On nothing in particular

Students seem amazed that the idea of "0" as a number is a relatively new concept, but students also see "0" as a numeral, as solid and inevitable as a "3" or a "7"--as most of us do. How could zero be invisible to the Greeks? (It wasn't, of course...but we only think so because we misunderstand our own concept of "0".)

XKCD rules!

That's not how we use it, though, at least not most of the time in the simple measurements or arithmetic we use in class. Zero serves mostly as a placeholder.

Take the number "306"--the "0" means there are no chunks of tens beyond those already captured by the three hundreds signified by the "3". We have 6 units, true, not quite enough to form a chunk of ten, so we need a way to show this: 3_6.

This is glaringly obvious to many of us who grew up without electronic calculators. The calculator we occasionally used in elementary school was the abacus, which requires grasping the concept of placeholders in order to manipulate the beads, and the slide rule, which requires the use of mental placeholders to make any sense at all.

If you do not grasp the very real difference between zero and the other nine digits, subtle problems arise, which become not so subtle when playing with significant figures ("sig figs"), those numerals that carry any real meaning when we apply math to the natural world.

We too often teach sig figs as a set of rules, which make little sense if the student has a poor grasp of zero, which many--through no fault of their own--do. The rules seem arbitrary and arcane, when, in fact, they define the limits of numerical reality when we measure the natural world.

"Numerical reality" comes off a bit too abstract. I should rephrase this: significant figures define the limits of observable mathematical relationships found in the natural world. The relationships are real, inasfar as they can be measured by independent observers. In class, however, sig figs become an exercise in futility for many.

I think it gets down to the zero. Electronic calculators add a level of abstraction to numeracy, leading to the ridiculous assertion science teachers hear at least once a week: "But my calculator says..."

And indeed, the calculator says exactly what it's supposed to say, when its operators plug in numbers without any feeling for what they represent.

And how could they? The Greeks had the concept without the symbol. We, alas, now have the symbol without the concept.

Yes, I know, the Mayans and Indians and Phoenicians and on and on and on had it down pat.
Thank Fibonacci for getting it to the western folk.

XKCD comic used wtihin his guidelines--very broad ones at that. Thanks! 

And yes, the sig fig rules are a tad arbitrary--range of error makes more sense.

NJCSA attacks the messenger

Another heads up from Tom Hoffman at Tuttle SVC--he's
the John Stewart of the education news world....

I like numbers. A lot.

I like numbers enough to have developed a healthy relationship with them. No one night stands. No extraordinary demands. We respect each other, stay open, and work to grasp each other's nuances.

I have long maintained a monogamous relationship with the decimal system. (OK, I once philandered with hexadecimals, and I toyed with binary when I was still a lad, but I've matured.)

I have been accused of seeing numbers better than the average bear. I maintain that most humans (if not bears) can see numbers as well as I can once they get past the torrid relationships from back in high school.

Data, on the other hand, are like spam--somewhere along the line numbers get processed.

Data is wonderful if you have a healthy relationship with numbers, and refuse to abuse them, no matter what glory or riches may tempt you.

If you love numbers, run over to Bruce Baker's School Finance 101 blog.

***

Dr. Baker loves numbers, too. He makes them friendly. He's also very, very smart. He could take data and mash it to his whims.

But he doesn't.

Dr. Baker plays with numbers with respect; he also respects his audience. He doesn't treat his readers as dummies, yet takes time to clearly explain his reasoning.



The New Jersey Charter Schools Association got upset with Dr. Baker's blog recently. His offense? He ran some numbers, and drew some conclusions:

[E]ven if we found that these schools produced greater gains for their students than similar students would have achieved in the traditional public schools, we could not sort out whether that effect came from school quality differences or from peer group differences (which doesn’t matter from the parent perspective, but does from the policy perspective).

Through a twisted piece of logic, the NJCSA insinuated that Dr. Baker gets money for shilling against charter schools.

Rutgers University Professor Bruce Baker is closely aligned with teachers unions, which have been vocal opponents of charter schools and have a vested financial interest in their ultimate failure.

I love teaching. I might work in a charter school some day; I've even dreamed of establishing one (the Margaret Donaldson School for Children--you heard it here first). I hope that the schools represented by the NJCSA have more respect for numbers and for truth than the shows NJCSA in its tirade.

The CEO Carlos Pérez flew in from Chicago to take over the reins here in Jersey back in June. He comes from the same town that produced Arne, a man with a flimsy relationship with truth data.

(See how the game is played?)

The comic is, of course, from Toothpaste For Dinner.

Pi what squared, Captain Hook?

Arrrrrr......Christmas 2009.

