Tuesday, March 30, 2010

Maths by Email and Drawing Eggs

'Maths' is the British version of the word math. 'Maths by Email' comes from CSIRO, which is Australia's counterpart to the NSF (National Science Foundation). This was recommended by someone in the Living Math Forum, and I just got my first copy. I am delighted!

There are two topics that are easy enough for kids to understand, and sophisticated enough that I learned something new - fair pizza cutting and the shape of an egg.

I have chickens, and was especially interested in this:
Eggs do not start out egg shaped. While they are forming inside the chicken, they are very round, and they don’t have a shell. When the egg is almost ready, the chicken pushes it towards the end of the oviduct (the tube that eggs travel down to get out of the chicken). The squeezing makes the egg into an egg shape. While it is being pushed, the egg grows a shell around itself!
But the math was even better. They show how to make an egg drawing using 3 pins, and they show how egg shapes are related to ellipses.

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Happy Spring!

Monday, March 29, 2010

On the Math Standards

Jason wrote a great post on the K-5 math standards, over at Number Warrior. My response:
I haven’t been willing to comment on these, because it’s just too big for me to wrap my head around. And I’m convinced that different kids need to do their learning on different timetables.
But I like what you’ve given me here.

I don't like the idea of standards. People I respect do. I feel strongly that we need to find our own solutions, locally, to our problems. Good ideas can spread online. I want students and teachers to have more power over their own classroom lives (extreme democracy).

Also, even if we think of the standards as guidelines, there are so many places where disagreement will happen - where perhaps no one has the One Right Answer. Jason points out the multiplication-as-repeated-addition controversy, but there are plenty of others.

Today I noticed one topic I'd move earlier (if I were writing the standard guidelines) and another I'd move later. We focus too much on 'addition facts' and 'multiplication facts', when kids might learn those much more easily if we waited a few more years (a la Benezet). If you are working on your multiplication facts, Maria D pointed out an interesting video some kids might like.

On the other hand, if you're having fun, why not introduce some of the squirrely topics earlier, so kids get lots of playful exposure before they have to worry about all the complexities? On Living Math Forum, someone asked about resources for negative numbers. My favorite was the card game Denise has on her blog. (I sent both these links along to the teachers at my son's school, and they get to decide for themselves whether to use them, and with which kids.)

Writing good standards would require:
1-Treating them as guidelines, not rules,
2- A clearer understanding of child development than most people have,
3- A deep understanding of mathematics, including a deep understanding of elementary mathematics, and how it's learned.

After 20 plus years of teaching, I'm still not ready for that challenge. I don't think many people are, regardless of whether they claim to be.

Teachers who have the freedom to make their own curricular decisions might find some good thoughts on what kids are ready for at different ages in Patricia Kenschaft's book, Math Power: How to Help Your Child Love Math, Even If You Don't.

Looking forward to reading other people's thoughts on all this...

Sunday, March 28, 2010

Steven Strogatz's Series in the NY Times

It started at the end of January with penguins ordering fish, fish, ... fish, in From Fish to Infinity. Now he's exploring functions and logarithms, in Power Tools. If you haven't checked it out, I think you'll enjoy it.

Saturday, March 27, 2010

The Cat In Numberland, by Ivar Ekeland

I love this book! I wrote about it before, in my post on A Dozen Delectable Math Books, but it was out of print then. It has come back out, and I've just gotten two copies, one for my son's school, and one for a friend. I think I'll need to buy about 5 more copies for friends with kids, and one for my niece, who's 15 and enjoys math.

Here's what I wrote before:
The Cat in Numberland, by Ivar Ekeland (ages 5 to adult)
The cat who lives in the Hotel Infinity gets confused when the hotel is full, and the numbers are all able to move up one room to make room for zero. This story is charming enough to entertain young children, and deep enough to intrigue anyone.

David Hilbert, a mathematician interested in thinking carefully about how infinity works, and different sizes of infinities, first made up the basic story. Many others have embellished on it. In this version, Mr. and Mrs. Hilbert run the hotel, and with a little help, find rooms for all the guests who come to visit, even when the hotel is already full. Mr. Hilbert gets worried when the fractions show up (doesn't everyone?), and Zero helps him see a solution.

John O'Brien's illustrations are delightful.

Friday, March 26, 2010

Expanding Your Horizons and Scratch

Expanding Your Horizons is a once a year conference for middle school girls, held at colleges across the country. It happened today at Contra Costa College, and I taught a workshop on using Scratch.

