I’m noticing all sorts of techniques for solving them.  In fact, I learned something from a recent submission and it inspired three new puzzles of my own!  I hope you enjoy.  I’d love your feedback.

I’ve been working the last two days on my own puzzles, and I’ve never had a better time working with inequalities in my life.  They have a way of sneaky way of revealing information that I’m really liking.  2x<y+z tells you some interesting stuff, for example.  Further down I’ll talk about how I write them, what makes for a good puzzles, and the puzzle-writing contest I’m having, but right now, why don’t you try some out?  I’m especially proud of 6, 9, 10, 11, and 12.  ENJOY!

In each case, order the three shapes by weight

Fun, right?  As I’ve said, I’m offering a prize for great imbalance puzzle-writing.  My favorite two puzzlists will receive a print of their choosing from the Stars of the Mind’s Sky series, up to 12″x12″.  Just post your problem(s) in the comments or email lostinrecursion@gmail.com.

So what makes a good puzzle?  It’s a matter of taste (like a lot of mathematics actually), but I tend to like my puzzles pretty simple.  A big, messy, just plain hard puzzle is just a hassle, but something compact and tricky to untangle, now that’s what I like.  I love when a puzzle requires me to think in a new way or exploit some clever little detail I hadn’t considered.  Nathan Chow sent me a really clever puzzle design that implements “entangled imbalances,” which I completely love.  Doesn’t it inspire you to write your own?  I could have cropped it so it didn’t look like an iPhone app, but I love art that reveals the creative process.

In case you’re wanting tips for writing these puzzles, I’ll tell you about my process.  I usually start with a single idea or part of the picture.  The key is thinking about what information it gives the solver, and what other information they’ll need to finish the problem.  After that it’s just a matter of cleverly revealing that information and piecing them together.  Maybe solving the problems above in order will give you a sense of the new ideas I had and was able to wrinkle in.

I hope you’re loving these as much as I am, and I’m dying to see your creations.  Mostly because I want some to solve!!!

While all this was going on, inspiration struck.*  What if the scales don’t balance?  What if one side weighs more?  Behold, my “imbalance problems.”

Unfortunately, problem 2 cannot be uniquely deduced.  It has two possible solutions.

These are just three I came up with for example, but I can see a whole world of possibilities.  I love how designing them is maybe a better problem than solving them.  How many pieces of information do I need to give (albeit, implicitly)?  [turns out it depends on what the information is! (maybe that's not surprising.)]

Imbalance Problem Contest

Inspired by this post by the incomparable Shawn Cornally, I’ve decided to offer a little puzzle-writing competition.  Spend some time writing great imbalance problems (the kind that push the state of the art), then share a link to them in the comments.  My two favorite puzzle-writers (based on indescribably subjective criteria) will win a print of their choosing from my Stars of the Mind’s Sky series, up to 12″x12″.  All of the rest of us will get a collection of excellent imbalance problems to solve and share with others!  Exciting right?  Who’s in?

+ + +

*not a ground-breaking, revolutionary, or truly novel inspiration, I admit.  For me, this is a clear cut Cohen-Ventorism.**  It’s such a slight tweak on an extremely common problem type.  Someone else (countless others) must surely have also come up with this.  And yet, a google search for “imbalance problems” yielded surprisingly little.

** Cohen and Ventor are two imaginary mathematicians who completely independently, perhaps simultaneously, discover (or coinvent) all sorts of mathematics.  (See Newton and Leibniz)

You can click on the Art tab at the top of the page, or you can hover for more specific pages.  I hope you’ll click around, because I’ve included lots of images and little descriptive blurbs.  I’ll post a few pics below to whet your appetite.  Click for more of the same.

I’ll continue to add to that section and use it to document and publish my work as an artist.  Speaking of which, I’m very excited to be included in the Saint Ann’s Faculty and Staff Biennial Art Show.  More on this another day.  I’ve also gotten access to a couple of very nice printers which can go as wide as 13″, so I’m excited to start making some beautiful prints of these images I’ve had lying around a while.

Here’s your bonus: a fantastic video that Justin put together showing off the three sizes of zome nodes used in our meta-meta-node.  Enjoy it all and leave a comment, why don’t ya!

* * *

My earliest love of mathematics came when I realized its utility in creating beautiful things. As a third grader (and ever since), I spent countless afternoons drawing stars and patterned shapes with a protractor or compass. In my work as a teacher I get to share that beauty in creation with my students. In my art I sometimes try to illuminate the complex structure and interconnectedness of simple, patterned objects. I’m compelled to understand complete spaces of related works and how my choices as an artist locate me within that space.

Stars of the Mind’s Sky
22.75 in x 22.75 in
digital print
2012

Stars of the Mind’s Sky is the title of a series of works exploring the space of regular star polygons. Here we see 300 stars “in orbit” along concentric circles. The number of points on a star increases with the radius, and stars of a given number of points are spaced evenly along their circle according to “density,” or the “jump number” used in generating them. Algebraically, these represent the subgroups and cosets generated by elements of a cyclic group. They have been colored on a gradient to indicate the number of cosets; a red star signifies a generating element. As a consequence of these structural choices, we may observe congruent stars with increasingly many cosets, shifting their way to blue along central rays through any red star.

A Zome node made from Zome nodes made from Zome nodes!

A student’s picture, found on instagram

To actually construct it, we built the top first, and then lifted it to build beneath so that the weight would hold the struts together during construction. In the end we held the dome up and slid the bottom underneath it. Amazingly, it held together in front of a big, cheering crowd.

I love MArTH Madness.  We’ve been thinking about this model for a while, and to see it actually finished feels amazing.  As one student put it, “It’s like we wanted to do it, and then we did it, and now it’s done.”  I love that.

Update: Justin posted a fantastic video showing the different node sizes.  Enjoy!

People who knew me in high school (perhaps even later) think of me much more as a musician than they ever considered me a mathematician.**  To me they’re the same.  Playing with pattern and beauty; structure and form; theme and variation.  Seeing what happens when you mess around around and see what you like and figure out what works – exploring what’s possible given a few restrictions: a set of givens, a chromatic scale, whole numbers, whole steps, some paper, a piano, or a string quartet.

