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:  http://bit.ly/MoreAwesomeTMC18.

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 Jed, James 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.

Narrative Equity

It should come as no surprise to anyone who spends more than 5 minutes on this site that I see a lot of connections among math, games and art.

My favorite game of all time is Magic: the Gathering. I love it in concept and in play. Amazing strategy, accessible at several levels and varieties and terrific flavor and art for bonus. But I'm not trying to convince you to try the cardboard crack - unless you're interested? - I'm just letting you know what I'm about to try to riff on. Mark Rosewater is the long time lead designer on MtG, and is very generous at sharing his design thinking, on Tumblr, in longer form blogposts and in podcast.  He is a serious student of game design, and focused on engaging play, so there are often connections to teaching and learning. A recent article is on narrative equity. One of the ways games engage players is the opportunity to make a story. It's a rich payoff, and can be significant to identity. I'd encourage you to read his post, but the examples after the intro story are in terms of Magic, so may not be accessible. Mark wraps up the introductory stories about his daughter and himself with this:

What do these two stories have in common? In both, Rachel and I prioritized having an experience. Our personal story carried enough value that it influenced how we behaved. It was an interesting concept, that people will give weight to choices based upon the ability to later tell a story about it. I call this idea "narrative equity."

The next step for me was applying this idea to game design. What does narrative equity mean to a game? Well, games are built to create experiences. I talk all the time about trying to tap into emotional resonance and capture a sense of fun. Narrative equity should be one of the tools available to a game designer to do this.

After thinking it through, I came up with seven things a game designer can do to help maximize narrative equity in their game.

What follows here is his list of game design connections to this idea, and why I felt like he was talking about teaching mathematics.

#1 – Create components with enough flexibility that players can use them in unintended ways

Math, to me, is ultimately about doing. We often make it about acquiring facts and techniques, and can lose track of why we are asking learners to do that. When learners are exploring these ideas, these powerful, culture changing ideas, which we are teaching, there are going to be ways to combine them to get new places. When we front load mathematical ideas, so that in the next section we can use them to solve this kind of problem, we're working against this.

The big shift for me on this was going from that linear learning curriculum model to a landscape approach like those in the Fosnot & Dolk work. (Image from this workshop.) They create a distinction among models, strategies and ideas, and realize there is a progression, but there are so many paths that learners can take from place to place. Formally or mentally, this is how I see curriculum now.

#2 – Create open-ended components that can be mixed and matched in unforeseen ways

To some extent, for me in math, this is about tools and representations. I am a deep believer that learners being able to represent (in the old NCTM process standard sense; create, move among and choose representations) magnifying their problem solving capacity. Given the ability to create graphs, diagrams, written/verbal descriptions, contexts, tables, equations or expressions... that creates excitement. I cannot tell you how often I learn something new or see a new idea and need to make it in GeoGebra or Desmos. And am delighted by the result. Or to write down a function to model a behavior. Or see a pattern in a table that was hidden from other perspectives...

Naturally, this dovetails with tool use. We live in the future, people, with free tech that gives capacities to everyone once reserved for super-geniuses. To some extent, I think why I stood out as a young math student was that I could do that in my head. Now everyone can! Why hide it from learners? Several of my Calc 2 students this summer had Calc 1 with NO TECH.  Augh! On the flipside, I felt like learning Desmos, GeoGebra and Wolfram|Alpha was a goal in my course, and was frequently happy to see them used in ways that we had not done. A good sign the learner is making it a tool of their own. We also had programmers making things, and a student from another U sharing his Mathematica programming, which they are required to use.

#3 – Design in unbounded challenges that allow the ability to create memorable moments

THIS. I want to get much better at this. The twist is that math does this naturally, so we've had to contort it to hide that aspect. I ask students to do this, but don't know how to support them. Especially when I see them, they have learned that the teacher always has an end in mind. Show us how! Show us an example! There are, of course, times for this. But when I ask you to see what you make with this, I really want to leave the door open.

