## Never Ask a Mathematician for Directions

I spent my formative years in a very rural county. Roads didn’t have names, or at least they didn’t have road name signs.

In college, my urban friends used to claim that if you asked someone in my hometown for directions, they’d say:

Go about a half mile, and turn left at Old Man Johnson’s farm. Then take a right at the huckleberry tree.

That very well may be true, but it’s no worse than the directions you might be given by a mathematician:

Well, you’re facing the wrong way, so do a reflection about this cross street. Go about 0.628397154 miles, then rotate π/2 radians and travel orthogonally to your previous vector for 600 light-nanoseconds. But they’ve got radar on that road, so keep your speed between 72 and 43 miles per hour. Turn left, and you should reach your destination in 2.4 minutes ± 0.3%, if you maintain an average speed of 46.8 mph.

Now, that’s bad. But at least I could understand all of it. Which is more than I can say for the directions that Google Maps provided yesterday. Traveling through the suburbs near Washington, DC, I had just crossed MacArthur Boulevard, and I was heading southwest on Arizona Avenue. As you can see from the screenshot below, Google Maps was suggesting that I turn right on Carolina Place, right on Galena Place, right on Dorsett Place, and then left on Arizona. In essence, it suggested that I reverse direction to take a 10-mile, 45-minute route.

I ignored that suggestion. Instead, I stayed on Arizona Avenue with the intention of turning right onto Canal Road in a quarter-mile. Just as I passed Carolina Place, Google Maps said that it was “Rerouting…,” and within 15 seconds, it confirmed that I had made the correct choice:

By ignoring Google Maps, I shaved 3.8 miles and 23 minutes from my commute.

WTF?

My speculation is that Google Maps attempts to avoid my chosen route because it follows Canal Road, which parallels the C&O Canal National Historic Park; it requires me to cross Chain Bridge, which offers a beautiful view of the Potomac River; and it then winds through an affluent neighborhood, where I can feel safe on tree-lined streets with elegant homes. Honestly, who would choose that when Google Maps is offering double the travel time and an opportunity to drive on the beltway?

I once asked Google Maps which highway I should take to California. It replied…

$\sqrt{1^2 + 4^2 + 7^2}$

Oh, yeah. Root 66.

The logic employed by Google Maps reminds me of a college friend…

He would always accelerate when coming to an intersection, race through it, and then brake on the other side. I asked him why he went so fast through intersections. He replied, “Well, statistics show that more accidents happen at intersections, so I try to spend less time there.”

## Don’t Drink the Flavor Aid

Okay, boys and girls. Time for a pop quiz.

Question 1. What cyanide-laced drink did 913 members of the Peoples Temple consume on November 18, 1978, as part of the largest mass murder-suicide in modern history?

If you said Kool-Aid, well, you’d be in good company. Most Americans think that that was the drink of choice in Jonestown, and it’s the reason for the idiom, “Don’t drink the Kool-Aid.” (More on that in a moment.) In fact, members of the cult consumed Flavor Aid, but media outlets at the time and revisionist history since have used Kool-Aid because of its status as a more genericized trademark.

Question 2. Why does Kool-Aid have a hyphen but Flavor Aid does not?

That question is rhetorical. Damned if I know. Ostensibly, the unnecessary hyphen in Kool-Aid is a marketing gimmick, like the removal of the period in Dr Pepper, the inclusion of the backwards R in Toys Я Us, or the use of the numeral 4 in Math Jokes 4 Mathy Folks.

Question 3. Why do educators, policymakers, and other adults continually offer Kool-Aid to math students?

Not literally. Let me explain what I mean.

“Drinking the Kool-Aid” is a figure of speech that refers to a person or group holding an unquestioned belief, argument, or philosophy without critical examination. Wikipedia says, “It could also refer to knowingly going along with a doomed or dangerous idea because of peer pressure.”

In math class, we offer Kool-Aid every time we tell students how important some mathematical topic is, how useful it will be to them in the future, or how knowing it will prepare them for a fantastic career.

C’mon!

When was the last time you were asked to add fractions at a job interview?

