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<

Chris Emdin #HipHopEd

Last night I got to hear Christopher Emdin in my own back yard. He was brought in to GVSU by the Black Student Union for Black History Month, without the College of Education or science educators even knowing about it. This is not going to be as much a recap as a response. I overtweeted during it as I think about that as my note taking now. (Here's the thread.) Saying he is a dynamic speaker is an understatement. He's the best presenter I've ever seen. It's a performance, it's heightened prose, it's preaching. Here's his SXSW keynote if you want a sample. And you want a sample. (Also his book, of course.)

So my response?


This is my vision of education, expressed better than I ever could. It is about acceptance of all the varieties of giftedness and personhood and a chance for them to do deep, meaningful learning as themselves.

Dr. Emdin's emphasis on story telling as a way to share your own ratchetness and enter into your learners' world really resonates with me. David Coffey and I have been talking lately about just how can teachers share what they do. A Teach Off giving a lecture? No. Telling the story of what they do and why they do it and with whom they are doing it? Yes.

Part of making space for that story is accepting the pain of those rejected and making space for it and the healing. I love his idea of swag/cool/ratchet as the in-between of wound and healing. It makes sense to me and ties in to some pretty deep beliefs I have about redemption.

Chris warns against going into the hood (which can be anywhere that people are marginalized) armed only with the pedagogies of oppression. Dewey and Piaget and Vygotsky are still heroes to me, but that means we must contextualize them as well.

He offers no panacea, but inspiration. Progress is possible. Learning is local. And embrace your own ratchetness.

Saturday, September 9, 2017

What is Math?

It's the first question I pose for my capstone students, and then I ask them for the five biggest discoveries in math. A good way to start their blog (or reanimate if I've had them in class before.) Here's two of the previous classes responses: Winter 14 and Fall 15. The responses often run similar courses. Total aside: one student claimed firstmathblog.blogspot.com. How was that availabl

There's lots of math is everywhere and everything. I've felt and said that myself. One of my favorite teaching memories is a Kindergarten class that I visited weekly, and the first moment was someone getting to challenge me: there's no math in... bridges! Are you kidding?! Bridges are all about math! Next week, there's no math in Batman!

But is it helpful? If math is everything, then maybe people are already doing all the math they need to know. Me telling them that they're doing math either makes it irrelevant, or invalidates my line of reasoning because they very well do know what math is, they had a decade or more of it, and it is not that.

On one blog, I asked is the math the thing, the mathematical description of it, or the making of the mathematical description?

Lauren said that math is tool, but it's also an opportunity.  That's new to me, but also familiar. Isn't that the spirit behind #wcydwt and #anyqs? I had a little experience this week like that. I usually ask some kind of data question (mine or a learner's) on my sign in sheets and then make a display. Usually then shared on Twitter. For me it's a part of immersion, making the classroom mathematical. In some classes it leads directly into making representations, or becomes data for an activity. Almost always a chance to notice & wonder. Wednesday I shared one without the label, and it got some fun thinking. Chance for a joke, chance for some figuring.

As for the milestones, I was struck by a few things this time.

  • Numbers - lots of mention of numbers, sometimes specific like π, i or 0, sometimes familial, fractions or negatives. I am too eager to move past these, often, but now want to embrace them. The abstraction of quantity - that is a big freakin' deal.
  • Pythagorean Theorem. Of course. But it is a big deal. I love its history, and continuing story, and think it must be one of the first examples of hey, this means this AND how can we use that? Thank goodness right angles are useful. Or are they useful because of this property?
  • Patterns. So glad they think of this as essential. But when Eugenia Cheng says that math is the logical study of logical things, I think that math might have been born when we realized that there were patterns of patterns. When we were first meta.
  • Euclid. One of the things that comes from the course is discovering the people in many cultures who took that step of writing and organizing what we know. There's something about math that makes it naturally becomes a system.
I love teaching this course, and learners who are ready to think about the meta-patterns are the main reason.


PS>  I was listening to Anne Lamott's TED interview yesterday where she was so encouraging about just write. Just write. It really made me want to blog, to sit down and write. So when Lauren's post made me think that I wanted to think, I wrote. I can't worry about my blog being a bunch of first drafts. I can't be held back by the two open tabs on my Twitter Math Camp post and my summer calculus post. I just have to write. If you're reading this, thank you. That's already too kind.

