Final Fantasy XIII-2 Clock Paradox and Hamiltonian Digraphs

I'm a long time fan of the Final Fantasy series, going back FF1 on the NES. In fact, I often cite FF4 (FF2 US) as my favorite game of all time. I enjoyed it so much that it inspired me to learn how to program! One of my earliest Java applets was based on a Final Fantasy game and now, 15 years later, I'm at it again.
I had a blast playing FF13, so when I heard about its sequel I had to pick it up. The game is fun and all, but I've become slightly obsessed with a particular minigame: The Clock Paradox.

The rules of the game are simple. You are presented with a "clock" with some number of buttons around it. Each of these buttons is labeled with a number. Stepping on any of the buttons deactivates that button and moves the two hands of the clock to positions that are the distance away from that button specified by the labeled number. After activating your first button, you can only activate the buttons which are pointed at by the hands of the clock. Your goal is to deactivate all of the buttons on the clock. If both hands of the clock point to deactivated buttons and active buttons still remain, then you lose and must start over.
See this minigame in action in the video below:


You may not know this about me, but I'm not a real big fan of manual "guess and check". I would rather spend several hours building a model of the clock problem and implementing a depth first search to find the solution, than spend the 5 minutes of game time trying different combinations until I find one that works. Yes, I'm completely serious. Here it is.
I think that the reason why I'm drawn to this problem is that it bears a close relation to one of the Millennial Problems: P vs NP. In particular, the Clock Paradox is a special case of the Hamiltonian Path Problem on a directed graph (or digraph). We can turn the Clock Paradox into a digraph with the following construction: create a starting vertex, draw arcs to each position on the clock and place a vertex, and finally draw two arcs from each positions following the potential clock hands from that position. The Hamiltonian path is a sequence of arcs that will visit each vertex exactly one. If such a path exists, then the Clock Paradox is solvable.

This little minigame raises several serious mathematical questions:

  • What percentage of the possible Clock Paradoxes are solvable?
  • Is there a faster method of solving the Clock Paradox? Can it be done in polynomial time, or is it strictly exponential?
  • Is there any practical advise topology can offer to help players solve these puzzles?
  • Is there anything these puzzles can teach us about the general Hamiltonian Path Problem?

I don't claim to know the answers, but I would offer the following advise: see if you can identify a node with only one way in or out. If you can, then you know that you'll need to start or end. If all else fails, you can always cheat by plugging it into my sim!
That's all I have for today. Maybe there will be some rigged chocobo races in the future... kupo.

The Three Axioms of Political Alogic

I find it rather interesting that the foundations of both logic and democracy can be traced back to ancient Greece. Here in the US, we've taken the Greeks' idea of democracy and brought it to a new level, but at the same time our political discourse seems anything but logical. We owe to Aristotle the "Three classic laws of thought", which are as follows:

  1. The law of identity. Anything object must be the same as itself.  P \to P
  2. The law of noncontradiction. Something can't be and not be at the same time.  \neg(P \land \neg P)
  3. The law of excluded middle. Either a proposition is true, or it's negation is.  P \lor \neg P

It's worth while to note that these statements are neither verifiable or falsifiable, qualities true of any "axiom". An axiom is supposed to be a self-evident truth, that gives us starting point for a discussion. The universe described by these axioms is one where "TRUE" and "FALSE" form a dichotomy. These axioms don't handle things like quantum particles or Russell's paradox in which things can be both true and false simultaneously. Nevertheless, they provide a useful tool for discerning truthhood. Politicians, however, are more concerned with "votes" than "truths". The following "Three Axioms of Political Alogic" are the negation of the "three classic laws of thought", and generally indicate situations where a politician is distorting the truth for personal gain. Although, that could change if Schrodinger's Cat decides to run for office.

The Three Axioms of Political Alogic

#1: The law of deniability

Just because something is, doesn't mean that it is.
First order (a)logic:  \neg (P \to P)
Sometimes politicians don't have their facts straight, but that won't stop them from proclaiming that a lie is the truth. The most common form of this seems to be the denial of evolution and climate change, despite the overwhelming scientific evidence. When the majority of the population is poorly informed about scientific issues, its much easier for a politician to appeal to these voters by reaffirming their misconceptions than it is to actually educate them. Just ask Rick Santorum.
There's a corallary to this rule, and that is that if you repeat the lie often enough then eventually the public will believe you. The right-wing media repeatedly refers to President Obama as "Socialist" or "Muslim", despite neither being true, in the hopes of eventually convincing the public that they are true.

#2: The law of contradiction

Just because two positions contradict each other, doesn't mean you can't hold both of them simulatenously.
First order (a)logic:  P \land \neg P
Politicians seem to have a natural immunity to cognitive dissonance, allowing them to hold two contradictory positions without feeling any guilt or embarrassment. Republicans like to call themselves "pro-life" while simultaneously supporting the death penalty -- something I never fully understood. How can one be pro-life and pro-death at the same time?
President Obama's 2012 State of the Union had a few subtle contradictions worth noting. President Obama begins by praising the General Motors bailout and goes on to speak out against bailouts near the end. He also called out "the corrosive influence of money in politics", while he himself was the largest beneficiary of Wall St donations during the 2008 campaign. When you consider that this President has built his position on the principles of compromise and cooperation, taking both sides of the issue seems to be his way of encouraging both parties to work together. Unfortunately, this strategy hasn't really worked out that well in the past.

#3: The law of the included middle

You don't need to choose between a position and its negation. You can always change your mind later.
First order (a)logic:  \neg (P \lor \neg P)
Politicians try to appeal to the widest possible base of voters. Since the voters don't always agree with each other on a particular issue, you'll often find politicians changing their stance depending on which voters they're speaking to. This law is the "flip-flop" rule of politics. Mitt Romney is a popular example, having changed his stances on abortion, Reaganomics, and no-tax pledges. These changes make sense from a vote-maximization point of view. Romney's earlier campaign in Massachusetts required him to appeal to a moderate voter base. In the GOP Primary, he now needs to contend with the far-right wing voters. If the votes he potentially gains by changing stance outnumber the votes he'd lose from the flip-flop, then he gains votes overall. Likewise, President Obama has also "flip-flopped" on some issues he campaigned on now that he's actually in office -- like single-payer healthcare versus individual mandates. Again, the President is dealing with a change in audience. "Candidate Obama" needed to appeal to the general population, while "President Obama" needs to appeal to members congress. He's still trying to maximize votes, it's just a different type of vote that counts now.

Parting Thoughts

This post started with a joke on Twitter about politicians' inability to do basic math or logic. After giving it some thought, perhaps they're better at math than I originally gave them credit for. They may not be able to answer simple arithmetic problems, but when it comes down to maximizing the number of votes they receive they are actually quite skilled. They may tell bold faced lies and flip-flop all over the place, but they do so in such a way that gets them elected and keeps them there. If we want politicians to tell the "truth" then we to start voting that way. We also need to start educating others about how to tell a "lie" from the "truth", and I hope someone finds these "Three Axioms of Political Alogic" a valuable tool for doing so.

Mathematics as a Foreign Language: a Tale of Two Classrooms

Last Thursday's #mathchat topic was "Is the spirit of mathematical thinking being swamped by a focus on technique?". One of the things that caught my eye during this discussion was a comment by David Wees suggesting that we teach math more like programming. I've proposed something similar to this before, but as the conversation continued into the details of learning how to program I started to think of the process like learning a foreign language. While I quickly came to realize that there were differing views on how foreign languages should be taught, I think there might be something to this idea. The human brain has built-in hardware to assist in learning language. Can math education take advantage of it?

Mathematics has its something of its own written language. A "conventional mathematical notation" has emerged through a variety of social influences. Some of those notations "just make sense" in the context, while others are adopted for purely historical reasons. As an undergraduate, college mathematics was like learning a foreign language for me. I had no idea what " \forall n \in \mathbb{R} " meant. Aside from "n", those symbols were not used once in any of my previous courses! It was culture shock. I eventually adjusted, but I now understand why mathematical notation can have such an intimidating effect on people.

