Math in Action 17

A highlight of February in these parts is Math in Action. Our local, 1 day math fest. Having been at the U for 20 years now, part of it is just great reunion, with our former students coming back to present and knock 'em dead. The last two years have felt stepped up, though, with a keynote from Christopher Danielson in 2016 and Tracy Zager, the math teacher I want to be, this year.

After taking a year off presenting last year, first ever, this year I was back at it to talk Math and Art with Heather Minnebo, the art teacher at a local charter that does arts integration. I've consulted with her, she's helped me a ton and we get to work together sometimes, too. (Like mobiles or shadow sculptures.) The focus this session was a terrific freedom quilt project Heather did with first graders. Links and resources here.

Next up for me was Malke Rosenfeld's Math in Your Feet session. Though I've been in several sessions with her before, I always learn something new about body scale mathematics. She ran a tight 1 hour session using Math in Your Feet as an intro to what she means by body scale math. One of my takeaways this time was how she made it clear how the math and dance vocabulary was a tool for problem solving. I often think about vocabulary in terms of precision, so the tool idea is something I have to think about more. Read the book! Join the FaceBook group!

On to Tracy's keynote. She was sharing about three concrete ways to work towards relational understanding. (From one of her top 5 articles, and one of mine, too.)

1. Make room for relational thinking.
2. Overgeneralzations are attempted connections.
3. Multiple models and representations are your friends. 

Illustrated by awesome teacher stories and student thinking. She wrote her book from years of time with teachers and students looking for real mathematics doing, and it shows.  Read the book! Join the FaceBook group!

Plus, just one of the best people you could meet. She gave her keynote twice, and then led a follow up session. One of the hot tips from that was the amazing story of Clarence Stephens and the Pottsdam Miracle. 

 The only other session I got to was a trio of teachers, Jeff Schiller, Aaron Eling and Jean Baker, who have implemented all kinds of new ideas, collaboration routines, assessment and activities, inspired by Mathematical Mindsets. I was inspired by their willingness to change and by the dramatic affective change in their students. We had two student teachers there last semester, and it was a great opportunity for them as well.

Only downside of the day was all the cool folks I didn't get to hang with, including Zach Cresswell, Kevin Lawrence, Rusty Anderson, Kristin Frang, Tara Maynard... So much good happening here in west Michigan. Check out some of the other sessions and resources from the Storify. 

See you next year?

Poetical Practices

#35! Angelou's #1, but there's no way she'll make
it through conference play without a loss.

I got a chance to hear Billy Collins last night, thanks to family friend Elizabeth.

I enjoy poetry a lot, but don't read as much as I would like.  I have virtually none committed to memory, despite thinking that would be very cool. I don't write it, but have been wondering this past year how you even get started.

He was charming, lovely voice, aware of the audience and built his set of poems like a jazz musician, making sure to hit the hits, but improvising based on conditions, inspirations and audience response. For example he read this poem, To My Favorite 17 Year Old High School Girl, which may be his biggest smash: (around 5:30)

The Colbert Report

We bought that book - it is a lovely retrospective with new poems including the one he and Colbert read.

In our reading, he had me right away; before his first poem he talked about how nice it was that we were there. That, in fact, it was nice that anyone liked poetry given the way most of us are introduced to it. By which, he meant, in school. Imagine if the first time we listened to music, it was someone picking a suitable piece, they played it for a whole group, and then sat us down to ask us questions about it.

By the same token, it's a wonder that anyone likes math, eh?

I really liked a lot of it. This poem, Aristotle, you can hear him read at the Poetry Foundation. It's about how Aristotle introduced or recognized the beginning middle and end structure for literature.

"This is the middle...

This is the bridge, the painful modulation.

This is the thick of things.

So much is crowded into the middle—

the guitars of Spain, piles of ripe avocados,

Russian uniforms, noisy parties,

lakeside kisses, arguments heard through a wall—

too much to name, too much to think about."

 If there is an official poem of Three Act lessons, this is it.

