For the Love of Math

math love

Photo courtesy of Keng Susumpow

Last Friday, my family adopted a sweet, little poodle puppy, named Zipper. The foster mother, Sally, had brought him from a Mexico shelter to her own home in Silver Springs, Md. During the home visit on Friday, we talked about our careers, and I mentioned that I write about math. That’s when she told me about her neighbor, the mathematician and novelist.

“You two should meet!” she said. Apparently, we have some of the same ideas about math.

Well, I did “meet” Manil Suri today, via the pages of the New York Times op-ed section. His excellent piece, “How to Fall in Love with Math” points out some ideas I’ve been extolling for years — along with a couple that I might have said were hogwash a couple of weeks ago.

As a mathematician, I can attest that my field is really about ideas above anything else. Ideas that inform our existence, that permeate our universe and beyond, that can surprise and enthrall.

Yes, yes, and again I say, yes! Mathematics is not exclusively about numbers. Hell, arithmetic is only a teeny-tiny fraction of what mathematics really is. Mathematics is the language of science. It’s a set of systems that allow us to categorize things, so that we can better understand the world around us.

Math is a philosophy, which I guess is what makes us math geeks really different from the folks who are merely satisfied with knowing how to reconcile their accounting systems or calculate the mileage they’re getting in their car. We mathy folks are truly interested in the ideas behind math — not just the numbers.

Last week, I attended a marketing intensive, a workshop during which I outlined my current career and explored how I want to take things to the next level. I’m ready to think bigger, and I need a plan to get me there.

The other entrepreneurs there thought there was real value in my creating a coaching service for entrepreneurs. My services would center around the numbers that these folks need to make their businesses survive and thrive. Marketing numbers, sales numbers, accounting numbers. They resisted the word “math” and advised me to really underscore the numbers.

From a purely marketing standpoint, I completely get it. I don’t have so much of a math wedgie that I can’t understand that the word “numbers” may be less threatening than “math.” So why not just go for it?

But the entire process left me thinking about what it is that draws me to mathematics. And ultimately what will drive me in a career, what moves me to get up in the morning and say, “Let’s go!” If you’ve been around here long, you know that it ain’t the numbers, sisters and brothers.

At the same time, I can’t say that I love math. But maybe that’s semantics, too. For the last two years, I’ve said that I’m attracted to how people process mathematics. But isn’t that just philosophy? So, isn’t that just math? This is what Suri had to say:

Despite what most people suppose, many profound mathematical ideas don’t require advanced skills to appreciate. One can develop a fairly good understanding of the power and elegance of calculus, say, without actually being able to use it to solve scientific or engineering problems.

Think of it this way: you can appreciate art without acquiring the ability to paint, or enjoy a symphony without being able to read music. Math also deserves to be enjoyed for its own sake, without being constantly subjected to the question, “When will I use this?”

At first, I disagreed with Suri’s thesis that math is worth loving — for math’s sake alone. But his analogy here is right on target. I couldn’t paint my way out of a paper bag, but each and every time I see “Starry, Starry Night” at MOMA, I catch my breath.

We come back to a failure to educate, as Suri so wonderfully elucidates in his piece. When we allow people who hate — or don’t appreciate — math to teach the subject, well, does anyone think that’s a good plan?

At any rate, I hope you’ll take a look at Suri’s piece. Meantime, I’m going to reach out to him to share my appreciation of math. Maybe there is a way — beyond teaching — for me to make a living as a math evangelist.

What do you think? Do you notice a difference between mathematics and numbers? Have you changed your mind about math in recent years or month? Please share!


Saving Face: Avoiding performance math

performance math

Photo courtesy of jin.thai

If there’s one thing most folks assume about me, it’s this: That I am some sort of mathmagician, able to solve math problems in a single bound — quickly, in public and with a permanent marker.

Nothing could be farther from the truth.

I don’t like what I call performance math. When I’m asked to divvy up the dinner tab (especially after a glass of wine), my hands immediately start sweating. When friends joke that I can find 37% of any number in my head, I feel like a fraud. I’m not your go-to person for solving even the easiest math problem quickly and with little effort.

Truth is I really cannot handle any level of embarrassment. And I’m very easily embarrassed. I’m the kind of person who likes to be overly prepared for any situation. This morning, before contacting the gutter company about getting our deposit back because they hadn’t shown up, I had to re-read the contract and literally develop a script in my head. What if I misunderstood something and was — gasp! — wrong about the timeline or terms of our contract?

