Imagine that you are at the center of a circular pool of unit radius. At the very edge of the pool, there is a monster who is trying to catch you. The monster cannot swim, so it can only run around the edge of the pool. You know that you can run faster than it while on the ground, so if you can manage to reach any point on the edge of the pool before the monster can get there then you will escape. However, the monster runs four times faster than you can swim. Is it possible for you to escape from the pool and the monster?

In order to clarify the riddle, suppose that the monster instead ran three times faster than you could swim. Then the obvious thing to do would be to swim in a straight line to the edge of the pool directly opposite from the monster’s starting point. To reach this point, the monster would have to travel π times as far as you (i.e., a distance of π*r* versus *r*). Since π > 3, you would thus be safe.

However, since the monster runs four times as fast as you, the simple strategy above will result in you getting eaten. Can you think of a more complicated strategy that does work in this case?

If that’s too easy, then can you escape if the monster runs five times faster than you? What is the maximum speed (e.g., five times, six times) that the monster can run such that you can still escape?

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According to the *An Inquiry-Oriented Approach to Undergraduate Mathematics *research paper in the *Journal of Mathematical Behavior*, Kwon and Rasmussen (2007) talk about the Inquiry-Oriented Differential Equations project as a collaborative effort to improve the undergraduate math education and to study how undergraduate math can draw on the theoretical and instructional advances initiated at the K-12 level as well as to create and sustain learning environment for powerful student learning. While there are several reasons for the initiation of the Inquiry-Oriented Differential Equations project such as the huge number of math departments at universities and colleges, the increase of student body diversity, and the decline in the number of math majors as mentioned by Kwon and Rasmussen, universities and community colleges need to implement this project as a fundamental way of improving the undergraduate math curriculum in general and differential equations curriculum in particular because this project can encourage the useful interaction between teacher and student through class activities and discussions in order to enable the teacher to measure the student mathematical thinking and how the teacher can form a new way of thinking based on what the student thinks.

In addition, Kwon and Rasmussen (2007) discuss several various characterizations of inquiry process in different research communities, for example, in general, inquiry can be identified as a set of assumptions using critical and logical mathematical thinking and considering alternative mathematical explanations, while in the philosophy of mathematics education, inquiry can be defined as the ability to learn how to speak and act mathematically, participate in mathematical discussions, pose conjectures, and solve new or unfamiliar math problems. I totally agree with all those different characterizations of inquiry process because they are located under the title of how teacher inquiry into student mathematical thinking.

For example, when I taught *Calculus II *in Fall 2015, I implemented this inquiry process using tools in my math book: *A Friendly Introduction to Differential Equations *because in my *Calculus II *class, there were some topics that are also relevant to differential equations such as finding the general solution of differential equations using the separable method. When an abstract topic in *Calculus II *such as partial fraction decomposition came up, I asked my students to think about a method that can save time and might solve fifty percent of partial fraction decomposition problems. Some students told me that there are no other methods to avoid solving systems of linear equations to find the required constants while others started giving me assumptions and suggestions about possible methods of doing partial fractions decompositions. In this situation, the students are learning how to mathematically investigate other methods built on their assumptions and suggestions and I encourage them to creatively think about that mathematical problems in search of solutions.

There are several advantages of applying the inquiry approach in our math classes such as differential equations and its positive effect on the improvement of teaching and learning is one of the fundamental goals that we are looking for in our math classes. We want to provide our students with the tools for successful advanced mathematical thinking processes and the methods of reinventing mathematical ideas and implementing IO and AMT has been a successful way for me to do just that.

References

Kaabar, M. (2015, January 5). A Friendly Introduction to Differential Equations. *Printed by CreateSpace, San Bernardino, CA*, http://www.mohammed-kaabar.net/#!differential-equations-book/cuvt. Accessed on August 29, 2016.

Kwon, O., & Rasmussen, C. (2007). An Inquiry-Oriented Approach to Undergraduate Mathematics. *Journal of Mathematical Behavior, 26*(1), 189-194.

