Go forth and multiply

“Six times eight?”  I asked.

I looked at Molly expectantly.  She didn’t look back at me – but that didn’t mean she wasn’t paying attention.  Her eyes roamed upward, then wandered left -- vast, blue, and unfocused as a summer sky.   Two seconds went by.  Five seconds.  Eight.  I could practically hear my kid’s synapses overheating. 

“Forty-eight!”

“Excellent!” I beamed at Molly with all the positive reinforcement that my bored, fidgety parent-self could muster.  But at the same time, I wondered – did my eight-year-old simply have a pathetically slow time-release on her powers of recall, or had she spent ten seconds actually calculating the answer?

On a math test, of course, it would only matter that ten seconds per answer is at least six seconds too long. Both my third-grade twins are required, by the end of the school year, to be able to answer (legibly and correctly!) fifty-six multiplication problems in under four minutes.  From the point of view of the unforgiving Speed Test clock, how they obtain the answers is immaterial.  From my point of view, however, the details of that invisible mental mystery make all the difference in the world.

(To the friend who has previously groaned, “Nancy, you’re mathing again” – yes.  Yes, I am.  My apologies.  Stop here, while you’re still safe.)

“So, Molly,” I asked casually.  “How did you know that six times eight is forty-eight?  Do you have that memorized?”

“No.”  She sighed.  Molly hates memorization.  Like, really, really hates it.  I try to push her on the things that matter – yes, you really do need to know your own address and how to spell “enough”– but when it comes to stuff like state capitals my heart isn’t in it.  I, too, suck at rote memorization.  Yeah, I’ll tell you where you can put all your state capitals.  You might think you want me on your Trivial Pursuit team, but, truly, you do not.  I will embarrass you.  I will embarrass myself.  I will not only fail to know anything about pro sports teams and soap operas (knowledge that I might snobbishly decry as plebeian), I will also fail to know anything about the Vietnam War, the Great Lakes, and any Vice President, ever.

“So, how did you figure it out?”

Molly’s answer was as rapid-fire as it was matter-of-fact. “Well, eight plus eight is sixteen.  Then I added another eight to make twenty four.  If three times eight is twenty-four, then six times eight is twice that much.  Twenty-four plus twenty-four is forty-eight.”

Yes.  Of course.  Because multiplication is not only commutative and transitive, but also distributive.  We all know that, right?  Right?

Molly’s line of reasoning seemed pretty good to me, as a ten-second effort by an eight-year-old – especially given the fact that math has been politely described by Molly’s teacher as “maybe not her best subject”.  Thus, I was moderately impressed.  But I was even more impressed with the math curriculum that got her there. 

I should pause here to admit that I know squat about curriculum development. I’m only a “teacher” in the way that many college professors are “teachers”: we earn a PhD in some esoteric subject such as “Species-Specific Effects on the Oxygen Isotope Ratio of Tree-Ring Cellulose,” and are then plopped down in front of a roomful of sleep-deprived teenagers and told to teach them Biology 101.  We have zero clue as to how to create the complex environment of excitement, engagement, and challenge necessary for learning.  We haven’t the faintest idea how to teach to different leaning styles, different skill levels, different majors, different genders, different socio-economic backgrounds, different shades of nail polish, whatever.  We’re also huge dorks, and the whole class knows it.  But I digress. 

In thinking about curricula, I can really only compare the way I was taught and the way my kids are being taught.  When I was in third grade, Mrs. Ricco tried to perk things up with cheerful bulletin-board displays depicting multiplication-airplanes flying across a colorful map.  None of us, however, were provided with any strategies for learning beyond rote memorization.  Purely as an avoidance strategy, I figured out ways to “cheat.” The nines and the fives were easy; I could multiply by ten and then either subtract, or divide the answer in half.  The twos and threes could be mastered via simple adding.  The fours were the twos doubled, and the ones and zeroes were obviously just silly.  The only facts I truly memorized were 6x6, 6x7, 6x8, 7x7, 7x8, and 8x8.  That I found it difficult to learn these six paltry problems shows just how dysfunctional I really am.  Nonetheless, I managed to be the first kid in the class to “fly my airplane across the map”.  I felt guilty about that achievement – because, secretly, I was cheating. 

A few months ago, Lizzy’s teacher sent an email to all us parents.  “Thanks for your concerns,” the message said. She explained that the school has a new math curriculum this year – EnVision Math, by Pearson Realize. “Yes, the bar has been raised. Kids' math grades are a direct reflection to the rigor asked for. They are learning....they surely are. It's just that we are so used to higher grades. . .”

