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When I first played the original Guild Wars beta, over 7 years ago, the mesmer came with a disclaimer on the character creator stating that the profession required advanced tactics and may not be suitable for new players. I was hooked on the mesmer from there on. In Guild Wars 2, this iconic class has undergone a substancial overhaul and is much more self-sufficient than in the original. However, the GW2 mesmer still has a somewhat steeper learning curve relative to some other professions. I posted a version of the following on the official forums during BWE3 to help new players more effectively use the mesmer’s starting weapon: the scepter. My hope is that this guide will help make the first 10 levels of the game as painless as possible, at which point you can really start to customize the mesmer to your preferred playstyle.

Every new character in GW2 starts in a story mode instance based on the selected race. If this is your first time playing GW2, you should use this opportunity to check out the game options, display settings and key bindings before heading off towards the opening quest objective. Soon enough you’ll encounter your first enemy or “mob”. You only have one attack unlocked, Ether Bolt, so use it. The basic auto-attack on the scepter has a 3 step chain. On every third attack, it creates a clone that looks like you and attacks your mob but doesn’t really do any damage. Now that you have a clone, let’s use F1 or F2 to shatter him for some extra damage. If the mob is attacking your clone or not attacking at all, hit F1 for Mind Wrack which deals direct damage to the mob. If the mob is attacking you or an ally, use F2 for Cry of Frustration which causes a condition called confusion that causes damage when that mob uses a skill. Try both of them out and get a feel for how much damage they do with a single clone. When you unlock your second skill, Illusionary Counter, you can use it to block an attack and generate a clone or tap it twice to blind your foe instead. Test out both of your shatters with two clones and see how they’ve improved. If you find yourself taking too much damage, use skill #6 to heal yourself. You may even unlock your third weapon skill which is a hard hitting channelled attack that applies several stacks of confusion.

Eventually you’ll get to an epic cinematic and huge boss mob. Look out for red circles or big attack animations and dodge out of the way. If you happen to get hit, don’t panic — you’re not dead yet. Start to use the #1 attack in the downed state on one of the low level mobs in the area. You can also use the #2 skill to create a decoy, the #3 skill to summon a phantasm to lay some extra hurt your target, or the #4 skill to heal yourself if you’re close to bleeding out. Keep it up until you score a kill and rally. Go back to working on the boss mob until you finish it off and save the day!

After killing the boss, you get another nice cut scene and a quest reward. This includes a bag, some XP, some coin, and your choice of two off-hand weapons. I’d highly recommend choosing the focus from the first quest reward. Not only is Temporal Curtain great for travelling, but it’s also great for snaring melee attackers so you avoid getting hit and acts as a light field. The Illusionary Warden gives you some AoE damage — which the scepter is lacking, provides defense against projectiles and doubles as a whirl finisher. This means that once you’ve unlocked both of the focus skills you can do your own skill combos! Start with using Illusionary Warden on a mob then place a Temporal Curtain on top of him. The resulting combo shoots Cleansing Bolts in different directions and remove conditions from allies that they hit. While testing this out, you may observe that the Illusionary Warden is a stationary phantasm, and won’t move until you use a shatter skill. I’d also recommend that you have at least one clone up before summoning the Warden so he doesn’t get killed right away.

Against melee attackers, you want to open with the scepter auto attack and use Temporal Curtain to cripple them. Your goal is to get at least two clones up before the mob gets close to you. When the mob starts to get close, use Confusion Images. While channeling, use Cry of Frustration. Depending on how many clones you had up, the mob should have 8-9 stacks of confusion. and should be just about to swing at you. Use Illusionary Counter to block the attack, and watch the mob take massive confusion damage while you get away scratch free! If that didn’t kill it, go back to auto-attacking, use the dodge roll to avoid the next two attacks, then use Mind Wrack when you get enough clones.

In dynamic events with lots of people, don’t be stingy with your shatters. The odds of you getting 3 illusions up in such a situation are pretty slim unless you’re fighting a champion, so don’t hestitate to do a single illusion shatter if that’s all you can get off before the mob dies. If possible, try to summon your Warden on a target in the middle of a group with high health, let him do his attack once, then shatter him. He’s going to die when your target dies anyways, so you may as well get the most out of it.

Once you’ve reached level 5, you’ll unlock your first utility skill. There’s a lot of really great skills to choose from here, but the one I would recommend picking up first is Blink. This is a ground targeted skill that teleports you to the selected location while breaking stuns. The reason I suggest this skill first is that it will help you get around the map a little bit faster when out of combat, while also helping you keep your distance from melee mobs. Look for skill challenges on the map, marked with the blue chevron, and complete them to unlock some other utility skills of your choice. My personal favorites on the 1st tier are Null Field and Signet of Domination. If you’re unsure of how a particular skill works, you can always use the PvP menu to go to the Mists and test it out on the target dummies there.

Eventually you’ll start to encounter some ranged attackers. Against these mobs you can open with Confusing Images and Illusionary Counter since they’ll generally start attacking right away. Since you get a clone on counter, this will give you a distraction to summon an Illusionary Warden on them and block their projectiles. Once you get a few more skill points, you can combine confusion with the Feedback utility skill to quickly take down ranged attackers.

I’d recommend sticking with scepter/focus until at least level 7 when you unlock weapon swapping. Weapon swapping really opens up a lot of play-style options for the mesmer. It’s much easier to survive as sword mesmer when you can weapon swap to a staff for Phase Retreat after blowing your defensive cool-downs. Likewise, the greatsword benefits from a weapon swap once enemies get close. You may want to keep the scepter/focus on a weapon swap until you unlock all the skills on your weapon of choice. These are just my opinions of course, so if there’s a weapon combination you really want to play with go right ahead! Regardless of what you choose, you should always keep a focus in your bag so that you can switch to it when you’re not in combat for the swiftness buff.

