Friday, December 30, 2011

Two Final Problems

Trig Problem 2
For my preservice high school teachers' "final" (really a last Standards Based Grading opportunity), there were two problems that while similar in many respects were quite different in results. All of the problems were listed by one standard, but typically could be used for other standards. It's the student's responsibility to describe what standards they are demonstrating, though I will help if it demonstrates something well that they need.

Trig Problem 2. (Standard: Law of Sines, Law of Cosines and applications)

Figure out some of the missing information in the diagram.

The pictures were made in GeoGebra, which I highly recommend for mathematical image creation, as well as more active uses.

Geometry Problem 1. (Standard Lines: parallel, perpendicular, properties of angles)

Find more angles.

Geometry Problem 1

Similarities: visual, finding connections, geometry, students have previously done and been assessed on similar problems.

Have to love easy-to-draw memes.
Differences:  throughout the semester students saw trigonometry as something difficult, and had much less confidence on them.  Students were very successful with the angles problem, able to find all the angles, and be able to justify their results. Why vertical angles are congruent, why there are 180º in a triangle, etc. On the "trig" they quickly resorted to visual inference (like the angles at A were all 60º), supposition, and ignored contradictions (such as finding that the length of CD was less than 6 units), and did almost no extension to other standards from circle geometry.

It was fascinating to read their work, and I wish we had more class time to look at the results. It felt like direct confirmation of the Van Hiele levels, and convicted me that as much time as we devoted to trigonometry, I need to find more ways to increase their experience.  While I thought the circle diagram was more subtle, I didn't realize the great difference in how students would see it. Only one student realized CD must be 6 units, which is the entry to me for many of the possible values that can be determined.

Tuesday, December 20, 2011

Extrinsic Motivation

It's been a weird semester and this is going to be a weird post. I'm trying to work through how I made a mess of it, and doing that publicly is odd... but it fits with how this blog has helped me develop as a professional.  I shared my portfolio at the beginning of the semester, which I put together to make a case for promotion to full professor, and I've been turned down for that by the college personnel committee and our dean, after a positive but slightly contentious department vote.  The reason for for the no was no peer-reviewed publications.  Serves me right, many say, applying for full without any.
I became interested in the teaching of math in a serious way when I got the chance to make over the math for elementary teachers at Penn State. We changed texts, and my friend Sue Feeley gave me some excellent reading recommendations in response to 'what do these teachers need to know, anyway?'  I got more interested when I realized how amazing and challenging it was to think about, and just fun besides. I was going to quit graduate school and go teach high school, but my advisor correctly urged me to finish. The nail in the math research coffin for me was realizing just how few people would care about the research I was doing.

At Grand Valley they hired me to be a math educator, in what I still consider to be a minor miracle. What were they thinking? I didn't think too much of publishing then because I was really just learning the field, and then I just never got around to it. I was also changing (hopefully growing) so fast that it felt weird putting something into print - who knew if I was still going to be doing that in a year or two? Plus work in the schools with students and with teachers in professional development was so much more satisfying.  That led me to blogging, as a way to share resources and post materials for teachers, and blogging led me to writing. (Such as it is.) It was ephemeral enough that I didn't feel chained by it, and informal enough that I could share my process and stream of thought, which I value over product.

Then I started getting positive attention at work for the blog. I had long accepted that the way I went about my job meant never being a full professor and I didn't mind at all. Several friends convinced me to consider applying for promotion, and when my chair mentioned to my wife Karen that I should, it became a home discussion, too.  I decided to try; I could be a test case, since I didn't really care.  But a funny thing happened on the way, and as I put together materials and considered the college criteria, I really convinced myself that I did fit the criteria. The one thing missing: peer review. I decided that my department would be the peers, and made the process about asking them about the quality of my scholarship. They felt it met the requirements, though some felt like that was the wrong question, and the right question was publishing. But our criteria don't require publishing.

So when the negative decision came, it was totally depressing. The dean made it clear that it's a "technical requirement," and, I'm sure he thought kindly, "if you had one paper accepted..." Which to me sounded like you're right, your work is deserving, but sorry, you forgot to check a box.  My negative reaction to this makes me feel foolish beyond measure, because my life is a constant stream of blessings. This is so totally a first world problem. It made me feel unappreciated at work, despite the great support I received from many people. Karen suggested my reaction came from a lack of previous failures, and that is part of it, too, I think.  It really mired me in negativity.

