A Mathematical Story by Gary Davis

He's posting Matrix Algebra, one chapter a day, at Republic of Mathematics. Chapter one tickled me with all its math references, but the main character (in that chapter) is not particularly likeable. The main characters in chapters 2, 4, and 6 are much more fun to meet.

I'm looking forward to finding out what happens next.

Math Circle Institute - Day 1

I swam at 7 am, in the Knute Rockne Pool. Ate breakfast in the most amazing college dining hall I've ever experienced. And made my way over to our daily meeting place. In prior years, we walked all those calories off getting between far-flung buildings. Not this year. Easy on the feet, hard on the belly.

The Kaplans started us off with this problem: A light ray leaves one side of a mirrored square. What happens. We discussed the physics of it, and tried to get away from that, back to math. We talked about modeling this as a billiard table instead. We kinda sorta proved that angle of incidence = angle of reflection. We never worked together on the question that pulled me: How many line segments will be made by the path of the light, before it gets back to its starting point? I figured something out, but I'm not sure yet why it would be true, so it could be wrong. It was a great problem to pull people in at lots of different levels - I think.

Then we talked about relating math circle stuff to classroom teaching, since most of us attending are teachers. Then lunch.

In the afternoon, the kids came. Bob, Ellen, Tatiana, Anand, Amanda, and Rodi did math circles with the kids, and we all watched. Afterward we discussed what went right and what went wrong. Plans for tomorrow, then dinner.

There's a lounge in our dorm. We met in the lounge, and Bob ran a session on sizes of infinity. Those of us who knew the 'answers' managed to keep our mouths shut. I showed off my manuscript to a few people. It's bedtime, but I wanted to do what Fawn did and offer you some notes, so here they are.

Why do social networks matter in teaching and learning?

Alec Couros asked this question on his blog, and is looking for lots of answers. Here's mine:

Hi Alec,

No time for video. I barely know how, and I'm on vacation. But the networking I do online has changed my life, and I'd love to share.


How the Internet Made Me a Mathematician and an Editor

I've been teaching math at college level for over twenty years (mostly here).
I have a son who was in a 'freeschool' (defunct now)
And I wanted to help out with math there.
So I started reading Living Math Forum, to think about how to teach kids.

On Living Math Forum
I learned about Out of the Labyrinth: Setting Mathematics Free,
by Bob and Ellen Kaplan.
I read it and loved it.
Their website, themathcircle.org,
Mentioned their summer Math Circle Teacher Training Institute.
I went. It was amazing.
I met Kate, Jesse, Ben, Amanda, and a few others (who don't do the online thing).

After following Kate's wonderful blog for a while, I decided to start my own.
I also started connecting locally with math circle folks, like Josh Zucker who runs the Julia Robinson Mathematics Festival,
And Paul Zeitz, who wrote The Art and Craft of Problem Solving.

I also began a monthly math gathering in my home,
Which was videotaped by Jeremy Stuart and Roy Robles
As part of a film on homeschooling they're producing.

If I think of a mathematical problem I'd like to work on,
I know all these people have my back.
When I get stuck,
They'll be there.

I've solved a problem I was stuck on for decades
(And got my solution process published),
And then a problem inspired by the game of Spot It.
(Each pair of cards has exactly one matching picture -
How do they do that?!)
I've explored Pythagorean triples.
I've built a stellated dodecahedron out of Polydrons.

And then there's our book,
Playing With Math:
Stories from Math Circles, Homeschoolers, and Passionate Teachers.
I've gathered 35 authors together to share their stories
(Which I mostly found through my internet connections).
Due out this fall.

None of this could have happened even ten years ago.
Math is in my life in a way it never could have been before the internet.
I saw that when I taught linear algebra this year -
I had a blast teaching some hard material,
And I think my students had fun learning it.

Travels Online (Fun Mathy Posts)

I'm traveling for 3 weeks, visiting family and (in 10 days) attending the Math Circle Teacher Training Institute. I'm not online much during my travels, and probably won't have much to say until I get back home. But I'm scanning through my stash on Google Reader, and have come across a few posts I'd like to share:

• The story of some home-made stuffed Platonic solids (a gift for a baby) on Moebius Noodles might inspire me to try my hand at sewing some up. Beforehand, I'd want to figure out how to make them all about the same size. Should I go for the same height? Figuring that out would be a fun challenge.
• Patrick Honner gives a TEDX talk suggesting some benefits of creativity in the math classroom.
• On her math circle blog at Talking Stick Learning Center, Rodi addresses the question, "Is everything related to math?"