What follows are my experiences with learning two foreign languages and how I think the difference between the two methodologies relates to the "math wars". I had 2 years of Spanish in high school and 3 semesters of Russian in college. I'm going to refer to the teachers as Mrs. T and Mrs. R respectively, for reasons that I think will be obvious later.

Mrs. T's Spanish class was held in a portable classroom at the edge of the high school. The classroom held about 30 students and the air conditioning barely kept out the 100-120 degree desert heat. I must give Mrs. T some credit for being able to do her job under such conditions. The classes often started with practice reciting words and phrases, followed by worksheets in groups and ending in a quiz. "Capitones, vengan aqui", she would say while slamming her hand down on the table in front of her, indicating that the students in the front row of the class were to carry everyone's work up to her. Everyday she would do the same routine, and everyday I wished that table would snap in half. We had done so many 10 point worksheets that at the end of the semester I came to the mathematical conclusion that the 100 point Final was only 2% of my grade. Being the little smart-ass that I was, I pointed out that I could skip the Final and still get an A. I don't think she liked that very much, because she threatened to fail me if I didn't take it. Aye que pena!

Mrs. R's class was much smaller, with only about 8 students. It was more like a conference room than a classroom. There was a U-shaped table that opened towards the white board, so Mrs. R could walk up to each person and engage in conversation. There was some rote memorization at first, while we learned the alphabet and basic grammar, but after the first few weeks of class Mrs. R started refusing to speak English in class. Class started with everyone saying hello and talking about his/her day -- in Russian. We role-played different situations -- in Russian. If I needed to know a word, I had to ask about it -- in Russian -- and someone would explain it to me -- in Russian. We watched Russian films and listened to Russian rock music. It didn't feel like a class, but rather like 9 friends with similar interests hanging out for an hour each day.

In both of the classes I learned much about the respective languages, but what really stuck with me in each case was the culture. I might not remember enough of the vocabulary to consider myself fluent in either language, but I'll still find myself singing along with Santana or Mashina Vremeni.

In the "Math Wars", the Traditionalists follow something similar to Mrs. T's method while the Reformers want math to look more like Mrs. R's class. Both methods "work", if test scores are all you care about, but there's a very subtle difference between them. In Spanish class, I always felt like I was always translating to and from English in order to communicate. In Russian class, I felt like I was articulating ideas directly in Russian. There's something beautiful about just immersing yourself in a different language until you learn it. I learned how to program in C by installing GNU/Linux and reading other peoples' source code. Sure I read a few books on the matter, but it was immersing myself in "C culture" that really solidified my understanding.

For students to really learn math, they need to be immersed in the "culture of mathematical thinking". I might not agree with the term "spirit", but mathematicians seem to display a common pattern of asking very entertaining "what if?"s and seeking out the answers. You can find beautiful math in something as simple as drawing doodles in class. There's more mathematical thinking going on when two kids make up a game during recess than there is in a thousand worksheets. Our body of mathematical knowledge is formed through communication and peer-review. It's is such a shame to see math classes run like a dictatorship, built around memorizing a list of "techniques". Sure, mathematics is an essential skill in finance, data, and engineering, but lets not underestimate the importance of "asking questions" in our focus on "problem solving".

Proceeding with the question "what if we teach math like a foreign language?", what might we do differently?

Mrs. T might argue that repetition seems to work, and there's a substancial amount of evidence it does (at least in the short term). Math class already has its fair share of repetitious worksheets, but what if we shift the focus of the repetition to learning the "alphabet and grammar" of mathematics earlier like Mrs. R's class? We could start with "set theory" and "logic" then work up from a firm foundation. The benefits could be substantial.

Mrs. R might also argue that students need to be immersed in the culture of math. Students should learn about the history of math and be exposed to "mathematical pop culture". Let's laugh together at XKCD or collectively gasp in bewilderment at the arXiv. It's moments like those that make us human. Lets embrace them.

Embrace the "culture of math".

Of course, it would probably be a lot easier to do such a thing with a student-teacher ratio of 8:1. One can only dream...

Unraveling Complex Systems: MvC3, Metagaming and Genetic Algorithms

Last year, I wrote an article about Street Fighter and Game Theory for Mathematics Awareness month. This year, the theme is “Unraveling Complex Systems” and I thought I would take the opportunity to expand on the mathematics of fighting games. Lately I've been playing a lot of Marvel vs Capcom 3, and in this article I'm going to attempt to show how the online community in MvC3 is a complex system. This article is intended for casual video gamers, but the mathematically curious might enjoy playing with included sample code. The sample code has been written in Scheme using Racket, formerly known as Dr. Scheme.

Marvel vs Capcom 3 and Rock, Paper, Scissors

In my last article, I made the case that fighting games in general can be thought of as a game of “Rock, Paper, Scissors”. In Marvel vs. Capcom 3, there are several different levels of “Rock, Paper, Scissors” going on within a single match. In addition to the “High, Low, Overhead” game discussed in my previous article, we also have games like “Attack, Block, Throw” and “Jump-in, Anti-Air, Projectile”. You can even see something of a “Rock, Paper, Scissors” game going on between different characters. What makes MvC3 different from other games in the fighting genre is that you have a roster of three characters playing simultaneously. This makes between individual character differences less important in the larger scheme of things, but what is more important is the strategy behind those three characters.

In MvC3, there are three basic strategies: “rush-down”, “keep-away”, and “turtle”. The “rush-down” strategy is simple, get up close to the opponent and attempt to dish out as much damage as possible. Some characters lend themselves to this strategy more than others, with a few notable ones being Wesker and Wolverine. The idea behind “keep-away” is to control the distance between you and your opponent using ranged attacks and projectiles. Some characters with a good keep-away game include Storm and Sentinel. The last strategy is “turtling”, which is playing a defensive game while waiting for an opportunity to punish a mistake from the opponent. Characters like Haggar and Hulk can make short work of an opponent once the right opportunity arises. While “turtling” can be highly effective against “rush-down” tactics, it tends to not do well against “keep-away” tactics. Thus, “rush-down” beats “keep-away”, “keep-away” beats “turtle”, and “turtle” beats “rush-down” – completing our game of “Rock, Paper, Scissors”. The pay-off matrix for this model might look something like this:

  Rush-Down Keep-Away Turtle
Rush-Down (0,0) (10,0) (0,10)
Keep-Away (0,10) (0,0) (10,0)
Turtle (10,0) (0,10) (0,0)

Consider a match-up between characters like Wolverine and Storm. Wolverine's set of moves might complement a rush-down approach while Storm's set of moves complement a keep-away strategy. The player playing Storm would likely attempt to “keep-away” from Wolverine as long as possible, chipping away at his health via block damage. Essentially, Wolverine has forced into a “turtle” position while the distance between the two is large because he doesn't have the tools to attack from afar. Wolverine's “rush-down” game doesn't start until he closes the distance between them. Once Wolverine is in close, it's going to be hard for Storm to shake him. Storm would need to switch into a “turtle” strategy until she can find an opening between the oncoming attacks to create some distance again.

Keep in mind that these are strategies, and not dispositions of particular character. While some characters in MvC3 may lend themselves to a certain strategy over others, you have three characters to choose from and all of them can be played in any of these three styles to some extent. A individual character's weaknesses can be compensated for with the appropriate assists. For example, the Wolverine player might choose a partner like Magneto with a beam-assist to help with his ranged game and the Storm player might choose a defensive assist like Haggar to help counter rush-down tactics.