I went from doing what I love because I loved it, to caring what someone else would say about it. And now I probably will try to submit for publication, though every obstinate bone in my body says to hell with it. Because it makes a financial difference for my family, though I hate that this matters. Which takes me back to having been so fortunate that I can be such an idealist at this advanced age.

Then it finally connected to me both how we do this to students all the time. Care about the grade! And how this is parallel to the new and developing teacher evaluation programs. With much higher stakes, where a no means you're out of school or out of a job. It's a nasty proposition, having to manage your professional life or academic life with someone else's criteria and interpretation of those criteria hanging over your head. 

Sympathy for the people really subjected to these extrinsic measures is helping me come out of my funk.  Plus to still be doing the job I love, with the constant amazing work that students do when genuinely learning.  Two #mathchats on math games this week! A new batch of student teachers to mentor next semester.  I want to re-evaluate how I'm trying to motivate students, and to be honest about it.

It's a Wonderful Life, when measured by what actually matters.

Friday, December 16, 2011

Holiday Game Design

#mathchat last night (twitter stream, wiki) was on "Games: Where's the math? How can we use games to teach mathematics?" One of my favorite topics, and a good discussion. There are so many things I like games for in mathematics: playing a game is quite like math, strategy is an excellent context for problem solving, engagement level for repeated exercises or tasks, etc. But one of the things I like best personally is making them. (That's definitely one of the appeals of collectible card games; building a deck is a lot like game design.)  The amount of math that goes into making a game can be quite a bit greater than playing it.

So today for the 5th graders I brought a half-formed game based on the Traveling Salesman problem. Georgia Tech has a nice Traveling Salesman Problem site, with a few games of their own, nice explanations and history of the problem. It was inspired by the ultimate Traveling Salesman: Santa Claus. Every home in a night? Mathematician Elves on the job.  I eventually changed the game to running Christmas errands in town here, and intentionally left it rather drab.

We played a few turns of the game to get the idea. Then I shared how I wanted them to be game designers today, and we discussed possible things to work on.

Game Design To Do:
1) Playtest
  • Are the rules clear? Do they need to be changed? 
  • Are the mechanics of the game okay? (Right number of destinations, how to move, placement of stop, dice to roll…) 
  • Is it fun enough? How can you make it more fun? 
2) Develop
  • Should there be obstacles on the map? 
  • Decorate the board; add fun details or pictures. 
  • Make nice game pieces. 
3) Create!
  • Make your own map. 
  • Change to the world map or the US map. 
  • Change the story of the game. Santa, UPS, mail carrier, … 
  • Completely new game idea: 12 days of Christmas, Christmas tree, Hanukkah Candles, Winter Break, … 

I also brought a blank grid, a polar map (to do a Santa Claus version) and a United States map. (Click for full size. PDF of the whole document on Google Docs.)

Nobody used the polar map - poor Santa!  The class had many people make improvements to Santa Haven, and several who made their own game.

Some of the improvements: new goals, like get all the presents to Grandma's house.  Board alterations, like road block, traffic jam, hazards, stop signs, school zones, etc. Some quite clever, like a gas station (you have to go in if you pass), or a muddy spot that divides your speed by 2 until you get to the car wash. Play alterations, like the bank after every present, or a specific chore list (home, school, presents, back home then school then home). One student made walking and driving rules; driving was double speed, but had school speed zones and roads they had to stick to. At the end we talked about how this was mathematical modeling, where they tried to take real life stuff and figure out what they would be like in the game.

The new games included 2 Risk variations on the US map, variations on the traveling salesman with all new maps, and a candy cane math game with problems on the spaces.

Quite a lot of creativity, so much enthusiasm. And, I think, a nice lead into designing some of their own games later.

Sunday, December 11, 2011

Rigor and Relevance in Parallel

(The math is at the end of this one.)

Last week I had one of those teaching collisions where it felt like every idea was dovetailing.  First some twitterer retweeted Terie Engelbrecht's post on rigor and relevance in the context of motivating students with respect not points. Her post was a riff on this International Center for Leadership in Education chart. My preservice high school teachers had asked for parallel lines, circles and proof for our geometry topic.  Elissa Miller tweeted about parallel lines in a way that also brought up relevance.

