Dr. Jeff on Mathematics Education

This is a Dr. Jeff’s Jeffism and a Dr. Jeff Speaks Out.

Math is the language of nature. If you yearn to know

how she operates, you must speak her language.

—Dr. Jeff

I wrote this essay because I needed to get something off my chest. It first appeared as a foreword to a Dr. Jeff’s Weekly Challenge posted on June 15, 2009, but I think it’s so important that I decided to commit it to a formal Resource Page here at Blog on the Universe. My Resource Pages are all found in the right navigation column under the section titled “Pages” and under the subsection titled “Dr. Jeff on Stuff – The BotU Resource Pages” (take a look at right.) I dedicate the Resource Pages to essays on important topics like: the Nature of Our Existence, the Art of Teaching, Scientists and Engineers as Heros and Role Models, and the Crisis in Science and Technology Education. I felt that an important essay on mathematics and mathematics education should be a dedicated Resource Page.

So here now is my sure to be viewed as an outrageous essay:

Dr. Jeff on Mathematics Education

Let me know what you think by leaving a comment on that page. Also—you can read more about this Blog’s Resource Pages HERE.

This post is a Dr. Jeff’s Weekly Challenge and a Dr. Jeff’s Jeffism.

Last week on BotU, your challenge was to take an imaginary, truly humongous piece of xerox paper—but with normal xerox paper thickness—and figure out how many times you’d need to fold it in half so the folded thickness is the height of you, the Statue of Liberty, the Empire State Building, and Mount Everest. For those that really wanted to challenge themselves, I invited you to keep folding so it would be thick enough to reach the Moon, the Sun, the nearest star, and beyond.

How’d you do?

BUT WAIT! If you haven’t yet read Weekly Challenge 3, DON’T LOOK AT THE SOLUTION HERE JUST YET! First read Weekly Challenge 3, or I’ll take back my paper.

First, a word from our sponsor—

You Want Me To Do What With a Bathroom Scale?

Weekly Challenge 4 to be posted Monday, June 29, 2009

Other Posts coming soon:

A Voyage in Corpus Christi

Twinkle Twinkle Little Star, History Tells How Far You Are

Lessons of Earth

MESSENGER: Target Mercury

And now the answers—

Remember that I began Weekly Challenge 3 by asking you to imagine a humongous piece of paper that, when standing in the middle of it, seemingly extends to the horizon in all directions. This is clearly NOT a real piece of paper. What I posed is called a Thought Experiment—an experiment done on the landscape of your mind. It’s a flavor of experiment that has been done by the likes of Einstein to revolutionize our understanding of space and time. A Thought Experiment often poses a problem that cannot be addressed with a real experiment because doing so is either impossible (as in this case) or clearly beyond current technology, or for that matter beyond any interest in doing it in the real world (e.g., where would you come out if you dug a hole right through the center of the Earth?) And this is precisely why it’s such a powerful type of experiment. It uses critical thinking coupled with what you think you know about the world, to hopefully produce a real conclusion—which often leads to a change in perception.

A note about trying to do this in the real world: You CAN’T actually fold a sheet of paper more than a few times because it quickly gets too small and all the paper is taken up in the curvature of the folds. Try it. If you want to increase the number of folds, you need a larger sheet of paper. But just a few more folds quickly requires a sheet that’s longer than your street, your city, your nation, and, in fact far longer than planet Earth is wide.

This problem was pointed out in the great comment by Maria Miller (see Weekly Challenger 3 page.)  She has a link to a story that made news a few years ago. In 2001, Britney Gallivan a high school student set out to understand the limitations imposed by folding as part of an extra credit problem in math class. She even came up with a limiting equation that defines the minimum size of the paper sheet you’d need in order to fold it a specific number of times. Here are more links to this very cool story about curiosity and drive: 1, 2, 3.

SO … IMAGINE a truly humongous sheet of paper—as long and wide as you need, and let’s explore what would happen to its thickness as you keep folding it.

The answers below are based on a very simple rule. You start with a single sheet that has the thickness of a regular sheet of xerox paper (0.1 mm for those that want to know), and every time you fold the sheet in half:

you double the thickness of the paper

No big thing right? You’d think that you’d need A LOT of folds to get a thickness equal to these large distances. Here you go—

1. How many folds would you need so the thickness is as tall as a ream of xerox

paper (500 sheets)?

just fold it in half 9 times and thickness is 512 sheets

2. How many folds so the thickness is as tall as you?

just fold it in half 14 times and thickness is 5 feet 5 inches (166.5 cm)

3. How many folds so the thickness is as tall as:

the Statue of Liberty?