The thoughts that led me to write songs and analyze Ben Folds and the Beatles and all of that music stuff, was mathematical thinking.  It’s the same I do now with everything.  I analyze, and I play – carefully and for pleasure.

- – -

“The trouble with knowing what to say and saying it clearly & fully, is that clear speaking is generally obsolete thinking.  Clear statement is like an art object: it is the afterlife of the process which called it into being.”

This is why it’s so important to do real mathematics with young people.  The process generates thinking.  Hearing the foolproof method and mastering the technique is not enough, because it merely replicates the fait accompli.  It’s the creation of the algorithm, the invention of the problem, and the mathematical process that need to be replicated, because these habits fuel creative acts.  So which is the math?  The object or the process?

“The problem with full statement is that it doesn’t involve: it leaves no room for participation: it’s addressed to consumer, not co-producer.”

There it is exactly.  If the goal of math class is to get this stuff into those heads, then who cares about having co-producer?  Is anything even to be produced?  My job, as I see it, is to turn the math on in the brains of these kids.  If I do it right, they come alive.  They interact.  They create.  They control.  THAT is participation.  And the best part is, it infects your brain.  Doing math affects how you see things, what occurs to you, what decisions you make.

See, this is what we work on in school.  This is what the students and I co-produce — their minds.  In math class, we make new mathematicians.

The logo for our all-school mathematical art event

Mathematical art has infected my school! It’s connecting students with mathematics in powerful ways, and it’s high time I write about it.  I’m going to tell the whole story, and there’s a LOT to say, so I’ll try to break it up into sections. Maybe I’ll condense later on, but I want to get it out there.  Enjoy the pictures!

Long story short – What started with a half year mathematical art seminar for high schoolers became a thriving community of artists and boiled over into a math art festival for over 650 students, called MArTH Madness, the largest, most successful math event in school history.

Justin and Anna have each written about it already, so I’m late to the game, but here goes.

* * *

A culture of mathematical creativity [Our Seminar]

By Hudson

This semester Justin, Anna, Max, and I are co-teaching a Mathematical Art Seminar, and “MArTH” is our pet name for the subject. Mondays, from 4:20-6:00, we get together with 10 or 15 students and explore the enormous world of mathematical art. We look at cool examples we find in books or on the internet. We discuss the mathematical concepts at play, but most of all, we spend time making and sharing mathematical art of our own. (We’re planning a gallery show this month.)

By Julien

We’ve played with photography, paper folding, music, video, computer programming, tiling patterns, polyhedral sculptures, and on and on. It’s been amazing so far how seemingly endless the world of mathematical art is. Better yet, the students just keep on digging it.  Their enthusiasm flows over into the facebook group we set up to share our work and amazing links. Students post. Teachers post. Students comment. Teachers comment. Everyone’s there to engage in our little MArTH community in whatever way they want. It’s been incredibly positive and very lively. (social media note: ours is a secret group. No one sees our posts but us, and membership is by invite only.)

* * *

An Opuning [All-school event?]

Geofix Rhombicosidodecahedron

At Saint Ann’s, we have a Reading Marathon and a Poetry Marathon and Scene Marathon, but no such math event. We’d been talking about a Polyhedra Marathon for a year or so, but just talk.  Given my belief that schools teach culture first and foremost, all-school events like these feel pretty important. Culture is built through common experience, after all, but apart from the SAT, we have very few shared math experiences.  This is especially true at Saint Ann’s, where the math classes are highly individualized.

Sensing the opportunity for a terrific pun, I suggested we do some kind of “MArTH Madness” event at school.  It was mid-March when I had the idea, so time was running out on this little pun, but Justin said, “we should at least do something this year.”  Within the hour we made a preliminary pitch to the math department.

Pattern block spiral

The idea was this: We all teach in the same 4 periods of the day, so let’s have a sort of in-school field trip.  Let’s bring our classes together and do all kinds of different mathematical art activities!  Sometimes great ideas are that simple.  They just need people to not say, “no.”  In this case, about half the teachers said they would bring a class, so we had our vote of confidence, and I got to work.

* * *

2 weeks of pre-madness [PLanning and Prep]

3D Construction in the cafeteria

{Where}: Space was a major concern, especially since the initial numbers showed over 100 students per period, and teachers kept asking if they could come. (Even some science teachers brought their classes.) Worst case scenario, we could repurpose our classrooms, but we were hoping for a huge space to fill with people. The gyms were entirely reserved for gym classes, but the cafeteria staff was refreshingly cooperative and let us have the space as needed. Phew! With only a few exceptions, all 6 rooms in the undercroft (our basement) were used for math during our periods, so we made a few room switches with study halls and other teachers, so we could hold all of the other activities down there. My own classroom (in the undercroft) has a removable wall, so we were able to combine the rooms and get another big space for about 70 students.

Quintuple Helix – side view

{When}: We ended up choosing April 3rd for the event date, all but killing the pun, but it was the date we needed.  Math periods were the first two of the day and the first two after lunch, so that was our time frame. That gave us 8:00-10:10 and 1:00-2:35. We had to move quickly in the lunchroom to clean up in between breakfast, lunch, and snack, but we made it work.

{Who}: Students from every grade 4-12 mixed together in the same places, working on projects of their own. Justin hung out in the cafeteria, Max taught in the large room in the undercroft, and students led every other session! Lots of seminar students helped out (9th-12th graders) and we even had a pair of 6th graders help lead a session on mathematical doodling. Anna and I floated around, and Sam Shah even came over from Packer. Other teachers were all around, following their students or just working on some art of their own.

Quintuple Helix – axis view

{What}: If we were going to be making art, we needed materials. We used money from the department’s supply budget to order scissors, straws, balloons, colored pencils, and markers from Quill, which came very quickly. We were able to get card stock, paper, and a few other things from the school supply closet. For computer-aided design we were able to use our netbooks and chromebooks as well as the department’s Makerbot. For the 3D construction space we used Geofix and some other stuff we already had, but mostly Zome. Zometool will mail out loaner workshop kits for a nominal fee, so we were able to get roughly 8,000 pieces of Zometool (2 kits of 4 boxes each) for about $150 after shipping. We also got to keep the Zome for the rest of the month and display the amazing works.  There was a small mix up with the order, but the kind people at Zometool were willing to ship on short notice with rush delivery, and we got everything in time! All together material costs came in under $300.

