Trying to find math inside everything else

Name That Solution

I was reviewing solving equations for my SAT Math class. It’s a tricky thing to do because “equations” includes linear, systems, quadratic, and exponential equations. A lot of different skills to go over in a short amount of time.

After working through the requisite problems, I wanted a little more practice, so I came up with a game that they could play, based on the Bid-a-Note sections of the old “Name That Tune” game shows. I called it Name That Solution. Gameplay goes like this:

• Start over with a simple equation, like “x = 2.”
• Each turn, a team can change the equation in one way to make it more complex. (For example, make it “x + 3 = 2” or “5x = 2”.) Only one operation and one term can be added at most per turn. The team finished by saying “I can name the solution of that equation.”
• On a team’s turn, they may challenge the other team to, in fact, actually solve it. (“Go ahead! Prove it!”) If the challenged team can, in fact, solve the equation, they earn a point. If not, the challenging team gets a point.
• First team to 5 points wins.

They played on whiteboards so they can change the equations quickly. The students quickly learned to not overextend themselves when making the equations harder, lest they find themselves challenged. So it leads to a nice exercise of constantly mentally making sure you know the steps to solve something before you take your turn, getting a lot of practice.

At the end of one of the classes, I did a big class-wide version, half the class versus the other half. But they wound up being very conservative, with neither team challenging the other and only take moves they knew they could solve. Which I guess was the point.

That final round.

Crossing the Transverse

Oh my god, I haven’t blogged since August! This has been a hell of a year, let me tell you. But maybe I’ll tell you in another post, because this one is about the new game I made in my Geometry class. (My first non-Algebra game!)

So the game is called Crossing the Transverse. The goal of the game (pedagogically) is to help identify the pairs of angles formed by lines cut by a transversal, even in the most complex of diagrams. The goal of the game (play-wise) is to capture your enemy’s flagship.

Here’s the gameboard:

I printed out the board in fourths, on four different pieces of card stocked, and taped them together to make a nice quad-fold board. Then I made the fleet of ships out of centimeter cubes I had, by writing in permanent marker on the pieces the letter for each ship.

Here’s the rules.

In the game, each type of ship moves a different way, which makes it feel a lot like chess – trying to lay a trap for the enemy flagship without being captured yourself.  Many of my students really enjoyed it when we played it yesterday. Today, though, to solidify, I followed up with this worksheet where they had to analyze the angles of a diagram much like on the game board. They did pretty well on it, so I’m satisfied!

Materials

Crossing the Transverse Rules

Printable Map (Prints on 4 pages)

No Stars Printable Map (If printing the background galaxy is not for you, here’s a more barebones version.)

Zip File with Everything, including Pages, Doc, and GGB files

Fighting for the Center

At the Math Games morning session at Twitter Math Camp 15, we’ve been created curricular games that hit on some topics that there aren’t really good games for. I came up with the idea for this one, and worked on refining it with the help of Paula Torres (@lohstorres1) and John Golden (@mathhombre).

This game is about measures of central tendency (and range for good measure). Not only do students have to determine all of those over and over as they play the game, but they can see how changing the data set changes the values, especially as the size of the data set increases or decreases. It seems really good because it drives the need to make those calculations.

All you need is two decks of cards. The game is designed as a two-player game, but it would definitely be best done as two pairs playing against each other, so they can talk to each other about their strategies and calculations. We also recommend having students keep a running tally of the values.

How to Pack Your Boardgames

Last year, before Twitter Math Camp, I was packing and trying to figure out which games to bring with me for the game night we were having before the conference started. I basically had three attributes I was considering: how big the game was, how good it was, and how many players could play it. I wanted to minimize the first one while maximizing the latter two.

So I tried to come up with a bunch of formulas for figuring it out, but nothing was quite working out. (I used BoardGameGeek ratings for “how good it was.”) At first I tried doing ${\frac{r \cdot p}{v}}$, but it was putting some games that just weren’t very good as top choices. The problem was that the volume was having too big of an effect – games could come in thousands of cubic centimeters of volume, but max at around 8 for rating and 12 for players. (I had to use amazon.ca to look up the dimensions because I wanted to use centimeters.)

