Don't Judge a Book by Its Cover

A closer look at the math in the popular facebook Common Core math meme

“This is Common Core” reads the caption on this image, which has over 87,000 shares and 33,000 “likes” on facebook.  Thousands of comments from applied mathematicians, engineers, and concerned citizens have declared the downfall of our educational system due to this “goofy, pointless” method of doing math. Teachers are commenting to declare they are still teaching the “old fashioned” method.

So what’s really going on in this image?

One way to solve thirty-two minus twenty-two is to count up from 12 on your fingers, “13, 14, 15, 16, 17…” all the way up to 32. If you keep track of the count correctly, you will get an answer of 20. Let's call this the counting up by ones method.

The “new method” shows counting up by groups.  Starting at 12, first you count up by ones“13, 14, 15” and write down 3.  Now you can count up by fives, starting at 15, and arrive at “20.” Write down a 5. Now you can count up by tens, starting at 20 and ending at a new current value of “30.” Write down a 10 for this step. At this point, we have to go back to counting by ones, so we think “31, 32” and write down 2. Now, add up the numbers you wrote down 3 + 5 + 10 + 2 and you get 20.

The method described above could be used to check you answer after you solved the problem via the traditional vertical column subtraction method we all know so well.

We don’t have any context on the photo, so we don’t know if the student and teacher were, in fact, checking their work when they made these notes.

Everyone solves math problems in their own way. I personally think counting up by ones (counting up from 12 to 32 on your fingers) is a lot quicker than counting up by the groups 3, 5, 10, 2. My brain works that way. But if you prefer to count by groups, go for it!

What if I asked you to solve $5,000 - $4,995? You can probably do this in your head and get 5. If we wrote out how you solved the problem, it would look like 4995 + 5 = 5000. That’s the so-called “new method.” If we tried to solve this by subtracting in columns (or the “old fashioned” method), we would be required to borrow in the ones, tens, and hundreds places. For some people, that’s no problem, but many people prefer not to do subtraction with borrowing. Counting by groups makes solving this second example problem a breeze.

What does this image have to do with Common Core?

Common Core suggest students master certain skills (say, subtraction) by certain grade levels. It does not force math teachers to use any specific form of math instruction.

This is one of many ways of explaining subtraction to elementary school students who have just learned addition. Just go with it!

A talented tutor will adapt to the method that the student prefers to use.

Follow the Math: The World Ends Today (May 21, 2011)

As you know, radio host Harold Camping has told us that the world is ending today. Please follow along with his ironclad logic:

• The number five equals "atonement", the number ten equals "completeness", and the number seventeen equals "heaven".
• Christ is said to have hung on the cross on April 1, 33 AD. The time between April 1, 33 AD and April 1, 2011 is 1,978 years.
• If 1,978 is multiplied by 365.2422 days (the number of days in a solar year, not to be confused with the lunar year), the result is 722,449.
• The time between April 1 and May 21 is 51 days.
• 51 added to 722,449 is 722,500.
• (5 × 10 × 17)^2 or (atonement × completeness × heaven)^2 also equals 722,500.

I think his math is correct, but how's his logic?

(Glad I published this before the Rapture! Whew!!)

Summer Reading Assignment

When I was a freshman in high school, I had a number of disagreements with my parents and teachers about the types of long-haired boys I idolized. For a final poetry analysis assignment, I chose to compare and contrast the poetry of Cream’s “Strange Brew” to Type O Negative’s “Black No. 1,” complete with footnotes about the Nosferatu references. My teacher reduced my grade and wrote at the top, “Good job although your choice of poetry is rather ‘dark.’” I was outraged at the censorship and the battle was on. The war ended with me getting a “D” for my final semester freshman year in English class.

The academic year was over, so my parents couldn’t punish me with an incentive system for increasing the grade. Instead, they devised a more constructive teenage rehabilitation program: 10 book reports over the course of the summer. They told me to drop the setting/characters/plot summary format from grade school and just write a couple of paragraphs analyzing the book or convincing them why they should read it. My mom showed me literary magazines and professional library journals so that I could see how librarians and academics were introduced to books.

I survived the most evil punishment in the world…
“Can I have a ride to Heidi’s house?”
“Sure, write a book report.”
…and turned it into job as a freelance writer and reviewer.

