## Friday, January 28, 2011

### Visionary School Design

Rebecca Zook just blogged about an amazing school in Indonesia, with lots of photos. I want to find out more. (I'm way too busy teaching to do that any time soon, but this will some day be my reminder.)

## Saturday, January 22, 2011

### Working in Groups

My classes start on Monday, and I'm still scrambling to get all my planning done, and materials ready. I want our first activity to set a good tone.

In beginning algebra, the first group activity I usually have them work on is my attempt to get them thinking seriously about fractions, instead of just grabbing for the procedure that pops up first in their brains. (I described my campground problem in this post.) It gets pretty noisy while they're working. That's not a problem for me, but I've wondered if I could start with something quieter first.

I've been reading

Each group has 4 people. Each person gets pieces of a circle, but they don't add up to one full circle. Everyone must be silent. You can offer your pieces to your groupmates, but you can't request (even silently) a piece. The goal is for everyone to get a proper circle. It just takes a few minutes, and then you ask them some questions about their process.

I've never talked with my classes about what is good practice in groups. I've just let them go at the problem I've posed. This will give me an opportunity to begin a discussion with them about how working in groups can help people learn, and what makes groups work well.

My biggest class will have about 50 people show up on the first day. (Our official cap is 40, but some of my students from last semester want in, and there are 5 on the official wait list.) So I need 13 sets of circles. I got them printed up at Staples, but now I'm cutting them. I don't usually do things like this, and I'm not relishing having to cut 52 circles. (I've done 14 so far...)

I'm doing a math salon today, too. (It would have been wiser to schedule it for next Saturday, but it'll be a good break, I guess.)

In beginning algebra, the first group activity I usually have them work on is my attempt to get them thinking seriously about fractions, instead of just grabbing for the procedure that pops up first in their brains. (I described my campground problem in this post.) It gets pretty noisy while they're working. That's not a problem for me, but I've wondered if I could start with something quieter first.

I've been reading

*Designing Groupwork*, by Elizabeth Cohen. (I don't remember who recommended it to me, but I know it was someone online - part of my 'Professional Development Network'.) There's an activity there that I've decided to try called Broken Circles.Each group has 4 people. Each person gets pieces of a circle, but they don't add up to one full circle. Everyone must be silent. You can offer your pieces to your groupmates, but you can't request (even silently) a piece. The goal is for everyone to get a proper circle. It just takes a few minutes, and then you ask them some questions about their process.

I've never talked with my classes about what is good practice in groups. I've just let them go at the problem I've posed. This will give me an opportunity to begin a discussion with them about how working in groups can help people learn, and what makes groups work well.

My biggest class will have about 50 people show up on the first day. (Our official cap is 40, but some of my students from last semester want in, and there are 5 on the official wait list.) So I need 13 sets of circles. I got them printed up at Staples, but now I'm cutting them. I don't usually do things like this, and I'm not relishing having to cut 52 circles. (I've done 14 so far...)

I'm doing a math salon today, too. (It would have been wiser to schedule it for next Saturday, but it'll be a good break, I guess.)

## Tuesday, January 18, 2011

### Vi Hart (I sure am late to the party, aren't I?)

Is there anyone reading this blog who hasn't already seen Vi Hart's videos? I guess the infinite elephants is my favorite.

But then the doodle games she mentions in the Snakes+ Graphs video sound too cool, too. I'm not much of an artist, but maybe if I watch her enough, I'll get inspired. (I tried transcribing the parts about the 3 doodle games, but she talks way too fast. Maybe she'll eventually transcribe this one, like she has some of the others.)

Great article about her in the New York Times today, too.

But then the doodle games she mentions in the Snakes+ Graphs video sound too cool, too. I'm not much of an artist, but maybe if I watch her enough, I'll get inspired. (I tried transcribing the parts about the 3 doodle games, but she talks way too fast. Maybe she'll eventually transcribe this one, like she has some of the others.)

Great article about her in the New York Times today, too.

## Saturday, January 15, 2011

### Fabulous Blog: Republic of Mathematics

**Definition of Blogcrush:**You've got a simple small task x that should take oh, about 15 or 20 minutes. You use Google to find some good sites related to x, and you find a blog that makes you forget your initial task entirely, spending the next 3 or 4 hours reading posts, following links, thinking deeply. You're in love, at least for the moment.