“Okay, Lizzy, your turn.  Seven times nine?”

“Sixty-three.”  Two seconds.  “I did ten sevens minus one seven,” she offered.  Lizzy is a better memorizer than Molly, but it seemed that she, too, was using other strategies.

I answered that email.  I wanted that hard-working, dedicated teacher to know that I didn’t give a rat’s ass what my kid’s grade was, so long as her brain was being stretched.  (I didn’t use those precise words.  I’m, like, a PTA member and whatnot.)  As far as I was concerned, the new math curriculum – despite some laughably awkward problems pertaining to Grover Cleveland and/or the booking of conference rooms -- looked a hell of a lot less lame-o than the previous one. 

What impressed me about it?  Well, I liked the fact that the kids were offered a host of different strategies for rounding numbers and estimating answers.  Not calculating – estimating.  In other words, they were being taught the kind of math that you might do as you toss items into your grocery cart, if you’re on a tight budget and don’t want to humiliate yourself at the register.  “What is a reasonable answer?”   Holy shit.  My brain exploded.  Because, teaching at the college level, I’ve faced students who chugged through all manner of complex calculations, and then gave me answers like “-8.4335” when the units in question were “fish”.

Second, I noticed that the twins’ worksheets used a lot of story-problems.  Yes, yes, yes!  In the real world, when you need to decide whether you have enough time to drop off the library books between the end of swim practice and the start of your dental appointment, the relevant numbers don’t appear printed on your dashboard, complete with +, -, and = signs. To me, it seems obvious that a kid needs to really, really understand what multiplication is – and some of the many situations in which it might be useful – before settling down to practice and memorize.  But when curiosity drove me to look up online discussions and comments about various math curricula, use of story problems popped up as a common criticism.  “EnVision Math doesn’t make sure kids know the basics before moving on to word problems!”  The basics?  Real life IS basic.   

“All right, Molly.  How about seven times three?”

“Twenty-one.”  The delay was too long.  She was adding.  But, heck, at least she knew enough to add seven three times, rather than adding three seven times.  “It’s an array.  Of course it’s the same either way.  You could just turn it around.” 

Third, the new curriculum demanded that kids write answers in words.  I don’t just mean answers like, “Tommy has six blocks.”  I mean answers like, “Tommy is wrong when he says that he will have ten blocks left after Susie takes two of his eight blocks.  He added instead of subtracting.”  Dubbed “Writing to Explain,” this is pretty much on par with rocket science, to an eight-year-old.  It is not a particularly popular activity at homework-time at our house -- but I love it.  Because… logic.  Because, kiddo, for the rest of your life, people will try to fool you with math, with graphs, with statistics, with numbers.  But you WILL NOT BE FOOLED!

How desperately do we need these skills?  Well, a New York Times article, succinctly titled, “Why Do Americans Stink at Math?” http://www.nytimes.com/2014/07/27/magazine/why-do-americans-stink-at-math.html?_r=0 gets into all the grim details of our comparative international patheticness.  At the core of the problem is the difference between doing math problems and understanding math. 

It wasn’t until I was in high school -- and a proud T-shirt-wearing member of the Metropolitan New York All-Stars Math Team -- that I started to have an inkling that maybe I’d been doing it exactly right, all along.  These ideas gelled when I was called upon to step up to my first paid teaching job.  I was sixteen.  My long-time neighborhood buddy – who far outshone me in social skills, creativity, knowledge of pop culture, and ability to get the damned basketball through the hoop – had bombed the math section on the practice SAT.  Her parents, pragmatically, decided to hire the cheapest tutor available:  the Insanely Nerdy Friend.  Sitting at my neighbor’s kitchen table, I soon realized that my smart and vibrant friend could do any problem she was asked to do – but only if she was told precisely what kind of problem it was.  This was how she’d been taught.  It was utterly useless.

Newsflash:  in the real world, figuring out how to set up the problem is not just important, it’s everything.   Want to bake only 2/3 of this gingerbread recipe?  Want to know whether you can afford monthly payments on the house, the car, AND that sweet new snowmachine?  Need to understand the relative risk of vaccinating your child versus letting them get polio, for chrissake?  We have computers for any really serious numerical grunt-work -- but if you don’t understand the question you’re asking, you have NO HOPE of understanding the answer.

