Math Hombre

Fraction Catch

2012-01-28T20:23:00.000-05:00

Sometimes it surprises me what I haven't written about here. Fraction Catch is one of my favorite games and it's been pretty successful in implementation from third grade to ninth grade. Partly the game, and partly the fraction cards.

I'm quite happy with the rectangle representation for the cards, as I have seen younger students use it a lot to do reasoning, and get a better sense of what the fraction is.

I often think of number sense having several parts:

understanding the number as a quantity
being able to flexibly represent the number
being able to compare two or more numbers
being able to compose and decompose the number flexibly

Whether the numbers in question are whole, integral, rational, real, radical, complex, or matrix. (Matrical?) The last bullet is, of course, the key to powerful computational fluency. Often I see students who have too little experience with the number as an actual quantity as opposed to a symbol, and it's positively frequent that students have no or limited ability to represent numbers other than symbolically.

It's very possible that the last two bullets are not actually part of understanding the number, so much as they are activities that deepen the first two characteristics, but I don't see the point in distinguishing them.

The game is very simple. Each player has a hand of three cards, plays a fraction onto the line of cards arranged least to greatest, and captures the lower adjacent card if they were able to play in between. Here's the rules and an example:

Playing this week with Mr. Schiller's class, I thought maybe this would be a chance to focus on the rules aspect of a game. I asked what might make the rules for a game good, or understandable, and they had no idea. It took a bit of rephrasing just to get across my question. I got the sense that I was not starting at the beginning, and switched tacks. Instead of demonstrating it first, I asked them to read the rules and then tell Mr. Schiller and I how to play.

It was challenging. Not very engaging, switch from the normal routine, and really communicated to me that I have to or maybe just should do some equipping to get them to be independent game players before working on teaching them designing. The class leaders got the idea, and taught the game to us and the rest of class. Mr. Schiller trounced me which they very much enjoyed. It was the terrible draws, I'm telling you.

Actually playing the game, though, students played pretty intently, though playing a couple games was enough for some. One interesting thing was what they went on to play. We suggested war or high-low-war for some, but one group used the cards to play their own version of Flower Power, a rational number ordering game from MangaHigh. (A lot of my favorite free computer math games are there; teachers register students and can track their progress.) Another group just wanted to put all the cards in order to make as long a streak as possible. Then they were noticing patterns about which cards were in the set and which weren't.

After the game, there was one good suggestion for a new rule: if you have two no-play turns in a row, you can swap in your whole hand for a new one.

Students made lots of good connections with the representations, and used them to compare fractions. Not too many got to the point where they were developing a strategy on where to play, but far enough that they would choose scoring plays over easy plays.

To summarize I put up some of the comparisons I had seen. All the students did well on comparing like denominators, like 3/8 and 5/8. They also were mostly solid on comparing like numerators, like 2/5 and 2/3. We talked for a bit about 2/3 and 3/4, and they used the nice strategy of how far from a unit they were, but the class couldn't figure out together how 7/10 compared to those two. Mr. Schiller let them know they'd keep the cards so they could play again later. He was impressed how well they played even though they had covered little of this in class beforehand.

Game Evaluation:

Goal(s) -spot on. Really addresses important ideas.
Structure - representation, ordering for the comparison, and some strategic depth that requires the numeric understanding.
Strategy - present.
Interaction - high in interaction, as what you are able to play and what you choose to play are both influenced by the opponent.
Surprise - the card game aspect helps with this and with catch up.
Catch-Up - check.
Inertia - the game ends quickly enough that most students want to continue playing. Because strategy deepens and fact knowledge increases with more play, it has pretty good replay value.
Rules - seem clear enough for the students to make sense of, but it was better modeled than read.
Context: Fun-Flavor-Hook. No context, not sure if it would help. More professional cards would be something; I had some paper decks and some cardstock, and the students preferred the cardstock. Talking about the rectangles as brownie pans was interesting to them... so maybe you could contextualize it. I think it's better as playing cards.

Strong in the yellow, green and blue makes this a good learning game. Give it a go and let me know what you think!

Out of Bread

2012-01-19T07:30:00.001-05:00

Out of bread this morning so ... tortillas for the kids' sandwiches. (Making tight little appetizer style rolls.) The first piece of salami for my Ysabela's funroll (marketing considerations) instantly prompted my math curiosity.

Zero

Any questions?

But then I wondered about what photo would be best for #anyqs?

One

Two

Three

And then just because...

Mmmm, ellipses. Did you see that eccentricity comic recently? (Xavier won't eat salami.)

I think the original photo (Zero) is the best for getting at the question I like here - how many pieces of salami to cover the tortilla? One gets at diameter comparison, Two does that even more literally, and Three might help create some dissonance. Which would you use?

I neglected to take pictures of the fun rolls (TM) despite my recent interest in spirals. May have to make a jelly roll.

Game Evaluation via NCTM

2012-01-14T17:54:00.001-05:00


The Product Game (pdf) aka Times Square (online) still my choice for best math game ever.

This is a quick post comparing NCTM's math game evaluation criteria with my design framework, based on Mark Rosewater's game design criteria.

I just stumbled across the NCTM's Tips for Teachers page on Math Games. Definitely check it out, they have solid links to games at the end. After a brief motivation about why to use games, they give their criteria for evaluating a game. I've reworded and reorganized them here, as the lack of any organization and structure was a wee maddening to me.

Mathematics:
M1. Does the game reward engaging in mathematical processes? (They connect with strategy. NCTM’s Process standards)
M2. Does the game's structure or context support the mathematics?
M3. Does the game promote conceptual understanding?

Game features:

G1. Does the game have a random component or choices to make with clear outcomes? Are students empowered?
G2. Does the game reward replay? (Variety in tasks or different pathways to the end.) Does the game have clear scoring?

Teacher and Student:
S1. Does the game give feedback throughout?
S2. Does the game support players through the most challenging parts? (Can they get stuck?)
S3. Does the game have teacher support for classroom use? (Extensions, connected lessons, chance to track students' progress.)

Learning environment:

L1. Does the game promote positive competition and a safe learning environment?
L2. Does the game promote social play? (Competition, collaboration, and communication.)

I want to compare it to the framework I've been using lately. Both to see

Goal(s) - M1-processes, M3-concepts
Structure - M1-processes (representation), M2-game mathematics, S3-extensions.
Strategy - M1-processes (problem solving)
Interaction - G1-choices, L2-social play
Surprise - G1-randomness
Catch-Up - G1-randomness
Inertia - G2-replay, S2-support as needed.
Rules - S3-teacher use.
Context: Fun-Flavor-Hook. G2-replay, L1-positive,

What the framework handles well:

Math goals.
A lot more clarity on gameplay.
Covers their characteristics in a usable format.

Do I need more?

I've got to think about the feedback throughout (S1). That feels important for an educational game. In some games it's just your success.
Similarly, support through the challenging parts (S2).
Classroom support (S3).

These are basically the education specific characteristics, though S1 is worth thinking about for games in general. It makes sense to me that if the framework is lacking it's in the context of educational games, since its origin was more general.

Note also the slides Maria Droujkova captured from Keith Devlin's math game webinar. His principles have a lot of overlap with the NCTM checks, but are expanded and better suited to multiple platforms. I think there is a place for skills mastery, though, as I would much rather have that in the context of a game than in drill and practice.

Spiral - So-So

2012-01-08T14:08:00.000-05:00

I've never seen a spiral board... wow!

Since Vi Hart released her Christmas time spiral celebration, I've been digging spirals again. I made a GeoGebra sketch to go with her video (link includes a link to her video), that I quite like. My tumblog is where I post one-off math and reblog other Tumblr math. So when I got the word from Mr. Schiller that this gameday the "Topic will be polygons/angles/rotational symmetry," it didn't take me too long to get to the idea of a spiral game. The way it worked out, though, has me wondering: how good does an educational game have to be?

Click for full size

There are lots of things to recommend it: kids think spirals are cool, it makes a nice race track, it allows you to see circle connections to angle, and that angles have the same measure whether small or big in size. I like race games for practicing with quantities, because it gives some repeated experience with a variety of the quantity, and gives you a reason to talk about the quantities. The GeoGebra I used to make the Archimedean Spiral (as opposed to Vi's logarithmic spirals) is posted on Tumblr, too. Then I just used GeoGebra's export as image to get the track into Word.

The problem with race games is that many of them devolve into chutes and ladders (American; snakes and ladders elsewhere). This one definitely did. I thought a one die game might be easiest, and after some practice settled on moving 15º times the die roll. It included right angles and gave some nice opportunity for mental multiplication. I justified the simplicity of the game to myself by adding a game-design objective. The winner adds a rule; that rule has to help with the catch-up characteristic of the game.

In addition - since the game was simple - I wanted to have another option. I brought some triangle grid paper and an eightfold diagram (links to Google docs) to support the students in making an art project with rotational symmetry. As I told the students, math art is as close to my heart as math games. I explained how to color a piece and then imagine it turning, or on the 8-fold to color in 1, 2, or 4 wedges and then copy it.

The students gave the game a good try, and they seemed to meet many of the objectives quickly. Watching them play, the game seemed a bit too long. One student who had gotten disengaged was willing to collect data for me on how long a game took. His results: 27 turns for a full game. 16 turns for 1.5 loops shorter. After one try, some people played on, many moved on to the math art, and a few pulled out the games they made in December. Mr. Schiller and I agreed it was too long, though the class was just barely in favor of okay-as-is. In terms of new rules, some students modified it to have 2 dice. Others added rules for if you land on someone +15º, an "if ahead, out one loop line," a -30º spot and similar. Several students were proud to share their art. Not many tried the triangle paper except for making free designs.

A revised, shorter game is at the end of this post. In general, it was so-so. The whole experience really raised for me the question of how good does an educational game have to be. These students have played some really good games so far, and I think they were disappointed that this one was more regular. I've definitely thought that educational games have a lower bar, since you're not interested in replay on many of them once your objective is met. My experience has been that any sort of game is a welcome change. But maybe if games are regularly played, the bar rises. I'd be interested in your opinions below, by twitter or email.

In terms of the game design framework I've been trying, and my rating of Spiral:

Goal(s) - good concrete objectives.
Structure - the spiral really fit the objectives well.My main question here is if the board should have angle measures on it. (Definitely, if polar coordinates are the objective.)
Strategy - no real strategy. Real room for improvement here.
Interaction - with no choices the interaction is limited to the racing. It's a hook, but no way to effect your opponent.
Surprise - not really relevant to this game.
Catch-Up - this game has it, both through randomization of the die and the board structure; but it's of the candyland/chutes and ladders variety.
Inertia - main reason for shortening the game. Overstayed it's welcomed. I think race games, in particular, probably need to be mindful of pace.
Rules - clean, simple. The add-a-rule rule was a big hit. I'll be using that again.
Context: Fun-Flavor-Hook. The spiral is a start to this, but some context for the spiral might have helped here. With all the spirals in nature, it shouldn't be too hard to add something. Maybe birds flying to the eye of a tornado? Hurricane?

The warning sign for this game is being weak in the green characteristics. Mr. Schiller and I were excited about it because of the strength in the yellow areas. If I was thinking about a commercial game, I think I'd make a deck of cards for movement, that would give more strategy and interaction. But that's a lot of printing for a one-off classroom game. Maybe you could simulate that with multiple dice? Make the rules a bit more complex, but worth it for gains in the green. Maybe roll three dice, pick one to use that you'll reroll next time. Trade for an opponent's die with one of yours that is higher. Worth a try! It will even increase angle use.

The modified game is up at Google docs.

Snakes & Ladders Image: Smabs Sputzer @ Flickr

Luddites on Facebook

2012-01-04T17:05:00.000-05:00

From London Permaculture @ Flickr

I tweet, blog, tumbl and I'm trying Facebook for a class this semester. I'm a huge advocate of Wolfram|Alpha and GeoGebra for math. But I'm secretly a Luddite. I only have an emergency cell phone for the car. I text once a month on my iPod just to keep the service, and I only got the iPod because I was trying to figure out how to use the technology in class. (Though I have grown fond of... my precious. Wait, what?)

Calvin College has a January series each year, accompanying a mini-term when students take enrichment classes. The talks are free to attend, some stream, and there are many remote locations at which you can catch the talks. (Around Michigan, across the country, Canada and ... Lithuania?) Today's speaker was Sherry Turkle. (Talk description, Dr. Turkle's website and infrequent Twitter.) Her new book on technology is Alone Together, which really captures the essence of her talk in a nutshell.

An outline of the problem:

Because of mobile devices we can bail out of reality at any time and moreso, we want to.
Technology is children's competition for their parent's attention, and now it is their turn to be distracted.
It's dangerous when technology's affordances meet our vulnerabilities. For example, if we're lonely, but afraid of intimacy, or so self-critical that we want to construct our own image.
Technology can turn distraction into busyness. Interviewed business people discussed being too busy to think or create, and ironically, too busy with communication to communicate.
People are comforted by keeping in touch with a lot of people but keeping them at a distance. Technology enables companionship without the burdens of friendship.
What was once collegial is now considered an interruption, but this also becomes a rationalization for avoiding real communication.
The essential paradox: the world is growing more complex, but - through social media - we're training ourselves to ask simpler and simpler questions.
Technology has interfered with the traditional separation process of adolescence. Separating from parents is harder, and forming your own identity in comparison to friends is harder. She captures this with a phrase: "I share therefore I am." Her subjects have made the validation of a feeling part of having it. A side effect of this is reducing people to something you use for your own purposes.

What's missing and what to do with it?

We are not cultivating the ability to be alone. They are missing the solitude that refreshes and restores. Creativity demands solitude.
This may be especially significant for teachers, who now have the burden of teaching students to work independently and how to be alone.
We need to be able to discuss the costs of new technology, without over-reacting or condemning critics as Luddites.
We need to ask the question: "does it serve our purposes?"
There's a tendency for us to oversimplify by assuming maturity of technology, when in reality it is still changing and malleable. In particular, because we grew up with the internet, we assume the internet is grown up.
Everything feels like it's about ramping up volume and velocity. Teaching might need to help students be able to slow down, reflect and isolate.
The language of addiction is not helpful, because it frames the discussion as addicted or cold turkey. Instead, analyze for how the technology is meeting your purposes.

Privacy

Young people have all but given up their right to privacy.
Mark Zuckerberg said: "People have really gotten comfortable not only sharing more information and different kinds, but more openly and with more people. That social norm is just something that has evolved over time. We view it as our role in the system to constantly be innovating and be updating what our system is to reflect what the current social norms are."
Intimacy requires privacy.
Democracy requires privacy.

Crux of the matter

Does the technology encourage us to inappropriately substitute fantasy for reality?
She contrasted immersion in a novel, which we know ends, and does not have relationships, with Second Life, where you build a home and can have a sort of imaginary family.

So what did I take from this?

From Intersection Consulting @ Flickr

It made me think about some of the progress I've seen with students accepting and participating in group work. I attributed this to shifting school experiences, but maybe it's due as much or more to developing interdependence. And was my frustration with students not doing homework this past semester in part caused by increasing difficulty in doing individual tasks?

