My First Professional Paper Published: On Doing Mathematics

The Journal of Humanistic Mathematics is an online professional journal. They accepted a paper I wrote, and it just came out today. I'm proud of myself.  :^)

Here's a summary of the goodies in this issue:

This summer issue covers a wide range of topics. Susan D’Agostino turns to Polya’s text How to Solve It to tackle a tricky problem: creating a new major in mathematics that is relevant in today’s world. Ilhan Izmirli gives us an overview of how culture affects our basic views of what constitutes mathe- matical knowledge. Meanwhile Sue VanHattum presents a personal, in-depth look at how mathematical problems actually get solved, and Kris Green con- templates how the teaching of mathematical thinking could impact public understanding of evolution. JoAnne Growney provides a friendly yet brief account of the diversity of the mathematical blogosphere and other online resources. Reuben Hersh presents a profile of Alvin White, the founder of our predecessor journal, the Humanistic Mathematics Network Journal. Marjorie Senechal reviews a book containing the non-mathematical writings of another hero of the JHM editorial staff: prominent activist, writer, and mathematician Chandler Davis. We also have a review of Joe Mazur’s What’s Luck Got to Do with It? written by Michael Lugo, and some fantastic math- ematical poetry, by Sarah Glaz, Florin Diacu, and Mari-Lou Rowley. The short story “Final Exam” by Robert Dawson wraps up this issue.

I've never put what I wrote about in the paper here on my blog, because it's about the solution to a very cool problem, and I'd rather not show people answers here. I'm more interested in enticing you into playing the game of math yourselves. The problem is:

On a circle, put some points. Connect each point to every other point with straight lines. How many regions are created for n points? (Check your prediction by working out the circle with 6 points.)

Play with it. It took me years to get to the solution. But it was always fun to play with.

Join Me: Math & Technology Workshop

Held near the shores of Lake Michigan, at Muskegon Community College, the fee for this I'm-sure-it's-amazing-because-Maria-Andersen-is-hosting-it week-long workshop is only $150. (Your hotel costs more - it's $65 a night.)

She's had two cancellations, so there are two open spots. It runs August 8 to 12. Wanna come play with us?

Sweet Site: mathcats.com

Hey, look at that! I just learned something from my own book.

I interviewed Denise Gaskins last year, and am editing the interview right now. I noticed a website she mentioned, called MathCats, and checked it out. I was delighted to find a River Crossing puzzle done up with animation, human body geometry, and something called obbl Architecture Blocks.

Lots more there to play with. Enjoy!

Math Teachers at Play #40

 Welcome to the 
Math Teachers At Play 
blog carnival — 

it's not just for math teachers! 

If you like to learn new things and play around with ideas, you're sure to find something intriguing here. Don’t try to read all 40(!) posts at once; take the time to enjoy browsing. Savor a few posts today, and then come back for another helping tomorrow or next week.

At my fortieth birthday party, I got a few of those gag presents meant to remind me how terribly old I was getting. Math Teachers at Play is less than 40 months old (it used to come out twice a month), but just imagine how many great math posts have been included over the months, in all 40 issues.



Forty: A Puzzle

In English, the number forty is spelled out so that the letters appear in alphabetical order. Can you find any other numbers that work this way, or is forty the only one? (If I couldn't find another, how would I prove it was the only one?) Does switching to another language help?

from You Can Count on Monsters

40's Trivia 

• Forty is a pentagonal pyramidal number.
• Forty is the atomic number of zirconium.
• Negative forty is the one temperature at which the Fahrenheit and Celsius scales correspond; that is, −40°F = −40°C.
• Forty is the number of thieves in Ali Baba and the Forty Thieves, from the Thousand and One Nights. (Both the numbers 40 and 1001 may have meant "many", rather than indicating a specific number.)
• 40 = 11000 (base two) = 1111 (base three) = 2*2*2*5
• Forty winks is a nice afternoon nap on a hot summer day.
  
