Thursday, October 27, 2011

Letters from Rapunzel

One of the benefits of having a daughter that is a voracious reader is that even when I don't have time to be looking for new books, she brings in a steady flow.  Literally only limited by the size of her book bag. Ysabela has a thing for fairy tale or myth inspired stories, which is what inspired her to pick Letters from Rapunzel (link to the author's site with a sample). (Y is also interested in talking cats, but that hasn't borne fruit yet.) A quick read, she tossed it to her mother afterward, with an "I think you'd like this." And Karen went, "wow!" and mandated my reading it. It's sadly out of print right now, but still available from libraries or the wonders of Amazon.

But why am I writing about it? This isn't Jen Robinson's wonderful reader's blog, after all.

School is a large part of this girl's life, so her letters to the unknown holder of Box #5567 have many observations about the experience of a bright but unothodox student, dealing with serious family issues.

The first homework we hear about is 'think of ten possible ways that Rapunzel might be rescued from her tower' using 'what we have learned about simple and complex machines.'  Not a bad assignment, I think.  But Rapunzel (not her real name) writes, "Aaargh. I don't want to learn about simple machines! When is the answer to a problem ever simple? When it's STUPID, that's when." She goes on to say how the assignment is better than what's a typical assignment. She then includes her list of 7 very clever ways to rescue Rapunzel.  "Note: Like I said, I'm NOT a slacker. The reason I only listed seven ways of rescue is because I don't want Mr. Cornally - that's my math and science teacher- to get too high an expectation of me this early in the year." (OK, I changed the name.) Later, she writes "I only got half credit for my Rapunzel homework. Mrs. Seisnik did not like the "frivolous" way in which I handled the assignment. "MORE SCIENCE. FEWER SILLY JOKES," she wrote in her crisp block letters. "AND WHERE ARE YOUR OTHER THREE ANSWERS?" As if the assignment were serious in the first place!"

That's probably enough for you to figure out whether you want to read it or not. If you're on the edge, I will tell you one of the 10 7 methods. "Weave a large trampoline out of native grass (try to avoid thorny branches) and convince Rapunzel to jump. (Not sure what kind of machine a trampoline is. A spring, I think., which might be an inclined plane of a sort.) Anyway, it doesn't matter because Rapunzel wouldn't jump onto an untested trampoline anyway, not if she had any sense of self-preservation at all. Or if she was wearing a miniskirt.)"

Along the way, she deals with her family problem, fighting her classification as gifted (which she refers to as being a deviant, because of the two standard deviation definition the school pschologist tells her), why you would study the most influential people of the last millenium when you could be thinking about who will be the most influential people of the next millenium, why teachers ask for creative work but don't actually want it, all day in school detention, unlikely and irrelevant math problems, writing what teachers want even when you're not interested, trying to accomplish community change, letters to an author and more.

herzogbr @ Flickr
It's an awesome story about reflection and understanding, really sticking to the student point of view. Finally her mom tells her about the deviant program, "You need to be challenged, honey, you need to DO something with that imagination of yours." I want every student to hear that from someone they'll believe! Once there, she finds out that their big assignment is independent study. "We pick a topic we are interested in and do a whole project on it. Anything at all. Not what the teacher wants us to learn about. What WE want to know."

I hope you have the chance to find it if you're interested. Regardless, may you help your students learn to make their own progress. As Rapun el writes at one point,

"P.S. It's hard rescuing yourself."

Monday, October 24, 2011

Ten Rules for Game Design

All rights (c) Hasbro.
OK, well, the first five. Mark Rosewater, the man behind Magic, is writing a 2 part article on game design principles. Worth a look. Especially for teachers who want to get their game on. I'll share his categories, with notes on educational games.

1) Goal(s). 
Usually easy for educational games.