PPS> If you don't watch the whole thing, you might watch around 25 min in (-15), where she talks about good writing is getting the reader to say "Ooh, tell me..." That set my teacher senses tingling. Her next part of that is that a confused reader is an antagonistic reader. That's exactly teaching, right? Where is the line between a learner wanting to know more, and not knowing enough to be interested. They need the beginning of a pattern, and to believe it's not just noise.

Tuesday, September 5, 2017

Top Ten Favorite Numbers

What numbers are the favorites of the people who have favorite numbers? I decided to ask on a lark, expecting a few responses, and it went crazy. (For my relatively quiet corner of social media.)

The idea had been bugging me since Joseph Nebus (who has a great weekly review of #mathcomics) linked to this comic from Cavna:

NO WAY are those the greatest, nor even the most popular. I can't even remember what tweet I saw that put this mild annoyance over the edge into asking out loud, but now I have a bunch of data on math teachers' favorite numbers.

This experience has taught me that our people care about numbers. They are more than quantities, they connect to ideas and stories.

Some things I noticed:

  • 18 is the first natural number not to appear.
  • Ironically, 2 is no one's 2nd favorite number, but is some people's 3rd or 1st favorite.
  • Having a symbol or name makes you a Big Deal number.
  • For about the top 20, number of mentions correlates to the Borda count (3 points for 1st, 2 for 2nd, 1 for 3rd).
  • No one loves negative numbers. Come on people. Transfinite numbers got more love.
  • 73 was the largest prime mentioned. Nope 163. Nope, 8675309. That number!
  • 6 was the last single digit to be mentioned. 
  • 42 did not show up for the first several hours, then stormed up in popularity.

I'm going to show this list to learners and ask them to think about why some of these numbers might be on here. In particular the larger numbers...

My top three (not included in the data) would probably be 4, 0, and Φ. 4 was my first ever favorite number. I explained to several adults how it was both 2+2 and 2 x 2. As a joke I'd get people to continue the pattern 2, 4, ... and if they said 6 I'd say 8 and vice versa. Little pain in the neck I was. (Except I was never little, as the family joke went.) 0 is the competitive spot. 10 - the first number to show place value? -1 - the huge discovery or invention? Something with a slick math history, like $sqrt{2}$, e, 1729 or 163? Something exotic, like Graham's number, a googol, or τ? In the end I have to go with 0. The digit that became a number, with cool Bahmagupta connotations. Փinally, the number about which I sometimes tell students that it was invented by my great, great, great grandfather. Even if there was no name connection, even if it wasn't so marvelously algebraic, even if I hadn't seen 3rd graders discover it through the amazing Fibonacci connection, I would have to pick it for the spiral connections.

The question elicited some great stories and tweets...

I can be pretty dense.

Bob Lochel shared this perfect kickoff to the top ten, from the show that made Top Ten a thing. Also, when asked early in my career for what I wanted to be like, I often cited David Letterman. I apologize to my students then, and to their grandchildren.

So from the home office in Grand Haven, Michigan,

Math Teachers' Favorite Numbers

10.  It's the answer to Life, the Universe, and Everything...

9. Not really...

8. Moving up one space,

7.  Lucky for us, lucky for you, this prime is one better than perfect. It's been up, it's been heavenly, it's been deadly, it is in 7th place with 7 points,

6. Often considered the first number, and still the...

5.  Pythagoras may have called this number the root of all evil, and it still gets a lot of hype. It is geometric and irrational, ...

4. Move on, folks.

Nothing to see here. Except the number that makes our place value system so craaazy good; Brahmagupta made it work. Often mistaken for a vowel, sometimes seen wearing a fashionable sash. Er, slash.

3. Fee or Fie? You won't fo-fum when you contemplate this 3rd place number, unless you go to the point where you're crazy and see it every where. Favorite of the Egyptians, the Greeks and God if you believe all the hype, it's....

2. Popular choice among mathematicians, who have denoted it after the greatest ever to be called one of their number. It turns up everywhere, and has all your base.
1. As surprising as Alabama football, we find here the number with not one, but two days dedicated to it. Half the number some claim it should be, but twice what it takes to be right. A great big slice oooooof - no. I hate pie jokes. And what's with everybody focusing on irrational, when it's transcendental?

If you want to dig more deeply, Carolyn Frye recommended the great RadioLab show on favorite numbers.

