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Don’t do algebra with numbers!

Posted in Rant, tagged , , , , , on November 30, 2012| 2 Comments »

As a physics professor, I have certain pet peeves.  For example, I cringe when someone says that “gravity” is 9.8 m/s2 when they mean the acceleration due to gravity.  I’m annoyed if someone says that an object “weighs” 7 kg.  And I stifle a laugh if someone says that a roller coaster is exciting because it goes so “fast”—humans can only detect acceleration, not speed, which is why we don’t notice that we’re traveling something like 67,000 mph right now in our orbit around the sun.

goose

“I feel the need for acceleration!”

But my biggest pet peeve may be students doing algebra with numbers.

Fellow physics professors will know exactly what I’m talking about, but for the uninitiated, here’s an example:

If you drop an object from a height of 20 m, how long will it take to hit the ground?

A student knows that a kinematics equation is needed, hits upon the correct one, Δy = vi Δt + (1/2) a Δt2, and then correctly identifies Δy = –20 m, a = –9.8 m/s2, and vi = 0.  So far, so good.  They’ve studied their physics, right?  What happens next is sheer madness:

algebra_with_numbers

Sigh.

Over and over again I tell students, “don’t plug numbers in until the end.”  But students love plugging in numbers.  They feel they’re actually getting closer to the answer if they’re manipulating numbers.  On some level, they still feel uncomfortable with letters—as if manipulating letters isn’t really “math”.

How does this problem look in my answer key?  Like this:

algebra 2

You can now plug in values if you like…and get Δt = √[2(-20)/-9.8] = 2.02 s.

Which of these approaches is more beautiful, more powerful?  The approach you pick indicates whether you “get” algebra or not.  If you do algebra with numbers, the answer you get is very narrow and very specific, even if you do it correctly.  That hypothetical student could have gotten 2 seconds as an answer, and I would have given them full credit.  But their answer would have been ugly.

The second approach is beautiful, because it is completely general and applicable to multiple situations.  I try to tell students “Look!  You found the time to fall a certain distance.  You now know the answer no matter what the height is, and even no matter what planet you’re on, since g doesn’t have to be 9.8 m/s2.”  This is usually followed by a blank open-mouthed stare, much like Kristen Stewart in a Twilight movie.

There is a more practical reason to avoid doing algebra with numbers.  It’s simply that when you do algebra with numbers, other people cannot follow your work as easily.  And then, if you make a mistake, it’s harder for someone else to spot.  Quick: what algebra error did the student make above?  It takes a while to find the mistake.

My ultimate point is that students need experience seeing the power of algebra.  It’s all well and good that algebra classes stress real-world applications—else, why teach algebra in the first place?  But real-world doesn’t only mean with numbers.  E=mc2 is certainly a real-world application of algebra, and it’s a lot more elegant than saying that 378,000,000,000,000 Joules is released when a teaspoon of sugar with mass 4.2 grams  is converted to pure energy, given that the speed of light is 300,000,000 m/s.  The hard part, for us physics professors, is to help this spoonful of algebra go down.

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Einstein’s “Imagination is more important than knowledge”

Posted in Uncategorized, tagged , , , , , , on November 26, 2012| 8 Comments »

Poor Einstein.  Is there anyone else who is misquoted more often?  Is there anyone else to whom more nonsense is attributed?

I have no desire to rehash things that Einstein said about “God”.  Einstein was by all accounts an atheist, an agnostic, or a pantheist—depending upon your definitions—and various religious apologists have been trying to co-opt the man for years by misquoting him.  Others have already discussed this at length.

My goal today is to tackle that old chestnut, “Imagination is more important than knowledge,” as seen on T-shirts, bumper stickers, and even on the packaging of the Albert Einstein action figure.  Did Einstein really say this, and if so, what did he mean?

Here’s the quote in context:

“At times I feel certain I am right while not knowing the reason.  When the [solar] eclipse of 1919 confirmed my intuition, I was not in the least surprised.  In fact I would have been astonished had it turned out otherwise.  Imagination is more important than knowledge.  For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. It is, strictly speaking, a real factor in scientific research.”  [From A. Einstein, Cosmic Religion: With Other Opinions and Aphorisms, p. 97 (1931).]

So Einstein did say this.  However, I maintain that the full quote in context has a different feel to it than the quote in isolation.

When I see “Imagination is more important than knowledge” on a bumper sticker, I think this: “Flights of fancy and imagination are more important than learning stuff.  So why should I study?  Einstein didn’t study.  He just sat around and daydreamed and came up with the most remarkable breakthroughs about the workings of our universe.  Imagination is more important than learning all the proofs and figures ranged in columns before me.  So I am going to follow good ol’ Einstein and daydream about being Batman.”

The New Age meaning of the quote is this: “I’d rather daydream than study.”  It’s Walt Whitman’s “learn’d astronomer” nonsense all over again.

