Showing posts with label story. Show all posts
Showing posts with label story. Show all posts

Thursday, December 3, 2020

Fractions vs Decimals

 From the things you forgot you wrote file...

Fractions vs. Decimals

The Battle of the Century



Ringside Announcer (RA): Welcome ladies and gentlemen to the Battle of the Century:  Fractions vs. Decimals!

Old Man Fractions has been king of the hill for so long he can remember the pharaohs.  But relative new-comer decimals has been rocketing through the ranks past previous contenders like Mixed Numbers and Percents, buoyed by the rise of science and handheld technology.  Tonight they settle the issue once and for all, mano a mano.

Color Commentator (CC):  That’s right, Jim.  And they have both clearly prepared.  Fractions has developed his upper body so much he looks positively improper.  Decimals has emphasized speed work, and is awfully quick to the point.  Hey, looks like they’re ready to start.

>ding<

RA:  They come out swinging!  Fractions looks like his strategy is to corner decimals and work his weaker visual representations.  Oh there’s a pie model and a fraction strip combo!  Decimals finally lands a 100 grid haymaker and gets back out to the center of the ring.

CC:  Looks like that speed work is paying off, Jim.  Decimals is coldly calculating without having to hit any special menu buttons on the calc, if you know what I mean.

RA:  Not really, Howard, but I’m used to it.  Oh!  Decimal made a rounding error and Fractions lands an uppercut.  

CC:  That’s exactly the answer, Kid Decimals!

RA:  The traditionalists are out of their seats, cheering on Fractions.  Even the French are into it!

CC:  He’s certainly got that je ne sais quoi, eh, Jim?

RA:  Huh?  Back to the action, Fractions is pressing his advantage.  But decimals sees an opportunity and – oh! The referee calls time!

CC:  I don’t think it was intentional, but that was definitely below the vinculum.

RA:  The referee gives Decimals a warning and they’re back in.  Fractions still looks a little wobbly, and Decimals presses the advantage, really working over Fraction’s arcane and misunderstood algorithms.  

CC:  Invert and multiply that!  Whew!

RA:  Fractions gives a nice example of unit fraction multiplication and is back in the fight.  Oh, and lands a nice left hand on a complicated long-division problem.

CC:  Decimals looks like he doesn’t know if his point is going left or right, Jim.

RA:  It’s back and forth at this point folks.  Fractions simplifies nicely, and catches Decimals a good one.  Decimals lands a nice easy comparison, but Fractions hits a unit confusion counter-punch.  

CC:  That’s half of something, alright.  

RA:  Then Decimals comes right back with a repeating combination!  Oh, and a non-terminating, non-repeating wallop!  Fractions has no answer for that.

CC:  Right in the Pi hole!  Practically transcendental ring work, Jim.

RA:  They’re really taking a beating out there.  Howard, I think the crowd’s getting confused about what’s important here.

CC:  I think you’re right, Jim, there’s kind of a baffled silence.   Not that unusual at a rational battle like this one, though!

>Ding<

RA:  That’s time.  The fighters move to their corners.  The judges communicate their decision to the ref.  It’s pretty close on my scorecard, Howard.  What do you think?

CC:  Did you double check your answer, Jim?  Nothing would surprise me –

RA:  The ref is ready and brings both fighter’s to the center of the ring.… he pulls up both fighter’s hands!  It’s a draw!

CC:  The judges have called them equivalent!  Oh the equality!  Looks like we’re in for a rematch.




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?


Monday, May 16, 2016

Quick Story over a Long Time

I'm at a church meeting for a couple of days. If you can't find teachers to hang out with, church meeting people are a good substitute. Selfless, friendly and some pretty good stories.

But what if you meet a teacher at one of these meetings? Boom!

John had many stories, most of which are not mine to tell. But he did tell about teaching in England in the 60s on a Fulbright year. On the ship over, they showed To Sir With Love. (One of my all time favorites. Must see teacher movie.) When he gets to East London, that's the school he's teaching in! Were they really that cockney? I ask. "Oh, yes."

At some point he became the first dean of students for a nearby high school. Since one of our greatest challenges with student teachers is building classroom culture, I asked what he advised. He said there's no short answer, but let him tell a story.

