Friday, December 18, 2015

Journey Through Genius Review

The Journey Through Genius
 by William Dunham is a fantastic overview of critical mathematicians and theorems that created the basis of the mathematics that we have studied since the beginning of our education. William Dunham was originally trained as a topologist but as his mathematical studies progressed, he became more and more interested in the field of mathematical history. He has published several books on this topic and his authority within the field shines through in Genius.

The book is composed of 12 chapters that each focus on a particular time period in mathematics or a specific mathematician's contribution. Mathematicians that are thoroughly covered in this text include Euler, Euclid, Newton, Leibniz, the brothers Bernoulli, and many others. Dunham does an excellent job of giving context to each of the mathematical contributions that he mentions. As a math major, I've often found that the historical context of what I am learning is often not mentioned. Dunham's Genius is an excellent source for filling in this educational gap. 

Although this book was incredibly interesting and eye opening to someone like me who plans to make a career out of mathematics, I do not believe that Dunham directed this novel towards any extensive audience. The historical background is very digestible and anyone with a basic understanding of world history would be able to place each of our mathematical heroes in the proper context.  However, the true appreciation of the geniuses that are covered in Dunham's book lies within their clever proofs which are outlined throughout the chapter. This part of Genius was incredibly interesting to me and the way that Duham outline the groundbreaking proofs was very helpful and insightful. However, this portion of the book would not be approachable for high school students or students very early in their mathematics career as the significance of the proofs may be missed. The book is excellently organized in a chronological fashion that pulls the reader along the history of these mathematical geniuses.

I would highly recommend Genius to anybody who plans to pursue higher mathematics for it provides an excellent narrative of crucial information that is often lost within our course of studies.

Overview of 495

My taking mathematics courses to satisfy my major requirements was incredibly beneficial in thoroughly learning material for each individual topic. Although my education has been excellent, oftentimes the course topics were rather disjoint. If making connections between fields was left up to my own creation, they often were not.

Over my semester in John Golden's Nature of Mathematics course, the connections between mathematical fields were presented directly to me in a very valuable way. Not only how the actual mathematics within the field were connected, but how these fields are connected historically.

Since this course was taught in chronological order, it was very clear to me how mathematical concepts were initiated, developed, and transformed into new ones. This view of the field that I have learned so much about shows mathematics as growing and dynamic. Seeing this development makes math seem much more personal and approachable. Mathematics in a historical context is much more human than the mathematics that is presented in a classroom. Topology wasn't created overnight and Lagrange multipliers weren't developed in a 50 minute block like they are taught.

Another important thing that I've learned during my semester in Golden's class is that mathematics isn't an individual activity. Sure I had done group lab assignments in Calc 3 or met up with my classmates to finish delta-epsilon proofs, but the field in general appeared to be one clever person solving a clever problem in a clever way. 

This course taught me that development in mathematics is the result of multiple sources of man power. For example: Fremat's Last Theorem. The BBC documentary on the solution to Fermat's Last Theorem that I watched as an assignment in Golden's class made me realize how many individuals it takes to make intellectual progress. Although the film focused on Andrew Wiles, the man who finally solved the proof, it also gave an overview of the people and efforts that contributed to the solution. This sort of collaboration wasn't presented in my previous math courses.

Overall, taking John Golden's course has placed mathematics into a context that is more approachable and more enjoyable than before. This class made me excited to continue my mathematics education.

Wednesday, November 11, 2015

Why aren't there more women in STEM professor positions?

Why aren't there more women in STEM professor positions?

As a female undergraduate math major interested in pursuing a career in higher education, this question consistently comes up in discussion. But rarely is this issue given explanations besides "men are naturally better at mathematics" or "women like subjective topics, not objective ones."

Firstly, many people have tried to explain the lack of women in STEM fields as a biological issue. Essentially, claiming that having XY chromosomes makes you unquestionably better at analytical reasoning. This claim would certainly give a definitive explanation of the issue at hand, however, it doesn't seem to be the case. 

As a student who attended a public institution in elementary and high school, a gender gap in mathematical performance has always been discussed. This disparity seems to rear its ugly head in every standardized testing situation: many claim that SAT and GRE scores indicate that male students consistently outperform female students on the quantitative portion of each exam.  For example, the following is a graphic of the trending test results of the math portion of the SAT from 1972-2012:

The true magnitude of SAT math sex differences

Many observers would claim that this 32-point difference in exam performance significantly and undeniably illustrates that the mathematical skills of male students is higher than those of female students. However, tests of significance can often be misleading. A statistical evaluation of the SAT data tells a different tale:

"Because significance tests can sometimes be misleading, scientific journals typically require other statistics to assess the importance of a result. The most common are assessments of effect size-tests that tell you how large the effect is.  Using the data released from the SAT board (Mean male = 521, sd = 121; mean female = 499, sd = 114), it turns out that about 3% of the variability in SAT math scores can be attributed to the sex of the test-taker; 97% is due to other factors—presumably differences in training and natural aptitude in math." (Cummins, 2014)

So,  there is no evident difference in the mathematical aptitude of male and female students. Men having a natural ability for mathematics over women has essentially been denounced by most educational and cognitive psychologists. Then why do women make up less than 25% of the STEM workforce? If biological sex does not seem to be a factor, gender disparities may be at play here.

