Thinking > Symbols

As if having to wear a mask while flying isn’t scary enough, last week I had a harrowing experience.

See Valentin Kirilov’s full video at https://vimeo.com/156775315.

It’s normally a three-hour flight from Yellowknife in the Northwest Territories to Portland, but about 20 minutes into a recent flight, there was an unsettling noise followed by an announcement from the pilot: “Ladies and gentlemen, this is your captain. Please don’t be alarmed by the noise you just heard near the left wing. Let me assure you that everything is okay. We’ve lost an engine, but the other three engines are working just fine. They’ll get us to Portland safely. But instead of taking 3 hours, it’s going to take 4 hours.”

She said this matter-of-factly, as if she were commenting on the color of my shirt or pointing to a squirrel that was enjoying an acorn. It was reassuring that she didn’t seem worried — but still, we had lost an engine! The cabin grew quiet as we waited to see what would happen next. But, nothing. We continued to fly as if nothing had happened, and the din of conversation slowly resumed.

And then there was another noise near the right wing, followed by the pilot: “Ladies and gentlemen, this is your captain again. Unfortunately, we’ve lost another engine. But the two remaining engines are functioning properly. We’re going to get you to Portland safely, but now it’s gonna take about 6 hours.”

Needless to say, you could hear a pin drop as we passengers waited with baited breath. And sure enough, there was another noise, and the pilot again: “Ladies and gentlemen, we’ve lost a third engine, but we’ll make it safely to Portland with the one engine we’ve got left. It’s just gonna take 12 hours.”

That was when the guy in the seat next to me lost it. “I sure hope we don’t lose that last engine,” he yelled, “or we’re gonna be up here all day!”

That story isn’t true. None of it. Not. One. Single. Fact. I mean, come on! Yellowknife? Why would I need to go to Yellowknife? To brush up on my Dogrib?

But the following story is true.

My honorary niece Ivy occasionally calls for help with math. Not surprisingly, the frequency of calls has increased with online learning during the pandemic. This is a modified version of a question about which she recently called:

A plane traveling against the wind traveled 2,460 miles in 6 hours. Traveling with the wind on the return trip, it only took 5 hours. What was the speed of the plane? What was the speed of the wind?  

It’s assumed that the plane flew at the same average speed in both directions and that the wind was equally strong throughout. On a standardized test, those assumptions would need to be stated explicitly. But on this blog, I make the rules, and I refuse to add more words to explain reasonable assumptions without which the problem would be impossible to solve.

As presented to Ivy in an online homework assignment, the problem stated that x should represent the plane’s speed and y should represent the speed of the wind. My first question was, “Why? What’s wrong with p and w as the variables for plane and wind, respectively?” But my second, and perhaps more important, question was, “Why are they forcing algebra on a problem that is much easier without it?”

Somewhere, many years ago, someone decided that the methods of substitution and elimination were critically important, if not for success in life then at least for success in high school. Anecdotally, this seems to be true; I was a bad-ass systems-of-equations solver in high school, and I was also the captain of the track team that won the county title, so… yeah. Sadly, that someone seemed to put less stock in a slightly more critical skill: thinking.

Here’s the deal. Against the wind, the plane covered 2,460 miles in 6 hours, so the plane’s speed on the way out was 410 miles per hour. With the wind, the plane covered those same 2,460 miles in just 5 hours, so the plane’s speed on the way back was 492 miles per hour. Since the wind is constant in both directions, the unaided speed of the plane must have been 451 miles per hour, exactly halfway between the against-the-wind and with-the-wind speeds. Visually,

The top portion of that diagram represents the plane flying against the wind; the bottom portion represents the plane flying with the wind. This makes it obvious that the the unaided speed of the plane is equidistant from 410 miles per hour and 492 miles per hour, and that distance is the speed of the wind.

Algebraically, the problem might be solved like this…

which is no different than the logical approach above: x ‑ y is the speed of the plane against the wind, and x + y is the speed of the plane with the wind, so the numbers are obviously the same. But using algebra just seems so much more… cumbersome.

Students in elementary classrooms often explain their thinking with drawings and diagrams. Unfortunately, some of that great thought is abandoned when students take a course called Algebra, and the mechanical skill of symbolic manipulation replaces the mathematical skill of logical reasoning. That’s a tragedy. As Jim Rubillo, former executive director at NCTM, used to say, “We have to stop being so symbol‑minded.”