In context, it’s clear that Einstein was talking about doing science.  Imagination is more important in making scientific breakthroughs than knowledge, but that doesn’t mean that knowledge is not important.  Einstein worked very, very hard to learn an awful lot of physics.  By all accounts, it took him almost 10 years to flesh out general relativity, during which time he had to acquire a lot of mathematical knowledge about Riemannian geometry and tensor analysis.  The “intuition” that Einstein developed during this time frame is what allowed him to be so confident of the results of Eddington’s expedition.  What Einstein calls “intuition” is just knowledge that has become so ingrained that you are no longer cognizant of it.

Einstein may have been more famous than most of his contemporaries, and it was probably due to his superior imagination.  But take Einstein’s imagination today and give it to a twenty-five year old high school dropout, and he’d be lost in obscurity, stocking shelves at Wal-Mart.  Imagination is more important than knowledge.  But only slightly more.

[Note: my book Why Is There Anything? is now available for download on the Kindle!]

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Missouri Proposition B: Know your algebra!

Posted in Uncategorized, tagged , on November 5, 2012| 2 Comments »

As I was driving to school today, there was a story on NPR about an upcoming ballot initiative in Missouri.  Proponents want to raise the cigarette tax from $0.17 per pack (the lowest rate in the nation) to $0.90 per pack.  The idea is to generate some revenue and at the same time discourage smoking.

Missourians love their cancer sticks.

Now, I’m not going to go into the pros and “cons” of the initiative.  I would be for it, based on some pretty common-sense data, but I don’t live in Missouri so I guess my opinion doesn’t matter.  However, I’d like to comment on some squishy reasoning put forth on the radio by an opponent of the measure.

In a nutshell: “The measure,” the opponent says, “wouldn’t raise money at all.  It would actually lower revenue.  That’s because any gains made by raising the tax would lower the amount of regular sales tax accrued.” [Note: this isn’t a verbatim quote but a recreation based on my imperfect memory]

You see, Missouri has a 4.225% sales tax which also applies to cigarettes.  And, the “logic” goes, if less people are buying cigarettes, the less sales tax revenue would be generated.

Really?  Is this person completely innumerate?  Let’s say that a pack of cigarettes costs $6.00.  Without the ballot initiative, a Missourian would pay $6.00, plus $0.25 in sales tax, plus $0.17 in cigarette tax, for a total of $6.42.  The state gets $0.42 cents a pack.

Under the new plan, a smoker pays $6.00, plus $0.25 in sales tax, plus $0.90 in cigarette tax, for a total of $7.15.  The state gets $1.15 cents a pack, obviously an increase.

To be fair, the state does actually lose money if the gain in revenue per pack is offset by the loss in the number of sales.  But when would this occur?  Let’s use algebra to find out.  Suppose that N1 is the number of packs of cigarettes bought in Missouri in a given year.  Under the current system, and for a $6.00 average price per pack, the yearly intake for the state is just N1 x 0.42.  Now let N2 be the number of packs bought under the new plan.  It’s easy to see that if the proposal passes, the yearly intake for the state is N2 x 1.15.

The problem is that N2 < N1 (presumably).  So the proposal loses money if N1 x 0.42 > N2 x 1.15, which, after one line of 8th-grade algebra, is the same as N1/ N2 > 2.74.  That is, translated into English, the new proposal loses money compared to the current system if state cigarette purchasing goes down by a factor of almost 3.

I don’t know about you, but I seriously doubt this initiative will cut smoking in Missouri by that much.

The numerically savvy will notice that there is a subtlety.  This result is for a $6.00 pack of cigarettes.  What if the price is much different?  This matters, because the cigarette tax is a flat number added to a pack, whereas the sales tax is a percentage.  If cigarettes cost the same as houses the ballot initiative would be ridiculous, since 4.225% of a huge number is much, much greater than a measly $0.90.  No one would buy cigarettes at all, and the state would lose a lot of revenue.

Is there a tipping point?  That is, is there a price for a pack of cigarettes for which the proposal loses money for any decrease in purchasing? Surprisingly, the answer is no.  Mathematically, we would say that N1/ N2 = 1 only when the price of a pack is infinite.

But setting the bar at N1/ N2 = 1 is unrealistic.  Any tax hike will cause some smokers to buy less.  So let’s make an educated guess:  looking at the first graph at this site, I can see that a price hike of $0.73 produced a corresponding decrease in consumption of about 25%.  (The actual drop would probably be less than this, since in the 1980’s and 1990’s $0.73 represented a higher percentage of the cost of one pack).  So we let N1/ N2 = 1/0.75 = 1.33, and solve for the price of cigarettes.

You get $48.33.  For one pack of cigarettes.

Conclusion: the proposed tax increase would increase revenue even with a plausible decrease in sales, unless cigarettes cost around $48 or more per pack.

Yay algebra!

Some close analogue to Mark Twain or E. B. White (with so many misquotes on the web, you can’t be very sure these days) said that analyzing humor is like dissecting a frog: you don’t learn much, and the frog dies.  So why did I dissect this frog, and show all the gory math details?

I wanted to point out that there is a lot more mathematical detail in most stories you hear, most issues you examine, than you suspect.  And I want to emphasize: if you don’t know math, if you are not mathematically literate, then you don’t know much of anything.

There.  I said it.

Go study your algebra.

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