Sunday, August 18, 2013

Once Again, Day One

We start classes tomorrow. No, I'm not ready yet. I've been learning how to use our online site, which comes from "Desire 2 Learn." (I hate that name. I feel like I'm spouting propaganda every time I say it. I'll be calling it d2l, and mentioning the problem of the name in each class. Gag.) I want students to be able to see material early, get copies of material when they miss class or lose their copies, etc. I'm having trouble uploading one particular file, and can't figure it out. Sigh.

I'll be teaching two sections of pre-calculus, one of calculus, and one of linear algebra. In each of my classes, I'll move the desks into groups of four, and hand out the syllabus and a sheet summarizing unit one and listing the homework. I'll also pass around a phone list for them to put their name and info on, which I'll copy and hand out the next day. And they will put their name on a 3x5 card, which I'll use to call on people randomly. Here are my planned day one activities for each class:


We'll be using (Thank you, Andrew Stadel.)
Our first problem: Breaths in a day (my guess: 8000, google, 17,000 to 28,000) I was 9000 off, 9000/17000 = 53% low

Visual Patterns
We'll be using (Thank you, Fawn Nguyen.)
Our first problems: #2 (easier) and #1, pretty hard

Equations for Graphs
We'll be using Daily Desmos. (Thank you, Team Desmos and all the contributors.)
Our first problem: 110a2 (use two eqns, or challenge yourself to find just one)

Graphing Stories
We'll be using and what I downloaded from Dan Meyer's blog. (Thank you, Dan Meyer.)

I have had endless trouble with the projection systems in my classrooms, so I plan to test it all out in both of them today. I plan to do one of these activities (or a quiz) with my students each day at the start of class. I think this will fill my hour up quite nicely.

Linear Algebra

Calculus (starts Tuesday evening)

On Screen:
Screen 1: What is the meaning of acceleration? (Write what you think it is on your paper.)
Screen 2: A rock is thrown upward. It reaches 11 feet, and falls back down.
What is the acceleration of the rock at the instant it reaches the very top of its motion?
Think about it on your own, without discussing it yet. We’ll vote, then discuss, then vote again.
Screen 3: A rock is thrown upward. It reaches 11 feet, and falls back down.
What is the acceleration of the rock at the instant it reaches the very top of its motion?
A. up B. down C. none D. not enough information

I don't want to bother with clickers, so I need to stop by the copy shop to get my vote cards made up. (I got this idea from Kate. Mine will be a bit different. I'll post more if it works out, and maybe even if it doesn't. Kate has a good calculus question at that post.)

Tangent Task
For the past two semesters, on day one, I had students carefully graph y=x2, and then show a line tangent to the graph at x=2. After that they were supposed to estimate the slope of the tangent line. It worked great in the fall. But in the spring, a bunch of them knew the derivative 'rule' and that destroyed the activity. I'm going to use a circle and a few graphs I've drawn this time, so they can't use 'rules'.

This class meets for 2 1/2 hours, so I'll have lots more planned, but I get to work on that tomorrow and Tuesday.

On the first day or two I also:
  • Explain how Donut Points work (Every time someone in class catches me in a math mistake, that's a donut point. When the class has gotten 30 donut points, I bring in donuts.);
  • Talk about the difference between what people think math is and what it really is;
  • Talk about mindsets, stereotype threat, and how neurons grow when we learn new things  (I like talking about myelin growth);
  • Explain how to get cheap textbooks (online, used, I allow older editions);
  • Explain that I will stamp homework each day;
  • On day one I ask them to find interesting things on the syllabus, on day two I ask them to share, giving me the opportunity to explain test retakes.

You may find other helpful ideas at my previous Day One posts, here, here, and here.

Saturday, August 10, 2013

Book Review: String, Striaghtedge, and Shadow: The Story of Geometry, by Julia Diggins

This book was recommended on Living Math Forum. I can see why. The storyline is very engaging overall, and gets you thinking about the history as if it were happening while you watch. But...

My biggest issue is that it is so gendered:
"A string can usually be found in a boy's pocket..."
"Ancient men discovered the ideas and constructions of elementary geometry ..."
"Through the ages, men have searched to find the secrets of the universe."
"A long, long time ago primitive men observed the lines and curves and other forms of nature."
"It was from this inner sense - man's sensitivity to the order and harmony of the universe - that geometry really began."