Jamie Radcliffe was a young visiting prof at Penn when I was at grad school there, and an all round good guy. (Now a full prof at Nebraska-Lincoln.) In addition to telling the best ever thesis joke, he had this great line about math and poetry. Even if he had only learned enough math to write doggerel, he was glad to have learned enough math to read the classic works of poetry from the all time great mathematicians.

Sometimes I think that this is the greatest sin of school mathematics. Making people think that the worst of the doggerel is all of math, and then making the students memorize it.  Not only missing out on many of the potential future poets of mathematics, but denying most students the whole art of mathematics.

But what would be the equivalent of poetry readings in school for math? The closest I've seen, I think, is Fawn's My Math exploration of Math Munch. (See also Sam's adaptation.) In my classroom, the day they bring in their patterns they've made and share their thinking and noticing is pretty close.

And it was crucial to let them talk. Just looking, I missed a lot of their intent. Other students noticed things that even the creators hadn't. There were several comments about "what if..." that were good math thinking. I also contributed a few noticings... I think that let them know that there was some real math here.

Afterwards he did a short Q & A. One of the first questions was about his process. He said, (paraphrasing from here out) take for example "I Chop Some Parsley While Listening To 'Three Blind Mice'" I was in my kitchen, chopping parsley, listening to Art Blakey. I was thinking, who hears three blind mice and thinks it's a good jazz tune. It's hot cross buns. But then I thought, how did they become blind? Was it congenital? Think how distraught the mother would be. Maybe an accident - an explosion! Mice covering there eyes. I take the pen out of pocket and now I'm at the office. If they became blind separately, how did they find each other? I mean how hard is it for a blind mice to even find another mouse, let alone two more blind ones? And then, what, the farmer's wife?! Now they've lost their tails, too.

And I start wondering how they came to be blind.
If it was congenital, they could be brothers and sister,
and I think of the poor mother
brooding over her sightless young triplets.

Or was it a common accident, all three caught
in a searing explosion, a firework perhaps?
If not,
if each came to his or her blindness separately,

how did they ever manage to find one another?
Would it not be difficult for a blind mouse
to locate even one fellow mouse with vision
let alone two other blind ones?

And how, in their tiny darkness,
could they possibly have run after a farmer's wife
or anyone else's wife for that matter?
Not to mention why.

Just so she could cut off their tails
with a carving knife, is the cynic's answer,
but the thought of them without eyes
and now without tails to trail through the moist grass

or slip around the corner of a baseboard
has the cynic who always lounges within me
up off his couch and at the window
trying to hide the rising softness that he feels.

By now I am on to dicing an onion
which might account for the wet stinging
in my own eyes, though Freddie Hubbard's
mournful trumpet on "Blue Moon,"

which happens to be the next cut,
cannot be said to be making matters any better.

He finishes this discussion by saying, it's about curiousity. I get curious about it, and then I just have to work it out. (Here's the song; I love Blakey, and know this album well - didn't hurt when hearing the story.)

I so get that - happened to me just this week with Justin Lanier's Star Fractal pattern. I just had to work it out.

 

Someone asked how old he was when he started. About 10. He saw a sailboat on a drive up the Hudson River parkway that he needed to write about. He figures everyone has 50 to 300 bad poems in them; high school is good for getting through a lot of them. Someone asked if poetry was good for expressing feelings. He told her that nobody cares. You're writing to get the reader to feel things. If you're good at it, they might start caring about yours.

One of our friends with whom we went, Joanie, was a high school lit teacher among other things (see her IB thoughts). I asked her how she taught poetry. Students try, and you just read it and give them feedback. A lot of it's terrible, but you let them know if they lost their focus or what they're writing about. I do want to be a reader for my students.

So I'm still processing it, but wanted to get these thoughts down. 

Do you have any thoughts on math as a liberal art? How do you teach to create an appreciation for the poetry of math, or to create a space for future mathematicians?

To close, I'll include one more of his poems. And ask if maybe we should be commiserating with poets more often.

Introduction to Poetry

I ask them to take a poem
and hold it up to the light
like a color slide

or press an ear against its hive.