Height is 305 feet (93 m)

just fold it in half 20 times and thickness is 350 feet (106 m)

the Empire State Building?

Height is 1,472 feet (448.7 meters)

just fold it in half 22 times and thickness is slightly short at 1,398 feet (426 m)

Mount Everest?

The summit is at 29,029 feet or 5.5 miles (8.8 km) altitude

just fold it in half 27 times and thickness is 8.5 miles (13.6 km)

4. How many folds so the thickness is equal to:

the distance from Earth to the Moon?

average distance: 238,900 miles (384,400 km)

just fold it in half 42 times and thickness is 277,650 miles (446,840 km)

the distance from Earth to the Sun?

average Earth-Sun distance: 93,000,000 miles (149,600,000 km)

just fold it in half 51 times and thickness is 142,159,000 miles (228,780,000 km)

the distance to the nearest star Proxima Centauri?

distance of 4.2 light years

just fold it in half 69 times and thickness is 6.3 light years

the width of our Milky Way Galaxy?

diameter: 100,000 light years

just fold it in half 83 times and thickness is 104,000 light years

the distance to the Andromeda Galaxy—nearest large galaxy to the Milky Way?

distance: 2.5 million light years

just fold it in half 88 times and thickness is 3.3 million light years

the distance from Earth to the edge of the observable universe?

distance: 78 billion light years

just fold it in half 103 times and thickness is 109 billion light years

YOU DON’T BELIEVE ME? I KNEW you were going to say that. So I created two Tables that show you how I got these answers really easily. One Table is in English units (inches, feet, miles), and the other Table is in Metric units (cm, km). Choose your system of units and be amazed!

Teachers:

Here’s a way to really do the challenge in the classroom by removing the pesky folding issue. Remember that the basic rule is each fold doubles the thickness of the paper. So instead of folding, just create a pile of paper using this rule, so you’re ‘simulating’ the effect that folding has on the pile’s thickness.

Remember the cartons of xerox paper we used for the post A Day in the Life of the Earth? Go get them out again!! I hope you know the person in charge of the copy room, or you’ll be banned.

Start by placing a single sheet of xerox paper on the floor. Then ask students in your class, one by one, to simulate each fold by doubling the number of sheets in the pile. Here’s the way it ‘unfolds’ (no pun intended)—

• First student sees 1 sheet on the floor, so places 1 new sheet on top: pile

thickness is now 2 sheets; they’ve simulated the effect of fold 1

• Second student sees 2 sheets in the pile so places 2 new sheets on top:

thickness is now 4 sheets; they’ve simulated the effect of fold 2

• Third student sees 4 sheets in the pile so places 4 new sheets on top:

thickness is now 8 sheets; they’ve simulated the effect of fold 3

Have them keep going and see how far they get. You already have detailed notes on how it will turn out. It’s my write-up in the two Tables, one for English units and the other for Metric units.

Important note: once you get up to 512 sheets, you can start using whole reams of 500 sheets (so again no need to open more than 1 ream of paper.)

Another note about how to start the lesson: pose the original challenge to the class, and have them see how many times they can fold a sheet of paper. After they quickly see the insurmountable hurdle that the folding requirement imposes, see if they can come up with this way to do the experiment by ‘simulating’ each fold.

Photo credit: NASA and STScI

This post is a Dr. Jeff’s Weekly Challenge and a Dr. Jeff’s Jeffism.

Math is the language of nature. If you yearn to know

how she operates, you must speak her language.

Before getting to the awesome challenge this week, I need to get something off my chest. It’s something very relevant to the challenge, but you might not think so at first—

My first language is English. I have very strong beliefs about how English should be taught in schools. I guess I’m a traditionalist. I also think that my views apply to how any language should be taught in schools around the world.

I think English belongs in English class. Period. You want to speak and read and write English, well do it in an English class. It doesn’t belong in a history class, or a science class, or for that matter a class on economics, art, sociology, psychology, or the law. Let’s keep English where it belongs. It’s just a language. So no English in those other classes. Just sit there and learn the concepts, nuances, big ideas, and emotional content of those subjects through …. osmosis. Think your thoughts toward other members of the class and share brain waves. And please, please … when you do this—DO NOT THINK YOUR THOUGHTS IN ENGLISH!