But wait... maybe not. Teachers are used to setting objectives, but the kind of objective makes a big difference in the game.  If in addition to content objectives, the teacher has process objectives, it can make a big difference in the game. The good news for mathies is that any game with strategy feels like it's connecting to the problem solving process.
2) Rules. 
Rules need to be understandable, but make things hard enough for the player. I think some ed games have trouble here, because of the old saw about about good teachers make things easy for their students. Goes well, however, with the resurgence of the 'be less helpful' mode of teaching. (We can't call it new if Dewey was onto it.)
3) Interaction. 
The game has to help what players do matter to each other. This is a major failing of Jeopardy and Bingo and Baseball, etc. where competition is the only interaction. Probably this is the best aspect of my most recent game Card Catch, with Nick Smith. Players set the goal for each other, and the longer the game goes on, the more information you have about your opponents' cards, which adds a whole second level of strategy and math.
4) Catch-Up
If a player who falls behind has no chance, it disengages them. I just recently noticed how much this matters to me. I think because as a game lover, this is one of the few things I loathe about them. Think about the slow grinding Monopoly death... (shudder) Within the game, players need to be able to catch up. It doesn't have to be likely - then it's Candy Land, where you can't keep a lead. You might as well be teleporting around that board. It does have to be possible, which will help create stories of the epic win.

In educational games this is a double danger, since so many ed games reward players who've already learned the material. If a math game is about who's fastest, there are students who start the game knowing there's no hope. Sometimes this is an easy fix by adding a bit of chance, but usually it requires structural design.  I think this principle is why so many games get pushed to review in the classroom, instead of being used to help learn.

5) Inertia.
Leave them wanting more. Get out while the getting's good. Dave Coffey is excellent at this with his lessons, always leaving students something to think about on the way home. I measure this by whether students are 'whew' or 'ohhhhh' when our time is up. In my experience this connects heavily with (2), Rules. Too easy or too hard shows up here.

Mark connects it with writing advice: make it as short as you can, then cut 10%.

Also tough for teachers, because we're trained to go until everyone finishes. Much better to have people sitting around doing nothing (quietly, of course) than to have anyone not have a chance to finish. That's murder to a game.

So Far So Good
I'd be really interested in hearing other people's ideas about this. Not sure where the best venue is ... maybe the Linked In game based learning group?

If his list next week is as good as this week's look for my part 2 next Monday! I'm enjoying wondering what the next five things must be...

It was 2 weeks! My 2nd part is up, with links to Mark's 2nd part.

Image credits: Usonian, Kathy Cassidy @ Flickr

Thursday, October 20, 2011

2nd Fundamental Theorem of Calculus

Our last calculus class looked into the 2nd Fundamental Theorem of Calculus (FTOC). We talked through the first FTOC last week, focusing on position velocity and acceleration to make sense of the result. Our interpretation was that the FTOC-1 finds the area by using the anti-derivative. How are those connected?

This week we wanted to peek at the 2nd part. (OK, it was me.) We looked up the result on Mathworld, and talked about barriers to understanding. The teachers identified that the conflation between the antiderivative and the integral (meaning area under a curve) is almost total, so that the theorem is just restating what we already think. Using this completely confusing notation and totally new way to define a function.  Perfect situation for a GeoGebra sketch to allow students to explore.

This sketch uses several GGB4 features.  It uses the integral[ ] command to find the area under a curve, which I would have had to cheat before, the input boxes to allow real freedom of entering a function, and buttons so that students don't have to know GeoGebra commands to refresh the view. This is my first sketch uploaded to GeoGebraTube, which is a huge improvement over the old webhosting at You can link to a teacher page or a student page, there's a link to download the file,  and there's easy to find embed code.  Best of all, the front page has a search, and shows recent uploads so it's fun just to check in.  And everything uploaded is CC3.0; darn near ideal resource.

If the embed code worked on blogger, I would have put it right here. (That's why only darn near ideal.)
Click here to go to the sketch on GeoGebraTube.

The teachers noticed all sorts of interesting things, recognizing anti-derivatives, seeing the +C (constant of integration) in action, and seemed to make sense of the integral definition of a function. They picked interesting functions to try, like increasing degrees of polynomials, trigonometric functions, functions without an analytic antiderivative (like cos(x^2)) and the fabulous e^x.