I say drop a mouse into a poem
and watch him probe his way out,

or walk inside the poem's room
and feel the walls for a light switch.

I want them to waterski
across the surface of a poem
waving at the author's name on the shore.

But all they want to do
is tie the poem to a chair with rope
and torture a confession out of it.

They begin beating it with a hose
to find out what it really means.

Geo-Albers-a

On Tumblr today, geometric aesthetic had this innoculous little image...

Wow. These are so elegant.  But there were no notes! What are these from? Google image search found this composite image all over, but also tipped me off that this was Josef Albers, a (literally) Bauhaus artist who moved to the US in 1933 (when the Nazis closed the Bauhaus) and became one of the 20th century's most influential artists (geometric abstract) and educators. Check out his quotations, frothing over with wisdom.  As much as I love art, there are these huge, gaping holes in my art education... I was even mooning over some of his paintings in the last year at the Hirschorn Museum in Washington! (By the way, the Hirschorn had a great retrosprective exhibition on Albers, and still has a lot online.)

The piece in the bottom left, ‘Structural Constellation, Transformation of a Scheme No.12′, particularly got my GeoGebra-juices flowing. (I found a nice image of it in a nice ArtBlart blogpost covering the Hirschorn retrospective.)  There is so much geometry to notice. But then to start to think about how to dynamicize it was the real mathart fun.

My first idea was to just make two overlapping rectangles, and see which of these proportions are consequences of that. It was much harder to make than I thought at first! This version is pretty robust; robust to me is about durability, keeping the constructed properties regardless of how you move the free points and whether you can make all the possible varieties. (For example, if you make a chevron, is it always concave, always a kite, and can it make any chevron?)

Play with the blue points to change the image. (This is an example of GeoGebra's new HTML5 embedding. Easy from GeoGebraTube on any platform that lets you edit the HTML. [I.e. not Wordpress.com, unfortunately.])



Here's the direct link to GeoGebraTube. I made the overlap by starting with one rectangle, then determining the angle of intersection for the second rectangle by specifying the intersection points. Hmm, now reflecting, I should have continued that! But I wanted the rectangle controls to be at the vertices, so it's probably unnecessarily complicated. I do like how it lets you play to find some of Albers proportions. In particular I got an appreciation for how the vertices of the colored parallelograms come in collinear sets of three, and the symmetries that makes.

My next effort was making a version that built outward from the innermost parallelogram. I thought that would still allow for some dynamic variation, but capture more of the symmetry that makes Albers picture so gorgeous.


I used the center of the parallelogram as a symmetry point for the corners of the rectangles. This one is at GeoGebraTube, too.


Because of adding in the rotational symmetry, only two extra points are needed to determine the rest of the figure. But the control over that inner parallelogram gives a large amount of variety still. (Especially if you turn it inside out.)

Lastly I wanted to figure out what exactly were the proportions that Albers used. As I looked, I realized that you could either build it from two squares divided into thirds, or build the whole thing outward from a single isosceles triangle. I also noticed his exquisite framing... just a flawless image in design and proportion. This one is also on GeoGebraTube, though it's not very dynamic.

Art is a such a good entry into some wonderful mathematics. Look at all the angles, shapes, similarity, proportions and symmetry in this Albers design. I hope you take the time to explore more of Albers work - you won't regret it!

Creative Pattern

So, like most semesters in most of my teacher prep classes, we started out by watching Sir Ken pose the question, "Do Schools Kill Creativity?" Especially for preservice elementary creatures, who often have trouble seeing themselves as math teachers, who often have had very negative math school experiences, and will even sometimes bust out with "I hate math" in front of their math teacher.

This semester's group got pretty into it: the story of Gillian Lynne was high impact, the idea that things need to change had traction, several recognized that they had been subject to this, and the desire to incorporate movement really resonated. (We have a drummer in class, so that might happen.) Some students wrote about creativity for their weekly work: Lauren and Kyrstin, for example.