I was looking for some good explanations of the distributive property. My personal professional development this semester will focus on what some people are calling the inverted classroom. (See Kate Nowak and Robert Talbot.) So I'm gathering online resources for our first topics in beginning algebra (which are reviews of pre-algebra toughies). I woke up early and thought I could knock out resources for a few topics before my son got up.

*Huh? What's that? He's up?! I haven't even started...*

I think my last blogcrush was Math for Love. Today I discovered Republic of Mathematics, by Gary Davis. I know I've seen it before, but I must have been rushing. It didn’t get into my Google reader and went off my radar. Today I savored it.

Here's a partial list of posts I loved:

- Depression and problem solving in mathematics: the art of staying upbeat
- The Rascal Triangle: how magic can happen if you listen to kids
- Doing mathematics in groups – a personal experience
- To work in groups or not to work in groups? Is that the question?
- Should students memorize the quadratic formula?

## Friday, January 14, 2011

### Open Problems

There are many problems in math that no one has been able to answer. A surprising number of these problems are easy to state, and even easy to play around with.Thinking about these problems can give a student more of an idea of what math really is.

I'd like to collect problems here that would be accessible to high school and community college students. James Tanton works with students on open problems, and they've gotten results. I think that requires finding little known problems. The problems I'm listing below have been worked on long enough that any new results from us amateurs are unlikely. But we can still have fun playing.

The word

1. Goldbach's Conjecture: Every even number greater than 2 can be expressed as the sum of two primes.

2. Twin Prime Conjecture: There are an infinite number of twin primes (n and n+2).

3. Collatz Conjecture: Play this game: Start with a positive whole number, n1. If n1 is even divide by 2, if n1 is odd multiply by 3 and add 1. You now have n2. Repeat the process indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1, which gives you a loop of 1, 4, 2, 1, ...

4. The Integer Brick Problem, to find a brick size in which the 3 lengths along the sides, the 3 diagonals along the faces, and the interior diagonal are all integers, or to prove that this cannot be done.

5. Is every prime larger than 3 the average of two primes? (from James Tanton)

6. A

7. The Lonely runner conjecture: if

8. Towers of Hanoi with 4 pegs: There is a stack of n disks on a peg, with the disks decreasing in size as you move up the stack. If you are allowed to move one disk at a time from one peg to another, and cannot put a bigger disk over a smaller one, what is the least number of moves it takes to move the stack. (seen twice recently, once here)

9. The Goormaghtigh conjecture: The only numbers that can be represented as 11...1 in more than one base are 31 and 8191. 31 can be represented in base 2 as 11111 and in base 5 as 111. 8191 can be represented in base 2 as 1111111111111 and in base 90 as 111. (We don't look 11 (two digits) in this problem, as any number n will be represented as 11 in the base n-1, making it easy to find numbers with two different representations that are all 1's. For example, 13 is 11 in base 12 and is 111 in base 3.) (thanks to Mary O'Keeffe)

10. Legendre's conjecture: There is a prime number between

Here's a good question: Which two problems on this list go together? If one were solved the other would be too. Are the two equivalent, or could one of them be solved and still leave the other open? (thanks to Christopher Sears)

What other open problems (simple to state and to play around with) can you add to this list?

[Edit date: 1/15]

I'd like to collect problems here that would be accessible to high school and community college students. James Tanton works with students on open problems, and they've gotten results. I think that requires finding little known problems. The problems I'm listing below have been worked on long enough that any new results from us amateurs are unlikely. But we can still have fun playing.

The word

*conjecture*means something that people believe is likely to be true, but which has not been proved.1. Goldbach's Conjecture: Every even number greater than 2 can be expressed as the sum of two primes.

2. Twin Prime Conjecture: There are an infinite number of twin primes (n and n+2).

3. Collatz Conjecture: Play this game: Start with a positive whole number, n1. If n1 is even divide by 2, if n1 is odd multiply by 3 and add 1. You now have n2. Repeat the process indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1, which gives you a loop of 1, 4, 2, 1, ...

