Dreyfus says that numerous mathematical objects such as equations, numbers, and functions can be expressed in the classroom in the context of generalization in order to make students more comfortable with upcoming math topics. But it can take more mental effort for students to generalize concepts, and according to my teaching experience, students tend not to try their best to generalize a mathematical concept if they do not receive good guidance from their teacher. I believe that students are not born as mathematicians, but they are born with a brain that can be creatively enhanced by continuing the practice of generalization that can then lead to abstraction.

For example, when I taught *Calculus II* in Fall 2015, in the *Telescoping and Geometric Series *course lesson* *I taught my students how to use generalization by starting with a simple example of finding the first partial sums for 1+2+3+4+5+.., and then I talked about the relationship between partial sums and infinite series. This method introduces students to the mathematical concept starting from something simple and easy and then moving toward the more general underlying foundations.

Similarly, in the example of the washer method I described in my previous post here on the AMS Grad Student Blog, I can start with a review about the volumes of disks, washers, and shells, and at the end use a real-life example to make it easy for them to find the volume of the given region. In this way, we can help students begin to form their own generalizations by teaching them how to reconstruct this particular concept in a way that is easy to understand.

There are several advantages to applying generalization in our math classes, and its positive effect on teaching and learning is a fundamental way to provide our students with the tools needed for successful advanced thinking in mathematics.

Do you have any examples of how you have helped your students better understand a tough abstract or general mathematical concept? Share your experience in the comments below!

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Not all results of medical tests are absolutely correct. Scientists can make mistakes when they conclude that something is true when it is actually false or that something is false when it is actually true. When something is concluded true and it is actually false, we have a **false positive **or type I error. On the other hand, when something is false and it is actually true, we have a **false negative** or type II error.

Should we really be worried about a positive medical test for a rare disease?

Mammography is one way to detect breast cancer at its early stage, which helps patients to increase their survival rates. In 2009, John Allen Paulos, a mathematics professor at Temple University, wrote an article, *Manmogram Math*, to discuss how frequent women should have their mammograms. If mammograms could help diagnose breast cancer, why wouldn’t we have them more frequently? According to Paulos, it is controversial whether women should have mammograms monthly since the tests could cause harmful effects resulting from radiation. Also, suppose you have a positive test, is this absolutely true that you have cancer? The answer is no.

So how worried should you be if you have a positive test of a rare disease? In other words, how accurate is the test? Lisa Goldberg, a statistics professor at UC Berkeley, briefly explained how we should approach this problem on a Numberphile’s video, Are you REALLY sick? (false positives). Goldberg discussed some possible outcomes that the tests can bring to us, which is summarized in the table below:

As we can see, there are two ways a test can fail to identify ‘real’ sick people. First, the test turns out negative while the test subject is really sick. In this case, it is a false negative. Second, the test can be positive while the test subject is really healthy, which is a false positive. Thus, it is important to measure the accuracy of the test when you receive positive test. In order to do that, we can find the probability of the sickness given a positive result, P(Sickness/Positive Result). Apply Bayes’ rule to this problem, we have:

and

Suppose we have a rare disease which happens in 1 out of 1000 people; the test is 99% accurate to tell real sick people and the test misdiagnoses 1 out of 10 healthy people as sick. According to our assumption,

As a result,

Thus,

Given the rare disease and a test that can mostly identify all sick people and give 1 false positive in every 10 test subjects, your chance of having the disease is 0.01, or 1 out of 100. So it is always a good idea to consult second opinion.

All medical tests can be resulted in false positive and false negative errors. Since medical tests can’t be absolutely true, false positive and false negative are two problems we have to deal with. A false positive can lead to unnecessary treatment and a false negative can lead to a false diagnostic, which is very serious since a disease has been ignored. However, we can minimize the errors by collecting more information, considering other variables, adjusting the sensitivity (true positive rate) and specificity (true negative rate) of the test, or conducting the test multiple times. Even so, it is still hard since reducing one type of error means increasing the other type of error. Sometimes, one type of error is more preferable than the other one, so scientists will have to evaluate the consequences of the errors and make a decision.

