Wednesday, August 27, 2014

Clap Hands - a motion pattern game

This game must exist in some form elsewhere, but it came to me yesterday and we worked out a good version of it with my preservice teachers this morning.

It starts with getting to do some of Malke Rosenfeld's Math in Your Feet this summer at Twitter Math Camp, and then subsequent discussions with her that have me thinking a lot about embodied cognition. The example of this in Math in Your Feet was knowing what I needed to do but the challenge of getting my body to do. Move left foot, move! In discussions, she connects this powerfully to research and writing of Seymour Papert. She said something like:
embodied vs “non-embodied” from the research: there is no non-embodied math. Either we’re pulling from previous lived-in-the-world experience to learn, or we’re actively constructing our understanding of self moving in space. We can harness that to give students an understanding of the world.
She's deep that way.

On our first day of class, one of the things we did was watch Ken Robinson's Do Schools Kill Creativity?  (If you haven't watched it, give it a go. He's a powerful speaker on creativity, and as close to Ricky Gervais as we're going to get in academia.) The student response from my class was really focused on movement. The Gillian Lynne story especially seemed to resonate; good omens for some of the learning I hope to do this semester.

So today we're studying patterns. First activity was pulling out the pattern blocks. We used that to model how to introduce math manipulatives to elementary students, and introduce the principle that with a new manipulative you need free play. Either immediately or promise the students specifically when they will get it. (Good management meets good pedagogy.) We used free play to introduce the question: is this free play doing math? Which we discussed in Elizabeth's Talking Points structure. (Fabulous, even the first time out.) Then in whole group used our examples to discuss the difference between a design and a pattern. (Is there a difference to you? I'd love to know what you think about that.)

Then it was time to go outside...

Clap Hands

groups of 4 to 7 people

Arrange people in circles of about 6. The game is pretty simple:

Building
  • One player starts, introducing a motion. Like, for example, a simple clap. Going around the circle, each player does the motion.
  • After the starting player does the motion, the next player adds a motion. Clap hands, raise right hand. Each player does the 2 part sequence.
  • After the second player does their two, the next player adds a motion. Clap hands, raise right hand, turn around clockwise.
  • And so on, until each player has added a motion and it has gone around. Clap hands, raise right, spin right, jump, snap fingers, shake right foot twice.
Survivor
  • The goal is to get the pattern to go around twice more. When it does, that pattern is complete!
  • If a player messes up the sequence, they step out. Try to get twice around from there.
  • If you get down to two people, the pattern is done.

I didn't get video because I needed to play this! Thanks to Jordan and other students who had great suggestions. Reaction to the game was very positive, and people were quite engaged. There was much laughter, too. Keeper!

I'm interested in your feedback on the game, and how you present patterns. So if you have time to tweet or comment, let me know.

Dan as Assessment

First day in a class on high school math with preservice secondary teachers. A rambly story, to be sure.

From The Duplex by Glenn McCoy
 We started making nameplates and forming groups of 3 or 4.  While that seems inconsequential, I leave the markers and materials up front. And it's just the tiniest bit of culture setting when someone asks "we do that now?" and "should we come up to get the stuff?" And letting them know if the groups don't meet the parameters: facing each other and 3 or 4 members. Am I really going to be a stickler? "We have 5 is that okay?" "Nope." Student 4k+1 walks in; I ask "so what are you going to do now?" I have several occasions to say, "good thing we're a room full of problem solvers."

The next activity, Piece of Me, is robbed adapted from David Coffey. I'm now calling it 2 Questions. Each person comes up with two questions to ask the person on their left in their group. Once everyone has them, ask away. Everyone has the right to decline to answer. Some of the questions were generic (how was your summer), some were trivia (Michigan or Michigan State) (Answer was Notre Dame and a story about her family) and some sparked good discussion: "why be a teacher?" or "middle or high school and why?"

The second phase is they each come up with two questions for me, one about me one about the course. Then the group picks two of those questions. As Dave documents, answering and discussing just what the students are interested in is a vast improvement over teacher ramble or reading the syllabus. (That's homework.) This day they asked about the observations they do in high school, what homework would be like, how much reading, etc. about the course. They asked if they would be teaching in class... that hasn't come up before. About me they asked about family, education, why be  a teacher, etc. Favorite question this time: "do people say you look like anyone?" They used to, when I was skinny, but fat now, so... Then the student said he asked because I looked so much like Francis Ford Coppola.

Alright!

Our first math activity was one of Sadie's Counting Circles. Regardless of how you think math should be taught, you must know what people think about math now. I shared how Jo Boaler and others have experimented with number talks to change people's beliefs about math. And how Sadie's counting circles are a good context for a number talk.