In a given game of MvC3, it's important to be able to change strategies on the fly. You might start the match with a “rush-down” approach, change to “keep-away” when the opponent starts “turtling”, then go back to a “rush-down” to finish off the match. Abstractly, we can look at MvC3 as a mixed strategy by assigning a probability to each of these three play styles. For example, lets consider two players in a hypothetical match. Player 1 chooses a rush-down heavy team mixed with a little turtling -- lets say 80% rush down, 0% keep away, and 20% turtling. Player 2 chooses a well balanced team, but leaning slightly towards the keep-away game – 30% rush-down, 40% keep-away, and 30% turtling. We can multiply these strategies with our pay-off matrix to find the expected outcome. In this case, the average pay-off is 3.8 for player 1 and 3.2 for player 2. Over the long run, we might expect player 1 to win roughly 54% of the time. By specializing in one strategy at the expense of others, player 1 has gained a slight advantage over player 2. However, player 1's strategy could also be foiled by a player that has chosen to focus on “turtling”. Consider a third player with a strategy of 0% rush-down, 40% keep-away, and 60% turtle. This player would have a 5.6 to 3.2 advantage over player 1, but be at a slight disadvantage to player two by a rate of 3 to 3.6.

Metagaming

In the example above, we've seen how it's possible to adjust strategies to gain an advantage over a particular opponent. In the event that you know nothing about your opponent's strategy, your best bet (from a mathematical standpoint) is to play each strategy with equal probability. However, the pay-off from this particular strategy is that you'll break even – win 1/3 of the matches, lose 1/3 of the matches and tie 1/3 of the matches. In order to win with any consistency, it is necessary to predict which strategy your opponent will play. This is where metagaming comes in.

Metagaming is the art of stepping outside of the game and using information external to the game rules to optimize the potential pay-off. In MvC3, we might examine the frequency with which each character is played and keep track of trends in player strategy. If the majority of the population is predominantly playing one strategy, then it's possible build a counter-strategy that will result in a favorable outcome. For example, Sentinel tends to emerge as a high-frequency character in MvC3 online matches. Sentinel's strong keep-away game (high beam, low beam, rinse & repeat) tends to shutdown a large number of beginning players. In order to win against Sentinel, it's necessary to be able to close that distance and rush him down. A character like Wesker might be particularly well suited for this role.

The metagame of MvC3 is constantly changing. As new strategies become dominant, new counter strategies emerge. On occasion, one of these new counter strategies will become dominant and new counter strategies will will start to develop. Each individual player is an autonomous agent. That player makes his/her own decisions about how to play. However, this player is not alone and may face a diverse range of opponents, each with their own individual strategies. Depending on what types of opponents a player faces, that player can learn from those matches and adapt a new strategy when appropriate.

From a mathematical standpoint, the metagame in MvC3 is essentially a “complex system”. We have a network of independent players connected together by matches played online. The game itself is highly structured with a fixed set of rules, but when we look at the system as a whole it can exhibit a variety of unexpected behaviors. To look at this system from a mathematical viewpoint, we might take a modeling approach. We set up a simple model of the system, add some players and connections between them, then let the model run and see what kinds of properties emerge.

Genetic Algorithms

One way that we might model this system is by using a genetic algorithm. A genetic algorithm is a programming paradigm based on evolution by natural selection. Natural selection dictates that the organisms that are best fitted for survival in a population are the ones that live to reproduce and pass their genes on to a new generation. In the context of this particular system, our environment is a population of players with varying strategies. Instead of genes, we have strategies employed by those players. If a player's strategy works relatively well against the population, that player will likely continue to use it. If a strategy doesn't work, it's back to training for a new one.

With this basic genetic algorithm, let's see what happens when we start with a small group of 5 players each playing a perfectly balanced game (1/3, 1/3, 1/3). After each play-off, the top 4 players keep their existing strategies and the loser goes back to the drawing board and chooses a new strategy at random. As we look at the changes in strategy over time, we can see that the top 4 players stay the same, generation after generation. We say that (1/3, 1/3, 1/3) is an “evolutionarily stable strategy”. As long as the majority of the population plays this strategy, no new strategy can take over the population. In gaming, this is not really a desirable thing to happen. The game isn't fun when every plays the same thing, and players generally refer to this as a “stale metagame”. It really shouldn't be that surprising that the system behaves like this, considering that (1/3, 1/3, 1/3) is the mathematically optimal strategy for this particular payoff matrix.

One of the interesting things about complex systems, is that you often see a high sensitivity to initial conditions. If we make a small change to the initial strategies in the previous example, say (.34, .33, .33) instead of (1/3, 1/3, 1/3), this increases the likeliness of a new strategy to infiltrate that elusive top four. There's an element of randomness as to when the new strategy will succeed, it could happen after the first generation or after the hundredth. Once it does, it starts to change the environment which allows other new strategies to succeed as well. In some sample runs of this population, only one or two of the original strategies were left after 10,000 generations – but there's a great deal of variance between trials. Playing a well balanced game is often a key feature of the new strategies, but might start to see a slight shift from “rush-down” to “turtle” emerging over time, countering the initial population's slight bias toward this strategy.

When MvC3 first came out, the metagame was largely dominated by a single character: Sentinel. Upon observing this, Capcom issued a patch reducing Sentinel's health. Some players criticized Capcom for this move, because it didn't change the gameplay mechanics that were being abused, but from our example here we can see how a small change can have large effects on the metagame. Overall, I think this was a pretty smart move by Capcom – it was just enough change to make the metagame interesting again.

The previous two examples have dealt with small populations that are mostly uniform to start with. In the real world of MvC3 online play, there are thousands of players with dramatically different strategies to start with. To attempt to model the MvC3 metagame, we need to look at larger player pools with a greater diversity of play-styles. To get a feel for this, let's look at a population of 20 random strategies, and replace the bottom 5 players with new strategies each round. With these changes, we see much more variance in the top players.

From one sample run with these conditions:

  • The top strategy after the first generation was (0.54, 0.27, 0.19).
  • The top strategy after 10 generations was (0.13, 0.81, 0.06).
  • The top strategy after 100 generations was (0.69, 0.06, 0.24).
  • The top strategy after 1,000 generations was (0.04, 0.81, 0.15).

There are many interesting observations to be made about the behavior of this model. First, we see that the metagame is much more dynamic when we start with random conditions instead of a uniform population. Balanced strategies tend to do well overall, but many of the top strategies are not necessarily balanced. Remember, its not the strategy alone that determines success, but the combination of the strategy and population. The fact that the top strategies after 10 and 1,000 generations are both “keep-away” heavy is a result of the population being “turtle” heavy during those particular generations. As the population changes, so do the winning strategies change. This ebb and flow from one strategy to another is what keeps the metagame interesting.

Conclusions

I think there's a couple of important lessons to be learned here for people who are new to the fighting game genre. The first lesson is the importance of a balanced game. If everyone is playing a balanced game, then the only way to be successful to play a balanced game. The second lesson is to not underestimate “gimmick builds” – strategies that focus on maximizing a particular play style at the expense of other. When there is a tendency for the general population to play a certain way, the right counter strategy can be highly effective. The third lesson is to learn from your mistakes. If your team strategy isn't working, don't be afraid to mix it up. You might find a new strategy works better against the general population.

As a footnote, I'd like to point out that this model is a very simplified version of what goes on in MvC3 games. For an example of what some “real” MvC3 games look like, I'd recommend having a look at Andre vs Marn's First to Ten. As you watch, see if you can identify when each player is playing which strategy. Does the rush-down/keep-away/turtle model fit with actual fights? How does changing out Akuma for Sentinel change Marn's strategy? Is this change predicted by our model?

Further Investigations

I've intentionally been a little vague with the definition of complex system, in part because most definitions are high level descriptions of behavior. Am I correct in the assertion that MvC3 single player is not a complex system, but the MvC3 multiplayer “metagame” is a complex system?

One of the things I find interesting about MvC is the assist system. In this system, its technically possible for a player to employ two different strategies simultaneously. How can we change our model to account for this?

