Most math teachers teach that that there’s one process for solving math problems, but this approach just isn’t very practical. Now that you’re a grownup, you can find your own way to the answer. I promise.
Today’s Film Friday is brought to you by one of my favorite teenagers in the world, Simon, who introduced me to Tom Lehrer earlier this year. This version was done by lipsyncORswim.
(Warning for those who are satirically challenged: this is supposed to be funny. So laugh!)
Math for Grownups blog readers tend to fall into two camps: grownups who are not parents and really hate math (or think they’re not good at it), and parents who are worried that they’re going to pass along their math anxiety to their kids. And so I thought I’d spend a little bit of time addressing some of the concerns of these parents.
Earlier this week, my friend and fellow freelancer, Debbie Abrams Kaplan forwarded the summary of a new bit of research on kids and math. Debbie is the author of two great blogs: Jersey Kids and Frisco Kids, and she figured that I might find some blog fodder from this study.
Boy did I! A couple of things jumped out at me:
No one has ever studied how the basic math skills of first graders affect their later understanding of math throughout elementary school. (Compare that with the many studies of early reading skills, and this fact will blow your mind, too.)
There are three basic skills that will help first graders become good fifth-grade math students.
I’m going to tell you those skills a little later, but first I want to introduce the concept of numeracy. Quite simply, numeracy is the ability to work with and understand numbers. When children are young, numeracy includes the ability to count, recognize the symbols that we use for numbers (which is akin to learning the alphabet), and even do some very simple operations (like 1 + 1 = 2). For high school students, numeracy includes more complex problem solving skills and properties of real numbers.Among math educators, there are big debates about how we can better teach numeracy. I guess this is like the debates about phonics vs. context support methods in reading education. But now that this study is out, it’s clear parents can help lay a firm foundation for our kids’ later success in math. According to this study, published by a team of University of Missouri psychologists, rising first graders should understand:
Numbers — I’m going to take this to mean whole numbers, since most first graders aren’t very familiar with fractions or decimals.
The quantities that these numbers represent — In other words, kids should be able to match a number with that same number of objects (five fingers, two cats, etc.)
Low-level arithmetic — And I’m guessing researchers mean things like adding and subtracting numbers that are smaller than 10 (excepting problems with negative answers).
If you’re like most parents, this is probably a duh moment. What’s so hard about recognizing whole numbers or understanding what five objects are? But I don’t think many parents spend much time emphasizing these ideas — at least not in the way that we commit to reading to our children every night.So here are a few ways that you can help instill numeracy in your pre- or elementary-school aged children.
Count things. Count everything — like the stairs that your climbing or the cars that pass your house or blocks as you take them out of the box or those adorable little toes!
Have your child count things. You can do this in really simple ways. Ask him to get you five spoons so you can set the table. When she wants some goldfish, tell her she can have 10 (and watch her count them). When you’re planning his birthday party, have him tell you which 10 friends he wants to invite. (Write them down for him, so he has something visual to count.)
Notice numbers. When she’s really tiny, ask her to say the numbers that are on your mailbox or on a license plate. Older kids can name multi-digit numbers, like 157 or 81. (And if you want to really be precise and prep your kid for school, don’t say things like “one hundred and fifty-seven. In math, “and” represents a decimal point, which is something most elementary school teachers will really drive home.)
Teach your child to count backwards. This can be a great way for kids to start understanding subtraction. If you know you have 10 steps in your staircase, count backwards as you go down the stairs. Then count frontwards as you go up!
Start adding and subtracting. Give your child 5 raisins and show her how to “count up” to 7 by adding 2 raisins to the pile. Then as your child eats the raisins one by one, “count down” to find out how many are left.
You don’t need to make a big deal about math. And for goodness sakes, skip the worksheets, flashcards and even video games — unless your kid really loves them. Integrate these basic skills into your daily life, and you’ll see your child’s understanding grow. (And you probably won’t feel so stressed out about it all!)What kinds of things do you do with your young elementary-age kids? Any teachers out there want to share their thoughts with the class? Post in the comments section.
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.
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.
Want to know what really burns me up? Telling a kid he’s not good at math. Know what’s just as bad? Telling a mom that her kid isn’t good at math.
This morning, I was early for my fitness boot camp class. (Sadly, I don’t get to work with Robert the personal trainer.) I chatted with a couple of the women who, like me, show up every morning at 6:45 for an hour of torture. Of course, I had to share that I got advance copies of my book, Math for Grownups, yesterday. Per usual, the conversation turned to the absolutely ridiculous and completely frustrating notion that people are either good at math and science or they’re good at language arts.
Now, for the record, I wholeheartedly disagree with that line of thinking. (Can you tell?) Keep reading to find out why.
One mom said that her son’s first grade class was good at reading but not math. How did she know this? The teacher told her. There are so many unbelievably wrong things about this situation, and thinking about it makes me want to scream:
They’re first graders! How can six year olds be bad at math? They might be missing some basic concepts, like counting up to add or the difference between a triangle and a square. But remember, nobody is born bad at math.
It’s the teacher’s job to make sure kids understand basic math concepts. Blaming the students is a cop out.
Telling parents that an entire group of kids is bad at math is a self-fulfilling prophecy. Already, parents (and other grownups) believe that math ability is like being tall or having blue eyes–you either have it or you don’t. This teacher may have unwittingly reinforced this idea, by making such a silly generalization.
If she told the parents this, what message is she sending the kids? If you’ve been a parent, you know that there are a trillion different messages that we send to our kids every day, without even knowing it. I would bet my last dollar that this teacher is somehow relaying to her students that math is just not their thing.