Have you ever been asked to hang out with a new friend, but maybe you didn’t know where they lived? Maybe you were going to an interview for your new job but needed directions. The solution to both of these problems is using a GPS system. We are very fortunate to be able to input an address and be given step by step directions to our destination. One rainy afternoon I began thinking about this and asked myself, “What was the world like before GPS? How did people navigate?” I began to think about how we can easily make it to the other side of the city, but how we now can make it across to the other side of the world. How did people navigate the sea without any technology? Thankfully George Christoph (view his talk https://youtu.be/AGCUm_jWtt4) and others give us somewhat of an insight on how ships were sailed way back when.

When sailors are sailing ships the vessels can be thrown off course very easily whether it was a human error, change in current, or a nasty storm. The slightest miscue could potentially cause the sailors to miss their destination. Since there was ambiguity in whether or not the ship was on the right course the sailors came up with an idea that would help them know if land was near. In the early days of sailing the Vikings would release a bird to fly above the ship. If the bird flew up above the ship and circled randomly then there wasn’t any land near. If the bird flew with a purpose away from the ship then the ship would follow in hopes of the bird finding land.

Sailing with the help of birds was a creative idea, but didn’t always lead to the results that sailors wanted. Another idea was discovered that included more mathematical reasoning. If the sailors knew the speed of the boat, then they would be able to figure out how far away they were form their port, which would give them their location at sea. The sailors would count the number of seconds that it took for a piece of rubbish in the water to get from the front of the vessel to the rear. They took the length of the boat and divided it by the time to get an estimation for their speed. This technique was clever, but proved to be too unreliable when attempting to navigate an ocean.

If sailors knew the longitude and latitude of their ship then they would know their location at sea which would greatly improve their chances of landing where they desire. Latitude wasn’t very difficult for sailors to determine. All that was needed was a compass, a calendar, and how high the sun should be at noon. However, what was more complicated was determining the longitude of the ship. The way that sailors determined the longitude included the sun and an accurate clock. The sun was important because of the rotation it appears to make in the sky. The earth rotates and makes it appear that the sun is making a 360 degree rotation around the earth. This means that each hour the sun would move fifteen degrees. Every four minutes is would move one degree. Every one minute it moves 15 arc degrees and so on. This information can be used if you have an accurate clock set to Greenwich mean time. For example, if the GMT time was 18:00 and the local time was 10:00 then you have a difference of minus eight hours, which means your angle measure would be fifteen times eight which equals 120 degrees. This means you are 120 degrees west. These calculations would prove to be very helpful for sailors who were navigating the sea. The only problem was the fact that the clocks only kept accurate time while on land. The movement, temperature difference, and humidity all disrupted the clocks while at sea. It wasn’t until 1775 when a man named Larcum Kendall made a clock based off the clocks made by John Harrison. The clock was taken on a three year voyage and they found that the clock had kept accurate time and only had an error of about eight seconds per day.

What the sailors were doing in the late 1700s and beyond is what we call using a GPS system. What is amazing is that our technology has developed over time to the point where it is taking the sailors from ships and keeping them on land while the ship takes its course. Technology is being created that would allow an unmanned ship to sail across the ocean to the proper destination by remote control and a GPS location. The technology is said to be a handful of years away from being put into action, but would be an amazing feat if this unmanned ship could actually sail across the ocean with no problems. The technology and systems that were created have been expanded upon to great lengths.

Works Cited:

*How Did Early Sailors Navigate the Oceans?* *YouTube*. N.p., n.d. Web. 23 Apr. 2016. <https://www.youtube.com/watch?v=4DlNhbkPiYY>.

*How does math guide our ships at sea?* By George Christoph. *YouTube*. N.p., n.d. Web. 23 Apr. 2016. <https://www.youtube.com/watch?v=AGCUm_jWtt4>.