Am I losing my Twitter and Facebook fans? Is this finally the real Dr. Jeff? What? You think what I said is just absurd? (Good.) You think that English, like any language, is the means by which we express and communicate the richness of our thoughts on all the subjects that address the human condition? Wow, that’s a mouthful. You’ve got me thinking. And please—don’t leave! In my defense, I just thought that English should be treated like we traditionally treat MATH in school. Addition, subtraction, long division, algebra, trigonometry, geometry, calculus, statistics … it often feels like the unwritten decree is “let’s only keep it in the math classes where it belongs!” Why isn’t MATH a natural part of all the subjects taught—as in the case of say …. English?! And the result? Kid to parent, or kid to teacher, or kid to friend: “What will I ever need this for?”

Answer: because without an understanding of and appreciation for MATH you’ll deny yourself the ability to see the richness and majesty of the world around you. A language like English serves as the foundation for our conversations about anything and everything—and so does math. And if that’s not a sufficient answer (it’s sad when it isn’t) then more practically speaking, without math skills, effectively competing in the job markets of the 21st century will be very difficult—because math is everywhere.

This blog is a good example. I’m trying to get across powerful concepts with a seamless fusion of English and math, and for many readers I bet the injection of math is jarring. But the math provides the insight into HOW I’m getting the high impact, “Oh Wow” answers. The math gives you the chance to take OWNERSHIP of the story at a deeper level—because I’m not asking you to take the conclusions on faith. We’re reaching the conclusions together. And the math should be embraced at the same subconscious level as is the English you’re now reading. Why? Because math and English have a great deal in common—

Mathematics is a language. It is the language of nature. If you yearn to know how she operates, you must speak her language. And nature isn’t just found in science class. A human being is a biological entity, and human society is a biological system. All of humanity is part of nature, so all those subjects of importance to human beings are richer if their study includes mathematics. And I’m convinced that our capacity for mathematics is an outgrowth of nature developing the means to understand itself.

English or Estonian or Japanese or any other spoken/written language models our very thoughts. That’s the point of creating them. Mathematics as a language provides a powerful means by which we can model the world around us so that we can understand it and navigate it successfully. (See The Power of Models page.) Imagine the power you have when you master a spoken/written language AND math!

Finally, math is the only language I know that transcends societies and cultures. It is the language that binds all humanity. So why do we teach our children to treat math as something that is difficult, disconnected, irrelevant, and something to be avoided?

I’ll have many blog posts that do a shout out to mathematics, and this is the first. For this week’s challenge I thought a math problem that coupled the size of you to the size and scale of everything else might be interesting. And it starts with something strangely familiar—a piece of xerox paper, albeit a humongous one.

Here now the challenge—

A friend gave me this sheet of xerox paper. It’s a bit unusual. It’s the same thickness as standard xerox paper, but it’s really long and wide. When I stand in the middle of it, the edge of the paper seems to extend to the horizon in all directions.

I needed to fold it in half. So I got the grounds crew at the nearby baseball stadium and asked them to pretend the rain just stopped and fold the paper the way they fold the tarp used to cover the ball field.

It took some time but they did it. Now the paper was twice the original thickness. This seemed like fun, so I decided to have them fold it again. After the second fold, the paper was 4 times its original thickness. Try it at home with a piece of paper. Fold it twice, and see how thick it is. Then fold it a few more times and you’ll see that—

every fold doubles the thickness of the paper

So here are the official challenge questions. But before setting out to really answer them, read them and take a wild guess for each.

1. How many folds would you need so the thickness is as tall as a ream of xerox

paper (500 sheets)?

2. How many folds so the thickness is as tall as you?

Hint: a ream of standard xerox paper has a thickness of 2 inches (5 cm.) I just

measured one.

3. How many folds so the thickness is as tall as:

the Statue of Liberty?

the Empire State Building?

Mount Everest?

And now, for those that really like a challenge, I dare you to keep folding and leave Earth. First read my The Nature of Our Existence page to get a sense of what lies beyond, and be sure to click on the photos for the captions.

4. How many folds so the thickness is equal to:

the distance from Earth to the Moon?

average distance: 238,900 miles (384,400 km)

the distance from Earth to the Sun?

average Earth-Sun distance: 93,000,000 miles (149,600,000 km)

for the next questions you’ll need that: 1 light year = 5.9 trillion miles or 9.5 trillion kilometers

the distance to the nearest star Proxima Centauri?

distance of 4.2 light years

the width of our Milky Way Galaxy?

diameter: 100,000 light years

the distance to the Andromeda Galaxy—nearest large galaxy to the Milky Way?

distance: 2.5 million light years

the distance from Earth to the edge of the observable universe?

distance: 78 billion light years

I bet you didn’t think THIS is what I was going to do with my piece of humongous xerox paper. (I promise to keep surprising you.)

Answer now posted here!

Photo credit: NASA and STScI