Source at

Fig 1.1 from this ASCD article
Some comments from my colleague Dave Coffey connected the Relevance framework to the Levels of Transfer, a professional development framework from Joyce and Showers. Before I switched to a communication framework, I tried to grade using that framework! Very ingenuous, as most students are not at an executive level of transfer, and it is not fair as an expectation for a large quantity of work. I did like the sense of ownership that it promoted, and the encouragement for students to show that they had made the learning their own. When it came time to convert to grades, Integrated Use was an A, and Executive an A+. Is that fair?

Terie's central question was: 'So why not change the "doing the work for points" idea into a "doing the work to improve my learning?" idea?' This has been an issue all semester with this particular group of students for me. Because this is a teaching class, we've even explicitly discussed from whence comes thier lack of responsibility. (As in the Condition of Learning, Responsibility; not as in a guilt trip from your parents. I hope.)

So I brought the framework into class and asked them to classify the kind of lessons they saw in observation... 10 to 15 lessons over the course of the semester.  I was seriously surprised how well distributed the X's were on the relevance framework, but not surprised by the low range of rigor. The students discussed how the ABCD was confusing, as D was the "best." They had quite an interesting discussion about whether math should be 'real world' or not. You all know the discussion: why do they ask us (math teachers)? Do they ask their history teacher? What about learning the math for math's sake? Sometimes the applications are so artificial.  The students hate the applications so much, why make them do them?

I think, from their discussions as well as their observation journals, I would have put most of their X's in A.  The application in the rigor framework led to a lot of the marks being along that line, as we haven't studied Bloom's, nor have they in other classes.  Personally, I do wonder about what the classroom should look like in terms of this scale. I like the idea of flexible access, and I want students to transition to analysis, synthesis and evaluation. The application framework seems like you do want a diversity of those problem types. Regardless, I'm really interested in what other teachers think about the framework.

To support the students in thinking about implementing this, I brought what I had thought of after Elissa's prompt. Their task:
Our question is: how can we plan a lesson or lessons that will support our students in moving towards being able to make proofs, understanding the required angle content, and engaging them all the while?
Consider the following and please add your own:
a) a puzzle made from parallels and transversals
b) a map of city streets
c) a classic Japanese problem using these ideas
Where do these ideas fall on the rigor/relevance chart? How should we sequence them or structure class to get to our objective?
Explore the activities in your groups, or design your own, and then we’ll come back together to discuss the ideas.
The puzzle: (Actually revised a bit from their use, based on play; changes were made to make it more accessible and focus on the mathematical properties. Click on the image for full size.)

The map: (Grand Rapids)

Tasks suggested for this were identifying parallel and perpendicular lines, vertical and transversal pairs of angles, combining measurement of street intersections, make informal arguments for congruent pairs, etc.

The Japanese angle problem:
(The top angle is 50 degrees, the bottom 30 degrees.)

Find the angle x. “Please solve the problem and if you can make an explanation that is amazing.” From the TIMSS video at (Love this problem; thanks to Rebecca Walker who first found it.)

Students tried all three, and they typically wanted a combination of them in their lessons for their hypothetical students.  The discussion of rigor in these problems gave us a lot of material (by which I guess I mean gave me a lot of informal assessment data) for a discussion of answer, solution, argument, proof in a later class.

I think these problems are pretty good samples of A, B and C lessons; I confess to being a bit stumped as to what a D lesson/problem/activity might be for this topic. The nature of abstraction in mathematics will lead to a lot of good lessons proceeding from B to C. Opportunities to venture onto D might be more rare. I'm very curious to hear what other people think.

Thursday, December 1, 2011

SBG Resources

From Rainbowcatz @ Flickr
I was gathering Standards Based Grading (SBG) resources for a colleague and thought that would be worth sharing.  Maybe this should be a LiveBinder? There's a definite math focus to my selections below, those it's not strict. Many people refer to SBG as Standards Based Assessment and Reporting (SBAR), which gets the whole 'grade' idea right out.

People: (Name links to Twitter)

Fundamentals and Further:
Twitter discussion
  • #sbar - find more SBG folk, or people trying it in your discipline, by a Twitter search.
  • #sbarbook - book group that 'meets' weekly for discussion about a particular book on assessment. Doesn't look like this semester's book is very engaging, though.

For completeness sake, here's my 2 (so far) SBG posts. Hey, this makes 3! If there are more examples of SBG in college, especially college math, please help me find them.

