Month: July 2016

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Ellen’s Choice: What If You’re Not Supposed to Enjoy Reading It?

At a conference in town earlier this year, I presented several panel discussions centering around the difficulty of defining and quantifying art. Our discussions in these panels got me thinking about literature, and how one of my main points could apply equally easily to much of the literature that students read in high school. The point in question is this: one of the defining characteristics of art, in my view, is that it is something that creates an emotional response in the viewer. Experiencing it changes you in some way.

This is easy to see when the emotions are ones we generally see as ‘positive;’ if a play makes your heart swell with hope for the future, or a ballet duet makes you flush with the excitement of new love, or an epic novel makes your heart race with anxiety over the safety of the main characters, it’s easy to argue that those works are art and have changed you. But what if the emotions you experience are more negative – what if a novel bores you, frustrates you, or drives you nuts? For many high school students, it can be hard to recognize that even if your reaction to the work is boredom or frustration, the fact that you’re having a reaction that strong means that the book is affecting you deeply – and it’s probably intentional on the part of the author.

Here are a few examples of what I mean:

Waiting For Godot, by Samuel Beckett

Waiting For Godot is a play where nothing happens. Literally. The entire play concerns two characters, Estragon and Vladimir, who are waiting by a single scrawny tree for the arrival of someone they refer to as ‘Godot.’ They have various aimless conversations, run into a few odd characters, and at one point spend a good three pages of stage directions trading three hats between the two of them. Reading the play straight through is interminably boring, as you might expect. Many a high school drama student has been tortured with this play, as they groan and read through yet another pointless conversation about whether they were supposed to meet Godot here or somewhere else, today or yesterday.

The important thing to remember while reading (or watching) this play, though, is that this boredom is completely intentional. The play is an exploration of waiting, and the kind of non-events that suddenly become very important when nothing else is at stake. When you’re stuck in one place with nothing to do but wait, you can see how it might become immensely important to figure out who should wear which hat or exactly how far away you should stand from the others – anything to avoid dying of boredom.

In fact, I recall hearing a story once about a group of prison inmates who read this play and had it resonate so strongly with them that they self-produced it and performed it for the rest of the prisoners. Why? Well, it’s a story about waiting – while incarcerated, that’s pretty much all they were doing. It spoke to them on a much more personal level than it might speak to the average high school literature student. Beyond that, though, it can also be seen as a commentary on the non-committal nature and essential cowardice of mankind, particularly in the repetition at the end of each act:

ESTRAGON: Well, shall we go?
VLADIMIR: Yes, let’s go.
[They do not move.]

After spending an entire act waffling non-committally about what they should do, Vivi and Gogo finally decide to leave – forget waiting for this man, he’s never going to come! And yet…they can’t do it. They can’t leave even after they’ve decided to go. How many of us have experienced this failure of will before in our lives? It’s universal. And how many of those prison inmates do you think spent their days dwelling on past decisions, cursing their own cowardice? Sure, it’s boring, but it’s a very human kind of boring.

That’s a lot to chew on for a play about nothing, eh?

The Good Earth, by Pearl S. Buck

I remember HATING this book when I read it in high school. Finishing each chapter of homework was a slog – it was one of the only times in my school career that I had to force myself to keep reading through my interminable boredom. And then – I recall one night, midway through our unit, I suddenly finished the whole thing, reading the entire second half of the book in one sitting. I didn’t think much of it at the time, but re-reading the novel a few years ago with one of my students, I suddenly realized what had happened: the boring slog of the first half was entirely intentional!

The Good Earth follows the life of a Chinese farmer, starting just before the revolution and beginning with the purchase of his wife from a wealthy merchant. His life for the first half of the book is boring and cyclical – it is entirely governed by the harvest seasons and his wife’s yearly pregnancies. It is justifiably dull, as the life of a farmer would be. Standout events in his life – a good harvest, a lean year, an upsettingly-ill-timed pregnancy – seem meager and uninteresting to our modern-day imaginations, but to him they are his whole world. And then, in the middle of the novel, the revolution happens. Things start to take off – he finds an abandoned store of wealth and becomes a wealthy merchant. Intrigue. Arranged marriages. Bigger houses. The cycle is broken. And in the final scene, we see a repeat of the opening scene, only this time, he is the wealthy merchant and another poor farmer has come to purchase a wife from him. The cycle begins again, but our protagonist has a new place in it. No wonder I read the entire second half in one sitting – that was the exciting part of the cycle! What better way to prime the reader for the breakneck pace of the second half of his life than by exposing them to the boring cyclical stuff first?

The Great Gatsby by F. Scott Fitzgerald

I remember reading The Great Gatsby in high school and having misgivings about it. I wanted to like it, I really did, but I kept finding myself hating each and every character. I couldn’t get into their excitement, share their joys, because I found them so frustrating. Even the young ingenue Daisy, who I genuinely expected to like, turned out to be vapid and careless and thought little of the consequences of her actions. Looking back on it now I realize that once again, that was the intention! You’re not supposed to like any of the main characters – the idea is that they are completely undeserving of their wealth and prestige. It’s a commentary on the faultiness of the idea of the “American dream;” the people who supposedly are living the dream are thoroughly unlikeable and leave a bad taste in your mouth.

Looking back on my own high school literature classes, I find myself wishing my teachers had impressed upon us the idea that not all books are written to entertain you, and you can be bored, frustrated, or upset by something in a novel and still see the merits of the novel itself. Sometimes those negative emotions are what the author intended to bring out in you, and in the process of trying to articulate why you’re experiencing them you can learn something valuable about the work as a whole.