My fear is the ease of distraction making it harder for students to learn to engage. People texting at stop signs shows the blurred boundaries. How much easier to text or update your status when you're really stuck on a problem? I don't want all solitude, but I want to help students find a balance. I'm definitely considering more solitary work as a precursor to group work. The power of our technology to support discussion and collaboration would have Vygotsky dancing, so I'm not giving up on it in any way.

I like that Dr. Turkle was against blind opposition, but, rather, in favor of intelligent use. The problem for me is that I am so different in my use of these things, that it's hard for me to know. My use of Twitter helps me get some insight, though, into the idea of monitoring to see if the tech is serving my purposes.

Two Final Problems

2011-12-30T14:41:00.005-05:00

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.

Extrinsic Motivation

2011-12-20T13:55:00.001-05:00

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.

Holiday Game Design

2011-12-16T15:25:00.004-05:00

#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.

Rigor and Relevance in Parallel

2011-12-11T14:00:00.000-05:00

(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.

Rigor, Relevance, Relationships

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 http://timssvideo.com/66 (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.

SBG Resources

2011-12-01T13:40:00.001-05:00

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)

Jason Buell - SBG resources that includes the SBG galas.
Shawn Cornally - collected resources at his blog
Riley Lark - SBG blogposts
Frank Noschese - SBG blogposts

Fundamentals and Further:

Ed Leadership article: Seven Reasons for Standards-Based Grading by Patricia Scriffiny
SBG Beginners Wiki, which also has math skills lists , started by Elissa Miller
SBG page at the MSmathwiki.
Blue Harvest - Shawn's SBG software
ActiveGrade - Riley's SBG software

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.

Skemp Discussed

2011-11-18T14:08:00.000-05:00

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? 

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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.

Make and Take

2011-11-11T13:05:00.001-05:00

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; How to embed a pdf via Google Docs; please let me know if it's not working!)

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.

Goal(s). See numbers as related by operations. This game is great for that.
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.
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.
Interaction. Choosing the target for your opponents and having to make their target offers lots of interaction.
Surprise. The cards you draw and the target you're trying to make.
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.
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!
Rules. Big win for this game. Very simple.
Context. No context, but the game did seem to have a pretty fun level of gameplay for students.

Image credits: qthomasbaker, ames sf @ Flickr

Game Design: 6-10

2011-11-07T11:16:00.003-05:00

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:

Goal(s). Easiest part for educational games.
Rules.
Interaction.
Catch-Up. Most subtle, maybe because so many games lack this.
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.

Goal(s). Design starts with objectives.
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?
Strategy. These three have to go into the primary design phase as well, or the game will just not have them.
Interaction.
Surprise.
Catch-Up. As you start to playtest, these two are important to attend to for good design.
Inertia.
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.
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.

Advice for Solving Equations

2011-11-03T10:31:00.000-04:00

Quick post to share the teaching idea. The preservice high school teachers read

“Advice for SolvingEquations,” Steuben and Torbert, MathematicsTeacher, 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

Letters from Rapunzel

2011-10-27T11:01:00.000-04:00

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."

Ten Rules for Game Design

2011-10-24T09:17:00.002-04:00

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

2nd Fundamental Theorem of Calculus

2011-10-20T18:40:00.000-04:00

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 geogebra.org. 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.

Gazillions

2011-10-15T16:40:00.001-04:00

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
One		Humans (meters)	10^0
Ten	deca-	Orcas, Anacondas (meters)	10^1
Hundred	hecto-	Redwood (meters)	10^2
Thousand	kilo-	Mountains' height (meters), Number of visible stars	10^3, 101010
Million	mega-	Width of USA (meters)	10^6
Billion	giga-	Diameter of the Sun (meters), Age of the universe (years)	10^9
Trillion	tera-	Diameter of the Solar System (meters), US national debt (dollars)	10^12
Quadrillion	peta-	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	10^21
Septillion	yotta-	Diameter of the Universe (meters), Number of stars in the universe	10^24
Octillion	hella- (petitioned)	Diameter of Universe (mm) Mass of the earth (grams)	10^27
Nonillion		Number of bacteria on earth	10^30
Decillion		Mass of the sun (grams)	10^33
		Number of atoms in the universe, Volume of the observable universe (m^3)	10^80
Googol		Possible volume of whole universe (m^3)	10^100
Centillion			10^303
Googolplex			10^10^100

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.

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!

Area Battle

2011-10-07T21:54:00.000-04:00

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.

Fraction Sense

2011-09-30T13:19:00.001-04:00

Watched a nice video about Dor Abrahamson talking about how math is, or should be, about making sense.

In this video he describes a device to let students play with a proportion, and I thought it would be simple enough to do in GeoGebra, so... here you go! I'm pretty happy with the sketch, it's clean and the controls should be simple enough for young students. Close means within 5%, and Wow! is within 1%, so it's easier with a large unit. There's a lot of research supporting introducing fractions with a comparison model before the typical part-whole model, so it's nice to have a tool for that.

Available as a GeoGebra file or dynamic webpage. I've mostly stopped embedding GeoGebra files on the blog because it adds to load time significantly, it seems faster as a separate webpage.What other fraction representations would you like to see in a dynamic representation?

Patterning

2011-09-29T14:56:00.002-04:00

I gave an algebra assessment this week (in SBG style) and a student really surprised me. So I thought I'd share.

The assessment:
Possibly relevant standards.
A. Algebra: representation, operations and modeling

Linear equations and functions
Quadratic equations and functions
Exponential and logarithmic equations and functions
Higher polynomial and rational equations and functions
Functional representation and operations.

The problems: take 10 to 15 minutes to consider the following problems. You may do 1 a couple or all. Your SBG score will not depend on a correct answer, but rather on showing your understanding of the ideas involved. (Of course, good understanding helps find solutions, so it’s not totally unrelated; but it’s easy to give an answer and show no thinking.) Since the answers are not the central issue, be sure to communicate your thinking, process and understanding. If you read these instructions, clap your hands once.

1. Using the parabolas at right, estimate the equations of the curves.
2. Discuss: how do you recognize a quadratic pattern from a table, and what does that have to do with the familiar parabola shape of the graph?
3. Use the idea of parallel and perpendicular slopes to give the coordinates for vertices of a square that has no sides parallel to an axis or to y=x. What is the area of your square?
4. Find a symbolic rule for one of the patterns below and look for connections between the symbolic and the visual. Extend the pattern one step to check your rule.

Analysis:
Problem 1 gives lots of nice information. Whether the student uses standard (harder) or vertex form, what the coefficients mean to them, and if they are consistent in applying that understanding amongst the different parabolas. Almost no one uses the roots, and I've yet to see someone apply regression to points.

Problem 2 pointed out that students over-identify parabolas with quadratics, as most students who struggled talked about increasing slope and symmetry as opposed to first or second differences. (Also that there was a homework that they didn't get to!)

Problem 3 was interesting in that no one started with the points. Everyone who tried it gave linear equations, established parallel and perpendicular, and struggled with how to get the sides to be equal length.

Problem 4 was the surprise. Most people chose the pattern on the left, and used visual identification of pieces to generate a direct rule. Only one student tried the pattern on the right. But instead of finding the quadratic pattern I intended, she noticed that each element was as long as the three previous elements summed. Definitely true for the picture... and got me to wondering if it would be quadratic. Nope. but definitely non-quadratic. I made a Google spreadsheet to compare, and this was unusual that the quadratic matched the sum of the first three pattern so well. I got interested in the ratios as they seemed to be converging. (Here's the spreadsheet if you want to play around for yourself.)

Finally I gave in, and looked up the sequence in the On-line Encyclopedia of Integer Sequences, where it is known as the tribonacci series. (Cute, eh?) This limit ratio is the solution of x^3 - x^2 - x - 1 = 0. I think I'll call it the Golden Ratio.

May your quizzes ever surprise you!

Math(chat)misconceptions

2011-09-23T10:45:00.003-04:00

I missed a good #mathchat last night. Chats are regularly scheduled twitter on-topic conversations. (Completish list; dizzying array of topics; mathchat is Thursday 8 pm (US-east) with a follow up on the same topic Monday at 3:30 pm.) You use twitters nice hashtag search feature to interact with people you may not typically follow (though it is a great place to find new follows). Mathchat, started by Colin Graham (@colintgraham), has a wiki page where topics are voted on and chats are archived. My current favorite way to follow is using TweetChat.

So if I missed the chat, how did I know it was a good one? There's the wiki archive, but also this time Bon Crowder (@mathfour) used Evernote to record and share the TweetChat record. Slick. It was good enough that I wanted to organize the list, which led me to figure to Tumbl it, but it turned into a blogpost when I realized the extent of the chat and that I had never blogged about mathchat before.

One thing that does not come across in this reorganized list is the conversational aspect. In Twitter you can click on a tweet and see all the previous tweets in that specific conversation.

People

lostinrecursion must change misconception: you're either an applied person or a pure math person. Bleh
mathheadinc Misconception: Students below 10th grade can't be in college algebra, calculus...
daveinstpaul Misconception: People with "math brains" learn math effortlessly.
LinaSouid misconception: Math teachers only know/care/love math.
LinaSouid misconception: Girls aren't good at math. Girls don't make good engineers.
lostinrecursion must change misconception: it's ok to say "I'm so dumb at math. I'm not just not a math person." especially for kids
delta_dc misconception: "I'm not good at math." form many of my preservice elementary teachers.
LinaSouid A majority of elem teachers at my school hated math. Spread it to all their students like a disease.
daveinstpaul If I have to pick just one misconception, it has to be "I'm not good at math."
MrHonner For me, this S misconception: "I'm good at math, so it's not important to work hard to become a better writer."

Teaching

lostinrecursion must change misconception: math is math class and homework should look like homework
lostinrecursion must change misconception: you really need to know algebra for the real world.
lostinrecursion must change misconception: practice is the best way to get better. (personal experience actually is)
LinaSouid #mathchat misconception: Math teachers know how to solve every problem, ever!
lostinrecursion must change misconception: you have to play the role of "teacher" to get class to work.
TenMarks Misconception: Memorization is the only way to learn math. (I'm looking at you, multiplication tables!)
lostinrecursion must change misconception: "they" write the problems. I just answer them. Can we tell what "they're" asking?
@lostinrecursion must change misconception: "there are problems I need to solve. I need someone to show me how to do that." - Salman Khan!
@lostinrecursion #mathchat must realize: math is made by humans like the ones in the classroom. So let's make math.
lostinrecursion must change misconception: stick to the book or you're in trouble (aka I don't trust you)
delta_dc Many of the misconceptions in math are result of learners trying to make sense w/o understanding. @graceachen has 2 nice posts.
LinaSouid misconception:" With a graphing calculator I can do anything"--especially from over confident high school students.
lostinrecursion must change misconception: you need the teacher to learn math. You need school to learn math.
lostinrecursion must change misconception: texting in class is bad. (I'm in field theory class enjoying this much more)
rvdemerchant misconception: there is an old math and a new math.
rvdemerchant old/new math? Did the old math think concept. understanding was bad? New math think procedural understanding bad? Nope
delta_dc misconception: memorizing multiplication table will make one better at math. A lot of parents of MS kids say this.
MrHonner I think the primary teaching misconception is "There is 1 right way to solve this problem."
cheesemonkeysf IMHO this is also the primary TESTING misconception!
MrHonner It is related to this misconception: "It is wrong for the teacher to admit they don't know something".
LinaSouid I don't know when math teachers (20 years ago) learned that part of their job was to be socially inept
mathheadinc Probably when they themselves were humiliated, probably at an early age. Vicious cycle

Field

ColinTGraham For me, if it was only one chance... I would want to change the idea that mathematics is about doing calculations.
MariaDroujkova Clear one misconceptions! math=arithmetic
ekendriss Misconception I'd like to clear: Statistics = boring.
MrHonner Major Math Misconception: There's only one right way to solve this problem. ColinTGraham Or: there *is* a right way....
msnorthrup Luckily this is fading with elem tchrs RT @MrHonner the primary tchg misconception is "There's one right way to solve a problem"
MrHonner Misconception: The most important thing is math is whether your answer is right or wrong
MrHonner Student Misconception: "If I try something and it didn't work, I've done something wrong."
shawn_ny That it lacks creativity.
delta_dc Misconception - being fast at math means being good at math.
LinaSouid misconception: You need to know how to prove something to use it and understand it.
LinaSouid: misconception: You can't guess.
lostinrecursion must change misconception: go with your first instinct. Your second is wrong (aka don't trust yourself)
delta_dc Misconception - math is only worthwhile if it applies to real world.
OoeyGooeyLady That u can "like" it but still struggle at it. I didn't need it 2 be easy. Just wanted to have chance to try!!
LinaSouid So true. People focus on perfection. Another misconception!

Specific topics

daveinstpaul Misconceptions: y is always a function of x. 0^0 cannot be consistently defined. √4 has two values. Everyone should take algebra.
daveinstpaul 4 has two square roots, which are denoted √4 and -√4. By itself √4 indicates the positive square root.
ColinTGraham By convention = accepted academic 'whatever'
lostinrecursion must change misconception: fractions are wrong if they're not reduced.
mathheadinc Misconception: This is not how to write a fraction (1 1/2)/4
daveinstpaul Oh, another misconception: All polygons are convex.
MathMatters2Me Misconception: radicals can never be left in the denominator
MathMatters2Me Misconception: Asymptotes are lines a graph approaches but never touches
MrHonner For purely math misconceptions, nothing beats sqrt(a^2 + b^2) = a + b.
msnorthrup As a 5th grade teacher - the misconception that "You can't take a big number from a small number" is rampant and problematic
daveinstpaul Misconception: When proving a trigonometric identity, you must work with one side of the equation at a time.
MrCHRISatCSI You don't have to carry a one!

Is Google cooking the books?

Then what?

LinaSouid Misconceptions will be resolved when math teachers are friendly, social, good role models
ColinTGraham "friendly, social, good role models" depends very much on how people arrive at teaching mathematics...
MariaDroujkova I am making a poster of today's #mathchat about misconceptions. So much concentrated math ed genius! You rock, #mathchat people!
MrHonner We need to remind ALL teachers that they teach kids, not math.

Think I'll have to at least lurk on Monday. I've been putting it on for background music during my preservice HS teacher class. (With occasional comments. Like you could resist.)

Photo credit: bfishshadow, petesimon @ Flickr

Pattern Generators

2011-09-03T11:41:00.004-04:00

I like the theme of the first week of my Math for High School course to be teaching for creativity.

This semester's group did a nice job with Dan Meyer's Toast video. I had them ask questions and record what they noticed. There was a neat dichotomy: the WCYDWT responses were all pretty traditional math textbook questions. The what they noticed branched far afield, wondering what would make a good soundtrack, wondering about darkness of toast and toaster design.

We followed that with a workshop (I think based on an Esther Billings and Pam Wells workshop) based on verbalizing and algebrafying number patterns.

Is this a pattern?

