Necessary math education

While I'm away, I thought this little opinion piece by William Falk, editor-in-chief of my favorite magazine, The Week, deserves some close attention:

I recall nothing of trigonometry and physics, except the feeling of nausea as the teacher filled the blackboard with a mystifying jumble of numbers and letters (the infernal, elusive x!). Higher math, I assumed at the time, was created for the sole purpose of adding to teenagers’ misery and self-loathing—the academic equivalent of acne. This awful memory comes to mind because my daughter Jessica is now taking trig and physics in her junior year of high school. Jessie is a better math student than I was, but she shares the family predilection for English, history, and verbal subjects, and will not be making a career in engineering or science. So why must she break her brain on quadratic equations?

Two brave mathematicians have stepped forward to argue she need not. In a column in The New York Times, Sol Garfunkel and David Mumford say algebra, trig, and calculus are wasted on those of us with no aptitude for higher math. Students clearly not headed for science or engineering careers, they propose, should track to courses that provide “quantitative literacy”—the ability to handle our own finances, understand percentages and probability and risk, and intelligently assess what “experts” like banks and doctors and politicians tell us about the mathematics of real life. The desperate need for literacy of this sort is indisputable: The average American carries more than $6,000 in credit-card debt; about half of all retirees have saved less than a quarter of what they will need; and our elected leaders convince the gullible it’s possible to balance budgets while preserving their benefits and cutting their taxes. Why keep fiddling with x, while Rome burns?

In case you want to see the original article to which Falk refers, check out How to Fix Our Math Education.

And, if you haven't subscribed yet, may I encourage you seriously to consider The Week? Sarita and I love it!

Math for the pure joy of it . . .

The visuals and commentary are about two times too fast for me to follow with great understanding, and I don't want to take the time now to re-watch these videos so that I can understand exactly what she is talking about, but Vi Hart reminded me of a joy I used to experience many, many years ago. For various reasons, I abandoned the pursuit. It gives me pleasure to see that others, however, think along these lines. I never doodled the way she does, but I used to find patterns like these rather fascinating.



[Please note that I have embedded YouTube versions of two of her videos here in my blog--just to attempt to entice you to actually watch one or both of them. BUT . . . if you want to understand them and really get "into" them, I encourage you to follow the text links . . . because she provides useful Wikipedia links that explain the more technical terminology and concepts. The text links will also bring you to her website that includes a lot of truly remarkable videos and technical (and not-so-technical) papers . . . about math and music and . . . well . . . art . . . and things of beauty . . . and lots of other stuff.]

Thanks, Luke, for the originating link that led to the video that led me to check out Vi's website:

Sociology 101: counting and calculating

I've talked before about my penchant for sociology or cultural anthropology.

As I've been reading through 1 and 2 Kings, I was struck by another cultural difference: How the ancient Hebrews counted--or expressed numbers--compared to how we would express them in our culture today.

See if this strikes you.

Let's start with the observation that Jeroboam and Rehoboam began to reign in their respective kingdoms at the same time: Jeroboam over the Kingdom of Israel; Rehoboam over the Kingdom of Judah.

1 Kings 14:21 and 31: "Rehoboam . . . reigned for seventeen years. . . . And Rehoboam slept with his fathers. . . ."

1 Kings 15:1 and 2: "Now in the eighteenth year of King Jeroboam, . . . Abijam began to reign over Judah. He reigned for three years. . . ."

1 Kings 15:9 and 10: "In the twentieth year of Jeroboam king of Israel, Asa began to reign over Judah, and he reigned forty-one years. . . ."

1 Kings 15:25: "Nadab the son of Jeroboam began to reign over Israel in the second year of Asa king of Judah, and he reigned over Israel two years."

1 Kings 15:27 and 28: "Baasha the son of Ahijah . . . conspired against [Nadab]. And Baasha struck him down. . . . So Baasha killed him in the third year of Asa king of Judah. . . ."

--I could continue the series. (I have done so in my own reading.) The pattern remains consistent. Have you caught what is going on?

• Abijam comes to power in the 18th year of Jeroboam.
   
