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 First

Jennifer Gibson writes at 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, (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.

Tuesday, March 13, 2018

Walk the Line

I keep a close watch on this number line 
I keep my eyes wide open for the sign 
Making sense, connections all the time 
For number sense, I walk the line

Apologies to Johnny Cash.

At Math In Action two weeks ago, I presented on number talks in the middle grades. (Here's the handout/resources, including a link to the slides.) I was quoting Pam Harris, then the keynote speaker was quoting Pam, so I made a comment on Twitter, badda bing, she turned out to be coming to my neighborhood the next week. Muskegon Regional Math Science Center let me crash, thanks Kristin Frang, so I got to crash for 1.5/2 days. 

Pam was presenting on secondary number strings (thread 1 and thread 2 for my notes). Mini-review: I wouldn't hesitate to bring Pam in to work with teachers. Great energy, hilarious, solid ideas presented in a way that invites teacher access, and great modeling of instruction. All building on a great central message about giving learners an opportunity to mathematize.

I'm teaching Introduction to Mathematics this semester, a gen ed math class with a lot of freedom. I'm using Anna Weltman's This Is Not a Math Book as a text, mostly as a resource for learners and introduction activities.  In my head it's about redeeming mathematics for this successful students who have (mostly) learned to dislike mathematics. While we're mostly doing math and art, emphasizing problem posing and solving. But we take some time to redeem arithmetic, and we need to do algebra before we do patterning.

So, Pam's problem strings (modified because I can't help), followed by some clothesline math... rare day when I didn't have a way to take pictures. Sorry! The theme of the lesson is what else do we know?, which is one of my main understandings of math.

I drew a line, put on a hashmark, and labeled above it x, and below it 3. What else do we know? Some discussion about putting on a scale, vs knowing left or right. Finally someone shares -x is -3, and says it would go somewhere to the left. The someone said that means we know zero, which I encouraged as a good mathematician question.

First I did Pam's coordinates problem string.

(-2,5) show in graph, table, function notation
(1, __) which first brought up approximating with the line, then the idea of slope.
( __, 0) which brought a lot of people to a halt as they tried to remember an algorithm, then a remembered algorithm for solving equations. I made a bit of a joke how I was not interested in a memorized method, sorry, but only making sense.

Next, x is -2. Where is zero, left or right? What is x? Some discussion but pretty quick.

x-4 is 6. Is x left or right? What is x? How do you know? Quick discussion.

x+4 is -6. Heated discussion. -10 and -2 mentioned frequently. More and less start getting used more than left an d right. Interesting symmetry comparison between the last and this.

Then I introduced the clothesline, strung across the front of the room, with just an x in the middle. What else do we know? I had cards to ask them about. 0, -x, x+3, 5. We know where zero is, then dissuaded. What if it's negative? We know where -x is, then dissuaded. Then x+3, that has to be to the right. We don't know how far, but definitely right. What else? x-3 we would know. Excellent! Finally I put 0 up (left of x-3 by less than the distance to x) and ask what x could be? 

Then I gave them cards. Make some cards, figure out the order you're going to reveal them and the questions you'll ask. Two really interesting situations came up. 

One group had a couple of variable expressions, and then x+200. Really nicely subverting the sense of scale, and a great numeracy discussion of what x could be then. They wanted to just toss off a big number, but other learners argued for more precision.

Another group introduced a new variable, g, and then blew our minds with g^2. Has to be to the right, because squares are bigger. But what if it's negative? It's still to the right? Always? Then a lot of discussion if placement of it was setting the scale. Not until 0 is placed. It was amazing.

So that's my story. Thanks to Pam Harris, Chris Shore and these great learners.

Saturday, March 10, 2018

Let's Discuss Professional Development

One of my favorite math ed profs is Sam Otten at Missouri (and the Lois Knowles Faculty Fellow). His research is interesting and situated, he holds teachers in high regard and listens to their ideas, and he illuminates research through the Mathed podcast. He has definitely enriched my practice. In addition, he's just a lovely and creative guy, as well as a world class expert on the DC Comics film universe. Beyond that, he's a GVSU grad, so I knew him when.

