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.

Friday, February 15, 2019

#AMTE2019 #MTEchat

Last weekend I was at the AMTE 2019 Conference. Paul Yu (GVSU colleague) and I were presenting a brief report on a project where we looked at how our different classes impacted preservice elementary TPACK. (Technological Pedagogical Content Knowledge.) Wait where are you going?

(The materials are here if you're interested.)

That's actually my reaction, too. I have a weird love hate relationship with research. I think it's important, I love to read it, but so much of it is irrelevant or over-applied. And almost all of us there have the job of preparing future teachers, and there's little talk of practice. I so much prefer going to meetings with teachers where we talk about teaching.  (Twitter Math Camp being the peak experience.)

There was some teaching talk. My colleague Esther was part of a session on mediated field experiences (being in the school with preservice teachers) that had a variety of people working in related contexts to talk together about what we're trying. But there were no resources to share or way to continue the conversation afterward.

The AMTE Equity Committee led a session on on addressing novice teachers understanding and readiness to teach diverse students. (Here's my twitter notes.)  At the end, there was a question would people be interested in a syllabus or reading list... Hell, yes! So maybe we'll get it somewhere, some how.

Headed into Denise Spangler's Judith Jacob lecture, I was pretty fed up with it. But I bumped into Joanne Vaskil, who was part of our brief report session. (She is part of a group using Twitter with their preservice teachers for responding to assignments.) Talking about it - she is an excellent interviewer - she got all of this out of me. And I was comparing this with the #MTBoS, which, to me, is built on sharing practice and questions about it. Joanne pushed for doing something about it. We thought about hashtags, and #MTEchat seemed about right. Being at the Association for Mathematics Teacher Educators, people were referring to us at MTEs. (#ITeachMathTeachers seemed a little too Sixth Sense.) Turns out people were already using it for this. Logical people are logical.

One of the values/ professional practices I try to instill in novice teachers is to not try to go it alone. Collaborate, find support, and share with others. Sometimes it happens in your school, which is the best option, but even then, think about being active at the district, state or national level. What is good for our students' learning is good for our learning. In turn, what is good for our student teachers' learning is good for our MTE learning.

Joanne already got the ball rolling. So... why don't you join us?  How are you preparing teachers? Share stories, questions, images, blogposts. If Twitter isn't your bag, help us figure out what the proper place will be. Glenn Waddell, researching teacher use of Twitter, could see that there was a healthy slice of people on Twitter at the conference. But they were most were isolated.

Let's get together!

Wednesday, August 29, 2018

Reading Cheesemonkey: Algebra Class

I'm teaching Intermediate Algebra this semester, which I haven't taught in a while, and so have been rethinking the course. My two big goals are:

  1. Redeem mathematics. These are students in a good university who are having to repeat content they've already seen and maybe more than once. I figure, for most of them, I can infer bad math experiences along the way.  I want them to appreciate math, to know why algebra was a big deal in the first place, and have opportunities to do math.
  2. Free them from the gatekeeper. With a new disposition, I want them to understand the content at a level that will equip them for success in further math classes (which most don't take), or statistics (which 60% do take), or even reconsider majors if they had eliminated something based on math requirements.
Dream big! 
Reality Check by Dave Whamond

As a part of Sam Shah's Virtual Conference on Math Flavors, Elizabeth Statmore wrote an excellent piece on math as a thinking course. So the first assignment included reading and responding to her post.