If you want to math more deeply, here's the data in a Google sheet. Thanks to everyone who participated, and sorry for clogging up your twitter feed.

I think sometimes I protest too deeply the stereotype that math is all about numbers. Maybe there are times to just go with it, and geek out.

Sunday, August 20, 2017

Wand Shopping

Preface: Justin Aion makes wonderful wands. I made a joke about wanting to write a blogpost, then this story came to mind. Encouraged by Justin and Audrey McLaren, I decided to post it. If you have comments or suggestions, I'd gladly take them.

Ollivander's Wand Shop

Charlie (probably not the one you’re thinking of, if you were thinking of any particular Charlie) had her heart set on Ollivander’s, though her family had always gone to the Wand Mart. She didn’t have anything against her older siblings’ wands, and they did well enough in school. But she wanted her wand from the same shop Mr. Potter had gotten his. As did many, as well as the many who wouldn’t get a wand from where you know, er, Mr. Riddle had gotten his. Wand Mart had the latest Wizards Instruments wands, all cleared for OWLs testing at the factory. But Charlie had her heart set, and she usually followed that.

She knew her parents were sure they were right. Mom went to Hogwarts with a hand me down, and was very happy to send her siblings with new wands of their own. Of course, Dad had no idea before Alpha went to Hogwarts. That was a surprise owl!  Wand Mart reminded him of muggle school shopping, with most things in one place. And it worked out for A & B, so why would they mess with a successful formula?

Which almost explains why she was in Diagon Alley without her parents, who assumed she was in her room at the top of the stairs packing for Hogwarts. Headmaster Granger had sent quite thorough list, organized by category and even with packing suggestions. (She recommended moleskin bags, but they were not required. The “quite simple” charm had given her parents fits. Charlie’s sister Trish had fixed it up, muttering about the arithmantic beading patterns.) Charlie mastered the packing charm in no time, only slightly illegal for a wizard of her age, and that gave her time to sneak out. If her parents thought they heard rummaging sounds, Charlie had no idea the cause.

It was her first time in the Dalley alone, and she was excited. She resisted the pull of WWW, and scouted out Ollivander’s. When there was finally a moment with an empty shop, she darted in. Mr. Ollivander’s son Mr. Ollivander was startled. He kept glancing at the door, waiting for the ubiquitous & reliably following parent or parents. Charlie blurted “Mr. ollivander my parents want to go to wand mart but i know that the wands here are better why else would mr potter have come here for a wand twice even and i thought that if you well i was imagining your father but you’re just as good i’m sure helped me pick out a wand then i could just tell them see its all done and if there’s any difference i’ll pay for it myself because i think it really matters-”
Handsome Kingwood wand, which Charlie
didn't look at twice.  (Justin Aion's, really.)

“Hello! What’s your name?”

“Um, Charlie. Carlotta if it bothers you to call me Charlie.”

“Why would it -”, Gerby thought and diverted, “So, you’re about to start Hogwart’s, Charlie?”

“Yes, sir.”

“I’ll tell you that many fine wizards started with a W.I. wand, and your parents have their reasons for picking that for you.” Charlie looked crushed. “But, if you want me to help you find a suitable wand, I’d be happy to help.” Sunshine returned to the shop.

“Yes! I need a Rosewood-Unicorn hair!”

“You do?! How do you know?”

“Rosewood is what Professor Sprout had and…”

“Mr. Longbottom, I suppose?”

“Yes! I’m a Gryffindor, too!”

“You know, I hope, that the wand chooses the wizard, but we can start there.” Gerby wandered out, muttered something, boxes fell, then “Ahah,” then “Bother,” then “Ahah” again. He returned with a reddish, short and rather thick wand with a plain handle, but intricate organic looking carving up the stem.

He held out the box, and Charlie oh so gingerly plucked it from the box. It felt… warm. Funny. Shaky. “Tell me what you think and/or feel,” Mr. Ollivander said.

“It’s funny.”

“Funny how?”

“Warm. Shaky.”

“Shaky or vibrating?”

Charlie thought about the difference. “Shaky. Do you want me to swish and flick?” She raised the wand as if to stop a marauding troll.

“No!” Gerby held up both hands. “Or, yes, but remember we’re in a wand shop. Lots of magic! Just gently, with your wrist” he demonstrated with his right hand “give a wave.” Though he didn’t have a wand in hand, Charlie thought she saw a fine mist of sparks trail his pointer finger.