OK, it's a cheap science math trick.
And, yeah, well, I'm Oirish, but me mum-in-law is descended from UK folk.

The Little Sump Pump That Could

Our electric bill skyrocketed. Our pump runs on electricity. Are we getting enough bang for our bucks?

Ah, finally, applied science! (One of the white lies teachers tell students is that what they learn in class matters. When was the last time you used the quadratic equation?)

Hmmm...the pump goes off when triggered by the float switch. I have about 9 inches of play there--lessee, square the radius times pi times 9" and we get 1385 cubic inches, .004329 cubic inch/gallon, and presto! we get about 6 gallons.

6 gallons does not sound like much--not even 3 cases of beer.

The pump, however, kicks on every 20 seconds. 18 gallons per minute.

18 gallons per minute-> 1080 gallons/hour -> 26,000 gallons/day. A typical Olympic sized swimming pool holds 660,000 gallons. Every 26 days (just under 4 weeks) my pump is moving an Olympic sized swimming pool.

But it's not just moving it sideways--it's moving it up. The bottom of the pump sits in a sump basin 28 inches deep. The outlet pipe is 76 inches above the floor. Lessee, 76" + 28" gives me 104", or 8 feet 8 inches.

Hope I haven't bored everybody with the numbers, but let me recap--every 4 weeks or so my little sump pump is lifting an Olympic sized pool over 8 feet.

A gallon of water weighs a smidgen over 8.3 pounds. An Olympic pool's worth of water, then, comes out to almost 5 1/2 million pounds of water! Over 2700 tons!!!

I am The Little Sump Pump That Could is jacking up 2,700 tons over 8 feet up every month.

***

Leslie pointed out a fundamental error in my calculations--the pump takes up volume. How much volume? Well, I could pull an Archimedes and see how much water it displaces, but it is busy working at the moment.

I could go outside and figure out how much is flowing per minute, and I might yet, but for the moment, let me assume that the pump volume is about 2 gallons, which would reduce the each load by a third.

The Little Sump Pump That Could is still jacking up over 1800 tons of water every month 8 feet 8 inches. That's over 30,000,000 foot pounds!

This month's bill was $40 higher than usual. For each dollar, I got 45 tons of water pumped up over 8 feet. That's about 2 cents a ton.

And that, my friends, is a good deal!

I weigh ~200 pounds. My little sump pump could take me over 28 miles up into the sky in a month.

Terminal velocity of mussels

Yesterday Leslie and I paddled along the edge of Cape May Harbor, again showing our prowess with tide charts by struggling against the ebbing tide on the flip side of our voyage.

Because pushing against the tide is tiring, I edged towards the beach and just drifted for a minute.

While doing pretty much nothing, I noticed a gull dropping something.

It followed the object, picked it up, the swooped back up, higher and higher, hovered a moment, then dropped the object again.

Gulls are marvelous fliers, despite the inane way they're pictured in Jonathan Livingston Seagull--I once saw a flock of them chasing down insects, rivaling swallows in their gyrations.

After its fourth try, this time successfully nailing a small bed of rocks on the beach, the gull had its meal.

Now, I know this has been reported, and I have found smashed shells on jetties, presumably from gulls, but I had never seen it done.

A few things struck me:

• The gull did not randomly drop the mussel--he clearly was aiming for the small pile of rocks.
• As the gull hovered upward, he went a bit higher than gulls usually do when just hanging around--it knew enough to get some air between its meal and the rocks below
• The gull consistently went to about the same height, and it took a bit of effort to get there.

So here are my bizarre questions of the day, which should be answerable, though I have about 37 other things I should be doing.

How far does it take a falling mussel to reach terminal velocity at sea level?
How close was that gull to that height?

I bet that gull was about as close to the answer through experience as I'm likely to calculate.

(Herring gull photo by Adrian Pingstone, 2003, via Wikipedia Commons; mussel photo by Joan Muller, Waquoot Bay National Estuarine Research Reserve, 1996, via NOAA)

Dyscalculia: If I were king...

Many of my students have little facility with numbers.


I'm not talking about algebra or geometry--I'm talking arithmetic.


While it's easy to lay blame on the kids, cell phones, television, their parents, the streets, vaccines, or perhaps a national conspiracy to raise better consumers, I think that schools can make some changes that might help.


First, make sure the curriculum works. This is the big one, and well-intentioned folks in charge of large budgets are wrestling with it in districts around the country.

(I first saw this video at Elona Hartjes' bloggsite, Teachers at Risk--it should be seen by every parent with a child in elementary school. I also recommend a look at Linda's thoughts in Technology in Teaching.)