One common practice for genetic algorithms is to mix the genes of successful players to create new players, rather than just randomly selecting a new strategy as done in this example. Typically this is done by using a “crossover”, which selects randomly selects genes from two parents. How does this change the results of the genetic algorithm?

Another way of looking at the players is to actually model each player as a program. This technique is often called genetic programming. What kind of programs do you think would be most successful?

Further Reading

Gintas, H. (2000). Game Theory Evolving. Princeton University Press: Princeton.

Mitchell, M. (1998). An Introduction to Genetic Algorithms. MIT Press: Boston.

Koza, J. (1980). Genetic programming: on the programming of computers by means of natural selection. MIT Press: Boston.

Dawkins, R. (1976). The Selfish Gene. Oxford University Press: Oxford.

Felleisen, M., Findler, R., Flat, M. & Krishnamurthi, S. (2003). How to Design Programs. MIT Press: Boston. Available online at HTDP.ORG.

VMATYC 25th Annual Conference: Day 1

Last weekend I attended the 25th Annual Conference of The Virginia Mathematical Association of Two Year Colleges (VMATYC), Virginia's chapter of the American Mathematical Association of Two Year Colleges (AMATYC). This was the first educational conference I have been to since I started teaching developmental math two and half years ago, so it was a very exciting event for me. What follows is my account of the seminars I attended at the VMATYC and what I learned from the experience. I've tried my best to summarize the events I attended from my notes, but please contact me if there are any inaccuracies.

I missed the early sessions on Friday due to class, but made it in time for the seminar I was most interested in: The Developmental Math Redesign Team (DMRT) Progress Report.

DMRT Progress Report

Virginia's Community College System (VCCS) has been in the process of “redesigning” the developmental math program for about two years now, and is now in the process of implementing some major changes to the way developmental math is handled at the community college level. The report was presented by Dr. Susan Wood, Dr. Donna Jovanovich, and Jane Serbousek.

Dr. Susan Wood began the discussion with a broad overview of the DMRT program. The DMRT began in 2009 with the publication of The Turning Point: Developmental Education in Virginia's Community Colleges, which highlighted some of the problems facing developmental math students. This document set forward the goal for the developmental education redesign, which is specifically targeted at increasing the number of students that go on to complete degree programs. The Turning Point also initiated the Developmental Mathematics Redesign Team. The following year, the DMRT published The Critical Point: Redesign Developmental Mathematics Education in Virginia's Community College System, which outlines the proposed changes to the developmental education program. Next, a curriculum committee began work on a new developmental mathematics curriculum, which is available here. These changes are slated for implementation in Fall 2011. Dr. Wood also made the point that these changes fit into a larger framework of the student experience, a cycle of “Placement/Diagnostic Instruments --> Content --> Structure --> Instructional Delivery --> Professional Development --> Student Support Services Assessment --> Placement/Diagnostic Instruments”.

Next, Jane Serbousek followed with more detail about the proposed DMRT changes. The content of the developmental math courses has been revised to better reflect what is needed to be successful in college. The content has also been reorganized from three five-unit courses, to a series of nine one-unit “modules”. The modules are competency based, and are intended to use a grading system of ABCF instead of SRU (Satisfactory, Reenroll, Unsatisfactory) which is currently employed. She noted that the question of “what constitutes mastery?”, is a difficult one. The intention of this modular framework is that students should only take the modules that are needed, as determined by the placement test, and work to improve their mastery of that topic before moving forward. This also allows for greater differentiation between students. For example, Liberal Arts students would have different developmental math requirements than students in STEM programs.

Part three of the presentation was led by Dr. Donna Jovanovich and discussed the goals of developmental math redesign. The three goals of the DMRT are (1) to reduce the need for developmental education, (2) reduce time to complete developmental education, and (3) to increase number of developmental education students graduating or transferring. Each of these goals has a related measure of success. For example, “reduced need for developmental education” can be measured by placement test scores and “reduced time to complete developmental education” can be measured by student success in developmental classes. One interesting statistic that Dr. Jovanovich mentioned was the following: only 1/3 of developmental math students that don't pass reenroll in the course the following semester, of those, only 1/3 pass the second time, but those that do pass through the developmental program successfully have a 80% of graduating or transferring. So while success rates for the courses are grim, there are long term payoffs for the students who do succeed.

Dr. Wood returned at the end of the session for some closing remarks. The steps for the DMRT program are to have the curriculum approved by the Dean's course committee and to find out how the modularization of developmental math will affect enrollment services and financial aid.

For more information, see the VCCS Developmental Education home page.

VCCS Reengineering Initiative

The second event I attended was a presentation from VCCS Chancellor, Dr. Glenn DuBois. The Chancellor began with an overview of the goals for the Reengineering Initiative, many of which are spelled out in the Achieve 2015 publication. The goals are to improve access, affordability, student success, workforce and resources. He noted that the VCCS is experiencing an increased number of students that register for classes, and increased number of these students are unprepared, a decrease amount of public funding, along with a call for more public accountability and more college graduates. Currently, about 50% of high school graduates require developmental education and only 25% of them go on to graduate in four years. He made the case that there is bipartisan support for improving the quality of education, using President Obama and Virgina Governor McDonnell as two examples. President Obama has stated that he wants to see 5 million more graduates in the US, while Governor McDonnell has stated that he wants to see 100,000 more graduates in the state of Virginia. This is the heart of the Reengineering Initiative: improving student success with sustainable and scalable solutions. Some of the funding for the Reengineering Initiative has been made possible by Federal funding, as well as the Lumina & Gates foundations.

In order to improve the 25% success rate of developmental education, the Reengineering Initiative is implementing major changes to the developmental math program. First is the opening of different paths for different students. Second is a revised business model which replaces a “test in/test out” philosophy with a diagnostics and short modules intended to improve mastery. To accomplish these goals, the Virginia Community Colleges are moving in a direction of more shared services, in areas such as Financial Aid and distance learning. The VCCS is also looking for ways to help local high schools better prepare students for college, such as making the placement test available to high school students and developing transition courses.

Best Practices in a Changing Developmental Education Classroom

The last event of the first day was a keynote presentation from textbook author Elayn Martin-Gay. Elayn's first major point was about the importance of “ownership” for both teachers and students, and how language can affect the feeling of “ownership”. For example, instead students' grades being “given”, they should be “earned”. She seemed very positive about the Reengineering Initiative, saying that it was “good to be doing something, even if it's wrong, [so that] you can tweak it and continue”.

She then proceeded into more classroom oriented practices, saying that it was important to monitor student performance and catch students “at the dip”. If a drop in performance can be corrected early, this can prevent the student from getting too far behind. She also talked about the importance of students keeping notes in a “journal”. This encourages good study skills, giving students a source to go to when it comes time for the exam. She suggested that teachers should “learn the beauty of a little bit of silence”. Teachers should not always jump right into a solution to a problem, but that waiting a extra three seconds longer will dramatically increase the number of student responses. She also said that teachers should “raise the bar and expect more from students”, and that “they will rise to meet it”. She recommended that disciplinary problems occurring in the classroom should be taken care of immediately, to maximize time for learning later.

After these classroom practices, she moved into some of the larger social issues affecting developmental education. She noted that the supply of college degrees has gone down, while the demand for experts has gone up. She jokingly called the first year of college “grade 13”, noting that many college freshmen have yet to decide on a long term plan. She cited seven current issues affected new college students: lack of organization, confidence, study skills, attendance, motivation, work ethic, and reading skills. She argued that reading is often the biggest barrier to earning a college degree.