Charlie complied. The wand felt as if she was pushing it. Sparks emerged, but big, with random direction and pacing. “May I?” Mr. Ollivander reached out for her hand. She went to put the wand in his hand but he said, “oh, no” and lifted his left hand to be under her right hand with the wand and held his right hand flat above hers. He leaned in and hummed, and the wand hummed back.

“The rosewood is fine, but I might like to try Japanese maple. It’s rare here, but would that bother you? But the unicorn hair is not the thing at all.” Charlie grimaced. “Sorry,” he said. “That doesn’t mean you’re not a Gryffindor, though I encourage you to be open about your house.”

He reached around under the counter, muttered what must be an inventory spell, and a foil covered wand box slid into his hand. When he pulled the box inside out, well, it wasn’t a lid, but more of a drawer, Charlie was confounded. “That’s not even a wand!” It was curved. Not like a bow, but, maybe a bit of a spiral? “How do you even know what direction the spell goes?!”

Gerby said, “I know. Unusual. Every wand is different, and I won’t make you try it. Maple, as I said, Japanese varietal, with the ruby red leaves, though the wand is quite blonde. It has a kirin scale edge core. Quite lucky, said the man who procured it but still had all his digits.”  Charlie set the rosewood wand in its box, and reached for the maple.

“Maple. Meh.” But when she touched it, it felt less warm than the rosewood. But comfortable. Picking it up - “ooh! This vibrates!” She swooshed it without being prompted, and it left no outward visual but instead made a sound. A fading single note?

“Ah, yes!” said Gerby. I had a good feeling about it. “Make a large circle, as perfectly round as you can.” She moved the wand as if drawing on a whiteboard, and made a shaky ellipse at best.

A disappointed, “oh.”

“Keep trying,” Mr. Ollivander said.

Charlie traced it again, and again, and again and soon the wand was drawing the circle by itself. Not exactly. Together? Somehow the circle just made sense. Mr. Ollivander rapped a knuckle in the center of the circle, and a gong sounded. Charlie giggled and Gerby laughed.

He leaned into Charlie. “The wand chooses the wizard, but the wizard learns the wand. How it is now, is good. But you will, or can, learn it. It’s a tool, which is how those W.I. … wands are made. But the understanding that works the tool is why it is your wand. I’m going to set this wand aside, and if you can convince your parents, fine. If not, know that it’s here waiting for you when you get a chance. Very nice to meet you, Charlie.”

“Thanks, Mr. Ollivander…” but he was already in the back of the shop. What an odd fellow. But - wow! What is a kirin? She was ready for whatever wand her parents picked, but she was already making the argument in her head why this was the wand. She wanted to reread Professor Granger’s list as well, with an eye for this learning idea. Somehow she had thought that Hogwarts would make her a wizard, but now she thought it might be more of her job.

Now what are the chances that Mom and Dad noticed her being gone?

Sunday, August 13, 2017

Sympathy Note

Trigger warning: learning styles will be mentioned.

I don't mean to mock trigger warnings or learning styles. I have colleagues who lose their mind at the mention of learning styles, because of the lack of research. And we've all seen people flip their lid at trigger warnings. (How do you warn them?) I like to think about learning styles as a framework that teachers use to make sense of what they see. And to think about what they might do in response.

I love to read, but don't have as much time for fun reading as I like. One benefit of driving to Twitter Math Camp (about 12 hours each way) was the chance to listen to a few books. I mostly read mysteries and science fiction or fantasy. On the way home, I listened to a Hieronymous Bosch mystery by Michael Connelly, usually quite good. This one was read by the actor who plays Bosch in the streaming series (thumbs up for that, too), so it was an interesting experience. I was using the Libby app, which connects to public libraries and was overall great. Through Libby you can search catalogs and make requests, and I requested the next book.

Turns out, I requested an audiobook. I don't have time to listen to a book... but, I had it, Titus Welliver (the actor) ... okay. I'm listening to it. But I have to be doing something else at the same time. I've never been able to just sit and listen. In school I was a doodler. For which I, and then later my kids, received plenty of disapproval from teachers. (Until grad school, when profs wanted copies of my notes!)