Oh yeah, and I hate being wrong. About anything.

In short, I’m not much of a risk taker. Unlike many of my friends and some family members, I can’t stand the thought of failing publicly. Imagine writing a math book with this hang up! Thank goodness for two amazing editors, who checked up behind me.

I’m also not a detailed person. Not one bit. I’m your classic, careless-mistake maker — from grade school into grownuphood. I’m much more interested in the big picture, and I am easily lured by the overreaching concepts, ignoring the details that can make an answer right or wrong.

For years and years, I worried about this to no end. How could I be an effective teacher, parent, writer, if I didn’t really care about the details or I had this terrible fear of doing math problems in public? What I learned very quickly in the classroom was this: Kids needed a math teacher like me, to show them that failing publicly is okay from time to time and that math is not a game of speed or even absolute accuracy. (It’s never a game of speed. And it’s frequently not necessary to have an exact answer.)

Two weeks ago, as I sat down with my turkey sandwich at lunch, the phone rang. It was a desperate writer friend who was having some trouble calculating the percentage increase/decrease of a company’s revenue over a year. (Or something like that. I forget the details. Go figure.) She really, really wanted me to work out the problem on the phone with her, and I froze. I felt embarrassed that I couldn’t give her a quick answer. And I worried that I would lose all credibility if I didn’t offer some sage insight PDQ.

But since I have learned that math is not a magic trick or a game of speed, I took a deep breath, gathered my thoughts and asked for some time. Better yet, I asked if I could respond via email, since I’m much better able to look at details in writing than on the phone. I asked her to send me the information about the problem and give me 30 minutes to get back with her.

Within 10 minutes, I had worked out a system of equations and solved for both variables. She had her answer, and I could solve the problem without the glare of a spotlight (even if it was only a small spotlight).

My point is this: Math isn’t about performing. If you like to solve problems in your head or rattle off facts quickly or demonstrate your arithmetic prowess at cocktail parties, go for it. That’s a talent and inclination that I sometimes wish I had. But if you need to retreat to a quiet space, where you can hear yourself think and try out several methods, you should take that opportunity.

Anyone who criticizes a person’s math skills based on their ability to perform on cue is being a giant meanie. And that includes anyone who has that personal expectation of himself. There’s no good reason for math performance — well, except for Mathletes, and those folks have pretty darned special brains.

Do yourself a favor and skip math performance if you need to. I give you permission.

Do you suffer from math performance anxiety? Where have you noticed this is a problem? And how have you dealt with it?

The Problems with PEMDAS (and a solution)

The Problem with PEMDAS

Photo courtesy of James Lee

If you’re on Facebook, you’ve probably seen one of a variety of graphics like the one below:

The idea is solve the problem and then post your answer. From what I’ve observed, about half of the respondents get the answer correct, while the other half come to the wrong answer. The root of this problem? The order of operations.

Unlike reading English, arithmetic is not performed from left to right. There is a particular order in which the addition, subtraction, multiplication and division (not to mention parentheses and exponents) must be done. And for most of us old timers, that order is represented by the acronym PEMDAS (or its variations).

P – parentheses
E – exponents
M – multiplication
D – division
A – addition
S – subtraction

I learned the mnemonic “Please Excuse My Dear Aunt Sally” to help me remember the order of operations.

The idea is simple: to solve an arithmetic problem (or simplify an algebraic expression), you address any operations inside parentheses (or brackets) first. Then exponents, then multiplication and/or division and finally addition and/or subtraction.

But there really are a lot of problems with this process. First off, because multiplication and division are inverses (they undo one another), it’s perfectly legal to divide before you multiply. Same thing goes for addition and subtraction. That means that PEDMAS, PEDMSA and PEMDSA are also acceptable acronyms. (Not so black and white any more, eh?)

Second, there are times when parentheses are implied. Take a look:

The PEMDAS Problem

If you’re taking PEMDAS literally, you might be tempted to divide 6 by 3 and then 2 by 1 before adding.

