## Monday, August 4, 2014

### What's on My iPad - Summer 14 Edition

What does a math ed prof keep on his iPad?  I thought I might as well just show you - though I did tidy up a bit before having company over.

Main Screen

Nothing too unusual here. Evernote is amazing even though I am a novice user. Really enables me to leave the laptop in my office a lot. Google Drive completes that picture, especially since I can make things available offline.

Main calculators: MyScript for computation and Desmos for graphing.

GeoGebra, OF COURSE.

Sketchology is a drawing program with near infinite zoom. You can really scrunch in and add detail.

Threes is my current game for a minute. Two Dots is the other one. Both have me stymied. iButtons is because I am still a class clown at heart.

The new Google apps are better, but still only for use in a pinch.

Paper is gorgeous.

Notability for marking up pdfs.

Skitch and Halftone for marking up photos.

Serviceable stuff here. Some of these would be more useful in a K-12 classroom than at university.

I go back and forth amongst Educreations and Screenchomp. Do you have a favorite app for this?

Voice Record integrates well with Google Drive, which has been handy for sharing and archiving interviews.

Three ring is interesting. It allows you to photo document student work and include it for a particular student in a class list. Feels like a piece of the SBG puzzle, and I'll be experimenting a lot more with it this year.

Tara Maynard and Caitlin Grubb have impressed me with their Nearpod and Socrative use. Need to be 1:1 for it to work, though, and we're not yet at the university.

When MyScript and Desmos aren't enough, I'll pull out Wolfram|Alpha. It needs wifi for full power, though. Sage does not, and can handle even bigger jobs. I think if I taught more upper level math I'd be using that a lot. It was handy for Number Theory, and can execute Python code, among other languages.  Quick Graph is a nice 3-D grapher.

The Common Core app is helpful and easy to use. Necessary these days.

Numbers is a gorgeous interactive book from Ian Stewart.

xFractal is a versatile fractal viewer with Julia and Mandelbrot sets.

Golly is a particularly nice implementation of Conway's Game of Life.

The Rekenrek (Number Rack) and GeoBoard are not as good as having a real one, but useful when needed away from the supply cabinets, or for recording demonstrations off the iPad.

I recommend all of these, but especially iOrnament, Isometric and Mandalar. Great feel and capabilities.

24 and 6 numbers are both for computational fluency with good structures.

I'm a fan of all of the Motion Math games. Meaningful representations and actions that helpf for constructing number concept.

Whole is a recent game which has a nice game context and mechanism for adding fractions to 1. The two Teachley games, Addimals and Mt. Multiplis, emphasize strategies for computation and have outstanding production values. (Think Cyberchase-level voice acting and animation.)

Devlin's Wuzzit Trouble is good game and requires problem solving.

The NCTM apps are good puzzlers and free.

I am not a huge fan of DragonBox or Math Evolve, but think both are as good as drill games are going to get.

These are separate because their goal is not to explicitly have the students do math. However they are essentially mathematical in structure, context or process.

I have spent way too much time over the years playing 2048, Entanglement, Flow Free, Number Addict, Dots and Two Dots. Thank the tech powers that there was not mobile Tetris when I was younger.

Scratch, Jr is a particularly nice, new coding app suitable for the quite young. Of the other three I think Hopscotch shows the most promise.

Of course, I do have just games on here, too.

Lines of Gold and Deck Buster are two great Reiner Knizia one person strategy games.

Ticket to Ride is a solid implementation of the board game (which is an all time great) which actually gets rid of the scoring (which can be onerous) and has pretty good AI to play against.

Risk is better in the app than on the board. There, I said it.

Doodle Jump I got when we were designing ParabolaX. It and Little Galaxy are are interesting combinations of dexterity and strategy. After taking Malke Rosenfeld's embodied cognition session, I think there's more here than I realized.

If you haven't played Plants vs Zombies, you're missing out. Fun game, surprising amount of intuitive math and strategy. It's all about rates!

Way too much sharing. If you're still here, I'd love to know what I'm missing, or what you find essential.

## Friday, August 1, 2014

The thing I was most afraid of about Twitter Math Camp was that it could not possibly meet my outrageous expectations. I had jealously not been able to go the last two years, and was so happy to go this year, meet so many teachers whose work I love, and get to experience this community that has become such a big part of my professional life, that there was no way it could measure up.

But I was blown away.