He is a part of the team that produced some new professional development materials, and I had a few questions for him about it. Mathematics Discourse in Secondary Classrooms, MDISC, is based on research and developed with teachers in the field. I'm a big believer in the importance of discourse in learning, and know that secondary mathematics has been one the places where traditional teaching has included the least discourse. I also think people need support to make changes, so something like this project is needed.

What inspired these materials? Was it an idea you wanted to develop or a response to situations you saw in the classroom?
The MDISC materials came from a group of math education scholars at Michigan State University and the University of Delaware, led by Beth Herbel-Eisenmann, who were passionate about the role of discourse in math classrooms. We all believed that there was profound value in students discussing mathematical ideas and building meaning together as a community. So at its core, MDISC is a set of professional development materials that are intended to help teachers increase the quantity and quality of discourse in their classrooms.

As we set out to create these materials, we tried to draw on other work that already existed in the math ed literature. Some of that work was Beth's own research with Michelle Cirillo. They had worked for years with a group of secondary teachers, examining discourse patterns and power dynamics. We also drew on the work of Chapin, O'Connor, and Anderson, who wrote a great book called Classroom Discussions that focused on mathematical discourse at the elementary level. They had some really amazing results with respect to student achievement scores that stemmed from a new emphasis on discourse. With MDISC, we tried to take some of those ideas from the elementary level and reinterpret them in ways that made sense at the secondary level -- focusing on middle school and high school classes.

Overall, the MDISC PD materials equip secondary math teachers to think about discourse in productive ways and it also provides them with specific tools for changing the discourse in their classrooms so that it really empowers students. It helps move us beyond teaching-as-telling.

What are some of the different ways these materials might be used? 
The MDISC materials include a physical facilitator's guide and then digital versions of all the participant materials as well as sample videos. It could be used by a teacher leader, facilitating sessions with secondary math teachers, or by a PLC of teachers who want to work through it on their own. It could also serve as a textbook for a graduate-level course, so a teacher educator going through the activities with practicing teachers, for example in a Master's course or an Ed Specialist course. The materials are designed to be a year-long study, with connections to everyday classroom practice, but it's flexible -- so with some adjustments, it could also be used in one semester. Or people could select which components they want to focus on.

There's also an optional follow-up where teachers can be guided through some action research, if they want to continue making purposeful efforts toward shaping their classroom discourse. There are several different options, and the MDISC team is very open to communicating with people if they have questions about enactment. We've also enacted the materials many times in many different settings, so we have a lot of experiences to share. 

As you piloted these materials, what were some of the changes you saw in classroom discourse?
We have piloted the materials and had others pilot them in both Michigan and Delaware, with several different groups of teachers. They have been very well received thus far, with some teachers willingly joining in for a second and third year because once they start, they don't want to stop thinking about their classroom discourse. Some of the teachers have called it the most important learning experience in their teaching career, and this even came from a 30-year veteran.

The most visible changes have been the number of students talking in class. They open up more and share their ideas, and the great thing is that they're sharing mathematical ideas. I think this comes from MDISC's dual approach of not only providing insight into the nature of discourse but also providing specific moves for teachers to use. For example, MDISC develops six teacher discourse moves that include inviting student participation and also probing a student's thinking. These are concrete ways to get the discussion going and keep it directed toward important mathematics.

Another big change that is noticeable is that more wrong answers come to the surface -- it's not that MDISC leads to student confusion (just to be clear), it's that an increase in discourse helps more student ideas to come to the surface. And of course some of those ideas are incorrect or imprecise, and that can lead to good discussions and good learning opportunities for the group.

What’s one feature of these materials or an example experience that might help teachers understand how they will support their teaching?
One feature of the MDISC materials is that they are practice-based and case-based. So teachers will get to make constant connections to their own instructional practices and their own students. Those connections are built right into the materials. And there is also the chance to see and discuss detailed cases of other teachers. Rather than lots of little isolated examples, MDISC instead is built around larger cases of real teaching. So for example, when you're learning about the transition from small-group work to whole-class discussions, you can actually see a middle school teacher as she circulates among her students and selects certain ideas to be shared later, telling the students that she'd really like them to bring it up in front of the whole class. Then you can follow the case to see how it played out in the discussion.

Another important feature is that the MDISC materials integrate an emphasis on equity. Powerful discourse means that everyone has an opportunity to be heard and to learn from the conversations. So there is a lot of attention paid to how teachers can use a discourse-based approach to reach more students, including those with traditionally marginalized backgrounds.