• 40 acres and a mule refers to the short-lived policy, during the last stages of the American Civil War in 1865, of providing arable land to black former slaves who had become free as a result of the advance of the Union armies into the territory previously controlled by the Confederacy, instituted by General Sherman. After the assassination of President Abraham Lincoln, his successor, Andrew Johnson, revoked Sherman's Orders and returned the land to its previous white owners. Because of this, the phrase "40 acres and a mule" has come to represent the failure of Reconstruction policies to return to African Americans the fruits of their labor. (Wikipedia)
  
• And last but not least, we have the forty hour work week, brought to you by unions.

 


Early Concepts

Nikki Olivier shares some of her family's math moments.


Jennifer Knopf brings us 4th of July Fun for beginning counters, with a sorting mat and graph to go with red, white, and blue star marshmallows.

In How to Teach Math Concepts at the Dinner Table, Bon Crowder looks through her young daughter's eyes at the commutative and associative properties, along with substitution. These concepts help form a strong foundation for first arithmetic and then algebra.



Arithmetic
Denise Gaskins is giving away a complete set of the Arithmetic Village picture books, by Kimberly Moore. The stories look beautiful and whimsical. They remind me of the Waldorf arithmetic stories, which are usually shared orally. I'm excited to see these stories in print. (The giveaway is open until July 17. Go comment now at Denise's blog if you're interested.)

Karyn Tripp has created an addition bingo game, and posted it at her blog.

Maria Miller suggests helping kids focus on the first step of a problem (and then the next) by bubbling it.

Jennifer Bardsley describes how using coupons to buy toothpaste can turn into a wonderful math lesson for young children in Coupons and Kids, Math in Action.

In Math or Magic?, David Ginsburg asks us to make sure students don't learn math as if it were magic, and walks us through an example involving multiplying fractions.

Rebecca Hanson offers a very simple arithmetic exercise that gets students thinking more deeply.



The Art in Math
Jenny got her first graders playing with Turtle Art (a simple programming environment that produces beautiful results), and they loved it. The turtle art software is free to download. To the left you see a 40-sided star, and to the right, my first art attempt in TurtleArt. (I loved making it.)

Luyi shows off her friend Justin's fabulous knot drawings.


Dan MacKinnon gives an example of what you can do with specialized graph paper, and links to a source for lots of free varieties of it.



Computers
If you haven't been following toomai's series on building a computer from first principles (using paper and then wood to create adders), you are missing out! Here's his first post in the series.



Algebra
John Golden has another cool game, called Linear War. I think of Algebra I as two-thirds linear and one-third quadratic; here's John's quadratic post.

Henri Picciotto writes about kinesthetic activities for secondary math; I'm looking forward to using some of these in class. (Henri maintains a math ed page as a website rather than a blog. There's lots of good material here; check it out.)



Probability
What's more likely, getting struck by lightning, or hit by a car? Which risky activities do you avoid, and which do you engage in? Take the BBC's Big Risk Test to learn more about your risk-taking profile.



Puzzles and Games

Check out Dan MacKinnon's Knight's Tour and King's Tour puzzle posts.

Jim Wilder brings us a magic square post.

Rachel Lynette offers a symmetry game called Guess My Grid, with a free game board available.





Teaching and Learning
Denise Gaskins is collecting quotes from bloggers. Which is your favorite?

Alexandre Borovik blogs about place value and the problem with not having a name for a number that has just one non-zero digit (which may be followed by any number of zeros).

Alexander Bogomolny blogs about the ambiguity of math words in English.


Geoff blogs about some benefits he saw with Problem-Based Learning.

Whit Ford blogs about Eight Attributes of Effective Activities, Problems, or Projects.

Terrance Banks used a menu system to allow students some say about how their quizzes were graded. He shows us how it works in this blog post.

Allison Cuttler posts about solving a hard problem and the lessons she took from that about how students feel about problem solving.

I went to a workshop on Complex Instruction, which involves using groups in math class, and blogged about it.



The Real World

Virtual Math Tutor blogs about Pyramid Schemes, with math coming to the rescue. I found this post surprisingly timely as I had just received a chain letter in the mail. (Chain letters are illegal in the U.S., so I tried to report it, but haven't succeeded yet.)