  • I really liked how Elizabeth emphasized that she does not care really about if you use the math you learn in her class or not. But that the point of her class is to, "learn how to think and communicate at a more advanced level than you are capable of right now."  She makes a very good point there. Most of us grew up hating or dreading math all together and she said that with the skills we learn in math by problem solving and implementing those problems and solutions in real life then we will understand the point of why she wanted them to learn math in the first place.
  • I like the way that Elizabeth talks about math as more than just numbers. She makes real-world connections that may interest others outside of the general math community. The article counters typical stereotypes about math, while building upon the idea that there is more to math than work. Math is problem solving, communication, another language to convey new (and old) ideas. Elizabeth teaches her readers that math is a tool of understanding that can be applied to many situations outside of mathematics. 
  •  I really liked how she made out math to be more than just a school subject, but rather a real world concept we use every single day. She relates it to the real world by saying, "you are going to need to make sense of things you don't initially understand." In this thought, she's saying that of course you aren't going to understand everything you learn right off the bat, but rather to keep trying until you do understand it and feel confident about it. I also like how she said in order to understand something, you have to WANT it and I totally agree with her on this because if you don't care about learning something, then it won't come easy to you. 
  • I  like how Elizabeth says the truth about things and doesn't really cover it. One line that spoke to me was "The fact is that math is a human activity. If you are human, you cannot escape it.." this really brought light to me that even though we may hate taking a class, us as humans need to go through it because its who we are. We need to learn how to communicate and think on a more border subject and open our minds to new concepts and ideas. 
  • I appreciated that she addressed the age old question of "will I ever use this again", as I will admit I have asked this before. I enjoyed Elizabeth's take on math as a way of learning how to communicate better, rather than a class to simply learn how to solve complex problems on a calculator. So while I may not necessarily continue to use every equation I learn this year, I will become overall a better thinker.
  • I really enjoyed that Elizabeth put a whole new spin on how math is used in life. I used to only think that I only learned it to apply to things in my life that needed math but she made me realize how many different ways it can be used. I really enjoyed when she said "What I care about passionately is that you learn how to think and communicate at a more advanced level than you are capable of right now." because it makes me appreciate not only my harder math classes but just all of my harder classes in general.
  • I appreciate how Elizabeth views math as a thinking course filled with discussion and collaboration. As someone who is going into the Hospitality field I value teamwork. It really is what will make or break a business and it really can be what can make or break success in Math.
  • I like how Elizabeth explains that math is more than just looking at meaningless numbers all day. But it takes teamwork and collaboration to explore how you can solve a system of equations using not only constructive thinking but also creative thinking to explore that there are many that you can solve these equations.
  • I like how Elizabeth describes math as more than just numbers and all that good stuff. She adds that if we want to be successful in math, then we have to want to understand it. I also like how she mentions that math can be used as a way to be able to make sense of things and think differently. 
  • I like how Elizabeth included how you have to want something in order to be successful in it. I agree that there is always many different ways to problem solve in math and real life, and we need to learn how to process these ideas. I love how she said we need to be persistent, strong and flexible thinkers in order to do well in life, we can't think the same way for everything or we won't get as far as we could in life. 
  • I personally like how Elizabeth says to understand things, you have to want to understand them. If you accomplish something, there was at some point a want to accomplish it. Elizabeth made math seem as if it was a part of human nature and not just a subject used in school to torture students. She thinks of math as a way to help individuals communicate and think more efficiently. I aim to use Elizabeths point of view, not only in this class, but also in the other classes that I have now and in my future.
  • I like how Elizabeth has her mindset and sticks to it. She believes that you have to want to do well to actually succeed and I agree with that. I also agree with how she makes math more about thinking, rather than just solving and numbers. That way it is much more relevant to the students. Also, she ties in communication which helps the students learn real life skills instead of only learning the course.
  • I liked how Elizabeth explained math and the teaching of math as more than a simple course, and how it is a language understood by all. I also like how she explained that she doesn't care if we liked math, she cared about us changing the way we thought and using it to advance our thinking and communication skills. The perspective she had on the subject was something I had never thought of before, but now makes so much sense. 
Sooo... thank you, Elizabeth!

Monday, August 6, 2018

Golden Triangles

Megan Lonsdale asked on Twitter about some mathart ideas to decorate stairwell panels and fuel a neat first week for her learners. Sam Shah shared ideas and resources, including cellular automata which I've got to try with kids. In the course of the conversation, I realized I hadn't blogged about one of my favorite lessons of the past year. This was for a Festival of the Arts with Heather Minnebo, an art teacher who's always welcoming to me and my preservice teachers. (We've presented together, too.)

I have this standard framing for types of mathart lessons. They start from thinking about art as problem solving.

  1. Art as the problem. (This lesson below.)
  2. Art as inspiration. (Look at the effect this artist got... lets do math on it.)
  3. Math as the problem. (Calder wanted his mobiles to balance perfectly...)
  4. Math as inspiration. (Escher extending his tessellations to the hyperbolic plane.)
The inspiring math here is the golden triangle. It's such a great structure... The acute isosceles (1, $$\phi$$, $$\phi$$) decomposes into into a smaller acute isosceles and an obtuse isosceles. Or, equivalently, the (1, $$\phi$$, $$\phi$$) acute composes with the (1, 1 $$\phi$$) obtuse to make a larger acute. Here's artist Dusa Jesih playing with the structure. Here's some GeoGebra so you can play as well. 

For me these lessons can spend time as all the different types, depending on your objectives. Show the artist pictures, math as a problem, what makes these triangles fit together like that? What else do they do? (2) What angles do we need so that they fit together like this? (3) Show the triangles, what can we make with them? (4) But since this was a festival of the arts, I liked the idea of presenting an art problem. Look at these triangles, how they fit together, what can we do to make the different kinds of triangles show up distinctly when we put them together? (1)

With the fifth graders I brought a few cut out to show how they fit, and then together we drew a big one and started decomposing, counting up how many of each type as we cut it up, then did some noticing and wondering. They saw 1 & 1, 1 & 2, 2 & 3, then the surprising 3 & 5... maybe a prediction? 6! (4 was an anomaly.) 7! (We were every one but now we're skipping.) 8! (Are they, like, adding?) Who knows! (We were surprised once, now we know it's not a pattern.)

Digression about the Fibonacci numbers because what mathy person could resist.

Now the art problem: We're all going to make some of these triangles, and we know we need more of the acutes, but how can we decorate them to make them visually distinctive when put them together? Each of the classes debated different options, but each gravitated towards the same solution. Lines and curves, pictures and words, two different patterns, two different colors... but ultimately decided on warm and cool colors. (That had been a topic in art class in the past couple of months.) Some class discussion on what qualified as which. I had printed enough triangles for each learner to do two. (PDF).

When we had enough or time was running low, we gathered to try to put the flipped triangles together. Once they were taped, turn the whole thing for the dramatic reveal. Learners were curious about the reveal, happy of the results, and proud to point out their elements in the whole class mosaic. The assembly process is not automatic, and you can see that there was some difficulty making a perfect tiling. All in all, this one's a keeper, and I'll be looking for opportunities to try it or a variation.