I hit a pretty good middle ground with the quarter the cross assignment in Calc 2 this summer. We used David Butler's examples to launch it and model, but then opened the calculus door by connecting to how we had great area calculating power. Many exciting results. Not all of them, but I don't think we can require creativity. Just make space for it, and celebrate it. For the assignment, we had a little experimentation in class, a bit more in the takehome and then a lot for the people who chose that for a writing assignment.

#4 – Create near-impossible challenges that can become a badge of honor

Mark sees #3 and #4 as related. And this is something I do not do much of in my teaching. I do give SBAR grades for good progress on hard problems, instead of credit for right answers. I propose extensions for writing, and have optional assignments that can be very challenging. Is that enough?

I think near-impossible is affecting me as a mathy type. The idea of a challenge, that a learner would remember solving or trying is probably the goal. How do we support them to give these a try, though? Much like #3, I think sharing student work on such things is probably a key part.

At Twitter Math Camp, Sasha Fradkin had a session on impossible problems. She didn't mean this kind of impossible, but I think by coincidence, it might fit the category. Something like: using three straight cuts, divide a circle up into 3, 4, 5, 6, 7, and 8 pieces. (Not all of those are possible.) One of her takeaways was to consider what do we want the learners to mean when they say 'this is impossible.'

#5 – Create alternate ways to win

In a game, of course, you're trying to win. If there is only one way to win, the game becomes boring and narrow quickly. If a multitude of strategies is available, the game is richer as people pursue different resources and strategies.

In class, this feels to me like assessment. The goal is demonstrated understanding. If the only way to do that is timed tests, I think that narrows the game. Now it's not competitive, maybe, and the people who are better at that don't necessarily bar others from success... unless it creeps into your test writing. Or you curve. Or you measure the middle and less successful students by those who are good test takers.

For me in college there was a strange thing. My first two years I was adjusting from high school's low expectation tests to honors courses where they wanted some version of deep understanding. I got some Bs. The high school tests just wanted recall, which due to no credit of my own was easy. I couldn't not know a lot of those things. But then, beginning of my junior year, tests just made sense. I wasn't any better of a student, but I think I went almost two years without missing a question. It was weird.  When I started teaching, this got me to include a lot about test taking strategies in my classes and review days.

Eventually, though, I realized that this meant that tests weren't doing what I wanted them to do. So now my learners know the standards they're being assessed on and there are multiple ways to demonstrate understanding. And they can reassess.

#6 – Allow players opportunities to interact with other people where the outcome is based on the interaction

I think this is a regular feature of classes that feature cooperative learning.  It does require communication that is not teacher <-> student. If your classroom communication is you talking or asking questions and people answering you or asking you questions, it is one dimensional in a three dimensional world.

#7 – Give players the ability to customize, allowing them opportunities for creativity

This is sooo hard. But, ultimately, necessary. Dave Coffey likes to say that if the only choice students have is to do something or not to do it, of course some will choose not to do it. Even if the choice is as simple as choose the even or odd problems to do can increase engagement. Is it possible to let students choose a topic? Form of an assessment? Application? Which question to investigate in a 3-Act?

I love Elizabeth Statmore's emphasis on returning authority to the learners. This is part of that. Give choices and ask them why they chose as they did. Math class does not have to be everyone doing the same thing at the same time. Choices imply there is self-assessment to do. To me, this is the holy grail of assessment: learners start to think for themselves about what do they understand and what do they not get yet. And what should they do about it.

Sometimes I describe Magic as chess where you get to build your own pieces and bring your half of the board. (Plus a layer of variability from being a card game.)

Endgame

Mark's last words:

Narrative equity isn't a lens you have to view every game component through, but it is something you should view some of them through. When putting your game together, be aware that you have a lot of control over what the end experience will be. By making certain choices, you can maximize those choices that lead to your players forming stories, which in turn will change how your players emotionally bind with your game.

I am left with questions. What stories will my learners tell about the course they had with me? Will they be the hero or at least the protagonist in those stories? Will it change their view of the mathematics genre? Will every learner get an opportunity to weave a tale?