The idea of the counting circle is two-fold. One is that it helps to build a positive learning culture where it's safe to speak up, mistakes are acceptable, multiple approaches are valued, and thinking is what we want to share, and the other is to develop number sense. Sadie would probably disapprove of my choice of starter value, but I thought too easy would actually disengage this group of math majors. So we did up by 97, starting at 235. It wound up being just a little uncomfortable for a couple of people - 99 might have been better. After 1.3 times around, at 2175, I put on the pause, and asked what would Amanda say (5 people further around the circle.) When everyone indicates they have an answer, I asked for volunteers for their thinking, and then just recorded it as they spoke it, eliciting details. And I didn't take a picture!The shared strategies included noticing that the ones place went down 3 each time, adjusting from adding 100 and multiplying 97x5 and adding it to 2175. Even the computation can be interesting. 97x5 used partial products to get 485, and then he added the 5, the 400 and then the 80.

With that set up, we moved to watching Dan Meyer's TEDx talk, Math Curriculum Needs a Makeover. (It's a classic for a reason.) They were very engaged, and had intense small group discussions after watching. One of the students led the whole class discussion, which became a good assessment for me. Their beliefs about teaching stood out sharply, thanks in part, at least, to their contrast with what Dan was saying.

My notes:
Not enough time!
-planning
-class time
Some time gained from students being able to do math practice at home. Tech helps with this. 
What stus know on tech is not always helpful.

Kids need too many basics to do this kind of work

Problems like this won't be on the standardized tests
I learned this way and the standardized tests were so easy. If you're taught this way, your understanding filters through standardized fluff.

Less memorization because you've created the formulas yourself.

We missed part of the process. This leads to more general methods because of questions like: How would you figure out for a tank ten times larger? Or is there a short cut for figuring it out more easily. In a class one time, for a question on the volume of a vase, why not just fill it and see? Prof answered: "what if it was 100,000 gallons?"
This kind of activity gets across the idea that you're not always going to be right. This is just one way to do it.

Grounds the math in reality. You know students are always asking 'when are we going to use this?'
I would summarized their talk as: "It would be more engaging, but..." I maintain that I do not want this course to be about me telling them how to teach, but giving them experiences that can equip them to construct their own vision of teaching and learning. Resources, reflection and a focus on student understanding lead to good teaching. But this experience helped me understand where we are starting and some of the barriers to where I want us to get.

We continued class by looking at 101qs.com and generating some questions. (Aside: read Pershan on learning to ask questions. We'll be tackling this later.) 

This led into launching the meatball three acts. We watched the video, and got to our estimates, and low and high guesses.



This was cut short because we had a presentation on Lisa Kasmer's excellent study abroad program in Tanzania .  I'm interested if anyone worked farther on it.

I was moved to write this up, because I tend to think of asking questions to get assessment data, and this wasn't intended to be assessment. I might not have noticed as well if I was running the discussion, but as an observer, it put me into notice-mode.

Postscript:
Part of the homework was to look for a 101qs prompt they found interesting. What they found interesting is in turn interesting to me! Here's a few:
  • Jennifer: Bowling for Pennies. "Would this be cheaper to make than buying a 'mirror ball'?":  Nicole Paris asked: Would this be cheaper to make than buying one?
  • Brittany: Waterkracht "centrale". Water power plant. How fast is the stream flowing?
  • Greg: Beatlemail "How many letters would each band member have to answer?"
    Ken Meehan asked: How many pieces of fanmail?? What is the first question that comes to your mind?
  • Sam: Brita How many water bottles are need to go around the earth once?
    Dan Meyer asked: How many times around the world would all those bottles wrap?
  • Kevin: NFL JumboTron HDTV's How many 60" televisions does it take to fit in the Texans television?
    Nathan Amrine asked: How many 60" TV's would equal one Texans TV?
  • Molly: Packing Box What is the total area of all boxes?
    Elaine Watson asked: What combination of small boxes and medium boxes can fill a large box?? What is the first question that comes to your mind?
  • Joshua: Angry Birds Can knowledge of quadratics improve your Angry Birds accuracy?
  • Jerry: Okay, this is the first one I found: Stuck Truck This reminded me of a story I once heard of about a semi-truck that came to the opening of a tunnel only to realize it was a couple of inches taller than the opening. Having to stop in the lane with no way to turn around, the truck had traffic backed up for miles as everyone had to consolidate down to one lane and everyone slowed down to gawk at the truck to see why it was stopped there. The police and the truck driver were all standing around the truck trying to figure out how to get the truck through the tunnel or turned around and had discussed numerous things but none of them seem to be the right answer. Then a little boy leaning out of the window as his dad drove by yelled out, "Why don't you just let some air out of the tires?" The question the original poster seemed to be thinking along the same lines, "How much beer would he have to drink to allow the driver to get the truck free?" However, my question had to do with how often this happens at this spot. Either it happens often and they should fix it or there is a sign and the driver just did not see it.
    Fred Jaravata asked: How much beer would I need to remove to help free the truck?? What is the first question that comes to your mind?
  • Dakota: Cylindrical Tunnel How many windows are in this tunnel?
    statler hilton How much glass is needed for this?
  • Jim: Pickle Stack How tall can you build a pickle tower?
    Krista Keats asked: What is the question?
  • Brody: Keep 'Em From Fallin' How tall is the main structure?
    Michele Thomareas asked: How wide apart are the columns?? What is the first question that comes to your mind?
  • Amanda: Waterworks at Legoland How far will the water spray?
    Rod Bennett asked: What happens to the trajectory of the water if she pedals faster?
  • Christopher: Roulette Wheel What are the odds of that many 19's in a row?
    Joe asked: What are the odds of the same number coming up 7 times in a row?
  • Anika: Circle Square How many circles are there?
    John Golden asked: What is the function for number of circles after each step?
  • Nick: 2010 Guatamalan Sinkhole How deep is the hole?
    Robert Kaplinsky asked: How much material will they need to fill the sinkhole?? What is the first question that comes to your mind?
  • Brooke: Firing Range How many lasers are there?
    statler hilton asked: How far away is the target they're shooting at?? What is the first question that comes to your mind?
  • Leesha: Perfectly-timed photo How high is the plane?
    Johanna Langill asked: How high is the plane?? What is the first question that comes to your mind?

