Vi Hart’s *Doodling in Math Class: Infinity Elephants* is a fun little video that brushes over many mathematical concepts without getting bogged down in technical jargon. Vi, a prolific recreational mathematician who also contributes heavily to Khan Academy, starts the video off by discussing infinite series such as ½ + ¼ + 1/8 + 1/16 + 1/32 + … = 1 and the issue of convergence of series. As you can see in the screenshot here, Vi draws elephants of length ½, 1/4, 1/8… of a page and relates this drawing to the idea of convergent series.

Watching the video reminded me of the dilemma with Gabriel’s Horn, a famous example of a shape that is infinite in length and surface area but finite in volume. Gabriel’s Horn is a useful example to employ in calculus classes to help students visualize integration in three dimensions while showing that some infinite shapes have finite volume.

This is often seen as an object that can be filled with paint but not painted, and it was pointed out that since the shape is unbounded it would take infinite time to fill it. This is where we connect to Vi’s problem: bounding an infinite shape in a finite space.

The way Mark Lynch decided to tackle this problem is by making it into a piecewise function: on is defined as if or if is a positive integer and on the interval . This makes a spiked looking shape that has the same principles as Gabriel’s Horn but is bounded. When bounding Gabriel’s Horn and shifting the focus to a piecewise function, our problem becomes a geometric series and we can explicitly compute the area.

The way Vi found her answer was a bit different; she looked to fractals for her approach. Problems like these have many applications and have many ways of solving them from integration to piecewise functions to fractals. However you look at these types of problems, the good news is that you will always have many further questions to pursue.

**Sources:**

- Fleron, Julian F. “Gabriel’s Wedding Cake.”
*The College Mathematics Journal*30.1 (n.d.): 35-38. Web. - Hart, Vi. “Doodling in Math Class: Infinity Elephants.”
*YouTube*. YouTube, 02 Dec. 2010. Web. 08 Apr. 2016. - “How-To Help and Videos – For Dummies.” How-To Help and Videos – For Dummies. N.p., n.d. Web. 08 Apr. 2016.

Lynch, Mark. “A Paradoxical Paint Pail.”*The College Mathematics Journal*36.5 (n.d.): 402-04. Web. - http://www.matharticles.com/ma/ma044.pdf
- http://www.skepticink.com/reasonablyfaithless/2013/10/02/painting-gabriels-horn/

Nate Silver is a statistics guru whose claim to fame has come from correctly predicting 49 of the 50 states in the 2008 presidential election as well as all 35 senate races. He then showed that this performance was far from a fluke as he gave another outstanding prediction for the 2012 U.S. election. A leader in the field of political statistics, he is the founder and editor-in-chief of his popular website FiveThirtyEight. Extending far beyond major political elections, the website also works with the statistics of sports, science, health, economics, and culture. Nate Silver can be referred to as both a star and a statistician—a combination that sounds more like a contradiction than a reality.

Being a mathematics major, I have learned to never blindly trust the statistics that show up in news reports, and Nate Silver is often one of the first to warn others to be wary of statistics presented in the media. A good intro piece to Silver’s statistical style and ability is the following video where Silver explores the question, “**Does racism affect how you vote?**”

In the video, Nate Silver breaks down the 2008 presidential race to find out if the ethnicity of a candidate was a big factor in the election outcome. While each person may have their own knee-jerk reaction as to whether or not there is an effect, Nate Silver does a fine job of objectively breaking down each piece of statistical evidence that is presented. By understanding the complexity of the scenario, Silver is very open about the level of accuracy of the data collected. Silver cleverly looks at many factors and has a good grasp of how various statistics should be used to understand the raw data. (And the final conclusion is that racism does in fact play a role in how the nation votes for the president.)

In another talk, Nate Silver explores the unique challenges and goals of political modeling and how to do it effectively.

Silver discusses the difficulty of using statistics as a means to predict the outcome of political campaigns. The first and most understandable complication is the lack of data. There have been few presidential races in recent history, so there is not a lot of past data to work with. Beyond that, the data collected for each campaign changes; for example, rapidly advancing technology continually affects the way we poll.