I just hope that these tips help some newcomers who might be turned off by the scepter after hearing horror stories from the first two BWEs. The scepter was vastly improved in BWE3 and I expect it to be even more polished by release. Overall, I think that using the scepter/focus will help you learn how to effectively utilize the core mesmer mechanics. There’s a lot to learn about playing a mesmer, but the class has a nice rhythm once you get used to it. Stick it out for a while and you might be pleasantly surprised!

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This weekend marks the 3rd Beta World Event for Guild Wars 2. I wrote a little bit about my general experiences in the first BWE, but this time I’m focusing on a very specific area of the game. In the first BWE, I was just playing the game and having fun with it. In the second BWE, I started to do a lot more “testing”. In particular, one of the things I was testing was the “Sharper Images” trait.

Sharper Images (SI) is a Dueling trait that causes critical hits from Illusions to inflict bleeding for 5 seconds. This trait was bugged in the first BW1 and didn’t work at all. In the second BWE1, it worked as described but a second phantasm trait called “Phantasmal Haste” was bugged resulting in some crazy damage output. This means that I didn’t get a very good perspective on how these two traits would work together, but that’s okay because I can do the math! In addition to seeing how the phantasm related traits would interact together, I also wanted to find out which stats to gear for in order to maximize my damage. In order to do this, we first need some information about how damage is calculated in GW2. Assuming a level 80 character:

  • Pandara_RA! at Team Legacy worked out the following formula for the base damage of an attack: $$ Base Damage = \frac{(Power) \cdot (Weapon Damage) \cdot (Skill Coefficient)}{Target Armor} $$
  • The chance of getting a critical attack is determined by the Precision above the base: $$ CritRate= \frac{4 + (Precision – Base)/21}{100} $$
  • When an attack criticals, it hits for 50% more damage plus any bonus to critical damage (Prowess). With this, we can find out the average damage of an attack using: $$ Direct Damage = (Base Damage) \cdot (1+(Crit Rate) \cdot (0.5+\frac{Prowess}{100})) $$
  • The last piece of information we need is the bleeding damage, which is dependent on condition damage (Malice). According to the GW2 wiki this is determined by $$ \frac{damage}{second} = 40+0.05 \cdot (Malice) $$. The bleed duration of 5 seconds can be improved through stats, but only pulses once per second. This means that we can round the duration down to find the number of pulses and find the total bleed damage: $$ \frac{damage}{second} \cdot \lfloor duration \rfloor $$

To get a rough estimate of Phantasm DPS, I put these formulas together with some various equipment set-ups and trait choices. You can download this spreadsheet here. To make things simplier, I focused entirely on “Illusionary Duelist” with SI because I knew it hits 8 times every 10 seconds. I also had to make several assumptions about how certain traits would stack, and all of this is subject to change when the game is released anyway. Despite these shortcomings, I found several interesting results:

  • Without any bonus condition damage, SI can add about 10%-20% damage depending on the target’s armor (best against higher armor foes) when used in conjunction with Phantasmal Fury. This puts it on par with most damage traits at the adept level.
  • With a skill coefficient of about 0.5 (a total guess BTW), the direct damage builds and condition damage builds I tried seem to even out in terms of potential damage. A lower skill coefficient tends to favor condition damage and a higher one favors direct damage.
  • Chaotic Transference bonus seems lack-luster relative to the heavy investment.
  • Phantasmal Strength and Empowered Illusions complement each other well in a power Build, but the investment for Phantasmal Strength doesn’t seem worth it in a condition damage build.
  • Phantasmal Haste tends to work better with a condition damage build than a power build. You don’t need to hit hard with SI, you just need to hit often.
  • Investing 20 points into Domination can have a big effect on condition damage builds because it extends bleeds for an extra tick. This makes Lyssa’s Runes a potentially interesting choice with SI because of the +10% condition duration, allowing you to spend 10 of those points from Domination elsewhere with minimal DPS loss.
  • The Rampager jewelry seems to be a better choice than Rabid for a condition damage build with SI. There’s no point to having strong bleeds if you aren’t applying them frequently enough.

There’s still a lot more analysis to be done here and some empirical data to collect in BWE3 to verify these findings, but the results look promising. As it stands, you can make SI work in either a direct damage phantasm build or condition damage build with the appropriate gear. Small tweaks to the skill coefficient can keep the two builds competitive if necessary. This fits with Arena.Net’s philosophy of having multiple play-styles be equally viable.

I’d encourage you to try out the spreadsheet with other gear and build combinations that I didn’t try. If you’re feeling adventurous, you might even extend it to include skills other than iDuelist or other traits I may have overlooked. If you find out any more information about how phantasm damage is calculated I’d love to hear about it in the comments!

Happy theory-crafting!

Update: BWE3

I did a little testing during BWE3, regarding the attack rates and skill coefficients of the different phantasms. This information should help give an idea of how much each phantasm benefits from stacking Power vs stacking crit/condition damage for Sharper Images. Please note that my recharge times were approximated, and Sanek over at GW2Guru came up with somewhat different numbers. I’m including both my attack rates and his for comparison:

illusion Hits Recharge Attack Rate (hits/sec) Sanek’s Recharge Sanek’s Rate (Hit/sec) Approx. Skill Coef. DPS Coef. (Mine) DPS Coef. (Sanek)
iDuelist 8 10 0.8 7.5 1.066666667 0.228956229 0.183164983 0.244219978
iSwordsman 1 3 0.333333333 5.5 0.181818182 0.734006734 0.244668911 0.13345577
iWarlock 1 5 0.2 6 0.166666667 0.080808081 0.016161616 0.013468013
iBerserker 1 5 0.2 6 0.166666667 0.281144781 0.056228956 0.046857464
iMage 1 5 0.2 6.7 0.149253731 0.397306397 0.079461279 0.059299462
iDefender 1 3 0.333333333 4.5 0.222222222 0.131313131 0.043771044 0.029180696
iDisenchanter 1 3 0.333333333 4.5 0.222222222 0.131313131 0.043771044 0.029180696
iWarden 12 10 1.2 14 0.857142857 0.033670034 0.04040404 0.028860029
swordClone 3 3 1
staffClone 1 1 1
scepterClone 2 3 0.666666667
gsClone 3 2 1.5

Knowing that the skill coefficient for iDuelist is only 0.23, stacking for condition damage seems to be the best method to maximize damage over time with Sharper Images given a high enough crit rate to apply it consistently. As a general rule of thumb, if your crit rate is less than 50% then you should be gearing for power and if your crit rate is greater than 50% then you should be gearing for condition damage.