Mahoney, Donna. “Underwriters get ready for crewless ships.” *Business Insurance* 15 Feb. 2016: 4. *Wilson OmniFile Full Text Select Edition*. Web. 4 Apr. 2016.

“Unmanned Ship Technology.” *Business Insurance* 15 Feb. 2016: 27. *Wilson OmniFile Full Text Select Edition*. Web. 4 Apr. 2016.

Ship with bird link- Included in the photo.

Clock link- http://yeoldecybershoppe.com/malahide/uncategorized/patrick-malahide-john-harrison/

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Anyone involved in the discipline of math can most likely recall one, if not multiple, instances of being questioned on the usefulness of math. Eduardo Saenz de Cabezon addresses this question in his TED talk “Math is forever” (which can be found here). He claims there are three types of responses. First, the attacking one, which states math has a meanings all its own without the need for application. Next is the defensive one, which replies math is behind everything from bridge building to credit card numbers. The third response is where Eduardo claims math’s utility stems from its ability to control intuition, thus making it eternal.

Is math forever? Eduardo seems to think so stating diamonds aren’t forever, a theorem is. Mathematicians spend their lives generating conjectures and searching for ways to prove them. Once a conjecture is proven true though, it becomes a theorem, which is a truth that will remain so forever. Therefore, concepts such as the Pythagorean Theorem and the Honeycomb Theorem will forever be true, regardless of whether or not we are here to acknowledge it. This idea is rooted from Platonism, which is the philosophical view that there are abstract math objects that exist independently from our thoughts. Thus, all math truths are waiting to be discovered and not invented.

There are two main contributors in the world of mathematical philosophy. The first is German mathematician David Hilbert (pictured to the left), creator of Hilbert’s Program. He claimed that all math is formulized in axiomatic form with a proof to accompany it; it is done so by using finitary methods only which gives proper justification for classical mathematic problems. Hilbert believed theories could be developed without the need for intuition and would generate a set of rules and axioms that are consistent so one cannot prove an assertion as well as its opposite. Hilbert, like Eduardo, believed the capabilities of math were limitless.

Hilbert’s work, in turn, inspired the work of Kurt Gӧdel (pictured right) and his Incompleteness Theorems. Gӧdel proved that Hilbert’s concept of a decision procedure that generates axioms cannot be possible; there will always be conjectures that need a proof that may not actually exist. Gӧdel’s first incompleteness theorem proved that math knowledge cannot be specifically summed up and identified. Even the soundest basic rules will have statements about numbers that can’t be verified. It is important to note however, that Gӧdel never had the intention of disproving Hilbert’s program but rather to offer a new view.

So this leaves the math community open to explore if math is created or exists regardless of human recognition. If a tree falls in the woods when no one is around, does it make a sound? If no one has been able to prove a conjecture, does that theorem still exist? Like many schools of thought, there is ambiguity and uncertainty. As an individual in the math community, we all are responsible for looking into the information and opinions and coming to our own conclusions. Yet one thing remains certain, intuition and creativity

are absolutely essential in mathematics.

Sources:

de Cabezon, Eduardo Saenz. “Math is Forever.” *TED.* TED, Oct. 2014. Web. 05 Apr. 2016.

Elwes, Richard. “Ultimate Logic. (Cover Story).” *New Scientist. *211.2823 (2011): 30-33. *Academic Search Complete. *Web. 4 Apr. 2016

Linnebo, Oystein. “Platonism in the Philosophy of Mathematics.” *Stanford University. *Stanford University, 18 July 2009. Web. 05 Apr. 2016.

Peterson, Ivars. “The Limits of Mathematics.” *Science News. *Society for Science & the Public, 2 Mar. 2006. Web. 5 Apr. 2016.

Zach, Richard. “Hilbert’s Program.” *Stanford University. *Stanford University, 31 July 2003. Web. 05 Apr. 2016.