4. The Integer Brick Problem, to find a brick size in which the 3 lengths along the sides, the 3 diagonals along the faces, and the interior diagonal are all integers, or to prove that this cannot be done.

5. Is every prime larger than 3 the average of two primes? (from James Tanton)

6. A

*perfect number*is a number that equals the sum of all its factors (including 1, but not including itself). Are there any odd perfect numbers? (from Wikipedia's list of open problems in mathematics)7. The Lonely runner conjecture: if

*k*+ 1 runners with pairwise distinct speeds run round a track of unit length, will every runner be "lonely" (that is, be more than a distance 1 / (*k*+ 1) from each other runner) at some time? (same source as #6)8. Towers of Hanoi with 4 pegs: There is a stack of n disks on a peg, with the disks decreasing in size as you move up the stack. If you are allowed to move one disk at a time from one peg to another, and cannot put a bigger disk over a smaller one, what is the least number of moves it takes to move the stack. (seen twice recently, once here)

9. The Goormaghtigh conjecture: The only numbers that can be represented as 11...1 in more than one base are 31 and 8191. 31 can be represented in base 2 as 11111 and in base 5 as 111. 8191 can be represented in base 2 as 1111111111111 and in base 90 as 111. (We don't look 11 (two digits) in this problem, as any number n will be represented as 11 in the base n-1, making it easy to find numbers with two different representations that are all 1's. For example, 13 is 11 in base 12 and is 111 in base 3.) (thanks to Mary O'Keeffe)

10. Legendre's conjecture: There is a prime number between

*n*^{2}and (*n*+ 1)^{2}for every positive integer*n*^{}. (mentioned in Bob and Ellen Kaplan's book, Hidden Harmonies)Here's a good question: Which two problems on this list go together? If one were solved the other would be too. Are the two equivalent, or could one of them be solved and still leave the other open? (thanks to Christopher Sears)

What other open problems (simple to state and to play around with) can you add to this list?

[Edit date: 1/15]

## Thursday, January 13, 2011

### Have You Heard of the P2PU? (Peer-to-Peer University)

Maria Droujkova is leading 2 classes as part of the Peer-to-Peer University (P2PU), and I'm signing up for both of them - Math-Rich Baby and Toddler Environment and Mathematics Curriculum Development.

Since it's all free, they had to come up with a way of making sure people are serious, so each class has a sign-up assignment. (Good call. I led an online study group a few years back and participation fizzled out pretty quickly. I had over 100 sign-up and then very little participation. This way, you can only sign-up if you start out by participating.)

For the first class I had to join AskNaturalMath. If you've read my blog for long, you know how highly I regard Maria's work. You'd think I'd have seen this before, but somehow I missed it. It's a lovely question and answer site. (Check out the question on making paraboloids with little kids.) From the FAQ:

My assignment for the Mathematics Curriculum Development course is to write a blog post on my "

My math education dreams start with my education-in-general dreams. I wrote a poem about that a while back. My Ideal School does not look much like a school. So my dreams of math curricula don't include much that looks like conventional curriculum.

My dreams include a Math Museum in every city. (Last night I attended an excellent webinar hosted by the founders of the soon-to-open Museum of Mathematics, who have organized the traveling Math Midway exhibit. Check out the bike with square wheels!)

What else? Fabulous mini-lectures, like those by James Tanton and Khan Academy, on any topic my students and I might want, provide a helpful background for spending our classroom time exploring concepts more deeply (check out Robert Talbot's post on inverted classrooms).

And then we need the curricula itself - lots of project descriptions that individual teachers can use without too much hassle. I think math circles and the blog world have offered me as much of that as I can handle so far. I need to make changes in the way I teach just a bit at a time. (I'll write another post later, pointing to projects I've used or developed.)

Quick summary: For the students I want freedom, and access to math-rich environments and good explanations. For the teachers I want access to lots of alternative experiences so we can retrain ourselves, and access to lots of great project ideas.

That's my dreams. What are my plans? In my paid job, I want to keep trying to do something sane in an insane system - to help my students enjoy math and learn to really think in math class. In my community work, I want to keep learning about how people interact with math in non-coercive environments, and eventually I want to found a Richmond Community Math Center. (My role model is Amanda Serenevy who founded the Riverbend Community Math Center in South Bend, Indiana.)