One of the ways I'm trying to encourage creativity is a work structure (syllabus) like this:

Daily Work: I’m asking you for 1 hour per class. Document what you did somehow and keep in a binder. It is not evaluated on correctness, but on percent completed. Keep an index/table of contents for which days you have work for. This work should either be doing math or learning about the teaching of math.  It is okay to double dip - use daily time for Family Math or weekly work. Just keep track of getting in your hours. I will offer suggestions, but this is your responsibility. It’s a good opportunity to practice generating ways to meaningfully work, which will be an important part of your work as a teacher.

Creating: from our work each week I am asking you to put an additional hour or two into deeper work of your choice. Revise or extend a daily work, play or make a math game, make some math art, find and read something in an area of interest, work on a math problem of interest or create a mathematical task… there is so much different work that teachers do. If you can connect it to our course work, it’s probably okay. Each week’s work will get feedback in terms of our rubric and qualitative.  But those aren’t grades. At the end of the semester this weekly work will be evaluated ⅓ on completion (did you complete work for each week) and ⅔ on exemplars. You will pick two examplars of your doing math, and two examples of your preparing to teach math.

There's a urli.st of their weekly blogs The list helps me in finding them all for giving feedback, but I ask them to link posts to our Facebook group as well. That gets more readership amongst the class than I've ever had before. One of the purposes of blogging their work is to increase their sense of audience. So if you do take a peek, please comment!

The math content we paired with this is patterning. Our first activity (close to this previously blogged one) got us playing with the appropriately named pattern blocks, trying to get at the idea of what makes a pattern a pattern instead of a design. Our ultimate idea was that it needs to be extendable. Not necessarily predictable, but when you see what comes next it should make sense with what came before. They built and then we talked about repeating patterns and growing patterns and then sequential patterns. To emphasize the extendable idea, we built patterns, then rotated to have someone else add on. Clearly - time for pictures.

Clear to everyone

No discussion

People accepted extension,
but felt like 3rd red block
could go "anywhere"

Generated interest because
the start was in a line, and the
pattern was extended 2-dimensionally

Patter creator admitted they
didn't know what came next, but
liked the extension. Next: 3 blues
top and bottom.

Arguments! Pattern creator wanted the trapezoids
double each step, extender focused on blues
"adding one" each time.

Is this a pattern? Designer claimed it was just a design.
Extender felt like the red-blue-green were lines
extending out each direction. All agreed: lovely!

Here's the handout, if you're interested.

The next day I wanted to build on the idea of the sequential growing patterns with explicit connections to algebra. My colleague Pam Wells has the best activity I know for this, adapted from a Mathscape activity. Here's my version. (As a Word doc, if you want to edit. Wasn't displaying correctly...)


Everytime I've used it the lesson has been engaging, provoking discussion, and very supportive of symbolic representation with the visual. Students wanted to work through all the letters on the front, though I only asked them to pick a couple. Many wanted to jump to building their own pattern immediately. Most glossed over the verbal description, so I pushed for that. In general with our pattern work, visual to verbal has been uncomfortable. This is a good activity for the connection between rate of change and the symbolic rule, as several students made that jump. Some students went from data to rule, and some from the visual.

A couple students extended this for their weekly work. I based my sample weekly work on the letter patterns, so I expected more, actually; but that's why we give students choice. Brett extended the letter idea to his whole name, which is actually a pretty nice context for adding functions. (File that one away!) Emily did a really interesting project, making some mathart that  had layers of patterns.

The lesson after this was dominoes - but that's clearly a story for another day. Later in the semester we'll do more patterns using ideas of perimeter, area and volume.

Sequential Circular Reasoning

I've been wanting to get better with the Sequence command in GeoGebra. It's a powerful tool for repetitive computation or construction, and math, of course, is full of the patterns. And then I saw...

#410 - Eights

From the perpetually fascinating Geometry Daily. Perfect opportunity for Sequencing.

First I dug into the geometry a bit. It seemed to me like the interesting bits were the 90º turns and the constant (looking) increase in scale. That means you could do them as a series of dilations - but it was complicated to think about the centers of dilation. Probably better to just figure out the radius and center of each circle.  It seems important that the circles osculate - the kissing is a big part of the visual effect.