The Problem with PEMDAS

Problem is, there are parentheses implied, simply because the problem includes addition in the numerator (top) and denominator (bottom) of the fraction. The correct way to solve this problem is this:

The Problem with PEMDAS

So in the end, PEMDAS may cause more confusion. Of course, as long-time Math for Grownups readers should know, there is more than one way to skin a math problem. Okay, okay. That doesn’t mean there is more than one order of operations. BUT really smart math educators have come up with a new way of teaching the order of operations. It’s called the Boss Triangle or the hierarchy-of-operations triangle. (Boss triangle is so much more catchy!)

The PEMDAS Problem


The idea is simple: exponents (powers) are the boss of multiplication, division, addition and subtraction. Multiplication and division are the boss of addition and subtraction. The boss always goes first. But since multiplication and division are grouped (as are addition and subtraction), those operations have equal power. So either of the pair can go first.

So what about parentheses (or brackets)? Take a close look at what is represented in the triangle. If you noticed that it’s only operations, give yourself a gold star. Parentheses are not operations, but they are containers for operations. Take a look at the following:

The Problem with PEMDAS

Do you really have to do what’s in the parentheses first? Or will you get the same answer if you find 3 x 2 first? The parentheses aren’t really about order. They’re about grouping. You don’t want to find 4 + 3, in this case, because 4 is part of the grouping (7 – 1 x 4).  (Don’t believe me? Try doing the operations in this problem in different order. Because of where the parentheses are placed, you’re bound to get the correct answer more than once.)

And there you have it — the Boss Triangle and a new way to think of the order of operations. There are many different reasons this new process may be easier for some children. Here are just a few:

1. Visually inclined students have a tool that suits their learning style.

2. Students begin to associate what I call the “couple operations” and what real math teachers call “inverse operations”: multiplication and division and addition and subtraction. This helps considerably when students begin adding and subtracting integers (positive and negative numbers) later on.

3. Pointing out that couple operations (x and ÷, + and -) have equal power allows students much more flexibility in computing complex calculations and simplifying algebraic expressions.

Even better, knowing about the Boss Triangle can help parents better understand their own child’s math assignments — especially if they’re not depending on PEMDAS.

So what do you think? Does the Boss Triangle make sense to you? Or do you prefer PEMDAS? Share your thoughts in the comments section.

Parlez-Vous Mathematics? Math as a foreign language

Photo courtesy of theunquietlibrarian

In redesigning my blog, I’ve read a lot of the posts I’ve written over the last year. In fact, take a look at this math: On average, I’ve written 13 blog posts each month or 164 posts (counting this one) since last May. And so I decided to repost this one, in honor of Math Awareness Month, which addresses the language of math.

When I was in college, majoring in math education, I learned that math is the language of science.  In fact, we called it the Queen of the Sciences.  (You’d better believe that gave me a sense of superiority over the chemistry and physics majors!)  And yeah, I think that the math I was doing then–calculus, differential equations, statistics and even abstract algebra–is mostly useful for describing some kind of science.

In some ways, everyday math is also the language of science.  Home cooks use ratios to ensure that their roux thickens a gumbo just right.  With proportions, gardeners can fertilize their vegetable beds without burning the leaves from their pepper plants.  And a cyclist might employ a bit of math to find her rate or the distance she’s biked.

But I think too often we adults get caught up in the nitty gritty of basic math and lose the big picture.  This is when many of us start to worry about doing things exactly right–and when math feels more like a foreign language, rather than a useful tool.

Earlier this week, I read a blog post from Rick Ackerly, who writes The Genius in Children, a blog about the “delights, mysteries and challenges of educating our children.”  In Why Mathematics is a Foreign Language in America and What to Do about It, he writes:

Why do Americans do so badly in mathematics? Because mathematics is a foreign language in America. The vast majority of children grow up in a number-poor environment. We’ve forgotten that the language of mathematics is founded in curiosity.  We too often think of mathematics as rules rather than as questions.  This is like thinking of stories as grammar.  Being curious together can be a really special part of the relationship in families.

These Stevendotted ladybugs are not wrestling. Photo credit: Andr Karwath

And I couldn’t agree more.  For all of you parents and teachers out there: how many questions do your kids ask in one day?  10? 20? 100? 1,000?  As Ackerly points out, especially younger children are insatiably curious.  They want to know why the sky is blue and what makes our feet stink and how come that ladybug is on top of the other ladybug.

A full 90% of the time, we can’t answer their questions. Or maybe we just don’t want to yet.  (“That ladybug is giving the other one a ride.”)  With Google‘s help, we can find lots of answers.  But how often are we asked a math-related question–by a kid or a grownup–and freeze?