It’s not like I’m the only person who knows this.  As noted in that NY Times article, Americans have so poor a grasp on what fractions actually are that when A&W released a 1/3-pound hamburger to out-compete the McDonald’s Quarter Pounder, people didn’t buy it.  They thought it was a bad deal – because it was, like, smaller.  “Focusing only on procedures… turns school math into a sort of arbitrary process wholly divorced from the real world of numbers. ..  Instead of trying to convey, say, the essence of what it means to subtract fractions, teachers tell students to draw butterflies and multiply along the diagonal wings, add the antennas and finally reduce and simplify as needed.” 

I cringe.  I mean, consuming more cheap ground beef is probably not the solution to America’s problems, but… death to the butterflies. 

I highly doubt that the new math curriculum at University Park Elementary School is perfect.  (Seriously, what was with the problem about Grover Cleveland?)  The teachers are struggling with it.  Some of the kids are struggling with it.  Clearly, many of the parents are also struggling.  I’m sure we could all fault it.  But if I do pick holes in it, it certainly won’t be for its valiant efforts to teach kids to understand, to think, to look at everything in more than one way, and to relate everything back to the real world.

Last week, when the twins bounced off the school bus in the afternoon, Lizzy immediately chirped, “Guess what?”  That’s how kids set up dramatic tension in a conversation.  “Guess what, Mom?  I passed the Speed Test!”

I congratulated her with genuine enthusiasm.  She’d memorized a bunch of math-facts to get there, and those facts are useful enough to be well worth the effort.  We do, indeed, all need to be capable of answering 56 multiplication problems in under four minutes. 

“Good job!  Nicely done!”

Then I looked over at Molly, who appeared to be examining her boots.  She can answer 56 multiplication  problems, too – but not in four minutes.  Maybe in eight. 

“Don’t worry, you’ll get there too.  You just need a little more practice.  Let’s see, what’s eight times nine?”

[Boot-crunch on packed snow.  Rumble of a distant truck.  Rustle of snowpants.]

 “Seventy-two.”

Think on, America.  Think on.

Timeless

Jay [jokingly]: “The end of the world is coming!”

Molly, aged eight [calmly spooning up another mouthful of soup]: “The end of the world has always been coming.”

I’ll admit, I love this no-nonsense take on The End Times.  I’m happy to credit my little pedant for knowing that the sun’s core is eventually scheduled to run out of hydrogen.  Nonetheless, I’d bet that (even if promised seventeen colors of Sculpey and The Swiss Army Knife with All the Things), she wouldn’t be able to guess, to within an order of magnitude, when.  

Is the sun is going to expand into a red giant and engulf the Earth ‘round about 5.4 billion years from now?  Or in a few thousand years?  Or in a few hundred, on a Tuesday afternoon?   Um…whichever.  Don’t ask Molly.  This is a child who still has trouble telling the difference between the “long ago” when many of her classmates’ ancient ancestors crossed the Bering Land Bridge, and the “long ago” when Ancient Mommy grew up without the Internet.  Then again, don’t ask the Average American, either. We’re not terribly good with billions.  Or even millions.  Or – let’s be honest here -- any number larger than, say, the combined total of fingers and toes possessed by all of our Facebook friends. I’d contend that most of us are more than a little lost in time – because most of us are more than a little lost in math.

Okay, granted. But is this actually a problem?  Really, when do most of us have to deal with astronomically large numbers? We don’t have annual incomes, odometer readings, bank account balances, or home values that take us past six digits.  If we’re lucky, we might enjoy, from birth to death, a grand total three billion seconds. (A few hundred of them may be wasted on this blog, but don’t worry – a billion is… a lot.) Is there any reason why (other than the fact that Nancy is clearly longing to take us on a Proselytizing Adventure in Math Nerdery) we need to haul our butts back to middle school to relearn the phrase “orders of magnitude”?

Well, yes.  For one thing, understanding honking huge numbers comes in handy when studying minor little topics such as, say, astrophysics, geology, computer science, chemistry, economics, the national debt… and, oh, pretty much everything else.  More particularly, applying this comprehension of bigness to the Grand Scale of History is pertinent with regard to two pesky topics that keep showing up in bastions of science such as Fox News online comment threads, Kansas school board meetings, and Congress.  I’m referring, of course, to the subjects of evolution and climate change.