What movie would you like to see DC make next?
Great question! When I'm not working in math ed or spending time with family, I love watching and analyzing DC superhero movies. I really loved Man of Steel and then I thought Batman v Superman took it up another notch, with great themes about immigrant experiences and the danger of overt masculinity having to face feelings of powerlessness. So although I'm excited about Aquaman and the Wonder Woman sequel, I would really like to see another Superman solo film make it onto the slate. And it would also be great if the Cyborg standalone would get the green light because I thought he was a really intriguing character in Justice League and I think his story could be used not only as a commentary about race in modern society but also about our increasing dependence on technology.

(Back to me) There's so much promising here. Use of real classroom discussions with connections to your own. The focus on equity. And the idea that in running it with teachers there's a measurable change in the number of kids participating in discussion, as well as the frequency and quality of discussions - that's a dream. I'd love a chance to work through this with teachers.

Find out more:

Funville Adventures

This has been a long time coming. Funville Adventures by A.O. (Sasha) Fradkin and A.B. Bishop is full of fun adventures.

Sasha is a Twitter acquaintance, an elementary math enrichment teacher with an amazing personal math journey, and I probably heard about the Kickstarter from there. I love to encourage these passion projects in general, but this book is especially delightful. (Sasha on Twitter & her blog.)

As a story, it may remind you in flavor of The Phantom Tollboth or Dragon Tales. Emmy and Leo are kids transported to an allegorical land, Funville. Kids in Funville each have a special ability. The story makes sense and is enjoyable without even knowing the math in a formal way, because the math is the idea behind the people they meet, but not how it's discussed.  These quirky characters are brought to life in quick vignettes and charming illustrations.

Part of the charm is that, since there are mathematical ideas behind the kids of Funville, the way they work and interact is surprising but logical.  Readers can predict what's going to happen or wonder what would happen.
“Yeah,” said Harvey as if it was the most natural thing in the world to have a power. “My power is to halve things in size.” 
“Halve?” inquired Emmy. “But he is no more than a tenth of what he was!” 
“That’s because ...
What do you think happened? Small mysteries like that. Big mysteries, like how will Leo get back to full size? Harvey has a brother Doug - will that relate? And the biggest: how will they get home?

Want to know more? There's a whole Funville Blog Tour, with lots of perspectives. As with a couple of those, you might find yourself wanting to make your own Funville characters. Even now, whom do you think Emmy and Leo might meet?

I have one here... Fan Funville Fiction!

Dylan's Dangling 

Emmy and Leo worked hard all week to get done with school work and chores, so they would have an afternoon free to visit their friends in Funville. They had developed the habit of tucking things into an old rucksack that would be interesting to see just how their friends' powers would work on them. This time the rucksack held a tiny Ant Man action figure, an elephant toy, a stretchable rubber snake, and an assortment of snacks.

Emmy was particularly interested in Fay's and Randy's powers and how they interacted with other powers. So she was always glad to see them at the other end of the slide down the Thief. But she and Leo were both surprised to see someone new in the playground.

He was sitting on one half of a see-saw, but was up in the air instead of down on the ground. Maybe Heather had been here? He had on a shirt that was way too long, but otherwise seemed to fit him well. Adding to his stretched out appearance was a very tall cylinder of curly black hair.

Leo ran over to him immediately. "I'm Leo!" he announced, and the boy answered him. "Oh, I know. I came down here to meet you two because I was so interested in the stories that everyone told about you. People without powers, but you're fun anyway? And Pencilvania? Wherever that is!"

"Pennsylvania," Emmy corrected, "but that's a good synonym! You probably know I'm Emmy, but who are you?"

"Dylan," Dylan answered, "and I'm stuck up here. I was playing with- "

"Heather?" Emmy interrupted.

"Exactly!" said Dylan. "But her mother was calling, and she hopped off, and didn't notice that she made that end so heavy, and ... long story short, here I am."

"How can we help?" asked Leo.

"Do you have anything small enough to stand up under the other end?" wondered Dylan.

"Sure!" said Leo, and rummaged in the rucksack for a perfect skipping stone he was hoping Cory would be willing to use his power on. He slid it under the down side of the see saw. "Like this?"