As some ways of addressing these issues, she presented a number of graphs relating college experience with employment and income. She said that she often presents these graphs at the start of the semester as a means of encouragement. She has students covert the statistics from annual income to an hourly wage so that they can more closely relate with the figures. She also included some ideas for asking “deeper” questions in math classes. One of the examples was “Write a linear equation that has 4 as the solution”. The trivial solution to this is “x=4”, then we can build off this to find others “2x=8” and “2x-3 = 5”. She says that students will typically solve these equations step by step each time, by the time she asks students to solve something like “-2(2x-3)/1000 = -10/1000” they start to look for an easier method – realizing deeper properties about equality in the process.

One of the things Elayn said that resonated strongly with me was that “students would rather be in charge of their own failure than take a chance on [asking the teacher]”. As math teachers, the general feeling of the audience was that study skills are not our focus, but as Elayn pointed out, those study skills can have a powerful influence on student success. By providing students with the skills necessary to “learn math”, those students can in turn take charge of their learning experience.

Next time: VMATYC Day 2

Stay tuned as I collect my notes from Day 2. Day 2 events include: “Online Developmental Math on the Brink: Discussion Panel”, “Developmental Mathematics SIG Roundtable”, and “The Mathematical Mysteries of Music”.

#MathChat Recommended Reading

This week's topic on #mathchat was "What books would you recommend for mathematics and/or teachers, why?". I offered several suggestions in Thursday's chat, but wanted to go back and explain in more detail "why". I've also added a few to help round out the selection. These books are listed in approximate order of "increasing density", with the more casual titles at the top and the more math intensive titles near the bottom. Of course, this ordering is my own subjective opinion and should be taken with a grain of salt.

Disclaimer: The author bears no connection to any of the publications listed here, nor was the author compensated for these reviews in any way.

Lewis Carroll - "Alice's Adventures in Wonderland", "Through the Looking-Glass"

Recommended for: all ages, casual readers

Charles Dodgson, perhaps better known by his pen-name "Lewis Carroll", authored a number of children's books including the aforementioned titles. What makes these books so special, is that Dodgson was also a mathematician and embedded numerous mathematical references in these works. Most people might be familiar with the many film adaptations of these works, but I'd highly recommend reading the originals with an eye towards the logical riddles and mathematical puzzles hidden in these classics. You can find these books online at Project Gutenburg. For little a taste of the mathematics involved, you might start here.

Charles Seife - "Proofiness: The Dark Arts of Mathematical Deception"

Recommend for: teens and older, casual readers, who don't think math is relevant to daily life

This book focuses on what I consider to be a important topic in the current socio-political climate. Ordinary people are repeated bombarded by "deceptive mathematics". Whether the source is trying to sell a product or push a political agenda, the inclusion of numeric figures or fancy graphs can go a long way to make a claim look more legitimate than it really is. Proofiness spells out some of the common warning signs of deceitful mathematics, so that the reader can be more aware of these practices. While somewhat lighter on the mathematical content that more advanced readers might expect, I think this book sheds some much needed light on an important social issue and was an enjoyable read. If you like this, you may also like How to Lie with Statistics by Darrell Huff

Apostolos Doxiadis, Christos H. Papadimitriou, Alecos Papadatos and Annie Di Donna - Logicomix

Recommended for: casual readers, comic book fans

Technically this is a graphic novel instead of your typical book, but that doesn't mean it doesn't cover some important mathematics! Logicomix presents Betrand Russell as the antagonist in a series of historical events that took place in the early 20th century, culminating with Kurt Gödel's Incompleteness Theorem. which shook the very foundation of mathematics. Logicomix makes superheroes out of mathematicians in an epic story, while exposing the reader to some amazing mathematics. Ties in nicely with Gödel, Escher, Bach below.

David Richardson - "Euler's Gem"

Recommend for: casual readers curious about topology

I was looking for a casual introduction to topology and found this little "gem". This is the book that I wish I read while studying topology in college! It covers everything from the basic principles of topology to the recently solved Poincaré Conjecture. Don't let all this mathematics scare you away from this title! The book is still written in a very approachable manner. It chronicles the life history of Leonard Euler, presenting the development of the field of topology in context that even the casual reader can enjoy.

Douglass Hofstadter - "Gödel, Escher, Bach: An Eternal Golden Braid"

Recommended for: semi-casual readers with diverse interests

When someone asks me for "a good math book", this is my go-to recommendation. This book has a little of something for everyone. Math, music, art, language, computers, biology, and psychology are woven seamlessly into a humorous and playful narrative, reminiscent of Lewis Carroll. It goes deep into mathematical concepts where appropriate, and uses visual material and metaphor to bring complex concepts down to Earth. I listed it as "semi-casual" due to the depth of mathematics involved, but a casual reader can skip some of the more math intensive parts and still get a nice overview of the general principles.

Jean-Pierre Changeux and Alain Connes - "Conversations on Mind, Matter, and Mathematics"

Recommended for: semi-casual readers, with interest in philosophy

This book spans several conversations between a Mathematician and a Neurologist on the Nature of Mathematics. One of the central questions is if mathematical ideas have an existence of their own, or if they exist only within the neurology of the human brain. Both sides present some fascinating support for their side of the argument. The material can be a little dense at times, making reference to advanced research as if it were common knowledge, and might not be appropriate for more casual readers. However, a reader willing to dig in to these arguments will reveal two very fascinating perspectives on the philosophy of mathematics.

James Gleick - "Chaos: Making a New Science"

Recommended for: semi-casual readers, preferably with some Calculus experience

Chaos takes the reader on a historical journey through the emergence of Chaos Theory as a mathematical field. An amazing journey through the work of numerous mathematicians in different fields, who came upon systems exhibiting "sensitive dependence on initial conditions". This book serves as an introduction to both Chaos Theory and non-linear dynamics, while shedding light on the process behind the development of this field. Some experience with differential equations would be beneficial to the reader, but more casual readers can get by with assistance of wonderful visual aids. A nice complement to A New Kind of Science below.

Roger Penrose - "The Emperor's New Mind: Concerning Computers, Minds, and the Laws of Physics"

Recommended for: more advanced readers with interests in physics and artificial intelligence

Roger Penrose is a well established mathematical physicist, and The Emperor's New Mind offers an accurate and well written overview of quantum physics. However, what makes this book interesting is that Penrose takes this physics and mathematics to mount an attack on what Artificial Intelligence researchers describe as "strong AI". Penrose makes the case that Gödel's Incompleteness Theorem implies that cognitive psychology's information processing model is inherently flawed -- that the human mind can not be realistically modeled by a computer. Whether you agree with Penrose's conclusions or not, his argument is insightful and is something that needs to be addressed as the field of cognitive psychology moves forward.

Stephen Wolfram - "A New Kind of Science"

Recommended for: more advanced readers with interest in computer science

Don't let its size intimidate you. If you made it through the titles above, than you should be ready to make headway into this giant tome. The central theme of A New Kind of Science is that complex phenomena can emerge from simple systems of rules. This is different from Chaos Theory (described above), in that this complexity can emerge regardless of initial conditions. A New Kind of Science takes the stance that we can learn a great deal about mathematics through experimentation, and makes the case that perhaps the vast complexity of the universe around us can be explained by a few simple rules.

A Rebel Math Curriculum

One of many insightful educators I follow on Twitter, Tom Whitby, wrote A Modest Blog Proposal asking for bloggers to post educational suggestions on October 17th, 2010. He proposed the acronym REBELS for “Reforms from Educational Bloggers Links of Educational Suggestions”. I found the idea of a "rebel" education very intriguing and if there's one place where educators need to resist authority, I think it's the mathematics curriculum.

Before we get into my proposed curriculum, it's important to have an idea of where we're starting. Paul Lockhart describes the existing system quite concisely in A Mathematician's Lament:

The Standard School Mathematics Curriculum

LOWER SCHOOL MATH. The indoctrination begins. Students learn that mathematics is not something you do, but something that is done to you. Emphasis is placed on sitting still, filling out worksheets, and following directions. Children are expected to master a complex set of algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part, and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables are stressed, as are parents, teachers, and the kids themselves.