A few other interesting things to note:

  • iSwordsman has one of the best skill coefficients of any phantasm. If you’re not using Sharper Images and have Power oriented spec, you may want to try out the off-hand sword.
  • iWarlock’s DPS is pretty pitiful without conditions. I’m not sure what the bonus per condition is, but I’d recommend having two staff clones up with iWarlock since they have a much faster attack rate. Edit: 10% bonus per condition
  • iWarden has quick attack rate and is has an AoE attack, but remember that this Phantasm is stationary. You’re very unlikely to get all 12 hits against a real player.
  • iBerserker has slow recharge AoE attack that moves down a line. It might be possible to hit an opponent twice with this if they’re running in the same direction, but I can’t be sure about it.
  • The Greatsword clones have the fastest attack rate of any illusion according to my tests. It seems kind of odd that the best clone for Sharper Images would be on a weapon with no innate condition damage.
  • iMage has a high skill coefficient but low attack rate. At first glance, this looks like it would be better for a power build than condition build, but you should remember that he also applies Confusion on attack.
  • iMage and iDisenchanter have bouncing attacks that hit three targets: 1 enemy and 2 allies. I couldn’t seem to get it to hit the same enemy twice, but this is something to check for on release.
  • Keep in mind that my original spreadsheet assumes that you leave your Phantasms out all the time. As of BWE3, this is no longer the optimal play-style. If you decide to go with a Power build, you’ll probably get the best burst damage by using Mind Wrack right after your phanstasm’s first attack cylce. Likewise, Cry of Frustration can now dish out some major hurt if you’re built for condition damage.

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In a previous blog post, I made the claim that much of the math curriculum is ordered based on historical precedent rather than conceptual dependencies. Some parts of the math curriculum we have in place is based on the order of discovery (not always, but mostly) and while other parts are taught out of pure habit: This is how I was taught, so this is how I’m going to teach. I don’t think this needs to be the case. In fact, I think that this is actually a detriment to students. If we want to produce a generation of mathematicians and scientists who are going to solve the difficult problems of today, then we need to address some of the recent advances in those fields to prepare them. Students should not have to “wait until college” to hear about “Topology” or “Quantum Mechanics”. We need to start developing the vocabulary for these subjects much earlier in the curriculum so that students are not intimidated by them in later years.

To this end, I’d like to propose 5 mathematical breakthroughs that are both relatively recent (compared to most of the K-12 curriculum) while also being accessible to elementary school students. Like any “Top 5”, this list is highly subjective and I’m sure other educators might have differing opinions on what topics are suitable for elementary school, but my goal here is just to stimulate discussion on “what we could be teaching” in place of the present day curriculum.

#1. Graph Theory (c. 1736)

The roots of Graph Theory go back to Leonard Euler’s Seven Bridges of Königsberg in 1736. The question was whether or not you could find a path that would take you over each of the bridges exactly once.

Bridges of Königsberg

Euler’s key observation here was that the exact shapes and path didn’t matter, but only how the different land masses were connected by the bridges. This problem could be simplified to a graph, where the land masses are the vertices and the bridges are the edges.

This a great example of the importance of abstraction in mathematics, and was the starting point for the field of Topology. The basic ideas and terminology of graph theory can be made easily accessible to younger students though construction sets like K’Nex or Tinkertoys. As students get older, these concepts can be connected to map coloring and students will be well on their way to some beautiful 20th century mathematics.

#2. Boolean Algebra (c. 1854)

The term “algebra” has developed a bad reputation in recent years. It is often referred to as a “gatekeeper” course, which determines which students go on to higher level mathematics courses and which ones do not. However, what we call “algebra” in middle/high school is actually just a subset of a much larger subject. “Algebra I” tends focuses on algebra as it appeared in al-Khwārizmī’s Compendious Book on Calculation by Completion and Balancing (circa 820AD). Consequently, algebra doesn’t show up in the math curriculum until students have learned how to add, subtract, multiply and divide. It doesn’t need to be this way.

In 1854, George Boole published An Investigation of the Laws of Thought, creating the branch of mathematics that bears his name. Rather than performing algebra on numbers, Boole used the values “TRUE” and “FALSE”, and the basic logical operators of “AND”, “OR”, and “NOT”. These concepts provided the foundation for circuit design and eventually lead to the development of computers. These ideas can even be demonstrated with a variety of construction toys.

The vocabulary of Boolean Algebra can and should be developed early in elementary school. Kindergartners should be able to understand basic logic operations in the context of statements like “grab a stuffed animal or a coloring book and crayons”. As students get older, they should practice representing these statements symbolically and eventually how to manipulate them according to a set of rules (axioms). If we develop the core ideas of algebra with Boolean values, than perhaps it won’t be as difficult when these ideas are extended to real numbers.

