creative math


You really don’t have to know or care what “binary trees” are to appreciate Vi Hart’s genius.  And I’m so excited to finally introduce you all to her.

Vi calls herself a “recreational mathematician.”  In other words, she plays with math, and it’s really amazing stuff.  Just a couple of years ago, she graduated from Stony Brook University, with a degree in music.  (Her senior project was a seven-movement piece about Harry Potter.)  Before that, she got hooked on math when her father took her to a computational geometry conference.  (George W. Hartis now chief of content for the soon-to-open Museum of Mathematics in Manhattan.)

In short, she’s not a trained math geek.  She just loves math.

She’s also funny and infectious.  I dare you to watch this video and not laugh.  And nope, you don’t have to know what binary trees are to get the jokes.  (Psst, you don’t even have to love math to love Vi.)

I’ll post more of Vi’s awesome videos in weeks to come.  Let me know what you think in the comments section!

When I was a camp counselor after my sophomore year of college, I had a standard response to kids who asked, “Do I have to?”  Whether they were complaining about sweeping out the cabin or taking a hike, I’d look them in the eye, smile and say, “No. You get to!”

I wasn’t a teacher yet, but I had this instinct to spin complaints into commendations.  Sometimes this worked.  The hikes were a good time, and even sweeping sometimes ended in fits of laughter or song.

But the more I think about math and grownups, the more I think that this flip response doesn’t apply.  I do think math is fun — well, some math.  I love proofs, from the two-column geometry proofs that I did in high school to proving properties of our real number system.  I also love doing some kinds of algebra, like solving systems of equations with two variables.

But I don’t love all math.  Try as I might, probability still screws with my head.  And I honestly and truly despise logarithms. (Those are to solve for x, when the variable is an exponent.  More than likely, you haven’t seen logarithms in decades.)

The realization that math doesn’t have to be fun really hit home twice this past year.  When I wrote my proposal for Math for Grownups, the publisher offered positive feedback, except for one thing.  “Don’t focus on the fun of math,” my editor said.  “Focus on the fact that we need it.”  That was a real wake-up call for me.  I couldn’t say to my readers, “You don’t have to do this math; you get to!”

And this spring, I also served as an instructional designer for two online, high school math courses, Algebra II and Probability and Statistics.  This meant that I reviewed the lessons, looking carefully at the pedagogy and mathematics.  I could tell when I loved the math.  I was ready to work every day and genuinely didn’t want to stop until everything was finished.  But when I hit a unit that was less engaging for me, I stalled.  I looked for anything else I could be doing — laundry, cleaning out my email, visiting my favorite blogs.

I didn’t love all of the math I was doing.  Why should I expect that of anyone else?

That’s why I say that math doesn’t have to be your BFF.  It’s like making dinner every night.  Some people can’t wait to get their hands into some fresh bread dough or chop up onions or heat up the grill.  Others are satisfied with take-out.  And then there are plenty of us who are very happy somewhere  in the middle.

But we’ve all got to eat, whether we love cooking or not.  And we’ve all got to do math.  You don’t have to love it, but you can learn to tolerate it.

What do you love or hate about math?  Share your ideas in the comments section.

Over the last year, I’ve come across lots of great math-related videos, and now that my blog is up and book is out, people are sending me links to many more.  I thought Fridays would be a great time to share them.  So, welcome to the first edition of Film Fridays!

Today’s little clip comes courtesy of my mother-in-law, who majored in math and then went on to have a seriously incredible career as a sales representative for American Greetings.  She uses math like it’s a second language — no big deal, thankyouverymuch.  (She also makes the most amazing pies ever.)

Still, this clip is a bit geeky — as many math videos are.  What I encourage you to do, though, is find the artistry and magic.  There will be no quiz.  This is just for fun. (Details are below the clip.)

So while this looks absolutely magical, it really does boil down to some very simple math.  The length of the pendulum determines how far it swings, and that in turn determines how many swings (or oscillations) it can complete in a given period of time.  In plain English: a short pendulum swings faster than a long one.  So the smarty-pants at Harvard built this pendulum based on the design of University of Maryland physics professor, Richard Berg.  Here’s the nitty gritty, if you’re interested:

The period of one complete cycle of the dance is 60 seconds. The length of the longest pendulum has been adjusted so that it executes 51 oscillations in this 60 second period. The length of each successive shorter pendulum is carefully adjusted so that it executes one additional oscillation in this period. Thus, the 15th pendulum (shortest) undergoes 65 oscillations.