Enjoy your summer!

New Page on Facebook for Playing With Math

Check it out. And 'like' it if you'd like to hear my frequent announcements about each current bit of progress on the book. I'll be setting up a newsletter soon that will have less frequent, more polished announcements about how the book is coming along.

Spot It Circle Today

The Bay Area Circle for Teachers is hosting a week-long workshop, and I got to do a math circle there this morning. I decided to do the Spot It analysis again. I had a group of 6 teachers, ranging from a 2nd/3rd grade teacher to high school teachers. They were all eager and persistent, and we'd did lots of good thinking together.

They played the game in pairs, and one pair started analyzing it before everyone was even done playing the game. I'd guess more than half the time was spent with participants working in pairs, with us working as one group together the rest of the time. I provided cards for them to make their own decks (half-size, 2"x3", fun colors, from Office Depot), which some of them used. Yesterday I was working on organizing those quotes from Bob that all involve keeping the discussion in the hands of the participants, so I was well-prepared when one person asked, "Does ___ work?", to reply, "That's an interesting question," and wrote her question on the board.

One person made a deck with 3 symbols - actually, we called them objects today - per card. A few made decks with 4 objects per card. We put a table on the board with 4 columns: # of cards, # of objects per card, # of objects total, total # of appearances of each object. We said a proper deck would have each card matching each other cards once, and found two different decks for 4 objects per card. Our smaller deck had 5 cards, 10 objects, each appearing twice. Our bigger deck had 13 cards, 13 objects, each appearing 4 times. We looked at whether our smaller deck was minimal (the smallest possible deck for cards with 4 objects per card) and whether our bigger deck was maximal. We wondered whether there were any proper decks with sizes between 5 and 13 cards (for 4 objects per card).

I recently heard the phrase Noticing and Wondering, and used it to start the Math Teachers at Play post. (Unfortunately, I don't remember where I heard it.) I'm loving it as a framework. Just two words. Just two simple questions: What do you notice? What do you wonder? I asked that today.

We had about 2 1/2 hours, and figured out some good relationships. We did not answer the question of 'How did they do that?' (How did the company that makes Spot It manage to create 55 cards, with 8 pictures on each one, with every pair of cards having exactly one match.) But we made some significant progress toward that goal, and it was exciting to see the thinking the participants did. I'd love to run a 6-week math circle starting from this game.

The Mathematics of Planet Earth

I really like Plus Magazine. If you have any interest in submitting something for this competition, I think it will be a good one.

Can you explain the mathematics of planet Earth?

Our planet is shaped by the oceans, the dynamic geology and the changing climate. It teems with life and we, in particular, have a massive impact as we build homes, grow food, travel and feed our ever-hungry need for energy. Mathematics is vital in understanding all of these, which is why 2013 has been declared as the year for the Mathematics of Planet Earth.
As well as encouraging research into fundamental questions about the Earth and how to meet the challenges it faces, there will also be many opportunities during 2013 for everyone to get involved including public lectures and workshops, competitions and exhibitions. The first such competition is now underway: the MPE 2013 competition to design an exhibit about the mathematics of Planet Earth.

_____
[At some point, I decided I didn't like writing posts that just linked to someone else's post. I do that seldom now. That's why I had so many goodies to share in the Math Teachers at Play blog carnival. There were about 20 submissions and over 30 posts I had saved over the past year. But this one felt worth it.]

Math Teachers at Play #51

Noticing & Wondering

I'm a pentagonal number, 

I have just two factors, 

and if you put me in base 2, 4 or 16, 

I'm a palindrome.

I wonder:

Is there anyone else like me in the number universe?

51 topics for your enjoyment...