MIDDLE SCHOOL MATH. Students are taught to view mathematics as a set of procedures, akin to religious rites, which are eternal and set in stone. The holy tablets, or “Math Books,” are handed out, and the students learn to address the church elders as “they” (as in “What do they want here? Do they want me to divide?”) Contrived and artificial “word problems” will be introduced in order to make the mindless drudgery of arithmetic seem enjoyable by comparison. Students will be tested on a wide array of unnecessary technical terms, such as ‘whole number’ and ‘proper fraction,’ without the slightest rationale for making such distinctions. Excellent preparation for Algebra I.

ALGEBRA I. So as not to waste valuable time thinking about numbers and their patterns, this course instead focuses on symbols and rules for their manipulation. The smooth narrative thread that leads from ancient Mesopotamian tablet problems to the high art of the Renaissance algebraists is discarded in favor of a disturbingly fractured, post-modern retelling with no characters, plot, or theme. The insistence that all numbers and expressions be put into various standard forms will provide additional confusion as to the meaning of identity and equality. Students must also memorize the quadratic formula for some reason.

GEOMETRY. Isolated from the rest of the curriculum, this course will raise the hopes of students who wish to engage in meaningful mathematical activity, and then dash them. Clumsy and distracting notation will be introduced, and no pains will be spared to make the simple seem complicated. This goal of this course is to eradicate any last remaining vestiges of natural mathematical intuition, in preparation for Algebra II.

ALGEBRA II. The subject of this course is the unmotivated and inappropriate use of coordinate geometry. Conic sections are introduced in a coordinate framework so as to avoid the aesthetic simplicity of cones and their sections. Students will learn to rewrite quadratic forms in a variety of standard formats for no reason whatsoever. Exponential and logarithmic functions are also introduced in Algebra II, despite not being algebraic objects, simply because they have to be stuck in somewhere, apparently. The name of the course is chosen to reinforce the ladder mythology. Why Geometry occurs in between Algebra I and its sequel remains a mystery.

TRIGONOMETRY. Two weeks of content are stretched to semester length by masturbatory definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and obsolete notational conventions, in order to prevent students from forming any clear idea as to what the subject is about. Students will learn such mnemonic devices as “SohCahToa” and “All Students Take Calculus” in lieu of developing a natural intuitive feeling for orientation and symmetry. The measurement of triangles will be discussed without mention of the transcendental nature of the trigonometric functions, or the consequent linguistic and philosophical problems inherent in making such measurements. Calculator required, so as to further blur these issues.

PRE-CALCULUS. A senseless bouillabaisse of disconnected topics. Mostly a half-baked attempt to introduce late nineteenth-century analytic methods into settings where they are neither necessary nor helpful. Technical definitions of ‘limits’ and ‘continuity’ are presented in order to obscure the intuitively clear notion of smooth change. As the name suggests, this course prepares the student for Calculus, where the final phase in the systematic obfuscation of any natural ideas related to shape and motion will be completed.

CALCULUS. This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned. To be taken again in college, verbatim.

So if Educational Rebels could have their way with the math curriculum, what would it look like instead? Certainly it would be different from what goes on in the Educational Empire.


Rebel vs. Empire terminology geekily borrowed from Star Wars. Image obtained from Wookieepedia.

Within the Educational Empire, there are Official Imperial Standards which teachers must adhere to or they will be fired and annual multiple-choice tests that students must take as if their lives depended on them. As Imperial teachers routinely state, students who do poorly on these tests will die poor and lonely, and students who do well on these tests will go on to an Empire approved Private Academy where they will accumulate massive debts which must be repaid to the Empire through decades of hard labor.

The Rebels would do away with all of this. Instead of multiple-choice choice tests, students would have authentic assessments where they create products they can be proud of. Teachers would have educational goals in mind for instruction, but the nature of the material covered is directed by the students. Students are free to learn about the things they are genuinely interested in, and often go home telling their parents how they “can't wait to go to school tomorrow”. Upon graduating from public schools, students are well prepared to start pursuing the career of their choice. Those who want to further the field of knowledge in their respective area of interest can go on to to a Rebel college for free, where they are guided in conducting effective scientific research.

The Rebels would need to rewrite the curriculum from scratch to accomplish all of this, starting with the math curriculum. The new curriculum would be nonlinear and individualized for each student. The students would be the ones to direct learning, and the teacher would be there as an experienced learning guide. The following tale describes the math curriculum of some hypothetical Rebel students:

Rebel Mathematics Curriculum

Elementary School

Student exposure to mathematics begins with games. Students start with simple games in the beginning, and the games get more complicated as the students progress. These particular students start with games like Rock-Paper-Scissors, Go Fish, Tic-Tac-Toe, Hide-and-Seek and Dodgeball. In the context of these games, students learn how to play within a set of rules and learn the basic language of logic and sets. Students learn what it means when the rules say “Do this and this”, “Don't do this or that”, “Move this from this group to that group”, “Combine this group and that group”, “Separate this group into multiple groups based on some quality”. These concepts form a solid foundation for mathematical thinking. As students progress, they get into games where counting becomes more important. Students play games like Sorry!, Chutes and Ladders, Hi Ho Cherry-O, and even sports like Tennis. As students become more familiar with counting, they get into games with more complex scoring methods like Yahtzee, Blackjack, Monopoly, and Risk where they further develop their arithmetic skills.


Dice games like Yahtzee can be used to introduce basic arithmetic skills. Image obtained from Wikipedia.

Middle School

In Middle School, students continue to play increasingly complex games. Students play board games like Chess and Go, card games like Poker, and video games like Sim City and World of Warcraft. Students are encourage to engage in meta-cognitive processes as they play, by talking about different strategies for optimizing how they play. The basic concepts behind Algebra and Probability emerge naturally from these discussions. Students start developing games of their own, beginning with board and card games and moving into programming simple video games. That's right, all students are encouraged to start programming in middle school. After the students develop a prototype game, they play-test the game, collect data from the play-test, analyze that data, and use the information they discover to revise the rules of their game. The middle school hosts a Game Convention at the end of the school year where each class puts on a demonstration of their game and the process they used to come up with it. The parents are invited to come play their kids' games and see how the students' critical thinking skills have developed.


Game record of a Go match between Honinbo Shuusaku and Gennen Inseki in 1846. Image obtained from Wikipedia. The game of Go is immensely complex, and one of few games where Artificial Intelligence research has yet to reach the level of professional human players.

High School

In High School, the class of students that worked as a group in middle school starts to diverge into different groups based on individual interests. Students with an interest in sports might have a math course that is focused on Geometry and Spacial-Reasoning, with a little bit of Game Theory on the side. Students with an interest in Music might combine Trigonometry and a little computer programming to produce new and interesting sound effects. A group of photography and art students start programming new filters in GIMP to create original effects for their images. A group of students interested in journalism learns about web programming as they put together a professional looking web site. A group of students that developed an interest in Racing games, is introduced to physics and some real-world automotive engineering. A group of students with an interest in programming starts learning about Calculus as they write their own First-Person Shooter. Another group of students uses a Lego Mindstorms kit to build a robot that sorts a line of objects by size, learning a variety of math and engineering skills in the process. Students graduate from high school with more than just a diploma, but a portfolio of work demonstrating their mastery in their subject area of interest. Students show off these portfolios at a convention where the local employers are known to stop by to identify potential job candidates. Most of these students will move straight into a job in their field of interest, but some will go on to pursue further research opportunities in college.


Products like Lego Mindstorms can be used in high school to develop practical engineering skills. Image obtained from Official Lego site.

College

All Rebel students have the option of furthering their education in a publicly owned and operated Rebel University. This education is provided at no cost to students. Exceptional students conducting research at the Rebel University may even be paid for their contributions. Rebel society recognizes the value of academic research, and considers the value of the knowledge resulting from student research to more than compensate for the costs associated with running the Rebel University. Students don't just go to college to learn, they go to further the existing knowledge in their respective fields.