One of the benefits of this community is that we write and reflect. The #tmc14 hashtag on Twitter might be overwhelming, but the wiki has all the presentation info and most of the slides, and the reflection blogposts (that's a partial list) capture a lot of what went on and some of what was learned.

It's a bit overwhelming to recap, so I just want to try to capture my big takeaways here. Names link to twitter accounts, more specific links spelled out.

Dance: Malke Rosenfeld
I have been a fan of her work for a while, and she was running a three day session with Christopher Danielson, a master teacher educator. Attending this session was no mistake, as it was fun, provided experience with a new-to-me concept of embodied cognition, and had so many teaching and learning things to notice...
 Also my excellent dance partner Melynee, who explained all things OK to me.

Malke taught us to dance and choreograph a dance, then set us challenges to solve. This involved making a dance pattern, learning it, and then transforming it. She wasn't using dance as a representation system for mathematics, she was teaching dance, to which mathematical ideas applied. She also established the Blue Tape Lounge by the ice machines at the primary conference hotel, where participants taught what they had learned to other people, and tackled extended challenges. Christopher provided the hand-scale mathematics, manipulative contexts that connected to the dance, like his beautiful triangle symmetry ... doodads.

It was stunning how the dance created a context with lots of motivation for communication, refinement and ownership. The product was a doing not a thing to have. (No typos in that sentence, but I'm not sure how to say it, either.) There was also a lot just to observe about the teaching. Christopher is a teacher educator and was on A-level meta-teaching game. There is enormous benefit in watching someone teach well outside your content, and Malke did many interesting things. Very positive, process oriented feedback.

More: Malke's TMC post 1 & 2, and her Storify of the relevant tweets.

Sadie led a session on Counting Circles. The class stands up in a circle, the teacher decides what they are going to count up by, and where to start. When the class is counting, everybody goes, the teacher records responses on a numberline on the board. ("Because number lines are awesome. And it's support for students."-SE) Count for the time you've got, then pose a prediction question to count up some number of spots more. When >everyone< has an answer, then solicit and record student thinking as they give it.

I'd watched the videos on her blog, but it was different getting to experience it. She shared how they tie into the building of classroom culture that she is seeking. In addition to the counting and number sense work, it is inherently collaborative, and leads to number talks at the end of the circle, when students make a prediction past the stopping point. ("If we kept going, what number would Judy say...") Participants quickly brainstormed a number of extensions, by extending the numbers counted, using integers, fractions, decimals or algebraic patterns. This ties in so well with Jo Boaler's research on resetting student beliefs about mathematics that I have to give it a go now.

More: Sadie's presentation page at the wiki has links to her other counting circle work, but also her first blogpost on it.

 so sturdy it mostly survived packing
Math: Edmund Harris
Obviously, the days were just packed with math, but among these Edmund stood out. He was the token mathematician, I guess, but added a ton to the proceedings. His literal bag of tricks produced laser-cut paper tiles for assembling 3-D models, laser-cut beautiful Penrose tiles with matching conditions, a plastic ratio proportion engine... and who knows what else. The TMC logo was his design and he is a serious mathartist in addition to mathematician. So much fun. He also gave a terrific My Favorites on the math in dots and arrangements thereof. I learned I must never be given access to a laser-cutter.

More: his blog and the dots.

Group Work: CheesemonkeySF
Elizabeth led the Group Work Working Group, which is what I would have attended if there were two of me. Thankfully they thoroughly documented their work. I did get to sit in on a flex session trying out the Talking Points structure, and it was everything that it had seemed from reading about it. These structures are a part of her effort to push authority towards the student, and be true to restorative practices. I really think this is essential. We do not have a new game to play, and we inherit students who have experienced a lot of inequity and been trained to helplessness.

More: GWWG at the wiki and Elizabeth's references.

Just meeting them:
There were so many people whose ideas and opinions I value that I was glad to meet. I also had a handshake list - people who directly or indirectly have inspired me to dive into the MTBoS, to tweet and blog, which has definitely improved my practice and enriched my understanding. If any of you read this, THANK YOU.

Special shout out: the organizing committee, especially Lisa Henry and Shelly T, without whom this would not have happened.

Challenges:
• Incorporate more and better structure in my groupwork based on the GWWG materials. In my classes and in the departmental diversity discussions this year.
• When can I have students moving purposefully solving an embodied challenge?
• How should I implement the counting circles? Which courses?
• Edmund's Dots. Build a representation for students to notice things, or another classroom routine that builds over the semester?
• Is there a way to incorporate Heather's Cut and Grow revisions or Rebeckah's Friday letters into a university environment? (probably; don't know)
• Tweet. Less.