Jonah Lehrer blogs about the problem with math, in sports. "Because it translates sports into a list of statistics, this tool can also lead coaches and executives to neglect those variables that can't be quantified."  The problem Jonah describes shows up in lots of other arenas (standardized tests getting you down, anyone?) - I'd love to see more bloggers write about this issue.

John Cook thinks about how different people would answer the interview question "What's the square root of 101?"

Plus Magazine has an article on the Unplanned Impact of Maths.

Summer Travels
Ihor found this sign in Italy, and wonders if numbering the floors this way makes for better math students.

Peter Price looks at Roman ruins to help him teach Roman numeration.

Katie Sorene brings us 7 more buildings of mathematical interest. I wish the math were explained more clearly - I'm not sure I believe all the hype about particular ratios.  I'd really like to see a floor plan of the Rushton Triangular Lodge; maybe I'll draw one myself...




Math Poetry
John Cook has posted a sweet collection of limericks. And Mad Kane offered up her limerick ode to Tau Day on June 28th (6.28).

Math Haiku for you, from Yan Kow Cheong.


JoAnne Growney posts math poetry regularly. In this poem, Robert Gerther asks whether math truly is a universal language.



Visit Our Sister Carnivals

• Carnival of Mathematics
• Mathematics and Multimedia
• Carnevale della Matematica
• Carnaval de Matemáticas

Exit the Carnival Here

And that rounds up this edition of the Math Teachers at Play blog carnival. I hope you enjoyed the ride. The next installment of our carnival will open on August 19 at I Hope This Old Train Breaks Down. If you would like to contribute, please use this handy submission form or email Mimi directly. (We've had trouble with the submission form these past few months. We hope they’ll get it fixed.) Posts must be relevant to students or teachers of preK-12 mathematics. Old posts are welcome, as long as they haven’t been published in past editions of this carnival. Past posts and future hosts can be found on our blog carnival index page.

google+

I got a bunch of invitations to google+ recently, and deleted them. I hate having to think about another new way of dealing with the internet. Then I read a post written by The Daring Librarian, added her blog to my reader, and read the review of google+ (and comparison to Facebook) she linked to.

I'm on Facebook but I don't care for it. I don't like the 40 character limit on updates, and I don't like that the only way to connect to someone is to 'friend' them. I also don't like following every comment from every person I've 'friended'. I think google+ will work better for me, though I'll stay on Facebook for now. Instead of 'friending', you add a person to one (or many) of your 'circles'. I created a circle called 'math buds' and have about 15 people in that circle already.

I'm hoping everyone I'd like to connect with will be on g+ soon.

Complex Instruction

In her book What's Math Got to Do With It?, Jo Boaler wrote about 'complex instruction' as a vital part of making groupwork function effectively. What I read intrigued me, so I searched for more detailed information online, but I couldn't find much.

This past week I was able to attend a workshop on Complex Instruction put on by the Center for Innovative Teaching, at the Urban School in San Francisco. The workshop was led by Laura Evans, who did a fabulous job of introducing us to these powerful ideas. I'm going to try to explain what I got out of this two-day workshop, but my head is spinning, so I might miss vital pieces or misrepresent parts of the theory. If you were there, and saw different aspects, please speak up. I was the only college teacher there, but there were only a few things I had to translate for my situation.

Here's how it works: Start with a group-worthy task, make sure the students are ready in their groups, and understand the roles they're expected to play, give clear and detailed instructions about how they should work together and what to do when they think they've completed their task. Then, let the math-play (play and work are equivalent in a context like this, right?) begin!


A group-worthy task ...

• is open-ended
• is based on discovery
• is challenging
• requires multiple abilities
• can be represented in more than one way

I'll feel like I really understand this when I can take a task/problem and evaluate it based on these criteria. I'm not there yet. I also need to be able to look at the curriculum we have and decide what sorts of group-worthy tasks would help the students learn each bit.