I don't think writers do that so much these days. (This book was written in 1965.) Writers sometimes still say 'he' when they mean all of us, and still sometimes say 'man' to mean people, but not often. Reading this book made me think modern writers must be avoiding this construction, even if unconsciously.

Why does it matter? Research shows that we need to feel a part of  a community in order to do our best thinking. Women and girls are shut out by this sort of writing. As much as I might like the content of this book, it sets me up as an outsider (even though the author is a woman!), and that's part of how stereotype threat happens.

I haven't read the whole book, but I did discover one error, I believe. The discovery of the fact that the square root of two is irrational seems to be described incorrectly:
"Then was it a ratio of whole numbers between 1 and 2? ... They tried every possible ratio, multiplying it by itself, to see if the answer would be 2. There was no such ratio.
After long and fruitless work, the Pythagoreans had to give up. They simply could not find any number for the square root of 2."

There are two problems here. One, they couldn't have tried 'every possible ratio', because there are an infinite number of possibilities. More importantly, it wasn't about giving up. If I understand the history correctly, they actually proved that no such ratio can exist. This notion of proof is a very important foundation - it's part of what mathematics is. So her version of this story takes away some of its drama.

The Pythagoreans believed, as she says, "that the universe was ruled by whole numbers." So to prove that a length exists which cannot be described by a ratio of whole numbers was extremely unsettling to them.

How do we prove that the square root of two cannot be a ratio of whole numbers? There is more than one way to do it. You might like Kate and Justin's way more than the one I usually use. It's less dependent on being comfortable thinking with variables. Here's the way I think of it:

If the square root of 2 could be represented by a fraction, we could writes that fraction in simplest terms as a/b. Then we'd have (a/b)2=2, or a2 = 2*b2 . Since the right side of this equation is even, the left side must be, too. If a2 is even, a must itself be even. Let's call it 2c. Then our equation becomes (2c)2 = 2*b2 , or 4*c2 = 2*b2 , or 2*c2 = b2 .  Now the left side of this new equation is even, so the right side must be too. And that means we can write b as 2d. But if both a and b are even numbers, then the fraction can be simplified. We started out with what we thought was a fraction in simplest terms, and found out that it could be simplified. This is a contradiction. It happened because we tried to write the square root of 2 as a fraction - it can't be done, and this proves it.

Proof by contradiction is a bit weird. I think I might like Kate and Justin's proof better myself.

Well, you might like String, Straightedge, and Shadow, even with its flaws. I might, myself. But I have decided not to include it in my Book Picks section.

Thursday, August 8, 2013

Book Review: How to Count Like a Martian, by Glory St.John

Once again, my work on the last bits of Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is getting me to write things that belong here.

Today I'm working on the Book Picks section, one of the resources you'll find at the back of the book. Much of it comes from what I've already written for the Math Books page of this blog. (You can see the tab for it above.) But there are some great books I hadn't written up yet.

Right now, I'm writing a description of How to Count Like a Martian, by Glory St. John. It's running too long for the Book Picks section, so I'm posting it here. I'll pare it down afterwards.

A really good way to understand place value is to work with other number bases. How to Count Like a Martian is a detective story in which the history of other number systems plays a starring role.

 “Out of the depths of the dark and starry night come the first of the faint and mysterious sounds … At your radio telescope, you are expertly tuning the dials.” You have just received a message from Mars. “You know that this is not a message in words. Martians and Earthlings would have too much trouble trying to find the same words to succeed that way. But there is another kind of language that both Martians and Earthlings understand.”

Numbers… And so you research the number systems that have been used on Earth, hoping that will help you decipher this message. The book proceeds to explain eight different counting systems, including the abacus, and computers. 

In the process, the concepts of place value (she just calls it place), base, and zero are explored. By the end of the book, you can see that the beeps and bee-beeps of the message you received are just the counting numbers, Martian style.

How to Count Like a Martian was written in 1975, when there were still dials and tape recorders. those two items may be the only evidence of its age. I wonder if any young kids will like it as much as I do. Please let me know if your kid loves this book.

Wednesday, August 7, 2013

KenKen: A Simple Puzzle That Goes Deep

In the conclusions I wrote for Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, I mentioned KenKen*. Our copy editor asked what it was. As I searched for a good reference to put in a footnote, a memory began to surface of an article I read long ago. Memory is funny. It turns out I had read the article over four years ago, and I still remembered a particular word from it - midnight. That led me to the article I had read, but it's unfortunately behind a paywall. Luckily, I had downloaded it the first time I read it, and finally found it on my own computer.