In other words: Pretty.

I am so excited to show you more videos!  I especially can’t wait to introduce you to Vi Hart, who does the most captivating math doodles you can imagine.  (Wait a minute, who else does math doodles?)  So check in next week. And if you have a video that you want to share, please send me the link: llaing-at-comcast-dot-net.

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.  [pullquote]We too often think of mathematics as rules rather than as questions.  This is like thinking of stories as grammar. — Rick Ackerly[/pullquote]

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.

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.

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

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.

I’m on the right track, baby

I was born this way

–Lady Gaga

It was day two of my second year of teaching high school geometry, and already I had been called for a parent meeting in the principal’s office. I was a bit worried.  What on earth could a parent have issues with already?

Mrs. X sat with her 14-year-old son across the desk from the principal.  I shook her hand and took the chair next to her.  The principal handed me a copy of my geometry class syllabus that I’d sent home with all of my students during the first day of class.  Like every other class syllabus at this particular school, mine included class rules, the grading system, a list of general objectives and the obligatory notice that I’d be following all other relevant objectives outlined by the Commonwealth of Virginia.

“Mrs. X has some questions about your syllabus,” he said, turning the meeting over to her.

“I don’t understand what this objective is,” Ms. X said, pointing to her copy of the syllabus and then reading aloud: “‘Students will use their intuitive understanding of geometry to understand new concepts.’  What does ‘intuitive’ mean?  Are you going to hypnotize my son?”

I instantly relaxed.  Clearly, I was dealing with an over-zealous, perhaps under-educated parent, who had been listening to too much right-wing radio (which in the early 1990s was railing against witchcraft in the classrooms).  I might think she was crazy, but I could handle this.

I calmly explained that all students come into my class with a basic understanding of shapes and the laws of geometry.  I needed my students to tap into this intuitive understanding so that we could build on skills they already had.

In short: These kids already knew something about geometry, and as a professional educator, I was going to take advantage of that.

What I didn’t realize was that my heartfelt theory was not proven fact.  But in April of this year, the Proceedings of the National Academy of Sciences published a study that does just that.  Here’s the gist:

Member of the Mundurucu tribe of Brazil (photo courtesy of P. Pica)

French researcher, Pierre Pica discovered that members of the Amazon Mundurucu tribe have a basic understanding of geometric principles–even though they aren’t schooled in the subject and their language contains very few geometric terms. In other words, geometry is innate.

In fact, Pica found that French and U.S. students and adults did not perform as well on the tests as their Mundurucu brethren.  Turns out formal education may get in the way of our natural abilities.

“Euclidean geometry, inasmuch as it concerns basic objects such as points and lines on a plane, is a cross-cultural universal that results from the inherent properties of the human mind as it develops in its natural environment,” the researchers wrote.

Bla, bla, bla, and something about points and lines.

Not to toot my own horn or anything, but what this means is I was right all those years ago.  We may not have been born with Euclid’s brain, but we do, at the very least, pick up his discoveries just by interacting with our world, rather than sitting in a high school classroom.

Actually, the philosopher Immanuel Kant said as much when he was doing his thing in the 18th century, so this isn’t a new idea at all.  But many students (and parents) didn’t get that memo.

The bottom line: aside from uncommon processing and learning differences, there’s no reason that you can’t do ordinary geometry.  More than likely, any obstacles you face are rooted in fear or stubbornness.

And I, for one, won’t let you get away with that.

Art and math are diametrically opposite, right?  Wrong.

Blossom, layering of enamel over silver. Photo credit: Hap Sakwa.

Shana Kroiz is a Baltimore-based, acclaimed jewelry designer and artist, whose work has been shown in the some of the country’s most esteemed galleries and museums, including The Smithsonian and the Museum of Arts and Design in New York City.  She’s kind of a big deal–and she does math!

When do you use basic math in your job?

Most days I contend with a variety of math problems, whether I’m measuring a piece or resizing a ring. I use wax to cast my designs, and so I have to convert the weight of wax into the the specific weight of the metal I am using. I also construct three-dimensional forms out of sheet metal, which requires some geometry. I have to know the sizes and weights of my pieces, so that they are not too heavy to be worn. When scoring and bending metal, I have to figure out the angle of my score lines in order to get the correct angle out of the sheet I am bending. Then there’s the business side of things: calculating the time it takes to make a piece with the cost of materials and the addition of any profit I need to make. Prices also have to be converted into a retail and wholesale values.