PS: Flavor Flav

I ended up submitting this to Sam Shah's Festival of Flavors, a blog conference of people thinking about the flavor of math in their classroom.  Just the keynotes he has lined up are spectacular, and I'd expect there to be many more worthwhile reads. So head on over. "Kicking the flavor, getting busy
You're going out, I think you're dizzy."

PPS:
I can quote several of Flavor's raps by heart, lest you think I take the name in vain.  

Year of Calculus

Just finished a year of calculus. And I fit it in between January and June.

So many things I wanted to write about along the way, but it just wasn't a year where that was going to happen. Maybe posts someday? Who knows, but I wanted to reflect on what I could remember now. Mostly I teach math for preservice teachers, elementary, middle and secondary, and some our senior capstone, which I do as a bit of a history of mathematics. But last summer we needed someone to teach calculus online, and I was certified, and I find it hard to resist a teaching challenge. (Our university requires that you pass a little course before you're certified to teach online. Ironically an in-person course. And, of course, no training required to teach otherwise. )

The online course was hard. I knew I would miss the day to day, getting to know the students, and the learning community, but I had no idea how much I relied on the students to make the work problematic for each other. The interaction is sooo much of the thinking, and even good participation on a discussion board is not going to hold a candle. The lack of the day to day formative assessment made me feel like I am teaching in the dark. The course was asynchronous, so the disconnect was maximal. Some people finished the course by week 8 (of 12) and some did the entire thing from weeks 7-12.  I required two video call/interviews, but not at any particular time. Now I think at least three, including one in the first week.

But when a teacher ed course got cancelled for low enrollment (we should be worried, I think), I got a chance to teach calculus 1 in person, and then in the summer eventually fell into calculus 2 for the first time in a decade. If you want to skip all the blah, blah, blah, and see the agendas and resources, here are the course pages (Google docs) for the Calculus 1 and Calculus 2. (Online calc 1 page, with more exposition as it was self-paced.) GVSU is a real hotbed of some calculus innovation, with Matt Boelkin's Active Calculus and flipping crazy Robert Talbert among others.

So what sticks out from the year of calculus? I think I'll try to write about (a) reordering calculus 1, (b) writing in calculus, (c) the interviews, and (d) some learner feedback. (I never got to (d)! Maybe next post.)

Reverse It
In discussions with #MTBoS folk about calculus (all of whom I cannot remember; Dan Anderson Paula Beardell Krieg, David Butler, Heather Johnson, Lana Pavlova...), the idea was floated to think about integration first.  Why do we teach derivatives first? What are the consequences of that? I'm already in favor of moving limits to the end of calc 1. Hand waving was good enough for Newton and Leibniz, so it's good enough for me. What are the problems that limits solve, and do learners know enough to know they are problems when limits are taught? For me, it's about precision, and precision comes late in a learning trajectory.  One of the problems with Derivatives First is that learners invariably think of them as antiderivatives, which makes the Fundamental Theorem of Calculus one big shoulder shrug. They see it as how to calculate definite integrals (first part) or never bother to parse it (second part).

The best thing about derivatives first to me is thinking about rate of change and the transition from velocity as an average to instantaneous velocity. But rate of change is difficult to visualize, I think, as it involves an abstraction even to get started.  But accumulation feels more visual. Someone suggested penguins coming into a room...  which led to this GeoGebra. It was a solid lead-in to Visual Patterns.  That was a good concrete start, gave us reason to do some algebraic modeling, and good use of table representations which lead to thinking about differences (first, second and so on).  We posed an accumulation question about the visual patterns, too: what if we wanted to build the current step and all the previous? You know, like we were just doing? That lets us get at the accumulation for the algebraic pattern we just found. And can we find a pattern for that? Always one degree higher, hmm. And the pattern for the first difference is one degree lower? And the number of differences to constant is the degree of the pattern? So much to notice. I tried a little, changing the patterns to bargraph, which also makes a nice connection to area under the curve. I want to explore this more.

One of the things about teaching calculus at college is that a fair number of students have had some calculus already. Not enough to test out, but enough to start a calc 1 class feeling like they've got all this already. Despite this really being algebra, it seemed familiar to no one. And their calculus connections helped intuition, but made them more surprised that it wasn't exactly the derivative and antiderivative.