I was also thinking about what you could generalize, as that is the point of making it dynamic.It's also easier to build those things in as sliders at the beginning of a sketch than editing them in later. (Though it's not that much harder.) The angle between circles, the number of circles, the dilation ratio... pretty good start.

To build in the angle I just rotated the the circle defining point by the slider and the opposite of that angle. And then I made vectors in those two directions from the center of the circle.  I was thinking I needed those directions to build the new circles.  The radii of the new circles would just be a, a^2, a^3 if a is the scale.

Then came the messy thinking. The center of the first circle is a* the original radius in the direction of one of the points. Then the 2nd circle is.. the 1st radius plus a times that radius in the 2nd direction. I was working it out symbolically, but now that we've got a picture...

So thinking about the centers, I had to organize my data. Usually I scribble on an envelope, but didn't have one handy, so I used Word.

Pretty neat once I sorted it out. I decided to separate the circle centers by the two directions since it was really massy trying to come up with a single sequence to describe the pattern. The key here was the Sequence command.  (Quotes are from the GeoGebra Wiki.)

Sequence[ <Expression>, <Variable i>, <Number a>, <Number b>]

Yields a list of objects created using the given expression and the index i that ranges from number a to number b.

Example:
L = Sequence[(2, i), i, 1, 5] creates a list of points whose y-coordinates range from 1 to 5: L = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5)}.

A list is what you would expect:

Lists
Using curly braces you can create a list of several objects (e. g. points, segments, circles).

Example:

• L = {A, B, C} gives you a list consisting of three prior defined points A, B, and C.
• L = {(0, 0), (1, 1), (2, 2)} produces a list that consists of the entered points, as well as these nameless points.

Note: By default, the elements of this list are not shown in the Graphics View.

But I couldn't get it to work. I often try to do too much at once, so I decided to make a list of the powers of a, and then the coefficients of u and v. The relevant commands in GeoGebra are:

Element[ <List>, <Number n> ]

Yields the n^th element of the list.

Example:

Element[{1, 3, 2}, 2] yields 3, the second element of {1, 3, 2}.

Append[ <List>, >Object> ]

Appends the object to the list.

Example: Append[{1, 2, 3}, 4] creates the list {1, 2, 3, 4}.

Append[ <Object>, <List> ]

Appends the list to the object.

Example: Append[4, {1, 2, 3}] creates the list {4, 1, 2, 3}.

 There is also Insert if you need more control than that.

So I defined my lists

• ays = Sequence[a^(i), i, 1, N]
• cofs = Append[0, Sequence[Sum[ays, k], k, 1, Length[ays]]] 

 It took a little monkeying around to get these next commands to work. I'm not sure if copying from Word was introducing extra characters or what. 

• Sequence[B'_1 + Element[cofs, 2n] u + Element[cofs, 2n - 1] v, n, 1, 10] (list1)
• Sequence[B'_1 + Element[cofs, 2n + 1] v + Element[cofs, 2n] u, n, 1, 10] (list2)

Then the circles are easy!

• CirclesList1 = Sequence[Circle[Element[list1, n], r Element[ays, 2n - 1]], n, 1, N]
• CirclesList2 = Sequence[Circle[Element[list2, n], r Element[ays, 2n]], n, 1, N]

I'm pretty happy with the results. If you'd like to play with the result or remix it for yourself, it's on GeoGebraTube for download or as a mobile-ready applet.

Geometry Daily is a good source for GeoGebra inspiration, as well as good geometry, and just beautiful art. (Usually I put them on my Tumblr.)

Pyth On

Mel Bochner, Pythagoras (4)
from wikipaintings

Arithmetical Design (quite a fun tumblr) posted this beauty today...

I thought that this was something that screamed to be dynamic. Off to the GeoGebra Cave, old chum!
