Our Library’s Summer Math Contest

Every summer, our local library runs a contest called The Great Big Brain Game. Young patrons who solve all of the weekly puzzles receive a prize. The second puzzle for Summer 2017 looked like a typical math competition problem:

Last weekend, the weather was perfect, so you decided to go to Cherry Hill Park. When you got there, you saw that half of Falls Church was at the park, too! In addition to all the people on the playground, there were a total of 13 kids riding bicycles and tricycles. If the total number of wheels was 30, how many tricycles were there?

A tricycle has 3 wheels. (Duh.)

First, some comments about the problem.

1. I dislike using “you” in math problems. I believe it’s a turn-off to students who can’t see themselves in the situation described. There are enough reasons that kids don’t like math. Why give them another reason to shut down by telling them that they went somewhere they didn’t want to go or that they did something they didn’t want to do?
2. Word problems are not real-world just because they use a local context, and this one is no exception. This problem attempts to show an application for a system of linear equations, but true real-world problems don’t have all the information neatly packaged like this.
3. Wouldn’t the person posing this problem already have access to the information they seek? That is, if she counted the number of kids riding bikes and the total number of wheels, couldn’t she have just counted the number of bicycles and tricycles instead? It has always struck me as strange when the (implied) narrator of a math problem wants you to figure out something they already know.

All that said, this was meant to be a fun puzzle for a summer contest, and I don’t mean to scold the library. I don’t know that I’d use this puzzle in a classroom — at least, not presented exactly like this — but I love that kids in my town have an opportunity to do some math in June, July, and August.

Now, I’ll offer some comments on the solution. In particular, the solution provided by the library was different than the method used by one of my sons. Here’s what the library did:

Imagine that all 13 kids were on bicycles with 2 wheels. That would be a total of 26 wheels. But since 30 wheels are needed, there are 4 extra wheels. If you add each of those extra wheels to a bicycle, that’ll create 4 tricycles, leaving 9 bicycles. So, there must have been 4 tricycles at Cherry Hill Park.

And here’s what my son did:

If you can’t see what he wrote, he created a system of two equations and then solved it:

2a + 3b = 30
a + b = 13

a + 2b = 17
13 – b + 2b = 17
b = 4

2a + 12 = 30
2a = 18
a = 9

That’s all well and good. In fact, it’s perfect if you want to assess my son’s ability to translate a problem and solve a system of equations. But I have to admit, I was a little disappointed. What bums me out is that he went straight to a symbolic algorithm instead of considering alternatives.

I think I know the reason for this. This past year, my son was in a pull-out math program, in which he studied math with someone other than his regular classroom teacher. In this special class, the teacher focused on preparing him to take Algebra II in sixth grade when he enters middle school. Consequently, students in the pull-out class spent the past year learning basic algebra. My fear is that they focused almost exclusively on symbolic manipulation and, as my former boss liked to say, “Algebra teachers are too symbol-minded.”

A key trait of effective problem solvers is flexibility. That type of flexibility comes from solving many problems and filling your toolbox with a variety of strategies. My worry — and this isn’t just a concern for my son, but for every math student in the country — is that students learn algorithms at the expense of more useful problem-solving heuristics. What happens when my son is presented with a problem that can’t be translated into a system of linear equations? Will he know what to do when he doesn’t know what to do?

The previous pull-out teacher said that when she presented my sons with problems that they didn’t know how to solve, their eyes would light up. They liked the challenge of doing something they hadn’t done before. I’m hopeful that this enthusiasm isn’t lost as they proceed to higher levels of mathematics.

Think of a Number

I love to create math games almost as much as I love to play them.

My favorite professional project was leading the development of Calculation Nation. And my favorite game on the site is neXtu, though other games on the site may promote more sophisticated mathematical thinking.

I have many reasons to love my wife, not least of which is her creation of the game Dollar Nim. While I can’t take credit for the rules, I will take credit for its analysis and its popularization. (What do you call a wife who makes up a game that gets you a publication credit? A keeper!)

Recently, I’ve been frustrated by the lack of games for teaching algebra. I’ll give props to the good folks at Dragonbox, which uses a game environment to teach algebra. But I’m not yet convinced that it leads to deep algebraic understanding; even they admit “to transfer to pencil and paper, children must be explained how to rewrite equations line by line.” They also claim that “in-house preliminary tests indicate a very high level of transfer to pencil and paper,” but that’s the fox watching the henhouse.