First graders who think they’re not good at math, grow up to be middle schoolers who think algebra is beyond them. These middle schoolers of course awkwardly morph into teenagers who are convinced that they won’t need (and can’t do) geometry, trig, advanced algebra, probability, statistics and calculus. And then of course, these acne-prone, love-sick adolescents become the smart, successful adults who tell me every day that they desperately need my book, because they can’t balance their checkbooks to save their souls.
But really, I’m not blaming this one teacher. I promise. She’s just the beginning of a long line of teachers and other grownups who buy the lies about math: that only people with “math brains” can comprehend the Pythagorean Theorem and no one uses math in everyday life anyway.
So, what’s the solution? The issue is not the students. The issue is that we somehow believe that there is only one way to teach math. Of course that’s not true. Teachers and parents have to figure out how our kids think and approach math in a way that makes sense to them. And we have to quit labeling ourselves and our kids.
All you parents out there, I’d be forever grateful if you’d do just one of these things:
If a teacher says your kid is bad at math–or worse, declares an entire class bad at math–please challenge him or her. There are only a few people in the world who have real issues with processing mathematical information, and I’m betting your kid isn’t one of them.
Stop telling people that you, yourself, are bad at math. Next time your dinner companion asks, “Can you help me figure out the tip?” bite your tongue. If you truly can’t find 15% of $24.68, pretend you didn’t hear or fake it. But please avoid saying the all-too-common, “I’m so bad at math!”
If you can’t help your kid with his homework, don’t declare: “I just don’t have the math gene!” Here’s the reality: You may not remember how solve a proportion, but that’s because you probably haven’t seen one for at least 15 years. If you were asked to diagram a sentence and couldn’t do it, would you say that you’re no good at speaking English? Of course not.
Replace your generalizations about math ability with messages like these: “I don’t remember how to do that. Let’s figure it out.” or “I remember doing these kinds of problems in school, and they gave me trouble. But I’m sure we can figure it out together.”
And just a quick footnote/disclaimer: I am the biggest advocate of schools and teachers that you will ever meet. Having been a public school teacher and been raised in a family of teachers, I’ve seen first-hand what our educators face on a daily basis. I just wish I didn’t hear stories like the one I heard this morning.
I’m going to gingerly climb off of my soapbox now. (My glutes are killing me after that workout!) But I ask you to share your thoughts on these generalizations about math ability and math education. What messages have you or your kids received? What do you think about them? How do you think we can counter them? And for all you teachers out there: how do you send the message to your students that they are good at math?
I’ve decided to start a regular feature about math education, called Summer School. In it, I’ll discuss some ways that parents can send the right messages about math. We don’t need another generation of grownups who think they can’t do math! 🙂
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.
So this apparently is big news in Myrtle Beach. A middle school math teacher actually took her kids out of the classroom to teach them math. In the school cafeteria, the students converted decimals to percents and found surface area and volume — as they were cooking up some healthy eats.
Photo courtesy of Jessica Masulli
Ya’ll, seriously. This is what how we use math as grownups.
(Okay, so the surface area and volume is a bit of a stretch.)
If you think doing math is about chalkboards and protractors, you’re
flat out wrong. (Besides schools use dry erase boards these days.)
Math is about getting your hands dirty, sketching a picture on a scrap piece of paper, doing some quick calculations on the iPhone. Most of all, math is about solving real problems — not those silly things that have something to do with trains in Omaha — and coming to these solutions in creative and sensible ways. (There. I said it: creative and sensible.)
Look, I like what this teacher is doing. And so do her students:
“You learn it better because you enjoy doing it,” said
Maya Bougebrayel, who made a vegetable chicken stir fry with teammates
Allison Klein and Carlisa Singleton. The girls, all 13, agreed that the
project put a creative spin on learning and made it easier for those who
are visual learners.
But if it wasn’t such a novel idea, wouldn’t grownups be better at math? Feel free to chime in 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.”
Peshaw!
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.
The biggest fights my father and I had were about math. I kid you not.
The year was 1984. I was a junior in high school, taking Algebra II. Radicals were kicking my scrawny, little butt.
(Remember radicals? They look like this: sqrt{24}. In Algebra II, you mostly learned to simplify them, as well as add, subtract, multiply and divide with them.)
My father wanted to help, and he had the patience of Job. But he was
not great at accepting that I didn’t understand. And I wasn’t great at
controlling my emotions. I hollered, cried and probably threw things.
Somehow, I got the impression that my dad thought I couldn’t do math, and I did what any strong-willed girl will: I dug in my heels.
That’s when I started drinking coffee, actually. I was so determined
to show my dad–and my Algebra II teacher, Mr. Gardner–that I got up at
4:30 a.m., sat in my dad’s easy chair with a cup of coffee and a stack
of sharpened pencils, and did problem after problem after problem.
I did every single radicals problem in the textbook. And then I did
them again. I took what Mr. Gardner and my dad taught me and figured the
darned things out. It took time, but I was determined not to give up.
Why on earth would I do this? Well, I’m stubborn, for one. But
probably the biggest reason is Mrs. Ivey. She was my geometry teacher
the year before, and she changed my perspective about math. You see,
before then, I knew I couldn’t do math. Mrs. Ivey convinced me that I was wrong.
She and my father are the reasons I majored in math. I found out I’m
a math teacher, not a mathematician. (Sometimes we’re one or the
other.) I’m fascinated by the ways people choose to do math, not by
complex computations or proofs.
Math geeks aren’t always born. Sometimes a teacher inspires us.
Sometimes we’re dragged kicking and screaming. And sometimes we just
learn to deal with math–because we have to.
What’s your math story? Share it in the comments section!