As a political statistician, it is difficult to make the determination of which polls will be most effective, considering each may inadvertently attract a different subset of the population to participate. As Silver looks at all of this data, he must analyze *statistics on statistics* to identify the best polls to use. This is where Silver’s job becomes particularly difficult and where experience plays a large role: he needs to determine how valuable each piece of information is and properly weigh each bit accordingly.

Nate Silver builds safeguards and rules into his statistical models which protect against intentional or unintentional biases or assumptions in creating the statistics. These rules consist mainly of how his models will weigh different kinds of data that is collected, making Silver a kind of numbers scientist who is very careful to not allow his hypotheses to affect the conclusions of his careful experiments. Instead of trying to make the data match a certain result, he lets the data speak for itself and he listens to what it has to say. The recent growth and success of quantitative analysis of social issues is nicely written up in The Guardian’s piece *Nate Silver: it’s the numbers, stupid.*

Beyond that, we have the issue of giving consumers context for the data in order to assist with, for instance, the distinction between causation and correlation. Silver gives an example of correlation with no proven causation: in 12 of the last 14 elections, if the Washington Redskins won the game immediately before the election, then the incumbent won the election. Taken out of context, this raw data could be fleshed out to produce plenty of misleading reports in the media. Along the same vein, the fascinating classic 1954 book *How to Lie with Statistics* by Darrell Huff helps to open the readers’ eyes to how easy it can be to produce a desired conclusion from a given set of raw data by using carefully chosen statistics.

Nate Silver’s high popularity stemming from his success in predicting political campaigns and social issues shows just how much the public appears to appreciate his objective style of presenting statistics amid a sea of overblown hype in the mainstream media.

]]>It is common in sci-fi literature to hear about higher special dimensions. In Star Trek and Star Wars, one is able to access faster-than-light travel by accessing warp speed or hyperspace, respectively. In both of these cases, space time is being bent or “warped” into the fourth dimension which allows for faster-than-light travel without actually breaking the speed of light. By adding a fourth special dimension to space-time, we are able to then “fold” space and instantaneously jump from one point to the next. Think about space-time like a piece of paper where you want to get from one point to the other. In normal space-time, the obvious answer is a straight line but by adding a third dimension of maneuverability, we are able to fold the paper and put the two points right next to each other and so are able to move from A to B instantaneously. This is how faster-than-light travel would work; we would bend space-time into the fourth dimension putting the points right next to each other.

Now we may find ourselves wondering more about the nature of this fourth dimension more than our original inquiry about faster-than-light travel. For instance, what would shapes in this fourth dimension look like? Are there dimensions higher than four? What happens to mathematics in these higher dimensions? These questions are far more interesting than our original sci-fi question so let us shift our focus to these more abstract topics. In Numberphile’s video “Perfect Shapes in Higher Dimensions,” Carlo Sequin talks about higher dimensional geometry. He begins his talk with a review of three-dimensional Platonic solids which are solid shapes where every face has the same number of edges, each edge is the same length, and each face has the same area. There are five of these displayed below.

Expanding on these, he delves into four dimensional regular polytopes, which are constructed from the “folding” of these shapes into each other in different ways. Because we have access to the fourth dimension, we are able to do what seems impossible by folding one of our shapes into another one. Because of the nature of higher dimensional geometry, it is impossible to visualize what the objects actually look like so we use wire projections to best represent them. Below is an image of a tesseract which is composed of eight cubes folded into the fourth dimension.

Even though it looks like a smaller cube within a larger one connected by six rhombuses all the cells are really cubes of the same shape. The weird shape is only due to our lower perspective. You can think of the inner cube as always being farther away than the outermost cube making our impression of the tesseract look the way it does because of our forced perspective. This kind of thought experiment could be expanded to include things like functions, vector fields, and all other manner of mathematical concepts. How strange do our day to day conceptions of mathematics get when we allow them play in higher dimensions?