'Smart in Math'
Before students work in groups, it's important to help them understand that we typically have many misconceptions about what it means to be 'smart'. Typically, people think that someone who is 'smart in math' ...

• answers questions quickly
• always gets the right answer
• doesn't have to work at it

But, really, people who are good at math ...

• are persistent
• wonder about relationships between numbers, shapes, functions, ...
• check their answers for reasonableness
• make connections
• are willing to try things out, experiment, take risks
• are resilient
• want to know why
• contribute to group intelligence by asking good questions
• notice and learn from their mistakes
• try to extend and generalize their results

Students may also need to know how synapses (the connections between neurons that are created each time you learn something new) are strengthened by repeated use. A new connection isn't strong until it's been used:

• multiple times
• in multiple ways
• after a time away

Roles for Group Members
Before the groups dive into a math task, the group members also need to understand the roles they'll take on. Any teacher interested in using these ideas can modify these roles, but the idea is to give students plenty of coaching in how to work productively, and much less coaching on how to do the math.

• The Facilitator asks if everyone understands what's been said, if anyone has a question, ...
• The Team Captain keeps the group on task, reminds people of how they're supposed to proceed, makes sure everyone's ideas are heard.
• The Resource Manager makes sure all conversations happen in the middle of the table, collects materials from the teacher, calls the teacher over when the whole group has a question, returns materials, ...
• The Recorder takes notes on the ideas, questions, hypotheses, prepares the group's presentation paper, makes sure everyone can explain the group's solution.

This list was part of the handout for the problem we were going to work on, which I'll describe next.


The Pile Pattern Problem

We needed to figure out the shapes for piles 5 and 6, and what their areas would be, and then to do the same for the 100th pile. We were also asked to think about the 1st and 0th shape, and if possible the -1th shape. While we worked on our mathematical task, Laura walked around and took notes on what we said to one another. She came up to us at an opportune moment and said things like, "Sue, it was really neat when you said 'I was thinking this, but it sounded like you were thinking about it this other way', you made connections between your thoughts and Rachel's." She had a very specific bit of praise for each of us, related to how we worked within the group, to solve the problem.

We all presented parts of our solutions to the whole group (of about 20). We were able to look at the pattern geometrically, algebraically, numerically, and graphically. We had a recursive formula for the area and an explicit one. We figured out what the 1st shape (#1) would have been, and hypothesized about the #0, #-1, and #-2 shapes. There were definitely some interesting twists to the problem.

Every step along the way, Laura would mention bits about how she'd do this with students. Make sure the group that only got one part gets to go first, have each group after that present one new way of looking at this problem.


Status
Within a group of high school students, each student has high or low social status and high or low academic status. (My question: How is this different among college students?)  If someone is quiet, it's generally because they don't expect their group to be interested in what they have to say, either because of past experience or because of subtle cues from other group members. Laura said, "Students hesitate to share as a way to hide or protect their status. High or low status is a great barrier to risk taking."

The teacher's job is to change that dynamic in a few ways. She has already told the group very explicitly what each person can do to help. She can also look for ways to 'assign competence' to students who have low status. If a low social status student has asked a question, she might mention how that was a great risk to take, and how it helped the group. Laura again, "When we raise their status, we give students excuses to take the risk that they deep down want to take."


I loved this workshop, and I hope to be able to implement some of the ideas. I wish the workshop had been longer. It would have been great to have a chance to practice finding or creating a group-worthy task, writing up instructions for it, seeing how groups work through it, and responding to the 'students' by commenting on how they're working together instead of offering them math tips.

 Edit on 5-30-13: When I wrote this, there were no complex instruction resources online. Now there is this website - it looks good.

The Math Teachers are Playing!

And the next edition of Math Teachers at Play is set to come out on July 15, in two more weeks. 

Do you have something you've been meaning to share? Would you like to write something by Wednesday, July 13, to submit? Have you noticed a new math blogger on your block you'd like to introduce to the rest of us?

The blogcarnival site may still be broken, so please send your submissions directly to me.