The last workshop of the day was planned to be this, which in the past has been a pretty good activity:

Objective: TLW develop mathematical patterns from the teacher’s perspective.

Schema Activation: when (if you do) do you notice patterns in real life?

Focus: What we talked about in Class 01 was really curriculum. What are we going to teach? Here’s the NCTM’s take:

The Curriculum Principle
A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades. A school mathematics curriculum is a strong determinant of what students have an opportunity to learn and what they do learn. In a coherent curriculum, mathematical ideas are linked to and build on one another so that students' understanding and knowledge deepens and their ability to apply mathematics expands. An effective mathematics curriculum focuses on important mathematics—mathematics that will prepare students for continued study and for solving problems in a variety of school, home, and work settings. A well-articulated curriculum challenges students to learn increasingly more sophisticated mathematical ideas as they continue their studies. (From the Principles and Standards for School Mathematics (PSSM for short), NCTM. All these principles have expanded information and explanation at the NCTM website.)

As we discussed, there are small shifts and large shifts. This activity applies to both: it’s an easy activity that allows for creativity (subtle shift) but may lead to you designing your own problems and questions for students (big shift).

Activity: we have blocks. Play with them!
1. Make a pattern of images with the blocks. A successful pattern for this task is one in which the next shape is determined, or is reasonable.
2. Describe your pattern in words. What’s happening, what’s changing? What can you say about the next step compared to the previous? What will the 10th step be like? A general step?
3. Describe your pattern mathematically. What can you say about the next step compared to the previous? What will the 10th step be like? The Nth step?
4. What connections do you see amongst 1 (the visual), 2 (the verbal), and 3 (the symbolic or mathematical)?
5. Repeat as time allows.

Reflection: choose 2
• Was this task too open-ended? Does it need more structure?
• Was this task engaging? Was it mathematically worthwhile?
• What were the strongest or most interesting connections you saw in a step 4?

But we didn't have time for that! (Not to butter them up, but they were jamming on the toast and I didn't want to cut that short.)

So what we did was:

Schema Activation: whole class discussion - what makes a pattern a pattern?

Focus: Pattern blocks, make what you consider a pattern.

Activity:
1. Make patterns.
2. Put a piece of scrap paper by your pattern with a Y and an N.
3. Gallery walk. Put a hashmark by yes or no if you consider that a pattern or not.
4. Stop by someone else's pattern, add three blocks.

Reflection: whole class discussion about what happened.Record your personal definition of pattern as it is now.

I'm kicking myself now that I didn't take more pictures. Most students made repeating linear patterns, some tessellation patterns, one person made a circular pattern, and then there was...

This was the only pattern that students did not see as a pattern. It was described as just random fitting together, and someone asked about the yellow. "They were supposed to be orange." And then the author shared how they were lines of blocks fit together. Another student shared how to her it was a skewed checkerboard. Then the class unanimously agreed it was a pattern. I thought that was really interesting and asked how they would verbalize it. Good descriptions followed. Could they capture it symbolically? No, not that I expected it. So I shared a bit about the wallpaper groups, and how mathematicians seek the power of good notation so they can symbolically manipulate. I also shared how I had thought the yellow was a pattern in a pattern.

The characteristics of patterns they thought most important was that it can be explained (there is an idea or structure) and someone who understands it can extend it or fill in a missing piece.

If one week determines a pattern, it's going to be a good semester!

Photo credit: The tremendous image up top is from Tanya Khovanova, in this post. The original question was rather brilliant: "Which one of these things does not belong?" Many thanks to Sue, who pointed out the author below.

Bamboo and Mac

2011-09-02T12:53:00.002-04:00

x-a-v-i.blogspot.com

Sounds like an awful sitcom... but it should be a killer tech combination. A colleague and I were discussing how to use a Bamboo tablet to write on pdfs (like scanned student work) on a Mac and I did not know. First response to questions like that: Twitter.

Why not Google? There was a Neil Gaiman quote I saw recently: "Google can bring you back 100,000 answers, a librarian can bring you the right one." Well, Twitter is like a few hundred librarians.

So far I've used Jarnal for writing on paper, and the Corel Painter Essentials that came with the board for more serious stuff. (Like my friend's Juvenile Diabetes Research Foundation team logo.) The Corel can import images but that would be slooow and big to save. Apple's native Preview has annotation tools, but nothing freehand. No support for math typing. Nice tool suite though, with stickies, side notes, highlighting and direct typing. The edits are visible in other programs that you can use to read pdfs.

Twitter says...

@scsocha named Adobe Pro. Obviously great if you have access, which I don't think we do...

I didn't realize Jarnal was a PDF annotator as well, until @bennettscience replied about it. Before getting the pdf, set your paper (Format > Paper and Background) to plain white. (You can do it afterwords, but Jarnal works page by page so...) You can have multiple windows open at a time... flexible and nice. Playing more with it made me realize that I am not using this program enough. It is a Java application, so it lacks some of the feel of a native Mac program. But it runs multiplatform.

@MitchKeller recommended Skim. It has great features, including stickies like Adobe Pro. A nice index of annotations. But the notes were only visible in Skim. When I opened in any other program it looked like the original. That doesn't make a lot of sense to me, so I'm probably missing something. Other than that, the functionality made it Preview Plus, which is a pretty good recommendation.

EDIT: Mitch gives advice in the comments about how he correctly uses Skim for this. "you do have to be sure to export the file as PDF with Embedded Notes and not just save it." Makes sense.

Brian Bennett also suggested Omnidazzle, which was entirely new to me. Turns out that it is a cursor enhancement program, enabling you to do funky stuff with your cursor, like spread pixie dust or sonar locate. However it also enables you to turn your cursor into a pen to write on screen and a magnifying glass. Won't help for this job, but seems useful for online teaching when you're showing your desktop. Especially the magnifier since Macs can't use the one that all my PC colleagues use.

OmniDazzle - Magnify mode

OmniDazzle - Cutout mode

Sketchbook Express was suggested by @kellyoshea. At first I couldn't get this to download - I got some kind of error about 'not available in US.' Finally opened in App Store (maybe it was trying to use iTunes before?) Good enough for drawing to make me think about the Sketchbook Pro for $60. But... I can't figure out how to open pdfs directly. Undoubtedly you could convert in preview, or screen capture what you want to comment on with Jing or the native ⌘-shift-4 (⌘-shift-3 for the whole desktop), but that's not suitable for heavy use.

Skitch was the last program I checked out; just bought by Evernote, so it works well on mobile devices, too. This is kind of like Jing+ on pictures, - Jing's video features. Definitely the best free screencap program I've used. It has already earned a regular spot in the starting rotation for me. Skitch has an online sharing feature (like Jing/Evernote) or can email directly the image you're making. It can open common image formats, including pdfs, but just shows you the first page. It can export to pdf, even starting as a screencap, though. The easy sharing options might make it worth using for student feedback, though. Skitch runs from the menu bar, which is a nice feature.

This has been fun. I'll definitely play more with Jarnal and Skitch, and test out Sketchbook Express to replace Corel Painter Essentials.

What do you use?

Portfolio Mine

2011-08-29T19:06:00.001-04:00

So the karma police have come calling, and after all the portfolios that I have demanded from students I have had to put one together myself. It is for promotion to professor.

I have long thought that my scholarship choices (presentations and inservices over publishing, not that many people don't do both) meant not being a full professor, but this blog has given me a reason to write in a way that fits me to a T. Let me take a moment to thank anyone reading this for whatever attention you've given my writing. It was also clear in writing just how important collaboration is to me, so I'd like to thank any colleagues, IRL or Twitter/Blog who might read this. I have truly been blessed in my vocation.

Since the heart of this blog is trying to share openly and honestly, here's the portfolio. If you have questions or comments, I'd love to hear them. The university still requires a paper portfolio, and it was a job figuring out how to capture it in a binder. Which takes me back to why am I having students make paper binders again...?

By the great Charles Schultz, of course.

Gradual Release of Responsibility

2011-08-20T12:27:00.002-04:00

xkcd: 894

How is it possible that I haven't blogged about this? The Gradual Release of Responsibility has been one of the key ideas to help me improve my teaching. It hails from literacy education, but applies to any kind of learning, and has helped me understand why some people's best learning stories are what they are. It's also very useful at the beginning of the semester/school year as I'm planning a course.

This is the first visualization I saw of this, from Margaret Mooney. Fisher and Frey are also strongly associated with this idea, and their Google book preview is extensive. I was introduced to this by Dave Coffey (of course) and he has a very nice explanation on his blog. Rather than repeat what he wrote, I'll share what it means to me. Sometimes it's a framework for a single class period, sometimes for an entire course, depending on the scope of the objective.

When I first became a believer in active learning, thanks to the reading provided by Sue Feeley when I was prepping my first math for teachers class, I became very extreme about never telling the students anything. I was proud to hear students pass on asking me questions and work further for themselves. On student evaluations, I took their comments about frustration as statements that this kind of learning was new to them. And I do think that's partially true. But one day when a student said, "why are you asking him, you know he won't answer..." it sounded more like I was denying my students support they needed. I realized that their telling me they were frustrated was because they were frustrated. (Crazy, I know.) And humans can't learn when they are frustrated.

What was missing was demonstration. I had stopped equipping my students for tasks because I equated it with telling. What the literacy education reading let me in on was the idea of a think aloud. Authentically sharing your thinking. But the key for me was the idea that while you're doing this, the students are active observers. You let them know what to watch for and debrief them on what they saw in your demonstration. They are still active! But they may need demonstrations on how to be observers. Early on in a class you might hear me say things like, "Oh. I hoped you'd notice how I..." Or just share my own observations. This is particularly relevant in math ed classes, but I use the idea and technique in content classes, too. Hopefully the demonstrations decrease in frequency and duration during a course.

The next phase for me I typically am too quick with, I think. Whole class work where I solicit ideas and suggestions from the class. It's a bit close to my novice teaching style for comfort, so I think I under use it. You can often gauge in a demonstration when students want more responsibility, or when you've seen some developing use of your objective process. The positive reason I skip more quickly to whole class-students lead-I support is because I want students to have the experience of trying without knowing how. If they can get used to that feeling of 'maybe I could...' it will make them so much stronger as problem-solvers. You can also do this phase in groups with active support from you, and I think this is easier with heterogenous groups.

This just makes so much sense to me as a structure, I sometimes feel guilty for having to be shown it. Of course I want my students to be independent, and of course they need to see and experience what I'm asking them to learn and of course there is an inbetween. Gradual Release also helps me keep focused on those big goals that I want the most, the problem solving and communication. It helps me with instruction because I better know what kind of responses are available for where I assess a student to be. A few classes have gotten to the point where they knew this for themselves; "could we get a demonstration of this?" That's an awesome place to be.

Hocking

2011-08-18T14:58:00.002-04:00


John Hocking From a newspaper article 1981 before a trip to China, the year before we met.

I attended the memorial of John 'Gib' Hocking this summer, with three of my undergraduate classmates from Michigan State. Ed Aboufadel, Terry George, and Jim Koss. By accident, my community college calc credit wasn't accepted at MSU, and it was one of the most fortunate accidents of my life. I enrolled in honors math, met people who are my friends to this day, and completely changed my career trajectory. He convinced me to continue on in math, and Ed is the one responsible for me winding up at GVSU.

To say Gib had a fascinating life doesn't do it justice. He went from working at Ford to the University of Michigan, served in army intelligence post-World War II, became an excellent fencer, sailor and race car driver... as well as a respected mathematician. His and Judy's home was famous for its hospitality and at the memorial many testified to the wonders of what they called Three and a Half. He bought the race car in England while on a Fulbright with the proceeds from his calculus text. (Bought by the publisher so it wouldn't compete with their top seller, said the author.) After retiring from MSU, he worked with a grandson repairing sailboat engines, designed the first new mechanical navigation device in a century, learned how to design heating and cooling systems (and designed them), and became an expert wood worker with the third nicest workshop in the world. (My guess is he knew the probability of meeting 3 people with nicer ones at the same time was effectively nil.) The Topology text he wrote with his advisor Gail Young (cf. the math genealogy) was the standard for years and is still respected though old-fashioned. (Dover edition still available!) As a retiree he rewrote large section of a sailing navigation text that had stood for 100 years to simplify and clarify the mathematics.

Ed and I chatted about him this week. Ed recounts how that first semester led him to decide to be a math major. "I want to be a math professor, like that guy." Renaissance man, well-rounded. A mathematician but not just a mathematician. Excellent teacher, Ed tells how he learned a lot about teaching just from being his student. Dr. Hocking really looked at his students and saw what was going on with them. Always made eye contact, and strove to have a sense of where we were. Further, he reached out to us, mentored us. Recommendations, advice, ... personalized our experience at Michigan State which can be rare with 40,000 students. He was a man who was very obviously intelligent but incredibly approachable.

4 spartans in Ann Arbor

We had him for five quarters in a row, and we had no idea how rare that was. He introduced us to topology, like the Alexander horned sphere, Möbius band, and Klein bottle. Shared what problems he was thinking about. Ed notes how he really gave the flavor of what was in store for him as a math student. He taught about the field of mathematics as opposed to just the topic of the day. Always used lots of applications, sailing examples. Ed says he somewhat recently was using the idea of bearings as examples while teaching trigonometry. I'm very grateful to Ed for both inviting him to speak at GVSU a few years ago for our seminar, and for finding out about and organizing our trip to the memorial.

I became a math major because by the time I finished all the stuff that Dr. Hocking had convinced me I needed to take, I was one class away from the degree. The appeal of teaching - based on his teaching - led me to pick the math TA position over the physics lab assistant position. He really formed us into a community. The four of us from the memorial (plus Amy Crammin Shao) were my first experience in a real study group, though we worked with many people from the class over the years. He had the whole class out to his house for Christmas. At the memorial, Bill Sledd, one of his colleagues from MSU (and another influential teacher from my past) remarked on how that group of students sent a remarkable number of people on to grad school.

Manipulative from Gib's GVSU seminar

Gib had tricks he'd pull out in the classroom, like a fancy move for drawing an extra large perfect circle, or writing behind his back while facing us. But mostly he was just tremendously authentic, sharing important stories from his life and his genuine thinking about his problems of the moment. We felt like we got to know him. A favorite pet memory of mine is the whole class chipping together to get him a Rubik's cube (expensive in the early 80s) for Christmas. He opens it, turns it this way an that, then says, "Oh," and solves it the first time in a minute. What a lightning quick, fascinating playful mind he had. Several people remarked at the memorial how much he loved games and I love to think what he might have done with games in the classroom.

Same sparkle at 90.

So raise a glass and toast to the teacher who inspired you most. Here's to Gib!

Special Thanks: Wendell Hocking, who shared the pictures of Gib for this post. And to the whole Hocking family for letting us share in the memorial.

Appendix: The American Mathematical Society shared the following notice on their "In Memory Of..."
John G. Hocking (1920-2011)
Hocking, a member of the faculty at Michigan State University from 1951 to 1987, died March 23 at the age of 90. He received his PhD from the University of Michigan in 1953 under the direction of Gail S. Young. Hocking and Young wrote a text, Topology, that was widely used. Hocking was an AMS member since 1951.