Reading a text, I have great recall and comprehension for names, plot, etc. I notice small details. I concoct theories. Listening... I'm still enjoying it. But will suddenly be 'wait, what?' The interface is handy, but it's been hard to back up. Whereas in a text, it's easy. I'm not following as well. Reading I can visualize the action like my own movie. Listening... not so much. "Wait, what?"

Is it a learning style? Am I a visual learner? Textual? Is that a thing? Inexperience vs experience? I don't know.

What I do know, is I'm committed to is remembering to say and write important points for my students. For having learning opportunities in a variety of modes, including motion.

Now I've got to get back. The bad guys have Bosch and the doctor cornered in the doc's office after hours.

Monday, July 31, 2017


(Have to read that post title Sixth Sense style.)

First do I write about: #iteachmath or Twitter Math Camp 17? ...  have to get the hashtag stuff off my chest.

I love that Dan is thinking about inclusivity, and it befits his problem solving orientation that he's willing to rethink any aspect of the situation.

I started blogging April 2009 with a 50 word post, just sharing a resource I liked. I thought I would use the blog to share the stuff I found around the web that I like or was thinking about how to use in my class. Ten posts later I finally shared an first activity that I did back then. A math game, of course.

This was a long time ago in internet years, and I understand that the world is different. I was inspired by what I was finding, and just wanted to join in.

First tweet, 2010.  (Find yours.)

I was at Maria Anderson's tech camp (@busynessgirl) and she suggested Twitter as a way to connect with student teachers. That's been great, but I wound up liking the math twitter/community plenty for myself.

When it was time, 2013, the community wondered how to refer to itself. They came up with Math-Twitter-Blog-o-Sphere, and I liked the silliness of it right off.

Is #MTBoS a barrier now? People are hurt by this suggestion, because they work like hell to make the community inviting and inclusive. And are always looking for more ways to do that better.

From where do the hard feelings come, then? I have theories. Basically this list is the consequence of people new to twitter don't know how it works yet.

  • some of the most followed people are friends. They take math with anyone, but also talk real life to each other. There's shared experiences, so they refer to things that not everyone was a part of. But because of the way Twitter is, we see some of those relationships. That could make you feel like (Justin Aion analogy) being at a party by yourself. As Justin says, at a party, they'd see you standing alone and approach you, but on Twitter, you can be invisible if you want.
    Remedy: new users can let people know they are there. 
  • People say 'Include #MTBoS and get your questions answered.' Sometimes? More people watch that tag and respond to new people than I think would ever happen in most communities, but not everyone gets responded to.
    Remedy: tweet @someone. If I see someone asking for a resource, I may not have a response, or know others who have better. But if you tweet @mathhombre, I reply. (I think?) I challenge you to find a community with a higher response rate.
  • #MTBoS is a community. We have relationships, shared values, and even meet when we can. If there's an in, there's an out.
    Remedy: come on in!
What I notice about these problems is that the remedies are all putting the burden on the people who feel outside, which is usually the hallmark of an exclusion problem. But that's where we need to see and popularize the efforts of Tina Cardone, Sam Shah, Lisa Henry et al. There wasn't an intention to brand anything, but having a name is part of making you a group, a tribe or a family. I would rather reassure people that this family wants you and is inviting you in, than worry about what the name connotes. 

It did feel autocratic, and like a dictate, but that's probably mostly because of his position in the community. He is the introduction to the MTBoS for most math teachers. He is going to hear the complaints the most, maybe? 

The timing was really unfortunate, as it distracted from the amazing keynote by Grace Chen. (Pts 1 and 2) (Which I will talk more about in the next post.) 

Dan is trying to connect people with #iteachmath. Great! I don't see how that solves any of the three bullet points above. Hopefully, the new tag will be successful. If it is, within a couple years people will feel like it's cliquish or there are rockstars and arguing for #mathlearnersunite. Great! 

I don't think of this as particularly important - or coherent - post, but this is a blog. I can work out my thinking here, and live long enough to be embarrassed of it. I can give a first take. No one else may read it or maybe it becomes the rare post that gets a comment. One of my Twitter Math Camp take aways, from Carl Oliver's sweet keynote (Pts 1, 2 and 3), is that it's important to push send. 

If you hear about the #MTBoS, my guess is that you will be curious enough to investigate. If you do investigate, you'll find things that will help your learners. If you value that, I encourage you to join in. The more you participate, the more you'll feel a part. If you can get to a Twitter Math Camp, you'll be stunned at the welcome. But nobody's going to make you.