The other thing that this helped with was keeping us away from rules to start. We started integrating for applications immediately, so we used tech to get answers. This helps put the emphasis where I want it, on the meaning of what is being added up or accumulated in the integral.

Writing Calculus
Waaay back when I was first teaching calculus, early 90s?, there was a book from the MAA on student projects. They were really artificial problems, but fun, and the first I had seen the idea of getting students writing in math. This was long before standards based grading was even a twinkle in my gradebook.

What I found this year was that student writing was amazingly synergistic with what I was doing for SBAR. It is generally hard in a 14 week college course to get students up to speed on the idea of SBAR, and getting them to take responsibility for what they have and need to show is a current challenge. I use a holistic grading scale that boils down to an A means you have given evidence that you can solve problems like this, and a B means you can solve this problem. So how do you show you can solve a lot of problems with one solution? You explain what you're doing, why you're doing it and evaluate your answer to see if it makes sense. (Why is it so hard to get learners to check their answers?! Hmm, if I want them doing it, it should be part of my assessment. So...)

What I noticed this year is that the writing, which was explicitly about putting words together to form coherent thoughts, helped the learners to make the jump to more complete answers on SBAR problems. (Also, the writings can be SBAR evidence.  Evidence is evidence.)  I ask for 6 or 7 writings, they get credit for completion, get feedback on the 5 C's rubric, and then pick 3 at the end of the semester to be their exemplars, which can be revised based on feedback. Everything I've read about teaching writing requires an opportunity to revise based on feedback. They post writing to the discussion board, and then give someone else feedback. Partly to just get them reading other people's writing, partly to get them evaluating in a way that gets them thinking about their own writing. The writing assignments (specifics on the course pages) come in a variety. Writing prompts, specific problems, choose from a short menu, make a miniguide to a topic (thanks Paula Beardell Krieg - great assignment), or open to their choice. Everytime I surveyed learners as to their preference, there was a great diversity, so I feel constrained to variety. What's a little weird is that you can't just give those all as choices each time, some people want the structure of a specific assignment.

Talking Calculus
My colleague Esther Billings has been using conferencing really effectively for years. So I've always meant to... but it's hard finding the time was my excuse. Despite seeing Esther somehow work it out literally next door. For the online course, I required two video conferences, but many left it until the end of the semester, limiting their effectiveness. In in-person calculus, I kept the idea of two interviews, and made one about differentiation and one about integration. I thought then people would start getting them done when we were done with the topic. Silly rabbit. So this past semester, I did two interview periods, done during that time they counted for an A, after a B.

So much good happens in these conversations. For one, getting to know the learners better. For two, many more people came for office hours afterwards than beforehand. Three, it is definitely my most accurate form of assessment.  Even with the writing that they are doing, it helped me know what they understand in a really specific way. Yes, that was an accurate mark in the grade book. But it was the best formative assessment that I have ever had.

Students came in, either with a problem they had worked on, a topic they wanted to talk about where I provided the problem, or something in between. Delightfully, a few students brought in something on which they were stuck - especially after the first experience.  The questions I asked further gave me an opportunity to model the kinds of explanations I wanted on SBAR problems. I gave A/B if they need my support, but were able to recap in a reflection. If I asked a question that they didn't know where to start, I'd share why I asked and my answer.

I will not teach without these in some form again.

While we're talking talking calculus, a frequent comment from learners was how helpful whiteboarding was. #VNPS if you're lingooey. These were very high engagement lessons, usually with a list of problems from which groups chose a problem or two. I get to watch and assess, interact with individual groups, and ask questions to prompt more in group discussion. We concluded with one group sharing in depth, me sharing comments about what I saw in common challenges, or each group giving a quick summary.

Next Time
So much went on in these courses, but I was not good at writing along the way.  I don't even know when the next time I'll get to teach calculus will be, but some of these lessons will come with me whatever is next. I'd love to hear your feedback, or questions, or how you think about these ideas. If I get to write more about this soon, I'll try to capture some of the learner feedback about these courses and features.