{How}: The day before, I gave the teachers a flyer to share with students explaining the activities. I split up the classes so that half of the teachers would begin up in the cafeteria doing 3D construction while the other half chose between the activities downstairs. Half way through the period we would switch. Each period, new kids.

The “menu” we handed out to students the day before.

MArTH Shirt

{Bonus}: We wanted to get MArTH Madness T-shirts for the presenters, but the best price we could find online was $18/shirt, and they wouldn’t arrive until well after the event. Luckily a talented colleague prints shirts at home! He agreed to print our shirts for $14 each and in time, with only 4 days notice!!! Shirts always make an event look legit. The design comes from some art I was making for the seminar exploring the structure of star patterns.  The back reads:

“Beauty is the first test. There is no permanent place in this world for ugly mathematics.” -G.H. Hardy

* * *

MARTH MADness! [The Big day]

The school was full of energy as kids were swept up in the mathematical creative process.  They came from study halls and asked if they could miss classes or come during lunch just to make more math art.  I walked around facilitating and squeezing in a little art of my own where I could.  I’ll describe what went on in each room.

Sam Shah, proud of his tiling

{Escher Tessellations} – This was led by Hudson, an awesome 9th grader, based on some stuff he noticed about how to create tiling patterns a la M.C. Escher. We printed out squares and hexagons on card stock, which participants could cut out. These shapes each tile the plane, of course, and Hudson explained how to convert them to new shapes that would create customized tiling patterns.

With a square for instance, if you cut a shape out of one side and glue it onto the opposite side of the square, the piece will fit nicely with a copy of itself. Repeat as desired until you have some beautiful shape that you want to cover the plane with. Then trace it over and over to cover some paper and color to your heart’s content.

A basic hexagonal tiling

You can do the same thing with a hexagon, and there are several other variations. On the square, for example, if you flip the cutout over before taping it to the other side, you get a tiling with glide flip symmetry. Lots of students wanted to try things of their own and explore what was possible. What if I flip them on the hexagon?  What if I do it like this?  Will it work?

Max teaches Sonobe

Best part, when they were finished, we stapled the patterns to the wall outside, just underneath an AWESOME sign Hudson had made that morning. My fifth graders stare at them every day before class, and some have continued working on new designs. We stapled a new tiling to the wall on Friday during free-choice time!

{Unit Origami} - Max (seminar co-teacher) led groups of about 40 or 50 students in some paper folding. He showed them all (and they showed each other) how to fold a Sonobe unit, a very basic design, as well as how to combine them to make a cube.

Icosahedral sonobe ball

With only about 20 minutes to work, it was hard to get into anything more than this, but Max had some other models on display like an octahedron and icosahedron, and the kids got a sense that a lot would be possible from this basic little structure. Amazingly, the excitement from this session carried over into the following weeks, and I helped several students put together larger structures, which we left on display in the undercroft.

A page of doodles

{Mathematical Doodling} – This session was led by 4 experienced math doodlers (junior, sophomore, and 2 sixth graders), who shared their work and helped students come up with their own clever doodles. Usually one of them would be explaining a specific kind of doodle to a few students, while another two helped with a large chalk doodle at the board, and the fourth worked on a fresh design at a desk. We also had chromebooks available so that kids could watch some of Vi Hart’s “doodling in math class” videos for inspiration and jumping off points.

Chalk board Apollonian gasket

Doodles included Apollonian gaskets, fractals, polygons, stars, coloring patterns, and whatever else occurred to the artists. Once again, we were able to staple some of the work to the wall for display.  There is something so simply satisfying about filling your school’s walls with student work.

Symmetry Artist piece 1

{Computer-Aided Design} – We had two rooms devoted to two two different activities using computers, both led by students. In one room we had chromebooks set up so that kids could use Symmetry Artist, a wonderfully beautiful applet. It let’s you select various rotational and flip symmetries, then it copies whatever you draw so that the result has the desired symmetry. The kids used the “line” option to make stars and polygons incredibly easily. As they drew, the image changed symmetrically, possibly suggesting mathematical dance.  When I stopped by, I showed off a few of my ambigrams, and a few played with the idea.  If you’ve never played with Symmetry Artist, you should give it a try. The symmetry itself is so beautiful you almost can’t go wrong.

Symmetry Artist piece 2

In the other room we had our set of 10 netbooks set up so that kids could do some algorithmic programming in Scratch, but the Makerbot stole the show. Our department got a Makerbot 3D printer earlier this year, and we’ve been playing with it since. We brought it down to the undercroft so kids could watch it print and print their own pieces, which they designed on 3DTin. We used that site mostly because we hadn’t yet installed OpenSCAD, and we didn’t yet realize how much better Tinkercad is.  (Which is A LOT better.)  Unfortunately, the ReplicatorG was glitchy after the move, so we didn’t get anything printed that day. Nevertheless, Noah, one of the seminar students, spent Monday afternoon in the computer center, printing out the MArTH Madness designs for the people that came to his activity.

I’ll just add here that the kids were unbelievable teachers for each other. They were patient, kind, and totally passionate.  They were an enormous help, and the event would not have been such a success without them. Students led 4/6 sessions, after all!

Artists at play

{3D Construction} – Justin was in charge of this space, but most of the work was completely unguided. After wiping off the lunch tables, we just set out boxes of Zome, Geofix, Pattern blocks, Unifix Cubes, and Polydron. Free build! Kids jumped in and built towers and solids and all sorts of spikey balls. A simple start often led to an idea and then a plan, at which point the student has to wrestle with the mathematical possibilities until they arrive at a satisfying end result.  Then the process spirals on, as one good turn yields another.  Against one of the walls we set up a display (just stuff we had already made) to give inspiration, and more pieces got added throughout the course of the day. I loved watching kids snap pictures with their phones. There’s nothing like the feeling of satisfaction and success you feel after building something you like!