MCTM thinking

2011-08-15T15:32:00.001-04:00

Marty Hogan @ Flickr

I didn't get to go to a lot of sessions at MCTM this year, as I only went one day and had to present a workshop twice. But the three I did get to provoked a lot of thinking.

One session was a teacher sharing her structure for long division. She has a very clear and concrete worksheet style sheet they fill in. Some people would call this terrible, some would call it terrifically clear. What I love is that she was sharing her work that made a difference for students. Her goal was to make visible the parts of long division that were transparent to her students. She works with students that have failed in math before, accumulated negative attitudes and feel sentenced to math class. With traditional long division instruction, she felt there were parts hidden from the students. Not knowing how to do those parts or even that those parts were there, they can't do long division despite being in high school. The teacher noticed with these sheets she had developed that the students were more engaged, felt like they could do it, achieved a higher success on these problems, and were able to move on. It connected for me with hearing Paula Lancaster talk about the benefits of structure in Universal Design. The teacher next to me asked lots of great questions: "were students able to use this structure without the sheets?" to "what was it about this structure that helped students?"

Another session was Danielle Seabold from Kalamazoo Regional Ed Service Agency. She connected the new Common Core standards for literacy to those for mathematics. Nice for me to see was that many of the strategies from Mosaic of Thought have appeared in the CCSS and I love how those transfer to mathematics. But more specifically, the Common Core has a Literacy in Technical Subjects component. Danielle did a great job leading a discussion about these, prompting us to look for connections and consider applications and opportunities. She links to many CCSS resources at her blog.

The last session I got to attend was about implementation of Standards Based Grading by Amber Cross and Jason Gubeno from Dansville High School. Very exciting. Entirely teacher-led, with great support from administration and buy-in from parents, they have replaced traditional grading over the last three years, inspired by professional development with Carol Commodore and reading including Marzano on grading practice and purpose. They are very happy with the changes this has helped bring about in their students' learning. It has correlated with modest increases in test scores, but more importantly has changed student attitudes and focus.

From PAEC SBG course

Jason and Amber referenced the parachute example to support their intuition about why make the change. They have a pretty clear system, one reassessment, a lot of emphasis on student accountability. And they have modeled some excellent reflection in considering what worked and why, and how they might adjust. I would love from them to write something up for the web about both their practice and their journey.

I was going to send them this great xkcd comic, but then Frank Noschese wrote a great little SBG post on it so I could send that instead.

Teaching: Hard or Easy?

2011-08-12T11:48:00.001-04:00

It's been interesting on twitter this week, in itself not unusual; a recurring topic has been teaching hard or easy. A colleague was almost politely flamed for a statement which caused the reader to infer that teaching was easy. It was funny to me because a long-running joke is that we want t-shirts with "TEACHING HARD." Then an internet colleague started tweeting and wrote about how teaching is easy.

I agree with both?

(Once again, I wish I was a graphic designer.)

On the Teaching Easy side is Aristotle's argument. (HT to Matt Wyneken who is a good spokesman for this.) Teachers, gardeners and doctors have natural work. People learn, plants grow, and bodies heal. These professions seek to encourage them to it better. The principles of learning are something with which we all have familiarity, since we have all learned. And I do think if my preservice teachers can just learn to assess student's understanding and reflect on their practice, that no matter where they start they will attain teaching excellence. That's a pretty simple recipe. Learning is exciting and furthers our purposes and people really enjoy it when it's authentic.

On the Teaching Hard side is the Problem. Each class is composed of 30ish (or more) individuals with completely different experiences, preferences and purposes that we are supposed to guide to an equal understanding of the same objectives. We often don't get to pick the objectives, or our materials, or our consequences, or our pacing chart or... And the people who have the most say about our conditions seem opposed to listening to teachers because they are partisans on the matter.

Furthermore, the individual problem of the most challenging students is , finding problems for them, determining effective support for someone who seems to learn so differently than you, overcoming years or decades of negative and false feedback in a culture that says our subject is impossible for all but a select few and irrelevant. You have to solve entry problems to get an opportunity to work on your essential problems.

Frankly, the difficulty of the problem puts most mathematical problems to shame. Reminds me of one of my favorite quotes: "If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is." ~John von Neumann

It is real work. A true vocation. I'm so blessed to both be a teacher and work with teachers.

same as it ever was

Love to hear what you think: hard or easy?

Photo credit: Richard Masoner @ Flickr

More GeoGebra 4 Teachers

2011-08-07T23:46:00.003-04:00

I presented two 1.5 hour workshops for teachers at the Michigan Council of Teachers of Mathematics annual conference ~~yesterday~~ Thursday, and thought I'd share a few quick thoughts. Here's my session page/lesson outline. All handouts are attached. Even in a computer lab with somewhat slow connections, running Internet Explorer, we were able to have the GeoGebra 4 beta installed and running in 10 minutes.

EDIT: Later I used those materials to share GeoGebra with the Muskegon Community College Math Tech Bootcamp and a workshop with at the Kent Intermediate School District. The materials are updated from that, and feedback from all three groups are below.

It doesn't take a lot of experience to introduce teachers to GeoGebra. I mostly just gathered resources, talked about the basics, then let them explore. It's worthwhile for me, too. In a room where many people had not even heard of it before (yet they are at the session - I love teachers' exploratory spirit) they found new corners and features for me to think about. I am no omega-class expert, but I've spent some hours with it. This is a rich program they're giving away for free.

The basics to me are:

understanding the main areas:

Tool bar, including pull down tools. Emphasize the selection tool, the move graphics view and zooms

Graphics area

Algebra view and the View menu for axes, grid and algebra view

Input Bar

The selection arrow
Undo
Object properties/Right-clicking objects

Just a few minutes and people are ready to roll. I made up a page adapted from the 2-day workshop of introductory algebra and geometry tasks, and then had options for further exploration of either. All the handouts are attached to the session page. By the end of each workshop, most teachers had made something that impressed them. A few just thought it would be valuable for making images for tests and handouts, and I think that's a proper way to start for some people. But most were digging in deep, and several got some math learning out of what they did. "Oh, that's why..." was overheard a few times.

The teachers recorded their comments on a Google doc (benefit of a computer lab session) and they're embedded below. If you are a GeoGebra user - share it with your fellows! If you are not, give it a try. You'll find it worthwhile within an hour.

Or I'll double your money back!

MCTM Feedback

MCC Math Tech Bootcamp Feedback

Kent ISD Feedback

Quadrilateral Diagnosis

2011-07-27T15:35:00.001-04:00

I was inspired to do this by the neat quadrilateral hierarchy sketch shared this morning on Twitter. But I got wishing they had made the types accurate - that you could only make squares in the square spot. And that their hierarchy used the inclusive definition of trapezoid. (Pet peeve of mine.) Then I thought what if the types lit up when you make the shape? That led to the sketch pictured below, available as a ggb file (EDIT: in GGB 4!) or as a webpage.

When I included it as an applet here, it just didn't work as smoothly as it does over at the geogebra hosting, or by displaying the file directly in a browser.

It was very fun figuring out the conditional tags to make the names show up. I think I've covered most of the corner cases. Figuring out a way to do convex/concave and quadrilateral or not

It was quite handy knowing multiple definitions of each type, which me wonder about a scaled down version of this as a problem to assign to students.

Where do you stand on the trapezoid definition? Is a parallelogram a trapezoid? (I say yes!)

Doodle Jump Math

2011-07-24T14:27:00.001-04:00

If at some point in this post you don't say "that's a bit of a stretch," I will not have done my job here.

I got an iPod Touch for work this summer. Despite being aggressively pro-tech, my personal tech level is loooow. No cell phone, no iPod, no iPad, no video game system ... ridiculous, really. But I'm working with Alejandro Montoya, a computer science grad student, this summer (following my colleague Char Beckmann) as he develops a cool iPhone quadratics game. (Due in the app store for free any day now.) One of the problems was a ridiculously low number of devices, so, time to invest. My colleague Paul Yu and I have an NSF proposal in to equip a classroom with iPod Touches (among other tech) and I'm a believer in really using things before asking students to do so. Paul and Dave Coffey use their iPhones well to support their class.

While developing the game, Jon Engelsma, director of my university's mobile development lab, wanted the game to use more of the iPhone specific capability. Each phone/pod is equipped with an accelerometer - which is why it can detect orientation and movement like tilting. In response, Alejo added a new aspect to the game where you're making the parabola in the air to show orientation. But along the way, Jon mentioned Doodle Jump as an example of a game that used it well. "Oh, only 99¢," I say.

Ruh roh.

At the time of writing this, I'm at a high score of about 20,000. I swear I don't use real time to play, just moments where I'm stuck somewhere.

There's math in that game?

My son asked that in surprise. As Alejo mentioned to some of the high school students playing ParabolaX, there's a LOT of math in programming and game design. Frank Noschese has written so much good stuff on the physics of Angry Birds. Doodle Jump has that. I've thought about how far up and across you can get on a jump, use constant speed estimation of moving platforms, etc. But other than doing some modeling (which would be interesting I think) of the jumps, there's not a lot of explicit math for players. There is, though, an understanding that the game is on a cylinder (go off the left, return on the right, etc.). That's made me wonder if anyone has developed an iPhone game that really uses the accelerometer to explore topologically interesting surfaces. A Möbius maze that you navigate with the accelerometer? A Klein bottle version of Othello? Who knows where that could lead.

There's math in that game
The Common Core State Standards Mathematical Practices:
1. Make sense of problems and persevere in solving them.
At its heart, like many games, Doodle Jump is a big problem (get as high as possible) composed of smaller problems (how do I get past the pink monster reliably). My perseverence, like most students, is very high for games.

2. Reason abstractly and quantitatively.
The constraints are simple: you can't stop jumping, and if you land on nothing or bump into an obstacle, game over. There's not much quantitative reasoning, unless you're fixated on a score. Then there's some nice linear programming on the fly to figure out how to improve your score and what's required. I have wondered why, being extremely right handed, I seem to be better with my left on this game. Specific game questions too, like why don't I see rocket packs anymore?

3. Construct viable arguments and critique the reasoning of others.
Hasn't happened for me yet, because I haven't discussed the game with anyone other than my son, who's even more of a novice than I am. But there's definitely opportunity, as we've discussed: Does a back lean enable higher jumps? That's my son's conjecture but I haven't experienced it yet.

4. Model with mathematics.
Also not yet. Though I am interested in trying to model the jumps, and I'm curious about how to even start. There's good reason to measure to improve your game play, but the modeling is the kind of idle mathematical reasoning that mathematicians love. I'd also love to know how the accelerometer data feeds into the shape of the jump. Constant horizontal velocity if tilted or does it depend on the angle of tilt?

5. Use appropriate tools strategically.
Not yet. But the modeling will definitely be helped by mathematical tools.

6. Attend to precision.
Hmmmm. Kinesthetic precision is definitely required. It matters how you move and how much. The engagement of this again has me wondering how to make math more kinesthetic more often.

7. Look for and make use of structure.
Very rich context for this. Even at my lower levels, there are many patterns to notice, and noticing them is crucial to survival in the game.

8. Look for and express regularity in repeated reasoning.
I think it's a new thought to me that this is a crucial part of video games, and I want to use this connection in math class. The skills required early on become automatic and no longer a problem. This is just natural as you get better, you gain automaticity with tasks that used to require thought. Even though you may make occasional mistakes. Does that describe math or video games?

What am I learning about teaching?
Half of learning about teaching is learning about learning. This game has been good for me to think about because I'm not very good. There was a brilliant teacher educator at Siena Heights, Sr. Eileen Rice, required her advisees to take a class in a subject with which they struggled. Great idea. Just like with a math problem, we can't problem solve if it's too easy for us. It's struggle that affords an opportunity for growth.

In the game I've had to figure out specific challenges, wonder about how things worked in general, identify areas where I need improvement, and make some realizations about my limitations. I can not shoot effectively. My videogame dexterity and reaction time is probably below average. But I've had nice moments of achievement. Figuring out how to get past a few beasts, then that I can jump on them, specifically working on using the cylindrical aspect, etc. Things I couldn't do before that I can do now.

The other half of learning about teaching is tackling the question of how do I support learners? That's the heart of instruction to me, and what motivates gathering data (assessment), giving feedback (evaluation) and problem and resource selection (planning). This game has made me wonder about all these.

Do I look up directions? Cheats? The game is popular and there are lots of tip sites out there. Even Doodle Jump cheats. This requires me to think about my purposes. Do I need the highest possible score? Do I want to figure it out for myself? What's the purpose of playing? Once I was a teacher that told students this information without asking. I was good at it, and got good evaluations, and most of my students did really well on tests. Then I was a teacher that would never tell this information, even if students basically begged for it. Students did some amazing work, found out things I hadn't known, and most were successful. But some students were frustrated, including some of the successful students. Now, I try to assess student purposes and provide relevant information. But it's harder than having a simple extreme policy.

Wow, I made it through that whole paragraph without mentioning how Khan Academy can be like those tips and cheats YouTube videos. (Shoot.)

Talk. It's also better with other people. I've been hampered by doing this game alone. Looking at the practices made me think about the richness I'm missing. I've seen this in Alejo's pilots of ParabolaX also - radical differences in what students get out of the game based on how much they discuss it. I want a healthy balance for my students between 'let me do it for myself' and 'how are you thinking about it?'

How do I measure success? I like the score as a measure of how far I've gotten, or as a measure of whether I've gotten better. But as I consider that, there's really better things to notice. The game territory changes as you climb, so that's a good measure of how far. (When I get to the bounce-once platforms currently, that's a good game! In the jungle setting.) Can I make difficult moves? Have I gotten better at getting better? It feels like I have better learned how to identify problems in the game and am more efficient at addressing them.

Then I got wondering what the scores even really mean. Is my high score the best measure of how good a player I am? Should it be my average or median score? Weighted somehow between the two? If the goal is the maximum score, it encourages high risk behavior that's bad on the average but when it works garners big points. Score-based thinking also pushes me towards tips and cheats, which are not in my best interest as a learner or enjoyer of the game. The best use of the scores is a nice mathematical problem, like a simpler version of trying to figure out what are the relevant baseball statistics. In general, though, it encourages me to go farther in the direction of SBG and portfolios.

So...
If you made it this far, thanks for sticking with the rambling. I'd love to know what you think about this, or other thoughts you have about teaching and learning from games or other odd contexts.

P.S. Yes the title is a little poke at the now ubiquitous Jump Math, which has some super-proponents. (There are samples at jumpmath1.org)

P.P.S. We need some games this engaging with more math content. Waker and MangaHigh are a start, but the accelerometer using games will be a big step up.

