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
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. 









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:  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 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.

Friday, July 27, 2018

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.  

Sunday, July 1, 2018

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.





Friday, June 1, 2018

Math Teachers at Play #117

Welcome to Math Teachers at Play117!

Only three months away from the 10 year anniversary. Where does the time go? How much playful math is that we've shared? Thanks, Denise! (Gaskins, the founder & still organizer of this here carnival.)


117 is a pentagonal number...
... what is the next Math Teachers at Play that will have a figural number?

117 is the smallest possible length of the longest edge of a perfect tetrahedron with integral edge lengths. Its other edge lengths are 51, 52, 53, 80 and 84. Perfect polyhedra are sometimes called Heronian. The faces of these have a special property. Can you guess? (I may have made a GeoGebra applet to get that image, if you want to play.)

I haven't had the chance to make it in GeoGebra yet. But Simon Gregg, the host of last month's carnival, has been making some spiffy tetrahedra in GGB.

117 is a difference of squares and a difference of cubes. Two cool!

I called out on Twitter for any new bloggers who would be interested in being in the carnival.

Jill Price is just starting to blog at diymiddlemath.blogspot.com. First

Jennifer Gibson writes at thinkthinkmath.blogspot.com. Lots of neat blocks and visual explorations.

Erick Lee suggested a Joel Hamkins post with some awesome ways to use orthoprojections (direct views) with learners to amp up the spatial visual learning.

Paula Beardell Krieg suggested 5 excellent Math Play activities, so she is my unofficial cohost of this carnival. Suggestion number 1: play with rhombi. Find more on all of these plus, at her blog, bookzoompa.wordpress.com. (Where I am enthralled with her calculus post.)

If you have suggestions for next month's, send to the Math Mama, my bud, Sue VanHattum or share via the form at the Carnival home page.  Sharing in the carnival, or hosting, is a great way to increase connections in the #mtbos/#iteachmath community.

Do you know about Joseph Nebus' Reading the Comics posts? He reads them all so you don't have to! Plus he's hosting MTaP in 2 months.

Denise shared a riff on a Marilyn Burns post, math debate on adding fractions. I think she means this post about how 1/3 + 1/3 = 2/6.  Contrary to the rest of the internet, read the comments!

Paula's 2nd: Pop Up Cards

Harley Davidson engine: Milwaukee 117
Here are some of my favorite posts from the past month.

Fawn Nguyen is cleaning house. Throw it all out!

Another Simon Gregg post, this on multiplication and Cuisenaire blocks.

Dan Ashlock has a nice connection between a class of puzzles and proof in math.

Solenne Abaziou on the connection between persistence and open ended tasks.

Paula's 3rd: fraction books. I think trying to make her zero to one image might be how I Twitter-met her!


Jenise Sexton writing powerfully on 'I hate math.' 
"At some point, it isn’t our words that change the behavior and mindset, it’s our actions, passions and desire to do the right work." - Jenise Sexton
Mike Lawler et filii investigating irrationality, inspired by a Mathologer video. (Part 2) Plus pentagons, so on theme.


Junaid Mubeen thinks Erdös is wrong: we do know why math is beautiful. Joshua Bowman in another thread shared his answer.


Paula's 4th: Rotate things!

This month included NCTM National and that generated some amazing content.

Tina Cardone on learning the language of math.

Mark Chubb hits all the big ideas as he's putting together an assessment workshop.

Sara Van Der Werf also going comprehensive, on engaging students. Her metaphors are always to die for; Ticket to Ride here.

Laila Nur with a hilarious, dynamic, poetic, challenging Ignite: Stay in Your Lane.

Annie Perkins writes about Danny Martin's Iris M. Carl Equity Address, and so does Wendy Menard. Watch it here, Taking a Knee in Mathematics Education. Discuss with us on Twitter afterward.