MetaZome Cube

Justin spent his time working with a coming and going group of students to build a metaZome structure. They used the zome pieces to build larger versions of the zome pieces.  (You know I love that kind of thing.)  He wrote about it here. I love that they didn’t quite know what it would ultimately be. The project had space to grow and build steam in whatever direction it might, and in the end they put together a beautiful metaZome Cube with diagonals and a center node.  After the Madness had subsided, we carried the structure to the undercroft, where it stayed on display for the rest of the month. We had to return the pieces last Friday, and student after student walked by saying, “Awwwww. Don’t take it apart!”

“Don’t worry,” we said, “next year we’re going to make something even better!”

* * *

MArTH on! [Reflections]

I am incredibly proud of the work we do with math at Saint Ann’s, and this newborn mathematical art program exemplifies a lot of my educational beliefs:  (I cannot speak for the department.)

1)  That math is something people do, and that we should be getting kids doing mathematics as much as we can. Well that happens every day in our seminar, and students are still riding the MArTH Madness doing-wave.

2)  That math can be done by everyone – regardless of age, grade, race, gender, and to some extent, regardless of mathematical preparation. Whether you’re a seminar student, a 4th grade girl, a 10th grade boy, or a 50 year-old teacher, you can make beautiful mathematical art, starting immediately.  And man, does it feel good!

3)  That math classrooms should be learning communities – groups of people that work together to learn and create.  This is exactly what our seminar is, and the all-school event was a summit that brought us together in unprecedented ways, around beautiful mathematical work.

4)  That the determining factor in the success or failure of many endeavors is personal significance, and that students can thrive in an environment of free creative choice.  Mathematical art is entirely about finding some part of the inexhaustible mathematical world that you can explore and call your own.  Original mathematical art provides personal attachment and identity with a subject that desperately needs it.

5)  That there’s more to an education than the transmission of data and technique, and that learning should be highly individualized. When we made Escher Tilings, we shared in the experience, but everyone was creating something unique, for themselves. What a student takes away from a room full of 3D constructors is highly nonstandard, beyond their own unique creation. Maybe they learn about Platonic solids, polygons, symmetry, or graphs or something. Or maybe they just learn that they are capable of bringing new things into being. They themselves can create mathematics.

The ideals of MArTH Madness can absolutely carry over into our students’ lives and into our math classrooms.  After experiences like these, a student’s relationship with math can change drastically.  When presented with new concepts, they may see a chance to play and create where previously they saw nothing but work.  Far too often, math is math class to our students, and it stops at the door.  It rarely extends beyond our own class, or grade, or school, but MArTH Madness showed us all that math can break these boundaries.  We can work together at any age.  We can all have original math ideas.  We can all create beautiful things.  We can make it on our own, we can notice it in the world around us, and we can all find ourselves in mathematics.

MArTH Madness teaches us that we are all mathematical.

* * *

Now, the seminar carries on with full-steam, in its basement room, even on beautiful afternoons. Next year we’re offering it full-year, as well as a middle school elective, which has heavy enrollment. Students and teachers are already looking forward to next year’s “Madness,” which may last over several days. We bring MArTH in and out of our classes, but best of all, it’s in the hands of our students, right where math belongs!

For more pictures visit the MArTH Madness Facebook page.

Anyhow, let me tell you about Mai Li. She’s an amazing young mathematician in our mathematical art seminar. She’s also opted out of Trig/Analysis for her junior year in favor of Intro Topology and Modern Algebra electives with Anna and a semester course called Fractals and Chaos. She helped lead the doodling sessions at MArTH Madness, and she just rocks in general. Well, as much as she would love to come help them share MArTH with the people at EDcampNYC, she can’t. She has to take the SAT’s.

No big deal. Whatever. She has to take them. It’s fine, but too bad. All of this is just the setup for some clever little thing that Justin said in our office the other day.

* * *

Q: You know what SAT stands for right?

A: The SAT Aptitude Test.

HA! Love it. It used to stand for “Scholastic Aptitude Test,” but I think they’ve abandoned that. (I wonder why?) I looked over some videos and stuff on the college board site, but I couldn’t find my answer. I did find this, however:

“the combination of high school grades and SAT scores is the best predictor of your academic success in college.”

hmm. Do you buy that?

The problem that I have with the SAT, in particular the math section, is that it cannot test for the applicants ability to do math. If you disagree, it’s simply that we have different notions of what it means to “do math.” I’m not gonna get too deep into this, because I’m mostly writing just to share Justin’s little meme, which might otherwise be lost forever.

“SAT Aptitude Test” is a fitting acronym, because the SAT is only really testing your ability to succeed on the SAT itself. Infer what you will. When students need to prepare, they don’t get a math tutor. They get an SAT math tutor. Could the inauthenticity be any more obvious? The biggest criticism of my department (accurate or not) is that we don’t adequately prepare students for this test. Perhaps that’s only further evidence of my point, since our primary objective is doing real mathematics with students as often as we possibly can. [disclaimer: I am not a spokesman for the school.]

What do the SAT’s demand for success? Technical training, perhaps, which is only one aspect of a mathematical education. Maybe most of all, SAT success requires SAT experience.

Want to prove that you’re SAT apt? Why not practice with the SAT aptitude test? Get it?

* * *

OK that’s it. This is just something I’ve been thinking over and really enjoying. Thank you, Justin for your unending nerdly wit.

Oh by the way, we can keep expanding SAT and get the following, all of which are the same thing:

The SAT

The SAT Aptitude Test

The (SAT Aptitude Test) Aptitude Test

The [(SAT Aptitude Test) Aptitude Test] Aptitude Test…

and on and on.

* * *

Good luck students, and try to remember it’s a collection of paper and ink. Don’t let it shake you too hard.

The question I’m a little more interested in, the one I posed in my first post on this stuff, is this one; If 9 is a meta-triangle (a triangle of triangles), is 27 a meta-meta-triangle?  I’ve often called it that, but I can see a different interpretation, so I asked the meaning.

