## 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

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

### ꓕWCƖ8

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 teacher.desmos.com/labs.

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

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.

## Sunday, July 29, 2018

### Words with Friends

The heart of my TMC18 was giving a keynote with Glenn Waddell and Edmund Harriss. It was an enriching and amazing experience. Though I present at conferences a fair amount, 2-4 times/year, this was different in a number of ways. I wanted to share a little behind the scenes as an encouragement to others to find ways to speak up. Here's the keynote, and the associated links:

Besides not having enough double consonants to appear with these two, I really didn't feel worthy, to be blunt, to keynote with them. At first it was just to present together, great!, but Edmund knew from the start that this could be a keynote.  But there's never been a group keynote at Twitter Math Camp (or any conference I've been to, now that I think about it) so how much did I have to worry? But then it was approved.  Still not too worrisome, but when the keynotes were announced, there was an explosion of interest. (Listen to Edmund if he has ideas for topics.) And, whoosh, there again were the feelings of 'what do I have to add to these two?' (Then Julie's keynote the day before was on this exact point.)

But I love to collaborate. And I've had the chance to do great things with people for TMC. GeoGebra with Audrey and JedJames Cleveland, Joe Schwartz,  and the whole Tessellation Nation experience at TMC16 (Joe's coverage). This time, preparation was spread out over almost a year. We met in Google hangouts, and exchanged posts and articles, ideas and tweets. These conversations were invaluable to me. As I think I said in the keynote, though maybe it sounded like a joke, when Edmund proposed the talk 'Mathematics isn't everywhere' my first reaction was that I say the exact opposite. A lot.

As we conversed, two themes emerged. The first revolved around what teachers are addressing when they say math is everywhere. We're justifying our courses. When am I going to use this? Why am I learning this? For me, teaching preservice teachers, the answer is the mathematical processes, the practices. It's doing mathematics. Dave Coffey and I gave a presentation once upon a time about verbing math. To math should be a verb, like to read. This has been a theme for me for decades. But still in discussing and preparing for this I realized the extent to which I still objectify the content.

At one point I told a story about one of my favorite pastors. (Aside: He never had printed sermons. Turns out when he was in seminary, he discovered he always froze if he had notes, fully written or even outlined. He was stiff and unnatural. Instead, he developed a routine of just reading, rereading and praying over the texts, and then just letting it go when it was time for the sermon. I think my teaching is a lot like that, actually. I really ruminate on what I'm teaching, how I think about it, why it matters and have the start of a lesson, but will follow it wherever it goes.) Jim gets up one sermon, and says, you may have heard, holding his hands up clasped, "here is the church, here is the steeple, open the doors and here's all the people!" Your grandmother probably taught you that. IT IS HERESY!
(Image source)

I felt like that is what we were doing. "Math is everywhere" is accepted doctrine, and we wanted to tell a room full of our friends that No, It Is Not. Fr. Jim's point was that the people were the church, not the building. And our point was it's your doing that is the mathematics.

Well, part of our point. One of my takeaways from this was thinking about mathematics as a way of knowing. At the moment I'm thinking about this as a kind of particle-wave duality. The wave is the doing of mathematics, and the particle is the field itself. One of the benefits of being in mathematics is how long a history we have of making progress and understanding. This side of our discussions became the Levels of Abstraction.

1. Seeing mathematics in the world.
2. Seeing the world through mathematics.
3. Finding aspects of the world through mathematics and mathematics through the world.
4. Finding aspects of mathematics through mathematics
5. Finding the limits of thought itself!

I love the verbing here - even when we're looking at the particle, we see the wave.

But at this point, the idea of the talk felt disjointed. We thought about a unifying theme of lines. But as we discussed, we found that there was a unifying theme. We wanted people to know we weren't adding to their burdens, this was something already present in what they were doing. Play is the element that connects the elements of great teaching.

Finally it was time for TMC18. We had slides, just. But had had no chance to practice. And TMC is busy. We talked it through twice more and planned out the timing. All while Marian delivered a quiet, intense, thoguhtful, heart felt challenge. Blew me away. Julie went full cheerleader/Oprah and encouraged everyone. How could we follow them? But the people were so generous and supportive, it all worked out.

So, I think you should present. Find people with whom to present, commit to it even if you're not ready, and give it a go. Regardless of the end, the work is worth it, and talking with other teachers or mathematicians is the way to get better. In your school, a local conference, NCTM regional or national. Ask someone you want to talk to, or say yes when someone asks you. I promise: you'll love it.