Paula's 5th: her favorite, origami pocket books. "my personal favorite to do with everyone because you can talk about squares, rotation, triangles, scaling triangles,(third step) pentagons, and then slide 2 pockets them together to make hexagon (so 5 + 5 becomes 5 + 5 -4= 6, very cool), and arrange into non-regular octagon."



Renamed from this excellence to
Tennissine. Sigh.
Some 117 end notes. Eight years ago, after element 118, they discovered 117. It's superheavy and was phenomenally named, Ununseptium. It has been renamed Tennissine, which, sorry Tennessee, cannot compare.

In this month, the world's oldest human passed away, at 117 years of age. Tajima Nabi was believed to be the last person born in the 19th century. Rest in peace.












Wednesday, May 2, 2018

Geometry Snacks

What a sweet book!


These are two amazingly creative and fun thinkers about math. I feel a bit like that theme in The Shack, where everybody is one of God's favorites, but these are two of my favorite people on math twitter.


I thought of Ed as @solvemymaths for years, with great resources, dead clever problems and, of course, the math Mr. Men.  Check his blog for all this and insightful writing on teaching and curriculum. He's inspired some GeoGebra work from me, too.


Vincent, @panlepan, I think, came to my attention through his math art tweets, but maybe it was in a Simon Gregg discussion, or possibly a tessellation deep dive... He's introduced me to many interesting bits of math history, art, and people on Twitter. He's always willing to think aloud and do problem solving in public.

So I was very positively predisposed to like a book from the two of them together, and I was not disappointed.

For more of a preview of the book, check Ed's Twitter feed, or an Alex Bellos introduction. But probably you should just go ahead and eat it, er, order it. Snacks this good will make you hungry for more.

These puzzles and problems are so good. It is composed of five sections, with the somewhat surprising order of What Fraction Is Shaded?, What's the Angle?, Prove It!, What's the Area? and Sangoku. Each section is followed by solutions. Not just answers, but real solutions that guide the reader through the thinking of one way to solve each problem. And these problems tend to admit multiple solutions each, so knowing way to solve a problem does not make it worth much less for thinking about.

The problems are all visually presented, in beautiful black red and grey, almost crossing the border into pop art. Often they are stark in their simplicity. "This can't be enough information to solve it!" But then one relationship occurs to you, then "if that's true, this must be the case, and what if I..." In other words, the visual problem posing invites connections and problem solving. In my classes, one of my favorite definitions of math is "Math is the study of what else do we know?" and this book exemplifies it. I am not sure about this, maybe it's just me, but there is almost a sense of humor in these problems. Maybe it's whimsy? Maybe just the authors' sense of delight in the mathematics coming through.

While I can conceive of another book of problems that are this accessible and engaging, what would still set this book apart is the organization. The sequencing of the problems is intuitive, almost curriculum-like, but in a good way. The principle that helps solve one problem is often applied in a new context in the next problem, or needs to be extended in an upcoming puzzle. The fraction problems familiarize you with the shapes in the angle problems. The sequenced reasoning about angles is a lead into the idea of other sequenced reasons in the proofs. The proof reasoning prepares the reader for computation with measurements in the area section. The Sangaku problems are not the classical how do you construct the image precisely, but problems posed about measurements and relationships in those harmonious arrangements.

A nice indicator of how accessible these problems are is that a similar problem of Ed's blew up into an internet sensation, the Pink Triangle. What fraction is shaded?
People have proposed roughly a jillion different ways to think about this, which explore traditional geometric techniques (add extra lines, transformations, similarity), estimates, discovering properties of partitioning fractions, and so on. Such a simple prompt, and I would argue even noticing the conditions is doing mathematics. This book encourages Polya's first phase: understand the problem, so often neglected.

So obviously I'm recommending this book. For yourself, for your role as a teacher or parent, or as an appreciator of the mathematical aesthetic. You will snack to satisfaction and return for more.

It was that or end with Bon Appetit, which I'm sure has been used in a healthy fraction of reviews for this book already.