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.

Thursday, August 2, 2018


My favorite professional meet of the year has come and gone. Here's what I'm still thinking about... divided into everything else and the equity session, Take a Knee, led by Marian Dingle and Wendy Menard.

Necessary proviso: there is so much good at a TMC.  The signal to noise ratio is unimaginable compared to any other meeting/conference I've been to. I'm not trying to represent everything, and I'm skipping good stuff. This is literally what I'm still thinking about.

Everything Else

Desmos preconference: this was all about computation layer for me. Despite Michael Felton's great introduction last year I did nothing with it. Sigh. Now I feel like maybe I could, if I get some time to just process. There's a help forum, an improved Scavenger Hunt (which are the learning activities) and some documentation. Look at Chase's and Madison's Estimation Stations for what is possible. (Or watch their My Favorite on it)Plus Eli's description that computation layer is really about connecting pipes to send data. Connect a source to a sink. Christopher led a design session that covered their principles for building an activity and showed it in action in the activity Marcellus the Giant. That was also the first peek of Snapshot, an amazing new teacher tool. Turn any of the Desmos tools on or off at

Marian's keynote. Quiet, intense and personal. This is directly a challenge to the community of math teachers. Are we on the side of equity? Are we doing what we can? Do we even see the problems, issues and concerns in front of us. Please watch.

Amie Albrecht teaches a problem solving course where she is doing so much fabulous pedagogy. The course has explicit goals of learning to problem solve, and to be able to share that verbally/presented or in writing. Feedback before grading, reiteration with wider and wider audiences... just beautiful. Folder of resources. Some things I'm still thinking about for our teacher education classes and for the redeeming mathematics class. Part of it, the Back of Mathematics, she shared as a My Favorite.

I caught Robert Berry's keynote at Desmos and his afternoon session on day 1 on the NCTM's Catalyzing Change book. Honestly, because I am terrible at reading programs ahead of time, I was just surprised he stayed! He really participated and was great about connections between the MTBoS and NCTM. One of the cool things in Catalyzing Change is that the NCTM is against tracking of students and of teachers. Are the most effective teachers teaching all the students? I do think it is a huge mistake for NCTM to paywall their essential high school content in this book. The 1999 Standards and Principles were so formative for me, and so hard to get into teachers hands. One lesson I'd love for NCTM to get from the teacher twitter community is that shared resources increases buy-in and participation. Teachers are naturally community-minded, and if you make them welcome and support them they will join. (Opinion.)

Julie's keynote. I was in two minds here. One, appreciative audience in need of the message, and two, person speaking the next day having to follow Marian and this. Wurg. The impostor syndrome message was timely. And if an old man who speaks regularly and has taught for 30+ years feels that way... sigh. But also, as a teacher educator, her message about being a teacher leader was perfect. It's not about doing everything, it's about finding what you love, doing that, and sharing. It reminded me of Dave Coffey's favorite Teaching Gap quotation:

The star teachers of the twenty-first century will be teachers who work every day to improve teaching—not only their own but that of the whole profession. -Stiegler & Hiebert
Sasha Fradkin presented on impossible problems. I love the idea of learners doing the work of mathematicians, and showing something can not happen is just as important as finding out what can. But how rarely do we ask them to do that? I'm still tossing over in my head what the difference might be between doing a general investigation, and specifically asking for outcomes that can't happen. Sasha is the author of Funville Adventures, which session I missed, but be sure to check it out.

Brian Bushart is still developing numberless problems with the teachers and learners of Red Rock.  It's really impressive to me, that they are making some great improvements to something that was already fabulous. But he realized that some teachers were using the structure in a deficit mindset. And thinking about Rochelle Gutierrez's ideas about mathematics identity, they reframed the problems with a story telling lens. Just amazing. (His slides.)

Some My Favorites: (all the TMC18 vids from Glenn Waddell)

Take a Knee
I spend a lot of time thinking about this, and trying to educate myself. I try to use the understanding I build as an inclusion advocate at the university, and in my teacher education classes, as well as some local work in the community. Despite all the time I spend reading about this, I am constantly humbled by how much more there is to learn and work to do on my own thinking. Last year's TMC session by Grace Chen, Brette Garner, and Sammie Marshall revolved around connections between equity and the Standards for Mathematical Practice. Personal work included developing a checklist to get past our internalized schema, and 'equity eyes' -training ourselves to see. (All three are/were Lani Horn's grad students. Never wrote it up for the blog, bad blogger.) It revolved around developing equity eyes.  This year I got to see Calvin Terrell who sometimes refers to this work as decolonizing. Then a 5 week workshop at work was titled Decolonizing White Consciousness, which seemed timely. That featured work of Robin DiAngelo (watch this on white privilege), adrienne maree brown (read Emergent Strategy), and a variety of readings and videos around the idea of identity.

So the morning sessions for me came down to Take a Knee or Islamic Art, and I couldn't not join Wendy and Marian. (Session resources. Twitter - #tmcequity) Both were a part of the TMC17 equity session and Wendy & José Luis Vilson's Racially Relevant Pedagogy session at TMC16 is the single most affecting hour workshop I've ever been to.

Day 1 started with us introducing ourselves with our identities. This feels very odd if you're part of a group or groups that gets to take this for granted. Straight, more white than not, male... naming has power and self-naming invites vulnerability. The day closed with an activity for trying to suss out how central all these identities are to you. It was gently brutal. In between, we tried to figure out what take a knee even meant in the context of our work in math education. A theme that continued over the three days started here: equity for our students and what did that mean, and using our lessons as a way to be relevant and real with our learners. Both are a part of the larger discussion of how teaching is political.

Day 2 revolved around standards and methodologies. Teaching Tolerance's Common Beliefs help us understand how what teachers bring to the classroom influences what we teach, and the Standards for Social Justice are as good a framework as I've seen for how we should aspire to teach. Rochelle Gutierrez's article on Creative Insubordination (in here from TODOS) provided a lot to talk about. And we had an awesome poster session on that.

It's insubordination because we are consciously trying to work against the status quo.

Day 3 was preparing to go back into our worlds. We began with powerful identity statements again. "Because of my race I can..." Says something about a group of people that can share such things. We then worked in small groups on what we can do, short, medium and long range.  My group was thinking about math lessons that reflect and think about the diversity of our schools, communities and country.

For me:

  • Short: diversify follows on Twitter. I got some great suggestions in responses to this tweet, and from the hashtag #disrupttexts.
  • Medium: incorporate SJ standards into teacher training.
  • Long: transform colleagues. Makes me woogly just to say it.
Further reading: Kent Haines - Pedagogy and Equity, Dylan Kane - Disrupt Math, Michael Pershan (not even there!) - Power Works by Isolating.

Next Year
Still thinking about this. I've been lucky enough to go 5 years in a row - is it time to make space for someone else? Selfishly, it is amazing to participate. But there won't be space if all the same people always go. I'm also conscious of not being a classroom teacher, and the thought of taking that spot is chilling. Maybe the TMC Midwest will happen? And absolutely no judgment on anyone else who is a repeat attender - I am only trying to process this for myself.