When I was a student teacher at West Mifflin High School, one of the teachers made the following statement to a student:

Unless you become a math teacher, you’re never gonna use most of what we teach you.

I hated that statement. Still do. But I’ve come to appreciate its honesty.

At the time, I was an optimistic 23-year-old. All I could think was, “Then why are you teaching it?” Now, I’m a cynical 44-year-old, and I know that he had little choice… students were going to be tested on it, whether he thought it was relevant or not.

How do we offer the Kool-Aid? Lots of ways.

An algebra student asks, “When will I ever need to factor a trinomial in real life?” Teachers often respond with answers like, “You’ll use it when you get to chemistry or physics,” or perhaps they’ll offer, “It can be used in area and construction problems. For instance, what if you know that the floor of a rectangular building is 1,344 square feet and its length is 20 feet greater than its width? Then you could use the equation w2 + 20w – 1344 = 0 to find the length and width.”

C’mon!

The simplistic trinomials presented in a typical Algebra I curriculum are not what you’ll find when you get to physics. There, you’ll be confronted with -16x2 + 25x + 42 = 0, which can’t be factored easily.

And if you’re ever on a construction site and someone gives you the math problem about length and width that’s described above, you should do two things. First, punch that person in the face. Second, while they’re tending to their bloody nose, ask them how they know that the length is 20 feet greater than the width and that the area is 1,344 square feet if they didn’t actually measure the damned building.

But it’s not just teachers who offer mathematical Kool-Aid to students. Policymakers do it all the time, too.

The Common Core State Standards for Mathematics (CCSSM) contain topics that should have been removed from the curriculum a long time ago. For instance, students are still expected to learn long division (6.NS.B.2), even though it’s an antiquated paper-and-pencil skill that — let’s be honest — no one uses anymore. Your smartphone can do the job just fine, yet CCSSM still gives it a place in the curriculum.

C’mon!

In defense of Common Core, the standard says that students should divide by “using the standard [long division] algorithm,” though it is referenced only one time in Grade 6 and then never mentioned again. Perhaps they’re not that serious about it being a staple in students’ mathematical diet. (Based on my experience, I’d bet that someone on the standards writing committee vehemently argued for its inclusion despite repeated objections, and it was included just to quell the discussion.)

And lest we be too quick to blame educators and politicians, your typical, run-of-the-mill, do-right-by-your-kids parents are equally at fault. What do they do?

• They buy books like Skippyjon Jones Shape Up, which does nothing more than teach shape names without understanding (a better option is this free shapes book from Chrisopher Danielson, Which One Doesn’t Belong?).
• They use resources like Bedtime Math to perpetuate the myth that story problems with keywords and no extraneous information is what math looks like in the real world.
• They tell their kids that fractions are important because you’ll use them in baking, yet they fail to realize that there are over 400,000 different recipes for German Chocolate Cake alone that can be found online, thereby proving that fractions aren’t that big a deal, since lots of different fractions result in yummy desserts and, if you need to double a recipe, you don’t have to know how to add or multiply fractions, you just need to use each measuring cup twice.

C’mon!

The counterargument isn’t that the specific skill will be necessary, but the way of thinking associated with that topic is an important life skill. I’m as much a fan of the phrase “habits of mind” as the next guy, but does anyone really believe that the thinking associated with finding common denominators will increase a student’s quality of life?

More Kool-Aid, as far as I’m concerned.

So, what can we do? First of all, use authentic problems in your classroom, and show students the value of creating a mathematical model (and then improving it) instead of just providing models into which they substitute values or assign measurements. Conversely, stop using fabrications that include a real-world context only to make students do math that no one in their right mind would ever do.

Next, stop teaching things that don’t need to be taught. Dr. Henry Giroux once told me, “To thine own self be true, even if it means being a troublemaker.” Be a troublemaker! Tell your principal that you refuse to teach trinomial factoring on the grounds that it violates the fourth amendment. Tell her that forcing kids to use substitution or elimination to solve a system of equations — just because that’s what creators of the state test are trying to assess — is lunacy if guess-and-check is a more reasonable strategy. And for Pete’s sake, don’t even think about pawning off Ceva’s theorem on unsuspecting geometry students as a potentially useful piece of information. (It’s not, and you know it.)