References:

[1] Paulos, J. A. (2009, December 10). Mammogram Math [Web]. Retrieved from http://www.nytimes.com/2009/12/13/magazine/13Fob-wwln-t.html?_r=2

[2] Haran, B. [Numberphile]. (2016, March 16). *Are you REALLY sick? (false positives)* [Video file]. Retrieved from https://www.youtube.com/watch?v=M8xlOm2wPAA

[3] kingscollegelondon. (2013, November 25). ‘*There’s A Math For That’-The Paradox Of The False Positive *[Video file]. Retrieved from https://www.youtube.com/watch?v=6WuTNMleuQI

[4] Fernandez, E. (2011, October 17). High Rate of False-Positives with Annual Mammograms [Web]. Retrieved from https://www.ucsf.edu/news/2011/10/10778/high-rate-false-positives-annual-mammogram

[5] MEDCRAMvideos. (2014, March 16). *Sensitivity and Specificity Explained Clearly* [Video file]. Retrieved from https://www.youtube.com/watch?v=Z5TtopYX1Gc

[6] Diagnostic Accuracy. (n.d.). Retrieved from http://ebp.uga.edu/courses/Chapter%204%20-%20Diagnosis%20I/4%20-%20Sensitivity%20and%20specificity.html

Images:

http://onlinemixingmaster.com/additive-or-subtractive-eq/

https://si.wsj.net/public/resources/images/BN-HB837_HJourn_F_20150223124818.jpg

http://pregnancytips.org/wp-content/uploads/2014/07/expired-and-defective-pregnancy-tests-lead-to-a-false-test-result.jpg

Having come to graduate school after teaching at a progressive secondary school, I questioned if it really had to feel this way. I moved out of the position of teacher—helping my students to feel positive about their progress and empowering them to actively participate in their education—and into the position of struggling student looking for assistance. It was during my reflection on the first semester that I began to give myself the same advice that I would give my students and take a more active stance in my education through self-advocacy.

At the start of the school year, it seemed that fighting through the negative aspects of graduate school was the most common attitude and exactly what I tried to do. I was having trouble communicating with one of my professors, which led to relatively negative experiences in class and equally unsuccessful office hours. If I had a question, I would ask other students or wait until my understanding of the material matured, which left me with many unanswered questions. While it was clear to me that I needed to do something to improve communication with this professor, I was unsure if self-advocacy was the right approach. I wondered if it would make me appear weak.

I tried many things in place of self-advocacy, including trying to better our relationship by asking questions in different settings, reading a variety of textbooks, asking older graduate students, and talking informally with the professor during departmental teas. While these steps did not prove to be a solution, I am grateful that I persisted. It was during an informal conversation at tea that I understood my professor from a different viewpoint. We stumbled into a conversation about education and he reflected, “I would do anything to help my students succeed.” I was frozen by an inner contemplation of the words that I just heard. Could this be true? I mean, sure, I expect my professors to have a desire to see students succeed, but that was by no means what I was experiencing. In that moment, I knew that I should step up and speak to this professor about my experience. I knew that he would be open to conversation and that we would be able to work together to create a more open and collaborative relationship.

I was able to connect with my professor as an educator. I recalled experiences that I had with my students and how important it was for them to take a self-advocacy stance in their education. By working with my students to create a positive learning environment that was right for them, we grew together and we respected one another on a new level. I looked forward to speaking with my professor in order to work more closely on making my experience a positive and successful one.

I entered into his office with an open mind, ready to change my behaviors and build a more collaborative, working relationship. We reflected on the semester, voiced our needs, and made a commitment to better our efforts. After this meeting, our relationship improved considerably. Now, when I think about this professor and our positive interactions, I end up thinking about the class as well. I am excited about the material and I am motivated to move forward. By being an active participant in creating my learning environment, I had established a space in which I am able to succeed. I removed the feelings of isolation and helplessness from my initial graduate school experience and I brought about positive change and growth. It is through self-advocacy and reflection that I was able to take progress into my own hands and create a successful learning environment.

]]>- September 24-25, 2016, Bowdoin College, Brunswick, ME
- October 8-9, 2016, University of Denver, Denver, CO
- October 28-30, 2016, University of St. Thomas, Minneapolis, MN
- November 12-13, 2016, North Carolina State University at Raleigh, Raleigh, NC.

The Sectionals are conveniently scheduled around weekends to minimize overlap with teaching and coursework. We enthusiastically encourage you to browse the program for each meeting (by clicking the four links above) and consider attending the AMS Sectional Meeting that is most appealing to you.

Best of all, the AMS is providing some **$250** travel grants for graduate students (click the link for eligibility details). The application deadline is **July 25th, 2016** at 11:59pm EDT (East Coast time).

Applications are only accepted through the MathPrograms.org link given below, and application results are given out nice and quickly by August 19th. The application is simple (**no reference letters!**) and well worth a little bit of your time. Remember: if you spend, say, half an hour filling out the application and you receive the award, then that’s equivalent to earning $500 an hour. Not bad!

**Click here to apply for the AMS grad student travel grant for the Fall semester Sectional Meetings**

Do you have any questions about the AMS Sectionals or the travel grant application process? Have you attended a Sectional Meeting in the past? Leave a comment below!

