Is Math a Myth?

There are those who are calling for scaling back mathematics education. One “public intellectual” (whatever that is), Andrew Hacker, has even written a book on the subject: “The Math Myth.”  Hacker loves to use words like “inflict” rather than “teach,” and wonders why we torture young Americans with math education in these days of computers and smart phones. (More on Hacker later).

Here is one reason. Mathematics is the science of reasoning. You might think that of little use, but you must use reasoning to weed out the arguments of political hacks and charlatans every election season. Here is an example.

On November 2, many newspapers ran a political cartoon by Jim Morin of the Miami Herald. The target of Morin’s partisan jibe was those who are concerned about the increasing expense of “Obamacare” premiums.

In the cartoon, a large, rotund loutish fellow, labeled “Health Insurance,” holds the message  “George W. Bush Years (up) 100%.” Next to him is a small, rotund fellow with the message “Obamacare (up) 25%.” Finally, a frenzied character, apparently Republican, is shouting “OH NO, WE NEED TO REPEAL IT!”

Here is Morin’s reasoning:

• Health insurance premiums increased 100% over the Bush years,
• Obamacare premiums are projected to increase only 25%,
• Therefore those concerned about Obamacare increases are hyperpartisan, hysterical idiots.

But, in truth, Morin is either preying on your mathematical ignorance or is a mathematical ignoramus himself. Neither interpretation is flattering.

Over the eight years of the Bush presidency, health insurance premiums did indeed increase about 100%. However, Obamacare premiums are projected to increase 25% this year alone. These two numbers can’t be directly compared because they occur over two very different timeframes.

It’s like saying that Sally made 25 dollars this year and Joe made 100 dollars altogether over the past eight years and then claiming that Joe makes a lot more money than Sally. If we annualize those earnings, Sally makes $25 per year while Joe makes only $12.50 per year ($100 divided by eight).

To compare the two health insurance rates of increase, we must find a common time scale. With a few simple calculations, we find that health insurance premiums increased approximately 9% per year over the eight Bush years. In fact, the Obamacare increase is nearly three times that of Bush on an annualized basis. Morin’s thesis is bankrupt.

Back to Andrew Hacker, who believes that your children are wasting their time in mathematical training. Let’s see how that works in reality.

In late August of this year, Hacker was interviewed on the weekly NPR show “Science Friday.” A political scientist by trade, Hacker is teaching a course called “Numeracy 101” at Queens College which is intended to impart a minimal, but adequate, amount of mathematical training. As a practical exercise, working with his students, Hacker calculated the answer to this question: “What is the ratio of black people killed by police as opposed to white people?”

Hacker breathlessly announced their findings: ” We’re the only ones who’ve discovered it. It’s a public statistic. For every 100 people killed by police, white people, 270 black people are killed. OK?”

Here is mathematical dilettante Hacker crunching numbers to support his liberal belief in racist police officers who kill 2.7 black people for every white person. The NPR audience, surely, ate it up.

But the truth may be a hard master. The Washington Post has been maintaining a database of police shooting statistics for several years based on “public information, news reports, and social media.” They believe it to be not perfect, but quite representative.

In 2015, the Post reports that 494 whites were killed by police. Applying the Hacker ratio, we would expect that 1,334 blacks would have been killed. But such is not the case. The WaPo reported 257 black deaths, a regrettable number, but an order of magnitude less than Hacker’s claim.

In this day and age, it is vital that citizens and voters attain and maintain a modicum of mathematical literacy. It is required to detect and debunk the claims of those aiming to sway you. These claims will be many, and you must question them if they don’t pass the smell test.

We may yet regret our collective decision refusing to expand charter schools. Match Charter in Boston, for example, serving inner city kids, delivered the astounding result of 97% of 10^th graders proficient or advanced in math, compared to 54% of district students.

We need more of that, not less.

It's Long Division

Arithmetic seems to terrorize a significant number us, students both young and adult. And nothing is more emblematic of that terror as long division.