(from mandarintools.com/abacus.html)

Arithmetic 

• Is counting on your fingers good, bad, or both? What do you think? Peter (Classroom Professor) analyzes the mathematical thinking in two classrooms, (giving finger counting a thumbs down, and visualizing a thumbs up) and Caroline (Maths Insider) says don't count on your fingers!  (In a series debunking math myths, I said go ahead, but these posts got me thinking. Now I'd say it's a starting point, and let's think about how to move on.)
• Liz (Homeschooling in Buffalo) posts on keeping it fun.
• Crazy Math Mom (Math Games ...) offers a fact family story.  
• An (Motion Math) offers 3 games for place value (one is app-based, two are non-electronic do-it-yourself games).
• Carole (Mathematical Thinking) shares her missing part cards to lay a foundation for subtraction.
• Christy (Just Another Step...) shows her kids binary, with cards that look fun to make. She throws in some interesting history and a magic trick.
• When my son was younger, I searched for learning ideas that were good for very active, physical kids. He would have loved Kath's (Kath is Math) 'Swat those numbers!'
• While Gord (quantblog) was doing multiplication using coffee beans on graph paper with his young son, he started thinking about an applet he wanted. If wasn't out there, so he built it.
• John (Math Hombre) has made a game where you multiply and divide by fractions to make the superheroes shrink and grow, Size the Day.
• John Henry is the legend of a steel-driving man who competed against a steam engine. David (Delta Scape) shares this story with school children, and then they time each other doing multiplication worksheets with and without a calculator. "Students are asked to predict which method will take longer, gather data, and compare the results using box-plots." Sounds like a good way to help students decide when it makes sense to use a calculator.
• Christopher (Overthinking My Teaching) writes about his daughter's request for a 'real' math question, and how context is what helps us think.
• Michelle (The Rookery) used Incredible Comparisons: The World in One Day (only available used) to help her students understand rates. I've visited Michelle's classroom, and it feels magical to me. Here's a quote from a post on playing class games, "A child who has fallen on his knees to plead with another child to 'smile if you love me' does not feel inhibited when it's time to raise his hand and take a guess at how to solve a math problem."

(from Christy's Game of Patterns)

Patterns & Logic

• Making up your own games is super-engaging. Christy (Just Another Step...) and her son made up a game of patterns that they had great fun with.
• Logic puzzles can be a great side door into the mansion of math. (Think about how Sudoku has swept the country.) The Island of Liars and Truthtellers is a classic setting for logic puzzles. Dan (mathrecreations) shares some background and 5 puzzles. 
• Dr. Techniko's game, How To Train Your Robot, sounds like a blast, suitable for very young kids, whose 'robots' are their parents.

 

(from britton.disted.camosun.bc.ca/jbsymteslk.htm)

Visual Math

(by Anna Weltman)

• Becky (Wide Open Campus) shares photos from her son Z's Escheriffic day, along with a link to a tessellation maker and some thoughts on the magic.
• In Not Just Shapes, Malke (The Map is Not...) continues her delightful series documenting her daughter's math discoveries. "As the structure of the universe continues to emerge in front of her very own (and open) eyes, how much more fun will her world be to play in, explore, put together, and then take apart again?"
• Justin (Math Munch) shares Star Art with the readers of Math Munch (a weekly math links blog), along with some puzzle news. (Links to directions for making this beautiful blue star are in the comments.)
• When you check out Emilio's (Triangulation) Interactive Triangulation, make sure you move your mouse over the pictures.
• Rachel (Plus Magazine) wrote Shattering Crystal Symmetries. If I understand correctly, chemists used to think crystals were always organized in a repeating pattern; Dan Schectman analyzed the structure of a crystal that could not have a repeating pattern, and won the 2011 Nobel Prize for chemistry for this work, which is based on the mathematical work of Roger Penrose. Amazing!  "Not only had mathematicians extensively studied symmetry, but, as mathematicians are prone to do, they were also interested in how to break it."
• Mike and Ian wrote another great Plus Magazine article, this one on Visualizing Probabilities.
• Erlina (Mathematics for Teaching) gives a number of visual representations of the difference of two squares.
• In Perspective in Math and Art, Annalisa (at Inside Higher Ed) writes about how learning to draw in perspective can be a bridge to learning math. "If you sketch a picture of the rails of the train track going into the distance, and you know where the first two railroad ties go, where do you put the next one?"
  