I've been a computer programmer in the past, so it wasn't hard to pick up enough to get started. Scratch is a free programming environment in which you use scripts to control characters called sprites. You can add sound effects and movement, and make cartoons, stories, or games.

I think the girls had fun. I promised to post on my blog so they could ask me questions here.  Any questions?

Friday, March 19, 2010

Math Teachers at Play #24

Check it out!

I'm looking forward to thinking about these puzzles Denise posted:

  • 24 can be written as the sum of three square numbers. How?
  • Can 24 be written as the sum of two consecutive integers? Can 24 be written as the sum of three or more consecutive integers? How many ways?
    [Which reminds me: Did you figure out the consecutive-integer puzzle from MTaP #22?]
  • How many ways can the letters M-A-T-H be arranged to form a 4-letter “word”? Okay, since there’s no 24 in the question, you’ve probably already guessed the answer — but can you prove it?
  • 24 is the largest number divisible by all numbers less than its square root. Can you find all of the other numbers for which this is true?
  • 24 is an abundant number, which means that if you add up all the numbers that divide evenly into 24 (except for 24 itself), the sum will be greater than 24 itself. How many other abundant numbers can you find which are less than 100?
  • What is the ones digit in the number 24^24?
    [That means 24 raised to the 24th power.]

I can't tell you my favorites, because I haven't had time to look through more than a few entries yet. (And if I don't post this link now, I might just forget.)

I think I'll savor it all on Sunday morning. Enjoy!

Thursday, March 18, 2010

What Are the Myths About Math?

Have I mentioned lately that I'm working on a book? The title has changed; now it's Playing With Math: Stories from Math Circles, Homeschoolers, and the Internet. I'm working with over 15 authors on chapters, and a bunch of my favorite bloggers are contributing posts. (If you'd like to contribute a blog post, email me at mathanthologyeditor on gmail - maybe I can fit it in. I don't have a publisher for sure yet. If the one I'm talking with doesn't work out, I'll probably go with lulu.com.)

I posted before on math myths, but that was a long time ago, and I've been giving it some more thought because I'd like to start the book out with this. I've rearranged and changed things up a bit. But I wonder if I've left anything important out. Can you help me?

(Some of the wording in my list below comes from Mind Over Math, by Kogelman and Warren, a great book for overcoming math anxiety. But I’ve added and changed things quite a bit.)


Who does math?
Myth #1: Some people have a 'math mind' and some don't.

Math #2: Math requires logic, not intuition. Math is not creative.

Myth #3: Men are better at math than women.

What do young people need?
Myth #4: Elementary school math is all about arithmetic.

Myth #5: It's bad to count on your fingers.

Myth #6: Gotta memorize those time tables.*

How is math done?

Myth #7: Learning math is about learning how to follow a procedure, and there's lots to memorize.

Myth #8: It's always important to get the answer exactly right, and you must always know how you got the answer.

Myth #9: Mathematicians do problems quickly, in their heads, and math is done by working intensely until the problem is solved.

Myth #10: There is a best way to do math problems.

So. What's missing from this list?

* I'd better put my reply to myth #6, or I'll catch way too much flak. It's a myth only because parents worry more about that than about whether their kids are learning problem-solving skills - hoe to really use math. Here's my draft response to this myth:

Sure, they'll need to know their times tables, for all sorts of reasons. But if someone doesn't memorize easily, give them something more intriguing to think about, where they get slowed down, but not stopped, by not knowing their times tables. The skill will develop in this need-to-know context.

Drill is likely to put a fact in the part of your brain that holds meaningless information like phone numbers. But our brains are much more adept at handling the things that have lots of connections. If you can get the times tables memorized in a way where they’re being used, that’s the best.

Sunday, March 14, 2010

Pi Day: What's Your Favorite Discovery?

Here's mine...

Maria D posted a link to this beauty on her Natural Math google group. This is my first time embedding a youtube video on my blog. I'm embarrassed to admit that something that turned out to be so easy was intimidating to me. (In case you're like me: Youtube has a box labeled embed on the right hand side of the page. Copy, choose "edit HTML" at your blog, paste.)

Here's another embarrassing admission: For many years, I thought of    and    as just "formulas". I only recently (last 10 years, that's recent for me) realized that the second of those is almost the definition of .


If we could measure perfectly, we'd measure around the edge of any circle (C is for circumference) and across the middle (d is for diameter), and then divide. It's always the same, and that's what is. "Ohh, now I get it!"

