# Cancer Scares and Polygraphs: What Can Bayes Do For You?

How many times have you seen a TV talk show mention how someone failed a polygraph, so they must have cheated! Lie detectors don’t lie, right? How many times have you had a friend get bad news on a medical test and see that emotional toll that took on them?

What could these two have in common? They could both be helped by Bayes’ theorem.

Bayes’ theorem is an idea in statistics that allows us to update the probability of something happening based on new evidence. You can see what it looks like in the featured image above. However, the best explanation I have ever seen comes from this image:

Let’s go through each part of this image individually:

• H: This is our hypothesis, or what we think is true. In the talk show example, this could be “I think my man is cheating on me.”
• D: This is our new information. For the talk show, it’d be the results of our polygraph test.
• P(H|D): This is the likelihood that the hypothesis is true given our new information. “What are the odds that my man is lying about cheating on me if the polygraph says he’s a liar?”
• P(D|H): This is the odds of seeing the new information if the hypothesis is true. Here, it’d be the odds that the polygraph would say someone was lying when they were lying. This turns out to be 88% (Rice, 2007).
• P(H): This was the probability that the hypothesis was true before the new information. For example, 22% of men say that they have cheated on a significant other.
• P(D): This the odds of the test saying the person was lying in every outcome. Here, you’d use the 88% it says someone is lying when they are, and multiply the odds that your man is lying (22%). You’d also have to add the odds the test says they are lying when they are not (false positive), about 14% (Rice, 2007), multiplied by the 78% chance your man isn’t a cheater.

This gives us the following result:

,

or 63.9%. This means that less than two out of three of those polygraph results are actually accurate. This low success rate, by the way, is one of the reasons most states don’t allow polygraphs in court.

But what about the bad medical test? How could it help us there? In statistics circles, the common example is the case of mammogram result that comes back positive for cancer. For our purposes, we’ll use numbers used by The New Yorker when they talked about this problem in this article from 2013. The numbers they quote are as follows:

• The chances that a women in her forties has breast cancer is 1.4% (meaning 98.6% don’t).
• The chances that a mammogram comes back positive when the woman has cancer is 75%.
• The chances that the mammogram comes back positive when the woman doesn’t have cancer is 10%.

Let’s imagine your friend in her 40s gets a positive result back on a mammogram. What are the odds that she actually has cancer? Bayes’ theorem tells us the probability would be

,

or only 9.6%. If that seems low to you, don’t worry: one study showed that 95% of physicians given similar numbers incorrectly estimated the likelihood to be about 75% (Rice, 2007). Your friend should be concerned and have further work done, but you can comfort her by saying that there is an over 90% chance it is a false positive for her.

What can Bayes’ do for you? It can help make talk shows more laughable, and it can help bring comfort when tests go south. It helps to give perspective and a more realistic view of the world. As it turns out, it’s pretty useful.

We’ll continue to explore these ideas in movie night this week with some videos that give even more examples about how Bayes’ theorem can help you every day.

The odds that those will be entertaining? 100%.

# Math and Music: What do you think?

It’s time to see how convincing I’ve been with these last two posts. What do you think? Can the study of math have an impact on music?

# Math Movie Night: Math and Music

Last time, we mentioned a speaker by the name of David Kung and his work comparing music and math. While I have not been able to find the lecture referenced in that podcast, I did find one that seemed to be fairly similar. While it is a bit long, it makes for fantastic listening as he has two other musicians on stage with him. His main point? “Math helps us understand music. Music helps us understand math.” If you’d rather not listen to the whole thing, I’d listen to the final portion of the presentation (starting at about 59:00) where he discusses how Bach used several math concepts in his music that were well ahead of the mathematicians of his day.

Now sit back, relax, grab some popcorn, and enjoy tonight’s feature math film, Symphonic Equations: A Mathematical Exploration of Music!

# Math and Music: Introduction

In this post, we begin a miniseries of posts concerning math’s influences on music. This audio selection comes from a podcast by the College of Physics and Math at BYU, and introduces a discussion that will be held about math and music.

Try as I might, I was not able to find the actual lecture by David Kung, nor any other that seemed to cover the same material. If you have knowledge of where it could be, please let me know.

# Exponential Growth and Decay

Occasionally you will find types of math that are not used very often, but are essential when they are. Exponential growth and decay are two examples of this. The odds of you using them on your next trip to the supermarket are fairly slim (more on using odds later), but some professions simply would not be able to do the things they do without them.

To get an idea of what a graph of exponential growth looks like, consider the following graph of the global human population:

Notice how it isn’t just increasing, but is increasing at a progressively greater rate? This is what is meant by exponential growth. If this graph were to be flipped the other direction so that the decrease starts steep but levels out as it approaches zero, that would be exponential decay.