Friday, November 18, 2011

Skemp Discussed

This semester we had the opportunity to discuss Richard Skemp's great article on Instrumental and Relational Understanding in class. (Relational Understanding and Instrumental Understanding,” Richard Skemp, Mathematics Teaching in the Middle School, September 2006; link goes to a pdf hosted at Portland State) Students read the article, with the following home workshop. When we came to class, they discussed at their table and then made one 'slide' presentations to the class on 2 ideas.  The questions are the ones I used for an online discussion before, recorded in this blog post.

Home Workshop 16 - Learning Math
Instrumental Understanding
How do I understand?
“Relational Understanding and Instrumental Understanding,” Richard Skemp, Mathematics Teaching in the Middle School, September 2006
Discuss in class Wednesday
Objective:  TLW use specific questions to better focus on and understand relational understanding.

Schema Activation: How do you multiply fractions?  How well do you understand the multiplication of fractions?

Focus:  Directed reading.
Below in the activity are a dozen discussion questions. As you read, keep notes on your thoughts about the questions.Read through the questions before reading the article.

Even though this article was written for teachers, Dr Skemp wrote mostly for researchers, and at times the language is a wee thick. Press on!

Activity: Read the article, jotting notes on the 12 questions below.  These are just notes, and you may find you have no thoughts on a couple of them. In the reflection you will expand on your thoughts for two of them.

1) What is the point of starting off with the Faux Amis story? 
(A faux amis are two words in different languages that sound similar but mean differently.  Sopa (soup) and soap (jabón) are my favorite from Spanish.  Skemp says that the ways we use "understanding" are as different as if they were faux amis.)

2) What is your favorite example of “rule without reason”? Why?

3) Does the author’s idea of looking for your own examples and his three reasons for it make sense? Why?

4) Explain Skemp’s two kinds of mismatches (in the classroom) in your own words.

5) Of his two kinds of mismatches, which is more common? Which is more of a problem for the teacher?

6) What are Skemp’s faux amis in mathematics teaching? Is either one an issue in your math major classes here in GVSU?

7) Would you add any advantages to his list for instrumental mathematics?

8) Would you add any advantages to the list for relational mathematics?

9) Do you agree with the advantages that he lists for the two types?

10) What’s an example of relational understanding in your non-math life?

11) What’s an example of relational mathematics understanding for you? How do you know?

12) So, what about your classroom? Will you teach for one, or the other, or both? Why?

Reflection: Pick 2 questions you would like to talk about in class, and write a thoughtful response to each.
  • Look over your notes/highlights/work from reading.  What did you take from it?
  • In your own words describe the ideas of instrumental and relational understanding.

The other thing that we've been developing in class is the idea of questioning, both as a teacher, and the benefits of student to student questions.  This class struggles with being quiet, but by the fourth group, they've hit full class discussion mode.  I really think that conversation is the only way to work towards understanding of big ideas. I filmed the first two groups and then handed off the iPod for recording.

In their groups I asked them to share their reflection from the workshop, they discussed a bit, and then to decide on two points to present to the class. Mostly they used the questions to frame their points. We talked a bit about presentation zen, and I asked them to make a 'slide' for their two points on the board, with the idea to not have a lot of text, but to support their idea with a succinct statement or even better, a visual. They did an excellent job, and I hope you enjoy sharing in their discussion.

Group 1 focused on questions 5 and 8.

5) Of his two kinds of mismatches, which is more common? Which is more of a problem for the teacher?

8) Would you add any advantages to the list for relational mathematics?

Group 2 

2) What is your favorite example of “rule without reason”? Why?

 5) Of his two kinds of mismatches, which is more common? Which is more of a problem for the teacher?

Group 3

5) Of his two kinds of mismatches, which is more common? Which is more of a problem for the teacher?

11) What’s an example of relational mathematics understanding for you? How do you know?

Group 4

12) So, what about your classroom? Will you teach for one, or the other, or both? Why?

Group 5

1) What is the point of starting off with the Faux Amis story?

10) What’s an example of relational understanding in your non-math life?
  (Nice because it was ambiguous whether their example was relational or instrumental.)

As I listened to their discussion, I was struck by how many of the concerns of inservice teachers they already have, which is a real testament to the idea from The Teaching Gap that teaching is a cultural activity.  If we do not do something to resolve the tension between what teachers feel they are expected to do (the job) and what they want to and should do (the vocation), we're not going to make any progress.  It's almost what Skemp is talking about with the faux amis about our two ideas of learning. The same two ideas are competing and confusing us when we use the word teaching.