I got a bite and this comment from Max Goldstein, but I wanted to share what I thought was in interesting answer, and once I started writing, the math started to push back, and I noticed new things, and on and on.   This is me doing math, so I thought I’d publish some of my work here.

* * *

Question: What’s a meta-meta-triangle?

Like most math, this depends on the meaning of the terms. Let’s take “triangle” as understood and consider “meta.”  A meta-movie is a movie about a movie.  When faced with the challenge of writing, young poets often write about writing.  That’s meta-poetry.  9, then, is a meta-triangle, because it’s a triangle of triangles.

Then what’s a meta-meta-triangle? It’s a meta-triangle of meta-triangles!  So in the powers of 3, that would be a 9 of 9′s.  That’s 81, not 27!

[Do you buy that?  Is it clear?]

* * *

Here’s an argument by notation:

Definition: meta(x)= x(x).
Examples: meta(play)=play(play)     —–>  Hamlet
meta(triangle)=triangle(triangle)     —–>  9

Let’s use m instead of meta.  It starts to sound weird if you say it too much.  Now m(string)=string(string), and m(asdf!3!)=asdf!3!(asdf!3!).

Note: The definition of meta(x) relies on x(x) making sense. If x is in meta’s domain, then x must be in its own domain! This is just spooky to me…

Then if “meta” is in its own domain, we know m(m)=m(m), by definition.  So meta(meta) is itself!  (I’m not making this up.)  As above, m(triangle)=triangle(triangle)=9.  In short, m(t)=9.  Get ready…

meta-meta-triangle is m((m(t))=m(9)=9(9)…  81?

[Does the notation make this clearer or obfuscate the ideas?]

* * *

Well that’s it.  It’s a little piece of mathematics that I spent time carefully wording for clarity and communication’s sake.  Choosing a single notation, while proper, can be a little austere and hard for the reader, so I picked and chose which representations to use on each line, in the same way I choose punctuation.  I hope that comes across.

There’s so much unreadable mathematics in this world.  It breaks my heart.  I’m convinced it’s helping to kill mathematics.  This means I have a responsibility to try and work on quality writing – concise, elegant, clear, and convincing mathematical arguments that can be read widely.  We’ve been putting extra effort towards this in my department.  In short, this is the essence of proof – “convincing” argument.  (note: unreadable symbology and jargon is often NOT convincing at all.)

Maybe you’ll share your own mathematics with the world.  The goal is the clear and simple communication of ideas.  Two-column proof is NOT the only way.

Anyhow, with MArTH Madness and everything else going on, life is really rushing straight at me right now.  Somehow it feels amazing!

* * *

These could serve as exercises for a student wanting to practice using exponent notation, but they could also serve as decent stand alone problems.  Something as simple as “How many dots?” maybe.  Perhaps I’ll post some on 101qs.com and see what kind of questions they get.  (I did.)

Problems and exercises can both be valuable, but they aren’t the same thing.  Math students should know the difference between doing math in its most free and open form, and completing exercises to build familiarity, skill, and speed.  I need to do a better job of including this in my teaching.

What do you make of these?

Perhaps you’re wondering “How much time did he spend drawing all those dots?

* * *

Through school and work, I’ve been able to download two copies of Mathematica, and I LOVE it!  It’s especially good for all of my mathematical art and imaging needs.  It also has an exceptional documentation system with hyperlinks and examples that make it very easy to teach yourself and play.  I’m amazed how much I’ve learned in two weeks.

Mathematica also handles recursive programming nicely, which is perfect for fractals.  I wrote some code this afternoon and generated these images.  Enjoy.

Now how long would THAT have taken me to draw?

Powers of Three

(Is it fair to call 27 a meta-meta-triangle?  Math up these comments.)

Anyhow, my students made the connection to Sierpinski’s triangle, and the whole “meta” conversation brought up Inception, video feedback, and other cultural references with which they are familiar, so the whole thing really connected to things they know. The exponent tells you how many levels deep the “inception” goes. Of course I talked about how 3^4=1x3x3x3x3, but these patterns are too rich to leave boiled down into something that simple.I hope you don’t think that’s too abstract or confusing for 8th graders.  These ideas are somewhat mind-boggling or perplexing, as Dan Meyer might say. They’re not impossibly difficult. They’re just weirdly fascinating and challenging. Ideas like that (and damn good problems) draw learners into the math world. The point here is that we had a chance to struggle with instantiated concepts, and by wrestling with things like meta-meta-triangles and by arguing about them, we did some math together. (Opinion: A mathematical conversation is math. Agree?)

If I tried to wrap the material up into a perfect little package that could be quickly delivered and easily understood or memorized, then there wouldn’t be any mathematics left at all.

* * *

The base ten number system is built on exactly these ideas, and scientific notation exploits that beautifully, using the exponent to indicate the level of depth. This was where I hoped to lead the class, and I had some (slightly dry) worksheets at the ready, so I handed them out.

While they were working, I pulled up an AMAZING visualization called Scale of the Universe 2. (If you’ve never seen this before, I wouldn’t blame you if you stopped reading and just played with that for 40 minutes.) I was happy to report that it was designed and programmed by some high school kids in California, but I was even happier that the room started to fill with energy. I narrated our little viewing a bit, but I could hardly get a word in between their questions! Their questions. A marathon is that big? How many central parks is Angel Falls? What’s “total human height?”

I answered a few, which was almost too much fun, and then we started researching. We pulled up wikipedia and google and found the most amazing information. Even I learned an incredible amount, and for the rest of the period we were tapped into the magic of curiosity and learning in the information age. Sometimes we would do computations in our head, sometimes on the board, and sometimes we would just pull up WolframAlpha and had the computer solve it for us.  Arithmetic, paper, and pencil have their place, no doubt, but using a great tool isn’t a sin.  It’s a virtue.  Especially today, with the most incredible tools readily available.

I didn’t care that we stopped working on the scientific notation problems. We were using scientific notation right then to understand our world. It was OK that the sheets didn’t get filled in. We could do that later. This was going to leave a mark. I went next door, grabbed 10 netbooks, and put them in their hands. “Keep going. Whatever questions you have, answer them.”