So I’ve been thinking about games I can play with my sons that will allow them to engage in algebraic thinking. But I don’t want them to know they’re engaging in algebraic thinking. I have two criteria for all math games:

• The game mechanics depend on mathematics. The math is not tangential to the game; it is the game.
• Kids don’t realize (or at least they don’t care) that it’s a math game, because it’s fun.

It pains me to write that second criterion, because math is fun. But I know not everyone shares that opinion. So I do my best to disguise any math learning in the game and then, when they least expect it — BOOM! — I drop the bomb and show them what they’ve learned.

So here’s a game I recently devised.

• Player A chooses a number.
• Player B chooses two operations for Player A to perform on the number.
• Player A performs those operations and then tells the result to Player B.
• Player B then tries to identify Player A’s number.

These rules leave something to be desired, since Player B could simply ask A to “multiply by 1” and then “add 0,” in which case finding A’s number would involve no work whatsoever. To be a stickler, an additional rule could impose that either addition or subtraction can be used exactly once and that no operation can involve either 0 or 1. In a middle school classroom, I suppose I would state such a rule explicitly; for playing this game with my seven-year-old sons, I opted not to.

We played this game three times on the car ride to school yesterday. One game went like this:

• I thought of a number (14).
• Eli asked me to add 3 to my number.
• Alex asked me to multiply by 3.
• I told them the result: 51.

Eli then guessed that my number was 16. He had subtracted 3, then divided by 3.

“No!” said Alex. “You added 3 first, so you need to subtract 9.”

“Why 9?” Eli asked. “Daddy only added 3.”

“But he multiplied by 3, so if you subtract first, you have to subtract 3 × 3.”

Eli then realized that my number was 14.

He thought for a second. “Oh,” he said. “I should have divided by 3 first, then subtracted.”

Wow, I thought. This is going even better than I hoped.

Though they didn’t use the proper terminology, the boys had a great discussion about “undoing” operations by performing inverse operations in reverse order. In 10 minutes, they taught themselves how to solve a two-step equation:

3x + 3 = 51

Grace Kelemanik once said that she knew she was being effective when she didn’t have to say a word. She’d watch from the back of the room as students carried the conversation and guided one another to correct mathematical thinking.

I will never claim to be half the educator that Grace Kelemanik is. But yesterday morning, I was pretty darn effective.

I’d love to hear about math games you’ve played with kids, whether you invented them or not.

Peanut Distribution

When we recently bought honey roasted peanuts at the grocery store, Eli speculated that there were 215 peanuts in the jar.

“I think there are less,” Alex said. “My guess is 214.”

“Okay, so now we have to count them,” Eli said.

“No,” I said, explaining that I didn’t want them touching food that others would be eating. I then showed them the back of the jar, which said that one serving contained about 39 pieces and the whole jar contained about 16 servings. They knew that 39 × 16 would approximate the number of pieces, and they estimated that the jar contained 40 × 15 = 600 pieces.

But then they wanted the actual value, and I wondered how we could use the estimate to find the exact product. More importantly, I wondered if it was possible to find an algorithm that would allow an easily calculated estimate to be converted to the exact value with some minor corrections.

My sons’ estimate used one more than the larger factor and one less than the smaller factor; that is, they found (m + 1) × (n – 1) to estimate the value of mn. A little algebra should help to help to provide some insight.

The product had a value of 600, so further refinement led to:

This led to an algorithm:

1. Find an estimate with nice numbers.
2. Add 1.
3. Add the larger factor.
4. Subtract the smaller factor.

This gives 600 + 1 + 39 – 16 = 624. And sure enough, 39 × 16 = 624.

This method works any time you want to find the exact value of a product when the larger factor is one more than a nice number and the smaller factor is one less than a nice number. Just estimate with the nice numbers, then follow the steps. The method can be modified if the larger factor is one less than a nice number and the smaller factor is one more than a nice number:

1. Find the estimate.
2. Add 1.
3. Subtract the larger factor.
4. Add the smaller factor.

So if you want to find the product 41 × 14, then the larger factor is one more than 40 and the smaller factor is one less than 15. The estimate is again 40 × 15 = 600.

Then 600 + 1 – 41 + 14 = 574. And sure enough, 41 × 14 = 574.

The same idea can be extended to numbers that aren’t the same distance from nice numbers. But that’s not the point. The intent was not to find general methods for every combination; instead, the hope was to use an easily calculated estimate as the basis for an exact calculation. I’m not sure this method completely succeeds, but it was fun for an afternoon of mental gymnastics.