Friday, November 11, 2011

Make and Take

I'm falling behind on blog-writing, but have to share this game. Definite keeper, with great potential. Easy rules, great mathematical situations and pretty fun.

The game grew out of a meeting with Nick Smith, one of our novice teachers with a good game eye. He was looking for a way to make a game with number operations and maybe order of operations that had us using cards and trying to make a target. Wasn't quite working out.

Finally it occurred to me that if you were setting the target for your opponents... started trying it with cards and BAM! A game. It's simple enough, probably someone else has come across it before.  Basically, you deal 5 cards to each player/team, each team picks one card for the other team to make by combining their remaining cards with operations.

(Direct link to document.)

To launch it with the 5th graders today,  the teacher and I started to play. I put up the values for Ace, Jack, Queen and King on the board - which was a good idea as students consulted it frequently. Today was 11-11-11, and Jake, one of the students, had a birthday... his 11th! (He was in the local paper last night.) So we renamed the Jack the Jake in his honor, since the Jack is worth 11. Sometimes stuff just works out.

After about three turns of demonstration, the students were clamoring to play. Who are we to stand in the way of a math game?  Students were engaged, making interesting combinations, and making more complicated combinations as play went on.  It adapted well to students at different levels, as they were choosing combinations, and I was able to see automaticity with subtraction improving in students who have some math struggles at the same time as self-identified math whizzes were challenged to find fascinating 4 card combos, like Jack - 4/4, divided by 2 to get 5.

I tried the game with younger learners previously, and they got good addition and subtraction practice, and think it would extend well to middle school as support for order of operations.  (Write down your combination and check it on a scientific calculator.)  It was nice that sometimes the game called for easier combinations and had moments of challenge. Students were actively searching for new people to play and telling stories about games and combos. Very fun.

To finish our time, we discussed the combos (I had recorded some of the better ones on the board, such as Q, 5, 4, 4 -> 10), the strategy and the name.  I like how the game becomes a context for some pretty good problems.  The students were split on what made a difficult target. The majority felt like middle cards were harder to make, but a few thought the smallest cards. I actually don't know! One aspect of the game that I like a lot is that you gain information about the opponent's hand as you play. A strategy that many came across was reusing target numbers that your opponent couldn't make.

There weren't a lot of suggestions for names... math war and math attack had some support.  More suggestions about the game would be welcome, also.

Having written those game design commentaries lately (one & two), I can't resist thinking about the game using it.
  1. Goal(s). See numbers as related by operations. This game is great for that.
  2. Structure.The game reflects the goal by using a shifting set of cards. The slow turn over allows students to build relationships and more and more complicated sets of computations.
  3. Strategy. The selection of targets and which cards to keep to make combinations is the first level. Taking into account your opponents' cards is a whole 'nother level.
  4. Interaction. Choosing the target for your opponents and having to make their target offers lots of interaction.
  5. Surprise. The cards you draw and the target you're trying to make.
  6. Catch-Up. This could be a weak area. Once kids are good enough, it's rare to miss the target, which means it's hard to catch up. That's when you switch to the four card variation, which can be very challenging.
  7. Inertia. Kids were divided on the 10 card winning condition. Some thought it should be lower. One student who loved the game suggested 13 cards!
  8. Rules. Big win for this game. Very simple.
  9. Context. No context, but the game did seem to have a pretty fun level of gameplay for students.

Image credits: qthomasbaker, ames sf @ Flickr

Monday, November 7, 2011

Game Design: 6-10

Mark Rosewater, head designer for Magic: the Gathering, has up the 2nd part of his intro to game design article, so it must be time for my 2nd part of my commentary thinking about educational games.

The first five principles were:
  1. Goal(s). Easiest part for educational games.
  2. Rules.  
  3. Interaction. 
  4. Catch-Up. Most subtle, maybe because so many games lack this.
  5. Inertia. Hard for teachers.
The second five:

6. Surprise. The game should have some unpredictability for players.

To me, this connects strongly to Interaction and Catch-Up. One way to get surprise is hidden information - which often can contribute to interaction amongst players.  Information can be hidden from both or the players can hide it from each other. The new game Flip Out has a good element of this with two sided cards of which each player sees different sides. A benefit for math and literacy is that this makes inference a part of the game.