I've added the grid in to help students see the area more clearly, and set the grid to distance 1 to keep it unit sized.  Linda Fahlberg and John Scammell helped me with the right script for the button with quick Twitter responses. ZoomIn[1] to get a CTRL-F (refresh view) effect, and UpdateConstruction[] to get a CTRL-R effect (recompute all objects).  (Written down so I will never forget again.)

Photo credit: ajlvi @ Flickr. He (?) said he tried to capture everything he needed to know for the GRE on the board and then take a picture, but it was illegible on his phone. If he's that clever, I'm sure he did fine.

Saturday, October 15, 2011


Thinking up an activity for the Common Core State Standards.
  • 8.EE.3. Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger. 
  • 8.EE.4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
I've been thinking about this recently because I seriously impressed some school kids by multiplying in my head a couple of numbers in the billions. Then when Char Beckmann needed an activity for the  Adventures in Mathematics 8th grade book - opportunity! (Or rationalization...) (These books are from the Michigan Council of Teachers of Mathematics.)

My first couple of ideas were: something based on the brilliant scale of the universe applet, or a game looking at different representations of these numbers (my love for rummy games), or an activity based on Fermi problems.

Walking the kids to school this morning I was thinking about the rummy idea, and came up with a new game mechanic variation on rummy:  instead of collecting sets, each turn you have to play a card out in front of you. Then opponent can capture that card with a match. Then you could capture the pair with another matching card... kind of a slow run building mechanic.  Don't think it will fit for the book, but I will definitely try it in a game later.

Thinking about the matching puzzle, we have:

Number Names Measurement Prefixes Things
Power of Ten
Humans (meters) 10^0
Orcas, Anacondas (meters) 10^1
Redwood (meters) 10^2
Thousand kilo- Mountains' height (meters),
Number of visible stars
Million mega- Width of USA (meters) 10^6
Billion giga- Diameter of the Sun (meters),
Age of the universe (years)
Trillion tera- Diameter of the Solar System (meters),
US national debt (dollars)
One light year (meters) 10^15
Quintillion exa- Number of grains of sand on earth 10^18
Sextillion zetta- Diameter of the Milky Way,
Number of water molecules in a drop
Diameter of the Universe (meters),
Number of stars in the universe
Diameter of Universe (mm)
Mass of the earth (grams)
Number of bacteria on earth 10^30
Mass of the sun (grams) 10^33

Number of atoms in the universe,
Volume of the observable universe (m^3)
Possible volume of whole universe (m^3) 10^100



Why aren't millions called unillions? Or just an Illion? Mil- means 1000! I've always thought it must be because it should be 1000 thousands. Would numbers be more comprehensible without the -illions? The US national debt is 15 thousand thousand thousand thousands!

In grad school we proposed (probably it was Richard) a number system where there would be big numbers (since everyone knows what a big number is), and then a really big number would be a number that the number of digits was a big number. A really, really big number, then, is a number whose number of digits has a big number of digits. Quite sensible.

So the activity for the book could be matching quantities in different columns, though that doesn't give any opportunities for computation. Maybe a bit of a matching puzzle with some clues that require computation and comparison.

I cut things out of my table until it felt a little challenging, with enough structure to serve as an example for deduction and learning. I then put together some clues to help students fill in most, but leave a few for research, deduction or guessing.
The chart on the next page needs to be completed. The researcher has the data to fill in but no idea where to put it. Solve the puzzle of where to put the extra information. There are some blanks in the table, and those are shaded in. However some of the open spaces must get two comparisons, because there are too many for just one in each open space. 

Unfortunately, these are NOT in order. 
Names to fill in: Trillion, Quintillion, Centillion, Decillion, Octillion, Nonillion, Quadrillion, and Googol. 

Prefixes to fill in: yotta, hecto, peta, zetta, mega, and tera. 

