Thursday, December 19, 2019

Playful Math 133

Welcome to the 133rd installment of the Playful Math Carnival.

133 is the 6th octagonal number. How many carnivals until the 7th octagonal? What kind of pattern do the 6th polygonal numbers make? (Play with some figural number GeoGebra.)

133 is also semiprime - the product of two prime factors. (Which two?) What is the next semiprime number? 133 is also a Harshad (aka Niven) number, a number divisible by the sum of its digits. It's an unusual stretch where 3 of four consecutive integers are Harshad - which are the other two? There's also a new to me fact: 133 is the number of partitions of 55 into distinct odd parts, which seems equivalent to the number of symmetric Ferrers graphs with 55 nodes. I haven't made sense of this yet, though! 133 is happy in base 10, so a good number for this holiday edition.

Last fact and shout out to Megan Schmidt: 133 is a square spiral corner number! Is that connected to any of the patterns discussed above?
 Literal math playing...

  • Dave Coffey gave a presentation on Math Play with a Purpose at Global Math. He talks about redeeming Bingo, among other things.
  • Speaking of Bingo, two of the groups at this semester's Family Math Nights made actual math games out of a Bingo premise. Meg and Madison made Food Bingo, which makes it about attributes of food. Erica and Claire had Star Bingo which used better number cards and some choice to enhance the game.
  • There was one game that I liked more than the PSTs who ran it! Give Hamburger a try.
  • Probably the breakout and most original game of FMN was You Must Cross the River. Eddie and Climie brought this D&D style game.
  • One of my HS preservice teachers tried to gamify Which One Doesn't Belong... I think Danielle is on to something.
  • Denise Gaskins, the founder of this here blog carnival, shares one of her many Hundred Chart Games.
  • Marilyn Burns shares her two dice sum game, which is a classic for a reason. She shares using it in 2nd and 7th grade!
  • Kent Haines assembled a Holiday Gift Guide for math games that might be too late for shopping this holiday from this post, but you'll want to keep this list.

The 133rd Space Shuttle mission 
was the last (39th) for Discovery. 
They installed the Leonardo Module 
to the International Space Station.

Some playful interactives...

  • NRICH shared a puzzle that is part about area, but made challenging through Cuisenaire Rods. Great lesson.
  • Kevin Forster shared Factris which is a multiplication/factoring version of Tetris. 
  • Scott Farrar made a cool GeoGebra activity implementing Always/Sometimes/Never with quadrilaterals. 

 The C-133 Cargo plane over San Francisco Bay.

Math stories...

Xenon 133 is an isotope 
that is inhaled to study respiration, 
among other medical imaging uses.

Math art to round us out...

  • Isohedral rounded up some of my favorite animated math artists in this post.
  • Very excited that Clarissa Grandi has a math art activity book coming out.  Look at her website and you'll see why I'm excited.
  • Nathaniel Highstein did an Islamic Geometry project with his students. Scroll down this thread to see their work and how he tiled them!
  • Paula Beardell Krieg has been killing it, but if I was picking one recent post it's this one about the gyrobifastigium. You heard me.
  • Simon Gregg tweeted some mathart that turned into WODB and latin squares and more, as only Simon can do. But what better captures the spirit of Playful Math?

Math art from the Public Domain Review.

The previous Playful Math was at Arithmophobia No More and the next is at Math Misery?. Would you like to host? Contact Denise Gaskins, or see the Playful Math homepage. People don't submit a ton of posts anymore, but I enjoy looking back at what I've found helpful from the math ed community and sharing it all together. So many resources and so much fun to be had.

Happy Holidays, or New Decade Blessings, or Sweet Playful Math!


Listening to Ibram X Kendi read his book, How to Be an Anti-Racist, and these are some notes along the way.

The introduction starts out with his humble admission of a speech he shared for a Martin Luther King Jr Day competition/celebration which he now views as racist. This leads into his focusing on the word racist, and how it has become viewed as an attack or a slur instead of a descriptor. Anything that blames a whole group for its problems is or can be racist. The struggle is to both be fully human and to treat others as fully human.