In February of 2008, Bill Warford of the Antelope Valley Press interviewed me about my career as one of the Top 500 book reviewers on Amazon.com. In the article, he cited my parents’ “creative punishment” for creating this book-reviewing monster. I’ve been having fun sharing my thoughts on books ever since the summer of 1995. Thank you, Mom and Dad.

Oh, and guess what? One of the students I mentor just finished her freshman year and her parents asked me for ways to keep her mind fresh over the summer…

Check out my website

I launched my website last month: http://www.luxtutor.com/

Alice's Adventures in Symbolic Algebra

British mathematician Charles Dodgson published nonsense fiction under the pseudonym Lewis Carroll. The fantastically absurd adventures of Alice in Wonderland are filled with Dodgson’s rants against the radical new math ideas appearing in Victorian universities.

Dodgson was a conservative, cautious mathematician who tutored at Oxford to earn extra income. He faithfully followed the principles laid out by Greek mathematician Euclid in the textbook Elements. You probably remember Euclid’s rigorous reasoning from your high school geometry class: start with accepted truths (axioms), use logical steps to build increasingly complex arguments, and sign off the mathematical proof with the Latin acronym Q.E.D. (for quod erat demonstrandum, or “that which was to be demonstrated").

You studied symbolic algebra in high school, but it was an emerging concept in Victorian times. Under the rules of arithmetic, if I have one apple and you give me two more, then I have 1 + 2 = 3 apples. With the power of algebra, we can think of apples as variables. So, x is number of apples I have, and you give me two more every day, so on any given day, I have x + 2*y apples, where y is the number of days that have passed. Variables are very useful but they are abstract. As abstract, say, as the Cheshire Cat disappearing gradually until nothing is left but his grin. Alice remarks that she has often seen a cat without a grin but never a grin without a cat, which is a commentary on this new abstraction in math.

The concept of imaginary numbers was once a radical idea. We all know that 2 * 2 = 4 and -2 * -2 = 4. So the square root of 4 is √4 = ±2. But what if we want to take the square root of a negative number? The square root of -4 doesn’t exist. So, we invent the imaginary number i, define it as i² = -1 and now we can solve the problem that had no answer. √-4 = √-1 * √4 = 2i. Just use the imaginary number i and now it works! You will have to accept the concept of i to pass American high school math standards, but Charles Dodgson was a traditionalist who wrote Alice in Wonderland to satirize absurd concepts like these.

Irish mathematician Sir William Rowan Hamilton took this imaginary number nonsense to an entirely new level by inventing quaternions.The complex number 2i has a real part (the number 2) and an imaginary part (i).Quaternions extended complex numbers into four parts instead of two. The algebraic properties like x + y = y + x do not apply to quaternions. Dodgson was understandably frustrated with new math that violated the commutative property!

Enter the Mad Hatter and his tea party. Alice is at a dinner party with three strange characters who have kicked the fourth character (Time) out of the room. Mathematician Hamilton could only describe rotations in a single plane (the movement of the characters around the dinner table) until he introduced the fourth variable of Time to his quaternion logic. When the Hatter tells Alice that “I see what I eat” is not the same thing as “I eat what I see,” he is telling her that the commutative property does not apply here. In this world, x times y is not the same as y times x! When the scene ends, two of the characters are trying to stuff the third into a teapot, so they can get out of this endless rotation and go back to being a complex number with just two parts.

Alice in Wonderland is Charles Dodgson’s satire of what he perceived to be semi-logic among his mathematical contemporaries. His fiction takes the new abstract math concepts to their logical conclusions using Euclid's rules for mathematical proofs, with sometimes mad results. In the chapter Pig and Pepper, Dodgson uses French mathematician Jean-Victor Poncelet’s continuity principle to turn a baby into a pig, demonstrating the absurdity of modern projective geometry. Dodgson’s lesson to Alice? Follow the rules of Euclid’s geometry to survive in Wonderland. Keep your ratios constant, even if your size changes after eating mushrooms with the Caterpillar.

Learn more:
The NPR story which first introduced me to Lewis Carroll as Charles Dodgson, mathematician
Melanie Bayley's PhD dissertation on Alice's Adventures in Algebra

Pi: Irrational but well-rounded

Happy Pi Day! What a good day to found my blog about private tutoring. I'll be back soon to write more about the hidden math arguments in Alice in Wonderland.