## Wednesday, July 16, 2014

### Super Mathio

Hedge wrote this fun post on Mario Brothers and math - specifically parabolas - that got me to tweet: I also now want a video game where you jump by clicking the vertex to make a precise parabola based on where you are. She asked the good question: Can you make that?

Should have thought that one through.

Fortunately, though, it's Twitter, so brilliant people to the rescue.

Desmos tweeted:

Pretty sweet! Find it at Desmos:

Beyond my current Desmos levels, but pretty amazing. Great image use.

Then Andrew Knauft tweeted:
Find it at OpenProcessing.

You click to do the jump. Beyond my current Processing skills, but excellent.

So I did -of course- have to try it in GeoGebra. I'm pretty happy with the result. The math wasn't too hard, though the scripting the buttons and resetting graphics is always tricky for me.

It's at GeoGebraTube for you to play with. There's a data mode that let's you try to calculate the best high point first - which is where I would want to go with it. It's an insufficient data situation as I purposely left off the coordinates of the bonus box.

There are some things to notice about parabolas as you play it, and for deeper work it would be interesting to think about how to add scoring.

GeoGebra Note: Andrew Knauft also helped with this! You define piecewise functions in GeoGebra using the If command, which also has an else variant. I.e. either If[<condition>,<then>] or If[<condition>,<then>,<else>]. I was getting errors with the inequalities to make the towers, and Andrew figured out that GGB didn't like an inequality with the else form. Redefined them as Ifs alone, and I was good to go.

This was very fun. I'm curious to know if or how you would use it, or what features you might add. As always, if you have an idea for dynamicizing, let me know!

## Friday, July 11, 2014

### Complex Instruction

Complex instruction isn't real.

OK. I think that's enough of that. (Complex instruction is an actual instructional approach for differentiation, check it out.) This post is part my general thinking about complex numbers and, in other part, a very specific visualization.

I. In General
This summer teaching the math-history-themed capstone class for our seniors, one of the themes that came out was how complex numbers are the natural setting for much of mathematics. In my own experience, I didn't really appreciate that until grad school, and our majors are graduating without enough experience to feel like they are really numbers.

My perspective: I think counting numbers feel like numbers to us because:
• we have an intuition for them as quantities,
• we can represent them in multiple ways, we can compare them fluidly,
• we are comfortable with what operations mean,
• and we have multiple ways to compute with them.
The more of those aspects are broken down, the less they feel like numbers to us. We see that with students at large whole numbers, fractions, decimals, radicals, etc. I've surveyed about 50 math major seniors and inservice math teachers now, and they are significantly less comfortable complex numbers than radicals and transcendentals. When I shared some of those results on Twitter, several mathematicians wrote back that this means the students have not developed an appreciation for how weird the real numbers are. Part joke, all truth.

Part of the power of being a mathematician is being able to make this transition for wide classes of objects: functions, matrices, operators, manifolds, algebras, etc.

Complex numbers came up in math history in the solving of quadratics with the great Islamic mathematicians; the solving of cubics with Cardano, Bombelli and Tartaglia; the great acceptance of them under Euler; and the deep understanding of them that followed Gauss. The students saw the connections amongst the lack of acceptance for irrationals, negatives and complex, but - in general - saw their discomfort with complex as justifiable, while rejecting irrationals or negatives was just a bit silly. The exception to this was the few students who had taken a second semester of abstract algebra and saw C as a beautifully complete field.

I think the problem might be that algebra is before we see our students, and the focus on calculus and linear algebra in our program has to support the engineering students as well as serve math majors. Algebra and linear algebra seem like the natural places to start seeing complex numbers as natural and necessary.

It was probably Michael Pershan who really got me thinking about this, as he started sharing his usual deep reflections on the topic a couple of years ago.  Bonus: Mike Lawler has a couple extensions (one and two) which my students this summer found interesting and valuable.

II. The Fundamental Theorem of ... Algebra

One of the morals of the capstone class was that if mathematicians labeled a theorem as Fundamental, it's worth your focus and understanding.

If you have time, please watch at least the beginning of this terrific Numberphile video with David Eisenbud, director of MSRI at Berkeley.