The alternative rock band Fugazi captures the sense:

It's a long time coming,

It's a long way down,

It's long division,

Crack and divide.

There is nothing more daunting to students we have tutored than long division. It seems a special torture invented in Hell for their personal torment.  But it need not be so.

Arithmetic was invented largely to keep track of human trade and commerce. How else would one keep things straight? Purchased 200 wheels of cheese, sold 192, lost 8 to vermin, made a modest profit. Without arithmetic, it would be impossible to track.

These early traders weren’t schooled mathematicians, but they weren’t stupid. They learned how to mark sticks or make imprints on clay tablets to record numbers. They added and subtracted to determine sales and inventories and profits and losses. Arithmetic was not an abstraction but the means to a very important end.

At some point they realized that when a customer ordered five crates of clay pots, each crate containing 10 pots, there might be a quicker way to determine the total of pots needed than to add 10 together 5 times. Hence was eventually born multiplication, with a variety of  creative solutions: the Ancient Egyptian method, the Russian Peasant method, the Gelosia method. Human ingenuity knows no bounds when trying to avoid tedious labor. These methods required a few brain cells but were better than adding a column of the same number, over and over and over.

But, then, technology forced us to take many steps back.

Imagine in the 1940s, the slowest, most unwitting arithmetician among us. That would be an electromechanical computer. The computing machines of Alan Turing’s day (of “Imitation Game” fame) were notably obtuse. Huge, hot, electric sparking and fumes, they could add a column of numbers not that much faster than you could. They had but two fantastic advantages: absolute accuracy and immunity to fatigue.

Oddly enough, though, these electrical behemoths didn’t know how to subtract. Or multiply. Or divide. All they knew how to do was add.

So how did the machines of Turing’s era perform subtraction and multiplication and division when all they knew was addition?

Addition is a basic, primitive operation. If you are given two apples, then three more, you will have a total of five apples.

                2 + 3 = 5

Even the Common Core methods will agree with that.

But say you are given five apples, then two are taken away. We know the answer is three – we did the subtraction in our heads. But Turing’s machine can’t subtract. So, brilliantly, we instruct it to add five and the negative of two:

                5 + (-2)= 3

Still using only the basic operation of addition, the machine has succeeded in subtracting by simply adding the negative of the subtrahend (the number being subtracted). We can now say that the machine knows how to subtract:

                5 – 2 = 3

…but in fact, we know that it is adding the negative to do so.

How about multiplication? That is simple for the sparking behemoth as well. After all, it is tireless, and simply replicates the feat of early humans  before the many multiplication methods were invented. It merely adds, repeatedly. Five cases of ten clay pots is calculated thusly:

                10 + 10 +10 + 10 + 10 = 50

Multiplication is just repetitive addition. (With some fussing with decimal points, but we can ignore that for the time being).

Now for the promised key to long division. If multiplication is repeated addition, might you suppose that division is repeated subtraction? Yes, you might, and you would be absolutely correct.

Let’s try to divide 49 by 12 given this method. Keep track of how many times we can subtract 12 and stop only when the balance is not greater than 12.

                49 / 12

                49 – 12 = 37         (1^st subtraction)

                37 – 12 = 25         (2^nd subtraction)

                25 – 12 = 13         (3^rd subtraction)

                13 – 12 = 1           (4^th subtraction)

                1 left over

The answer is 4 with a remainder of 1. That’s division, using only subtraction. And remember that subtraction is just the addition of negatives. This is how Turing’s useful idiot, the sparking monster that could only add, was able to subtract and multiply and divide as well.

The moral of this story is that arithmetic is ancient. It is rooted in the real world, not an abstract implement of student torture.  And, hopefully, by realizing that a quite stupid computer can do this, you will have the confidence that you can, too.

Now, to those of you who found this all very simplistic, it’s time to volunteer as a tutor. Some yearning student needs your insights. This is how you give back.

The Hitchhiker's Guide to Mathematical Balderdash

By the time you read this, the midterm elections of 2014 will have passed. You will be either ecstatic with the peoples’ decision or deeply disappointed. In either case, it’s time to take a breather and think of more ethereal things.