  
  

(from Fawn's area of a circle lesson)

Algebra, Geometry, & Trigonometry

• Kids are never too young to do some algebraic thinking with the Function Machine or Guess My Rule game. Denise (Let's Play Math!) spells it out carefully, and John (Math Hombre) writes about using it with college students, "7 to 1 and then 1 to 7 drew an audible gasp."
• Smruti (Maths Study Blog) shows a method for finding simple side lengths when you have one side of a right triangle. One side is not enough to establish just one possible triangle, but if you'd like to play with finding Pythagorean triples (3 whole numbers giving the lengths of sides of a right triangle, like 3-4-5), then this technique is intriguing. [His site has flashing ads and brought up a pop-up window. I believe it's safe, but can't be sure.]
• Mimi (I Hope This Old Train...) does estimating areas of circles, and Fawn does circumference and area of a circle, along with dissecting polygons. (I've been marveling over how circles and triangles are so tightly connected, and may use these with my trig students.)
• Terrance (So I Teach Math...) gives us a 'relay race' for polynomial functions.
• Feanor (Jost a Mon) has translated a marvelous story of a boy solving a word problem.
• David (Questions?) has a good puzzle that blends algebraic and geometric thinking.
• Nat (Musing Mathematically) did a marvelous project in his Workplace and Apprenticeship class, on how to package soft drinks, that I hope to use with my pre-calculus class. Each of the five posts was exciting to read. (Nat's posts: the framework, the brainstorm, the design, the math, the show.)  

(from Haggis' puzzle)

Puzzles & Games

 

• Haggis (Knot Your Average Sheep) helped design some activities for an interactive evening at the museum (National Museum of Scotland), and included this:  "Can you colour the lines [on the star above] with 3 colours so that at each star 3 different colours meet?"
• Robert Abbott (inventor of the card game Eleusis) has shared some great online Logic mazes.
• Mike (Spiked Math Comics) asks, "What's wrong with this contest?"
• Here's a puzzle from Alexander (Cut the Knot): Given a sequence of numbers, pick any two, say A and B, randomly and replace the two with the result of A×B+A+B. Repeat the procedure until only one number remains. Try to predict the final result. You can play with it online. What's happening?

(from Rick's blog banner)

Notation and Language

• Sometimes the notation makes a math topic harder than it needs to be. Take logarithms, for example. Where'd that word come from, anyway? Kate (f(t)) uses power₂(8) = 3 to invite her students to figure out what the new function is. I used her idea, but changed it to  P₂(x); it worked great.
• Rick (Exploring Binary) wanted a word for the portion of a binary number after the ... umm, "decimal" point. You know, the part that represents a fraction. He wants to know if 'bicimal' works for you.

  

(from Brent's post)

Breaking News 
(The MT@P Times)

• The Museum of Mathematics will be opening in New York City on December 15, 2012.
• Peter and Christian (The Aperiodical, a math links blog) found a CNN story on an advance that may change public transportation, based on linear algebra. If a bus will be running more often than every 10 minutes, passengers can wait less if there's not a schedule. Each bus stops at each end of the line the right amount of time to average its time between the bus in front and behind it. Bus bunching (where one bus ends up right behind another) has always been a big problem, and this solves it. Most of the mathematical paper is quite readable.
• Mayan Artwork Uncovered in Guatemalan Forest, Includes Numeric Calculations
• Are Sharks Doing Math?! (The headline says they are, but we may not want to call it 'doing math' when it's unconscious behavior.)
• Mathematicians Win $289 Prize for Constructing 17x17 Rectangle in 4 Colors With No Monochromatic Rectangles (as reported by Brent at The Math Less Traveled)
• Egyptian Tomb Mystery May Be World's First Protractor

(from Bon's Geogebra post)