[Edited on 3-15 to add:]  Jumping from basic to advanced, here's a 6-part series working through a proof that is irrational (by Brent at The Math Less Traveled). I did fine for the first 3 or 4 parts, and then lost steam when there was a delay between posts. I've wanted to understand this for years, so I'll go back soon and work my way all the way through it. Unfortunately, this proof is not at all intuitive - understanding the proof is not the same as really having a feel for why must be irrational.

My son just woke up. I found my compass, and showed it to him. ("I've seen that before," he says, trying to seem bored.) Then I tried to draw 6 circles around a center one, but I guess I squeezed the compass as I went, because the outer circles didn't meet up like they should have. Time for Geometer's Sketchpad (or geogebra, for those of you who've learned it) ...

What's your favorite Pi Day discovery? 

[Formulas and    symbol created at codecogs.]

Tuesday, March 9, 2010

You Can Count on Monsters, by Richard Evan Schwartz

Schwartz is a mathematician and an artist, and his monsters are the numbers from 1 to 100. Number one is sad because it can't play the factoring game with the other numbers. He shows why, with factors trees that have 1's included, and can just go on forever. (His factor trees have the branches going up; mine have always gone down.)

The prime numbers are the basic monsters, and the other monsters are made from strange conglomorations of their prime factors. Searching for the factors in the composite monsters is sometimes easy (that's the 10 monster to the right) ... 

Sometimes hard (that's 99 to the left)...

And always interesting.

My 7-year-old son was confused by 8. He thinks of 8 as four 2's, but there are only three 2's in the picture. I think it's a good confusion. My guess is that he'll eventually get how the composite monsters are made from other numbers, and then the 8 monster will suddenly feel right.

At the end, Schwartz gives lovely explanations of how to find the primes, and why they go on forever.

Available for $24.95 at AK Peters (the publisher); cheaper at Amazon (but check that shipping charge to make sure).

[Transparency: I got my copy of this book for free, as a review copy. But I think I might just buy 2 or 3 copies to give as presents. Yum!]

Saturday, March 6, 2010

It's a link kind of day.

John Spencer has a good way to think about one of the troubles with textbooks and other imposed curricula.

Rebecca Zook tells how talking about fractions "the Chinese way" helped one student learn and like fractions better.

For John Conway fans, his recipe for success. Thanks, Tanya!

Science Teacher quotes Diane Ravitch, who spoke on Democracy Now (transcript down the page). Here's another Ravitch quote:
The Obama administration appointed somebody from the NewSchools Venture Fund to run this so-called “Race to the Top.” The NewSchools Venture Fund exists to promote charter schools. So, what we’re seeing with the proliferation—with this demand from the federal government, if you want to be part of this $4 billion fund, you better be prepared to create lots more charter schools.
And the discussion over at Kate's f(t), about how to convince folks of the right answer to the 3 door problem, is great.

Thursday, March 4, 2010

Probability: Behind one of these doors is a new car...

Perhaps you've all heard of the "Monty Hall" Problem? I hope this is new to a few of you. I'm writing about it today because I just learned about a twist in the controversy over it I hadn't heard before.

Monty Hall was the host of Let's Make a Deal, and would often play a game with contestants where he would show them 3 doors.

Behind one was a brand new car, and behind the other two were stinky old goats. You pick a door, and he shows you a goat behind one of the other doors. He now allows you to switch. Do you do it?  (We'll assume you prefer new cars to old goats.)

Marilyn Vos Savant wrote about it in Parade Magazine in September, 1990. She got piles of letters in response, mostly from people who disagreed with her analysis. (I'm trying to avoid giving away the answer here, so you can play with the problem yourself.) She was right, but a number of mathematicians told her she was wrong. How is that possible for such a simple little problem?! I think it's because we trust our intuition too much.

I remember reading that column when it came out, and getting the answer 'wrong'. But that's because I made an assumption that she didn't address one way or the other in her statement of the problem. I assumed the game show host would try to mislead you. To do this as a math problem, it's important to add one thing to the statement I gave above. You need to know that the host will always show another door with a goat behind it, and offer you a chance to switch.

And that is what I just read an article about. Monty Hall himself pointed it out in an interview with John Tierney in the New York Times. (It was published way back in 1991, but a discussion of probability problems this morning led me there.) That article will tell you how to solve the problem, so don't click until you're ready to see their answer.

Have fun!
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