The internet is fascinating. The topics which ebb and flow on social media are a revealing window on the subject of human thought, susceptibility, and superstition. Perhaps a lesson or two to learn here.

For instance, a recent post making the rounds is typical, piquing our interest and suggesting some magical properties. It proposes that our shoe size can predict our age.

One of a class of such postings, this one posits that your shoe size can predict your age as follows:

1.       Take your shoe size (whole number, round up if necessary)

2.       Multiply it by 5

3.       Add 50

4.       Multiply by 20

5.       Add 1014

6.       Subtract the year you were born

And voila, the result is your shoe size as the leftmost digits and your age as the rightmost. “It’s magic!” proclaims the post. Balderdash.

Let’s take this apart, understand it, and identify its limitations.

Our first task is to express this as a simple expression:

(SHOE x 5 + 50) x 20 + 1014 – BIRTHYEAR

Let’s try it assuming a shoe size of 9 and birth year of 1971.

(9 x 5 + 50) x 20 + 1014 - 1971

The result is 943. Shoe magic, indeed! This person’s shoe size is 9 and age 43!

Dang – how did it know?

Rest assured, there is no mystery here. Let’s apply a little math. Starting with the first expression, we can reduce and represent it as follows:

((SHOE x 5) + 50) x 20 + 1014 – BIRTHYEAR

SHOE x 100 + 1000 + 1014 – BIRTHYEAR

SHOE x 100 + 2014 – BIRTHYEAR   

                                                                             

Now it begins to make a bit more sense. SHOE x 100 shifts the shoe size to the left and 2014-BIRTHYEAR yields the person’s age. Adding them together gives us the result:  943 in this case.

A few things might become obvious to you at this point. The use of shoe size is completely arbitrary. We could use hat size or the number of cups of oatmeal in your breakfast or any other number. Shoe size, per se, has nothing at all to do with it.

If you play around, you will also find that if the birth year is over 100 years ago, the calculation breaks down. Also, once we get to next year (2015), the age calculation is no longer valid: it only works for 2014 because of the constant “1014” in the original expression.

No magic at all, this is a cheap algebraic parlor trick.

Another recent math problem making the rounds of social media raised more acrimonious debate than that of the supporters of Senator Elizabeth Warren vs. those of Senator Ted Cruz. My goodness, math has only one right answer, what is the grounds of debate?

This one is based on mistaken assumptions, perhaps a cautionary note in all of our dealings.

Take a look at this expression:

36 / 6 x 3 + 2

In other words, 36, divided by 6, times 3, plus 2. But in which order do we apply these operations?

One option is to divide 36 by six first, then multiply by 3 , finally adding 2. The answer would be 20.

18 + 2 = 20

Another alternative is to multiply 6 by 3 first, the divide into 36, finally adding 2 yielding 4.

36 / 18 + 2 = 4

Two distinct answers, 20 and 4. Which is correct? The wrong answer could blow up the next resupply mission to the International Space Station (math applied to the real world can be really important).

There is a guide known by the acronym of PEMDAS which describes the order in which operations are to be performed:

1.       Parenthesis

2.       Exponents

3.       Multiplication

4.       Division

5.       Addition

6.       Subtraction

This guide is a gentleman’s agreement meant to remove ambiguity. But the prescription leaves itself open to misinterpretation.

A common (incorrect) assumption is that multiplication comes first relative to division but, in fact, multiplication and division are of equal weight  (as are addition and subtraction). When operations are of equal weight, they are processed left to right as encountered. The correct way to interpret PEMDAS is PE(MD)(AS), meaning;

1.       Parenthesis

2.       Exponents

3.       Multiplication and division, equal weight, left to right

4.       Addition and subtraction, equal weight, left to right

So the proper way to evaluate the above expression is as follows:

36 / 6 x 3 + 2

6 x 3 + 2

18 + 2

20

If you guessed 20, you win the prize!