In the meantime, here are a few things we might consider:

*Ban calculators until a child hits physics class

No excuses, no excuses, no excuses--calculators kill number sense.

If you happen to wander into some trignometry, I'll make allowances for a slide rule. Heck, I used to be able to calculate physics problems faster than the calculators we had back in the 1970's, which actually had a noticeable lag when doing some functions.

I have children today who do not grasp that 100 divided by 20 is not the same as 20 divided by 100, and who will argue the point because when they use their calculator their way they get their number.

*Return analog clocks to the classrooms

I'm old enough to rant about digital music--I only caved in to the CD thing because Aerosmith's Pump was not released on vinyl, and, well, I liked their elevator music.

Still, digital music loses something in the process, beyond the romanticized snack, crackle, pop of vinyl records.


Digital clocks give less information than analog faces. That I even have to
explain this saddens me, but here it goes:

1) Kids need clocks--blame the Benedictine monks if you will, but time matters. A digital clock is a number that flips now and again.

Time was divided into minutes and seconds long before the monks developed the timepieces that plague us today.

Hours and minutes and seconds plunges us into radians. Blame the Babylonians, too , but for whatever reason (perhaps 360 was close enough to the number of days in a year), circles are divided into 360 degrees. Western civilization is encumbered by circles divided into degrees--and despite (or maybe because of) a handy-dandy button on your calculator that flips radians into degrees, folks venturing into physics and engineering need a smattering of understanding of how degrees work.

An analog clock takes you halfway there

2) "It's quarter past one" doesn't work in digital land; 1:15 flashing in LED red cannot convey any sense of quarters or "half pasts" or any fractions at all.

An analog clock makes fractions visual. Daily.

*Teach baseball

Baseball is our national game. It's being eliminated from the Olympics, true, and the NFL now dominates our national psyche, but baseball remains an integral part of who we are.

Baseball is all about numbers. Batting averages, slugging percentage, earned run averages.

I bet half the kids in my neighborhood grasped the concept of 0.2 vs. 0.3 (a "200" versus a "300" hitter) via baseball.

*Use an abacus

Alas, abaci have already been commoditized --you can own your own franchise!

The Japanese no longer spend a whole lot of time in class studying the abacus (though they do spend a little, likely more for historical interest), but abacus schools outside of class in Japan is all the rage. Studies have shown that kids who master abaci have a better comprehension of math.

An abacus makes the decimal system tangible.

(Along the same lines, I think kids should learn a number system besides our base 10; binary might be helpful, or maybe even hecidecimal if you're feeling flush--kids need the mend-bending antics of using a different base. It helps students grasp both how arbitrary a system as well as how powerful. Might even help them with programming.)

*Count to 100, or 1,000, or maybe even 10,000...at least once a year

OK, this sounds cruel, but bear with me. Sometime this year I will (again) challenge anyone to count to a billion (a second at a time), and several college bound sophomores will jump at the opportunity. In the three plus decades it takes them to finish, I will have likely have died.

Kids (and adults) have little sense of numbers.

Four or five times a year, have the class count aloud to a thousand, a second at a time. Figure it will take almost 17 minutes.

Once a year, try 10,000, and figure on using almost 3 hours. Now it may seem like a waste of time, but certainly no more deadening than a lot of the worksheets prescribed by our fancy textbooks.

Now a story.

My 7th grade teacher was Mr. Knukowski--he wore a bow tie and plaid jackets, and he had a crewcut in 1972.

If we got caught talking in class, he would make us write the word "quiet" a thousand times. Turns out he didn't care who actually wrote it out, just that it was done by hand.

A cottage industry developed in class, kids trading pencils, Twinkies, kisses, whatever, for sheets of paper holding a thousand "quiets. We learned a lot about economics, and the concept of thousand was forever ingrained in our minds, particularly among the gregarious.

*Play with American wrenches

In 1866, the metric system became legal in the States; in the 1970's, there was a huge push to make everything measurable metric. Today, you still need to know the difference between a 7/16 and 1/2 inch wrench.

Now it sounds trivial, but those of us who spent part of our adolescence under a hood got pretty good at knowing which wrench to call for if the 7/32 was a hair too big (or small).

If you have an old bag full of socket wrenches, which is the way most of us carried them, trying to dig through the bag without knowing which wrench was the next step up from a 7/32 cost a lot of time.

The metric system is way too easy--and while the American system is a testament to the stubbornness of our ways ("we don't need no stinkin' metric") and and to how strongly the inch is imprintedin our minds, it also provided immediate rewards for having some clue about fractions, at least the sixteenths and thirty-seconds sort.