The sketch started with a right triangle, and then the regular polygon tool to make the squares on the side. I wanted the triangle connecting the next squares to be similar to the original, so I made the side of a square to be the new hypotenuse, rotated it by one of the non-right angles then used the perpendicular tool to make the similar right triangle. Finally, I constructed  the first two additional squares.

Clearly too much work to repeat in the dozens. To use the Create New Tool command you select item or items in the sketch. Then select the command from the tools menu. My first try I forgot that I would need the points to make subsequent squares. Delete the bad tool from the Tool Manager. (Can also rename there if you're trying for something more pythy than Tool 1.)

When I had the squares and vertices selected, the second step of the Create New Tool dialogue was to determine the inputs. GeoGebra will select some ancestors to start, but you can modify the inputs. In this case, GeoGebra selected my first two free points, which doesn't suit. I wanted the inputs to be the the endpoints of the hypotenuse. At the last step you select a name and can attach a custom icon if you're being tricksy.

Once I had the tool it was quick to construct the spirals, and then aesthetics like a coloring scheme and positioning. From the GeoGebra color dialogue you can click the plus, which brings up an RGB color input. (For those times when you need beige, 255-245-235.)

I was going to stop there, but decided that people needed to be able to make their own spirals how they wanted, so added a checkbox to go back to the beginning. (If you make something send me the pic and I'll add it to the post.) Sadly the new points show up with labels - I don't know how to turn that off. Maybe if the labels are off before I make the tool? Tried that and it works!

Here's the finished sketch at GeoGebraTube: teacher page or applet. Sadly, the custom tools don't seem to show up in the HTML5 mobile applets yet.

Bochner has several mathematically influenced paintings, as well as the first three Pythagoras painitings. Check them out at wikipaintings.

Exponential Potential

It really struck me listening to Shawn Cornally in this week's #globalmath session on SBG (click on the Recording tab) how his perspective as a physics teacher leads him to approach his math lessons as experiments. Starting with an experience that makes us want to model or makes modeling useful is definitely the start of some of my favorite math lessons. (While I'm writing this he tweets: "Leaf-Blower soccer went over *really* well today in physics. (vectors, f=ma)")

Starting exponential functions with my preservice teachers, I love to use this lesson adapted from a 5th grade Math in Art lesson. (From my pre-blog webpage.) The idea is the multiplicative patterns present in a Sierpinski Carpet.


(Also in Word format if you want to edit.)

One of the interesting discussions in the initial exploration is the 9 or 17 issue. 9 squares if we count the number of squares as distinct shapes, 17 if we unitize to the smallest level square. For algebra students there's some good opportunities for equivalent expressions, regression and even deduction of function rules. This is a good opportunity for sharing how recording how you're getting your answer can be more powerful than recording answers. The 17, for example, is 1·9+8, then the next level is 9·(17)+8·8. But later, most write it as \( 9^n–8^n \) - which can lead to a pretty neat binomial expansion. Maybe even more interesting and accessible is the 9=1+8, so the next step is 73=1+8+8·8, and gives them a way to generalize this pattern besides recursion.

Once we get to the design your own carpet, there are so many new patterns to find. Here are some samples from this week:

Note that these last two aren't really Sierpinski patterns - but they still raise interesting patterning questions that are extensions of what we already noticed.

Probably easy to see why this is one of my favorite lessons. I also have seen the power of adding in places where students who have not traditionally been strong in math class can do amazing work.

The next lesson to follow up this one has some other opportunities to gather/generate multiplicative data.



It always strikes me how even college math majors find things to be surprised about in this data. Especially the penny balancing one. This class made some neat displays of their data - but I haven't taken the pictures yet. (Didn't know I would be blogging this, as I thought I already had! Maybe I was thinking of the quadratic simulations?) I'll add them at first opportunity.

First Opportunity:

Next we'll look at modeling this data symbolically using technology, and asking questions that raise the need for logarithms. Since, of course, every exponential data set is logarithmic when seen through the looking glass.

Some Sum to One

Another quick activity. Put this version of it together for the 5th graders, but didn't get a chance to do it with them. This would be suitable 5-8, and my preservice teachers got a lot out of it also. I added some more support (in structure) on the square for middle school age kids. The grid divides well into thirds, fourths, twelfths and the like.