#3. Set Theory (c. 1874)

Set Theory has its origins in the work of Georg Cantor in the 1870s. In 1874, Cantor published a ground breaking work in which he proved that there is more than one type of infinity — the famous “diagonal proof“. At the heart of this proof was the idea of thinking of all real numbers as a set and trying to create a one-to-one correspondence with real numbers. This idea of mathematicians working with sets (as opposed to just “numbers”) developed momentum in the late 1800s and early 1900s. Through the work of a number of brilliant mathematicians and logicians (including Dedekind, Russell, Hilbert, Peano, Zermelo, and Fraenkel), Cantor’s Set Theory was refined and expanded into what we know call ZFC or Zermelo-Fraenkel Set Theory with the Axiom of Choice. ZFC was a critical development because it formalized mathematics into an axiomatic system. This has some suprising consequences such as Gödel’s Incompleteness Theorem.

Elementary students probably don’t need to adhere to the level of rigor that ZFC was striving for, but what is important is that they learn the language associated with it. This includes words and phrases like “union” (“or”), “intersection” (“and”), “for every”, “there exists”, “is a member of”, “complement” (“not”), and “cardinality” (“size” or “number”), which can be introduced informally at first then gradually formalized over the years. This should be a cooperative effort between Math and English teachers, developing student ability to understand logical statements about sets such as “All basset hounds are dogs. All dogs are mammals. Therefore, all basset hounds are mammals.” Relationships can be demonstrated using visual aids such as Venn diagrams. Games such as Set! can further reinforce these concepts.

#4. Computation Theory (c. 1936)

Computation Theory developed from the work of Alan Turing in the mid 1930s. The invention of what we now call the Turing Machine, was another key step in the development of the computer. Around the same time, Alzono Church was developing a system of function definitions called lambda calculus while Stephen Kleene and J.B Rosser developed a similar formal system of functions based on recursion. These efforts culminated in the Church-Turing Thesis which states that “everything algorithmically computable is computable by a Turing machine.” Computation Theory concerns itself with the study of what we can and cannot compute with an algorithm.

This idea of an algorithm, a series of steps to accomplish some task, can easily be adapted for elementary school instruction. Seymour Papert has been leading this field with technologies like LOGO, which aims to make computer programming accessible to children. Another creative way of approaching this is the daddy-bot. These algorithms don’t need be done in any specific programming language. There’s much to be learned from describing procedures in plain English. The important part is learning the core concepts of how computers work. In a society pervaded by computers, you can either choose to program or be programmed.

#5. Chaos Theory (c. 1977)

Last, but not least, is Chaos Theory — a field of mathematics that developed independently in several disciplines over the 1900s. The phrase “Chaos Theory” didn’t appear in the late 1970s, but a variety of phenomena displaying chaotic behavior were observed as early as the 1880s. The idea behind Chaos Theory is that certain dynamic systems are highly sensitive to initial conditions. Drop a shot of half-half into a cup of coffee and the resulting pattern is different every time. The mathematical definition is a little more technical than that, but the core idea is relatively accessible. Chaos has even found several notable references in pop culture.

The other core idea behind chaos theory is topological mixing. This could be easily demonstrated with some Play-Doh (or putty) of two or more colors. Start by combining them into a ball. Squash it flat then fold it over. Repeat it several times and observe the results.

The importance of Chaos Theory is that it demonstrates that even a completely deterministic procedure can produce results that appear random due to slight variations in the starting conditions. This can even be taken one step further by looking at procedures that generate seeming random behavior independently of the starting conditions. We live in an age where people need to work with massive amounts of data. The idea that a simple set of rules can produce extremely complex results provides us with tools for succinctly describing that data.

Conclusion

One of the trends in this list is that these results are easy to understand conceptually but difficult to prove formally. Modern mathematicians seem to have a tendency towards formalism, which is something of a “mixed blessing”. On one hand, it has provided mathematics with a firm standard of rigor that has withstood the test of time. On the other hand, the language makes some relatively simple concepts difficult to communicate to younger students. I think part of the reason for this is that the present curriculum doesn’t emphasize the rules of logic and set theory that provide the foundation for modern mathematics. In the past, mathematics was driven more by intuitionism, but the math curriculum doesn’t seem provide adequate opportunities for students to develop this either! It might be argued things like “new math” or “Singapore math” are helping to develop intuitionism, but we’re still not preparing students for the mathematical formalism that they’ll be forced to deal with in “Algebra I” and beyond. Logic and set theory seem like a natural way to develop this familiarity with axiomatic systems.

Observers might also note that all five of these proposed topics are related in some form or another to computer science. Computers have been a real game-changer in the field of mathematics. Proofs that were computationally impossible 500 years ago can be derived a in minutes with the assistance of computers. It’s also changed the role of humans in mathematics, from being the computer to solving problems using computers. We need to be preparing students for the jobs computers can’t do, and my hope is that modernizing the mathematics curriculum can help accomplish this.

Do you have anything to add to this list? Have you tried any of these topics with elementary students? I’d love to hear about your experiences in the comments below.

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Gary Davis (@republicofmath) wrote an article that caught my attention called What’s up with pre-calculus?. In it, he presents a number of different perspectives on why Pre-Calc classes have low success rates and do not adequately prepare students for Calculus.

My perspective on pre-calculus is probably far from the typical student, but often times the study of “fringe cases” like myself can provide useful information on a problem. The reason why my experience with Pre-Calc was so atypical, is because I didn’t take it. After taking Algebra I, I had started down a path towards game programming. By the end of the following year, where I had taken Geometry, this little hobby of mine hit a road block. I had come to the realization that in order to implement the kind of physics that I wanted in my game I would need to take Calculus. I petitioned my counselor to let me skip Algebra II and Pre-Calc to go straight into AP Calculus. They were skeptical at first, but eventually conceded to my determination and allowed me to follow the path I had chosen.