Conclusion

In this Rebel Education, gone are the days of Algebra, Geometry, More Algebra, Trigonometry, and Calculus. Gone are the days of lengthy multiple choice tests. Teachers assess students by analyzing the products they create and encourage the students themselves to critically reflect on their own creations. Students are not pressured to meet Imperial standards, but instead are responsible for setting their own goals for improvement each semester. The students don't feel like they are competing to score higher than their classmates, but instead learn to recognize that each of their classmates has a different set of skills and that by cooperating they can achieve things that they could not do alone. While the Empire is pumping out clone after clone, the Rebels are producing a diverse array of students with varying sets of knowledge and skills.

Which students do you think would be happier and more successful in life? Those with their Empire prescribed cookie-cutter education? Or those from the Rebel academies?

I must confess that there has not been enough research to predict what the long-term effects of such a Rebel education would be. However, I do think there is a substantial amount of evidence indicating that the traditional Imperial curriculum is failing. Educational research provides incremental improvements to the existing curriculum, but perhaps the system's assumption that everyone should have the same education is fundamentally flawed. At some point in the future, it may become necessary for "Educational Rebels" to overthrow the "Educational Empire" and challenge this assumption. The mathematics curriculum proposed here may not be perfect, but it might provide a starting point that educators can revise and improve over time.

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(d/dt) Nature Of Math = 0?

Over at the Republic of Math, Gary Davis launched a preemptive strike on Thursday's upcoming #mathchat topic: "Does the nature of mathematics change as students get older or is it only the teaching methods that change?". His conclusion? Neither. I would tend to agree with this, but I'm going to play devil's advocate here and argue a different perspective.

Looking at this from the overall historical perspective, the "nature of math" has not changed very much over the years. The concepts at the core of mathematics, like quantities and patterns, are the same as they were thousands of years ago. The teaching methods for encouraging development also remain fundamentally the same as students age. Teachers seek to identify a student's present level of understanding and design learning activities that will bring that understanding to the "next level". From this perspective, neither the nature of math nor methods of teaching change over time.

However, there's something interesting in how this #mathchat topic is phrased. The question is not "does the nature of math change over time?" but rather "does the nature of math change as students get older?". If we concede the former point about the nature of math remaining the same over time from a historical perspective, the phrasing of the #mathchat topic hints at an alternative interpretation. Redirecting this question to the reader:

Did the "nature of math" change as you got older?

My answer to this is "Yes". The "nature of math" became more and more refined as I was exposed to new mathematical ideas. In particular, my preconceived notions about "the nature of math" were shattered into pieces when I learned about this guy:

Kurt Gödel

Around the turn of the 20th century, Russell, Whitehead and Hilbert were attempting to build a solid foundation for mathematics using logic and set theory. This is more or less how I thought of the "nature of math" prior to college. I thought that if I only understood the basic rules of the "math game", I could figure out anything I needed to from applying those rules in a logical manner. It turns out I was wrong.

In 1931, Gödel published his famous Incompleteness Theorem. This result proved that the kind of system that Russell, Whitehead and Hilbert were attempting to create would either be incomplete or inconsistent. Likewise, this theorem single-handedly destroyed the concept of "the nature of math" that I had built up over the years. In its place I started to form a new concept about the "nature of math". Gödel had taught me to start looking at mathematics from outside the box.

Some other works that altered my notions about the nature of math were Turing's proof that the Halting Problem is undecidable and Cantor's Diagonal Argument. There are many parallels between all three of these results. What really resonated with me was the metacognitive component to these proofs. The nature of math had shifted from thinking about how I could play within the rules of the game, to thinking about how those rules could be bent or broken. Instead of "thinking about math", I started "thinking about how I thought about math".

This change in my perception of the nature of math not only influenced my future learning, but also changed how I thought about previous topics. The Republic of Math article mentions trigonometry, which is one of those areas that I was forced to revisit with this new perspective. As Gary notes in this and previous posts (here and here), there are lots of problems with trigonometry's reliance on triangles. Under my new "nature of math", the assumption that triangles needed to lie in a Euclidean space was no longer a safe assumption to make.

My background as computer programmer also altered my conception of trigonometry. Because trig functions and square roots can be computationally expensive, I developed a habit of avoiding them whenever possible. Most of the time I can get the data I need from a dot product instead of working with angles. Instead of using the distance between points, I'll often use the square of the distance as my metric. I'd learned to not just solve the problem, but to reflect on how I was solving the problem and try to optimize that process.

This brings me to the second part of the #mathchat prompt, which is "do teaching methods change as students get older?". Earlier I discussed the notion that the teacher should identify a students level of comprehension and guide them towards the "next level". With this trigonometry example, we can see that this model is overly simplistic. Math is not linear. Instead, the teacher must not only identify the student's current conception, but also which path that student is following so they can encourage them in that direction. If students are starting to think about trigonometry in terms of vector dot products, guide them towards linear algebra. If students are starting to think about triangles on spherical surfaces, guide them towards non-Euclidian geometry. Experienced educators like Gary are probably very adept at this. However, I think the reality of the situation is that most math teachers are not.

From the perspective of this student, the "teaching methods" did not change as I got older. With one exception, my math teachers adhered rigidly to the following procedure:

  1. lecture to blackboard for entire class period
  2. assign dozens of homework problems
  3. test on material
  4. rinse and repeat

How many math classes have you taken that followed this pattern? How many did not? There is an urgent need for the kind of teacher training Gary describes, where the focus is on personalized student development. Too many teachers are caught up in teaching the content, when they should be facilitating student learning instead.

In conclusion:

Does the nature of mathematics change as students get older or is it only the teaching methods that change?

Yes, a student's model of the "nature of math" can change as that student grows older and discovers new results from the field of mathematics. I also think it's arguable that the "nature of math" is not necessarily static, as presumed above, but that Gödel's Incompleteness Theorem fundamentally changed the "nature of math" by using mathematics as a tool for analyzing itself -- giving birth to metamathematics.

No, teaching methods do not change as the student grows older, but they do vary from teacher to teacher. In general, I think the "typical math teachers" need to take a cue from Gödel and start thinking more outside the box. The mathematics classroom needs to shift from its lecture/homework/test/repeat cycle, where math is essentially taught using an "argument from authority", to an experimental environment where students are encouraged to question the information they receive.

Reflections on #mathchat: Mathphobia

Today's #mathchat was a repeat of last Thursday's discussion on “Mathphobia”. One of my questions in Thursday's chat prompted a very insightful commentary from Gary Davis, a.k.a. @RepublicOfMath. With this new evidence in mind, I tried to observe today's #mathchat with a fresh perspective. I couldn't quite condense my thoughts into 140 characters, so I'm taking this opportunity to summarize what I learned from the experience.

First, I think its important to clarify what is meant by “mathphobia”. For the sake of clarity, I'll use the term “mathphobia” in the same sense as @RepublicOfMath's article. Mathphobia is a condition where an individual is terrified of mathematics to the point of feeling physically sick at the thought of math. I'll use the term “math anxiety” to refer to a lesser version of this condition, where an individual experience a fear of math that interferes with mathematical performance but is not as completely disabling as mathphobia. In general, moderate symptoms of math anxiety are highly prevalent in society. As @ColinTGraham noted, research studies have shown that simply telling adults that they're going to take a math test will cause their blood pressure to rise! I realize that this distinction between “math anxiety” and “mathphobia” is somewhat fuzzy, but for the sake of argument these labels will suffice for now.

Reviewing Thursday's #mathchat archive, I think you can see two different conversation lines taking place. One conversation about mathphobia and another about math anxiety.

With regards to @RepublicOfMath's proposal that mathphobia is the result of abusive teachers, this makes a lot of sense from the standpoint of classical conditioning. If a student repeatedly has painful experiences with mathematics instruction, then the student will gradually learn to associate the two. As a consequence, experiencing any subsequent mathematical instruction will automatically trigger a painful response.