Finally, be honest to yourself and your students. When a freshman in your algebra class asks, “When are we ever going to use this?” look him in the eye and say, “Probably never, but it’ll be on the state test.”

Yeah, that answer will leave a bad taste in your mouth and in your student’s mouth, too. But you can wash it away with a nice, tall glass of Kool-Aid.

Glug, glug.

## Prime Time at NCTM Minneapolis

This afternoon, I’ll be presenting “Experience the Math Practices with Games and Online Tools” at the NCTM Regional Conference in Minneapolis. So if you unwittingly find yourself in the Minneapolis Convention Center at 1:30 p.m. today, please stop by.

But how cool is this? My session is #210, and today is Friday, November 13. Awesome, huh?

Wait, maybe you don’t see it:

210 = 2 × 3 × 5 × 7

Friday, November 13 = 11/13

Yeah, that’s right! The factors of my session number combined with today’s date are the first six prime numbers. You don’t have to be a math dork to appreciate that! (Though it doesn’t hurt.)

Why is 6 afraid of 7?

I assume it’s because 7 is a prime number, and prime numbers can be intimidating.

Thanks to Castiel from Supernatural for that new twist on an old classic.

## Wait, Wait… I’ve Got a Math Question

“Not My Job” is a segment on the NPR game show Wait, Wait… Don’t Tell Me! During the segment, host Peter Sagal asks a celebrity three questions, on a topic about which they likely have no clue. For instance,

• Rob Lowe was asked questions about bratwurst, not the Brat Pack;
• Leonard Nimoy was asked questions about the other Dr. Spock (you know, the celebrity pediatrician).

My favorite of these segments, however, was with singer-songwriter Will Oldham, better known by his stage name Bonnie “Prince” Billy. Sagal explained, “You sing mostly sad or mournful songs, interspersed with the occasional tragic one. And we were thinking, who’s the singer least like you? […] So, we’re going to ask you three questions about Ms. Doris Day, the sweet-faced, sweet-voiced singer most famous in the 50s and 60s.”

As Sagal congratulated him, Oldham pretended that Doris Day was in the room with him. “Hey, Doris, you were right!” he said. “All the questions were about you!”

Now, that’s funny!

(As an aside, let me share with you a slide that I sometimes show during presentations about classroom technology:

Yep, that’s Doris Day in the 1958 movie Teacher’s Pet. Hopefully that image looks a little odd to you in 2015. Honestly, if you’re teaching math to adolescents and still using chalk and a wooden pointer, I’ll kindly ask you to consider a different career. There are other options for you. For instance, you could become a dentist and specialize in square root canals. But, I digress.)

So, back to the point.

The celebrity quiz contains three multiple-choice questions, each with three choices. If the guest answers at least two questions correctly, he or she wins. Which got me to thinking…

What is the probability that a celebrity guest will get at least two questions correct if she guesses randomly?

Brute force is definitely an option here. Write down all possible answer choice combinations, randomly decide which configuration will be correct, and then figure out how many of the possible combinations would yield a winner. Not pretty, but it works.

Speaking of combinations, here’s a joke that just has to be shared:

Courtney Gibbons’s comics used to appear at Brown Sharpie, but then she got a job.

## The Life and Times (and Fractions) of Eliza Acton

Title Page

You may know Eliza Acton as an English poet or Victorian cookbook author, but I prefer to think of her as the Queen of MP.6.

CCSS.Math.Practice.MP6:
Attend to precision.

Her bestselling Modern Cookery for Private Families — which was first published in 1845, ran through 13 editions by 1853, remained in print until 1918, was reissued in 1968 and 1974, and then resurrected again in 1996 by Southover Press, with two more editions since — might very well be the greatest cookbook ever produced.

The prose within the volume is magnificent. She included recipes for “Poor Author’s Pudding,” “Printer’s Pudding,” and “Publisher’s Pudding,” the last of which “can’t be made too rich.” The directions for the “Publisher’s Pudding” explain that it should be covered with a “sheet of buttered writing paper,” which no doubt gives recipe readers some idea about the thickness of paper to be used, but also implies something about the publishing industry. (The complete text is available from Google Books, if you’d like to check it out for yourself.)