Image credits: there were no good CC images for this post, so if I used one of yours and you wish it not, just let me know. All images click through to original source. The ragecomic was too true not to include.

iPad Brainstorming

2011-07-22T13:22:00.003-04:00

Brett Jordan @ Flickr

In general with technology, I think there are few must haves. I much prefer tech that is usable in pieces rather than by wholesale adoption. For example smartphone use vs TI-Inspire. This is written with iPad in mind, but would apply to any tablet computer, I think.

iPads and tablets seem like that kind of technology. For my school several of us were asked to think about how teachers could use iPads. Most of the uses involve students having the iPad. So I thought I would put down what I think of, and then troll for other suggestions in comments. I'm looking for ideas that involve the teacher bringing one iPad into the room. Also if there are ideas to distinguish it from a smartphone or laptop.

Ideas

Give it to students. Do they need to look something up, compute something, watch a video, capture a video, read a reference text... The larger interface makes it more suitable for group work.
Constructive use of social media in the classroom. Backchannel, Google Groups or Plus, Edmodo, etc. Recently did #mathchat on Twitter with a class and it was a drag bopping back and forth to the classroom computer and taking dictation on class comments. Instead I could have said, "here's the iPad."
Document your experiences to share with students. My colleagues Dave Coffey and Sean Lancaster do this already. Twitter, Evernote and I suppose now Google+. Portability > laptop, interface > smartphone.
Portable reference library. Mobility > laptop, readability > smartphone.
Data collection in class. Check off attendance, notes on student work or participation. Hard to carrry your laptop around to each group, easy to carry iPad. I used to so some of this on a palm but it was quite clunky. Apps for this are developing rapidly. BlackBoard Mobile is being pushed hard and may figure in for GVSU.
Assistive technology. Differentiate your lessons for a student with distinct requirements for access to a lesson. (Spellcheck suggested Assertive Technology - I like that, too.) See for example this post on iPad assistance (via @langwitches on Twitter).
Moves towards paperless workplace. All those meetings where there's handouts for each teacher that move immediately to the recycle bin...
Meta-use: model new technology adoption and integration for your students. By trying out new tech and sharing your process, you are modeling towards their Technological-Pedagogical-Content-Knowledge development (for teachers; see http://www.tpack.org) or the similar structure in other fields.
Who knows? Apps are being developed at a break-neck pace. Putting these devices in teachers hands will expand their capabilities in ways we don't even know because they being created tomorrow. Cf http://www.mobilemouse.com/ that will allow you or students to control the classroom computer from anywhere in the room. There's a dedicated iTunes room for teacher apps now. (Warning iTunes link opens in iTunes.) Quite an opportunity for innovation in scholarship of teaching or even collaboration with the University's own mobile development lab. Or for students who are budding developers.

bbspot via forevergeek.com

References/Resources:

Apple/Education/iPad - the Mothership
iPad for Teaching thread at macrumor
Innovative Educator blog (also several other posts)
iPad in Education blog
Teach with your iPad wiki
Jim Spirialis' (@adelaidelad) iPad for Education LiveBinder

Storytelling

2011-07-14T14:06:00.001-04:00


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.

GeoGebra 4 Teachers

2011-07-07T14:25:00.001-04:00

My current logo attempt.

I had a great experience this summer co-leading with Michelle Bunton a GeoGebra workshop for teachers. We decided to do a loose structure, emphasizing algebra one day and geometry the second. People were free to register for one or both days. We got about 22 teachers, with 14 for both days. One teacher was bitterly disappointed because they wanted premade activities, and our focus was on learning the program. Though we tried to connect people to plenty of resources and the wide-world of GeoGebra sketches, this teacher left before that. Most of the teachers took the freedom and ran, and it reminded me of what a joy it is to be in a room full of independent, motivated learners working on stuff that matters to them. I learned a lot, which is typical of such situations.

We decided to run the workshop using GeoGebra 4, as it's being released at the end of the summer. Unfortunately - I was a novice on it. I'm an enthusiastic GGB 3 user, but have been weak on spreadsheet use. Adding more novel features was a little scary. Turned out well, as it made us co-investigators with the teachers. And the program is great. I mean it was great, but now it is greater. They've managed to add features without making it perceptibly more complex. That's rare; I love this software and its developers. Guillermo Bautista's GGB4 sneak peek series was very helpful. (Of course!)

We structured the workshops with time to get software loaded. And I wanted the teachers on Twitter to converse on backchannel through the day. That was hit or miss, due to Twitter's tendency to ignore new users as an anti-bot measure. I wound up having people follow me so I could follow them, then retweeting their tweets. This made some people show up but not others... mystery. I thought it was important because I get some of my best GeoGebra support on Twitter. And, in fact, during the workshops we got some excellent input from @mike_geogebra and @lfahlberg .

So startup, a demonstration sketch to show some of the potential for classroom use, an overview of the program parts (tool bar, menus, views, etc.), specific tasks to figure out how to use, and then free explore time. The workshop website has all of our materials (see the workshop page), plus the sketches created by participants. One exciting feature is that participants spent time sorting and classifying sketches by standards strand in a Google spreadsheet with links to the sketches.

One problem were still working on is the "now what?" We're trying a Google group and want the website to morph into a place for teachers to share sketches. (If you are interested in joining the page to share your materials, just drop me a line.) Teachers also wondered aloud what made this experience useful in comparison to some other professional development, and that's worth looking into.

As I said, I learned a lot: about professional development, workshop development with a new partner, and a lot of GeoGebra. The GeoGebra knowledge I could put into a sketch! Note that the sketch links below use GGB4, which can download as a Beta.

Dave made a nice height vs time projectile sketch that used text box inputs. (That is my favorite new feature so far.) Cristine got me thinking about more subtle show/hide condition with her beautiful Unit Circle. One of the participants made a face sketch for which he figured out how to limit the domain of a function. (Which he seems to have never uploaded!) @lfahlberg taught us how to make a reset button for a sketch. Chris and Jason just pushed and pushed and explored in general. I can't remember why we figured out that sliders can have calculated min and max now.... I was seriously impressed at what good use everyone made of their time.

After the workshop I made this projectile sketch to practice. It's a projectile in the x-y plane, where the slider advances or animates time. You put in the initial height by textbox, set the vector of your throw by dragging the vector, and can do target practice to a garbage can or by throwing at a seagull. (No actual seagulls were harmed yotta, yotta.) The button resets the targets. This was fun to make.

The big screen shot is an animated gif - one of GGB's new export formats. Go ahead and click on it! The garbage can was inspired by Dan Meyer's WCYDWT ball toss, which is also a sketch.

GeoGebra is a supertool, which even has the power to get more powers, and I definitely want my students and preservice teachers skilled in its use.

Centering

2011-07-06T17:09:00.000-04:00

I've been loving the posts people are writing for this year's virtual conference responding to the prompt: What is at the center of your classroom?. The conference is an amazing collection of writing put together by Riley Lark, blooger at Point of Inflection. (Which I've always liked because it sounds - and reads - like he's synthesizing intention and reflection.)

This is an attempt to emulate my betters, which is actually not a bad strategy for improvement.

I want the center of my classroom to be empowerment. As a bad beginning teacher, emulating David Letterman of all people, I realized that I loved teaching math. I'd tell people that there weren't many things you could teach where the student would literally be able to do something at the end of the class that they couldn't do at the beginning. While I still like that, I now think it can happen in many more places than math class, and have a much different idea of what I want the students to be able to do.

On my own I got to realizing that problem solving was what I really wanted to teach, and my friend Sue Feeley introduced me to Polya (figuratively) and the other NCTM process standards. Both helped give me language to describe things I had realized, and both indicated a path to set out on. Vygotsky helped me understand why students responded so differently to the same task.

Mosaic of Thought, Cambourne's Conditions of Learning and other literacy education work helped me start to understand how to teach processes, and I don't know that I would ever have found that were it not for Kathy and Dave Coffey. The conditions are the heart of what I want for my students, and creating or nurturing those conditions is what I see as the main responsibility of my work.

Engagement is first and foremost. Cambourne notes that engagement requires learners to believe:

they are potential doers of what is being taught. This fits with and explains the idea that the students need to be the ones working in the classroom. (See also expectations)
what they are doing matters to them. This is why teaching the processes is so important to me. Polynomial division does not matter to 99% of students. Problem solving will make 100% of my students' lives better.
they are safe to try. This is why classroom community is so important. (See also approximation.)

Cambourne links engagement with two conditions:

Immersion. Learners need to experience real and rich mathematics of all kinds. Still one of my measures of how rich a question is is to consider how many processes it invites learners to engage in.
Demonstration. Learners need many and authentic demonstrations of what they are learning. This was huge for me. I had become a hardcore discovery-based teacher. I was proud of my students saying things like "Why ask him, you know he won't tell us." (Oh, that is painful to me now.) I took students' feedback about their frustration and decided that it was because they weren't used to this mode of learning. But really, I was asking them to do things they'd never seen. Asking them to learn how to dance when they'd never seen one. Coupled with a lack of good feedback, it's a real testament to those learners that they got as much out of it as they did. Adding demonstration let me back into the discussion. Not to tell the learners how to do it, but to share with them authentically how I think about it. I make space for them first, as I prefer if they're demonstrating to each other, but I watch and assess for when they need demonstration as a support.

The conditions that make engagement more likely are:

Expectations. This goes hand in hand with the Equity Principle from the NCTM, which is near and dear to me. I believe all people can do significant, important mathematics. I really do. I try hard to communicate this to my learners.
Responsibility. Learners make their own decisions about when, how and what to learn. THIS IS ANARCHY. I know. It's dangerous, especially when our learners have been trained for dependence and helplessness. Most students are not ready for full freedom immediately, but it is my goal for all of them. It's also my ongoing struggle to get students to understand that I both mean this and it doesn't mean that work is optional.
Employment. Learners need time to try out their learning in authentic situations. This connects with real life mathematics, with project based learning, with discovery learning and more. The students need to be the ones working if they are to be the ones learning.
Approximation. Learners need to be able to make mistakes without fear of punishment. If there's one area that math has completely screwed up on in the past, it is this. I do it, now you do it perfectly. This is crazy. We all know that no one learns anything important this way. And that the mistakes people make are crucial for learning in the first place.
Response. Learners need real and meaningful feedback about what they are interested in working on. I now have my students put stickies on any work they turn in with what questions they have for me. Cambourne: "response must be relevant, appropriate, timely, readily available and nonthreatening." Grades are not feedback in this sense, and can only strive to be timely and readily available. (Although that does make grades better so far as it goes.)

Though I have made progress in creating the conditions, I have a long way to go. (Although that implies an end to the journey and there may not be one.) They help determine what I teach, what my classroom is like, what I want to know about the students and their learning and how I evaluate their work. That I would not have found them without colleagues in my professional learning community speaks to the heart of why that is important. So thank you, to Sue, Dave and Kathy, for helping me get this far.

Linear War

2011-07-04T15:01:00.001-04:00

So many things to write about to catch up... but it's been a while since I posted a game, so with an impending Math Teachers at Play at Math Mama Writes. The submission form seems to be wonky, so submit directly to Sue. Plus this kind of fits with U.S. Independece Day, as we have been known to fight a war or two.

This game is good already, but could be great. So if you have feedback, let me know, please! As is, it's probably best used as a review game, but I'll comment afterward about how it could be used as a framework for a unit.

Set up: Make your own deck: 11 lines. Each line should be drawn so that it passes through at least two points with integer coordinates, such as (-2,4) or (5,5).

Claim your deck! Mark each line card on the graph side with your insignia. Initials, emoticon, math symbol, etc. – your choice. Tip: make your cards NICE and personalized. Decorations and alterations that do not obscure the line or the math are not only permitted but encouraged.

War: 2-4 players. Each player needs a deck of 11 face down cards, shuffled or not – it’s up to you. Set aside any extras, make one more if you need it.

Players roll the die for the combat. (2nd roll and beyond, the winner of last battle rolls.) Flip over the top card of your deck and follow the combat rule. On a tie, flip over one more card to determine the winner of the battle. If more than two are playing, this is only on ties for best and only the people who are tied.

Play through the deck once. The winner is the player at the end with the most cards. Give cards back to the owner. Except for the Spoils of War.

Spoils of War: Out of the cards the winner captured, they take one card from the opponent’s deck to keep. Add your mark and cross out theirs. This may mean the loser needs to make a new card for their deck for the next game.

Example: The first roll is a 2. Least slope. -2 < 1

(The X and lemniscate are the players' personal marks.)

Math notes:

Use the cards for sorting activities before playing.
Have players keep track of hard to determine battles.
Discuss card design strategies.
What about undefined and zero slope lines?
What other combat rules could you have?

Handouts: as a Scribd file, and the graphs template. At the right is an image (larger when you click on it) that you could also print for the graphs.

Discussion: Ted, one of the excellent summer grad student/teachers, tried this cold as an end of year activity with a small group, and they struggled with it. He felt like it had a lot of promise, but that the math requirements kept students from the game since they were rusty with it.

Trying this with teachers convinced me of it's potential, as it even uncovered math for them to discuss, and generated situations they had to think about.

I could see this game being at the end of a linear unit, where students have been generating graphs as examples as they go through the topics, using them for activities like finding slope, sorting from least to greatest x-intercept, y-intercept and slope. Use them to construct tables or find equations. Non-contextual, but strong on representation. What do you think?

I'm trying hard not to use too many unlicensed images but this is too perfect. God bless you Bill Waterson, wherever you are.

Grading

2011-06-30T15:12:00.001-04:00

It was disconcerting recently to realize that my grading policy is still dysfunctional. As much as I've moved towards standards-based grading for content, end result evaluation, and student-chosen exemplars, there's vestiges of teacher control and obedience.

With my spectacular grad class this summer (see here and here), there's still an attendance policy. I felt a conflict because going by my own policy would mean giving a grade or two below A to students who exceeded my goals for the course. I asked the students how they felt about it. All inservice teachers, they asked "were the goals met?"

I was lucky in that this was about an A, A- or B+. No heart-rending decisions. But in another situation, it easily could be.

In the end I did the right thing. But now I'm considering how much coercion I want in my syllabi. I make class worthwhile, as much as I can. (This class's most frequent student evaluation comment was wanting more classes to explore what we did in more depth.) Do I have to try to make my students come to class? Is it coercion or support? You know, to help them make the "right" decision.

As it is, I get requests to have more due dates, and require things from them more frequently. Do they need it? What can I do instead, with which I can live? Help!

Images: from the excellent but on hiatus wetherobots.com.

A Teacher Talking

2011-06-19T10:37:00.000-04:00

It's a little hard to share this because he speaks so nicely about our class. But my realization is that it's really his work that he's talking about.

Our end of semester assignment is one I cribbed from Maria Anderson at her college math/tech camp. She asked us each to do a Little Big. That is, a little presentation that gets at the big ideas. I ask the teachers to make it available over the internet, as I see that as an important professional participation tool. Slideshare, Prezi, Google presentations or even a PowerPoint file shared via box.net. Bill is the first person to share by webcam, and the result seemed pretty powerful to me. He gave permission to share it more broadly, so you get to see what you think.

Bill is an experienced teacher who speaks his mind directly and is very free to share what he does and listen to others. Follow him on Twitter at @breenw77.

Post a comment here or tweet him directly with your response.