Which I love as basically the central problem of human existence. In a recent Zadie Smith interview, she responded to a question:
"You recently wrote about Toni Morrison that the thwarting of human potential was her great theme. What is yours? My own feeling is that it’s about the failure to be human. Everybody’s born and everybody exists. But to be fully human takes a little bit of effort. I think my novels are about the challenge of actually being human and not avoiding the responsibility of being human, which is very heavy. There’s a responsibility of the single person, the responsibility of the married person and of the person with children, the person without, of the dog lover — each tiny path has its kind of demands upon you, which are incredibly hard to fulfill."

Dr. Kendi points out that racist acts or statements are often followed with denial. When we say we're not racist, we're joining in denial and warping the meaning of the term as a descriptor. The distinction is not between racist and not racist, but between racist and anti-racist. Not racist, Dr Kendi points out, is a denial, and racists are the first to deny. So denials are no way to distinguish. Denial is akin to colorblindness, and antiracists aren't ignoring important characteristics of people that have affected their lives. They are seeing the effects and working to counteract racist thought and action. 

Each chapter covers a different specific kind of racism and antiracism. Starts with definitions, tells stories, often personal, sometimes historical, and supports with science and statistics. Then he illustrates the definitions with examples of what a racist and antiracist do or say. It's a robust structure that really supports the book's aim, which is really just the title.

Group vs individuals is a major theme of the book. Racism is the historical most harmful way of grouping individuals that we have manufactured. Dr. Kendi makes clear that every time we think or use 'because they are <fill in race>' we are being racist. To be anti-racist is to break those narratives, to treat people as individuals, to work against the consequences that racism has caused. One of the major shifts in this way of thinking is that black people can be racist if they are engaged in this kind of thinking and action. His motivating example is his own anti-black racism, and he shares anti-white racism from his story as well. Including himself in this analysis is humility and truth speaking in action, and it is powerful. 

In most of the diversity and inclusion learning I have had up until this point, the focus has been on the inequities produced by individual and systemic oppression of non-white (even as that definition has shifted) people. In this view, the minority groups can not be racist because they have no authority or power to oppress. Bias has come to be the identifier for individual racial preference, explicit or implicit.  Dr. Kendi's vision is more powerful to me because it addresses the cause of the oppression and fights against the core of what went wrong as racism was constructed.

Recently, a friend and colleague asked me for resources for anti-racist math education resources and I couldn't really think of any. I made a Google Doc to gather the resources I do now about: Please feel free to add or comment. We do have people in the math ed community working toward this and I know I don't know the half of it.

As Dr. Kendi discusses education, he is particularly concerned with the "racial achievement gap". The whole concept is, necessarily, in his framework, a racist idea. To be antiracist is to believe that individuals face greater challenge in schools and each learner is capable of achievement. Here is a blogpost where he details the enraging history of the idea and the tests which maintain it today.

I've taken long enough to write this that Mindshift has a post today about these ideas applied to education.

This book was specifically helpful to me this semester. One of my classes was driving me a little crazy. Pre-service teachers who were not engaged, who didn't listen to instructions, who didn't seem to care even when it involved working with kids. But this book made me realize I was treating them monolithically. I was not treating them as individuals, I was not seeing and encouraging the work of those who were engaged, and I was lowering expectations. I am a spoiled college teacher with low numbers of classes and small class sizes and I was struggling with this most fundamental of my responsibilities. This realization helped me have a better attitude, helped me individualize my thinking towards the learners. 

I love the synergy between this view of antiracism and call to action. It feels of a piece with the call to rehumanize mathematics from Rochelle Gutierrez from Sam Shah's and Hema Khodai's Humanizing Mathematics Conference

Friday, November 8, 2019


Quick math game post. 

My preservice teachers have been working on Family Math Night games, to somewhat mixed effect. ( ) Camille and Jada found a game where you used playing cards to put them in order. As I was thinking about a game for betweenness and number comparison, this popped up whole sale. Though Camille and Jada have not been wowed, I was impressed with some of the thinking I saw.

We're using Tiny Polka Dot cards, but you could play with playing cards, or my preference, number cards the kids make themselves. (Like Paula's awesome cards.)

Start: Each play draws two cards and puts them in low-high order. Whoever has the highest high card goes first. (Tie, highest low card goes first. Both tied... figure it out!)