Hopefully as you read these examples, your passion for this math topic will grow exponentially. I hope your interest won’t decay as the conversation goes on. [Editor’s Note: I know the puns are bad, and I’m not sorry. You knew what you were getting into when you read a blog by a math teacher.] Here are a few key uses for these concepts:

• Radioactive half-life: As radioactive material decays, it does so exponentially. For example, if you have a 100 g of a radioactive material that has a half-life of 5 minutes, after five minutes you’d have 50 g. After another five minutes (so 10 total), you’d have 25 g left, and so on.
• This is vitally important to fields such as archeology because of a process called carbon dating. Carbon dating works by tracking how much carbon-14 has decayed into nitrogen-14 and using the half-life to see how long it would take to do so.
• Nuclear engineers also need to account for radioactive half-life when designing storage containers for the waste created by nuclear energy.
• Medical professionals use technetium-99m in imaging tests by tracking the radiation before it completely decays. This requires them to know the half-life of technetium-99m so that they know the time-window they have to run the test before the material decays away.
• Compound interest: Either your best friend or your worst enemy, compound interest forms a cornerstone of the modern financial system. While simple interest adds a percentage of the principal amount every time, compound interest adds a percentage of the principal amount, but then adds a percentage of that new amount. If the compound interest is applying to your investments, that’s good news! Things aren’t so rosy when it is applying to your debts. As the population graph above shows, things can get out of control fairly quickly if you aren’t careful. Luckily, you can now use your interest in exponential growth to help interest work for you and not against you.
• Population growth: As the Commonwealth of Australia could tell you, rabbits with food and without predators reproduce quickly. Exponentially, in fact. All populations will reproduce exponentially until they approach the carrying capacity of the area, or the maximum number the area can support sustainably. Basically, as long as they have food, water, and are free from disease or predators, life will continue to increase in large numbers.
• The growing human population, shown in that graph above, is an interest to anthropologists, who study the history of human civilization. Political scientists (and other policy makers) use it to predict population growth in the future, and how that will affect supplies. Currently, the carrying capacity of the world’s human population is not known and is a topic of debate. One organization polled scientists and found that carrying capacity estimates ranged from 500 million to 1 sextillion, depending on how well you assumed people wanted to live.
• Biologists also look at population growth for everything from rabbits in Australia to bacteria in a Petri dish. Things reproduce, and math helps to predict how often.

While not the most commonly used tools in algebra, many fields simply wouldn’t be possible without these functions. Hopefully your love of them grew exponentially as you learned about their use. [Editor’s Note: Essentially the same bad pun twice? I’ll apologize for the lack of creativity, but not the joke!]

# Sidebar: Generational Differences in Education

For my EDTECH 537 class we read three papers about generational difference in education, with one arguing that “digital natives,” or people who were raised with technology, fundamentally think differently than other learners (Prensky, 2001). A second paper looked at Prensky’s arguments and described in turn why each were poorly researched (McKenzie, 2007). While some of his arguments weren’t entirely logical (e.g., some video games are violent, therefore no video game could be useful for education), his arguments were far more convincing and research-backed than Prensky’s. Finally, we read an article that found that while difference between generations existed and should be understood, they were not significant enough to warrant a change to the instructional design (Reeves, 2008).

As McKenzie points out, Prensky’s unsupported arguments are often repeated and have spread quite far. However, many of these comments and arguments sound like the comments and arguments that are always leveled at the next generation:

Simply put, people always see the next generation as having a short attention span, poor conversational skills, and a lack of drive. This has always been the case, and probably always will be.

Should a colleague of mine suggest changing our content to match a generational difference, I would first listen to what they would say. For example, younger students tend to need less in the way of explanations for how to use digital tools, and some analogies (e.g., referencing the maps that used to be in phone books) no longer make sense to younger students.

If instead they were to suggest a fundamental shift in the tools or methods of instruction based solely on a perceived generational divide, I would have to disagree. Simply telling a colleague that is rarely enough to make a difference, and it shouldn’t be. After all, just saying things it what got these ideas started. Instead, I would point them towards data-backed literature like Reeves & Oh. Once they are convinced that the differences between generations are not as substantive as they thought, I would help them choose the tools they wanted to use based on the subject matter or learning theory rather than as an attempt to appeal to  “digital natives” or “digital immigrants”.

After all, kids these days aren’t as unfocused as we give them credit for.