In fact, there are six four-dimensional regular polytopes which we create from “folding” our platonic solids into the fourth dimension. In order to do this, we must conjoin at least three of our platonic solids around a shared edge and then we are able to “fold” them into our four dimensional polytopes. At this point it is rather difficult to explain in further detail without the aid of several dozen pictures so I will leave the creation of the other five regular four-dimensional polytopes to the video. The video also looks into normal polytopes in *n *dimensions and found that there are only three regular polytopes for all dimensions higher than four. It is in this sequence where I will continue my inquiry. Why only three? What about the nature of space restricts the number of regular polytopes to such a specific number? Usually when we discover the nature of a space to be some numerical value or idea, we obtain zero, one, or that there are infinitely many. It seems very strange that the nature of reality forces there to be exactly three regular polytopes for dimensions higher than four. This brings up some interesting philosophical questions as well about the epistemology of mathematics. How is it that we are able to know things in mathematics which are unable to be experienced? How can we know the nature of a space which cannot be found in our own universe but nonetheless follows universal laws? Perhaps most significantly, what is mathematics and what exactly is the relationship between mathematics and reality?

Citations:

Sequin, C. (2016, March 23). *Perfect Shapes in Higher Dimensions*. [Video file]. Retrieved from https://www.youtube.com/watch?v=2s4TqVAbfz4

Image 1; http://www.21stcenturysciencetech.com/images/fall2000/moon1.jpg

Image 2; http://www.daviddarling.info/images/tesseract.jpg

“Like you use sentences to tell a person a story; you use algorithms to tell a story to a computer” (Rudder 2013).

In today’s day and age, we have the world at our fingertips. The internet has made many things easier, including dating, allowing us to interact and connect with a plethora of new people–even those that were deemed unreachable just fifteen minutes beforehand.

*Inside OKCupid: The math behind online dating *talks about the math formula that is used to match people with others on the website OKCupid, the number one website behind online dating. Christian Rudder, one of the founders of OKCupid, examines how an algorithm can be used to link two people and to examine their compatibility based on a series of questions. As they answer more questions with similar answers, their compatibility increases.

You may be asking yourself how we explain the components of human attraction in a way that a computer can understand it. Well, the number one component is research data. OKCupid collects data by asking users to answer questions: these questions can range from minuscule subjects like taste in movies or songs to major topics like religion or how many kids the other person desires.

Many would think these questions were based on matching people by their likes; it does often happen that people answer questions with opposite responses. When two people disagree on a question asked, the next smartest move would be to collect data that would compare answers against the answers of the ideal partner and to add even more dimension to this data (such as including a level of importance). For example- What role do the certain question(s) play in the subject’s life? What level of relevancy are they? In order to calculate compatibility, the computer must find a way to compare the answer to each question, the ideal partner’s answer to each question and the level of importance of the question against that of someone else’s answers. The way that this is done is by using a weighted scale for each level of importance as seen below:

Level of Importance Point Value

Irrelevant 0

A Little Important 1

Somewhat Important 10

Very Important 50

Mandatory 250

You may be asking yourself ‘How is this computed?: Let’s say you are person A and the person the computer is trying to match you with is person B. The overall question would be: How much did person B’s answers satisfy you? The answer is set up as a fraction. The denominator is the total number of points that you allocated for the importance of what you would like. The numerator is the total number of points that person B’s answers received. Points are given depending on the other person’s response to what you were looking for. The number of points is based on what level of importance you designated to that question.

This is done for each question; the fractions are then added up and turned into percentages. The final percentage is called your percent satisfactory – how happy you would be with person B based on how you answered the questions. Step two is done similarly, except, the question to answer is how much did your answers satisfy person B. So after doing the computation we are a left with a percent satisfactory of person B.

The overall algorithm that OKCupid uses is to take the n-root of the product of person A’s percent satisfaction and person B’s percent satisfaction. This is a mathematical way of expressing how happy you would be with each other based on how you answered the questions for the computer. Why use this complex algorithm of multiplication and square-rooting when you can just take the average of the two scores? Well, a geometric mean, which is “a type of mean or average which indicates the central tendency or typical value of a set of numbers” (Rudder, 2013), is ideal for this situation because it is great for sets of values with wide ranges and is great at comparing values that represent very different properties, such as your taste in literature and your plans for the future and even whether or not you believe in God (best of all, the algorithm can still be useful even when there is a very small set of data). It uses margin of error, which is “a statistic expressing the amount of random sampling error in a surveys results” (Rudder, 2013), to give person A the most confidence in the match process. It always shows you the lowest match percentage possible because they want person A and person B to answer more questions to increase the confidence of the match. For example, if person A and B only had answered two of the same questions margin of error for that sample size will be 50%. This means that the highest possible match percentage is 50%. Below I have included a table that shows how many of the same questions (size of s) must be answered by 2 people in order to get a .001 margin of error or a 99.99% match.