Play: draw a card. You use it to replace one of your cards, your choice. If it was between them (the hamburger in the bun), you score the replaced card. Otherwise, discard the card.

Winner: first player to five scored cards. (Flexible to adjust length.)

Example: you have a 3 and a 6. The next card you draw is a 7. You could replace the 3 (having 6 & 7) or the 6 (giving you 3 & 7, definitely better for fitting in between.)

Now you have a 3 and a 7 and you draw a 4. Replace the 3 and put it in your score pile, one point. You have a 4 & 7 for your bun.

The homemade or Polka Dot cards give an opportunity to work on cardinality, and comparison is an often overlooked part of number sense. The subtle thing here is deciding which card you replace which works on difference. The catch up mechanic is that scoring a point narrows your window, which makes it harder till you score your next point.

Quick, fun, but I guess not for everyone.

Friday, June 28, 2019

Playful Math 129

From affinitynumerology?
Welcome to the 10000001st (binary) Playful Math Carnival (aka Math Teachers at Play). We often try to start these with some interesting facts about the number, and I was surprised to see many lists of 129 different this or that. Maybe because of the special properties of 129...

It's on the internet, must be true. Regardless, I'm hoping for some good creative, optimistic and inspirational ...ah... vibrations? ... below, so we'll roll with it.

129 is, of course, not prime (3x43, making it semiprime), and I got to wondering which 2 digit primes were and were not prime when you add 100. What's the longest string of primes you could make xx, 1xx, 2xx, ...? 129 is also, fabulously, the sum of the first 10 primes. Which other consecutive sums of the first n primes are prime? Is that common? Why do we only worry about even n?

129 is a Blum integer. People evidently used to think Blum primes were good seed for RSA moduli.

$$129 \to 1^2+2^2+9^2 = 86 \to 8^2+6^2 =100 \to 1^2+0^2+0^2 =1$$
This makes 129 a happy number. (That procedure either leads to 1, happily, or to an 8 number repeating cycle.

The place holder name for the undiscovered element 129 is unbiennium. I'm not sure why I was surprised to find out that there are names for those...

Shakespeare's Sonnet 129 has a bit that makes you think maybe it's mathematical...
Mad in pursuit and in possession so, 
Had, having, and in quest to have, extreme; 
A bliss in proof and proved, a very woe; 
Before, a joy proposed; behind, a dream. 
But it ends, "All this the world well knows; yet none knows well To shun the heaven that leads men to this hell." so I think it was probably another relationship saga.

On to the posts!

129 Antigone is large main-belt asteroid about 114 km in diameter.
Submissions and Shares

Karen Campe jumped in with the perfect starting post. It could have been the carnival post, I think! Summer Math Refreshments has links to online events, chats and includes puzzles.

Manan Shah, who hosted Issue 128, shared a post Summer Excursion #6, which has three playful activities linking words and numbers. See also his Primal Words, a completely new-to-me idea. A little bit crypto. Manan's current challenge is: Give me math words (lower case) except for "integral", "derivative", or "calculus" and I'll give you a score A. Your tasks? (1) Find words with A / word length >= 4 (2) Find A for the three excepted words"

Denise Gaskins, the originator of this here blog carnival, has a free summer service: 8 weeks of fun math activities. All by just an an email sign up. See more at her Playful Math Sampler.

Paula Beardell Krieg is really looking for a math summer. She designed a calculus beach towel, which you can buy, too!

Some of My Recent Favorites

James Propp on Mazes, Puzzles and Proofs. (He also hosted the June Carnival of Mathematics.) His posts are proper essays - always well written and intriguing.

Try the NY Times tile game.

Tina Cardone on crochet patterns.

Mike Lawler's Family Math dug into angles in Zometool polyhedra.

Sarah Carter's latest Naoki Inaba puzzle share. If by some chance you are unfamiliar with Sarah's Monday Must Reads, they are a great sampler of each weeks math twitter and chock full of math play.

Loved this mathy post for Pride Month.

A Lee Sallows Magic Pyramid puzzle.

Chase Orton's address on using play to heal a broken relationship with math.