He's teaching chemistry, and a student came in to find out if he really needed the class. "What do you want to do?" "I'm going into biological research." "Well, you're almost certainly going to need chemistry." So he takes the class, and does brilliantly. One day they're working with condensors, and John warns the student that this is expensive glassware. The student accidentally lets it roll off the table and it shatters. The student is broken up about it and worried, so John consoles him. If you do what you're planning on, you'll pay society back many times over.

At this student's 50th reunion, John gets to see him after his PhD in biology, and a career working in medical science all over the world. He got to collaborate with two Nobel prize winners. John shares the story from high school, and all are amazed and touched.

It turns out John only became a teacher because of getting a second chance when he was in high school. But the moral is the same, I think. We're investing in people. Teach your students, love your students, and you have no idea where your adventures will lead you or them.

Good night!

ps. if you ever get to meet John, ask him about the dynamite.

Sunday, January 10, 2016

Story Teaching

Quick post thinking about Dan Willingham's post on The Privileged Status of Story, which I got to via Dan Meyer's post Study: Implicit Instruction Rated More Interesting Than Explicit Instruction

What constitutes a story?
"The first C is Causality. Events in stories are related because one event causes or initiates another. For example, "The King died and then the Queen died" presents two events chronologically, but "The King died and the Queen died of grief" links the events with causal information. The second C is Conflict. In every story, a central character has a goal and obstacles that prevent the goal from being met. "Scarlett O'Hara loved Ashley Wilkes, so she married him" has causality, but it's not much of story (and would make a five-minute movie). A story moves forward as the character takes action to remove the obstacle. In Gone With the Wind, the first obstacle Scarlett faces is that Ashley doesn't love her. The third C is Complications. If a story were just a series of episodes in which the character hammers away at her goal, it would be dull. Rather, the character's efforts to remove the obstacle typically create complications—new problems that she must try to solve. When Scarlett learns that Ashley doesn't love her, she tries to make him jealous by agreeing to marry Charles Hamilton, an action that, indeed, poses new complications for her. The fourth C is Character."

At the end, Willingham challenges us to incorporate these C's into lessons. In particular, the most important C, Conflict. "Teachers might consider using 10 or 15 minutes of class time to generate interest in a problem (i.e., conflict), the solution of which is the material to be learned."

I think this is compatible with several MTBoS approaches, in particular & obviouly, 3-Act lessons.

Character - my biggest question after my first read was who are the characters? Not in a heavy handed Life of Fred way, but in the story. I think it must be teacher and students for us. We resolve the conflict, after all.  Probably one of the inherent advantages of inquiry teaching is making the students the central characters. Not that we teachers can't be involved - I think we have to be ready to jump in, too. But we can't be Deus Ex Machina everytime, and let the students know there's always an out.

Math lessons are well set up for storytelling otherwise, I think.

Causality - why does this work is a great basis for an investigation. Add up the digits - if that's divisible by three the original number was, too. What? How could that work? Look - these three centers of a triangle are always on the same line. Why on earth...? Of course, if we make it out that knowing the fact is more important, we're killing the story. This is historically a great spark for mathematical developments as well. While I was writing this Sam Shah posted this image which got my mind wandering, making me go off and do some GeoGebra.

Conflict - I have no idea if this is unusual, but I try to get good math arguments going every chance I get. I usually refuse to be the authority. ("Is this right?" What do you think? "I think so..." Well, let's ask the class!) Plus anytime I ask for an answer, I always ask if there are any other answers. And when the students propose answers, there's a chance for a math argument. It also makes me think of Chris Luzniak and his Math Debates.  Even whether a particular topic is math can be a great argument. There's a course I start with Sudoku, and the last question is, were we doing mathematics? I have never had a class agree on this answer.

Complications - is there anything more mathematical than this? Oh, that worked. What if we added this? Could we do it still if we didn't know that? Messing around with conditions is prime mathematical behavior. And if the problem is problematic enough, this happens by itself. I could solve it if I knew that, now how do I find that out? Or you're trying all the cases and get to one where the freak out lives. Or you're practicing the very mathematical habit of mind of trying to find counter-examples to your own idea.