Wonders of the Math World

It was Dr. Seuss who said…

Think! Think and wonder. Wonder and think.
How much water can 55 elephants drink?

These are the things about which I think and wonder.

Ever wonder why pizza is circular, delivered in a square box, and served in triangular slices? Weird.

Divorce is like algebra.
Have you ever looked at your x and wondered y?

Ever wonder why textbook authors write problems about people who buy 58 dozen eggs at the grocery store? Or why their editors don’t edit them?

How Much Does Your Name Cost?

Here’s a contrived yet fun math problem that I shared with my sons recently:

A local hardware store sells bronze letters. However, the letters vary in price; some are more expensive than others. When I was at the store the other day, four people purchased the letters in their names. Their names and the prices they paid were:

Aiden $491  •  Ned $225  •  Dane $399  •  Ed $135

The price of a name is equal to the sum of the prices of its letters. The price for uppercase and lowercase letters is the same, and there is no additional surcharge or tax. How much would the following people pay to buy the letters in their names?

Edna  •  Ian  •  Nadine

Those of you who know a little algebra will have no trouble with that problem. Those of you who don’t shouldn’t have too much trouble, either.

But then, I realized I could extend the problem for some added fun. And who am I to keep fun things to myself? So, here ya go.

I saw this sign in a window the other day:

At first, I thought the store was engaging in human trafficking. But then I realized that $269 was the price for the bronze letters that had been used to spell the name Eli. Inside the store was a price list for other names:

AIDEN – 491	AL – 248	ART – 267	BEA – 290
EARL – 415	DANE – 399	ED – 135	ELI – 269
FAY – 220	GABI – 289	HAL – 284	IVY – 143
JACK – 234	JAY – 232	KO – 60	KAI – 283
LEXI – 272	MAVIS – 363	MAX – 215	NED – 225
PAT – 210	PERRI – 330	QI – 93	QUIN – 199
SAMMY – 338	WILL – 243	ZENO – 243

The store didn’t have a list of prices for the individual letters, but then I realized that I didn’t need one. From the table above, I could figure out how much my name  would cost.

Can you figure out how much your name would cost?

You can download both of these problems for use in a classroom (or at a mathy party) from the following link:

Name Letters (PDF)

And while I don’t believe in answer keys, you can check your work by using the form found on this page.

Name Letter Form

For what it’s worth, the longest name ever — according to Wolfe + 585, Senior, who has a pretty long name himself — is Rhoshandiatellyneshiaunneveshenk Koyaanisquatsiuth Williams. Her entire name name would have cost $4,073 at this store — an astounding $2,359 for her first name, $1,119 for her middle name, and a veritable bargain at $595 for her tame-by-comparison last name. (Incidentally, this is the name that appeared on her birth certificate. As the story goes, her father later increased her first name to 1,019 letters and added an additional 36 letters to her middle name. You know… just in case the name wasn’t long or unique enough already.)

Mathematical Finances

Got a bead of sweat running down your forehead as you frantically race to complete your 1040? Here are a few math finance jokes to relieve the stress.

Financial Trigonometry: If someone asks you to cosine, don’t sine! Instead, go off on a tangent! That’ll save you $40,000!

Financial Algebra: My wife leaves Houston at 8:39 a.m. on a plane bound for Albuquerque. She arrives at 9:42 a.m. and spends the next three days at a hotel with my best man. If she then decides to leave me for him, how long will it take me to pay off the Visa bill from this trip of infidelity, assuming an annual percentage rate of 18.5%? 

Financial Formula: Easiest way to determine your cost of living? Take your income, then add 10%.

And just in case you needed another reason to never trust a financial mathematician…

A pure mathematician asks, “Would $30,000 be too much?”

An applied mathematician asks, “How about $60,000?”

And a financial mathematician says, “How about $300,000? That’d be $135,000 for me, $135,000 for you, and $30,000 for a pure mathematician to do the work.”

A New Tattoo

I was both nervous and excited when I walked into the tattoo shop. I had been working extra hours in the tutoring center to save for this, and the moment had finally arrived. Sure, I could have played it safe and asked for a tribal pattern on my upper arm. But I’ve always lived by the mantra, “Go big, or go home.”

“What are we doing today?” the tattooist asked.

“How about x + 6 across my left cheek?” A blank stare. “You know, like from algebra,” I explained.

I got the sense he wanted to ask some follow-up questions… but then decided against it. Instead, he started inking. Twenty minutes later, he held up a mirror so I could see his handiwork.