]]>Erik Stern and Karl Schaefer discuss the cross-curricular possibilities with math and the art of movement and dance in their video, Math Dance. I think it is appropriate to ask in what ways can art benefit the math classroom? Is dance the only method that we should consider? What benefits do students gain from movement in the classroom? Not only am I on track to become a math teacher at the secondary level, but I also am a dancer. How perfect is it to come across a Ted talk that addresses two of my greatest passions in life? Stern and Schaefer begin by simply dancing and moving on stage and transition into the ways these moves can relate to mathematics and can happen in the classroom. They were able to use movement and dance to work with students on counting combinations. They did this by having students figure out the different combinations their hands could make when shaking hands. A simple problem, but one that becomes much more interactive and fun for students as they come up with their own fun handshakes. They continue to go through multiple movement-based activities that can be done in the classroom to teach mathematical concepts. They also address the benefits of movement and learning in the classroom. In regard to the benefits of movement in classroom, there is a book titled* Teaching with the Brain in Mind* by Eric Jensen that provides and array of reasons that movement when learning is good for the brain.

As we have looked at the benefits of movement in the classroom, I would now like to on the idea of math and dance. What ways can we use dance to teach mathematics or to simply learn mathematics? Can we do so at a college level? One book titled *Discovering the Art of Mathematics: Dance* connects ideas and patterns in mathematics to dance. The entire book is based on movement. One of the most interesting aspects of this book is that it is an inquiry-based book. It allows the person reading the book to do the work and activities to discover how different types of dance can be related to mathematics. The book states, “this is a not a regular textbook. This is a book which makes you move and think and write and discuss” (p. 7). It is not simply a book that gives you questions and asks you to solve for an answer; you must work through and discover new ways to find solutions or answers. In order to find those solutions, you must use movement and dance. The book addresses multiple types of dances and the connections that can be made to mathematics. Between the video and the book, there are two different approaches to combine dance and math. The video takes a mathematical concept and finds a ways to make movement work in order to teach it. On the other hand, the book takes a type of dance and finds the different connections that can be made to mathematics. Both the Ted talk and book offer great tools and resources for those teaching mathematics and both address the benefits of using the creativity of movement to teach a mathematical concept.

As I continued to look for more information on how to connect the ideas of mathematics with movement, I came across a very impressive mathematical dance discussed in the article,“Dancing Triangular Squares—The Process of Creating a Mathematical Dance,” by Corinne Wolfe. Not only did I find an article about the dance that was performed I was also able to find a video. The name of the article was “Dancing Triangular Squares—The Process of Creating a Mathematical Dance.” This article takes a completely different approach to how dance and mathematics can be combined. In this article the author “uses movement to illustrate mathematical concepts in lessons” (p.11). Wolfe also uses movement to discover links between different shapes. The ultimate product of the performance was to demonstrate “how two consecutive triangular numbers make a square number” (p. 11). The article shows just how the dance itself works.

As I compare the original Ted talk, the inquiry-based book based on dance, and the article where students actually perform the dance, I begin to recognize the differences among the three. In our original video, we used movement in the classroom to teach students different mathematical concepts; in the book, we discovered that we can learn math from different types of dances; and in the last article with the corresponding video, we are looking at teaching an audience a concept by dancing/performing for them.

We have seen from two places that we can indeed use math and dance as one. What about other arts? Can we use drawing? Acting? The answer is yes. I came across a article titled “Improvisation in the Mathematics Classroom” by Andrea Young which discusses the possibility of taking the art of acting and improv and using that to teach mathematics. In this essay, the author discusses the idea that students get to think creatively when doing these improv exercises. The same goes for when we combine dance and math. Improvisation also pushes students just a little outside of their comfort zone and allows them to take risks (p. 469). Young uses improvisation in her classroom not only to review concepts and move forward with lessons (as the other resources addressed with their own creative methods) but also as a tool and technique to get students to know each other.

Works Cited:

Http://www.youtube.com/channel/UCqIy4eFpZwJ23O7a7Y-lFgA. “MATHS DANCE: TRIANGULAR SQUARES by Corinne Wolfe.” *YouTube*. YouTube, 21 Feb. 2014. Web. 06 Apr. 2016.

Jensen, Eric. *Teaching with the Brain in Mind*. Alexandria, VA: Association for Supervision and Curriculum Development, 1998. Print.

Ornes, Stephen. “Math Dance”. *Proceedings of the National Academy of Sciences of the United States of America* 110.26 (2013): 10465–10465. Web.

Renesse, Christine Von, Volker Ecke, Julian F. Floran, and Phillip K. Hotchkiss. *Dance: Mathematical Inquiry in the Liberal Arts*. N.p.: Draft, n.d. *Art of Mathematics*. Draft, Sept. 2015. Web. 6 Apr. 2016.

TEDxTalks. “Math Dance: Erik Stern and Karl Schaffer at TEDxManhattanBeach.” *YouTube*. YouTube, 18 Nov. 2012. Web. 06 Apr. 2016.