Enough of the numbers. Revel (or commiserate) the recent election results, and embrace a bold new confidence in debunking the Internet’s mathematical puzzles. You can do it!

Math is the Path to the Middle Class

It was great news for Rhode Island recently. General Dynamics Electric Boat has been awarded a $17.6 billion contract to build ten Virginia-class attack submarines and the Quonset Point facility will be adding 450 jobs. Some commuting workers from Massachusetts may benefit as well.

This is consistent with President Obama’s pledge to build the economy from “the middle out.” The wages from these jobs would inject an additional $30 million into the local economy with beneficial ripple effects (the “multiplier”) boosting barbers and bakers and candlestick makers.

But not so fast. Once of the most common positions on the Electric Boat website is a “QP Inside Machinist.” This job is described as follows:

“Set up, program editing and operation of CNC milling and turning centers. Verifying part configuration to plan requirements using various high tolerance precision measuring tools; must be able to work independently and with minimum supervision.”

From the list of qualifying requirements, this one stands out: Strong mathematical skills in geometry & trigonometry preferred.

Oops. How skilled are our recent high school grades in math? Are they ready for the rigors of the workplace? Or, rather, have they succumbed to our cultural aversion to math?

In Hollywood, only geeks and geniuses (e.g., Matt Damon in “Good Will Hunting”) are good at math. The cool kids steer clear. It’s too hard. It’s not cool. It’s the butt of jokes.

But as can be seen, math may be a qualifying requirement for a well-paying middle class job. And as our information-based economy continues to unfold, this will be increasingly true.

Long gone are the days when one can make a good wage based on the sweat of one’s brow. Lifting 50 pound bags of flour is now relegated to pallet jacks, with a single operator displacing dozens of Italian immigrant mothers who had previously been paid to stack tons of goods. We hate technology, but each of us with a smart phone is embracing it. The workplace has changed.

But have our cultural and educational systems changed apace? Has mathematical literacy, numeracy, become increasingly desirable and culturally acceptable? Alas, it has not.

It is shockingly apparent that we have not prepared our kids for the new workplace. They believe math is hard, uncool, geeky, and hence, avoid it. But if the alternative is a minimum wage job as a barista, are we serving them well?

A plethora of studies have shown that high school math skills are correlated to higher earnings later in life. And not just earnings, math ability also eases our way through the increasingly complex thicket of everyday life. A 2013 study done by researchers at Princeton University found that in the financial meltdown of 2008, poor basic math skills correlated strongly with mortgage defaults. Controlling for all other factors (age, ethnicity, education, household income), the researchers studied hundreds of subprime mortgages across New England. Their findings were surprising in that it wasn’t specifically the choice of mortgage contract that led to default, but rather other life behaviors indicating poor overall financial decision making.

How to motivate kids to learn math and teach them more effectively?

One whimsical thought is that if only the media, Hollywood, and sports idols could embrace this cause, things might be different. We have shifted culturally against smoking tobacco and in favor of gay rights, why not a campaign to make math acceptable? It’s too easy to laugh at math geeks, as witnessed by “The Big Bang Theory” whose innumerate character played by Kaley Cuoco wins our affection.

A more serious route being debated by educators is to teach math in context. For instance, high schoolers should be taught basic financial skills and, in the process, exponents and logarithms. That is how interest calculations and amortization tables are made, why not learn how in context and not as part of an abstract course in algebra? This could be amplified by having chemistry and biology teachers, for instance, explaining the math required to understand their subject matters.

After all, math was not invented as an abstract topic. Fractions were an outgrowth of commerce, where early merchants needed to portion out fractional bushels of grain or wheels of cheese. Multiplying 2/3 times 4 was a practical exercise, not something dreamed up to torture a fourth grader.  Likewise, geometry and trigonometry were developed from the building trades and nautical navigation, not as an abstract brain teaser for high schoolers.

This won’t be solved anytime soon, but we must make numeracy a top goal. Those well-paying jobs at Electric Boat are awaiting.