“You don’t like it, do you?” he asked.

“No, no, I love it!” I said. “What makes you think I don’t like it?”

“I can tell by the expression on your face.”

Math Dreams

“I can’t get the numbers to stop.”

That’s what Eli told my wife tonight when he woke from a bad dream.

I once had a dream where I was afraid of numbers, too. During the week leading up to my midterm in Linear Algebra, I was reviewing problems from old exams. Though I thoroughly understood all that had been covered in class, and though I was able to complete all the exercises from the textbook with aplomb, I was only getting about 25% of the questions on the old exams correct.

It was freaking me out, and the night before the exam, I went to sleep very nervous.

My sleep was broken by a nightmare in which numbers were flying past my head like cannonballs from a numerical howitzer. They were coming from every direction. To make matters worse, my head was being squeezed by the brackets of a matrix as if it were in a vice.

I woke in a cold sweat. It was 5 a.m. Too anxious to sleep any more, I went to a study carrel in the library where I read and re-read the textbook and continually tried the problems from the old exams. I skipped my morning classes and studied for five straight hours. Yet I was still only able to get a quarter of the problems correct.

The midterm was at 12:30 p.m. At 11:55 a.m., I saw a woman from my class. Though I had never spoken to her before, I approached her and said, “Excuse me, aren’t you in Dr. Sibley’s linear algebra course?”

“Yes,” she replied skeptically.

“Would you mind helping me?” I asked, embarassed by the question.

“Uh… sure,” she said.

I asked her a few questions about the topics we had covered. She confirmed that I understood the material correctly. “So why am I missing so many of the questions from the old exams?” I asked rhetorically.

She took a look at my work and confirmed that it, too, was correct. Then she revealed the punch line: The professor had given us a packet of six exams, and the answer keys for all six exams were copied onto a seventh sheet. But the answer keys and the exams were not in the same order — each answer key had to be matched to an exam by noting the semester and date on each.

Arrgh.

I was relieved. Yet frustrated. I managed an 85 on the midterm… yet I’m certain I would have aced it had I gotten a good night’s sleep.

Mark Jason Dominus has much better math dreams than I. He posted the following problem to the mattababy listserv. He claimed that it came to him in a dream; when he woke, he thought it was still a good problem, so he decided to share it.

The volume of a 3 × 3 × 3 cube is 27 cubic units, and the volume of a 2 × 2 × 1 prism is 4 cubic units. Theoretically, six prisms should be able to fit inside the cube, with three cubic units empty. But can you arrange six 2 × 2 × 1 prisms so they fit inside a 3 × 3 × 3 cube?

Friends of mine who taught at the Center for Talented Youth claim that M. J. Dominus would often arrive late to the evening study sessions. As the story goes, he would take naps after dinner… if his alarm sounded while he was in the middle of a math dream, he would shut off the alarm and return to sleep so he could finish whatever mathematical proof his subconscience had been working on.

Wow.

What’s the scariest or coolest math dream you’ve ever had?

Algebra, a Symbol-Minded Pursuit

It is reported that when Augustus DeMorgan was asked his age, he responded algebraically:

I was x years old in the year x².

From personal experience, I can assure you that responding to a simple question with an algebra problem is no way to make friends. But perhaps DeMorgan had better success with this tactic than I.

As it turns out, there is a similar fact regarding my age and year of birth.

In the year x², my age will x years with the digits of x reversed.

The following anonymous quotation would, I suspect, meet with DeMorgan’s approval:

The human mind has never invented a labor-saving machine equal to algebra.

Almost everyone has an opinion about algebra, and most people have expressed their opinion without anonymity.

• Stand firm in your refusal to remain conscious during algebra. In real life, I assure you, there is no such thing as algebra.
  Fran Lebowitz
• One person’s constant is another person’s variable.
  Susan Gerhart
• [My algebra teacher] kept putting problems up on the board. I just kept following her and erasing the problems. Then she yells at me. I’m like, “Number 1, I like to attack the problem, not the person. That’s the first rule of problem solving. And B, you kinda seem like you’re a trouble maker, because you got to come up with all these fake problems, and it’s really cutting into our pizza time.” And she’s like, “You can’t list things 1 and then B. It’s 1 and 2, or A and B.” And I’m like, “Oh, you don’t like it when I mix numbers and letters together? Like you do in algebra, you hypocrite?”
  Mike Vecchione