"Mmmmmm hmmmmm," said Dylan, who was already concentrating. Slowly the see saw seat lifted up, pushed by the stone, which was growing. But not getting like a bigger stone - more like a tree. Once the see saw got level, Dylan hopped off.

"So your power is growing things?" asked Emmy.

"Not exactly..."

Tuesday, January 30, 2018

Book Club Fall 17

One of the many fun parts about teaching our capstone course, the Nature of Modern Mathematics, is the reading. Instead of me mandating a book (most instructors choose the excellent Journey Through Genius, by William Dunham), the learners can choose a book. Comes a day, we then have a book club class where people get to discuss with others reading their book, then share with the class what they thought. I try to keep notes, demonstrating poor steno skills.

The possibles this year are on this Google doc, which gets revised year to year.

How to Bake π, Eugenia Cheng

Connected all our classes, abstracted ideas but then super concrete accessible examples. Everything came together. Author is a little scatter brained: 15 subsections in each chapter. Even the toughest of concepts can be broken down. Two parts: what is math? What is category theory? Good connections.

Is God a Mathematician?,

History of math, Newton, Aristotle, Descartes… not proving that God is a mathematician, but looks at the beliefs of all these people. How does math intertwine with science, physics, biology… Example, knot theory. Is math discovered or invented?

Journey through Genius, Dunham

Goes through theorem by theorem. Some was over my head, but the writer makes it very understandable. Example, quadrature of the lune. Most interesting was about Archimedes proof of the area of the circle.  Recommend it because it ties into a lot of things throughout our math classes, but you learn something.

The Teaching Gap

Compares German, Japanese and American lesson plans and how we teach. But mostly contrasting Japanese and American. In Japan they encourage more struggle. “US teachers are just not smart enough to teach the way researchers recommend.”

Joy of X, Steven Strogatz
Brian, Angel

Not especially challenging, written for a general audience. Longest chapter, 10 pages. Covers a lot of different areas of mathematics. Example, dating life. First half, playing the field, 2nd half find someone better than the first half… Snell’s law, ‘light behaves as if it was considering all possible paths … nature seems to know calculus.’ The focal points of the ellipse of Grand Central Station. Infinity. Is it odd or even? Recommend it. Even makes Hilbert’s Hotel understandable.

Fermat’s Enigma, Simon Singh
Proof of Fermat’s last theorem. Left so many conjectures, but the last one was a doozy. Made it as understandable as possible.

Genius at Play, Siobhan Roberts
Kelsey, Tony
More of a biography. He hasn’t published a lot, but his ideas are everywhere. He doesn’t like being known for the game of life. It’s hard to read, because the math problems are so hard. But you get to know his personality. See and say sequence from a student was frustrating, but then a source of great mathematics.

Quite Right, Norman Biggs
A history of time, … money. But 70% math. Start with caveman, then follow it forward. How to divide evenly, then follows through other cultures to modern math. Gives a sense of where math came from, but not all of it.

Finding Fibonacci, Keith Devlin

Story of Devlin finding the history of Leonardo of Pisa. Not recognized for his accomplishments. He didn’t really discover anything, but introduced real arithmetic and algebra. Son of a merchant. Really started a revolution. Only 14 copies of Liber Abaci in the world. Fibonacci sequence was just a puzzle in the book. Golden ratio, limit of the Fibonacci sequence. Does appear in nature, but not as much as people say.

e: the Story of a Number, Eli Maor
Most of the chapters don’t even mention e, but then it brings it back. Funny stories about many mathematicians (Bernoullis, Napier, …) Just a general  history, with some more focus on math. e is discovered, transcendental number…

Math Girls, Hiroshi Yuki
Math, but always in a story. Girls solve problems that have an easy access launch.  Someone who read this for a second book said it's mostly about the math content, but that content is deep and interesting.

The Man Who Loved Only Numbers
About Paul Erdös, an interesting, different, cool guy. Never owned many possessions, traveled from host to host working on mathematics. Took a lot of espresso and drugs, proved that he wasn’t an addict by stopping, but his math stopped, too. So he started back up. I really liked the prime number section.

I was able to entice a couple futute teachers to read José Vilson's This Is Not a Test for their 2nd book, and they were captivated, with strong recommendations.

...much time passes ...

I just now, in the next year, realize I never pushed send on this one! So >push<