Do you use any tools to help with this math?

Yes, I use calculators, calipersdividers, scales and, of course, computers. They all help with precision and time management.

How do you think math helps you do your job better?

Without math, it is almost impossible to do precision work. I work with a lot of potentially dangerous chemicals, and the math involved keeps me safe.  Plus, if I mix the chemicals incorrectly, the result won’t be what I need.  Being precise with my math means that I can avoid having to do things over again.

How comfortable with math do you feel?

I do most of the same sorts of problems over and over, so I feel comfortable in the studio, and can teach to my students. But there are times when I wish I had a deeper or broader understanding of how to use math. Sometimes I think I take too long to find the answers to calculations.  If I understood how to use a different formula I might get to the answer faster.

Did you have to learn new skills in order to do this math?

Yes, but I had to work it out on my own. When I had a tangible need, I figure things out.

What kind of math did you take in high school?

I went through algebra and some geometry. And I didn’t feel like I was good at it at all! I could follow a problem if I had a model, but I did not have a good enough conceptual understanding of math to work out the formulas on my own. So I would say I was average at best, but I think if it had been taught in a way that I could understand I would have been much better.  I do think if math was taught with more useful applications, students would have an easier time learning, understanding and being engaged in math as a useful tool for life.

Each Monday, I feature someone who uses everyday math in their jobs.  If you would like to be featured (or if you know someone who you think should be featured), let me know at llaing-at-comcast-dot-net.  You can also catch up on previous Math at Work Mondays.

So tomorrow the world is supposed to end. Okay, not quite. The rapture will begin.  Apparently Earth won’t be destroyed until October 21. So you have some time to get your [email protected]*% together.

But this isn’t the only reason that the number 21 is significant. Here are some other interesting (in a pocket-protector kind of way) facts about this Very Important Number.

  1. 21 is the sum of the dots on a die: 1 + 2 + 3 + 4 + 5 + 6 = 21
  2. 21 is a Fibonacci number: 1, 1, 2, 3, 5, 8, 13, 21, …
  3. 21 is the third “star number.”
  4. 21 is smallest number of differently sized squares that are needed to tile a square.
  5. 21 is the legal drinking age in the U.S. (Not so math geeky, eh?)
  6. Blackjack baby!
  7. [pmath]{2^21}-21[/pmath] is a prime number.

Anything you want to add?  Or are you too busy packing your bags?

In last Friday’s Open Thread discussion, Gretchen posted this question:

My husband’s company does not provide health insurance for me and the kids, which is a $12,000 value. In his field, there is a salary scale based on education, number of years experience, geography, etc. The salary scale assumes that the employer provides health insurance for the family. His salary is currently at 79% of the scale, and his employer wants to eventually get him up to 100%. But that doesn’t include the insurance, so it won’t really be at 100% and is not now really at 79%. But I can’t figure out which way to do the math so he can show them the actual percentage. They’re saying he’s at 79 percent. I’m saying it’s lower because they aren’t accounting for that $12K.

All of that boils down to this: What percent of the salary scale is Gretchen’s husband actually making, given that he, and not his employer, pays the $12,000 bill for insurance? There are two steps to this problem:

1. Find the actual salary that is at 100% of the scale.

2. Find the actual percent of Gretchen’s husband’s salary, minus the cost of insurance.

I’m going to tell you up front that we’re going to use a proportion here.  What is  proportions?  A proportion is two equal ratios.  So, if you have two fractions with an equal sign between them, you have a proportion.

And how did I know to use a proportion?  Well, my big clue was that we’re working with percents.  Percent means “per one hundred,” and per one hundred means “out of one hundred,” which just means, “put the percent value over 100.” In other words:

[pmath]79% = 79/100[/pmath]

The tricky part is figuring out what the proportions should be.

Step 1:

[pmath]salary/x = 79/100[/pmath],

where “salary” is Gretchen’s husband’s salary, and x is the top salary on the scale.

That’s because the company assumes that your husband’s salary is 79% of the scale. (Notice this: “salary” and “79″ are in the numerators — or top values of the fractions.)

