Posts in: Mathematics & Computer Science

Art and Math – a perfect pair

I attended an arts school for 7 years before college. Starting in the 6th grade, I spent an hour-and-a-half every day honing my skills as a filmmaker on top of taking my other classes for the normal middle/high school core curriculum. My peers were all artists as well, each focusing on one of 11 different art majors. Our teachers recognized the power of having a room full of artists and used this to enhance the learning that took place in each classroom.

It would be impossible to count the number of projects I had to do where I was tasked with taking the classroom material (be it the novel Jane Eyre or the Krebs Cycle) and converting it into some sort of artistic presentation. At the time, I think a lot of us scoffed at being told to mix our academics with our arts, but the skills I learned by doing this work still benefit me today.

In college, so many of our summative projects are papers or exams. A dry regurgitation of the material achieves the aim of assessing how well the content was absorbed, but often doesn’t task us to really contextualize our learning. When professors assign something more creative, they make space for students who process differently to flourish, and this often leads to the creation of really cool final products.

The final for this class is not an exam or a report but is instead an open-ended choose-your-own-topic presentation of material. I love it.

I jumped around a lot when it came to picking a topic and medium for my project. I wanted to do something that was fun and would introduce my peers to new information. At first, I figured I’d make a film since that’s where I’m most practiced. But without a film crew or time to animate, I thought maybe a website would be easier. As I started considering topics to study, I realized something visual was going to be extremely necessary if I wanted to tell my information in a way that was easy to understand. I ultimately decided to put together an illustrated book (I hesitate to call it a children’s book since the content is a bit complex).

This process has been so much fun. I spent an hour on Thursday just drawing detailed depictions of various shoe types (I promise it’s related to my project). I had the opportunity to play with art, narrative, and parody while also coming up with ways to explain a fairly complex mathematical concept. It was definitely a challenge trying to take what could easily have been a 5-10 page paper and distill it into less than 1000 words, but, as a result, I was able to gain a really solid understanding of the material for myself as well.

I wish more CC professors would offer open-ended projects like this. I don’t mean that we should get rid of exams and papers all together, but I do think more emphasis on finding the intersections between ideas would only help with student learning. That’s the point of the liberal arts anyway, isn’t it?

The title page for my book

If we make Jill our dictator, will she not leave for Nat Geo?

 

 

 

 

 

 

Why I Love Math

Growing up, I always liked math. I enjoyed the challenge of being faced with a problem and trying to figure out the tools and method needed to solve it. It was cool getting to see how the world around me could be described through the lens of mathematics. Unfortunately, as I reached higher levels of math, the problems we were solving stopped being grounded in real-world problem-solving. Where math used to be asking questions about the speed of a rock rolling down a hill or the number of games played in a baseball tournament, it was now verging into higher dimensions and abstractions that had lost touch with the everyday problems facing people.

While coming into college I had considered being a math major, after taking upper-level classes, I found myself put off by math that seemed so devoid of life. I didn’t care about the volume of a 15th dimensional sphere or how to solve a Diophantine equation. Instead of pursing the traditional theoretical math track, I found refuge in the world of applied mathematics. Here, I was still allowed to ask questions about problems that were relevant to the world around me. I decided to pursue a mathematical-economics major in order to find compromise between my joy for math and my need to have my learning grounded in reality.

These days, I spend a lot of time thinking about the types of problems that people are faced with and the tools that they use to make decisions. I am given the task of trying to use existing information about people and predict their future actions. Math has begun to reprove to me that it is important to the world of real people. With a few relatively simple mathematical tools, I have been able to explore the value of getting a college education for future life success and the value of using the Electoral College to conduct US elections.

I think it is unfortunate that one of the few CC classes specifically focused on applying math to the world is billed as “Not Recommended for Math Majors”. What has been really cool about this class is the variety of experience that everyone brings to the material. Some of my classmates are Political Science majors, others are writers, others of us are here for the math, and some people have no idea what they want to study. Because we all have different ways of viewing the material, we are able to have really interesting conversations that would not happen in other math classes where people only care about the mathematical processes.

The prevailing narrative that math is a defunct subject that tries to be as confusing as possible is just not true. Anyone can benefit from thinking mathematically as is evidenced by this class. For me, the world of applied math has rekindled my love of mathematics, and I hope that for others it can show them that math can be relevant!

Voting isn’t fair

If you’ve spent any time talking about American politics recently, you’ve probably heard about how the electoral college gives priority to certain voters or how some states make it harder for some people to vote than others. You may have even signed a petition promoting a national popular vote or rallied to enfranchise citizens who have been traditionally not allowed to vote. The idea that voting isn’t exactly fair is probably not surprising. What might be surprising is that, no matter how hard we try, we can’t actually design a voting system that is fair.