Marvelous. One of the things that so infatuated me with my advisor, Nigel Higson, was his ability to motivate and moralize the mathematics. The dog on the leash is the nice metaphor here. I think the power of this representation is not in the static image he draws, but his ability to manipulate and use this imagery to connect ideas. For most of history, this has been a significant barrier in mathematics.

III. The Visual
Of course this has to end up in GeoGebra...

The ability of GeoGebra to have two graphics windows allows us to dynamicize Eisenbud's picture. Here's the GeoGebraTube page for this sketch. In works online (the student page), but is better if you can download and open it. GeoGebra does not compute in complex numbers natively - yet - so this sketch has a lot of workarounds to treat points as complex numbers. This post isn't about how to make the ssketch, though, it's about using it.

GeoGebra tip: when a point or slider is selected, you can move it with the arrow keys. Hold down shift while clicking an arrow for smaller changes.

The sketch has one window for input to a polynomial, and the second window for the output. For example:
To get that feel for what Eisenbud was doing, I turned on the trace for the points. Complex valued functions and analysis is very much about paths and what happens to them under transformation.

We can use this sketch to find real and complex roots. When you click circular path, it plots all the points in the input plane with the same magnitude as A, the blue point. The output plane shows the result. So when the orange transform of that circle goes through the origin, there's a root with that magnitude. The green slider rotates around the circle, allowing you to findthe exact value. Above is a real root for that function, $$f(x)=x^3+2x^2+x+1$$, near $$x=-1.76$$. Where are the complex roots?

One more example: Find the first root.

So, since this is a real coefficient function, the conjugate must be a root. I made this cubic to have the root $$x=2$$, so I show that root also. The transformation of the rectilinear path by the function is quite fascinating, no?

Studying different kind of roots can be pretty interesting. What do you think is going on here?

We can investigate the winding number phenomena as well:

There are so many things to notice once we have a visual. It supports more people in developing the kind of intuition and acceptance that might some day see accepted status for our complex brothers and sisters. After all, the reals are complex, too.

Seriously, I'd like to think that we're coming into an age where technology makes these ideas more accessible to a wider number of people. Can you think of other features or visualizations that we should try to make? What are the key concepts of complex numbers that need the support of a dynamic visualization?

Notes:
• GeoGebra's gif export only does the active graphics window, so these are done with the RecordIt app for Mac - highly recommended.
• It's a little weird that the function has real coefficients but complex constant, but it seemed worth it to get at some of the points that Dr. Eisenbud raises in Numberphile video.
• The work arounds consist of making separate real and imaginary functions from the input function. It looks like this:

## Friday, June 20, 2014

### Playing with Math

Today's the day! The crowdfunding for Sue Van Hattum's book Playing with Math opens up. I'm excited about the book, proud to be part of it in a little way and so happy for her.

If there's one phrase that captures my approach to mathematics learning and teaching, it's 'playing with math.'  So I'm really cheesed that Sue has stolen this title for memoirs... wait. That's not where I was going with this. Besides, the full title is Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers

I didn't meet Sue until after she had moved away from here (West Michigan), but got to know her via what is now the Math-Twitter-Blogosphere, and then in real life on one of her return visits. In this book she has gathered together many of my favorite aspects of the math community and culture, plus more that I have yet to know.  I got to be a realatively early reader of the manuscript (found a v1 file on my computer!), and have seen it start good and get better from there. This is going to be an amazing resource. Her philosophy in building the book was very much about building a community, sharing the people and math of which she is so fond with you. I think you'll find it intriguing, entertaining and helpful. Bloggers, math circles, living math forum... Sue is great at connecting people.

I'm a great believer that teachers get better through conversation, and every piece in this comes across as powerful teacher or learner sharing. It's a rare anthology where you feel like you wouldn't cut a thing, but this is one of those. The pieces I have returned to more than once already include Bob and Ellen Kaplan's reflection on a prison math circle, Maria Droujkova's rejoicing in confusion, Malke Rosenfeld's mapping the territory, and Allisson Cuttler's putting herself in her students' shoes. And... it could easily become the table of contents. In editing, Sue worked hard to preserve the author's voice, make the book very inclusive of student and teacher diversity, and to represent each of her three communities.

And each teacher story finishes with a puzzle or game. Tanton, Halabi, Gaskins, Salomon... Van Hattum. In addition to editing, Sue is a great and reflective teacher, and her own writing and games are an important part of the book. It is very much like a teacher weaving a lesson together from student work and responses, the way she tells her vision of mathematics learning from such a wide variety of different authors.