• Abijam reigns for 3 years and Asa, his successor, comes to power in the 20th year of Jeroboam and will reign for 41 years.
   
• Nadab comes to power in the 2nd year of Asa's reign.
   
• Nadab reigns for 2 years before he is killed in the 3rd year of Asa's reign. . . .

Catch that?

• A three-year reign takes place when a king comes to power in the 18th year of another king's reign and dies in the 20th year of that other king's reign.
   
• And a two-year reign occurs when a king comes to power in the 2nd year of another king's reign and dies in the 3rd year of that other king's reign.

Normally, the way I look at things, if someone reigns from the middle of the eighteenth to the middle of the twentieth year, I would say he reigned for two years. Similarly, if someone reigns from the middle of the second year to the middle of the third year, I would say he reigned for one year.

But, according to how the writer of 1 Kings records it, I would be wrong. The correct answers are three years and two years, respectively.

Best I can make out: If you reign in the 18th year, then you have reigned for one year. If you reign in the 19th year, then you have reigned for a second year. And if you reign in the 20th year, then you have reigned for a third year: three years, total.

Similarly, if you reign in the 2nd year, then you have reigned for one year. And if you reign in the 3rd year, then you have reigned for a second year: two years, total.

And so forth.

Check it out!

The pattern continues at 1 Kings 15:33; 16:8, 10, 15, 23, 29; 22:41-42, 51; 2 Kings 3:1; 8:16-17 . . . --And that's as far as I've gotten.

Bible math

As I committed myself last year, so I've committed myself this year: I want to read through the Bible in one year.

And on Monday I came across some numbers that simply "don't make sense" to me. I'm wondering how anyone else might reasonably interpret them.

In the Book of Numbers, chapters 1 and 2, if I'm reading the text correctly, we get an enumeration of all males "from twenty years old and upward, all who were able to go to war" (1:20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, and 45-46; also 2:4, 6, 8, 11, 13, 15, 19, 21, 23, 26, 28, 30 and 32). --I actually took the time to align all the specific numbers mentioned in chapter 2 to the numbers mentioned in chapter 1; and I checked the math to see if all the numbers added up. (They did.)

As I read this unbelievably repetitive portion of text, I kept wondering:

Why does the author keep repeating himself? Why not simply say something along the lines of, "The number of men from twenty years old and upward able to go to war, numbered by clan, by their fathers' houses, were as follows: From the the tribe of Reuben, 46,500; from the tribe of Simeon, 59,300; from the tribe of . . . "? Why, instead, this lengthy repetition of verbiage:

The people of Reuben, Israel's firstborn, their generations, by their clans, by their fathers' houses, according to the number of names, head by head, every male from twenty years old and upward, all who were able to go to war: those listed of the tribe of Reuben were 46,500.

Of the people of Simeon, their generations, by their clans, by their fathers' houses, those of them who were listed, according to the number of names, head by head, every male from twenty years old and upward, all who were able to go to war: those listed of the tribe of Simeon were 59,300. . . .

And so on and so forth.

Unbelievable repetition!

But I slogged through. And the ultimate summary of both chapters--chapter 1 and chapter 2--is that there were 603,550 men 20 years old and upward who could be expected to go to war in behalf of Israel at that particular point in history.

"But the Levites [i.e., the tribe of Levi--the priestly tribe] were not listed among the people of Israel" (2:33).

Okay.

So then I came to chapter 3. And in chapter 3 we have the census of the Levites. And I checked the numbers there, too, and I agreed with the author that, according to what he had written in 3:21ff, if those verses are accurate, then "All those listed among the Levites, . . . all the males from a month old and upward, were 22,000" (Numbers 3:39).

Good.

But then I ran into a problem.

And [YHWH] said to Moses, "List all the firstborn males of the people of Israel, from a month old and upward, taking the number of their names. And you shall take the Levites for me--I am [YHWH] --instead of all the firstborn among the people of Israel, and the cattle of the Levites instead of all the firstborn among the cattle of the people of Israel." So Moses listed all the firstborn among the people of Israel, as [YHWH] commanded him. And all the firstborn males, according to the number of names, from a month old and upward as listed were 22,273.