• McKenzie, J. (2007). Digital nativism, digital delusions, and digital deprivation. From Now On: The Educational Technology Journal, 17(2).
• Prensky, M. (2001). Digital natives, digital immigrants part 1. On the Horizon, 9(5), 1–6.
• Reeves, T. C., & Oh, E. J. (2008). Do generational differences matter in instructional design? In Instructional Technology Forum (Vol. 17, p. 2014). University of Georgia.

Image Source: http://illustratedcourtroom.blogspot.com/2012/06/

# Teachers and Math: A Guest Post

Today’s guest post is by Alan Liddell, one of the lead instructors for AcerPlacer in Ogden, Utah, a private company that teachers college math courses for its students. I asked him to write this post because of one response on an informal survey I conducted where one teacher said that they didn’t use any math because they didn’t teach it.

As a math teacher, one would think that it is obvious that I would use math in my everyday life for my career. While it is true that I do teach college-level math on a daily basis to students, my uses of math go far beyond the classroom.

For instance, I use math for statistical analysis for the company I work for. I am in charge of calculating the percentage of student test out rates and to perform statistical analysis to determine lead causes of percentage results and to predict future percentages based on previous years. I am also required to use pivot tables to compare different sets of data to determine causality and possible relationships between them.

In addition, I create spreadsheets in Google Sheets and Excel that require algebraic expressions to be inputted to auto-populate cells and columns. Although the language of Excel and Google Sheets maybe be different than traditional math, the concepts are the same.

I also use math to quickly number crunch various menial tasks around the office. I may have to make copies of a certain homework packet, so I will use mental math to determine how many copies to place into the copier queue.

As a final anecdote, I used the Pythagorean Theorem to help me at the post office for work. I was tasked with picking up boxes to ship our books to online students, and when I got to the post office, they had various sizes to choose from. The one I thought might work was a box that had the proper height, but had width and depth dimensions of 8.5 in and 1.5 in respectively. I knew that our book had a width of approx. 9.5 in, so I used the Pythagorean Theorem to determine if the book could fit on a diagonal in the box (it turns out that it could not). Using math saved me an additional trip back to the post office.

# Women and Math in the Media: A Commentary

As a math teacher, one of the biggest obstacles I face when teaching students is math anxiety. Many students have it to varying degrees. Specifically, I’ve been thinking about how math anxiety tends to fall along gender lines and, of course, The Big Bang Theory [TBBT].

Before I start, I just want to be clear: I don’t hate TBBT (I actually find it pretty funny), and I certainly don’t think that it’s the cause of the problems that I want to talk about. However, I do feel that it is a very visible symptom of a larger over-arching problem. Besides, no one will have read the research articles I’ll be quoting, but most people will have seen at least one episode of one of the most popular currently-running TV shows, so it gets to be the unfortunate lightening rod for my disappointment.

For those who may have not heard of TBBT, it is a show about scientists hanging out and being nerdy. It starts out with the beautiful Penny moving across the hall from Leonard and Sheldon, two physicists doing research at Caltech. That’s a fairly straightforward sitcom setup. However, several criticisms have been leveled at the show because of its use of stereotypes as part of its humor. Penny is the beautiful, normal person. The scientists are men, socially inept, and unable to talk to Penny easily. Why is this a problem?

This reinforces some harmful stereotypes about women in science (and scientists generally, but we’ll put those aside for a moment).  Penny is often shown as completely clueless about various science and math topics and has to have them explained to her by the shows male cast members. In this, the writers cast Penny as the audience, who the writers assume know nothing about the topics on the show. For the first several seasons, there are no other female leads. This is alleviated a little bit with the introduction of Bernadette and Amy, a microbiologist and neurologist respectively. However, for their first few seasons, both are portrayed as socially awkward and admirers of Penny’s beauty. In short, it went from saying that “girls can’t do science” to “pretty girls can’t do science.” Finally, in the most recent seasons, Bernadette and Amy have taken on greater roles and shed some of their questionable character traits to become better female role models.

Maybe you’re wondering why I’m making such a big deal of an admittedly small part of the show. It’s all in good fun, right? I point it out because the way women are portrayed in media affects how girls and women feel about their abilities to perform in math and science. Multiple studies have shown that women report higher levels of math anxiety than men, and while the reason for this is up for some debate (Jameson & Fusco, 2014), studies have found that examples and role models can have positive or negative effects. For example, elementary-aged girls are more likely to have math anxiety and believe in the stereotype that boys are better at math than girls if their female teacher also has math anxiety (Beilock et al., 2010). Negativity about math is contagious, even if it is left unspoken.