So then I tried cube rooting the volume, or doing an exponential functions like ${\frac{p \cdot e^{r}}{v}}%s=2$, or finding the geometric mean of the three numbers, but still nothing came out right.

I was basically using three games as test cases: Dominion, which is one of the best games I own but it really big; Pixel Tactics, which is one of the smallest but is only 2 players; and The Resistance, which is small-ish, really good, and can go to 10 players. I figured that any good method should tell me to leave the first two games at home, but to bring the Resistance. If they didn’t, it wasn’t right.

Eventually, after doing some research, I determined that a common technique used in psychology when comparing variables of different ranges of values is called standardizing the variables. Basically, for each attribute, I would find the mean and standard deviation. Then, for each game, I would subtract its value from the mean and divide by the standard deviation to get a standardized value. Then I just needed to add up the three standardized values and the ones with the highest score would win. And, as predicted, The Resistance came out on top.

Productive? Failure

The next chapter of Reality Is Broken starts off with this question: “No one likes to fail. So how is it that gamers can spend 80 percent of the time failing, and still love what they are doing?”

It’s an interesting question.  Many games, such as Demon’s Souls, as known for their fiendish difficulty – as that is often portrayed as a positive aspect, not a negative. Dr. McGonigal notes that in one bit of research from the M.I.N.D. Lab, the researchers found that players felt happy when they failed at playing Super Monkey Ball 2 – more so than even when they succeeded. Why would that be?

One thing they noted was that the failure itself was a kind of reward – when the players failed, the scene the played was usually funny. More importantly, though, players knew that their failure was a result of their own actions and symbolic of their own agency – they drove the ball off the course. Because everything was in their control, the players were motivated to give it “just one more try.” I know I’ve certainly had that feeling before – intense concentration on a hard task and then, “Aaaaah!” Coming just short of success, I immediately leap back into trying again.

Of course, that’s not true of every game. There are many games where failing makes me want to give up. There’s two main elements that differentiate the two – agency and hope. If failure is random and feels out of our control, it is demotivating. (Think Mario Kart when you get slammed with a slew of items right before the finish, when you were in 1st place.) But if we see that the failure was fully within our control – and another attempt shows us getting ever so slightly closer to that goal – then the hope of success can feel even better than success itself.

This feeds off of the idea that learning is inherently interesting. When you win at a game, you are successful – but then what do you do? But when you fail, you are learning how to play the game well, and that learning and the act of mastering the game’s mechanics is what is so motivating.

As math teachers, we often talk about Productive Failure – the idea that our students learn better by attempting something themselves, failing, and correcting, than by simply being instructed on the correct method ahead of time. The theory of this is matched by many of our observations (and by research) – but we often have the problem of people being shut down by failure. It ties in a lot with math anxiety and attitudes about math – if I think I am bad at math and that’s just the way it is, failure if just reinforcing that idea, not motivating me to try again.

In the book, Dr. McGonigal doesn’t talk about productive failure – she talks about fun failure. The key factors she mentions – a sense of agency and hope – are what’s so often missing from our math-phobic students. Math feels out of their control – and so any success is accidental, and any failure is predestined.

What can we do? Our main goal is to be a guide – because failure is productive for learning, we want to help the student overcome it themselves. And that means doing what we can to provide that sense of agency and hope.

For a gaming example, Rob was playing a game and was struggling against a particularly frustrating boss (Moldorm from A Link to the Past) – a single false move in the fight would knock him out of the room and he would have to start the whole thing over. Even though he had been having a lot of the fun with the game, this single frustrating experience was enough to make him consider giving up on the game altogether. I knew he would enjoy the rest of the game and wanted him to keep playing, so I stepped into action. One thing I did was provide him with the locations of some fairies – while they would not directly help him defeat the boss, they would lower the frustration of dying and having to repeat the dungeon. The second thing I did was just to watch his attempts.