Now that we know how the computer comes up with this algorithm, it makes you wonder how do these match percentages affect the odds of person A sending one or more messages to person B. It turns out that people at OKCupid had been interested in this question as well and had messed with some of the matches in the name of science. It turns out that the percent match actually does have an effect on the likelihood of a message being sent and the odds of a single message turning into a conversation. For example, if person A was told that they were only a 30% match with person B (and they were only a 30% match), then there’s a 14.2 % chance that a single message would be sent and about a 10% chance of a single message turning into a conversation of four or more messages. However if person A was told that they are 90% match (even if they are only a 30% match), then the odds of sending one message is 16.9% and the odds that the one message turns into exchanging 4 or more is 17% .

I believe that the future of online dating is very broad and exciting. However I have some concerns about the algorithm and that it relies heavily on a person’s honesty and self-assessment. If I was to further analyze this topic I would look into how the length of the first message affects the response rates. Also, how it affects the odds that the conversation will continue for four or more messages and whether those messages would the same length or longer/shorter than the initial message sent. The extent of the questions that have yet to be asked about this particular set of data and the idea of online dating/ matching with people who are possibly oceans away are enormous; however, the data will linger on the Internet for many years to come and I’m sure will analyzed hundreds of times more to answer many many more questions.

Citations:

Hill, K. (2014, July 28.). *OKCupid Lied To Users About Their Compatibility As An Experiment*. http://www.forbes.com/sites/kashmirhill/2014/07/28/okcupid-experiment-compatibility-deception/#4cbde4745eb1

Match Percentage. (n.d.). Retrieved April 26, 2016, from https://www.okcupid.com/help/match-percentages

Rudder, C. (2013, February 13). I*nside OKCupid: The math of online dating. *[Video file]. __https://www.youtube.com/watch?v=m9PiPlRuy6E
__Rudder, C. (2014).

Cost functions are linear functions describing a relationship between an initiator and its results. They are often used in business and accounting to predict and minimize future costs. A worked-out example of constructing and minimizing an average cost function is in the YouTube video below by James Sousa.

I was interested in looking into how these cost functions are used in the analytical approach to ergonomics, which is an applied science to configure the work space so that it is both safe and functional. In this mathematical approach, researchers use cost functions to model kinetics, which is the energy and motion of human limbs. Instead of measuring a financial cost, the results measure reachability, dexterity, stress, potential energy, etc.

**The goal is to maximize reachability while minimizing stress at each joint.** To do this, researchers determine the area of reach (boundary envelope) and move the work space to optimize these cost functions (Abdel-Malek, Yang, 2004). The points that we need within reach are called* target points*, whereas the contact point on the body (i.e. the tip of a finger) is called the *specified point*. Imagine a coordinate system originating at each joint and motion constraints depending on the magnitude and range of the vectors (limb portions). Similar coordinate systems on the human body are used in other areas of research such as physical therapy and robotics.

The end goal is to configure the work space where all the target points are inside the boundary envelope but the cost function is optimized; to do this, researchers create 2D and 3D graphs of these work space envelopes.

It’s important to note that people are both physically and mentally affected by their environment. The productivity, efficiency, and safety of someone’s work can be maximized if careful decisions are made regarding the worker’s environment, and the analytic approach to ergonomics can help make these decisions.

Are there any workplace changes you have made over the years that have made it much more comfortable to work long hours at your desk? Share any ergonomic tips in the comments below!

References:

- http://www.merriam-webster.com/dictionary/ergonomics
- Abdel-Malek, K., Yang, J. (2004). A mathematical method for ergonomic-based design: Placement. International Journal of Industrial Ergonomics, November.
- K. Abdel-Malek, J. Yang, “A mathematical method for ergonomic-based design: Placement” International Journal of Industrial Ergonomics, November 2004.

My solution: give a one or two question quiz at the start of each class period written to take around five minutes or maybe a little longer depending on the day.