I did it again with my 10th graders and again with my 5th graders. Amazingly, the Algebra 2 kids would ask me questions and not even realize they were staring at the screen of the most powerful information tool ever available. “Google that!” All of a sudden this was a lesson in using the internet. Whatever you find, whatever questions you answer, record it. Tomorrow we’ll share. “Where?  You could use Google Docs…”

If you believe that school is where we equip students for the world, is there any doubt they need these skills and tools in their kit?

* * *

I’m telling this story for three reasons:

i) That’s cool math up there. Exponents and fractals are amazing, and they actually connect to my recent post, quite possibly the coolest problem I’ve ever come up with.

ii) To show how computers can help us meet the enormous demands of true interest. 100 years ago, when our school system was designed, I think this was literally impossible. Schools today, however, have an amazing opportunity if they can shift their mindset.

iii) I love when a class spins on a dime in the most unpredictable ways. It tells my students are awake. It tells me they are conscious and active, processing and guiding our work. I love it. I think of the transfer of power (not just information and technique) as a central goal of my teaching, so when stuff like Scale of the Universe 2 falls into my lap, my job can get very easy.

I’m so thankful to be at a school where we can follow our whims and simply study whatever is most fascinating.

Google loves women.

News to many (mostly delivered by Google’s home page), but today is International Women’s Day!

Fun fact: On March 8, 1917, Russian demonstrations marking International Women’s Day initiated the February Revolution! (Thanks to Bill Everdell, with whom I share my classroom, for that little tidbit.) More would know the holiday if we lived in Russia, of course. Even places from Afghanistan to Zambia have made this a national holiday.

Well, I’ve been thinking about gender equality for a long while, especially with regards to math, so I thought I’d share some thoughts.

* * *

I would say we have a real problem with girls and math. I won’t site studies, but let me share some observations:

Graduate mathematics students are overwhelmingly male. This term, for instance, we have four women in Complex Analysis. Algebraic Topology has none.

Top scorers on math contests are routinely male. Why?

I teach a class of 8th graders that segregate themselves by gender every single day, without so much as a word about it. What is that?

At Saint Ann’s I created a course called “Algebra 2: Functions and Abstract Algebra.” In its first two years, only three girls have taken the class. Another two dropped the first day. Am I the problem?

We also now have a fine spread of one semester math electives for high schoolers. They’re buried here, but we offer incredible courses like Intro Topology, Non-Euclidean Geometry, Fractals and Chaos, and The Complex Plane. And yet, registrants are overwhelmingly male. My Complex Plane course hadn’t a single girl in it!

Even our incredible Mathematical Art Seminar (a group of more than 20 students) has only four girls!!! What is going on?

This is killing me.

* * *

Especially, since I also teach 5th grade, where it is plain to see that my students are equally apt. Few will disagree that boys and girls look and behave differently (how else could we tell them apart?), but I don’t see even the slightest tendency to mathematical weakness in my female students. Some of the most delightfully playful, thoughtful, and powerfully-minded 11 year-old mathematicians I know are girls.

What gives? “What happens to girls in math class?” (I was asked that by a parent of two incredible girls this year.)

I don’t know…. I don’t have the answers…. I can’t fix the problem, on my own, but I’m sure that consciousness is the first step. (It almost always is.)

So I think about it everyday. How do I treat girls differently? Do I call on them as much? Do I expect the same from them? Do I talk to them the same way? Do I look at them the same way? Do I fear creeping them out? Does that differently shape the male-female teacher-student relationship? Is it not possible to have the same relationships with female students as I have with males?

It’s very easy for me question my actions, but extremely hard to know what it’s like for my female students, or even my male ones for that matter. I simply stay conscious and make every attempt, big or small, to encourage female mathematicians. We’ve made extra effort on Math Munch, for instance, to include stories about females in mathematics. Check out this and this.

The math question is only part of a much larger set of societal issues. As the spring clothes come out, think about what’s going on. What do we expect of our girls?

* * *

A turning point for me came when I read Douglas Hofstadter’s incredible tome, Metamagical Themas. I greatly encourage you to read the section called “A Person Paper on Purity in Language.” In it, Hofstadter argues against our gender-based language habits, and by analogy (as usual) to racial language, reduces it to absurdity. Let me reiterate; This paper is totally worth reading. It’s had a great influence on me.

It’s the reason I feel completely awkward every time I hear or say, “you guys.” Sometimes I almost can’t stop myself, even when I’m speaking to a group of all girls. What is that about!? They’re not guys. Five years ago, I would have said, “it’s fine. Who cares? Even girls do it,” but now it actually makes me cringe.

And so, I have made a very small change.

I say “y’all.”

* * *

My mom’s side of the family is from Tennessee and Kentucky, so I’ve heard y’all plenty of times. Younger me found y’all entirely repugnant. The plural “yous” is also an option, but unfortunately both often carry low-class connotations.

Nonetheless, I say y’all.

Saying it means confronting and denying a strange male-default. I take some pleasure in sounding a bit more like personal hero, Ben Folds, but I keep on, because I believe it’s right. In any case, it certainly sounds awkward at times, especially in private school NYC, but I couldn’t care less.

The more I say it, the more naturally it flows. Better yet, every time someone comments or questions me, I have the perfect opportunity to discuss gender equality!

* * *

I’d LOVE to hear from readers about what you do to take on these issues. Please comment!

In any case, thanks for reading, y’all!!! Happy Women’s Day!

I bring it up, because I had a fantastic exchange with an Algebra 2 student yesterday, and it couldn’t have happened if I refused or was prohibited from using Facebook with my students.

* * *

An Algebra 2 student of mine sent me a message last night saying, “Those Vi Hart videos are pretty crazy, The doodle games can get so absorbing.”  I of course loved this for lots of reasons.  Math was seeping its way into his life, and he was spending his own time thinking mathematically and scratching the math itch.  This, more than anything, in my opinion, is the critical trait present in the lives of mathematical thinkers.