Graphic essay on a Hilma Af Klimt exhibition. (She's in my #mathart category.)

And I'll throw in my post for World Tessellation Day, which comes every June!

Summer Foxtrot

Professionally Developed

Summer is, of course, when teachers dig deep and retrench and so often engage in self-professional development (when not off in formal PD). Dylan Kane has been doing deep thinking and reflecting on race and teaching. And Wendy Menard recommended this anti-racist reading list at the NY Times. Maybe Marian Small's post on building your mathematical confidence.

Sunil Singh's Math Recess book is having a slowchat this summer.  Can't get more playful than summer recess, right?

'Til 130

The numerology folks also seem to feel 129 is an "angel number," which tells you to look to put your talents to the services of others. Right now, there's no designated host for Carnival 130 - which usually means Denise will make sure it's hosted at her place. But maybe 129 is telling you that you should host? Contact Denise or I to make it happen. Writing these is always fun, and helps you realize how much good stuff teachers and math afficianados are putting out there.

The Hindenberg was airship LZ 129.
ps> So there's this from Archimedes' Lab... Napoleon & Hitler!

  • They were born 129 years apart;
  • They came into power 129 years apart;
  • They declared war on Russia 129 years apart;
  • They were both 129 cm tall;
  • They were defeated 129 years apart!

I only made up one of those.

Monday, June 17, 2019

World Tessellation Day 2019

4th annual World Tessellation Day. Begun in 2016 by Emily Grosvenor for M. C. Escher's birthday. (More of that story and Emily's book.) Don't think I'll get a chance to make a new tessellation for today, but here are some of my favorites from the past year. (Mine followed by the real treats from others.)

In Isometric

GeoGebra applet
GeoGebra applet

GeoGebra applet

GeoGebra applet


Not nearly everything you could see. Do you have a favorite - list it in the comments, please! If you have ideas for a GeoGebra applet, I'd love to hear it. 

To explore further, there's a tessellation page on this blog, or a tag on my tumblr, or GeoGebra books for (mostly) tile altering tessellations and Islamic Geometric Patterns.

And happy WTD. Go cover something with a finite set of shapes and no gaps or overlaps! Might need to work on a better slogan.

Sunday, March 17, 2019

Post Pi

My preservice elementary course this semester is an embedded field experience. Each week I write or find some lesson for the 3rd graders, and they teach in groups of 2 to 3 or 4 3rd graders, and then reteach in our next class period.  Each class I sit in with a group, and everyone has some time to assess and reflect themselves and the learners while I debrief with that group. Before the first time teaching, at least, we try to rehearse together. As a whole, this is how I want to teach teacher prep from here on out. We're getting to less content, but I see so much more learning.

This week I had a lesson planned for Thursday, Pi Day, that had nothing to do with π. It was on these terrific Naoki Inaba place value puzzles that Jenna Laib shared. But with Pi Day approaching, and #MTBOS talk of activities and Scrooges, how could I help but think of a lesson?

Sunday, March 10, 2019

Model Citizen

I've been thinking a lot about mathematical modeling this semester. I'm teaching a preservice secondary class, and we're trying Catalyzing Change as a text. It mentions modeling 20 times in 100 pages, and definitely indicates it as one of the primary motivations for mathematics' importance.

"A mathematical model is a mathematical representation of a particular real-world process or phenomenon that is under examination, in an attempt to describe, explore, or understand it. When students engage in mathematical modeling, they often have the opportunity to leverage mathematics to understand and critique the world. Mathematical modeling is the creative, often collaborative, process of developing these representations. Modeling always requires decision making that involves determining which aspects of the phenomenon to include in the model and which to suppress or ignore and what kind of mathematical representation to use. As noted by the mathematician Henry Pollak (2012), throughout the modeling process, both the real-world situation and the mathematics must be taken seriously."

I rewrote my content standards for the course trying keep this in mind.  I was recently getting observed for a personnel thing, and it was a day we were thinking about modeling. The observer was David Austin, who's teaching a new modeling course that we're offering as a part of developing an  mathematics emphasis for the math major. I asked him to share the idea of the course and we tried to connect it to them to encourage future teachers to take it. Part of how we addressed that was a classic trigonometry problem of fitting a cosine model to the height of a seat on a ferris wheel. But they weren't using video of a real Ferris wheel. Why would we need a trig function for a real Ferris wheel? So I'm pretty sure that this was not real world modeling.