Skipping from Geometry to Calculus meant that there were a lot of things that I needed to learn that first month that many of my peers had already covered. I had never even heard the word “logarithm” before, had no idea what e was, and had only a cursory understanding of trigonometry. These were the topics I had missed by skipping Pre-Calc, and I was fully aware of that, so I “hit the books” and learned what I needed to know about them. By the end of that first month I had caught up to the rest of the class and by end of the semester I would be helping other students with those very same topics.

I think the most obvious difference between myself and the “typical Calculus student” was the level of motivation. Many of the students in Calculus were there because “it would look good on a college application”. I was there because I wanted to be there. A common problem throughout math education is the “When am I ever going to use this?” attitude. I already knew where I was going to use the math I was learning. I had an unfinished game at home that needed a physics system, and every new piece of information I learned in Calculus made me one step closer to that goal. If you had ever wondered why a 4th order Runge-Kutta method is better than Euler’s method, try writing a platformer.

The second difference was a little more subtle, but there were some conceptual differences in how I thought about exponential, logarithmic, and trigonometric functions. The constant “e” wasn’t just some magic number that the textbook pulled out of thin air, it was the the unique number with the property that $$ \frac{de^x}{dx} = e^x $$ and $$ \int e^x dx = e^x $$. When it came to sine and cosine, I would think of them like a circle while my other classmates would picture a right triangle. They would hear the word “tangent” and think “opposite over adjacent”, but I thought of it more like a derivative. Sure, I had to learn the same “pre-calc” material as they did, but the context of this material was radically different.

A couple years ago I suggested that Pre-Calc should be abolished. The trouble with Pre-Calculus (at least in the U.S.) is that the course needs to cover a very diverse array of questions which includes exponential, logarithmic and trigonometric functions. I would argue that these concepts are not essential to understanding the basic foundations of Calculus. The math curriculum introduces the concept of “slope” in Algebra I, which is essentially the “derivative” of a line. There’s no reason why we should be sheltering students from language of Calculus. The concepts of “rate of change” and “accumulation” can and should be connected with the words “derivative” and “integral”, long before students set foot in the course we presently call Calculus. As students become more comfortable with these concepts as they relate to lines, parabolas and polynomials, then gradually step up the level of complexity. When students start to encounter things like surfaces of revolution, then they’ll actually have a reason to learn trigonometry. Instead of trigonometry being the arbitrary set of identities and equations that it might appear to be in pre-calc, students might actually learn to appreciate it as a set of tools for solving problems.

I think this issue of Pre-Calc is really a symptom of a larger problem. The mathematics curriculum seems to be ordered historically rather than conceptually. I’ve heard Pre-Calc described as a bridge to Calculus. This makes sense when you consider the historical development of Calculus, but not when considering the best interest of students in today’s society. Leibniz and Newton didn’t have computers. Who needs bridges when you can fly?

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At long last, I finally got my chance to play Guild Wars 2! An ambitious title to say the least, GW2 is leaps and bounds ahead of its predecessor. I clocked in as much time as I could over the beta weekend, and am now going through withdrawal so I thought I’d take this opportunity to share my experiences. I had hoped to record some video over the weekend, but the performance impact was a little more than I expected. A choppy YouTube video would hardly do this game justice, so you’ll just have to settle for a written account.

As a mesmer primary in the original Guild Wars, it should be no surprise that I gravitated towards this class in GW2. I had my concerns about how this class would carry over, but any doubts that I had are now gone. The GW2 mesmer uses completely different game mechanics than the GW1 Mesmer, but it still captures the essential feeling of the class. I played the mesmer class through the end of the BWE personal story line at level 18, and spend the rest of my time playing PvP with various classes.

Coming from GW1, I had a lot of preconceived notions as to what GW2 would be like. It makes sense to organize these into the parts I overestimated and parts I underestimated. These are not necessarily pros and cons, but simply differences between how I thought GW2 would work and how it actually worked. Keep in mind that the game is still a work in progress, and I’m trying to give an honest opinion so that it can be made even better in the future.

Since I spent the most time playing a mesmer, that’s probably the best place for me to start. From the very beginning of the game, it became clear that I had overestimated the shatter skills and underestimated the phantasm skills. Don’t get me wrong, a Mind Wrack with both traits is definitely a force to be reckoned with, but it’s just that I thought that the Mind Wrack traits and a high level of Guile would be essential to any mesmer build. It turned out that this wasn’t the case and there were other viable ways of playing the class. In fact, the phantasms were so awesome that I was usually hesitant to shred them. The three illusion limit also meant that I had to be careful about when I used my clones to not overwrite my phantasms. Since the illusions are locked on to a single target, this added an additional incentive to shred them before the target dies. All of these considerations made the shatter skills something that required careful timing and not just another skill to spam.

As far as the mesmer weapons are concerned, I think I overestimated the scepter and underestimated the staff. The scepter was great at pumping out clones, but I ended up relying on phantasms for most of my damage and it was hard to keep up confusion in PvP. On the other hand, I found myself enjoying the staff a lot more than I expected. The idea of random conditions and boons was something of a turn off for me when I first read the skill descriptions, but the way that it worked was that offensive buffs/debuffs were tied to offensive skills and the defensive buffs/debuffs were tied to defensive skills. Even though the results were random, they were still something that was useful in the situation.

Slightly related to the points above, I think I overestimated the confusion condition and underestimated poison. I think that part of this was that I was trying to compare confusion with hexes like Empathy and Backfire in GW1, where a whole stack of hexes becomes problematic to remove. A whole stack of confusion in GW2 still only counts as one condition, making it relatively easy to remove — if it was even worth removing at all. On the other hand, I somehow missed the memo about poison reducing healing by 33%. This healing reduction coupled with a sometimes deceptively long duration made poison into one of the more threatening conditions for me. I distinctly remember this one thief that I fought where I had to keep spamming my healing skill until my condition removal was up again to avoid dying from poison long after I had killed him. By the time I was able to remove it, he had respawned and was back for more while I was still at 25% health — ouch.