After a while, when he was ready to give up, he said something to the effect of how he had tried over and over again but had gotten nowhere. But I told him that was not true – when he first tried, he would maybe get 1 hit, or perhaps none, off on the boss before being knocked off the ledge. But in later attempts, he was getting around 4 or 5. He had greatly improved in his tries – and so if he kept trying, he might succeeded. He conceded that might be true, but still took a break, frustrated and tired.

The next morning, I looked up some info and found that the boss only required 6 hits to be defeated – so that meant that in the last attempt, Rob had been very close to success! When I told him that, he was filled with hope (and well-rested), and upon loading up the game, proceeded to beat the boss on the first attempt of the day.

Our goal is productive failure, not frustration. When we are following the mantra of “be less helpful,” I think we still need to help in a different way – help dispel frustration and provide the tools for success, even if we are not telling the students the path they need to take. Be less helpful seems like a hands-off policy – but it’s quite the opposite; we need to devote even more attention to our students when we are letting them struggle on their own.

Satisfying Work

Earlier this year, Justin Aion wrote a post about how he tried to make his class boring on purpose by just giving silent independent work, to make them appreciate what he was normally doing, and how it backfired gloriously. At first, he wondered what he can do to break them of this preference for what they are used to and what is easy. About two months after that, we wrote about a similar situation, and wondered the following:

I’m beginning to wonder if my attempts to give them more engaging lessons and activities have burned them out.  I’m not giving up on the more involved activities.  I want them to be better at problem solving, but I think by trying to do it every day, I haven’t done a good job of meeting them where they are and helping to be where I want them to be.

As I read more of Reality Is Broken, though, I encountered an alternative explanation. In the book, Jane McGonigal wonders why so many people play games like World of Warcraft and other such MMORPGs where the gameplay is not, shall we say, the most thrilling. Many people find enjoyment in what other players call “grinding,” playing with the sole purpose of leveling up. In general, it’s a lot of work to level up in the game to get to what is considered the “good” part of the game, raiding in the end game.

But it’s work that people enjoy doing, and that’s because it is satisfying work. Dr. McGonigal defines satisfying work as work that has a clear goal and actionable next steps. She then goes on to say –

What if we have a clear goal, but we aren’t sure how to go about achieving it? Then it’s not work – it’s a problem. Now, there’s nothing wrong with having interesting problems to solve; it can be quite engaging. But it doesn’t necessarily lead to satisfaction. In the absence of actionable steps, our motivation to solve a problem might not be enough to make real progress. Well-designed work, on the other hand, leaves no doubt that progress will be made. There is a guarantee of productivity built in, and that’s what makes it so appealing.

Well, now, doesn’t that sound familiar? It kinda hit me in the gut when I read it. As math teachers, we are often preaching that we are trying to teach “problem solving” skills – but the thing is, people don’t like solving problems! It made make think of those poor grad students who are working towards their PhD – grad school burnout is a big issue, and one of the major contributing factors is that grad students are trying to solve problems, and so often feel like they are getting nowhere. Their work is inherently unsatisfying, which makes those that can finish a rare breed.

Our students, of course, are not all made of such stuff. But I’m not at all suggesting we drop our attempts at teaching problem solving and only give straight-forward work. Rather, I feel like we need to find a balance – for the past year, as I embraced a Problem-Based Curriculum, I may have pushed too far in the problem-solving direction, and found my students yearning for straight-forward worksheets, just as Justin did. But they also enjoyed tackling these problems, especially when they solved them, and I do think they had more independence and problem-solving skills by the end of the year.

So what should I do? Dr. McGonigal ends the chapter by noting that even high-powered CEOs take short breaks to play computer games like Solitaire or Bejeweled during the work day – it makes them less stressed and feel more productive, even if it doesn’t directly relate to what they are doing. (This reminds me of the recess debate in elementary school.) So even as I go forward with my problem-solving curriculum, I need to weave in more concrete work, and everyone will be more satisfied by it.