The other easy way to add surprise is random events - which can contribute to making catch up possible.  The only thing that makes Monopoly playable is the dice rolling.  In Euchre, no matter how good you are, you need cards to play. The math benefit is the addition of probability, even if informal, to game play. It's no surprise that the two most common game pieces are dice and cards.

7. Strategy.

Interesting to me that this is so low on the list, which makes me wonder what he was ordering them to achieve.
by mhuang @ Flickr
This is the biggest add-on for educational games over other activities. The problem solving inherent in any game with strategy is such fantastic grist for mathematics.  Mathematicians often see math as a game because of this strong connection. How do we achieve a result with allowable moves? Using games with K-12 students,asking for their strategies always makes for an amazing summary and unearths most of the math content of the games. It also helps build Inertia as then students are more interested in playing again, trying our others' strategies or designing ways to beat them.

There's a natural tension between Surprise and Strategy.  If things are too random, strategy loses all impact. If things aren't random at all, it is chess or go.  Both great games, obviously, but also both games that struggle with Catch-Up and Inertia for many players. Plug for Magic: the balance of these two elements is a large part of what makes the game so bloody amazing. Also applies to Bridge, to a lesser extent. (Yes, I'm claiming Magic > Bridge.)

8. Fun.

I struggle with this. Because I find interaction, surprise and strategy so engaging, I love games in general. I'll play anything. But what makes a game fun to kids is often a surprise to me. It's not uncommon for me to take a game to kids, and let them add the context. I wrote about this a bit with my Division into Decimals game.  Games like Decimal Point Pickle and Power Up had this in spades. Probably this is the difference between a game being good, and the game being a smash hit.

To some extent I think the last two principles are really subcategories of this one. Did they get pulled out to make ten or - more likely - is there something I'm missing that makes them truly distinct?

9. Flavor.

10. Hook.

Flavor is about the context and setting for your game, which heavily influences the fun aspect for players, in my experience. At least on entry, and Mark connects this to the barrier or entry cost to your game. The other principles determine long term fun. In our house, this gets us to play a game fr the first time, but won't sustain interest. One neat point he makes about flavor, though, is how it can influence design. My youth Bible study is making a return of the Lord card game based on the 10 bridesmaids parable. (Yes, really.) But the context for the game is inspiring a four horseman of the apocalypse feature that will definitely add interest to the game. Probably shouldn't have shared this story.

The idea of constructive flavor reminds me of my colleague Jacqui Melinn talking about integrated units.  A marine biology integrated unit is not when you put your math practice problems on a whale-themed sheet, it's when your questions about whales require math to think about and solve. Good flavor isn't an add on, but supports the game mechanics. For a math game, this gets at the structure of the game supporting the mathematical objective. Cheap flavor is the hallmark of flashcard/drill math games. "Look you're doing lots of multiplication, but it's on a baseball diamond!"

Hook is what gets people to try your game. This is less important in educational games to me as we have a built-in market (students), but I'm also not trying to sell my games to a publisher. (So maybe my hook is that my games are free?)  However it does remind me of Dan Meyer talking about a hook for a lesson, and could well be linked to engagement. I just don't know how to tease it out from flavor and fun. Maybe hook is a measure of whether the game has things that make you wonder?

My Nine
Looking over the list, I think I'd order them more like the following to get at my process.

  1. Goal(s). Design starts with objectives.
  2. Structure. (Not in his list!) What is the essential nature of your set of learning objectives and how can that show up in the game?
  3. Strategy. These three have to go into the primary design phase as well, or the game will just not have them.
  4. Interaction.
  5. Surprise.
  6. Catch-Up. As you start to playtest, these two are important to attend to for good design.
  7. Inertia.
  8. Rules. For me this comes late; kind of a synthesis step as you think about how to communicate the game. It will often result in design revision, though.
  9. Context: Fun-Flavor-Hook. To me this can't really be evaluated fully till you're out with the intended audience. You need a first take on this before that, but should be open to major changes in this area.
Boy, I enjoyed these two articles. Thanks, Mark, for  writing them, and giving us fledgling designers something to think about. Note that Mark has a Tumblr where he answers many questions and gives the behind the scenes story at Wizards of the Coast. I asked, for example, about the order of his list, and he wrote: "I left it up to my subconscious. That was the order that felt organically correct. I’ll be honest, that I can’t exactly explain why."

Images: All Magic: the Gathering stuff is very heavily (c)'d by Hasbro.