  About Teaching Math

• Denise (Let's Play Math!) says, "What better way could there be to do math than snuggled up on a couch with your little one, or side by side at the sink while your middle-school student helps you wash the dishes, or passing the time on a car ride into town?" Mmm, tell me a math story, please.
• Colleen (Mathematics, Learning and Web 2.0) offers David's Powerpoint collections for Number, Algebra, Proof, Geometry, and Statistics.
• Erlina (Mathematics for Teaching) considers what a teacher needs to know to teach fractions and decimals.
• Bon (Math is Not...) asks teachers to reconsider the ways they use Geogebra. She hopes teachers will open lessons up so students can make their own discoveries. She says, "I discovered math when I used GeoGebra. Math I never knew."
• If you want students to learn math through projects (Project-Based Learning has its own acronym of course, PBL), you need to come up with projects that fit your subject and your teaching style. Bryan (Doing Mathematics) brainstorms some enticing ones. Geoff (emergent math) makes a plea for more inquiry-based lessons (is that the same as project-based?) He has set up a google docs repository for each course from algebra to calculus, and lots of folks have contributed ideas. You can use theirs or add some more.
• Sue (Math Mama Writes, that's me) posted on a way of structuring learning situations as games. Not the competitive sort, more like a treasure hunt where everyone can win.
• Paul (Lost in Recursion) on the inadequacies of grading, "The product of mathematical work is mathematical thinking. Trying to grade it is useless." 
• Maria (busynessgirl) knows that getting students to participate actively is vital, and whiteboards are a great tool, but what if you don't have enough whiteboards? Betty solved that problem!
• Caroline (Maths Insider) shares some inspiring quotes.

And now we've come to the end of the 51st Math Teachers at Play blog carnival. Here's one last parting thought... I once read that, among the Tsilagi (Cherokee), you become an adult at 51. (Perhaps that's a bad translation, and you become an elder at 51?) That idea really stuck with me, and when I turned 51 I thought often about how much I'd matured since I was 18. With a 10-year-old in my life, I'm still working hard at maturity... What's 51 mean to you?

And one last question: "Are there coincidences in math?"





The next Math Teachers at Play blog carnival will be hosted at Denise's Let Play Math! blog in the 2nd week of July. If you'd like to be a host of this monthly carnival, check here for open dates. Until then, take your time to savor all these goodies, and when you're done here, check out the 87th Carnival of Mathematics at Random Walks.

Interviewed for npr.org

Nope, I won't be on the radio; it's for their website. I was asked questions for an article about "helping children develop a good attitude toward math". I'll let you all know when it's published. (I'm very excited. I hope she mentions our book, Playing With Math: Stories from Math Circle, Homeschoolers, and Passionate Teachers.)

Teaching for Understanding?

"When I use a word," Humpty Dumpty said in rather a scornful tone, 

"it means just what I choose it to mean — neither more nor less."

If someone says they 'understand' something, what does that mean? Elementary teachers often think explaining why means giving a cute rhyming 'reason',*  a shocking thought to me. And a stark reminder that we each have our own ideas about the meanings of the most basic words.

In 1976, Richard Skemp wrote Relational Understanding and Instrumental Understanding (Mathematics Teaching, 77, 20–26, 1976). Here's the heart of it:

Stieg Mellin-Olsen**, of Bergen University, suggests that there are in current use two meanings of the word 'understanding'. These he distinguishes by calling them ‘relational understanding’ and ‘instrumental understanding’.  By the former is meant what I have always meant by understanding, and probably most readers of this article: knowing both what to do and why. Instrumental understanding I would until recently not have regarded as understanding at all. It is what I have in the past described as ‘rules without reasons’, without realising that for many pupils and their teachers the possession of such a rule, and ability to use it, was what they meant by ‘understanding’.

Suppose that a teacher reminds a class that the area of a rectangle is given by A=L×B. A pupil who has been away says he does not understand, so the teacher gives him an explanation along these lines. “The formula tells you that to get the area of a rectangle, you multiply the length by the breadth.” “Oh, I see,” says the child, and gets on with the exercise. If we were now to say to him (in effect) “You may think you understand, but you don’t really,” he would not agree. “Of course I do. Look; I’ve got all these answers right.” Nor would he be pleased at our devaluing of his achievement. And with his meaning of the word, he does understand.

[The whole article is at Republic of Mathematics. You might want to read it now.]

So when I ask my students to show me, with thumbs 'up, down, or sideways' whether they understand, they may not be telling me what I think I'm asking. Yikes!

How do we teach for 'relational understanding'? John Golden had his math ed students read the article and respond. And in this excellent article, Grant Wiggins says we can't teach this sort of thing, we can only design an environment that helps the students come to it on their own.

No, there is no way around it. If you want students to have meaningful learning experiences that culminate in transferable insight and know-how, then you have to lose time to gain it. You have to slow down the teaching to speed up the learning.

_____
* In the experience of a wise friend who works with many elementary teachers, trying to help them improve their mathematical understandings.
** Also of interest, The Politics of Mathematics Education, by Mellin-Olsen.