On the evolution front, I promise not to draw dozens of cladograms and start babbling about retrotransposon markers, plesiomorphies and synapomorphies.  However, I will assert that understanding how we got from primordial soup to microwaved leftover minestrone is a heck of a lot easier if you start with a sense of proportion.  A quick primer, for those who grew up in red states:

1.      Not only were more than seven days involved in the process, but single-celled organisms were The Only Show in Town for about two-and-a-half billion years of our planet’s three-and-a-half-billion-year flirtation with life. 

2.      It then took another three-quarters of the remaining billion years before the first dinosaurs showed up (with the express purpose, of course, of leaving their bones lying around just to trick us). 

3.      Although dinosaurs were a super-recent innovation compared to blue-green algae, they nonetheless ruled the earth for almost a thousand times longer than human beings have existed -- and have been extinct for a million times the run-time of The Price Is Right.

So much for evolution.  As for climate change… Yeah, I’m one of those scientists who gets paid to study this stuff.  Therefore, I can’t possibly be trusted when I suggest that Hummer exhaust and cow farts are causing various important bits of our planet to burn, sink, desiccate, or wash into the ocean.   Meanwhile, I’ve heard head-in-the-sand climate arguments that mix orders of magnitude like smoothies in a blender. 

Yes, there was waaay more carbon dioxide in the atmosphere 200 million years ago than there is today – and even more, 500 million years ago.  Um, yeah, climate-change naysayers, that’s right.  You win.  Except that 500 million years ago, there weren’t even any plants in North America.  And, for that matter, there wasn’t any North America.  Heck, we hadn’t even gotten to Pangaea yet.  Okay, let’s simplify things.  Yes, Earth has been much hotter in the past than it is now. Yes, there has been a lot more carbon dioxide in our atmosphere in the past.  Both of these statements are true.  Both are immaterial -- because humans are not (to the nearest order of magnitude) blue-green algae.

We are, however, within one (genetic) order of magnitude, chimpanzees.  Or (hey, close enough) rhinoceri.  This, of course, amuses me immensely -- but it also pleases me mathematically. To me, being asked to express a mathematical answer to “within one order of magnitude” is positively delightful.  It appeals to my utterly lazy and vague nature.  I don’t need to know whether we’re talking about 43,896 red-backed-voles or 27,981 board-feet or 80, 001 heavily armed Klingon invaders; I just need to off-handedly scribble “n x10⁴”. 

However, in assigning homework and tests to undergrads, I quickly noticed that while answers in the tens or hundreds usually came back unscathed, things started to get ugly in the tens of thousands, and by the time we hit the millions, it was a crapshoot. (“Now, for the grand prize of the Jacuzzi Spa AND the Home Salon, what is the dollar value of the gross domestic product of Cameroon, plus all extant Google stock?”)  On one hand, most calculations don’t require the level of precision that computers and calculators churn out. (If you are measuring your height to eight decimal places, you don’t understand that “your height” is not generally considered an atomic phenomenon.)  On the other hand, once you start losing track of where that crucial decimal point belongs, you find that you are have the bodily proportions of a Playmobil person, or the Statue of Liberty.  Those undergrads were thus subjected to irksome tirades about why understanding orders of magnitude is a Very Big Deal. The good news, as I told them, is that it’s really not that difficult a deal. 

The bad news is… well… Fox News online comment threads, Kansas School Board meetings, and Congress.  Our nation has a bit of a reputation – and, to be honest, it’s getting kind of embarrassing to constantly wear the International Dunce Hat of Mathematical Imbecility.  Do we have any hope of redeeming ourselves?

Well… maybe some. Now that Molly and Lizzy have reached the lofty academic pinnacles of third grade, their math homework is getting to be a bit more fun than that old second-grade chaff (“Rodney wants to buy a pencil for five cents and a ruler for twelve cents, because Rodney lives in an alternate 1950s universe in which pennies are still actual money.”)  The other day, I asked Molly to explain her multi-digit subtraction problem, just to make sure she really knew what she was doing.  Her answer sounded so reminiscent of the lyrics of Tom Lehrer’s classic “New Math” that I giggled with parental pride (“Don’t worry, base eight is just like base ten, really -- if you’re missing two fingers.”)

The superb teachers at University Park Elementary are clearly doing their utmost to make a bunch of snot-nosed, fidgety, four-foot-high people ready to control the stock market, program your virtual-reality holodeck, and give you a prostate exam. (Please take a moment to think about this: that person who is telling booger jokes, dressing up in a plastic Princess Elsa outfit, and accidentally Silly-Puttying the cat is going to be flying your plane or performing your quadruple bypass. Okay.  Now take a deep breath and carry on.) 