Wolfe, Corinne. “Dancing Triangular Squares – The Process Of Creating A Mathematical Dance.” *Mathematics Teaching* 242 (2014): 11-13. *Academic Search Complete*. Web. 6 Apr. 2016.

Young, Andrea. “Improvisation in the Mathematics Classroom.” *PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies* 23.5 (2013): 467-76. *Taylor & Francis Online*. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 28 Jan. 2013. Web. 06 Apr. 2016.

Images:

http://i1.ytimg.com/vi/Ws2y-cGoWqQ/maxresdefault.jpg

https://i.ytimg.com/vi/9HdDRh0Ohvk/maxresdefault.jpg

http://d.gr-assets.com/books/1347315034l/3280.jp

In his Ted Talk, *The Art of Puzzles*, puzzle master Scott Kim walks the audience through his career of designing puzzles, displaying his passion and the meaning behind his work. Puzzles can be described in different ways and come in a variety of forms. In his talk, Kim defines a puzzle as a “problem that is fun to solve and has a right answer.” Not only does he think of them as a kind of hobby or entertaining activity, he states that puzzles are a form of art that he has dedicated himself to for over 20 years. Starting at a young age, Scott Kim gained an interest in creating puzzles and turned his passion into a career. His main goal in his work is to create puzzles that are memorable and leave a lasting impression.

When we think of puzzles, we might initially think of the typical puzzles like jigsaw, crossword, or games like chess. Scott Kim’s work falls outside of this mold. As a puzzle master, he has created a wide variety of puzzles. In this talk, he breaks up his journey into multiple parts. He begins with his first time creating an original puzzle with the “folded letter.” His interest shifted to the perception-altering form of puzzles called “figure ground,” then to games and investigative reporting for publications like *Discover* *Magazine*. Most recently, his focus has turned to online games connected to social media, specifically through his website *shufflebrain.com* that he created with his wife, Amy Jo Kim.

Puzzles are an important part of the history of mathematics. Further research on puzzle creation makes it evident that many famous mathematicians showed interest in the subject. Scott Kim explains in his talk how he is not interested in working from someone else’s matrix (like crossword puzzles or Sudoku), but instead prefers to create original and noteworthy things. Besides providing a mentally stimulating form of entertainment for people of all ages, how can puzzles be used in constructive ways in society? Kim aims to answer this question by channeling his intelligence and creative energy in productive ways through his work. His approach appeals to the aging population, especially those interested in maintaining brain health. Studies have shown that engaging in puzzles and stimulating games improves brain function. Kim published a book with neuroscientist Richard Restak titled “THE PLAYFUL BRAIN: The Surprising Science Of How Puzzles Improve Your Mind,” which explains the reasons why activities like these mentioned have a positive effect.

In this talk, Scott Kim points a few ways the world of gaming is changing. First is that the online computer game audience has expanded. He also mentioned the increasing popularity of games that are healthy for your brain and the social media craze. Bringing these ideas together, Kim aims to create games and puzzles that fit the modern world. Combining his wife’s expertise in social media and his in puzzles and software development, they created their website. The website caters to online gamers and incorporates puzzle games that exercise your brain. An example that he shows in his talk is a game called “Photograb,” which is meant to enhance visual skills. The game tests your ability to recognize a magnified piece of an image by looking at the regular sized image. These types of puzzle promote brain health and may help players to avoid diseases like Alzheimer’s and dementia. As more people learn about and begin using puzzles as a tool to maintain brain function, the effect could be very beneficial. Overall, I think that Kim’s progressive approach to puzzle creation is very interesting and something that could make a huge impact on the future of online gaming and beliefs about brain health.

Sources:

“A Younger Brain.” *Good Housekeeping* 257.3 (2013): 74. *Academic Search Complete*. Web.

ALTER, ALEXANDRA. “The Mind of a Master Brain Teaser.” *Wall Street Journal Eastern Edition* 14 May 2011: C11. *Academic Search Complete*. Web.

Restak, Richard M., and Scott Kim. *The Playful Brain: The Surprising Science of How Puzzles Improve Your Mind*. New York: Riverhead, 2010. Print.

“THE PLAYFUL BRAIN: The Surprising Science Of How Puzzles Improve Your Mind.” *Kirkus Reviews* 78.19 (2010): 983. *Academic Search Complete*. Web.

“Tribute To A Mathemagician.” (2005): *MathSciNet via EBSCOhost*. Web.

Winkler, Peter. “Famous Puzzles Of Great Mathematicians.” *American Mathematical Monthly* 118.7 (2011): 661-664. *Academic Search Complete*. Web.

Image:

https://tedcdnpi-a.akamaihd.net/r/tedcdnpe-a.akamaihd.net/images/ted/133970_800x600.jpg?quality=89&w=800

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).