I'll look forward to comments from my new classmates, and of course from anyone else reading my blog. [There are 426 people now subscribed through Google Reader, and over 40,000 page views. I am honored.]

Since it's all free, they had to come up with a way of making sure people are serious, so each class has a sign-up assignment. (Good call. I led an online study group a few years back and participation fizzled out pretty quickly. I had over 100 sign-up and then very little participation. This way, you can only sign-up if you start out by participating.)

**Math-Rich Environments**For the first class I had to join AskNaturalMath. If you've read my blog for long, you know how highly I regard Maria's work. You'd think I'd have seen this before, but somehow I missed it. It's a lovely question and answer site. (Check out the question on making paraboloids with little kids.) From the FAQ:

The goal [of AskNaturalMath] is to help you do family math: at home, with friends, on trips, in math clubs, and in online communities. In turn, you can help other people here. This community ... values similar things in math: playfulness, beauty, personal meaning.I was also supposed to post a question there about making young children's environments more mathematical, but I didn't have a real question to ask, so I asked Maria to give me an alternate assignment. If anyone here has a question in that realm, please ask it, here or there.

**Curriculum**My assignment for the Mathematics Curriculum Development course is to write a blog post on my "

*dreams and plans for mathematics education, in the context of curriculum development".*So here it is...My math education dreams start with my education-in-general dreams. I wrote a poem about that a while back. My Ideal School does not look much like a school. So my dreams of math curricula don't include much that looks like conventional curriculum.

My dreams include a Math Museum in every city. (Last night I attended an excellent webinar hosted by the founders of the soon-to-open Museum of Mathematics, who have organized the traveling Math Midway exhibit. Check out the bike with square wheels!)

What else? Fabulous mini-lectures, like those by James Tanton and Khan Academy, on any topic my students and I might want, provide a helpful background for spending our classroom time exploring concepts more deeply (check out Robert Talbot's post on inverted classrooms).

And then we need the curricula itself - lots of project descriptions that individual teachers can use without too much hassle. I think math circles and the blog world have offered me as much of that as I can handle so far. I need to make changes in the way I teach just a bit at a time. (I'll write another post later, pointing to projects I've used or developed.)

Quick summary: For the students I want freedom, and access to math-rich environments and good explanations. For the teachers I want access to lots of alternative experiences so we can retrain ourselves, and access to lots of great project ideas.

That's my dreams. What are my plans? In my paid job, I want to keep trying to do something sane in an insane system - to help my students enjoy math and learn to really think in math class. In my community work, I want to keep learning about how people interact with math in non-coercive environments, and eventually I want to found a Richmond Community Math Center. (My role model is Amanda Serenevy who founded the Riverbend Community Math Center in South Bend, Indiana.)

I'll look forward to comments from my new classmates, and of course from anyone else reading my blog. [There are 426 people now subscribed through Google Reader, and over 40,000 page views. I am honored.]

## Tuesday, January 11, 2011

### Starting New Courses? Don't forget the Soft Skills Convention Center

Back in July, Riley Lark founded a Soft Skills Convention Center. There are 17 great articles there, and now is a good time to re-read them if you're starting new courses like I am.

## Monday, January 10, 2011

### Good Article: Neyland's Playing Outside

Generally speaking my best teaching has not resulted from a kind of effortful deliberateness. It has come when I have allowed modes of attentiveness and intuitiveness to come together in a kind of play that goes in its own direction; a direction that, when viewed in retrospect, turned out to be propitious. This is when I was the most effective. This is also when I was the least focussed on predetermined pedagogical paths and on self-conscious carefulness. Good jazz players and good teachers make mistakes, and have bad days, of course. This does not alter my firm belief that we are at our best when we are improvisers, not when we are corporate linear planners.Jim Neyland wrote this toward the end of his delightful article Playing outside: An introduction to the jazz metaphor in mathematics education.

To me, good teaching is like conducting an orchestra of learning. The teacher and students are responding to one another in such an organic way that it's hard to imagine how the teacher makes her decisions.