One of the other things that caught me off guard was that I overestimated Vitality and underestimated Toughness. In GW1, gearing for PvP was quite simple: stack as much health as you can. GW1 had a large number of armor ignoring damage sources and having enough health to survive a large damage spike was essential to surviving long enough for the healers to react. In GW2, there are no “healers”. I found that rather than dealing with a single large damage spike, sustained damage was much more of a threat. This made it more practical for me to focus on improving damage mitigation rather than just stacking health. Of course, this might also be a function of beta players not using voice chat to coordinate spikes.

In terms of the PvP modes, I think I overestimated “Structured PvP” and underestimated “World vs World”. I spent a lot of time playing Random Arenas in GW1 and it seemed like the “Structured PvP” was going to be right up my alley. I enjoyed the Structured PvP in GW2, but I think that was mainly because it gave me a chance to play with a fully leveled and unlocked character. The problem I had with Structured PvP it was that it felt too much like an FPS match to me. Win or lose, you were queued for the next match on the same server, fighting the same opponents with the same team over and over until someone leaves. Despite the best efforts of the game to auto-balance the teams, they were often off by one player which makes a big difference with smaller numbers. Nothing like fighting a 2 on 1 battle to ruin the fun. I also spotted a couple of “leechers” in the beta, which I thought was kind of odd. Its not like it was worth farming Glory in the beta, so I’m assuming that they just went AFK and forgot about it. However, “World vs. World” was a completely epic experience. Despite the fact that none of us really knew what we were doing, the sheer number of people on the battlefield was a sight to behold. In comparison to World of Warcraft, World vs World seemed like Alterac Valley on steroids with added benefits outside of the battleground. The only downside to World vs World that I saw was that the repair costs could rack up quickly if you were careless.

All in all, I think Guild Wars 2 met and then exceeded my expectations in a lot of areas. If I were to make a comparison, it felt like Guild Wars 2 was like the MMO equivalent of an Elder Scrolls game. The starting “tutorial zones” seemed less like a tutorial and more like a story “hook” — my favorite of which being the Charr. The dynamic event system made questing feel less like a grind and more like a sandbox. There’s a main story if you want to follow it, or you can just go explore the world and do whatever you like.

I also felt like I had a great amount of freedom in how to play my class. The build I finally settled on with my mesmer in PvP was this one. The Illusionary Warlock and Illusionary Duelist combination packed some serious damage while the Chaos traits and crippling clones made me pretty difficult to kill. Chaos Storm and Null Field allowed me to skill combo by myself for extra conditions. The Portal skill was simply amazing in Conquest mode for feigning retreat from a node, only to return later with reinforcements. The only downside of this build was that I had to be careful when swapping the scepter so that I didn’t overwrite my phantasms with clones — I would just use the two pistol skills and the channeled confusion skill then immediately swap back to the staff. In PvE I replaced the pistol with a focus and Portal with Blink to get around faster, since the burst damage wasn’t as important.

Another hidden surprise was the crafting system. I didn’t get much time to play with it, but it seemed to be a nice balance between the “discovery” system of Final Fantasy XI and the “recipe list” system of World of Warcraft. You combine random materials to learn new recipes, but once you learn they are added to the recipe list. As an added bonus, it seemed to accelerate the crafting process when I was creating items in bulk! It’s the little details like that which made me feel GW2 was respecting the time I spent playing.

Are you a long time Guild Wars player that participated in the GW2 Beta? What did you underestimate or overestimate about the sequel?

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Is “rationality” a measurable quantity?

In a previous blog post, I discussed some common logical errors that often arise in political discourse. This led to a rather interesting discussion on Twitter about political behaviors and how to model them mathematically (special thanks to @mathguide and @nesa_k!). One of the questions that came up this this discussion was how to define “rational behavior” and whether or not this is a measurable quantity. What follows is my hypothesis on “rational behavior”: what it is and how to measure it.

Please keep in mind that this is just a hypothesis and I don’t quite have the resources to verify these claims experimentally. If anyone has evidence to support or dispute these claims, I would certainly be interested in hearing it!

Defining “rational behavior”

Before we can begin to measure “rationality”, we must first define what it means to be “rational”. Merriam-Webster defines “rational” as “relating to, based on, or agreeable to reason”. The Online Etymology Dictionary describes the roots of the word in the Latin rationalis, meaning “of or belonging to reason, reasonable”, and ratio, meaning “reckoning, calculation, reason”. It’s also worthwhile to mention that ratio and rational have a distinct mathematical definition referring to the quotient of two quantities. Wikipedia suggests that this usage was based on Latin translations of λόγος (logos) in Euclid’s Elements. This same Greek word lies at the root of “logic” in English.

Based on these definitions and etymology, I think its fair to define rational behavior as “behavior based on a process of logical reasoning rather than instinct or emotion”.

Even this definition is far from perfect. In the context of game theory, “rational behavior” often defined as the process of maximizing benefits while minimizing costs. Note that by this definition, even single celled organisms like amoeba would be considered to exhibit “rational behavior”. In my opinion, this minimax-ing is a by-product of evolution by natural selection rather than evidence of “reason” as implied by the typical usage of the word “rational”.

I should also clarify what I mean by “logical reasoning” in this definition. In trying to quantitatively measure rational behavior, I propose that it makes sense to use a system of fuzzy logic rather than Boolean logic. By using the Zadeh operators of “NOT”, “AND”, and “OR”, we can develop an quantitative measure of rationality on a scale of 0 to 1. In logic, we say that an arguement is considered sound if it’s valid and its premises are true. Since we’re using the fuzzy “AND” in this model, the rationality measure is the minimum truth value of the logical validity and base assumptions.