Thursday, November 3, 2011

Advice for Solving Equations

Quick post to share the teaching idea. The preservice high school teachers read 
“Advice for Solving Equations,” Steuben and Torbert, Mathematics Teacher, April 2006

The reflection was to give your advice. While I think the advice they give is solid, what I like is the giving of advice.

John G:
1) remember the meaning of the =. These things are the same thing, though they look different.
2) remember your purpose and for what you’re solving.
3) think about the best representation for solving this equation.

Ellen B:
1. Don’t forget about substitution! I wouldn’t have thought about this in the problem in the reading, but it made the problem so much easier.
2. Could you change the form of the equation to make it easier to solve?
3. Know what you’re solving for; what should the solution be in terms of?

Alyssa B:
1. Look at the problem before you start; what do you already know or what is important to note?
2. Make sure your answers make mathematical sense!
3. If you’re stuck, try rearranging/changing the form of the equation.

Greg O:
1. Never give up on a problem, if your stuck try rearranging it to make more sense to you.
2. Always look for things that you know in a problem so that you can maybe substitute something for it.
3. Make sure you can reverse the process so that you can get the original answer, a way of checking to make sure your answer is correct.

Jordan D:
1. It’s alright if you can’t get it right at first. You learn by spending time on the equation and from mistakes.
2. Experience helps you solve challenging equations, there are many steps that do not seem natural until we use them.
3.Always check your solution, make sure that it makes sense and works.

Emily W:
1. “You will get much more out of a problem if you work on it for 15 to 20 minutes and fail than if you turn to the solution after only 3 minutes.”
2. When you first start a problem, notice everything you can about the problem (Can x = 0? How many solutions will this equation give? Etc.).
3. If the method you are comfortable with does not come up with the correct solution, don’t be afraid to use one you are less comfortable with.

Mike Simon
1.) When solving a problem that looks difficult, begin by stating what you know.
2.) Work with what you know and if you get to a point where you are not sure if there is a next step, look over your work.
3.) After checking over your work, a make a guess and see where it leads you. There is not shame in being wrong.

Matt York
1. The whole purpose of the problem is defeated when you give the solution after just 3 minutes.
2.  As you are solving, always keep in mind the goal... never forget the purpose of the problem.
3. If you have no idea where to start, try some numbers which make sense in the context of the problem, it might lead you somewhere.

Joe Freiman
1. It’s better to work for 15 minutes and fail, than to look at the answer after a minute.
2. Using your past experience is the best way to solve difficult equations, so by practicing constantly and gaining more experience is the best way to get better at solving complex equations.
3. Check your work, make sure your answer makes sense.

EDIT: some end-of-the-semester catch-up-additions...

Brandi Stewart
1) It is ok to be wrong and have to try many different ways.
2) Check your work
3) It is not bad to ask for help, especially if you have tried the problem on your own and do not know how to figure it out.

Courtney Johnson
Mine are kind of specific but...
1) Try putting all variables on one side and setting equal to zero to see if using the quadratic formula is a possibilty
2) See if any substitutions can be made (e.g. trig identities)
3) Try removing a common factor to simplify

Amanda Hoezee
1) You will get much more out of a problem if you work on  it for15 to 20 minutes and fail than if you turn to the solution after only 3 minutes.
2) One does not need to be talented to solve challenging equations; one only needs experience.

Mitchell Brady
1) If you are stuck on a problem keep it simple and work with what you already know about the problem a solution that may help.
2) If you are stuck go back through your work to see if something sparks your memory to continue.
3) Finally, do not be afraid to continue and be wrong, just learn from your mistakes and try another method, but first figure out why it was wrong so you do not make the same mistake twice.

Ryan Warner
1) Failing isn’t always the worst thing, you can learn a lot sometimes by failing
2) There is almost always more than one way to solve a math problem, when solving equations this is definitely true so try different ways if you can’t figure it out right away.
3) Learn everything you can about the equation first before you try to solve it.

Shannon Penix
1) It is important to look at the problem all the way through before trying to solve it, noticing things you already know.
2) It is alright to take a while to get through a problem and even not succeed. At least there is learning involved.
3) Take the time to go back over your work if you get stuck. There may be earlier steps that could trigger your thinking to continue.

Jeremy Sheaffer
1) Be fair to both sides of the equation.
2) Check your work
3) The harder something is to learn, the more chance there is that you will remember it.

Image credit: dullhunk @ Flickr

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!