Saturday, August 23, 2014

Elementary Read

Planning my fall pre-service elementary math course, I was thinking about books. In the distant past we've read Deb Schifter's What's Happening in Math Class? (strong teacher narratives), and more recently Jo Boaler's What's Math Got to Do It. (Here's a recount of one of our book discussions about it.) But in my other classes, it's been very good to offer choice to students. (Here's a post about that.) I'm a big believer that teacher-to-teacher reflective conversation is the best PD, and book discussions make good context for those discussions. (A pdf of some research on this by Burbank, Kurchauk and Bates in The New Educator.)

I was finalizing my list for them to choose among, and thought to ask on Twitter. As usual, unexpected generosity in people thinking and answering. (I don't know why it's still unexpected.) Here's the responses:




A Mathematician's Lament.

I don't have a long list I'm drawing from, but Marilyn Burns' "Math for Smarty Pants" comes to mind. 


@j_lanier I second this. Have ordered to share with my elementary teachers in the district.


Rudin! Go big or go home ;-)

Children's Mathematics. 

Euclid’s Elements, because it’s comprehensive :P


Powerful Problem Solving. Lots of great examples.

 
Young Children Reinvent Arithmetic: Implications of Piaget's Theory by Constance Kamii


maybe "Creative Problem Solving in School Mathematics" by George Lenchner.


I second but I also like 10 Instructional Shifts by @steve_leinwand

making sense:teaching & learning with understanding by James Hiebert - geared k-8 but great for all math teachers


#1 for me is What's Math Got to Do With It? by , #2 is Knowing and Teaching Elementary Mathematics by Liping Ma ... #3 is Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School by Carpenter, Frankl, and Levi


Van de Walle, Teaching Student Centered Mathematics 

What a great bunch of suggestions. So my final list for them to choose from is below. I'm requiring at least two people per book, at most four. (24 students) In addition to the benefits of choice, I'm hoping that a variety of books enriches our classroom discussion.
  • Accessible Mathematics: Ten Instructional Shifts That Raise Student Achievement, Steven Leinwand, (Amazon) [Practical, pre-service teacher approved)]
  • Intentional Talk: How to Structure and Lead Productive Mathematical Discussions, Kazemi & Hintz, (Amazon) [Applies to more than math; good support for helping students learn to converse productively]
  • Making Sense: Teaching and Learning Mathematics with Understanding, Carpenter, Fennema, Fuson, Hiebert, Murray & Wearne (Amazon) [Writers and researchers of the best elementary math curricula around tell what they think is important.]
  • Math Exchanges: Guiding Young Mathematicians in Small Group Meetings, Kassia Omohundro Wedekind, (Amazon) [Similar to intentional talk, more strongly based in literacy routines.]
  • Math for Smarty Pants, Marilyn Burns (Amazon) [Collection of entertaining problems across all kinds of math from a master math teacher.]
  • A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form, Paul Lockhart (Amazon) [Not sure about putting this on. Many readers are disappointed in the 2nd part, but the 1st part people see as a powerful argument for why math teaching has to change.]
  • Powerful Problem Solving, Max Ray (Amazon) [New book from a very deep thinker about how to teach math.] 
  • What's Math Got to Do with It?: How Parents and Teachers Can Help Children Learn to Love Their Least Favorite Subject, Jo Boaler (Amazon) [If I was picking one book for everybody this would be it. Dr. Boaler is doing a lot to research and share how to make math better.]
Last cuts: Moebius Noodles,  The Math Book by Clifford Pickover (beautiful history of math), and Deb Schifter's What's happening in Math Class.

5 Practices by Smith and Stein (dropped for Intentional Talk and Exchanges) and the NCTM's Principle to Actions were just not accessible enough in this structure. I think if everyone was reading the same book, those would work better.

This course focuses on pattern, geometry and statistics, with number and operation in another course. Otherwise CGI would be on for sure. The Young Mathematicians at Work books are a fine series we use with our elementary teacher math majors.

P.S. And then, like any modern story, it ends with a sequel invitation...

Good question, extend. If you could get your child's HS math teacher to read one book, what would it be?


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.

Now we start with the games!

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.

ParabolaX is our GVSU quadratic game app. Not bad!

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

Twitter Math Camp!

Edmund Harriss' logo explained
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.

Counting Circles: Sadie Estrella 
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!