Using this definition, we can also define irrational behavior as “behavior based on an invalid logical argument or false premises”. I’d like to draw a distinction here by defining arational behavior as “instinctive behaviors without rational justification”, to cover the amoeba case described above. An amoeba doesn’t use logic to justify its actions, it just instinctively responds to the stimuli around it.

Rationalism and Language

There’s an implicit assumption in the definition of “rational behavior” that I’ve used here, and that is that this requires some capacity for language. First-order predicate logic is a language, so the idea that “rational behavior” is language dependent should come as no surprise. In fact, the same Greek word “logos” from which “rational” is derived was also used as a synonym for “word” or “speech”. The components of language are necessary for constructing a formal system, by providing a set of symbols and rules of grammar for constructing statements. Add a set of axioms (assumptions) and some rules for inference, and you’ll have all the components necessary to construct a logical system.

A Dynamic Axiomatic System Model of Rational Behavior

A this point we can start to develop an axiomatic system to describe rational behavior. Using the operators of fuzzy logic and the normal rules of first-order logic we can create an axiomatic system that loosely has the properties we would expect of “rational behavior”. It’s very unlikely that the human mind uses the exact rules of fuzzy logic, but it should be “close enough”. We also have to consider that the basic beliefs or assumptions of a typical person vary over time. Thus, it’s not enough to model rational behavior as an axiomatic system alone, we must consider how that system changes over time. In other words, this is a dynamic system.

As we go through life, we “try out” different sets of beliefs and construct hypotheses about how the world works. These form the “axioms” of our “axiomatic system”. Depending on whether or not these assumptions are consistent with our experiences, we may decide to keep those axioms or reject them. When this set of assumptions contains contradictions, the result is a feeling of discomfort called cognitive dissonance. This discomfort encourages the brain to reject one of the conflicting assumptions to reach a stable equilibrium again. The dynamic system resulting from this process is what I would characterize as rational behavior.

One particularly powerful type of axiom in this system is labeling. Once a person takes a word or label and uses it to describe him or herself, the result is the attribution of large number of personal characteristics at once. The more labels a person ascribes to, the more likely it is that a contradiction will result. Labeling also has powerful social effects associated with it as well. Ingroups and outgroups can carry with them substantial rewards or risks depending on the context.

Rather than rejecting faulty axioms when confronted with cognitive dissonance, some individuals develop alternative methods of reducing the discomfort. The general term for pattern of behavior is called cognitive bias. This behavior can take a variety of different forms, but the one that is most relevant to this discussion is the confirmation bias. One of the ways in which the human brain can reduce the effects of cognitive dissonance is by filtering out information that would result in a contradiction with the base assumptions. Another relevant bias to consider is the belief bias, or the tendency to evaluate the logical validity of an argument based on a pre-existing belief about the conclusion.

Whatever form it may take, cognitive bias should be taken as evidence of “irrational behavior”. Not all cognitive biases are of equal magnitude, and some arguments may rely more highly on these biases than others. The goal here is not a Boolean “true” or “false” categorization of “rational” and “irrational”, but more of a scale like the one used by PolitiFact: True, Mostly True, Half-True, Mostly False, False, Pants on Fire. The method of applying truth values in fuzzy logic makes it highly appropriate for this purpose.

Examples in Politics

Consider this clip from The Daily Show. Using this clip may seem a little biased, but it’s important to remember that John Stewart is a comedian. Comedians have an uncanny knack for walking the fine line between “rational” and “irrational”, providing an interesting perspective to work with.

In the first example, we have the issue of Rick Santorum and JFK. After reading JFK’s speech on religious freedom, Santorum says that it made him want to throw up. In order to defend this statement, Santorum uses a good ole fashioned straw man argument by claiming that JFK was saying “no faith is not allowed in the public public square” when in fact JFK was saying “all faiths are allowed”. I think Santorum’s behavior here is a prime example of irrational behavior. Taking this position may very well earn him some votes with the deeply religious, but it’s clear that Santorum has some problems finding consistency between his personal beliefs and the First Amendment. His position is not based on a valid logical argument, but on a physical response to the cognitive dissonance resulting from his conflicting beliefs. This example also shows the power of deeply held self-labeling behaviors like religion.

Mitt Romney made some headlines with his “NASCAR Team Owner” blunder. It would appear that Mitt Romney had gone to Daytona to try and score some points with “average Americans”, but a slip of the tongue showed how out of touch he really is. To Romney’s credit, his behavior here is about half-rational. His assumptions are probably something like this:

  • I want people to vote for me.
  • People vote for someone they can relate to.
  • Most people know someone who likes NASCAR.
  • I know someone who likes NASCAR.

It makes sense from a logical standpoint, but it turns out that the person who Romney knows that likes NASCAR just happens to be a
“team owner” instead of a “fan”. This small detail makes it unlikely that people will relate to him, but at least the foundation of a logical argument is there.

This brings us back to Rick Santorum again. This time, Santorum calls President Obama a “snob” for “[wanting] every American to go to college”. Not only is this comment blatantly false, but he’s employing an ad hominem attack in lieu of a logical argument. This example draws a nice dichotomy between President Obama and Rick Santorum. The President is making a rational argument in favor of higher education which is well supported by evidence. By opposing this rational argument on a faulty premise, Santorum comes out of this situation looking mostly irrational. His behavior makes sense if you consider the effects of confirmation bias. Santorum believes that the President is trying to indoctrinate college students to become liberals. He believes it so thoroughly that he simply filters out any evidence that would contradict it. While most observers can hear the President say “one year of higher education or career training“, Santorum doesn’t. He hears the part confirms his beliefs and filters out the rest. I’d imagine that for Santorum, listening to President Obama speak sounds something like the teacher from the Peanuts cartoons: “one year of higher education wah wah-wah wah-wah-wah“. To Santorum’s credit, at least he had the mind to retract his “snob” statement — even if only partially. This shows that the underlying mechanisms for rational behavior are still there, despite his frequent leaps of logic.