Then Dan Meyer took on modeling last week. He was at a panel with the folks who produced THE report about this, GAIMME. (People just say 'game'.)  They identify four aspects to mathematical modeling. It's real world that bothers Dan. He said/writes:
If your definition of “real world” labels the US tax code as real and polygons as non-real, your definition is not useful. To most US K-12 students, the US tax code is very non-real and polygons are very real. 
If you define “real-world” as a property that is binary rather than continuous, that is fixed across all cultures and time rather than relative and mutable, if your definition doesn’t account for the ways (per Freudenthal) that contexts become real in someone’s mind, it isn’t useful. 
And if your distinction between “mathematical modeling” and “learning” depends on “real world,” a descriptor without a definition, it isn’t a meaningful distinction.
I love Dan's willingness to wade into a fight, and to constantly rethink and refine his ideas about teaching and learning. But I don't see how he's helping here. Real world may not have a sharp mathematical definition, but the idea that we can bring things from the real world into the math class is helpful. Many of Dan's first big ideas were about this, and I don't think he's denying it now. He also wrote a lot about pseudo context, which was helpful in getting at the idea that real world is not the be all end all.
Dog Eat Doug by Brian Anderson

I don't care about labeling or not labeling tasks as real world or not. I do care about the process the learners are engaged in with these problems.

GAIMME outlines this as the process:

  • Formulating the problem or question
  • Stating assumptions (often requiring simplifications of the real situation) and defining variables
  • Restating the problem or question mathematically
  • Solving the problem in the mathematical model
  • Analyzing and assessing the solution and the mathematical model • Refining the model, going back to the first steps if necessary
  • Reporting the results
Paula Beardell Krieg was sharing some of her usual awesome mathart, and it led to a quick conversation. She made an image that was interesting to her with trigonometric functions then did her arty magic to make it into paper that she then origamied into boxes. I commented that I'd been wondering if the modeling aspect was part of what I found so engaging about engaging in art in math class. (I was working on a David Mrugala idea, and, though not real world, I tried a lot of different functions trying to get different effects.) Getting the math to make what you want is messy, open and genuine. She said: 
This idea of starting with something fuzzy then refining resonates with me in two ways. One, mathematically, is that more and more it seems like (from the glimpses I've had of higher level math) is that what mathematicians are doing is always trying to get more and more accurate models. ... What the artist does is similar in that we start with a sketch then keep refining until we decided we've gotten close enough. When I am playing with these functions to get images that I like, what's really awesome is that I get to use the math that I know as well as learn new, or refresh myself, with new stuff. It's an amazing process for me, as I have to push my math envelope to push my art envelope. It is so satisfying, but it does require so much mucking around. That's the mess that you talk about. It's also the play that I think about for my own work and for the work I do with kids. So much discovery in mucking about. I know you know that this is where so much learning happens. I am convinced that imposing too much structure (which is different than NO structure) sucks the potential out of learning. No structure can do the same. Finding that sweet spot, which is actually quite large, is a good place to aim for.

For Dan this leads into a big idea "Teachers need fewer ideas about teaching." Whoa! I'm definitely not ready to get into that. But I want one of those ideas for math teachers to be "learners need to be doing mathematics" and some of what doing math is is encapsulated in this process.

The end of the conversation (for now) with Paula was a direction for me to pursue these ideas.

  1. Definitely we're on the same wavelength here. The advantage of art over modeling is that the learner does more of the problem posing. When I pose the problem or provide the data, maybe some are interested, others not. But art is them finding their own problem. The down side is that not everyone's interested in making art, and I don't know enough about selling that.
  2. When I'm working with kids I find it's helpful not to think of what we're doing as making art or of having fun. What is more useful is to ask them to see what they can do with what I give them, and to try encourage a sense of play. The results I'm looking for has less to do with art and more to do with discovery. I did something like this with 2nd graders this past week. I may tweet about it ....

So it comes back to play! I think this is inherently tied up in the first step: the mathematician poses the problem.