If you think math is the one subject that just can't work this way, perhaps Neyland's opening will change your mind a bit:

The trouble with mathematics is that it looks a little more logical and consistent than it is. Mathematics has a universally recognised exactitude. It also has an inexactitude that tends to remain concealed. Mathematics contains what we might call ‘illogical truths’; that is, truths whose apprehension requires, not logical reasoning, but insight and imagination. These ‘illogical truths’ are experienced as surprises; as malformations. ... The person who studies mathematics is more than a logician. The study of mathematics has an inescapable element of unpredictability. And, mathematics endlessly escapes being captured within the logical frames of any mathematical structures.[Thanks to Malke Rosenfeld of Math in Your Feet for pointing me to this inspiring article.]

## Sunday, January 9, 2011

### New Toys & Games: Perplexus & Trango

When I go to Michigan to visit my family and friends there, I always stop by Mackinaw Kites and Toys in Grand Haven. They have the best game selection I've ever seen, and always have lots of interesting games and puzzles out, with salespeople ready and eager to demonstrate.

This time I got another set of Katamino for our friends up north (whose daughter loved it), a 3D puzzle for the boy I tutor (it reminds me of Rubik's cubes, which he loves), a magnetic monkey jumping toy which my son is enjoying, and Perplexus, which my son and I are both puzzling over. Perplexus is a clear plastic ball with an amazing 3D pathway inside that you try to keep the ball traveling along on. Because it involved eye-hand coordination, my son is better at it than I am. He likes that.

My brother got my son a game called Trango. My son hasn't played it yet, but I played it with our friends up north, and enjoyed it. Each turn you play either 2 or 3 pieces, according to the roll of the die, by connecting them to the other pieces already played. You get points every time you make one of the 4 shapes shown on the box (triangle, hexagon, chevron, and spade). You can add one piece to a patch already on the board and get points for the new shape you made. Playing this really got us looking for shapes.

This time I got another set of Katamino for our friends up north (whose daughter loved it), a 3D puzzle for the boy I tutor (it reminds me of Rubik's cubes, which he loves), a magnetic monkey jumping toy which my son is enjoying, and Perplexus, which my son and I are both puzzling over. Perplexus is a clear plastic ball with an amazing 3D pathway inside that you try to keep the ball traveling along on. Because it involved eye-hand coordination, my son is better at it than I am. He likes that.

My brother got my son a game called Trango. My son hasn't played it yet, but I played it with our friends up north, and enjoyed it. Each turn you play either 2 or 3 pieces, according to the roll of the die, by connecting them to the other pieces already played. You get points every time you make one of the 4 shapes shown on the box (triangle, hexagon, chevron, and spade). You can add one piece to a patch already on the board and get points for the new shape you made. Playing this really got us looking for shapes.

## Saturday, January 1, 2011

### Happy New Year

My celebration of the new year has often involved writing a list of the things I'm grateful for. (I've slacked off a bit in recent years, partly due to single parenting, partly due to the fact that I shifted from paper journals to writing on the computer.) Here are some of my gratefuls for this past year. I'm grateful..

- ... to Jeremy Stuart and Roy Robles for the lovely video they made of my math salon.
- ... to everyone who commented here, and to the 39,350 readers of this blog.
- ... to everyone who submitted a chapter for the book
*Playing With Math: Stories from Math Circles, Homeschoolers, and the Internet*, and to Maria Droujkova and others who have helped with the book. - ... to everyone who attends my math salons.
- ... to my morning class in the fall, for their determination and good spirit.
- ... to my college district (and union!) for the chance to have a year sabbatical - it was marvelous!
- ... to James Tanton for his book,
*Math Without Words*. - ... to paperbackswap.com and betterworldbooks.com, for making my book-buying habit more sustainable.
- ... to Julie Brennan, for the wealth of sharing that happens on Living Math Forum, the community she created, and to Maria Droujkova for the same at Natural Math.
- ... to Chris Evans for reading and commenting on ever chapter in the book.
- ... to Linda Palter for drawing marvelous pictures for the book.
- ... to DawnMarie, who watches my son when I go off to do my math projects.
- ... for the internet.

Subscribe to:
Posts (Atom)