Where I think these C's might be helpful to me is in being more intentional about the type of math the students are working on, and using this structure to help design how I'm going to try to get my lead characters to find the problem.

For my first math for elementary teachers tomorrow, I want to create the conflict for my students between what they know about elementary mathematics and what they need to know. I loved Graham Fletcher's progression of multiplication, so I'm going to try to use that in contrast with their native ideas about teaching multiplication. (Also such a nice synthesis of understanding to model for them.) In the past I've mirrored mathematics development in children and schools, starting with number concept and building up. This will essentially be going in reverse, but will hopefully be a more obvious need to know that will motivate the deconstruction on supposedly simpler topics to follow. Wish me luck!




Thursday, July 14, 2011

Storytelling

Not me, sadly.
(by Travel Aficiando @ Flickr)

We just finished up our Vacation Bible School, which had Joshua as a theme.  I'm the storyteller, and my thoughts were much on teaching throughout the whole process.  Dave Coffey and I often talk about teaching as story telling (EDIT: turns out he was writing about storytelling at the same time as I was - shocking), and love drawing lessons from Harry Potter as we both love the books.  (So much so that when my otherwise-entirely-admirable summer class revealed they had not read them I was at a bit of a loss when it came to examples for questioning in literacy. Stunned, I was.)

The process started with learning about the subject.  I'm not a bible scholar, but I am willing to write bible studies.  Mostly I just share the questions I wonder about as I try to make sense of the readings. I use BibleGateway.com as a tool, as it has many translations and a robust search feature.  I copied the relevant bits of scripture out, and made it into a study.  (Shared here.) I got to discuss it with three groups of people before writing the story.  In particular, with a men's group where (rather unbelievably) I'm a junior member.  Finally it was time to write the story. (Shared here.) I wrote the narrative, following the facts of the story. I went over it again, thinking about the teaching point(s) of each of the three days.  I'm not always explicit with those, but it's going to be harder for kids to notice if I don't put it in there to notice.  I went over it again, strengthening and adding connections. Things like: if the ark is going to be used on day 2, make sure it's mentioned where I can on day 1. Go over it again - does it include what's important and is what's important emphasized?  Finally, practice the telling.

My daughter performed with me to help with the illustration and comic relief, and we'd go through the story right before telling it. Originally she was going to be Joshua the first night when young, then Israelites later, but we wound up keeping her as Joshua because the kids seemed to identify her really strongly with the part. As we told the story, we responded to the kids' response. We monitored both their general engagement and asked questions about what they would do, or what thought might happen or what they knew about things in the story.  We told one version to the K-4 crowd, and another to the age 3 and 4 group (with lots more marching around). We used the props and set pieces we had and figured out how to do things with limited time and resources.

Sounds a lot like teaching, right?

I've been really interested as some teachers share on Twitter their summer planning process.  It seems some are frustrated by trying to plan without their students, and I agree that this is questionable. But I also got to thinking that it offers an opportunity to think about your objectives in another way.  As a story.  Your plans won't be able to be set until you know your students, and assess what they're bringing to the story.  But you can think about what's important and look for the narrative.  To me it's quite like what Dan Meyer's been writing about when he's considering how to set, describe and pitch the problems you find.  I also think that's the kind of work we need to find better and better ways to share with each other, as it's very portable amongst schools and students.

Sunday, January 16, 2011

Quest for the Holy Snail

We are beginning our middle school math for teachers course thinking about integers.  What a miserable topic, on the face of it. Trying to think about why K-12 students might need or see negative numbers, these pre-service teachers came up with three spot on suggestions:
  • Answers unanswerable problems: 5-7
  • Gives relative position or quantity
  • Alternative way to think about subtraction
Really sharp thinking, I thought.  And if you think about why mathematicians invented them (or worked out how they work), this is definitely it.  Completeness and closure, and then a more sophisticated understanding of operations.  My suspicion is that these make poor justifications for middle school students.

As I thought through contexts, I thought that the relative position or quantity situations were the strongest.  Something got me thinking about keeping track of change as the most interesting of these.  That and our heavy family diet of fantasy led to this story.  Apologies to Mr. Python and to you, the reader.