This block I am enrolled in MA110 – The Mathematics of Social Choice. This class focuses on how groups of people make decisions. The first week of the class, we examined a variety of different voting systems (which is a fancy way of saying ways to take people’s preferences and decide who wins an election). What we found out through exploration of these systems is that for elections with more than two candidates, no voting system is truly fair.

So what do I mean by fair? Before I list the criteria that we outlined in this class, I suggest taking a couple minutes to think about what you think a fair voting system might look like….. Okay, got your list? Now let’s compare.

Here are some of the things we decided should be fulfilled by a fair voting system:

  • Voters should be allowed to rank their preferences for candidates however they’d like
  • Every voter’s vote should be worth the same amount
  • Every candidate should be treated the same
  • Getting more votes should never hurt a candidate
  • If a majority of voters like a candidate the most they should win
  • If everyone likes one candidate more than another, the candidate that is less liked shouldn’t be able to win
  • If a candidate would lose an election, removing them as a candidate shouldn’t change who wins

How does this list stack up to your list? Did we come up with any criteria that you hadn’t thought of or do you have any you want to add to our list? Hopefully, you’ll agree that these are some important things to have in a voting system. Here’s the thing though – there doesn’t exist a single voting system (no matter how hard you try) that will satisfy all of these criteria at the same time. There’s a whole Theorem (called Arrow’s Theorem) that proves why this is the case. Ironically, one of the systems that is closest to meeting these criteria is a dictatorship! (It only violates the ideal that every vote should matter the same.)

At this point, maybe you’re resolved that it might be okay to drop one or two of these criteria (though not the one that lets it be a dictatorship!). If you’re okay with that, then I do have some solutions for you! Check back for a later blog post with some ideas for voting systems that seem okay.

Real Analysis: Week 2 (and doing things with real numbers)

Since learning about some of the basic properties of the real numbers, about how they form a line that is completely continuous, we’ve begun to talk about what those properties allow us to do with the real numbers. When we think about and talk about math, we tend to think and talk about manipulating numbers, not simply admiring their existence. The idea that “proving calculus” is important rests on the fact that our society requires (and our minds enjoy) this kind of numerical manipulation, and that we want the results of these endeavors to have some real meaning.

The first type of active manipulation of numbers that we talked about was the act of arranging real numbers into a sequence. A sequence of numbers is a subset of real numbers which are arranged in a particular order. Further, each element of the sequence can be constructed from some formula; that is, if you want to find the fifth term in the sequence, you can plug “5” into a given formula and acquire that term. For example, we can create a sequence of all numbers of the form {1/n} where each “n” that we plug in is a natural number (the kind of number that you can count on your hands). In the sequence, the first element is 1, the second element is 1/2, the third element is 1/3, and so on.

This sequence, like all sequences, contains an infinity number of terms. This is because there are an infinite number of natural numbers, and therefore an infinite number of “n”s to plug into the given formula. An infinitely long list of numbers all sharing a common formula is a pretty astounding, but it turns out that many sequences have an even more incredible property. This property is called convergence, and it means that after a certain term in the sequence, there are an infinite number of terms which are all practically equal to each other.

The sequence {1/n} is a sequence with this property. Eventually, as the natural numbers which we plug into the formula get very large, the term as a whole gets very small. And when n gets extremely large, when we start dividing 1 by numbers like 1,000,000,000, the elements of our sequence get very close to each other and to zero. Because of this, we say that the sequence {1/n} converges to zero. Interestingly, the terms are never actually equal to zero, and never actually equal to each other. 1/1,000,000,000 is a different number than 1/1,000,000,001, although they are very, very close together in value.

There are actually a lot of sequences that converge, and which converge to a variety of real numbers. This property of convergence is deeply tied to the structure of the real numbers- that line which we learned about at the very beginning of class. Since that line contains every possible number, it is possible for it to contain an infinite number of numbers which are as close together as we could want, without those numbers ever actually being equal. This allows for convergence: if the real line were missing values, there would be gaps in the line which inhibited the closeness of terms in a given sequence.

Sequences, while fascinating, might seem on the surface to not actually “do” much in terms of manipulating numbers. They hold a lot of power, however. They allow us to not only order information, but a potentially infinite amount of it. We can add terms of these sequences together to get new sequences, or to get sums of numbers that approximate complicated functions. These are all things we’re going to talk about in the next few days of class, as we keep looking at what the line of real numbers allows us to do.

