Guest Post: Reformers in Math Education

Today’s guest post comes from Madison Hodson, an instructor at AcerPlacer who studies mathematics/statistics education at Utah State University.

In this article I would like to highlight a few educational reformers. I would also like to discuss a brief history of education and the impact that it had on mathematics and mathematics education.

Johann Pestalozzi was an educational reformer of his time. He believed that all children deserved a fair chance to attend school and be educated. He tailored the curriculum to meet the needs of each of his students, especially the underprivileged and poor children. Johann’s influence was not limited to the 1700’s alone; today we have education and teaching criteria similar to Johann’s approach to education. Examples of Johann’s influence in today’s education include: an emphasis in children taking an active role in their learning, having a well-rounded education system, and a student-centered teaching approach. Johann also emphasized that school and home are both places for learning to take place and that parents and teachers must work together to make that happen. He also introduced the focus on not only what subject is being taught, but how that subject is being taught. These are just of few of the valuable contributions made by Johann that influence our educational system today.

Another reformer that I would like to mention is Franz Gall. Prior to Franz Gall’s work, the world believed that the intellect and all learning came through the heart and the soul. Franz Gall changed that belief to the knowledge that is still held today- that learning and intellect stems from the brain. With this newfound discovery, Gall divided and categorized the brain into separate sections that each serve a specific purpose. Gall’s findings are helpful in teaching mathematics because we can target specific concepts or learning ideas that engage various parts of the brain. We can also recognize that everyone’s brain works a little differently, meaning that individuals learn differently. As math teachers, there are many methods and examples that we can use to convey the same concept to our students. We have the resources and knowledge available to us to target specific teaching methods that will yield the best long term results.

As educators, we have the responsibility to not only help our students understand how to solve complex math problems, but also to teach them how these concepts can be useful and applied to real-life problems. Critical thinking and problem solving skills are indispensable assets that all individuals must learn in order to succeed. Math teachers are fortunate because we can combine our curriculum with applicable examples that will give students useful skills for life.

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Math Movie Night: Probability and Gambling

This week we talked about probability (specifically Bayes’ theorem) and how it could help you in everyday life. What if I told you that probability could even make you a better gambler?

Probability and gambling have always had a symbiotic relationship; studies of gambling improve probability, and probability improves the ability of people to gamble. Even if you aren’t a gambler at heart, very similar mechanics permeate many types of games today, making it an interesting study.

Without further ado, our feature presentation: How Science is Taking the Luck Out of Gambling!

 

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:

CodeCogsEqn(1),

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

CodeCogsEqn(2),

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 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!]