Task:  Archaeologists definitely did not unearth this journal from the middle ages.  But as you read through the account of this thrilling fictional adventure, please keep track of the number of knights.  Make two graphs:
  1. A graph of days on the quest vs. the number of knights.
  2. A graph of days on the quest vs. the change in the number of knights from the previous day.
A mathematician would probably mark Jan. 11th  as day zero, but what you do is up to you.  As the graphs are finished, label the important points with what was happening in the story.  Which graph do you think shows most clearly what happened?  Why?

Quest for the Holy Snail
Diary of Sir Vaysez

Jan. 11th, year of our Lord 1011.
We set out for Gudtonoya today. Our company numbered 100 good knights, in search of the Holy Snail. It is hoped that recovery of the Holy Snail will bring peace to our land.

Jan 12th. As we topped the hill outside Dentite-on-Wails we encountered a party of Orcs. 10 men lost. Ran away screaming at the sight of the beasts. The orcs waved us on, wanted to know if we had extra biscuits. Not for their ilk!

Jan 13th. Faced an enemy party of evil knights from Notusistan. The blocked our path and challenged us to battle.  We shouted as loud as possible, they were overwhelmed and joined our party. 25 converts to our cause!

Jan. 14th. Sad morning. Turns out Notusistannis were playing a trick. They left and convinced 50 gullible men to go with them. Silly knights, such tricks are for children!  King seems depressed.

Jan 15th. Wandering through the Moors of Lesthan. Doing nothing for morale. More knights returned home today. The company is down to 35. Still mostly mighty!

Jan 16th. Surely this day shall be remembered always. Camped at the edge of a pond, the King heard a voice calling to him. When he looked into the water, a maiden stared back at him! (Not a reflection, as he is quite, um, "rugged" might be polite.) This maid said that the heavens did find favor with him, and she reached out from the water holding the Spear of Justice! Hurrah!

That said, 10 more men left. Mumbling something about “watery women handing out mystical weapons is no basis for any kind of representational government.” Good riddance, say I! 

Jan. 17th. 12 new knights have joined. We marched today to the Castle of Awwshux. The nobles there, inspired by our majestic liege and Expointsalot, have joined our quest. (That’s what he’s calling the spear; doesn’t seem like good namesmanship. What’s wrong with ‘Spear of Justice’?) They also had information that …

Jan. 18th. Sorry I never finished yesterday. Dragon attack!  Lost a few brave warriors, but the dragons were repelled. Our company numbers 33 hale and lightly toasted fellows.

Jan 19th. Marching on Notusistan, as the Shuxters are sure these dark knights hold the Snail. We’re worried, as we hear they eat snails. Might as well be French, right? Thrilled by promise of action, many knights have joined our company. We are 51 knights strong, and should enter Notusistan on the morrow.

Jan 20th. Exhausted.  This was a major battle.  Fully armored knights clashed, bashing sword on shield, lances driven forward. So glorious! Surprisingly, we still have 51 knights. Don’t think they lost any warriors either. But it was glorious, I say!

Jan 21st. Turns out Notusistannis are good cooks.  Might as well be French, right? They invited us in for a meal. We recounted the glorious deeds of the battle from the previous day. 8 men left because the food was "too spicy."  Have a palate, man! But 5 brave Notusistannis have joined us. Turns out they didn’t even know the Snail was here. They said that if it’s anywhere, it must be in the castle of the Wizard King of Wartshog.  It is rumoured we shall face the undead.  Then 6 more men left.

Jan. 22nd. I write these words weary, but victorious. 10 more knights joined us for the final assault. But the legion of skeleton fighters claimed heavy tolls.  Relentless, they were.  Then the Wizard King smashed 5 brave knights as we rushed him together. But 22 knights, including our King, Herbert, made it through. The King smashed the evil warlock's dread terrarium and recovered the Holy Snail!

Rejoice all you lovers of Blessed Mollusks!  Once more our land shall know peace.

Chapel of the Holy Snail

Writing this made me think of Denise and her adventure math stories.  Check them out at Let's Play Math!

Photo Credits (Flickr): GraphicReality, Ton MJ, ElitePete, estherase, modowd, greyloch, rogersanderson