The Problem with Gamification in Education

(I suppose I shouldn’t say “the” problem, because there are many problems that I won’t be directly addressing, like extrinsic vs internal motivation.)

I’ve read a lot about gamification in the classroom, and while I’ve often thought about it and borrowed some elements from it, I’ve never gone whole hog. The motivation aspect is one of the reasons, but today, as I started reading Reality Is Broken: Why Games Make Us Better and How They Can Change the World, by Jane McGonigal, I realized there’s more to it.

In the first part of the book, Dr. McGonigal provides a definition of games. A game has four defining features: a goal, a set of defined rules, a feedback system, and voluntary participation. And if you think about gamification, you can easily pick out which of those elements is missing.

Because schooling is mandatory and, if you are taking a particular class, the gamification of that class is also mandatory, gamification of ed itself is not a game. If I gamify my chores by playing ChoreWars, I am choosing to take part in that game (even if the chores need to be done regardless). But if my teacher chooses to use a system of leveling up and roleplaying in my class, it is no longer a game; it is a requirement.

When I tried to think, then, about what in education would best fit these four requires, the first thing that came to mind is BIG, Shawn Cornally‘s school in Iowa. There students choose to participate in some project of their own devising, creating the goal and the voluntary participation. Then it is the school’s job to provide the feedback and the rules.

(An aside on the importance of rules – Dr. McGonigal quotes Bernard Suits who said, “Playing a game is the voluntary attempt to overcome unnecessary obstacles.” The rules are those unnecessary obstacles, and the excellent example given was golf. The goal of golf is to get the golf ball in the hole, but if we did that the most efficient way (walking up to the hole and dropping it in), we would get little enjoyment from it. But by implementing the rules of the game, we make the goal harder to achieve and thus much more fulfilling.)

So the big warning to those who want to gamify their classroom is this: if you require it, it’s not a game, no matter what game elements you include.

The Factor Draft

Last year at #TMC13, I ran a session called Making Math Games. I stared off with an overview of what makes a game a good game, while still being good math pedagogy as well. Then we spent most of the session in two groups brainstorming idea for games for topics that are somewhat of a drag to get through. The other group worked on something in Algebra 2, though I don’t recall what – I must say both groups were supposed to write up what we did and neither did. (But I do think Sean Sweeney was in the other group, so maybe he remembers.)

My group worked on a game for factoring, focusing on Algebra 1. I took the ideas from the session and made a mostly operational game. Then, about 2 months ago, Max Ray came to visit me on the day I was unveiling that game in class. He saw it and it worked out…okay, but here was definitely improvements to be made. So we talked over lunch (about many things, not just the game – he’s great to talk to!) and then tried out some changes with my lunch gang. The changes seemed to work and I went forward with the new version in my afternoon classes to great success. By the end I think I had a really wonderful game, and so I wanted to share it with you.

The Materials

A set of Factor Draft cards includes 3 differently-colored decks. Mine, pictured here, were green, blue, and yellow. One deck (green here) is the factor cards, with things like (x + 2) and (x – 1) written on them. Another deck (blue) is the sum cards, with numbers like 10x or -4x. The last deck (yellow) is the product cards, with numbers like +36 or -15.

The Set-up

Lay out the cards as follows: make a 3 x 6 rectangle of factor cards, a 4×3 rectangle of sum cards, and a 4×3 rectangle of product cards, all face up. Place the remaining cards in separate piles next to the playing area.

In the cards I printed, I didn’t put the Xs on the blue sum cards. Max suggested I do because it’s easy to be confused on which is which.

The Objective

The goal of the game is to collect 4 cards that can be used to complete a true statement of the following form: (factor card)(factor card) $= x^2$ + (sum card) + (product card).

Gameplay

Each turn, a player may select any card from the playing field and place it face-up in front of them. They then replace that card with a new card of the same color from the deck. Play passes to the left. A player may have any number of cards in front of them, and may use any four cards to build a winning hand.

The cards I collected after turn 5. There’s two possible cards I could pull to win the game – can you see which ones?