Teachers Talk

2011-06-17T18:05:00.003-04:00

A recent blogging assignment in my grad class was this:
Blog: Your choice. What about your thinking or practice do you want to share with the world at large. This can be a record of something you’ve done, a particular activity to share, work from this course to share, or an opinion piece on an issue of the day (Khan Academy, standardized testing, Michigan education funding, ...)

They wrote interesting pieces, and I love hearing teachers' voices about that which they care most. Four of the teachers have public blogs, so I'll point to theirs with this post. Leave them a comment, encourage them to keep blogging! Those who were blogging on Blackboard, I'll quote more extensively. Post a comment here for them on how they should export and continue their writing!

Bill: Credo about what it takes to teach mathematics.
Also see: Bill's take on the 3 Act Story (I'm quite interested in this as a structure. Here's a short urli.st of blogposts I've found on it.)

Eric: Textbooks. "The secret to being a lazy math teacher? Good textbooks!..." Good hook!

Also see: HS grading

Melissa: only assignment she missed all course! Plenty to read at her blog, though.
See: Her description of her lesson planning process.

Ted: quick take on the public's perception of teachers and Summer Vacation.

Also see: his outstanding concept map for linear intercepts. (Not a blogpost, but wowser.)

Amy: Traditional Teaching
Taking this class has really got me thinking about the way that math is traditionally taught, and the way that I teach math. It has opened my eyes to the importance of teaching students how to problem solve and think critically. This has caused me to feel uncomfortable in my classroom for the past week or so. I feel that I want to make some changes, but yet it seems so overwhelming. The math department I work in is very concentrated on "everybody doing the same thing." This includes assessments and lessons. Also, the time that it takes to develop new tasks for student also is a daunting task.

So, as I reflect about changes I want to make for next year. I am thinking about making small changes. I think my first focus is going to be on formative and summative asssessments. Providing more opportunities for mastery rather than completion. I would also like to incorporate reflection on a daily basis to get students thinking about their problem solving and being able to put their thoughts into words.

In the future, I would like to work towards having more discovery and problem solving activities related to the concepts in my classes.

Erin: the switch from Michigan's High School Content to the Common Core State Standards
I have to complain for a minute about the switch from Michigan standards to the new Common Core standards. I have no problem with switching to common standards, I have no problem with the content of the standards, and I don't even have a problem with standardized tests based on the standards. My problem is this...

We are supposed to begin teaching to the new standards this year with testing based on the new standards to begin in 2 years, however, in the meantime, we are still being tested on the Michigan standards. That doesn't make any sense. The standards really are different in some ways and we are in the process of designing our curriculum based on the new standards. We have to be careful for the next two years that we also teach the HSCEs becuase that will be on the MME.

Someone tell me why we are rolling out the new and testing the old. Is testing really that important that we can't miss 2 years? Perhaps at least we can remove some of the consequences in the meantime so that we can develop a coherent curriculum without fear of the government taking over our schools.

Monica: Differentiation
When I took my Curriculum Development Class, we focused on Understanding by Design and Differentiated Instruction. Prior to that class, I had believed that differentiated instruction was synonymous with individualized instruction. And it wasn’t until the last 4 weeks or so that we started to talk more and more about what DI was, and how to adapt lessons and differentiate them.

First, some things I learned about DI:

Quality differentiation begins with a growth mindset, moves to student-teacher connections, and evolves to community. In a growth mindset students persist in the face of setbacks and see their effort as a pathway to mastery. Students embrace challenges, learn from criticism, and are able to find inspiration and motivation in others’ success. Rather than plateauing with skills and knowledge, students with a growth mindset reach higher levels of achievement.
Quality DI is rooted in meaningful curriculum (not fluff!)
DI is guided by on-going assessment which is used not for grades, but for instructional planning and providing feedback.
and DI addresses students’ readiness, their interest, and their preferred method of learning.

Different methods of differentiating instruction include using

Choice Boards--like a tic tac toe, where you have 9 activities listed, one in each box, and have students choose whichever 3 activities they’d like to do, as long as they make a tic-tac-toe. These activities should be rooted in the same learning objective, but address different learning types (multiple intelligences).
Cubes or Think Dots--cubes would be using a net for a cube and having one question on each of the six sides. Think dots follow the same idea, but the students would roll a die or number cube, and then do the problem underneath the number they rolled on a worksheet (the worksheet would have 1 through 6 on the top, and the 6 problems listed under it—rather than making the cubes, now you just make a worksheet, and use a number cube).
Sternberg’s Tri-Mind--list three different sets of directions to address the same objective. One way of addressing the objective would be analytical, one way would be practical, and another would be creative. There are tests, similar to multiple intelligence tests that students can take to see of the three they prefer.
You can also have two similar worksheets, where one is more advanced and the other is more basic, depending on the level of the student. Students don’t know they have different leveled activities, but this is a great way to address the “just-right” problems for students on a case-by-case basis. You can make more than a basic and advanced, setting up four or five levels, but remember, differentiation is not individualizing.

In this class, I wish we would have learned earlier what differentiation was all about, and how to differentiate activities. This was most beneficial part of the class for me, and I wonder why “stuff like this” hasn’t been around longer. I feel that DI addresses issues that have been around in schools longer than solutions have, and this is something that should be included in all undergraduate teacher prep classes now (which I’m hoping it is)!

Monica's digital decimal differentiation designs are available at Scribd or by email from me. There's quite a bit of work done on tic-tac-toe, tri-mind and cubes.

Closing Thought
Powerful stuff when teachers start sharing on Twitter or writing for sharing.

So why don't you join the conversation, you?

Photo credits: Search Engine People Blog, Cliff1066 @ Flickr

Carpet Questions

2011-06-16T21:25:00.000-04:00

Went to get on the elevator at work, and like a bad stereotype of the absent-minded prof almost said "Oh! Excuse me" to this roll of carpet. I thought, 'I should take a picture of that!' Got my student Ted to pose with it, and I'm asking: any questions?

Ted said, "I thought that's what that was about."

Here are more photos. The label probably has the answer to many questions that could be asked. The original image is on the left, a shot of the edge of the roll, and then the shot of the label. Ted is 6 feet precisely, so perfect for the occasion.

One further thing I'm wondering is that when the carpet is laid out it looks like it's actually in squares instead of from rolls. So now I'm wondering if that's an illusion or ...

Teachers Investigate Quadratics

2011-06-11T10:00:00.128-04:00

In our spring grad class we've spent the week looking at assessment and instruction in the context of quadratics. The pedagogical side of class was spent talking about Skemp, making a assessment concept map on Mindmeister, a rare mindmapping tool that allows realtime collaboration...

...a questioning framework (shared in this old blogpost), a video of a teacher leading a lesson on solving quadratics (from the new to me resource of Inside Mathematics, with videos, coaching, lessons and problems), sharing a variety of articles on assessment, and watching Shawn Cornally's TEDx talk. The assessment articles included one new to me, "Using Assessment for Effective Learning," Clare Lee, Mathematics Teaching, Jan 2001. That led me to two books (that are available online through our library), Language for Learning Mathematics, by Lee, and Learning to Teach Mathematics in the Secondary School, in which she has several chapters. Shawn's talk is worthy of a post of it's own... really has me thinking.

Between the classes, the teachers tried to find quadratic data of their own. One speculated about pulse rates in fight or flight situations (concave down quadratic?) but couldn't find a good set of data. Another found a neat class of problems about water draining in a quadratic pattern. Sadly, my remaining physics doesn't let me know why that would be. One of the students did a great little think aloud using Jing (click for the short video; I don't know how to embed someone else's screencast) with a picture of a record dirt bike jump.

He did a quadratic fit to this using GeoGebra. The picture does not seem to be from perpendicular to the plane of the jump, which raised the nice question: is a parabola seen from askew still a parabola? (The other nice tip he had was to use SmartNotebook for within the screencast... interesting.)

To me, one sign that students are genuinely investigating is that they get to questions that I have to think about or don't know the answer to.

With the focus being on deepening their understanding, I asked the teachers to choose from these investigations, with the addition of the water draining.

1) Galileo Galilei (essentially that means his father was named Galileo, too; so he’s Galileo Jr.) conclusively disproved Aristotle’s idea that heavier things fall faster. He took Aristotle to task for never figuring out a good way to test it. Next he wanted to study how and if speed changed during a fall. But it was too fast for his available tech. So he devised several clever ways to slow it down. One way was to roll a ball down an inclined plane rather than dropping it. (Why would this have the same information as falling?) He did one experiment by building bumps on the ramp and spacing them until the sound of the clicking over them was periodic. That’s hard to replicate in class. But another experiment was to time a rolling ball down a plane, and then find a point where it took half or a quarter of the time. (Read more at http://bit.ly/lwcK1h but after you experiment.) Try that out, take some measurements and discuss what you find. How can you get a variety of data to look for patterns?

2) There’s a famous relationship between quadratics and second differences in the output, if the inputs havea constant difference. Choose a couple of different quadratic functions and experiment, generating and organizing data.

a. What is the relationship?
b. Why is the relationship?
c. How does the relationship connect to either the standard form or vertex form of a parabola?

3) We often care about the roots of quadratic functions. Is it easier to solve for the roots a function in standard or vertex form? Why? What would the quadratic formula look like for vertex form? Could you use that to find or derive the quadratic formula for standard form?

4) Not much for visual learners here. Can you devise a pictorial growth pattern that has a quadratic relationship between input and output? Well probably it’s easy!

But there must be more interesting patterns than that. Can you make a pattern with a ≠ 1? b and c ≠ 0? How would you visualize first and second differences in such a pattern? Would the function for your pattern have roots? What would they mean?

I felt choice was important, especially for experienced students who already knew a lot about quadratic functions. They did an excellent job with their choices, looking for something new to do. Here are some results of their investigations. Afterwards, they came together and shared the results and had a great discourse.

(1) Not to gender stereotype, but the boys were rolling things even before they finished reading the first question. The first attempt was rolling a tennis ball on about a -1/3 slope. They got this data at right. Barely any difference between linear and quadratic. They were surprised by the small size of the a value. They thought it was quite interesting that this was a situation with distance as an independent variable and time as the dependent. (I thought that was cool but hadn't noticed or expected that.) They noticed that if they didn't know to get a quadratic, they probably would have stopped at linear, because r=.985 is clearly good enough.

I asked if they could slow it down at all. So they immediately changed the slope and switched to a hot wheels car that would run truer.

An in between occasion, before (0,0) was included.

This interesting question came up: is averaging the data and finding the regression curve any different from finding the regression curve on all the data? (For this one it turned out the same.) One teacher included (0,0) as a point, which raised the question should the other one using all the data use one (0,0) or three? Should they add (0,0) - because if there was some experimental effect it would not be in a theoretical point like that. How can you tell from the data that the car is speeding up? How did Galileo notice that falling wasn't uniform?

Seems to me that real investigations always raise interesting questions. Canned investigations raise many fewer. The big question came up: how did Galileo know that rolling was essentially like falling? While this was the most dramatic investigation, each of the others raised great questions and connections, too.

(2) Only one student looked into differences, with some consultation. She quickly found a pattern relating the second difference to a. But I put up the following GeoGebra sketch. (File here, or as a webpage.) The teacher checked her work because of the 4.5 2nd difference. She realized it was because of the delta, and set to work finding the relationship between the 2nd difference, a, and delta. Excellent math detective work. We discussed the new applications of difference equations because of computers, and the weird connection with derivatives. They noticed that b and c had nothing to do with the second difference, and how that made sense if the second difference detected quadratic behavior.

(3) This was the one I thought was the least interesting, but the teachers found several cool bits. First of all, they did it by rewriting the vertex form into the standard form, and then used the quadratic formula to get a vertex form quadratic equation. But then they liked it well enough that they wondered why they had never seen it before. It was noticed that the work they did identifying the standard form with the vertex form gave an equation for the x-location of the vertex for the standard form, -b/2a.

I mentioned that I would never have thought of doing it the way they did, and they started wondering what I meant. Another teacher saw a possibility, and solved directly from the vertex form, getting another quadratic equation that was even simpler.

(4) Two groups tried this. They captured their work in photo, so it's probably just best to show.

The teacher on the left was surprised that there was a 3-dimensional way to think about it. The teachers on the right were surprised that there was a non-3-dimensional way to think about it. The rightside teachers liked the idea of colorcoding from the leftside, and made an adjustment.

Left as an exercise for the blog-reader what the function rules are for these sequences. The discussion revolved around trying to make the pattern understandable to a teacher that finds it very hard to visualize. The unanswered question was: what would roots mean in this situation?

Our closing reflection was on what elements of the investigation allowed them to deepen understanding. They brought up the closing discourse as well as collaboration, the manipulatives, the new questions, and trying to collect their own data.

Pretty good lessons to draw, and some quality mathematics.

Galileo Photo Credit: from Flickr, tonynetone

Graphing Stories: Balloon and Tower

2011-06-08T17:06:00.000-04:00

Yeah! I have rarely been so excited to get an email. "We just pulled your Graphing Story out of the oven!" The only downside is that my name is prominent, when I really just incited other people to make them. But I'm still geeked. Huge props to Dan Meyer and BuzzMath for doing this.

First, the Balloon. Filmmaker - Anna Minnebo, Balloonist - Gregg Minnebo.

Second, The Towers of Hanoi. Stacker/Graphist - Eric Thuemmel; Camera - Monica Leneway. In the original he just got the fourth stack complete on the 15th second, but it just barely gets cut off here. He was quick.

Still to come: four simultaneous tower builders... can understand why the graph for that is a special problem.

Grading: SBG and U

2011-06-06T23:16:00.001-04:00

Math Monster
by Mister Awesome @ Flickr

Standards Based Grading to me is the idea that the teacher lays out what students are responsible for demonstrating ahead of teaching, and students have a long period during which to demonstrate them, possibly up until grades are finalized. And students have multiple opportunities to demonstrate. (This is a Part II to the previous grading post.)

Other people describe it better and more thoroughly. Especially Sam Shah and Shawn Cornally. Also please check out the beginner's wiki started Elissa Miller and the SBG gala hosted by Matt Townsley. (Note that you could be interacting with these outstanding professionals on Twitter: @samjshah, @thinkthankthunk, @misscalcul8, @mctownsley) Frank Noschese is thinking about it powerfully, too, in physics, but I haven't had the chance to interact with him about it.

I will say that I've only used it with preservice teachers so far, but they were mostly an appreciative audience for it, and would like to see it in their content classes. I will be doing it in my content classes, starting with a graduate calculus class in the fall, but we're so pinched for math educators right now that I don't get to teach any straight content courses.

The preservice teachers have been helpful for improving my practice of it with their feedback. If you're making the change, I'd encourage you to discuss it with your students, give your reasons, and involve them in the process. I was only going to do it through in class assessments and similar things in office hours, but I added an SBG option to portfolio submissions and added an interview option for office hours. The biggest remaining thing is how to communicate it better at the outset, with which the resources in the second paragraph will help.