The idea was probably Mondrian inspired, mixed with a geometry activity that I like: divide a geoboard into four non-congruent equal parts.

The task for this one is to make different fractions that add, or fit together, to one, or a whole square. Had very interesting conversation with Dave Coffey today about fractions and introducing operations, and representation... and then that conversation spread to Lisa Kasmer.  Nice to work with such interesting colleagues.

EDIT: oops! had the permission set incorrectly on this. If it's not appearing below, here's the link to the pdf.




Two examples from my preservice teachers. The second one was made with flaps that lifted up to reveal the value of the fraction shown. Nice!

You can have your students simplify or not - both create interesting situations.

A nice extension challenge is to do this Egyptian style, with all unit fractions.

To Do + Math on Vacation

The Mathniños at the Modern Wing

Calder Mobile

Math Teachers at Play 22
I'm hosting this months blog-carnival, and I'd love your submissions from your own blog or nominations from blogs you read. Submit nominations at the blogcarnival site, or using the widget at the bottom of the right column on this page.

I never had a chance to write about Carnival 21 at Math Mama Writes... but Jonathan (JD2718) had a nice post about a subtle shift that produced a good effect.

An aside: I got to meet Math Mama over break and Sue was as nice and interesting as you'd think from her blog. We were going to meet up with Maria from Teaching College Math, but the weather prevented it (sorta).

Biggest Math Story of the Year
Time Magazine has submitted that the proof of Langlands' Fundamental Lemma was the 7th Biggest Scientific Discovery of 2009. A quite technical exposition of the Lemma (not the proof) is here. (Hat tip Peter Woit of Not Even Wrong). I do not have an understanding of the math, but it concerns a connection between Galois group theory (about solutions to algebraic equations) and general linear algebra representation theory (think matrices on steroids). One of the most famous theorems that uses the Lemma has Femat's Last Theorem as a corollary. Powerful stuff.

It's nice for students to know that math is ongoing, and also to hear terms like lemma, theorem and corollary used genuinely. I also like how this emphasizes the power of connections in mathematics, and how even (or especially) the world's most powerful mathematician's start problem solving by solving simpler problems or finding another way to put the problem. It also should be noted that Ngo's proof was submitted in 2008, but took until this year to verify.

Math (Art) on Vacation
My wife and I recently won tickets to the Cirque de Soleil show Banana Shpeel (which I would recommend.) On the quick trip to Chicago this week sans kids (hat tip: grandparents!) we had more time in the Art Institute than we would have had otherwise.

In the geometry class I've been writing about, we do one constructive project in which students build a polyhedron for the following context:

Hoity Toity, the upscale chain of Haute Couture for the masses, is having a competition to design new and original knick-knack boxes. Being an accomplished mathematician, you have an unfair advantage, which you intend to exploit to the maximum. Boxes must have a volume of at least 1 liter. (1000 cm3).

Students do a terrific job in general. One student this semester made a dog-shaped polyhedron. Cool. Some classic Archimedean solids, as well as prisms and antiprisms and such.

But at the Art Institute I had two ideas for new variations. The Modern Wing in itself is an inspiration as it is filled with beautiful proportions and rectangles with more connections and relationships than you could ever hope to count.

The Chess Set

This wonderfully sculpted set filled me with visions of two people working together to make a set of polyhedra vs non polyhedra. They could do found or constructed objects. It will require connections (what makes the difference between the king and queen, how will players know this is a rook, what's the difference in the bishops between the sides...) and analysis of the solids and their volume.

The Building

I so wanted to see the isometric drawings for this. I wouldn't insist that the building be built, which gives me pause, but I would want two different representations that would help someone build it. Students should design their own advanced building and think about it in scale

I'm curious to see where these ideas go, and what students make of them. I would like to hear more from other teachers about from where their ideas come, and the process they follow to implement them and refine them in the classroom. I think we teachers do a lot of our sharing as a product exchange.