails, nobody wins. The kids suffer twice—first on their transcripts, and second in their blessed little hearts, where they’ll lose faith in themselves, or in the fairness of schooling, or (most likely) in both. The teacher suffers, too—facing worried administrators, outraged parents, and inner doubts about being “cut out” for the profession.

So let’s take it as a given that you can’t fail ‘em all, and explore some possible causes.

**Pacing.**

My first year teaching geometry, I sampled the whole platter of teacher mistakes. But I really gorged myself on one in particular: bad pacing. I marched through one topic per day, with no breaks for water or rest. Who needs synthesis? Who needs review? Isn’t this stuff obvious?

Well, math you know is always obvious. Math you’re learning never is. It’s healthier to learn one thing well than to “see” five.

**Assessment.**

When I write a test, I’m often tempted to overload it with interesting, challenging, novel questions. To make it a true test of wits and wills. But a test should assess basic skills, too. It needs a good mix of easy, medium, and hard, if it’s going to hold an honest mirror to students’ abilities (rather than a horror-movie mirror in which they can’t see their own reflection).

A bad grading scheme can create problems, too. It’s easy to turn a test into an all-or-nothing affair—by asking lots of similar questions, or many short questions with no partial credit, or questions where 1c is impossible unless you nailed 1a and 1b. That’s a recipe for whole-class failures.

Grades aim to reflect the quality of a student’s work, and “quality” is one of life’s most flexible concepts. Mathematics—despite its reputation as objective—leaves teachers a lot of latitude.

**Background**

There is no “first rule of math teaching,” except perhaps “talk about math teaching.” (This is math education’s primary difference from Fight Club, which it resembles in all other respects.) However, there is a truth that I find to be almost universal: Whatever you expect students to know upon arrival, they probably know less.

That’s okay. Learning is hard. Teaching is hard. Summer wipes a giant eraser across all of our mental whiteboards.

Every student has gaps: concepts they’ve never learned, techniques they’ve never nailed, anxieties they’ve never addressed. I can’t pretend those gaps reside in the past, that they’re debits on the ledger of some prior teacher, irrelevant to my work. Those struggles are here, in the present, weighing down my students, derailing their learning. I’ve got to survey the landscape of their thinking—to see what’s there, not what I want to see.

When you ask Google something you should already know—“how much do library books cost” or “who is Tom Hanks” or “what is pasta made of”—the search engine doesn’t snark or seethe or pass quiet judgment. It just shows you the answers: “they’re free”; “the mayor of Hollywood”; “pasta molecules.”

A good teacher is like a sentient Google. No finger-pointing. No recriminations. Just a benevolent omniscience, helping everyone to take the next step.

]]>*And in case you were wondering, here’s how that first question (“Thanks for taking this survey!”) turned out:*

*Original Post*:

Today, I have a request for you: I’d like some waffles, please.

But in lieu of a crenelated syrup castle, I would gratefully accept your taking 3 minutes to fill out this quick reader survey.

I am super grateful for those who take the time to look at this blog and verify that its drawings are, indeed, bad. I would love to know more about why folks come here, what they seek, and if they’d like T-shirts.

All survey participants will be entered into a drawing for the prize of my affections, in which 100% of entrants shall win.

Link here, or you can find it embedded below:

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