Image 1 retrieved from: https://www.bing.com/images/search?q=Incompleteness+Theorems

Image 2 retrieved from: https://www.bing.com/images/search?q=david+hilbert

Image 3 retrieved from: https://www.bing.com/images/search?q=Incompleteness+Theorems

It is said that the winners write history. While usually this is reserved for the perspectives in history textbooks and other writings, it also finds true in the evolution of mathematical history as well. Beg to differ? Ask Recorde. He has Leibniz and his winning calculus notation to thank.

Find x. Well, we have found it several times as mathematicians, used it several times in problems, and assumed it as the universal unknown. The unknowns span from just a mathematics variable and into popular culture, spanning the X-files, the X-factor, and Project X. But where did we get x? In a TEDx (again, why x?) talk, Terry Moore presents a brief explanation of what he has found is the reason we use x, and the reason is perhaps more comical than expected.

Moore explained that he undertook learning Arabic to better understand the history of mathematics. He explains that the word “shalan” translates to “something,” as in something unknown or something arbitrary. He goes on to explain that when the Europeans, mainly the Spanish, came to translate the Arabic mathematical findings, they were presented with a problem: the Spanish did not have a sound for “sh.” Therefore, they picked a hard “ch”/”ck” sound, as in the Greek “Chi”, symbolized as “X”, which, when translated to Latin became “x.” Moore jokes at the end, “Why is it that X is the unknown? X is the unknown because you can’t say ‘sh’ in Spanish.”

Having watched this video, I was inspired to ask, “What other notation have we taken for granted?” I then went and thought about why we use symbols. While yes, mathematics could be done with entirely words and documented arguments, the use of symbols stems from our use of language. Terrence W. Deacon explains in her book “The Symbolic Species,” that we, as humans, have developed symbols that “don’t just represent things in the world, they also represent each other. Because symbols do not directly refer to things in the world, but indirectly refer to them by virtue of referring to other symbols, they are implicitly combinatorial entities whose referential powers are derived by virtue of occupying determinate positions in an organized system of other symbols” (99). The use of symbols (variables) to describe other symbols (words, or other variables) has become a part of our general nature.

Besides from x, a symbol perhaps most used by mathematicians is the equal sign, =. What does this mean and how did it evolve? Until about 400 years ago, there were numerous symbols that meant equal. At first, there was no sign for equal, but rather it expressed rhetorically with such words as *aequales*, *faciunt*, or *gleich*, taking form in a variety of languages. At one point, just the abbreviation *aeq *was used. As a variety of other mathematical notations had formed, so did a multitude of ways to write an equals sign. Buteo used [ to show the function of equality, and Diophantus used two parallel lines, ||, to show that two quantities were equal. Descartes suggested that be used to signify equality, and, for a while, he had begun to gain some popularity for this notation—as well as developing a widely, used coordinate system. (Cajori 297-301)

Up to the 17^{th} century, the = symbol had a plethora of meanings, including parallel lines, difference, or even plus or minus. In Berlinghoff and Gouvea’s *Math through the Ages*, it is noted that the symbol is suggested by Recorde in the 15^{th} century, but not adopted for a couple hundred years later until Leibniz preferred = to other symbols in his calculus notation, which had proven to be more successful than Newton’s. Just think, if we had used Newton’s calculus notation, we would still be using instead of =. (100)

So Leibniz wins! And therefore so does Recorde and the traditionally accepted sign, =, is now used in all of the textbooks.

References:

Berlinghoff, William P.Gouvêa, Fernando Q.*Math Through The Ages: A Gentle History For Teachers And Others*. Farmington, ME : Oxton House Publishers ; 2004. Print.

Cajori, Florian. *A History Of Mathematical Notations*. New York : Dover Publications, 1993. Print.

Deacon, Terrence William. *The Symbolic Species: The Co-evolution Of Language And The Brain*. New York : W.W. Norton, 1997. Print.

Moore, Terry. “Why is ‘x’ the unknown?” *TED*. Feb 2012. Accessed 28 March 2016. http://www.ted.com/talks/terry_moore_why_is_x_the_unknown

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