 

~Anna

 

Real Analysis: Week 1 (and one really important line)

One week ago, I didn’t know what Real Analysis even was. I knew it must be important (it’s a requirement for the major, after all), and that it was probably a theory based course (since we’re “analyzing” how numbers work, rather than learning the methods that these workings allow). Going into the first day that way was exciting- after a summer spent mostly at home, it felt like I was throwing myself into some far off corner of the universe and waiting to find out what it looked like.

As it turns out, Real Analysis is a little bit more like every corner of the universe. Our professor, Molly, introduced the class as a course that would allow us to begin “proving calculus.” And calculus- that’s huge. Calculus may be only one branch of mathematics, but it’s the branch that most of math education in the US most naturally leads to. That makes sense, too, since calculus has so many applications in so many important fields. Proving the validity of calculus implicitly assures the validity of fields in economics, physics, chemistry, and pretty much any natural or social science which uses complicated statistics to understand a data set. Real Analysis, then, seemed like some pretty powerful stuff.

We started our first class with some of that powerful stuff, right away, talking about what the set of “real numbers” even is. It’s hard to wrap my head around it, to be honest, because here’s what the real numbers are- they’re the line of every possible number that exists in either the positive or negative direction of zero, with not a single gap in them. The numbers we deal with most frequently in our day to day life are “natural numbers,” the kinds of numbers you can count on your fingers. They’re every number you can get by starting with 1 and adding 1 to itself as many times as you could possibly want. It’s an infinite number of numbers, but compared to the real numbers, the natural numbers are missing a lot. To get to the real numbers from the natural numbers, you first step through the integers. The integers include the natural numbers, every negative version of a natural number, and zero. But we’re still not even close to the real numbers. Next we step through the rational numbers- these are every number you can get by taking a ratio of two integers. So numbers like 13/167, or 2/9, those are rational numbers. And as you can probably imagine, that’s also a really big set of numbers.

But it turns out that, for as many rational numbers as there are, there are far more numbers that can never be expressed as a ratio of two integers. So if we were to lay all of the rational numbers out in a line, there would be spaces, empty spots of numbers which we couldn’t express using that set. The real numbers, though, fix that problem. They include every rational and irrational number. Essentially, any number that you could find out in the world, only the set of real numbers is guaranteed to include it.

So the real numbers are pretty special. We learned that in the first fifteen minutes of class. And it turns out that because the real numbers are so special, mathematical systems that use the real numbers have some cool properties. We’ve been exploring those kinds of properties since them. For example, any interval of numbers (like the interval from 0 to 1, written as (0,1)) is guaranteed to have a single largest element, one unique number bounding the entire set below it. But if you don’t use the real numbers, you can’t guarantee such a thing. That’s just one example, but the essential thing I’ve learned so far is this: the simple fact that the real numbers exist allows for and dictates the behavior of entire fields like calculus . The rules of most math that I’ve ever seen in my life are themselves governed by a line.

For only the start of a class, that’s a pretty amazing thing to have already learned, and I can’t wait to see where in the universe we end up in the next few days.

 

~Anna

 

Machine Learning: Classifying Films

In the era of netflix, there’s a wide array of available films to watch online. One of the features that made Netflix so successful was its ability to recommend new movies. This ability is equivalent to answering the following questions:

If a person likes a particular movie, what are some similar movies? If a person likes a given set of movies (and rates them accordingly) what is a good estimate of their rating of another movie?

One method for answering these questions is called K-Nearest-Neighbors, or KNN. The way this works, basically, is that we use a function that calculates the ‘distance’ between two movies. Distance is is quotes because it’s not entirely clear how to do this – and in fact, there are multiple methods. Our method was to compare how similar the ratings were for those two movies over all users. So if movie1 was rated 3 by user1 and 4 by user2, and movie2 was rated 2 by user1 and 4 by user2, then the distance between movie1 and movie2 would be sqrt( (3-2)^2 + (4 – 4)^2) = 1.

Once we have all these distances (from a given movie), we just return some of the movies that had the lowest distances as our recommendations!

For our homework assignment yesterday, we had to write a program that performed this sort of analysis. Here’s an example of the output:

 

Toy Story:

2.69947506562 Star Wars (1977)
2.85147058824 Return of the Jedi (1983)
3.01114649682 Independence Day (ID4) (1996)
3.1107266436 Rock, The (1996)
3.37230769231 Fargo (1996)
3.41573033708 Mission: Impossible (1996)
3.43060498221 Twelve Monkeys (1995)
3.46153846154 Willy Wonka and the Chocolate Factory (1971)
3.5 Star Trek: First Contact (1996)
3.60740740741 Jerry Maguire (1996)
3.72161172161 Raiders of the Lost Ark (1981)
4.171875 Men in Black (1997)
4.18067226891 Back to the Future (1985)
4.18571428571 Empire Strikes Back, The (1980)
4.2012987013 Twister (1996)