The Good:

- I had an idea about their mastery of the concept by the time they turned in the quiz and once I graded it, I knew where they stood and which issues to address. Some days the students finished way faster than I thought they would and other times the anxiety was palpable and eraser dust was flying. Once I knew where the problems were, I could fix them a day or two after the initial exposure to the content instead of a week or more later (as was the case with weekly quizzes). The daily quizzes fixed the problem I started with.
- The frequency of quizzes gave students a tangible reason to stay on top of material but the number of them made any individual quiz fairly low pressure. You might say that serious students will keep up to date on the material anyway, but I think this strategy motivated more students to study regularly rather than cramming.
- Almost all of my students came to class almost every day. This might not seem like a big deal but I think it’s awesome! In fact, a fellow grad student implemented this idea a few semesters ago with increased attendance as a primary objective. In general, students who go to class do better (though there are exceptions) so I was really pleased with student attendance. I didn’t have a huge jump in attendance from when I was doing weekly quizzes, but there was an increase nonetheless.
- Here’s the selfish one: grading was a breeze. Somehow grading a couple of one-question quizzes throughout the week seems way better than grading a four-question quiz over the weekend.

These outcomes paint a pretty picture about giving daily quizzes but there were plenty of drawbacks that came along for the ride. In my next post, I’ll go through some of the bad and the ugly parts of my daily quiz implementation. Stay tuned!

]]>In Conrad Wolfram’s video, Teaching kids real math with computers, he discusses the interesting topic of mathematics in education versus real world mathematics. He argues that math education is all about calculating and that this “chore” can be done effortlessly with computers. He wants us to stop teaching calculating and start teaching math. Math in the real world is done by virtually everyone that is solving a problem or looking for the answer to some question. In school, math consists of lower level questions that require a lot of by-hand calculations. Problems seem simple and trivial compared to more challenging and rigorous problems in the real world. His answer to this problem is to have students use computers more for the computation aspect and to focus on posing the right questions, translating those questions into math problems, and verifying those results to their real world applications. His model for what math is and what it should be within our education system looks like this:

One might ask, ‘why do we not do this?’ or ‘why don’t we add more computers in our education system and have students engage more in steps 1, 2, and 4?’ According to Mike Thomas’ *Teachers Using Computers in the Mathematics Classroom: A Longitudinal Study*, there are many more computers that can be used within schools, but they are not within the math departments. Many schools have computers within computer labs or IT rooms, but the access in the math classroom is limited. Teachers in this study also think that learning is not increased from the enhanced use of computers which could be the case in two ways. First, students could simply look up or “Google” the question and answer they are searching for in which case the students are not learning, they are simply coping and pasting answers mindlessly. Second, they are using the computers to plug and chug to find an answer. In both cases, the computers are being misused rather than being used to deepen the understanding of the math content at hand. This scenario comes back to Wolfram’s notion that math problems within the current classrooms are dumbed down and do not require students to do much when given a calculator or computer to complete all of the computations. This notion brings in the fact that math curriculums should be changing to fit Wolfram’s notion. In order to have students engage deeper with the mathematics, the problems should be more like the real world and have applications that students can see. Dan Meyer gives his personal account of how this could change in his video, Math class needs a makeover.

Dan Meyer argues that high school math classes need a makeover and that it will start with problems we are addressing in the math classroom. He points out that math problems give us way too much information that real world problems would never provide and we just take those numbers, plug them into a formula, and an answer is produced. This never happens in real life and if we want to teach our students how to solve real world problems, we need to start giving them real world problems. These problems should start with a general question about a problem that the students then use to ask sub-questions to decide what matters and what they need to solve that problem. This is where students are posing the right questions then taking information and formulating the problem into mathematics. Computers come in to do the “chore” of computation and then the answers are verified and applied back to the real world problem at hand. Meyer’s video gives an example of this process that he does with his students and it is something that all math educators should do with their students at the secondary level to promote more real math in the classroom and less direct computations.

Math classrooms are changing and for the better. More computers are being used whether they are handheld devices such as the Texas Instruments graphing calculators or actual desktop/laptop computers. These devices, according to Sue Garner and Michael Edwards in *The CAS Classroom* and *Symbolic Manipulation in a Technological Age* are being used more so than ever and are providing students with a more conceptual understanding of the mathematics at hand and are allowing teachers to ask more conceptual questions about real world problems in which students are allowed to use computers to calculate answers that the students can then verify and interpret to their real world application. Computers even offer students multiple representations of the same problem that otherwise would not be available. Computers allow students to engage in those real world problems on a conceptual level that prepares them for the ever changing world they are about to enter. Students that can formulate their own questions, develop sub-questions, use computers to do the computations, and then translate those answers into verified applications in the real world will learn more math than ever. This is what real math is and what we as future educators should be teaching our students. Students will retain the ability to formulate questions and how to answer them more so than being able to calculate them by hand once they encounter mathematics in the real world.