But what makes this work mathematically interesting is that it is the first cookbook that used precise measurements in its recipes. Without Modern Cookery, a middle school word problem might look like this:

The recipe for a loaf of bread calls for some flour, a dash of salt, and enough water to make the dough pliable. How much salt would you need to make two loaves?

I suppose the answer is “two dashes,” though printers would likely call that an em dash. (Cue cheeky, all-knowing editor’s laugh here.)

It was the recipes of Ms. Acton — like the one for a disgusting drink known as Milk Lemonade, which calls for 6 oz. sugar, ¼ pint lemon juice, ¼ pint sherry, and ¾ pint cold milk — that paved the way for the wonderful word problems that students enjoy today:

My recipe calls for ⅔ cups of white flour and 2⅕ cups of wheat flour. How much flour do I need in total for my recipe?

Oh, wait… did I say wonderful? I meant awful.

Who the hell measures flour in fifths of a cup? And why would anyone need to know the total amount of flour? Just dump it in a bowl and mix!

The word problem above without specific measurements is purely speculative; it’s almost certain that someone else would have thought to include exact measurements had Eliza Acton not come along, and students would have still been subjected to unrealistic fraction-containing word problems. But the purported imprecision within recipes is spot on, as shown by this recipe taken from an early 18th century English text:

Fill yr pott halph full of wien & [a] good share of sugar. Milke in as much cream & stirr itt once about very softly. Let itt stand two houres before you eate itt.
[from MS Codex 753, compliments of rarecooking.com]

Admittedly, that recipe is for an Ordinary Sillibub, which is basically a red wine float, and hence the recipe is very nearly useless. But it is typical of the imprecision that was commonplace before Ms. Acton’s arrival.

All hail Eliza Acton, Queen of MP.6!

## Fractional Eggs

I search for new recipes at allrecipes.com all the time. This morning, a search yielded a delicious recipe for pumpkin pancakes, which sounded like the perfect breakfast for a crisp fall morning.

One of the things I love about allrecipes is the ability to customize the number of servings. The default number of servings for the pumpkin pancake recipe was six, but I could adjust it to four, a more appropriate number for our two-adult, two-child family:

So I did. And as you’d expect, each item in the ingredient list was reduced to ⅔ its previous amount. Sort of. Two cups of flour was reduced to 1⅓ cups. One cup of pumpkin puree was reduced to ⅔ cup. But 2 teaspoons of baking powder was reduced to 1¼ teaspoons, and 1 teaspoon of cinnamon was reduced to ¾ teaspoon.

The reduction in the number of servings was 33⅓%, yet the range of reductions in the ingredients varied from 25% for salt (from 1 teaspoon to ¾ teaspoon) to 50% for ground ginger (from ½ teaspoon to ¼ teaspoon).

But I get it. It’s not typical for most kitchens to contain a spoon that measures ⅙ teaspoon. So there’s clearly some part of the algorithm that completes the conversion but then finds a “nice” fraction that’s in the right neighborhood. Fair enough.

But what the hell’s going on here?

Is it really better to display ⅝ egg instead of ⅔ egg? Couldn’t the algorithm recognize that fractional eggs just aren’t all that common and leave it as a whole number?

My guess is that the programmer is one of the folks to which this statement alludes:

5 out of 4 people aren’t very good with fractions.

That joke represents one-fifth of my favorite fraction jokes. Here are the other four:

Why won’t fractions marry decimals?
They don’t want to convert.

I’m right 4/5 of the time. Who cares about the other 10%?

There’s a fine line between a numerator and a denominator.

Sex is like fractions. It’s improper for the larger one to be on top.

If you find a store that sells ⅝ egg, please let us know about it in the comments.

## One, Tooth, Ree, …

Were a state assessment item writer to get his hands on the puzzle that I discuss below, I believe that this is what the problem might look like:

Fifteen congruent segments labeled A-O are arranged horizontally and vertically to form five squares, with five of the segments shared by two squares each. Which three segments can be removed to leave three congruent squares?