Conclusion

I hope I’ve at least managed to present a definition of “rationality” that’s a little more precise than the everyday use of the term. I’m sure some people out there might disagree with the way I’ve rated the “rationality” of these behaviors. Different people have different experiences and consequently have different assumptions about the world. If we were to use multiple “rationality raters” and average the results, perhaps we might have a decent quantitative measure of rationality to work with.

Part of the problem with measuring rationality is the speculative nature of trying to determine someone else’s assumptions. We can generally use what a person says as an indication of what they believe — at least for the most part. It’s also important to consider not only the statement, but the context in which the statement is made. In political discourse, we implicitly assume that politicians are being honest with us. They might be wrong about the facts, but this idea that they are honestly representing their own views is something that voters tend to select for. Perhaps this is why Romney is still struggling against Santorum in the primary. Santorum may have problems getting his facts straight and presenting a logical argument, but he has a habit of saying what he believes regardless of the consequences. Romney, on the other hand, says what he thinks will win him the most votes. Many voters do not vote “rationally”, they vote according to how they “feel” about the candidates. Romney may be more “rational” than Santorum, but his calculated responses cause him to lose that “feeling of honesty” that Santorum elicits from voters.

In the next article, I’ll attempt to explain the origins of rational and irrational behavior. I think the key to understanding these behaviors lies in evolution by natural selection. I would argue that both rational and irrational behaviors contributed to the survival of our species, and this is why irrationality persists into the present. Stay tuned!

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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.

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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.

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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…

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Dear Mr. Stephen Colbert and Americans for a Better Tomorrow, Tomorrow,

I understand that you are looking for suggestions for what to do with that money that I gave you. I’m sorry it’s not much, but now that Congress fucked up America’s credit rating I’m expecting my students loan rates to go through the roof. While it is no longer my money to spend, I think the answer for what to do with it lies in the very same reason why I gave it to you in the first place:

I would like you, Stephen Colbert, to turn the 2012 Presidential Elections into a Media Circus.

If there’s one person that can do it, it’s you, and I’m pretty sure you were going to do it anyway.

In fact, Americans for a Better Tomorrow, Tomorrow has already started drawing much needed attention to campaign finance issues. As ABTT continues to create media headlines, it’s benevolent ringleader can take this media coverage and convert it into Strategic Humorous Indoctrination Transmissions, or SHIT. Once the SHIT is produced, ABTT can seek volunteer monkeys to hurl the SHIT in every direction. The SHIT throwing will no doubt generate more media coverage, drawing new contributors to ABTT and providing the necessary resources to produce more SHIT. The next step is to find a presidential candidate that has what it takes to make our political leaders to look like a bunch of clowns. SHIT throwing monkeys are nice, but it just wouldn’t be a circus without the clowns.

My first idea was to hijack the Republican primary with a write-in campaign. Put out a few advertisements suggesting that everyone write-in the name of their favorite Conservative pundit in the GOP primary. If the votes for rest of the Republican candidates are somewhat evenly split, a social media driven write-in campaign might be enough to make it “first past the post”. Of course, there might be some potential legal and viability issues to consider with this approach. In the long run, perhaps this unnamed individual’s talents might make him better suited towards influencing public opinion of the candidates than being one himself.

As an alternative, I’d like to suggest Jello Biafra (Eric Reed Boucher). Yes, that’s right. The former Dead Kennedys front man: Jello Biafra. Jello has been involved in the Green Party presidential campaigns since 2000, and although he’s hasn’t announced any intention to run in 2012, there are some who think he should.

Now, hear me out here. The US Green Party is at a huge disadvantage in the upcoming election due to the Citizens United ruling. You see, like your pal Buddy Roemer, the Green Party takes issue with unlimited corporate contributions and candidates rely on small value donations from ordinary citizens. This means that Jello wouldn’t be able to run on the Green Party ticket and still accept PAC contributions. This makes it practically impossible for any Green Party, or other 3rd party, candidates to compete with the corporate financed campaigns. For a 3rd party to influence the election at all, it needs to be the center of a media circus. This is where ABTT comes in. The ABTT doesn’t have to support Jello, but rather it just needs to vilify him.

You see, Jello isn’t really your “tip of hat” type. He’s more of the “wag of the finger” type. He’s the kind of guy that will suggest radical ideas like instituting a maximum wage, abolishing the military, or lowering the voting age to 5.. The media needs a crazy-socialist-liberal to keep the news interesting! Jello’s political views should put him on the top of the “Threat Down”, but most people have never heard of him or the Green Party. It’s really too bad this is the case, because every superhero needs a supervillain and Jello would be a perfect arch nemesis for Steven Colbert. He even comes complete with built-in product placement and catchy potential campaign slogans like “Put a Former Dead Kennedy in the White House” or “There’s always room for Jello”.

If Jello were to run and selects a well-known Green Party figure as his running mate, like Ralph Nader, Cynthia McKinney or Howie Hawkins, the result could be a strong Green Party ticket. Jello’s punk rock attitude and anarchistic tendencies could be just the thing to pick up the disenfranchised youth vote. Regardless of whether or not Jello can win the election, a voice like his is much needed to shift the course of debate in the 2012 elections.

The plan is simple. Musical guests are no stranger to the Colbert Report, so there would be nothing out of the ordinary about inviting Jello on to talk about “The Audacity of Hype”. Ask him why he hates Obama and let the SHIT throwing monkeys do the rest.

For a little taste of Jello, check out the following videos:

Jello Biafra (public speech) – LIVE

Jello Biafra on Anarchism

Green Party 2000 Convention – Interviews with Key Figures 2

Jello Biafra on Net Neutrality & the COPE Act

Jello Biafra at Open The Debates Rally at DNC in Denver 2008