We started talking about this video, my personal favorite, all about different kinds of stars.  I said something about how all stars with a prime number of points come in one piece, and he said “something i was thinking about with stars that might not go anywhere, what happens if you arrange the dots that map points into a shape other than a square or a circle? do the shapes change or do they just get a bit warped?”  A question of his own – the sure sign of a mathematically active brain.

I spent a ton of time thinking about this last Spring, so I started telling him all about my research and a few of the intriguing questions I worked on.  In fact my own weird path led me to a remarkable fact about triangular numbers in modular rings, and that somehow led me to write some pieces for a DIY music box I bought last year.  I played it for our class one day, and he was amazed to hear the mathematical design.  For me, this is “math in the real world.”

We went on for a while talking our way through a few things.  He was clearly ready to play with the ideas on his own, so I tried to give him just enough to fuel his curiosity and send him on his way.  Here’s my favorite part: “yeah okay i feel like theres a piece of this that im missing, and with the increasing skipping of points when do you stop? is the star closed or composed of polygons.”  I didn’t have to wait for our next class to chat (something I can never find enough time for anyway).  I just went to his profile page, hit record on the webcam, and drew him a few diagrams!

[Have you noticed his grammar and spelling were less than impeccably proper?  Can you think of any reason why I should care at all?!  I can't.  Not when we're in the middle of great math thoughts.]

This whole conversation was a perfect lead-in to the Mathematical Art seminar I’m leading with Justin Lanier and two other colleagues starting next week that this student will be taking.  Talk about bringing motivation and interest to class.  He also posted a picture of his own “string art” drawing earlier this week (deeming school Facebook worthy), so he’s clearly ready to play around with mathematical projects of his own.  I’m thrilled and honored to mentor that process, wherever that may occur.

* * *

My school puts the relationship between teacher, student, and subject at the center of its philosophy, so I feel rock solid about stories like this.  In fact, I wonder if it gets any better.

We have a policy about email that says teachers and students are free to communicate this way, but feedback on classwork should come during class time. To me, handing back a piece of paper with written comments seems rather equivalent to emailing it, but whatever.  I suppose I get it.  That’s fine.

Luckily, our policies don’t yet preclude me from interacting with students through other social media, though I know some administers are worried about having “official classwork” populate there and would probably wring their hands at me.  I’ve heard a new social media policy is coming down the pipe, but I’m just praying my students and I can continue to connect as successfully as we have.

We are friends.  Not the kind that enable your bad habits or exist for status in the often uncomfortable school social scene.  We’re friends with shared interests, and Facebook is where we show them off and connect around them.

I’m not on Facebook to gossip or read whiney statuses or relish in school drama.  I’m there to share content.  Posts with real substance.  Quotes worth reading.  Pictures worth seeing.  Links worth clicking.  Ideas worth thinking about.  The kind of content worth filling your life with.  This is what a life of learning is about.

By interacting with students through Facebook, I can help them fill their life with good stuff and play the role of their intellectual friend – the one who challenges them and points to great stuff they can explore on their own.  Isn’t that teaching?

I just hope whatever policy comes down the pipeline lets me continue to do what I’m already doing well – making real connections with students around content.

_ _ _

A surreal teaching video recently put into focus just how artificial class can be, and in particular how inauthentically the role of teacher is often played. I want my students to see me as human, deserving of their respect. I don’t want my authority in the classroom to arise from the power of grades, or sending kids out, or threats, or yelling, or just being an adult, so how do I gain authority with my students?

I can backtrack this for a long time, stopping thought after thought, thinking “yeah, but how do I do that?” How do I show them that respect is important? How do I get students to give each other their attention? How do I get them to deserve each other’s attention? How do I get them to do anything without already having authority?

I try to do this a little differently every year. I’ve tried “norming.” Several times, I’ve included, “just don’t piss me off.” (pathetic in hindsight) I’m looking for something more sustainable, a potential motto for my students and me alike. Here’s what I’m thinking.

Don’t be Times Square. Be the Flatiron Building.

* * *

Times Square

I live in New York City, and I love the architecture. One place I really hate, though, is Times Square. As a new yorker you almost need an excuse for being there, because no one wants to be caught dead in such a gaudy tourist trap. Times Square is screaming at you with its lights, sounds, and overactivity. It’s almost oppressive to the senses. Times Square demands your attention, like all caps – LOOK AT ME!!!! I’M WORTH LOOKING AT!!!! SEE!?!?!?!?!? COOL HUH?!!? Every time I see it, I want out.

I was talking today about a teacher who typifies the “Times Square” approach to authority. He is overpowering, dominant, and very very loud with his elementary school students. Many students come to love him, but some, so I hear, are traumatized each year. My approach to teaching relies on student values, but his method is all about pressing value and compliance down from above. It’ll never work for me and my students.

Times Square is sensational, but completely unsustainable. How much time can you spend there, in the lights and the crowd, before being completely overwhelmed, fatigued, and disinterested? Who grows thoughtful in that environment?

* * *

The Flatiron Building

The Flatiron Building, on the other hand, is my personal favorite. One of the first skyscrapers, and once the tallest building in the world (at a mere 285 feet), there’s no question it was originally an attention grabber. Even still, its design is striking and has stopped me in my tracks several times, but it’s much quieter architecturally. It gives you room to stand and admire it. Every time I see it, I want to stare.

What makes the Flatiron so compelling and inviting is its simplicity and beauty. If I can just show my students beautiful and inviting mathematics, and give them the space they need to respond and take it on, they might stay longer in its presence. This means taking time to appreciate content, and it means quieting down to share in the experience together. If I can model this behavior and bring them the Flatiron building, perhaps they’ll follow suit.

Everyone knows Times Square for the lights and the crowds, but who knows what the buildings actually look like?

It’s short, but sweet, and it comes to us via John Burk, writer of an amazing blog, called Quantum Progress.  (Another trusted stranger I found on Twitter.)

It’s a clip of Steve Jobs sharing “one simple truth” that will change your life forever.

When you grow up you tend to get told the world is the way it is, and your life is just to live your life inside the world. Try not to bash into the walls too much. Try to have a nice family life, have fun, save a little money.