I had saved it back then because I was using KenKen with the kids at Wildcat Community FreeSchool. The puzzles I was sharing with the kids seem pretty easy, but to solve them requires holding addition facts in your head while also thinking logically about the relationships. This is a good way to deepen your hold on those facts. I wanted the parents to understand how valuable this simple puzzle was, so I copied the article.

'Midnight' was in the first paragraph, in the dramatic opening of the story written by Leo Lewis for The Times of London:
At one minute to midnight every September 30, the decrepit, cluttered schoolroom of Tetsuya Miyamoto stands frozen in time. Breaking the sepulchral silence of the Yokohama side street, the clock ticks over into the first day of October and a fax machine in the corner shudders to life. 

Throughout the rest of the night, page after page spews out of the machine, each one representing a different seven-year-old child, each one an application form pregnant with parental hopes and fears. 

The class these parents so desperately wanted their kids in consisted of puzzle-solving sessions. Tetsuya Miyamoto provided the KenKen Puzzles he had invented, and the children would then work alone for 40 minutes on up to three puzzles. The first is a 4 by 4 grid, the next is a harder 5 by 5 grid, and the third one is harder yet. After they've worked on the puzzles alone, the group works together on a puzzle Tetsuya Miyamoto puts on the board. He calls on a student for a number, then says right or wrong. That's it (according to Leo).

Before I go any further, I'd better share a KenKen puzzle with you.  Here's today's puzzle from the New York Times. If you like it, go there for more puzzles. Since this puzzle is 4 by 4, each row and column will have the numbers 1 to 4 in it. The clue on the middle top, 6X, means that the two numbers for that outlined box must multiply to 6. We know we can't use 1x6, because 6 won't be used in this puzzle. Is that enough to get you started?

Tetsuya Miyamoto designed his KenKen puzzles to draw students in and get them sweating:
Every puzzle, says Mr Miyamoto, contains a “trick, a discovery – a story”. The puzzle works in his classroom, he says, only because the children want to root out the clues and persevere with the discovery process. “As the feeling of achievement increases, so too does the level of concentration,” he says.
By combining the four main mathematical functions of addition, subtraction, multiplication and division, the brain is forced to dart between competing theories. The puzzle, he says, is impossible to solve without the scientific process of trial and error. 
The puzzle, Mr Miyamoto says, draws out the primal, self-starting learning instinct of human beings – an instinct that is notoriously suppressed by the fact-cramming teaching methods of the Japanese education system, but which he says needs to be encouraged in people of all backgrounds.

I think U.S. schools are headed toward that same sort of fact-cramming. It was never a good idea, but it seems clear to me that we need particular facts less than ever with the internet at our side. What we need are understandings of how it all fits together.

Mr Miyamoto’s theory is that the brain – of a child or adult – is failed by conventional teaching. By concentrating on a “third way” of problem-solving, he believes that the mind becomes a more potent tool for dealing with the rest of life...
I wish I knew what second way is implied here. I'm assuming fact-cramming is the first way.
For both children and adults, runs Mr Miyamoto’s theory, the brain feeds on what it has worked out for itself rather than what it has been told to focus on.
This important idea has been stated many ways by many excellent teachers. I am reminded of the quote the Kaplan's use to define their math circle philosophy:
"What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises."    --G. C. Lichtenberg

There are other ways to make arithmetic challenging and appealing, some of which you'll find in Playing With Math. But KenKen is a particularly easy one to bring into your life. Enjoy!

*The name KenKen is trademarked. Because of this, you can also find these puzzles under other names. Calcudoku seems to be the most common.

Saturday, August 3, 2013

Self-Referential Puzzle, by Jack Webster

If you've been following our Facebook page, you know that Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is nearing the end of copy editing. We still need to add some artwork and get the pages laid out nicely. Then we'll be done.

Today I'm working on solutions to all of the puzzles in the book. I just spent two or three hours lost in the craziest puzzle. I had solved it a few years ago when I first saw it, but of course I don't have notes from back then.

If I hadn't solved it before, I'm not sure I would have believed I could do it. When you first look at it, it doesn't seem possible that the few clues Jack gives could be enough. They are. I have a page-long explanation of the logical steps I took to solve it. I wonder if it's possible to solve it in any other order.

If you like challenges, this is a good one. And if you like this, Jack has others on his site.

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