To solve this proportion, we need to plug in Gretchen’s husband’s salary and then solve for x. In order to make this easy to explain, I’m going to assume that his salary is $100,000.

substitute:   [pmath]{$100,000}/x = 79/100[/pmath] cross multiply:   [pmath]{$100,000*100} = 79x[/pmath] simplify:    [pmath]{$10,000,000} = 79x[/pmath] solve for x:    [pmath]$126,582 = x[/pmath]

So if his salary is $100,000, the top salary on the scale is $126,582.

Step 2:

[pmath]{$100,000-12,000}/{126,582} = p/100[/pmath],

where p is the actual percent of the scale.

Let’s look carefully at this proportion: The first ratio is just the salary minus the cost of insurance, over the max salary in the scale.  (That’s what we found in step 1.)  The second ratio is just like the second ratio in step 1, except that we don’t know what the percent is.

Now, pay close attention to this.  Check the top numbers to be sure they match. We want to know the actual percent of the scale that Gretchen’s husband is making — and that’s what’s represented in the top number of each ration.

Check the bottom numbers to be sure they match.  Do they?  Why yes!  Yes they do!  That’s because $126,582 is 100% of the salary scale.

(Unlike my 10-year-old daughter’s outfits, math is very matchy-matchy.  Knowing that will help you organize your problems and check to see if they’re set up properly.)

Now all we need to do is solve for p.

simplify:    [pmath]{$88,000}/{126,582} = p/100[/pmath] cross multiply:     [pmath]{$88,000*100} = {126,582p}[/pmath] simplify:       [pmath]{$8,800,000} = {126,582p}[/pmath] solve for p:      [pmath] 69.5 = p[/pmath]

So what does this mean? If Gretchen’s husband makes $100,000 a year and is paying $12,000 for insurance, he’s earning 69.6% of the salary scale.

If you made it this far, you get a gold star!  Pat yourself on the back, and take the rest of the day off.  This is a complex problem that depends on an understanding of proportions and how to solve for a variable in an algebraic equation.

Never fear!  I’ll unravel some of these mysteries in later blog posts.  And of course, if you have a question, ask it in the comments section!

The more I talk to people about math, the more I hear this refrain: “I don’t like math, because math problems have only one answer.”


Okay, so it’s not such a crazy idea.  Most math problems do have one answer (as long as we agree with some basic premises, like that we’re working in base ten).  But math can be a very creative pursuit — and I’m not talking about knot theory or fractals or any of those other advanced math concepts.

I have a friend who is crazy good at doing mental math.  She can split the bill at a table of 15 — even when each person had a completely different meal and everyone shared four appetizers — without a calculator, smart phone or pencil and paper!  This amazed me, so I asked her how she does it.  And what I discovered was pretty surprising. She approaches these simple arithmetic problems in ways that I never would have thought of.  She subtracts to solve addition problems, divides to multiply.  And estimation? Boy howdy, does the girl estimate.  In other words, she gets creative.

(She also has a pretty darned good understanding of how numbers work together, which is probably the biggest reason she can accomplish these feats of restaurant arithmetic.)

While there may be one absolutely, without-a-doubt, perfectly correct answer to “How much do I owe the waiter?” there are dozens of ways to get to that answer.  Problem is, your fourth grade math teacher probably didn’t want to hear about your creative approach.

See, when we learn math as kids, we’re focused on computation through algorithms.  (In case you’re not familiar with the word, algorithms are step-by-step procedures designed to get you to the answer.)  You did drill after drill of multiplication, long division, finding the LCM (Least Common Multiple) and converting percents to fractions.  But nobody ever asked you, “How would you do it in your head?”

The good news is that now you’re all grown up.  There’s not a single teacher who is looking over your shoulder to see if you lined up your decimal points and carried the 2.  You can chart your own path!  And when people are given this freedom, they often find really interesting ways to solve problems.

Don’t believe me?  Try this out: Add 73 and 38 in your head.  How did you do it?  Now pose the question to someone else.  Did they do something different?  If not, ask someone else.  I will guarantee that among your friends and family, you’ll find at least three different ways of approaching this addition problem.

So, let’s do this experiment here.  In the comments section, post how you solved 73 + 38 without a calculator or paper and pencil.  Then come back later to see if someone else had a different approach.  If you’re feeling really bold, post this question as your Facebook status, then report the results in the comments section.

And while you’re at Facebook, be sure to visit and like the Math For Grownups Facebook fan page!