And [YHWH] spoke to Moses, saying, "Take the Levites instead of all the firstborn among the people of Israel, and the cattle of the Levites instead of their cattle. The Levites shall be mine: I am [YHWH]. And as the redemption price for the 273 of the firstborn of the people of Israel, over and above the number of the male Levites, you shall take five shekels per head; you shall take them according to the shekel of the sanctuary (the shekel of twenty gerahs and give the money to Aaron and his sons as the redemption price for those who are over."

--Numbers 3:40-48

"What's the problem, John?"

The problem is the number of firstborn!

You've got 22,273 firstborn males a month old and upward among a male population of 20 years old and upward of 603,550! If the 603,550 included all males a month old and older, the math would be slightly better. Indeed, assuming the kinds of population numbers we see in various parts of the world, where the average age is, say, 15, then the statistics would be somewhat better.

But let's take the Bible at its word. There were 603,550 men aged 20 and up, and there were 22,273 firstborn males a month old and older.

That means that, among a minimum of 603,550 men, there were a maximum of 22,273 firstborn males.

603,550/22,273 = 27.1

!!!!

Do you begin to "feel" the problem I felt?

If these numbers are to be believed, and if I have interpreted the Bible correctly, then, at best, only one in 27 men was a firstborn. On average, each father had to have had 27 sons. Add in the daughters (who weren't counted), and you'd have parents averaging somewhere above 50 children per couple.

--Anyone want to suggest some solutions to this conundrum?

Musical Mystery Tour Ends

I didn't realize there was a 40-year mystery up for grabs until I read Beatles Unknown "A Hard Day's Night" Chord Mystery Solved Using Fourier Transform:

It’s the most famous chord in rock 'n' roll, an instantly recognizable twang rolling through the open strings on George Harrison’s 12-string Rickenbacker. It evokes a Pavlovian response from music fans as they sing along to the refrain that follows:

"It’s been a hard day’s night
And I’ve been working like a dog"

The opening chord to "A Hard Day’s Night" is also famous because, for 40 years, no one quite knew exactly what chord Harrison was playing.

Professor Jason Brown of Dalhousie University’s Department of Mathematics decided to apply a mathematical calculation known as a Fourier transform to solve the problem . . . and came up with the idea that George Martin, the Beatles producer, added an F note played by a piano.

The resulting chord was completely different than anything found in the literature about the song to date, which is one reason why Dr. Brown’s findings garnered international attention. He laughs that he may be the only mathematician ever to be published in Guitar Player magazine.

For the full technical presentation of Brown's triumph, see CHAAAAAAAAG...It's been a hard day's night! and, perhaps, Wired's How Math Unraveled the 'Hard Day's Night' Mystery.

Fun!

(By the way, in case you need a reminder about the opening chord, here is a copy from the Wired article:

)

My kind of mathematician!

I needed some stats for a letter to my investment advisor. I did a Google search and found Political Calculations.

First Time Visitor to Political Calculations? asks one post:

What makes Political Calculations unique in the world of blogs is the majority of our posts are focused on answering questions for which we don't already know the answers. We then do our best to answer those questions using all the problem solving skills we've gained over the years. Along the way, we create tools to help answer them and what's more, we post them here so you can see how we got to the answers we found.

What's really cool is that the tools we build can answer a lot more than just our own questions. You can change the input factors to better agree with how you see things, to update them with the latest data available to make the answers as relevant to the world today as possible, or if you just want to make one of our tools solve a problem that directly affects your life. And they're always capable of answering our favorite question: "What if ...?"

The author then provides a list of all his "tools to date" (November of 2007).

Wow!

I found one tool that intrigued me. (There were many more. But this one just caught my fancy.) What If We Bought Cars for the Poor Instead of Light Rail? --Yeah! Good question!

And the answer?

--Well. You'll have to check it for yourself.

Assuming he's done his calculations correctly--and I think that's a reasonable bet--his answer agrees with the thought I've had.

I'm afraid, however, that he has not included all the "externalities" that would have to be properly accounted for.

His calculations do make me think, however.