If at any point a player achieves victory, if they had more turns than the other players, they must allow the other players additional turns to attempt to tie. Upon a tie, discard the winning cards and continue play as a tie-breaker.

A winning hand.

My co-teacher, when we were testing the game, said that it felt like Connect 4, in that with each move you have to decide whether to go on the offense to try and complete your four cards, or go on the defense and block the other players’ sets. But as each player gets more and more cards in front of them, it’s hard to see all of the connections and effectively block, so the game will always eventually lead to victory.

I may need to adjust the number of cards and type of cards in the decks, but I think what I currently have works well – if you have any feedback on the card distribution, let me know. The sum cards go from -10 to +10, with the numbers closer to 0 more common. The product cards go from -60 to +60, with each product card being unique. And the factor cards go from (x-10) to (x+10), also with the ones closer to 0 being more common. (There are no (x+0) cards.)

I did a whole little analysis to determine how many of each type of card to include…but maybe that’s a post for another day.

Sum:Product Deck – The first four pages are the sum deck, the next four are the product deck, the last four are the factor deck.

Factor Draft Play Mat and Rules – Players can use these mats to place their cards and check for a win.

A Boss Fight?

One of the things about arranging your grading system like a game, as well as being a math game aficionado, is that it is pretty easy to combine the two. While yes, students can take quizzes or write essays to gain levels, they can also beat me in a math game. Of course, I’m not easy to beat, so winning against me would really show some mastery. (I do, though, allow them to gang up on me when the game is more than 2 players.)

The only students that really challenge me are the ones that hang out in my room at lunch, even though I’ve offered the challenge to everyone. And it’s cute because when they do lose they get even more determined, often because they may lose by a very small margin. (This is occasionally by design.)

The only game I’ve lost so far is Blokus, where the two Kevins beat me (but my score was still above the 4th player). As a reward, I gave them a level in Visualizer, as I figured that was the most applicable skill to winning the game. Planning ahead and visualizing paths in your mind is a useful skill. That same skill is the reward if they beat me in Ricochet Robots. In that game a team of Jane and Kevin tied me, so I still gave them reward, but they didn’t win.

It’s interesting trying to match games with skills. For example, the reward for winning at 24 is Tinkerer (since you need to play with numbers and try different things to succeed). It’s easy for games I made myself: if they can win at the Factor Draft (an upcoming post, I swear), they are a master of factoring. I have considered giving some points, not quite mastery, if they win against their classmates or my co-teacher, but to be a master, you gotta beat the final boss.

I’d love to have a bigger collection of games that I can use as assessment of skills, not just algebraic skills but the Standards of Practice as well. Any suggestions?

Scrabble Variant

(inspired to post by Anne’s 30-Day Blog Challenge)

So I was playing Scrabble last night (I lost – it’s one of those board games I’m not the best at) when we talked about how, when you are playing with good competent players, the board often winds up with knots of small words close together.

Kinda like this one.

So we talked about how we could promote long and fun words instead of those same short words all the time, and thought you could have a variant where you get bonus points based on how long your word is, regardless of which letters you use or where you place it.

Such a bonus somewhat already exists – you get a 50 point bonus if you use all 7 of your tiles. So we thought we could add other bonuses for other lengths. We agreed we should keep the 50 point bonus for 7, and that you shouldn’t get a bonus for only using 1 letter. As well, we thought a 2 tile go should get 1 point as a bonus. So I said I could definitely model it from there.

I tried to feed those data points into Wolfram Alpha for a fit but they provided linear, logarithmic, and period fits, all of which were terrible. I then forced them on a quadratic fit (after all, 3 points make a parabola), which was alright, although maybe too many points for a 6 tile play. Then I did an exponential one (though I had to use (1,0.1) since Wolfram didn’t like using (1,0) in an exponential fit, as if we couldn’t shift the curve down.) Then I just fed them into Desmos and rounded.

Below are the graphs and the tables for each fit. What do you think of this variant? Which point spread would be better? Of course, we’d have to play it to see….