With math anxiety, there are similar mechanisms at work. For example, the rise in blood pressure in preparation for a math test can be interpreted as a conditioned response to the need for an increase in cognitive processing. The high prevalence of math anxiety symptoms suggests that math anxiety can develop with or without “abusive teachers”. I think that a variety of the “mathphobia causes” discussed in Thursday's #mathchat may contribute to math anxiety in some form or another, but may not be a cause of the more extreme mathphobia as described above.

With today's #mathchat, I saw something a little bit different happen. The conversation took a turn towards “math avoidance” – the lack of participation in mathematical activities. Here I think we see the crux of the problem. When a student develops math anxiety or mathphobia, that student begins a behavioral pattern of math avoidance. This behavior is self-reinforcing because it allows the student to avoid the painful stimuli associated with math. In order to undo the association that underlies the math anxiety or mathphobia the student needs to be presented with stimulus-response situations that are positive, but when the student avoids math altogether this becomes a difficult task.

The other complication that math avoidance presents is that it becomes difficult to distinguish between students who suffer from math anxiety or mathphobia, and those who are avoiding it for other reasons. Those reasons might be a lack of perceived relevance, a negative social image of math, or a lack of self-confidence. Many of these issues were identified in Thursday's #mathchat, but the focus of today's chat really tied them all together for me.

In conclusion, I think we need to address math anxiety and mathphobia from two directions. First, the classroom needs to be a safe environment where students are free to make mistakes and learn from them rather than being punished for them. Secondly, the behavior of math avoidance needs to be addressed. In order to facilitate the extinction of the conditioned stimulus-response to math, students need to be exposed to math in a positive environment. At first glance, it may seem like this is “treating the symptom rather than the cause”. However, if teachers do not provide temporary relief for the symptom of math avoidance, it won't be possible to “treat the cause” of math anxiety or mathphobia.

Some questions for further discussion:

  • Where does one draw the line between "math anxiety" and "mathphobia"?
  • How can educators address "math avoidance" behaviors?
  • What are the best practices for creating and maintaining an empathetic and non-threatening mathematics classroom?

Rationalize This!

It's been awhile since I blogged, so I thought I'd take a moment to talk about couple interesting math problems that came up in a conversation I, @SuburbanLion, had on Twitter with @MathGuide, @RepublicOfMath, and @GMichaelGuy about a month ago. The topic of discussion was the role of rationalization problems in Algebra II and whether or not the current curriculum addresses the “conceptual core” of these problems.

A common example that one would see in an Algebra II course is:

  • Rationalize the denominator of \frac{1}{\sqrt{2}}

Or simply:

  • Rationalize \frac{1}{\sqrt{2}}

The expected answer for this problem can be obtained by multiplying the numerator and denominator both by \sqrt{2} to get: \frac{\sqrt{2}}{2}. Students might be assigned dozens of such problems in Algebra II. The question to be asking is “Why?”.

I suspect that the obvious answer is historical tradition. These math problems have been passed down from generation to generation as “standard Algebra II problems”, and even the new Core Standards includes “rewrite expressions using radicals and rational exponents” as an objective. Instead of treating these problems as a “means to an end”, these problems have become something of an end in themselves. Algebra II students learn to rationalize expressions because that's what they're going to be tested on. End of story.

The real reason for having these problems in Algebra II goes deeper than that. The Core Standards hits on this reason (at least partially) with one of the additional objectives: “Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, and division by a nonzero rational expression.” While this objective is not specifically talking about radical expressions, the core concept is the same. More concisely, we might say that the important point of rationalizing radical expressions is to come to the conclusion that \mathbb{Q}+\mathbb{Q}\sqrt{p} is a field.

Ironically, the 93 page Core Standards does not even mention the word “field” even though this is essentially the core concept that students are learning about. Students shouldn't be rationalizing expressions just for the sake of rationalizing expressions, they should be exploring the intermediate steps between the field of rational numbers and the field of algebraic numbers. By the way, the phrase “algebraic number” doesn't appear in the Core Standards either!

To make things interesting, G. Michael Guy presented a couple rationalization problems that are typically not included in Algebra II problem sets. These are actually really great examples of the potential complexity involved in rationalization problems.

Let's warm up with the first one, which is significantly easier:

  • Rationalize \frac{1}{1+\sqrt[3]{2}}

This one has an elegantly simple solution using the sum of cubes factorization:

  • a^{3}+b^{3} = (a + b) \cdot (a^{2}-ab + {b^{2}})

Making the substitutions a = 1, b = \sqrt[3]{2}, we can take advantage of this product to rationalize the denominator:

  • \frac{1}{1+\sqrt[3]{2}}\cdot\frac{1-\sqrt[3]{2}+\sqrt[3]{2}^{2}}{1-\sqrt[3]{2}+\sqrt[3]{2}^{2}} = \frac{1-\sqrt[3]{2}+\sqrt[3]{2}^{2}}{1+\sqrt[3]{2}^{3}} = \frac{1-\sqrt[3]{2}+\sqrt[3]{2}^{2}}{3}

This solution is simple enough that this could almost pass for an Algebra II problem. Rationalization problems in Algebra II are something of a gimmick: the problems chosen are special cases designed to have an easy answer. The methods taught in the textbook will solve the given problems, but they don't generalize well to the larger class of problems. The techniques used in Algebra II provide very little help in rationalizing an expression like \frac{1}{1+\sqrt[3]{5}+\sqrt[3]{3}^{2}+\sqrt[3]{2}}!

Before we attempt something like this, lets go back to \frac{1}{1+\sqrt[3]{2}} and come up with a method that will generalize well. This is where easy problems come in handy as a test bed for discovering the broader patterns. The general case of rationalizing \frac{1}{x_{0}+x_{1}\sqrt[3]{2}+x_{2}\sqrt[3]{2}^{2}} can be reasonably done by hand.

Rationalization of 1/(1+2^(1/3)) by hand

The method used here is not likely to be seen in a typical Algebra II classroom, as it relies on concepts from Linear Algebra which typically aren't addressed until later. Seems a little backwards if you ask me. The trick here is to think of the product (x_{0}+x_{1}\sqrt[3]{2}+x_{2}\sqrt[3]{2}^{2})(y_{0}+y_{1}\sqrt[3]{2}+y_{2}\sqrt[3]{2}^{2}) as the product of a matrix and a vector:

  • M_{x}\cdot\overrightarrow{y} = [1, 0, 0]

Solving this equation is then simply a matter of multiplying both sides by M_{x}^{-1}. This method extends nicely to even harder problems, including \frac{1}{1+\sqrt[3]{5}+\sqrt[3]{3}^{2}+\sqrt[3]{2}}. Inverting a 27 by 27 matrix is not something I'd want to be doing by hand, so this is where computers come in handy. Here's an algorithm for rationalizing \frac{1}{1+\sqrt[3]{5}+\sqrt[3]{3}^{2}+\sqrt[3]{2}} in Sage. Compare this with Wolfram|Alpha's result.

I'm a strong believer that computing should play a larger role in mathematics education than it is presently. Not only should the curriculum be addressing the fact that the algebraic numbers form a field, but also that all algebraic numbers are computable numbers. By shifting the focus of discussion from solving problems to finding an algorithm for solving those problems, we can reveal a better picture of the mathematics behind the problem. Simple problems worked out by hand play an important role in the process of designing an algorithm, but an ability to generalize the solution should be the larger goal. If the rationalization problems that students are completing by the dozens do not lead the student in the direction of a general solution, then those problems are not doing their job. Perhaps some “harder” problems are necessary to encourage that generalization.

For the record, “computable numbers” are not referenced once in the Core Standards. It's hard to give students a 21st century education when the math curriculum is trapped in the early 1900s.