References:

Thomas, Mike. “Teachers Using Computers In The Mathematics Classroom: A Longitudinal Study.” *New Zealand Mathematics Magazine* 43.3 (2006): 6-16. *Academic Search Complete*. Web. 8 Apr. 2016.

Edwards, Michael Todd. “Symbolic Manipulation in a Technological Age”. *The Mathematics Teacher*95.8 (2002): 614–620. Web…

Garner, Sue. “The CAS Classroom.” *Australian Senior Mathematics Journal* 18.2 (2004): 28-42. *Academic Search Complete*. Web. 8 Apr. 2016.

Images from:

https://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers#t-269409

Needless to say I failed to meet just about all of these goals – not just a handful of times, but pretty regularly. And yet I think I have succeeded in becoming a healthier and happier version of myself, just not in the ways I expected.

The biggest change in my emotional well-being came from really establishing friendships. I started making more time for friends and the activities that make me happy. I sometimes went to karaoke on a weeknight. I went to cookouts to take a break from endless homework on the weekends. Grad students are not robots. We need more than just fuel to be productive. Having fun times to look forward to (and reflect on) makes the hard work of being a grad student so much more bearable.

Part of making time for fun activities meant learning when to put my work away and say “no more.” A younger me hated the thought of handing in an incomplete homework assignment. Accepting the fact that I might not finish my homework actually made me much less anxious about working on it. It no longer seemed like a looming task that would never be done. A friend reminded me that the point wasn’t getting a good grade; it was understanding the material. I started to see value in struggling with problems, even if I couldn’t solve them before the deadline. When working on research, too, it can be very beneficial to step away from it when it starts to become frustrating; when you come back to it with fresh eyes you’re likely to better understand the problem at hand.

Time management seems to be one of the most challenging aspects of grad school for me. As math grad students, we often have the freedom to decide when and where to do our homework, grading and class preparations, and research. That freedom can make it difficult to fall into a regular self-enforced schedule, like getting to the office by 9 am every day – something I accomplished just a few times this semester. My failure to adhere to that schedule told me that perhaps it was not the best schedule for me. In fact, often I worked at home in the morning before heading to campus for my afternoon classes, enjoying the quiet solitude. I learned that I could be most productive by finding the times and places that made me actually want to work.

Lastly, I started exercising regularly this semester (something I had done as an undergrad but let slip when I got to grad school). While it can be difficult to make time for exercise, it has many benefits to offer – including improved sleep, a better mood, a break from academic work, and a sense of accomplishment. I got into new activities by signing up for phys ed classes and by exercising with friends, both of which made exercising more fun and held me accountable for actually working out.

I am by no means a perfect grad student (after all, nobody is). I have bad days and weeks when I’m really stressed out. I still procrastinate and wish I’d managed my time better. And toward the end of the semester I didn’t get many workouts in. But I’m learning that an important part of being a successful grad student is being a happy grad student, and a happy grad student I most certainly am.

]]>In his humorous 2015 Numberphile video, Matt Parker discusses a remarkable formula by Jeff Tupper of the University of Toronto whose graph is the letters, numbers, and symbols in the formula itself. More precisely, this formula:

To find out how it’s done, check out Parker’s video, plus this background explanation and generalization by Shreevatsa R. You can use Tupper’s formula to plot your own name (or anything else you like) using this Python code provided by Kaito Einstein.