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Mathematical Journeys: An Exercise in Averages

A few summers ago I wrote a blog post about finding math in unexpected places as a way to keep skills sharp through the summer break. One of the unexpected places I talked about was the world of tabletop Role-Playing Games (RPGs) such as Dungeons & Dragons. Such games are essentially communal storytelling exercises which use chance elements to help guide the story via a set of polyhedral dice.

I’ve been running a D&D game for a group of friends for several months now, serving as “Game Master.” As Game Master I serve as lead storyteller for the group, while the others each create a character to experience the story firsthand. My job is to create the framework for the story. I devise and flesh out the world that the story takes place in, present challenges for the players to overcome, and rationalize the effect their actions have upon the world. Overall, my goal is to create circumstances that will allow the players to be heroes. Today I’d like to delve a little deeper into the math applications involved in a D&D game, through the use of an example from the game I’m currently running.

In the current story arc, my players recently made the acquaintance of a tribe of goblins led by a goblin chief by the name of Skizaan. Goblins are a very classic antagonist in the setting of D&D, and they are often seen as little more than cannon fodder – weak enemies who can be bowled over easily. However, these particular goblins are highly organized and skilled at group tactics, and as the brains behind their operation I wanted Skizaan to present more of a challenge to the group. In the rulebook for the game there are statistics for the average goblin, as well as a “Goblin Boss” – both of which were a bit too weak for my tastes. I decided to amp up Skizaan by altering some of his statistics to increase his ‘Challenge Rating’ (known as ‘CR’ – the system the game uses to help Game Masters determine which antagonists will present an appropriate challenge for a player group of a given power level).

As part of this alteration, I needed to increase the amount of damage Skizaan dealt with weapon attacks to bring it in line with my target CR. The rulebook contains a handy table listing the average values for various statistics at different CRs, so looking at the table entry for my target CR shows me that the average damage value per round should be about 33. Great – that’s my target. However, there’s a bit of a wrinkle – D&D uses dice rolls to determine damage dealt, so every enemy lists a number and type of dice to roll when an attack hits. This adds another element of chance, where combat can lead to drastically different outcomes depending on the luck of the dice – which is one of my favorite aspects of the game. The table only lists average damage output, though, not dice ranges, so I needed to figure out what combination of dice would result in an average damage of 33.

This is starting to sound like a word problem from a math test!

Now, there are several ways to do that, but here’s the system I worked out. To figure out the average result of a die roll, I just need to take the average of the lowest possible roll and the highest possible roll. Since I know my target average, reverse engineering the average formula is the quickest way to get there algebraically:

(lowest (l) + highest (h))/2 = average

(l + h)/2 = 33

l + h = 66

Now it’s time to think about the way dice rolls work for a moment. The lowest possible outcome would be all 1’s on the dice, the highest would be every die rolling its maximum. The game uses dice that have 4 sides, 6 sides, 8 sides, 10 sides, 12 sides, and 20 sides, so I have a few options. The thing that immediately hits me, though, is that 66 is 11 times 6. If I went with six dice, my lowest possible roll would be 6, regardless of what kind of dice I was using. That would leave me with 60 points left to divvy up between 6 dice, so they’d need to be 10-sided dice. In game terms, I’d write that as 6d10. So rather than the original 1d6 of the Goblin Boss, my new amped-up goblin chief could do 6d10.

So that’s one option, but there’s another aspect of the luck of the dice that I want to acknowledge: the spread of possible outcomes. The lowest and highest values of that 6d10 roll would be 6 and 60, which means that this attack could vary wildly in terms of how much damage it actually does. Sure, on average it’ll be around 33, but if my dice are having an off day when we actually play, Skizaan could do practically no damage and the players would steamroller him, which would be pretty anticlimactic. I’d like to find a way to make the rolls more consistent, with less variation. The way to do that is to change up the type of die rolled to decrease the spread – make the highest and lowest values closer together.

To do that, I’ll need to shift the composition of the roll to be more dice with fewer sides each. More dice increases the lowest value, and less sides each decreases the highest value. So let’s take our target of l + h = 66 again. We can split it into the same 6 and 10, but switch their placement – instead of six ten-sided dice, let’s use ten six-sided dice. That would make the lowest value 10 and the highest value 60. That would increase the average value a bit, from 33 to 35, but would condense the spread a bit in the process.

However, there might be an even better option – we do have four-sided dice at our disposal. Let’s say we used as many four-sided dice as possible to get the same average outcome. At this point, I use a slightly different logical process. Take a look at our previous two examples. With ten-sided dice, we ended up with six times eleven. With six-sided dice, we ended up with ten times seven. In each case, we end up with the number of dice rolled multiplied by one more than the number of sides on the dice. So if we have four-sided dice, we can figure out the number of dice to roll by dividing our target of 66 by one more than the number of sides on the dice – 66 divided by 5. Rounding down to avoid decimals, we end up with 13 four-sided dice. Plugging that back into our main formula to check our work, we get

(13 + 52)/2

32.5

So we’ve decreased the average just a tiny bit, but in exchange we’ve given ourselves a much more condensed spread on the dice. Minimum damage for Skizaan’s attacks is now 13, which is much better than 6, and his maximum damage potential is a still very threatening 52.

This is just one example of the many ways in which serving as Game Master for a game of D&D involves quite a bit of math and logic. Running this game has been a great experience for me in problem solving with algebra, and I thought you might enjoy hearing about it.