Another Speedbump Classic

The most powerful concept to the shift has been giving the students a clearer purpose on the assessments: to demonstrate what you understand by communicating your thinking. Much of the emphasis on the right answer is gone, as is the expectation that test questions will be trivial repeats of tasks already done. Not that my tests were like that lately (have to go back over 20 years for one of those), but it was a bone of contention with students. Now it makes (more) sense to them that they couldn't show understanding on a question like that. I've had a few students reject a problem because they knew how to do it already. (That's not the majority, but some day...)

It's different from K-12 use because in the university we see the students so much less. We give up class time for independent work outside of class, which minimizes time for summative assessment. I struggled to provide multiple assessment points. Put lots of former standards on assessments as choice, and polled students as to what previous standards they wanted on. My standards were much broader than they would be in a content course, as math ed classes wind up covering things like "all of high school mathematics." So I made my standards pretty broad, but we looked at examples of more focused grade level standards. In the future as I reuse, I'll try to add some of those specifics as ways to demonstrate the broad standards. I also let them know that the final grade would take into account which standards we had covered and assessed in class. Some of the content I don't set until the preassessment is in, so it's hard to know ahead of the semester.

Here's the policy on my middle school math syllabus.

Standards Based Grading: SBG is a relatively new way to assess students that seeks to get a higher correlation between grade and understanding. On each of the objectives below, you will have opportunities to demonstrate your understanding. These objectives are a bit broader than you would expect in a secondary classroom, since we are seeing content from three years of schooling. In a secondary classroom, the teacher identifies the standard demonstrated, but in this preservice teacher preparation course you will also be trying to identify which of your work is evidence of which standard.

Scores do not mean an answer is right/wrong, but are meant to reflect how much understanding was demonstrated. It is possible to demonstrate good understanding of a concept without even finishing a particular problem. The score for each category is the average of the 2 highest scores. If there is only one score it is discounted by 1; a single A becomes a B, etc. You can reassess on specific objectives during office hours or at arranged times.

A+ complete understanding and can extend on your own
A complete understanding, can apply when appropriate
B some small difficulty applying or missing a small point of understanding
C significant difficulty in application or missing a major point of understanding
D mechanical application of ideas without understanding
F little to no understanding or evidence of understanding

Mathematical Content Objectives

A. Number: representations and operational concepts
1. Integers
2. Operations on integers
3. Rational numbers: fractions
4. Operations on fractions
5. Rational numbers: decimals
6. Operations on decimals
B. Algebra: representation, operations and modeling
1. Patterns: recognizing and generalizing
2. Variable: as unknown and changing quantities
3. Linear and exponential relationships
C. Geometry
1. Similar figures and proportional reasoning
2. 2-D figures: characteristics and sorting
3. 3-D figures: characteristics and sorting
4. 3-D representation

After a messy fall semester of trying to run parallel SBG and traditional, and a messy winter semester of struggling with full implementation, I'm very happy I came down this road. I have four basic goals for my grading:

From Comically Vintage
Don't be a Dodo!

fair - reassessment helps this.
measures real understanding - move away from non-problems helps this.
not fear or anxiety inducing - students said this was a big improvement.
measures where the student is at the end of the course - clear improvement.

While I didn't get a lot of out of classroom reassessment until the end of the semester, I did get people using the in class assessments to reassess. Students were more responsible for their own marks than ever before, and rather than tracking grades, they were attending to objectives. Broad over-generalized objectives, but I had to start someplace!

I strongly recommend you consider SBG, whether you be K-12 or 13-19. If you do, let's talk!

Watermelon: any questions?

2011-06-06T08:44:00.000-04:00

Dan Meyer, purveyor of pedagogical principles and master of mathematical memes, has popularized or coined the idea of #anyqs on twitter. A math teacher will post a photo (or video) and ask, "Any Questions?"

Here's my photos:

Post questions in the comments or I'll record any twitter questions I see.

Grading: Road to SBG

2011-05-31T12:53:00.000-04:00


From the excellent comicallyvintage.tumblr.com

So, my son and I (X=11) have been discussing starting sentences with "so." It appears that it may be genetic. In real life I'm quite awkward in conversation and transitions are especially difficult, and I think "so" is an all purpose connector. Leads to something that fits or to a change in topic.

Of course, transitions being awkward is something that I think is true in general. How to get students to transition from what they've known to something that may be completely new is the crux of Standards Based Grading, or more wholesome and holistic ways to evaluate as a whole. Back When I Was a Worse Teacher (tm) I could sometimes be heard to joke (sortofjoke) that I wished I could offer my students a deal: take a C and leave the class, stay and take the class for a chance - no guarantee - at better. I was known to be an easy grader though, and I wanted it to be the case that if students made a genuine effort that they would get at least a C. I also use to jokingly propose a Survivor style grading system where we vote people out of class starting at the third week. First out fail, but get the rest of the semester off. Thought the tests could be like challenges where multiple people can earn immunity.

Strange Brew
my current favorite comic strip
does my favorite movie line to quote?

It was a joke! The point of the joke was to get people to think of why they were in the class, and the grade might not be the point.

Working with Dave Coffey, who is our local assessment guru, I saw him use a portfolio to assess our Math for Middle School class, and we have long used a portfolio for our student teacher assistants. In our introduction to mathematical reasoning class there's a summative assessment called the proof portfolio. One time I was teaching it, at the end of the course one of the students asked: "I'm so much better than I was at the beginning of the semester - why does that writing count in my grade?" Whoa. My only response was that THAT was a very good question.

I realized that more of my grading had less to do with where the student was at the end of the semester than I had wanted to think. Mostly because of the reason I hate the most: that's the way it's always been done. Furthermore, I realized I hadn't ever really thought about what I wanted grading to achieve, let alone whether it was accomplishing the job. It's usually pretty quick for a room full of students to agree on some characteristics of good grading:

fair
measures real understanding
not fear or anxiety inducing

It usually takes some discussion to get to:

measures where the student is at the end of the course

My portfolios shifted to feedback only up until the end of the semester, and then the last time they are graded. The grade is about half on completion (as a percentage) and half on exemplars. The students choose the exemplars. In early turn ins, the students ask for feedback on what they want to know, and I share what the grade would have been, and give feedback on their issues. The would-have-been grade allows me to identify big issues that the students don't seem to recognize. I collect at the 1/3 and 2/3 point, though some students want more frequently. (So they can be more responsible.) The final grading is relatively easy though visually daunting; how many boxes of grading? The final exemplars need annotation about what makes them exemplars, and have been the best way for me to evaluate problem solving and communication.

What are we assessing?
brad_holt @ Flickr

So what about the content? That was hard for me. How to evaluate content in a way that allowed for improvement right up to the end. Getting involved in twitter around the same time I was reading more and more educational blogs exposed me to standards based grading. Especially Sam Shah and Shawn Cornally. They supported with good stories and honest difficulties, links to several educators trying or mastering the practice, and good resources, like the SBG wiki. I definitely wanted my preservice teachers to be exposed to it, and felt like it was the missing piece of the puzzle. Or at least better. In the fall I tried running parallel systems - wow, was that a bad idea. This past semester I went full out, and did better. Learned a lot for the fall.

Next time: what I did, how it went, and how I'll adjust.

Who Are the New Teachers? The Long Story

2011-05-29T16:42:00.002-04:00

At our university, content educators are mostly in their respective content departments, which is why we have a dozen or so math educators in our math department. In our secondary teacher prep, we have three courses that are our "Math Ed" courses: Math 229, which is HS content focused, Math 329 - which is MS content focused, and Ed 331 - which is our content seminar for teacher assisting, when the novice teachers are in schools for the mornings and teaching at least a unit. We are in negotiations to see them during student teaching, which will be excellent.

This is another guest post from a student assistant: Brock Walsh. He paused school for a bit, but then came back with a very clear motivation about wanting to be a teacher. Dave Coffey already posted a bit from him, where he used the NCTM process standards as an outside resources.

In his teacher assistant portfolio, he reused a bit from his 229 class, and I thought it was a neat opportunity to follow a student from early on until later in their teacher education. As an add on, I also included his piece from this past semester on the Conditions of Learning, which Dave recently posted in the Learning Museum.

Equity - Insights from the Past

(The following is a paper that was written for MTH 229, in which I had looked into the principle of equity as I related it to my experience of a nine week observation.)

Articles: Excellence in the high school classroom is something that teachers strive for. Sometimes conducting a learning filled classroom can be easy, but other times a teacher might not fully see and take advantage of teachable moments for all students. Being aware of and preparing for these teachable opportunities for all students to learn at a higher cognitive level is vital and defined under the Equity Principle of the high school principles and standards.

The article “Focusing on students’ Mathematical Thinking” by M. Lynn Breyfogle and Beth A. Herbel-Eisenmann focuses on trying to understand the thought processes of a student’s reasoning instead of relying on a student’s answer. Reasoning occurs when a student has time to think and then explain their thoughts. The time that is given after a question and before an answer is known as “wait time” and within this time, a student’s cognitive thoughts will increase. In the article, the authors emphasize an important detail. They quote from their findings, “Although most teachers are aware of the importance of waiting after they have asked a question, the importance of waiting after a student responds has received less emphasis (Rowe 1986). This all relates to students maximizing their learning by having the time given to them so they can process ideas for themselves.

The article goes further to say that when a student has given a correct answer, we as teachers should question them as to how they arrived at that. Students will often learn the most from themselves or when another student explains their reasoning. Asking for justification is a great way to evaluate not only a student, but the class as a whole when they respond and get involved in the discussion. Putting both “wait time” and “justification” together strongly represents the idea of equity and its importance in class.

The article “Unveiling Student Understanding: The Role of Questioning in Instruction” by Azita Manouchehri and Douglas A. Lapp relates to the Equity Principle directly by emphasizing the point that we as teachers need to ask the right questions for optimizing a lesson. Our questions need to facilitate learning and with the right questions being asked we can pull out conceptual reasoning from the entire class.

Magoo0311 @ Flickr

Personal: The only class in high school that ever truly challenged my reasoning was AP Calculus. Not because it was a hard class, but because the teacher invested so much into our learning and asked questions that forced us to explain ourselves. The mathematics classes that I have taken in college act the same way. The professors ask questions that require my justification. Sometimes I don’t fully know how to justify my answer and that can be blamed on the fact that I never had to do it through grade school. The in-class illustrations from the articles represent teachers asking questions that facilitate class, but do not emphasize reasoning like the classes that I have taken in college.

One specific instance of equity that I can remember my AP Calculus teacher applying was related to group work. The class was split into groups of fours and had to present on asymptotic behavior. Each person in the group had to focus on one specific aspect to present to the class and the groups had to hold each other accountable for their work. I can remember that there was a ton of questioning that occurred which felt like a debate. Within that debate, a lot of reasoning was taking place and uncertainties were being explained! The class as a whole was involved, and that is something special when an entire class is participating in discussion. In general, any time a teacher is at the front of a classroom instructing or going through a worksheet, and maybe only asking questions that a few students answer is a case of poor equity and should be avoided.

Observation: I conducted my observation at a well funded school with nice facilities. I observed a teacher and her freshmen/sophomore Geometry class during sixth period on Tuesdays and Thursdays. The class was primarily of white ethnicity, but there was one black boy and girl, and a hispanic girl. There were 15 females and 12 males in the class. The three learning objectives that I observed were review on algebraic properties, theorems about angles, and the last day was devoted to preparing for an upcoming test.

The questions posed in class were probably 50/50 for being open or closed. I noticed that when a question was presented in open form, there would generally be justification with the response. I compared notes with Susie K. from class to find a comparison between in class questioning. She told me that from her observation at Jenison High School, students were asked open questions about half the time in an algebra class but in the geometry class there were generally more open-ended questions. This makes sense to me as it seems fit that the higher level class should be challenged by the questions they get asked. A teacher should expect that as a student’s cognitive level grows; then they should also be able to reason more in depth. The wait time in the class I observed was generally around 3-4 seconds. This is reasonably good, but like the article proposed, there was really no wait time after a student responded. A good way that the teacher made sure each student in the class would have time before a response was by saying, “Everyone think about the problem by yourselves and then compare with a neighbor.” This way, students have time to learn by themselves and from their peers.

Further identifying the questions asked in class, about 22 percent of them required justification. I consider this to be a relatively adequate amount for a Geometry class, but would be something I would like to see get higher in preparation for more advanced math classes. A good example of an open-ended question that required justification was, “What justification do we get for AB+BC=AC?” I realize this seems obvious but sometimes that is exactly what is needed. She also asked, “If both L1+L2=180 and L2+L3=180, then shouldn’t they equal each other? Explain how you know this.” This question set-up made the students think about the properties and theorems that apply to these statements. These types of questions force students to think about possibilities. When called to answer, then the student can explain their best reasoning for an answer. Justifying yourself will sometimes correlate directly with equity if an explanation is clear and insightful so that the whole class learns from the response to the question. All of this stems from the question though, if a good open-ended question was not asked to begin with, then the opportunity for and learning in general has been lost. A good open-ended question posed in class was, “When I say adjacent angles, can you picture that in your mind?” Another one was, “How do you prove something true? What does it take to accomplish this?”

Interview: I conducted my interview questions by simply asking a few questions after each class to get a general sense of what she expects form her students. About instruction and questions Cristina said, “When I generally ask questions, I expect the students to think before giving a response. I’ll ask for understanding from the entire class and if nobody has a question then we move on. I expect students to ask if they don’t know. For communicating, specifically in Geometry, I expect students to use the correct language and correct theorems/properties. It’s important for students to have this foundation.” For the workload she said, “Homework happens every night, and each student is expected to complete their work or at least give a good attempt towards answering the question. Of course, I want all of my students to do well. It is really up to the student to provide the effort, and I am here to help each individual student as much as possible.”

Outside Resource - Conditions of Learning

One Laptop Per Child @ Flickr

This is my opportunity to share my understanding for the “outside resources” portion of my portfolio. During the exit interview, I was asked to explain my reasoning for using the Process Standards from NCTM, and Cambourne’s Conditions of Learning. Somewhat confused by this inquiry, I responded that I included them because they are both a framework that I feel needs to be implemented in the classroom everyday. These are both a resource that I want to keep a focus on when I teach because when using them, I feel my learners will effectively learn more. I was told that these were not the usual types of resources that are used, but upon my explanation, John and Dave understood my intentions of having them included and commended me for seeing these outside resources as a means for having a framework that benefits me in the classroom. Including them in this portfolio is a way for me to have a constant reminder of them.

Spirograph 1

2011-05-23T13:31:00.002-04:00

from Robert S Donovan @ Flickr

I have always loved Spirographs. When I inherited my sister's I was in heaven - I think the exact box pictured here, supposedly from 1968.

Seeing this tumblr entry with an animated Spirograph picture when I had a few minutes sorta free, inspired me to finally put together a sketch. So I thought I'd share it. It's definitely good for experimenting to see if you can figure out what is going on. Very fun to make. The webpage lets you see the patterns, but doesn't give nearly the control that the GeoGebra file does. It succeeded insofar as you can make some pretty pictures, and I didn't have to cheat by parameterizing the curves..