That’s a very limited life. Life can be much broader once you discover one simple fact: Everything around you that you call life was made up by people that were no smarter than you and you can change it.  You can influence it.  You can build your own things, that other people can use.

Once you learn that, you’ll never be the same again.

* * *

John shared it with me because it so echoed my thoughts from this post, called “Humans did that! (and you could too)”  You can read John’s thoughts here and see the cool “made by humans” sticker he designed.

Point is, Steve Jobs had it exactly right!

The world is run by humans, just like you and me.  This life is yours and you can influence it.  You can push at it.  You can shape it as you like!

Learning this is a genuine paradigm shift, like waking from the Matrix or realizing its rules can be bent.  The moment you realize that is the moment you are conscious enough to take hold of your life and design its trajectory.

I desperately want my students to have this same awakening.  I want them to realize our class time is theirs and mine, and we are free to shape it as we please.  If we can be conscious and intentional enough to take control, together we can plot the unique course of our learning.

I teach stuff like Algebra and Calculus, but these are the courses I really want.

Well these damned grades are still around, and they continue to drive me crazy.  Here’s a few more stories about how silly, misguided, misleading, harmful, and uninformative grades often are.

* * *

How many points is a photocopy worth?  I mean, if you assign a homework problem, and someone photocopies the proof from the book, how many points is that worth?  I only ask, because this just happened.

A couple friends are working on their Masters at NYU, and this term they are in the same Complex Analysis class.  They’d like to discuss homework together, but Justin tends to put things off to the last minute, while Liz is a more meticulous student.  When Liz wants to chat, Justin usually hasnt started, which drives her a bit crazy, especially when he gets better homework scores.  This week it came to a bit of a head.

Put off and put off again, Justin’s homework was overdue.  The last problem said something like, “feel free to use your text and the following property.”  Looking through the text, he found a solution to the exact problem, property and all.  ”I could copy this down by hand, verbatim, or I could just photocopy the proof,” Justin thought, before heading to the copy machine.

So how much is a photocopy worth?  FULL CREDIT!

Justin got full credit for photocopying a proof from the book and citing the reference.  Despite Liz having worked this problem out and thought her way through, she earned a 4.9/5 on the homework to Justin’s 5.

Liz in a half-dejected tone – “I’m just not as smart as you.”

* * *

I’m wrapping up my own Masters program, though the luster and appeal of academic training are certainly fading.  For whatever reason, I have little motivation for the two classes I’m taking this Fall.  Note taking has been minimal, homework pushed off, and I spend most classes on my phone, texting, tweeting, and playing Whale Trail.

In keeping with mathematical tradition, most exams ask for replicated proofs from lecture.  I’m usually very good at this kind of thing, because I listen to the lecture and make sense of it in real time.  When the exam comes, I simply retell myself the story of the lecture and write it down, but having paid little attention this term I sensed impending doom.

As I walked in to my Field Theory exam, I was confident it would go terribly, and looking over the questions, I was certain.  Unlike the good students who had memorized the notes, I knew none of them.

I just wrote down whatever I thought could resemble proof and put down any shred of detail I could recall.  For one I simply included all of the premises and said things like “this contradicts the maximality of n, hence we see that f is irreducible over F,” without much attention to the absurd lack of mathematical rigor or value.  About an hour later, I was the first to leave.

The professor must not have read the thing, because I got an 80, while friends I know to have strong command earned 60′s.  Worse yet, I have an A at midterm, despite knowing for certain that I only sort of know the field theory.

* * *

How much do I need to say about how ridiculous these two stories are?  Obviously Justin’s points were not earned for clear knowledge or mastery, not even for hard work.  The very same is true of my exam score.

The first thing people ask about schools without grading is “if you don’t give grades how can colleges tell how good the kids are?”  (The answer is obvious, you just tell them by writing about each child.)  My question is this – If schools are going to continue giving grades as meaningless as these, then how can we tell anything?

We can easily blame these professors and say they’re “grading poorly.” Perhaps Justin deserved a zero.  Perhaps I should have failed.  If there’s any case to be made for grading, I’m certain these teachers have got it wrong, but these two stories point to a fact that is true across the spectrum of graded environments.

The central lesson: It’s about doing the work and not about the learning.

When we ask students to memorize and replicate for tests, this is surely the message.  Even worse, we equate the work with learning, when they are plainly distinct.

It’s not like putting iPads in the hands of seven year-olds makes everything easier at school, but man, when you’re interested and want more of something, the internet has got it going on.  Here’s a small ode and the story of a new website, “Math Munch.”

* * *

A Magic Square

First, a quick story of something I found and shared with some colleagues.  In a magic square every row, column, and diagonal add to the same number.  These have historical origins, known to the Chinese as early as 650 BCE, but the geomagic square is an amazing recent twist on the idea.  (see below)  My colleagues were blown away and decided to share these with kids.

I see incredible things like this everyday, and just like that, by putting my friends in contact with it, their teaching changes.  Their classrooms become just a notch more “content rich.”  For this I was called “Ambassador to the Internet,” a role I am happy to play.

Can I do the same for my students simply by putting them in contact with rich math sources more often?

* * *

This year I’m working harder to show my students how much they can learn from the internet if they take it seriously.  It’s important to me that they know math exists outside of classrooms, and that I am not their only source for math expertise.

Justin had the brilliant idea to start a blog and share with our students, so three weeks ago we started Math Munch, “a weekly digest of the mathematical internet.”  I wrote my first entry today, so I thought it was time to share.

Each week we try to post three good math sites covering all kinds of math.  We feature creative and artistic aspects of math as well as connections to number concepts.  We’re also thinking about trying to showcase the people behind the work and include a diverse group of mathematicians.  Hopefully we do more than amaze our readers; we aspire to spark creative work.

Justin asks his fifthies to check the site once a week and write a journal reflection on what they’ve seen.  I haven’t shown it to all of my fifth graders yet, but I did have one girl check it out during her free choice time on Friday.  She heard about it by word of mouth and after playing around said “this website is so cool!”

Nothing better than that.

* * *

I hope you’ll take a look around Math Munch and enjoy the huge and fascinating mathematical internet.

Bon appetit!