(image courtesy of Weisstein, Eric W. “Tupper’s Self-Referential Formula.” From *MathWorld*–A Wolfram Web Resource. http://mathworld.wolfram.com/TuppersSelf-ReferentialFormula.html)

]]>

The start of summer is a great time to get organized: perhaps you finally have time to focus on something other than the next problem set, or perhaps you’re too burned out from finals to do anything else productive. Back in the good old days of undergrad, mathematics was a mysterious substance that emanated from textbooks, whiteboards, the mouths of my classmates, and occasionally, Wikipedia. Now that I’m in graduate school, more and more of what I learn comes from journal articles, which can prove difficult to organize. When I wrote my undergraduate thesis, my papers were all printed copies stored in an accordion folder, but as a graduate student reading hundreds of articles, this system is hardly practical. Luckily, there are several **reference management programs **that keep track of papers you’ve read and generate bibliographies for you. After searching through many options last winter, I finally settled on Zotero, which I highly recommend (and no, I am not being paid to write this).

I chose Zotero [zoh-TAIR-oh] because of the following features:

- It’s free
- It only takes one click in Chrome, Firefox, Safari etc. to download a new paper to your collection
- You can store the actual PDFs of the papers themselves, not just the bibliographic information
- You can write notes about each item or tag them with keywords
- You can highlight or annotate the PDFs using another program (e.g. “Preview” on a Mac) and import these highlights/annotations into Zotero
- If you search for a term, Zotero will look through the titles, tags, notes, and even the highlighted passages and annotations you’ve imported for all the documents in your library
- Zotero will generate bibliographies and in-line citations in both BibTex and MS Word

In addition, Zotero lets you share your collections with others, import collections you may have already started in other software packages, and access your collections (well, the citations at least) from multiple computers. The PDFs themselves are stored on your original machine (by default, in your “Downloads” folder), unless you choose to put them in Dropbox, OneNote, Google Drive, or some other cloud storage. You should probably do this anyway to back them up.

My workflow so far looks like this:

- Search for a paper or author on Google Scholar
- Follow the link to the website of the journal article and click the Zotero button to download both the citation and, if your university can get you access to it, the PDF. The Zotero download button circled in this picture is available for all webpages once you install the Chrome extension, which I found to be quick and painless. Other browsers are also supported.
- The article is happily stored in Zotero. On the left, you have the option of organizing your library into folders. On the right, the article metadata is listed. The menu above the metadata allows you to add notes or tags, which are then searchable.
- Using the Mac “Preview” program, I highlight and annotate the PDF.
- When I’m done, back in Zotero I extract the highlights and annotations. I might do this once a month or so, since I can do it for many documents simultaneously.
- Extracted annotations and highlights now appear in the righthand panel.
- I can search my entire library for the word “heck,” and Zotero will find the annotations where I wrote “what the heck?”
- If you use LaTex, simply export everything to BibTex and do your bibliography and citations from there. If you need to write something in MS Word, install the Zotero extension in Word, which will allow you to add citations in just a couple clicks:
- Upon clicking Word’s “Zotero Insert Citation” button, you’ll be taken to Zotero where you easily search for the paper you want, and it will be added in whatever citation format you prefer.
- And with one more click, you can generate a bibliography.

I’ve found this system extremely helpful this semester, and if you don’t have a system of your own that you like, I suggest you give it a try. Zotero is quick to learn, and I have yet to run into any bugs. Granted, things don’t always go as smoothly as the example above, so here’s what to do in some other situations:

- If you just have the PDF (say, it was emailed you to) you can have Zotero extract the citation information (e.g. author, title, year) from the PDF metadata to allow you to make citations.
- If the PDF doesn’t have any metadata encoded (perhaps you have a scanned book chapter, for instance), you can download the citation information from the internet and then attach your PDF to it.
- As a last resort, if you can’t get the complete citation information online in a downloadable format (as is sometimes the case with books or websites), you can enter the missing information by hand.
- Conversely, if you find the citation online but not the PDF (or if you don’t have free access to it), you can still download the citation into Zotero. You can always add the PDF later if you get access to it.
- The most annoying situation is when I can’t highlight or annotate the PDF (this often happens if it’s a scanned copy that was laid slightly diagonally in the scanner). This isn’t a Zotero issue of course, but Zotero will still allow you to type notes about the document as a whole in Zotero’s own notes section, even if you don’t have a way to annotate specific passages.

Lastly, if you’re still not convinced and want to try out other reference management software, here’s a good place to begin: https://en.wikipedia.org/wiki/Comparison_of_reference_management_software

Do you have a reference manager that you particularly like? Let us know in the comments below which one is your favorite and why!

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