1. {A, B, C}
2. {A, C, E}
3. {B, K, N}
4. {D, E, F}
5. {G, I, N}
6. {H, A, L}
7. {I, B, M}
8. {K, L, M}

You’ve likely seen this problem before, but in a simpler and more elegant form.

Remove three toothpicks to leave three squares.

Eli shared this puzzle with us the other night at dinner, after hearing it during his math enrichment class that day. (It’s my sincere hope that every family has similar conversations at mealtime.) This was a great problem to discuss at the dinner table, since a collection of manipulatives was close at hand.

As you might expect, Eli and his brother started by removing toothpicks randomly and seeing what happened. But it turns out that random toothpick removal is not a great strategy. There are 15 toothpicks in the arrangement, so there are 15C3 = 495 ways to select three of them.

Of course, some of those selections are obviously wrong, such as selecting the three toothpicks in the middle row, or choosing the three highlighted in pink:

But even ignoring the obviously wrong ones, that still leaves a lot of unobviously wrong combinations to consider. Yet I let my sons randomly select toothpicks for a few minutes without intervention. Why? Because of a primary tenet for effective problem solving:

Get dirty.

You can’t get any closer to a solution if you don’t try something. Even when you don’t know what to do, the worst thing to do is nothing. So, don’t be afraid to get your hands a little dirty. Go on, try something, and see what happens. You might get lucky; but if not, maybe it’ll shed some light.

After a few minutes, they conceded that random toothpick selection was futile. That’s when they modified their approach, leading to another important aspect of problem solving:

Look at the problem from a different perspective.

“Hmm,” said Eli. “That’s a lotta toothpicks.” He thought for a second. “But there are only five squares.” That’s when you could see it click, leading to the most popular problem-solving principle on the planet:

Solve a simpler problem.

There are 5 squares in the arrangement, and there are only 5C3 = 10 ways to select three of them. If you first choose three squares, then you can count the number of toothpicks that would need to be removed. For instance, you’d have to remove five toothpicks to leave these three squares:

That doesn’t quite work. But the problem is a whole lot easier if you focus on “leave three squares” rather than “remove three toothpicks.” There are only 10 possibilities to check.

From there, it was just a hop, skip, and jump to the solution.

So, maybe this problem, as well as the million or so other remove-some-toothpicks problems, isn’t very mathematical. It only contains two mathematical elements — basic counting and geometry. In the vernacular of Norman Webb’s Depth of Knowledge Levels, the kinds of questions that you could ask about this problem — “How many toothpicks are there?”, “How many are you removing?”, “What is a square?” — would be DOK 1.

But they’re fun, and they’re engaging to kids, and they develop problem-solving skills, such as those mentioned above as well as perseverance.

For more problems like this, check out Simply Science’s collection of 16 toothpick puzzles.

Speaking of the number 16, here’s my favorite toothpick problem.

Remove 4 toothpicks to leave 4 small triangles.

And finally, because I know you came here looking for jokes, here is the only joke I know that involves toothpicks. It’s rather disgusting. Continue at your own risk.

One night, as a bartender is closing his bar, he hears a knock at the back door. It’s a math graduate student. “Can I have a toothpick?” he asks. The bartender gives him a toothpick and closes the door.

Five minutes later, another knock. “Can I have another toothpick?” asks the student. The bartender gives him another one and closes the door.

Five minutes later, a third knock. “Can I have a straw?” asks the student.

“Sure,” says the bartender, and hands him a straw. “But what’s going on out there?”

“Some lady threw up in the back,” says the grad student, “but all the good stuff is already gone.”

The Math Jokes 4 Mathy Folks blog is an online extension to the book Math Jokes 4 Mathy Folks. The blog contains jokes submitted by readers, new jokes discovered by the author, details about speaking appearances and workshops, and other random bits of information that might be interesting to the strange folks who like math jokes.

## MJ4MF (offline version)

Math Jokes 4 Mathy Folks is available from Amazon, Borders, Barnes & Noble, NCTM, Robert D. Reed Publishers, and other purveyors of exceptional literature.