On the other hand, positive portrayals of women in careers and science in magazines have been shown to help improve women’s performance on a math exam (Luong & Knobloch-Westerwick, 2017). Notice that wasn’t long-term exposure to those stereotypes, that was simply reading a few pages from a single magazine immediately before taking the exam. So if those good stereotypes helped after just a few minutes, even if that effect was temporary, imagine what constant exposure over the course of an entire TV show could do for a child?

As I mentioned, TBBT has definitely taken steps in the right direction in this regard. I am certainly not trying to get people to hate the show, but rather point to it as one of the most visible examples. However, the myths and stereotypes about women and math persist across many shows and stories in culture today. While having more positive role models of women in math and science won’t magically fix every issue faced by women in STEM fields, it could help alleviate at least the math anxiety faced by girls becoming women.

I’m not asking for anything drastic. All I am asking for are a few more Bernadettes and a few fewer Pennys.

• Beilock, S. L., Gunderson, E. A., Ramirez, G., & Levine, S. C. (2010). Female teachers’ math anxiety affects girls’ math achievement. Proceedings of the National Academy of Sciences, 107(5), 1860–1863. https://doi.org/10.1073/pnas.0910967107
• Jameson, M. M., & Fusco, B. R. (2014). Math anxiety, math self-concept, and math self-efficacy in adult learners compared to traditional undergraduate students. Adult Education Quarterly, 64(4), 306–322.
• Luong, K. T., & Knobloch-Westerwick, S. (2017). Can the Media Help Women Be Better at Math? Stereotype Threat, Selective Exposure, Media Effects, and Women’s Math Performance: Media and Stereotype Threat. Human Communication Research, 43(2), 193–213. https://doi.org/10.1111/hcre.12101

# Discussion: Algebra vs Statistics

For this discussion, we will be talking about the article Down With Algebra II! In it, the author argues that, based on a book by Andrew Hacker, algebra II should be eliminated as a required course and replaced with a class on statistics.

Let me start by saying I strongly disagree with the author that advanced algebra has little worth. If I felt that way, I wouldn’t be writing a blog to help people see how math is used. However, the discussion to be had here isn’t whether algebra is good for students to study, but rather if statistics would be better. Statistics are indeed everywhere, and many people are forced to understand things like medians or means to understand news discussions about political topics. However, is that of more worth to students than learning the abstract thinking that accompanies algebra?

What do you think? Would algebra or statistics work better as a general education course?

# Matrices and Linear Algebra: A List

Linear algebra is the study vectors and vector spaces, while matrices are essentially grids of numbers used to solve some problems in linear algebra. If that sounds hard and complicated, that’s because it can be. Many colleges dedicate entire courses to the study of this one topic. However, the basics of matrices and linear algebra aren’t too bad, and can even be fun.

Here is a list of just some of the professions that need to have a knowledge of this math topic:

1. Aeronautical Engineers – This profession designs aircraft, and matrices are incredibly important to that (assuming you want your plane to stay in the air). Some of these matrices can get fairly large (12×12, for example), and while they would be very difficult to solve by hand, they are fairly easy to solve using computers (Phillips, 2010).
2. Computer Programmers – Speaking of solving things using computers, programmers heavily rely on matrices (though they often call them arrays). Many programming languages use them, and not just for science; most graphics are essentially matrices containing color information for each pixel.
3. Ecologists – It may be hard to believe, but ecosystems are complicated things. To prove changes over time and distance weren’t due to random chance, ecologists often use a statistical test called PERMANOVA to test their hypotheses (“Permutational analysis of variance”, n.d.). This test relies on matrices to properly set it up.
4. Civil Engineering – In the same way that people don’t want their planes to suddenly stop moving, most people prefer their bridges to not suddenly start. To accomplish this, each individual beam that is part of the bridge has to be evaluated for how much weight it carries, and this is most effectively done using linear algebra.
5. Robotics – When working in robotics, it is often easier to think about multiple pieces of the machine having their own coordinate systems (Hiob, 1998). To go back and forth between them, you can use matrices.

This is just a small sampling of the different uses of matrices and linear algebra. While they may be tricky at first glance, they are powerful tools in many fields.

Now are you glad that you took the red pill and learned just a bit more about the uses of a matrix?

Hiob, E. (1998, June 11). Robotics: Using transformation matrices to change from one coordinate system to another in robotics. Retrieved July 16, 2017, from http://commons.bcit.ca/math/examples/robotics/linear_algebra/index.html

Permutational analysis of variance. (n.d.) In Wikipedia. Retrieved July 16, 2017, from https://en.wikipedia.org/wiki/Permutational_analysis_of_variance

Phillips, W. F. (2010). Mechanics of flight (2nd ed). Hoboken, N.J: J. Wiley.