I'm interested in developing some lessons on ratios that could be visualized in interesting ways with the curves traced. If you have feedback on the sketch or on the lesson idea, I'd love to hear it.

The View menu should allow you to refresh views in the applet below, but it can get hung up. I sometimes have to click pause several times before it pauses. As I said, on this one I recommend the GeoGebra file, if you're going to do any serious playing around.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Algebra's Tiling

2011-05-18T23:03:00.004-04:00

I don't think algebra tiles or blocks are a panacea for what ails student algebraists, but I do think they are powerful. Even though as a mathematician I am pretty comfortable with symbolic reasoning, at heart I am a visual thinker. Having a way to visualize algebra opens up many possibilities for learners, even when they had a good symbolic understanding beforehand. The goal for me is not to replace symbolic manipulation, but support the concepts.

I designed the activity for students that may have not done a lot of investigation before. So it starts with a lot of modeling, and then letting them try. Students did an amazing job.

Even before doing the mental math (making the point that even number operations can be visualized, plus setting the context for the follow up activity), I asked the students to take a look at the blocks to see what they noticed. Mr. Boeve had had the students play with the blocks the day before, which is an excellent idea. They noticed corresponding dimensions, colors, different designs. These were the tiles from Algebra Lab Gear, so there's 1/2 x blocks, 1/4 x blocks, 5 sticks and 25 sticks.

None of the students had seen the visual multiplication before, but they were willing to give it a try. They made connections as to why the pieces represented what they did. The verbal connection, x squared, was biggest, but then a few students recognized the x times x relation. They picked up the symbol to picture representing quickly, and that gave an opportunity to talk about how there are many different ways to write things in math. They had 2x+3+1x+2 and 2x +3 +x+2 and 3x+5. We introduced what mathematicians call simplifying, which they connected to fractions.

I raised the problem of negatives - how could we show negatives, because algebra has a lot of those. They thought we could have two color blocks, or use some kind of design. How could we do it with the blocks we have? Maybe we could separate the positives and negatives. Nice thinking!

Sometimes I think of mini-lessons like these as equipping the students for problems. The problem list offered practice, light extension and serious problems. After 15 minutes to do their choice, which they loved having, we came back together and I asked if there were any they wanted to see me do? They suggested problems, and if there was a student to explain them, they gave it a try. One of the themes throughout was "give it a go." I made sure to ask some students who had incomplete or incorrect thinking so we could talk about that, too. Because the whole situation was different, it helped with making that safe.

In discussing the subtraction problems, they got to three separate ideas: taking away, zero pairs, and adding the opposite. The students exploring these ideas were able to give reasons for why it made sense with the blocks.

We just worked with the document camera, but if you want virtual algebra tiles, there's the National Virtual Manipulative Library tiles, a two color version from the Michigan Virtual University, and NCTM's Illuminations Algebra Tiles. None of them are ideal, but all are serviceable. It's very possible to make homemade algebra tiles, and there was a good article about that in the Mathematics Teacher: "Algebra for All: Using Homemade Algebra Tiles to Develop Algebra and Prealgebra Concepts," Annette Ricks Leitze and Nancy A. Kitt, September 2000, Volume 93, Issue 6.

Thanks to Mr. Boeve and his classes for the nice opportunity! Jill Beauchamp came along for experience with the algebra blocks, and she was a great support to the kids, so thanks to her, too.

Playing With Blocks

Photo credits: Eamonn @ Flickr

Who Are the New Teachers?

2011-05-17T12:11:00.000-04:00

Guest post today from one of our student teachers from this past semester. Sarah Cavazos will be student teaching in Fennville, MI this fall and is - to steal a phrase from Sir Ken Robinson - exceptional but not an exception. She is bright, dedicated and passionate about teaching. I find that many of our novice teachers have much in common with her. This past semester I got to see her try a game of her own design in a classroom that had not done much of that, and adjust it on the fly to improve both gameplay and better address learning outcomes. Then she gave a terrific Little Big at the end of the semester about teaching all students.

Triangle Rummy

Object of Game: Obtain a set containing the picture, the name and the definition of the same triangle.

Triangle Rummy is a game made for two to four players that contains 24 cards (7 pictures, 7 names, 7 definitions and 3 Free cards). There are 7 different triangles in the deck of cards, each demonstrated by a picture, a name and a definition. There are also three Free cards (Free Picture, Free Name and Free Definition).

To start the game, the dealer deals out three cards to each player and places the rest of the deck face down between the tables. The youngest play starts by taking the top card from the deck. In order to keep three cards in their deck, the player must discard one of the cards in their hand face up next to the facedown pile. The player to the left can chose to take the face up card in the discard pile or choose to take a chance and take the next card in the discard pile. In order to win, a player must obtain the same triangle demonstrated by three different cards.

If any player obtains a Free card, they can choose to use that card in place of a triangle name, picture or definition depending on the name of the card. For example, say a player had the picture of an isosceles right triangle and the name of the isosceles right triangle and then obtained the Free Definition card. In order to win the game, the player must say the definition of an isosceles right triangle. If the Free Picture card was played, the player must draw the correct picture as well as have the matching name and definition cards. If the Free Name card was drawn, the player must name the correct triangle and have the matching definition and picture cards.

The Unreachables

Photocredit: qthomasbower @ Flickr

Power Up

2011-05-16T15:04:00.002-04:00

A math game for exponents.

A colleague gave me a game created by one of her students Steven Reynolds. The idea was having the power as an exponent connected with the power of a superhero. I liked the premise, as I'm a comic fan myself and have a son who's obsessed. The timing of playing this with the 5th graders was perfect, the day before Free Comic Book Day, the closest thing the superhero community has to a holiday.

The fifth graders had had no experience with exponents before, so this was the introduction. The game was good ground for getting them talking to each other about things like 3^4, how to calculate it, and noticing some of the properties. The main confusion was typical: 2^3 = 2x3. But they went from me reminding them to reminding each other to the vast majority being comfortable with the notation.

The most exciting thing about the game was how much people got into the context and then wanted to play. Some students that had never gotten this much into a math game were engaged; not just in the creative side of it, but in playing the game, also. The students asked if they could choose to work more on their origin stories for their writing time later and Mr. Schiller gave them that option. Two students gave me permission to post their stories, which are after the revised game below. (As usual, good suggestions from the students.)

I launched the game by creating my character on the overhead, at the same time as they were creating theirs. While people were still working on different steps, I had a few people share what they had so far. My super hero was the Time Keeper, who got his powers by finding a watch that turned out to belong to Father Time. He got super strength (base 4), super speed (base 3), and the magic watch (random). Then I played a game vs Shadow the Hedgehog in front of the whole class in which he pounded me. The game play is pretty quick, so kids were all over the room playing each other. Overall, they rated this game a definite keeper.

PowerUp.v3

If Scribd gives you weirdness about paying, you can always email me for the pdf or the docx. I'm grateful to my son, budding comic artist at almost 11 yr.s, for the following illustration of the example from the game sheet. (Click for full size.)

Here's two of the origin stories.

Shadow the Hedgehog by Jim (created by science, doesn't know if he's a hero or villain)
At a space complex ark, he was looking out the window. "Shadow, you've been looking out that window all day," Maria said. Shadow replied, "I just want to know what it's like down there. I've seen pictures, but... I just want to know what it's like."

(3 years later) "Doc, why am I nervous?" asked Shadow curiously. "You will see," Doctor Robotnik said.

A blaring alarm sounded. Doc cried, "Intruders! Shadow, guard Maria with your life!"

Shadow and Maria were running away from the GVN soldiers when one yelled, "Stop or I'll shoot!" Shadow told Maria, "run! I'll try to stop them." One of the soldiers pulled out a pistol, fires, and Shadow yelled, "Maria!" Shadow causes chaos control with Maria, and she says, "Shadow, before I die... Be friends with the people on earth." Shadow yelled, "NO!!"

The Doc says, "take out the gun soldiers and their leader and see how they feel when their loved one is lost." Shadow said, "yes, sir."

When the Doc was arrested, they did not know about Project Shadow.

The Changer, by Katrina
"Ah, the sweet smell of safari!" I said to the driver. "Now don't go too far from the jeep, 'cause you'll get bit," the driver said.

I walked for hours. I could not find the jeep, when all of a sudden a cheetah was face to face with me. I ran and ran. I could already see the blood coming out of me.

The cheetah bit me. I screamed for my life. A couple of minutes later, I was home in an instant. I went to my lab, and mixed some potions together to get the cheetah out of me. But instead, I was on fire.

I could feel the flames. I felt like I was off the ground. I looked ..."What the heck," I said, " What is this, a joke?" Something in my head told me it wasn't. "Now I know what to do," I said with an evil laugh. I lifted up my car with no trouble. Then I knew that I was now a super hero with super-strength, super-speed and firepower. I help fight crime. I help police catch villains.

And my name is the CHANGER.

Photo credits: The Tick I have no rights to, but he is my favorite. Legal use, Fern R and Spencer77 @ Flickr.

Triangle Mosaic

2011-05-04T19:49:00.001-04:00

Holy cow, have I been busy. Sorry for the lack of new posts. What makes it worse is that I have had several guest posts to get up that students were kind enough to send me weeks ago. In addition to scads of things that I want to write up for myself!

The first is from a preservice secondary teacher named Jill Beauchamp. She is active in coaching cheer, and in our local Dutch culture. (And it's almost tulip time.) I'm pretty sure she's a licensed wooden shoe dancer.

On an assignment that gave a choice of follow up options after playing with Pierre Van Hiele's mosaic puzzle in class (from “Begin with Play,” by Pierre van Hiele, Teaching Children Mathematics, Feb 1999), Jill chose to make a activity based on my triangle puzzle. And she was willing to share it! I like how she really captured Van Hiele's idea of beginning with play, and uses the puzzles to get at the triangle properties. She makes the most of what I was designing the puzzle to do, have one triangle of each type.

The assignment:

Teaching Math – Mosaic Making

Choose one or more of the following to do for this:
Analyze Van Hiele’s mosaic. What geometric properties of the pieces permit all the combinations we saw in class?
Create your own mosaic puzzle and document your design process.
Create a new lesson using PvH’s mosaic or my 7 triangle mosaic at http://mathhombre.blogspot.com/2010/12/triangle-puzzle.html
Document your work, and be sure to include a reflection.

Schema: I decided to take a look at your 7 triangle mosaic – nice work! This would be difficult for me to create on the computer, so I am very impressed. When first thinking about a lesson in regards to the mosaic, I could only consider it being a fun puzzle. With our exposure in class to different workshops regarding the original mosaic, I began to think about the properties each triangle in your mosaic possessed. You have:

-Two right triangles
-One isosceles triangles
-One right isosceles triangle
-One equilateral triangle
-Two scalene triangles (One acute and one obtuse)

Fabulous! You have one example of everything.

Focus: With my class, I would want to explore why these triangles fit together the way they do. Assuming the students have not yet learned about triangles, this could be used as an introduction. Let’s say I have this class for 60 min. Here’s how my day would go.

Lesson: Properties of Triangles

Introduction: (5 min) Talk about puzzles!

What kinds of puzzles do students like to do?
What makes a puzzle puzzling?
What are some mathematical properties of puzzles?

Introduce Mosaic

Mosaic Play and Record: (15 min) Allow students to play with the pieces and try to create the mosaic. As they do this, I would like them to document their actions:

What they tried
What pieces worked together?
What didn’t work together?
Qualities they notice about the triangles

**If students solve the mosaic, they should focus on:

Is there another way to solve it?
Why do some types of triangles fit together and others don’t?

Discussion: (20 min) I would ask all students to pull apart their mosaics and separate the individual triangles. Then I would ask them if they saw any similarities between any of the triangles?

*As this is happening I will write up student ideas on the board. If need be, they may come up to the board and illustrate their thinking.

Assuming they already know terminology for a line, angle, point etc. I will have students pull out the rulers and protractors to assist them in drawing more comparisons. Once we have a pool of properties, we can begin to group the triangles accordingly. Once we are able to do this accordingly by the deduced properties, I will write the names of the triangles on the board (but not yet with their corresponding group). Instead I will ask students what they think goes with each.

Properties: (With any luck, we get some or all of the following, although I’m sure I’ll get some other interesting thoughts!)

3 equal sides
2 equal sides
No equal sides
3 equal angles
2 equal angles
No equal angles
Right angle
Obtuse angles
Acute angles

Hopefully, they will see comparisons between the word “Equilateral” and the same angle and side measures, “Right” and the triangles with 90 degree, or right angles, “Scalene” and the triangles that depend on their individual scale/measure, although “Isosceles” doesn’t work too well, but it can be the odd guy out.

I will want to pay special attention to that sneaky little purple “Right Isosceles Triangle.” This guy is important because he shows that two properties can hold for one triangle. Maybe we could explore which properties can hold together and which ones don’t (As I’m writing this these ideas are just kind of coming…)

A scalene can be a right triangle. Why? Because one angle may be 90 degrees, the other two differing, and all sides of different lengths. A scalene cannot be isosceles or equilateral because it goes against the definition of scalene.
An isosceles triangle can also be right, but can an equilateral triangle also be isosceles? No, the definition of isosceles is EXACTLY two sides of equal length. Although it can be either acute or obtuse depending on the size of the angles
A right triangle can then be isosceles or scalene. It cannot be equilateral because one angle must be 90 degrees, thus going against the fact that all angles in an equilateral triangle must be 60 degrees.

Teacher Question: So then, are triangles actually right triangles? Or does the word “right” just classify a specific type of isosceles or scalene triangle? A right triangle cannot exist outside of one of the two classifications.

Sorry for my tangent. The above discussion over the “right isosceles triangle” may be something for another day! My hope would be to get to the last part of my lesson…

Discovery: (15 min) The students would then need to reassemble the mosaic (I will show them the put together puzzle if they need it). With their protractors and rulers I would like them to work on:

Measuring the divided angles in the corners of the square. What is the sum of these angles? What type of triangles have an angle like this?

Measuring the divided angles along a straight line within a puzzle. What is the sum of these angles? What do they notice about all of these sums along a straight line? How does this compare to the sum of the angles within a triangle?

Lastly, I would like them to paste their mosaic together on a piece of paper and write out the angle measures, side lengths, and classification for each triangle. Students should make a note of anything else they notice.

Reflection: (last 5 min of class) What is one realization that surprised them today? Can they put anything they’ve seen into another context? How might it relate to something else?

MY Reflection: Wow, This was wonderful. I had the initial idea for the lesson because I thought it was so cool how the angle measures across a straight line will add up to 180 degrees. A simple concept, but it helped a lot of things make more sense when I recognized it. I think a lot of times we have this subconscious knowledge that we utilize everyday but don’t fully recognize. Once I started planning out how I would eventually get to a measuring activity, ideas just lead into one another, making this a lot longer lesson that I intended. There is no way I would get through the discovery part in 15 min! For me, this order of events seemed to make the concept clear. Perhaps it should be a day and a half sort of lesson?

Do you have any feedback for Jill or I about the lesson? What would you try?

Photo credits: Jill Beauchamp, quinn.anya and bjornmeansbear @ Flickr