The twins have moved on from adding and subtracting tens of things (“Sally has 35 toy cars and Juan has 18 toy cars.  How many toy cars will they have after they cram 11 toy cars down the bath drain and Sally’s big sister tells on them?”) to adding and subtracting hundreds and even thousands of things (“Fairbanks Alaska is 1521 miles from Seattle.  From Seattle to JFK New York is another 2418 miles.  However, you can only fly on free miles if you go to goddamned Newark Airport instead, and take three trains.  If the TSA spend 12.3 minutes inappropriately palpating your teddy bear and the Long Island Railroad is suffering unexplained delays at Hicksville, when will you get to see Grandma and Grandpa?”) In short, they are starting to be expected to understand orders of magnitude.

In truth, I’m pretty happy about the elementary math curriculum.  Not only does it feature an ethnically diverse and unfailingly cheerful cadre of cartoon children getting excited about the relative heights of buildings and the capacities of conference rooms, it also emphasizes understanding numbers, rather than simply manipulating them.  Some problems include space for kids to write out, in painfully smudged dull pencil, answers to such stumpers as, “Is your answer similar to your estimate?” and “Explain why your answer is reasonable”.

What?  The answer has to make sense?  The pigtailed people have to be able to elucidate – in actual words – why it makes sense?  Holy subtraction, Batman.

Moreover, the kids are being offered a more holistic perspective on time, space, and history that I recall from my own elementary years.  Not only is Molly ruminating on the age of our solar system, but Lizzy is excited about the evolution of equines.  As I tucked her in a few nights ago, she asked me, “Do you know about Lascaux Cave?” 

Yes, this piece of history takes us back only 17,300 years, not anywhere near the full 200,000 spanned by Homo sapiens -- but I’m pretty sure that when I was her age, I was taught that history began in 1492, and didn’t really get going until 1776.

There are ways to teach this stuff to grownups, too.  A few years back, UAF was temporary home to a wonderful display on the history of Earth.  It was, at one level, a fairly standard collection of large, museum-quality informational boards, each one detailing some aspect of evolution.  What made the display unusual was not so much the content of any one board, but the ratio of that content, and the placement of the boards in relation to one another.  The display was a walk through time – set up to scale.  It’s amazing how much you can learn about RNA, prokaryotes, eukaryotes, and photosynthetic cyanobacteria when they are given so much elbow room to accommodate their blatant lack of elbows.

As for the college students who suffered through my courses, they were not let off the hook without a whole side-stepping foray into the world of back-of-the-envelope calculations – a subject so delightful to me that I’m going to save it for another lexical outpouring in which I will babble about Enrico Fermi, piano tuners, and the Drake Equation.  I’m sure you can’t wait.  It will include some spectacularly humongous numbers.

Trillions!  Quadrillions! Time and space don’t just require large numbers, they deserve them.  And we deserve the joy of wallowing in all those order of magnitude.  We -- even the little people among us -- deserve the thrill and the challenge of at least trying to comprehend.

Jay [still joking]: “But the end of the world imminent!”

Molly [with patronizing patience]: “I don’t think you understand what ‘the end of the world’ means.”

Neither do you, Molly… but you’re getting there. 

Yesterday, on our walk to the school bus stop, as we were admiring the almost-full moon hovering above the treetops, Molly asked me if I’d caught a glimpse of the “blood moon” the night before.  “No,” I admitted, “But my friend Otto got some amazing photos.  He’s way down in Idaho, but it’s the same moon, of course.”  Teasingly, I sang, “There is just one moon and one golden sun…”

Molly looked at the moon again.  “Well, just one moon and sun that are ours,” she clarified.  “Really, there are lots.”

I had to laugh. 

That’s right, kiddo.  And I’m going to keep blathering away to you about the enormity of space, and the enormity of time.  Perhaps it’s idealistic of me, but I want you to get it.  Really get it in a way that’s impossible without wrapping your head around Big Numbers.  We are pinpoints, mere asterisks in an intergalactic-sized universe– and yet, if we turn the telescope around and look for other orders of magnitude – 10^-4, 10^-5, 10^-6 – we are impossibly vast.  Photons!  Electrons!  Quarks!  Hadrons!  Indeed, there are so many more insanely nerdy blog posts hidden in all that delicious numerical space…

“Yup,” I agreed.  “Billions and billions.”