Nix the Tricks and Moebius Noodles are both great examples of books that are from and for the math community, and this is a great next step. Please consider supporting it; I think you'll be glad you did.

Some other resources, reviews and comments:
A family favorite to which we were introduced by Sue. Our semi-annual gaming get togethers are now pretty highly anticipated!

## Tuesday, June 17, 2014

### Capstone Book Club - Summer 14

Time for another installment of Read 'Em and Weep... no, that doesn't sound right. See a previous edition here.  Students grouped with others who read the same book, plus a group of readers with unique books. After they have a chance to discuss for a good bit, I ask the groups to think of what they want to say to the whole class about it. In a whole class circle, we discuss each book in turn. These notes are from the whole class discussion.

Accessible Mathematics:
shifts emphasize how to adjust your classroom… eg. multiple representations. Liked it b/c it wasn’t just changes. Like minimizing what is no longer important; calculators for example. A lot of specific examples… real life. \$1.89 v easy numbers…
Review:
Love and Math:
Good for non-math majors so they can understand what we’re talking about. Example of symmetry: which is more symmetric a circular table or a round table? If you left the room and I turned. More of a story and Frenkel’s search for knowledge.
Reviews:
e: the Story of a Number
We agree that the beginning - the history of the mathematicians who worked on it - was very accessible. Could not follow the shifting representations of problems in the last few chapters. The end was significantly harder to understand. The part about the personal life of mathematicians was cool, and who winds up getting credit for achievements. Publish or perish keeps some people who deserve credit from getting it. Qualified recommendation. Music and logarithms. Some review that was helpful. Interesting to see how they discovered what we study, and how hard it was to find some of what we take for granted.
Reviews:
Euler: the Master of us All
Is Euler human? Today you have to be very specific, but Euler did so much in so many areas. He goes by topic, what was known before Euler, hen what Euler did. Proofs, but open to people who can handle logs and series. Highly, highly recommended. Popularized complex numbers. His proof writing was like Romeo and Juliet; the balcony just works. Blind for a lot of his life…
Review: Kyle Ferguson: plus some extenisve book notes.
Gödel, Escher, Bach
Really about Gödel and the incompleteness theorem. He’s trying to explain to a general audience. Tortoise and Achilles, a fable, then ties it into a serious discussion of the mathematics. He also ties in art, music, zen philosophy. Not a clear path to the Incompleteness Theorem, but about what is interesting along the way.

A Journey through Genius
Paint a picture of mathematics that is logically sound and aesthetically beautiful. He picks the best proofs, a bit of history about the man, then an explanation. You could read this in sections to understand something specific and the culture it came from. Not just about the applications, but appreciating it for what it is. Logic still holds true centuries later.
Review: Jason Lohman

The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Prob-
lems, and Personalities by William Dunham
Also by Dunham. Proofs and problems of mathematics, arranged alphabetically. Which is confusing vs historical order. Chose some really interesting proofs. Any natural number as distinct integers and odd integers.
Review: Nate De Maagd. Includes some of his proofy highlights. (click through to Gdoc)
Vision of Elementary Mathematics
Good for non-math majors teaching elementary school. Good for visual learners, lots of diagrams. Also for getting kids to discover math on their own. Gives alternate perspectives that could help in talking to students. It’s a little repetitive. But we’d really recommend it.
Reviews:
Mathematician’s Lament
Problems with math teaching as it is right now in schools. The differences between teaching art and teaching math. The creativity is taken out of the math. There’s nothing to explore or discover. Like teaching art with only paint by numbers. Couple issues with some of the arguments: like teachers try to make it interesting but it already is interesting. Or you learn when it’s relevant to you, but he had argued against relevance. Also doesn’t describe how to make it better. Says math should be a free for all - against all structure. Everything he’s telling us is pointless. He’s against drill but favors real world problems. We teach definitions for no reason. (Quadrilaterals, for example. But then how do we communicate math?) Has a mathematician’s perspective, but he didn’t have a teacher’s perspective. What he offers doesn’t seem appropriate to most schools. The conversations at the end of each chapter were confusing. “What to do with elementary students?” “Just have them play games.” Strong opinions but no evidence. You should read it but it will make you frustrated.