For whatever reason, many people are afraid to be curious about math.  Or they’ve had that curiosity beaten out of them.  I think that’s because don’t want to be wrong.  As fellow writer, Jennifer Lawler said to me the other day:

It’s funny because when I make a mistake in writing—a typo, etc.—I let myself off the hook (“Happens to everyone! Next time I’ll remember to pay more attention.”) But if I misadd a row of numbers I’m all “OMG, I’m such an idiot, and everyone knows I’m such an idiot, I can’t believe they gave me a college degree, and why do I even try without my calculator?”

The same goes for answering our kids’–or our own–calls of curiosity.

So what if we decided not to shut down those questions?  What if it was okay to make some mistakes?  What if we told our kids or ourselves, “I don’t know–let’s find out!”  This could be a really scary prospect for some of us, but I invite you to try.

What’s keeping you from being curious about everyday math? What do you you think you can do to change that?  Or do you think it doesn’t matter one way or the other?  Share your ideas in in a comment.

Our first Math Treasure Hunt winner is Marcia Kempf Slosser! Congratulations Marcia, you’ve won a copy of Math for Grownups (or if you already have a copy, I’ll send you a gift card). Want to enter? All you need to do is find an example of the daily clue, which is announced on the Math for Grownups Facebook page each day. 

Homework Help: 4 middle school math facts you probably forgot

Photo courtesy of .raindrops.

Every so often, at around 7:00 p.m., I’ll get a call from someone I know.  “I don’t understand my kid’s math homework,” they’ll say.

These folks aren’t dumb or bad at math.  But almost always, they’ve hit a concept that they used to know, but don’t remember any more.  And those things can trip them up — big time. So, I thought it might be helpful to review 4 middle school math facts that may give parents trouble.

Every number has two square roots.

This is the question that prompted this blog post.  I got a call from a friend who didn’t understand this question in her daughter’s math homework: “Find both square roots of 25.”  Both?

Most adults have probably forgotten that each number has two square roots. That’s because we are typically only interested in only one of them — the positive one.

Yep, the square roots of 25 are 5 and -5.  In other words:

[pmath]sqrt{25}[/pmath] = 5 and -5

It should be pretty easy to see why this is true.  (You just have to remember that when you multiply two negative numbers, your answer is positive.)

5 · 5 = 25

-5 · -5 = 25

1 is not prime.

This question came up in my own daughter’s homework last week — a review of prime and composite numbers.  Remember, prime numbers have only two factors, 1 and the number itself.  So, 7 is prime.  And so are 13, 19 and even 3.  But what about 1?

Well, it turns out the definition of a prime number is a little more complicated than what we may assume.  And I’m not even going to get into that here.

But there is a way for less-geeky folks to remember that 1 is not prime. Let’s look at the factors of each of the prime numbers we listed above:

7: 1, 7

13: 1, 13

19: 1, 19

3: 1, 3

Now, what about the factors of 1?

1: 1

Notice the difference?  Prime numbers have two factors, 1 and the number itself.  But 1 only has one factor.

0 is an even number.

This idea seems to trip up teachers, students and parents.  That’s because we tend to depend on this definition of even: A number is even, if it is evenly divisible by 2.  How can you divide 0 into two equal parts?

It might help to think of the multiplication facts for 2:

2 x 0 = 0

2 x 1 = 2

2 x 2 = 4

2 x 3 = 6 …

All of the multiples of 2 are even, and as you can see from this list, 0 is a multiple of 2.

Anything divided by 0 is undefined.

Okay, this gets a little complex, so bear with me.  (Of course, if you want, you can just memorize this rule and be done with it.)

First, we can describe division like this:


Using a little bit of algebra you can get this:

r · b = a

So, what if b = 0?

r · 0 = a

That only works if a is also 0, and 0 ÷ 0 gives us all kinds of other problems.  (Trust me on that.  This is where things get pretty darned complicated!)

So how many of you have thought while reading this, “I will never use this stuff, so what’s the point?” You may be right.  Knowing that 0 is an even number is probably not such a big deal.  But at least your kid will think you’re extra smart, when you can help him with his math homework.

What are your math questions?  Is there anything that’s been bugging you for ages that you still can’t figure out?  Ask your questions in the comments section.  I’ll answer some here and create entire posts out of others.