Comparisons to fill in: Possible volume of whole universe (m3), Age of the universe (years), Mountains' height (meters), Width of USA (meters), Anacondas, Diameter of observable universe (mm), Mass of the earth (grams), Number of water molecules in a drop, One light year (meters), Number of bacteria on earth, Number of grains of sand on earth, Diameter of the Solar System (meters), Number of stars in the universe, and Redwood Trees’ height (meters) 

There were some weird facts the researcher remembered – maybe it will help you fill in the missing information! 
1. A googolplex has a googol zeroes. 
2. Thinking about word connections like tricycle and quadrilateral might help. 
3. The researcher remembers thinking that the number of grains of sand was exallent. 
4. Number of bacteria on earth is so big that there is about a sextillion for each human. (And there’s billions of humans!) 
5. A weird science measure is a mole. One mole of water is about 18 g, and has about 602 sextillion atoms. 
6. It would take about a million earths to have the same mass as the sun, even though the sun is made out of hydrogen and helium, mostly. 
7. It’s about 2000 km from Michigan to Florida. 
8. An average grain of sand is about 1mm wide. If you made a line out of all the sand on earth it would stretch for a light year! (The distance light can travel in a year.) 
9. The biggest official distance measurement is a yottameter, which is a billion times bigger than a petameter. 
10. In computers, a terabyte (TB) is 1000 GB, and a gigabyte is 1000 MB.

My favorite scientific notation/order of magnitude problems are Fermi problems, so I did put in a few of these for extensions.

The brilliant physicist Enrico Fermi used to love posing crazy questions to his students and colleagues, so that now sometimes people call crazy estimation questions ‘Fermi Problems’ in his honor.

For example, he’d ask how many piano tuners there are in Chicago. He’d make a guess as to how many people, how many pianos, how many times they needed tuning and how many pianos one tuner could tune.

Try these Fermi problems and then make up your own! A tip is to think mostly about the powers of ten.
1. How many jars of peanut butter to fill up the Empire State Building?
2. How many photographs are in all the houses in your town?
3. How many middle schools are there be in the United States?
4. How many songs are downloaded in Michigan each day? 

Dr. Fermi said if you make enough guesses, some are over and some are under, and you would be surprised how accurate you might end up!

Friday, October 7, 2011

Area Battle

Yeah! Back in 5th grade today. Been too long. The fabulous Mr. Schiller invited me back in, and we hit the ground running with a nice new area and perimeter game: Area Battle.  It's close to Area War with a few new wrinkles that make it a better learning game.

I launched the explanation that it would be sort of like War, but comparing area instead.  I showed them these two cards:

And asked which had the larger area. Unanimously the class agreed on the one on the right. What is the area? And someone explained how they got 3. "Do you agree or disagree?" Loud agreement. On to:

Students disagreed about which was bigger here, so we counted carefully. Finally all agreed that the cross was bigger.

The last set took a little longer to count, and then had a tie. The students knew that in War you play more cards and then compare. I described how we have a small deck, so for our ties we'd just play one more card, and then highest perimeter wins the tie breaker.

We talked about how in this game you make your own deck, but it had to have one card each with areas 1 square up to 9 squares, and then they could make two free cards however they wanted.

We then explained the full rules of the game. Each team draws two cards - always keep two cards in your hand. Then the team who's turn it is picks a card and declares 'high' or 'low' - whether the highest or lowest area will win this turn. The other team picks a card to play and the cards are revealed. The winner takes both cards. If it's a tie, each tied team plays another card, and highest perimeter wins.  The next team takes a turn going first.  Play until at least one team has played all its cards. The team that won the most cards wins.

Students did a great job working in teams of two to creae a variety of interesting shapes. We chose the restriction that shapes had to be made of squares and half squares, or half rectangles. Once they had a deck check (1 to 9 + 2), they were free to find a team to battle. there were some rule clarifications to clear up, and a couple people got too competitive, but they really gave it a go. One group played a three team game and that went well, too. After playing they came back together to discuss strategy.  They liked having a variety of cards, that hit both high and low. They talked about recognizing a neat idea from another team and then using it. One team started out with a 1/2 square area card but several had them by the end. Universal thumbs up as to whether they would recommend it to other teachers.  They were requesting time to play in class later, which is surely a good sign.

Below are some images of student cards, and then a link to the handout if you want it.