I feel like how we’ve been taught to teach does help handle a lot of these issues. It would work if everybody loved math like he does.
Reviews:
The Joy of X
Insightful and helps make math more accessible. Analogies from Sesame Street. “Fish, fish, fish, fish, fish, fish…” all the way up through complex numbers. Split up into 30 short chapters. From numbers to infinity. I liked the way he uses personal stories. Very readable. Not for diving into math but there’s a lot of stuff that helps clarify. Even for someone who struggled with math. But it won’t change your mind. Great book for college freshmen who have to start making some sense of math. Also good for middle school and high school teachers.
Reviews:
The Math Book
Looks intimidating but it’s an easy to read. A page and a picture. Goes through history including lots of things that you would not think of as math, like tic tac toe or mancala. Also see a lot of mathematicians come up multiple times, which is neat to see
Reviews:

Students choose their own books and that really pays off. Very positive feedback this time around. We follow it up with a book swap, so you get to read a second book that is of interest to you, or at least to skim it. (That's why some people have more than one review linked.)

Some of the reviews are fabulous; if you're interested by the blurb here I'd really recommend following up. And of course, if you have a chance to comment on a student blog, that's excellent.

## Friday, May 16, 2014

### I See Number Theory

We had a really interesting week in my number theory class. We are really a seminar, seven teachers and I investigating elementary number theory together.  I hope they're learning half as much as I am.

This week we were exploring primes and modular arithmetic. The first day we were thinking about the $$4x \pm 1$$ and $$6x \pm 1$$ structures, and the results that there are an infinite number of each type of prime.

To gain modular arithmetic practice, we played Modular Skirmish. (Cf. this post on Gauss.)

Then we started looking at this GeoGebra sketch:

The numbers increase from the bottom left corner up the column. My first attempt was a growing square, but that let you see asymptotic distribution of primes more than the modular structure.

We put this sketch up on the front screen, and advanced n. Teachers noticed the empty top rows (multiples of the modulus) and how some values separated the primes into rows: 2, 4, 6, 8, 10, 12... while others seemed to form diagonals: 3, 5, 7, 11, 13... We wondered about which were the most consecutive primes or gaps in a row, and whether that would change as m increased. (Personally I got wondering about where are the largest square gaps.) Teachers connected many of the patterns to the rows in 6. For example, in 7:
The diagonal really means the next prime is +7 -1 or 6 apart.So it goes back to that 6 structure.

Modulo 10 is really just looking at the last digits. We noticed that no digit seemed more or less common out of the four possible. Also, no consecutive dots more than 2. Is that always true?

The two coolest structure theorems are with respect to four and six. I think these helped in understanding why primes are of the form $$4x \pm 1$$ or $$6x \pm 1$$. Which may have also helped with the proof that there are an infinite number of primes of the form $$4x - 1$$ (or $$6x - 1$$ ).

We did find a modulus where there was a row of 8 consecutive primes, but I can't rediscover it!

Understanding the six structure also helped us understand a diagram that we were looking at the previous week, from a designer who was really impressed with a 12 structure. (Source in reddit/r/mathpics. The picture isn't super precise, but did offer a lot of making sense opportunities. And colorful!)

Rather than make the course a tour through the great theorems of number theory, my hope is that it can be an opportunity to do math ourselves. So instead of necessarily illustrating a theorem, I'd rather find a way to notice things that might lead to the theorem. Since we're interested in K-12 applications, divisibility tests and primality tests are of interest; that means exploring the ideas in Fermat's Little Theorem.

So the idea came - given the success of the modality/primes visualization - to visualize exponential patterns in the modular context. This sketch is what I came up with.
Oh! The patterns they found!

Definitely a lot of things that I had not noticed. Not, interestingly, Fermat's Little Theorem, but there were many observations that will lead there.

A lot of our discussion was about pairs of cycles. The visualization made it clear when two different bases created the same path, up to direction. Eg. $$2^m \mod 5$$ and $$3^m \mod 5$$.

Furthermore, they noticed this awesome pairing within the cycles. Here's the nicest mod 13 pair.

Look back at the other data... there's a lot to notice. And it definitely has me wondering. (Copyright, trademark and kudos to Max and Annie from the Math Forum.)

It's hard to imagine that introducing a theorem and sharing a proof would have resulted in building any more understanding, and there's no way it would have led to doing any more math. And this will make the theorem so much more meaningful when we get there. If we do, with such a fine boatload of conjectures to explore.