Women receive approximately half of the STEM doctorates awarded in the United States, in fact, a study of top STEM graduate programs revealed that most have a nearly 1:1 ratio of male to female students participating in their programs.( Then, approximately 45% of entry-level STEM professor positions are held by women. All of these steps to obtaining a teaching position in a STEM field seem to be somewhat evenly dispersed between men and women. Yet women only make up only 21% of science and engineering full-time professors. What happens to the gender disparity on the tenure track? The tenure track to become a full-time professor has no pit-stops, once you get off track, you cannot reenter. The pressure (internal or external) for women to start families occurs at approximately the same age as the start of a tenure track. Many professional women are encouraged to forego their careers in exchange for being the primary caregiver of their families (a gender issue that affects all fields of study, not just STEM). Many women who plan to start families abandon the pursuit of a full time position and opt for adjunct, annually renewed positions. So, the pressure of starting a family occurring simultaneously with the tenure track process explains the extreme drop in full-time, female, STEM professors. 

Something else to think about:

Are women actually underrepresented in STEM fields? This graphic implies that this is not the case, but that women tend towards the biological and social sciences. What could be causing this?

Monday, September 28, 2015

What are Amicable Numbers?

Two numbers, k and j, make are amicable if the sum of the proper divisors of k is equal to j and if the sum of the proper divisors of j is equal to k.

History of the Amicable Numbers:

Although the equation that derives pairs of amicable numbers was popularized by Fermat and Descartes in the early 16th century, the concept and derivation of these unique numbers was first investigated by an Iranian mathematician named Thābit ibn Qurra (826, 901). Qurra was also one of the first mathematicians to describe the phenomenon that we now know as the Pythagorean Theorem. 

After Fermat and Descartes studied these unique numbers and rediscovered the pairs, Euler extended their work in the 18th century and discovered dozens of new pairs. After years of extension on these pairs of numbers, as of 2007, there are 12,000,000 known amicable pairs.

Interpretation of the Amicable Numbers:

Let's try to better our understanding of this definition by looking at an example. Firstly, the "proper divisors" of a number n are all of its divisors excluding n itself. Normally, we would list the divisors of the number 6 as

1, 2, 3, 6

However, the proper divisors of 6 (for this example we will denote this by p(6) ) would be listed as
p(6)= {1, 2, 3}

So, now that we know what a proper divisor is, we can continue interpreting the definition of an amicable pair.  Using our proper divisor notion, it follows that two numbers, k and j, are amicable if
Σp(k)=j and Σp(j)=k

The numbers 220 and 284 are  amicable, let's see why that is.

p(220)= {1, 2, 4, 71, 142}
p(284)= {1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110}

From the definition, it should follow that Σp(220)=284 and Σp(284)= 220, let's double check:

1 + 2 + 4 +71 + 142 = 284
1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 220.

So, we can see that 284 and 220 are amicable numbers.

Here is a great NumberPhile video for more explanantion on amicable numbers:

Finding Amicable Numbers:

So many great mathematicians studied amicable numbers because of the difficulty of finding these numbers. Part of this difficulty lies within the relationship between amicable numbers and prime numbers. This can be seen in Euler's rule for finding amicable numbers as follows:

"Euler's rule is a generalization of the Thâbit ibn Qurra theorem. It states that if

p = (2(n - m)+1) × 2m − 1,
q = (2(n - m)+1) × 2n − 1,
r = (2(n - m)+1)2 × 2m + n − 1,
where n > m > 0 are integers and p, q, and r are prime numbers, then 2n×p×q and 2n×r are a pair of amicable numbers."

Where Do We See Amicable Numbers?:

 Although there is no direct application of these pairs of integers, amicable numbers have a significant historical impact. Not only did the search for these pairs attract so many recognizable mathematicians over time, but they were representations of "friendship." In Genesis 32:14, Jacob gifted Esau with 220 goats, a number chosen to exemplify his love. 

Another instance in which amicable pairs were used to portray positive relationships was in Medieval horoscope projections. If two people were given a horoscope containing the numbers 220 and 284, then this indicated a future romance between them. These whimsical applications of amicable numbers further emphasizes the natural attraction that people have for these pairs.

Sunday, September 13, 2015

The Influence of Greek Mathematics on Modern Proof Methods

As mathematics progressed from simply being a field of computation and quantitative observation, the techniques of proof that we use today began to develop. The first historical observation of what we today consider a mathematical proof was during the time of ancient Greek mathematicians. The contribution of proof was incredibly vital in the progression of the field of mathematics.

Editor of the Ancient History Encyclopedia, Cristian Violatti, claimed that there were three proof techniques that arose during the development of ancient Greek mathematics:

  1. The technique of abstraction
  2. The technique of generalization
  3. The art of deductive reasoning
The first technique contributed by the Greeks, abstraction, is a key quality of communicating in modern mathematics. Before the concept of abstraction was introduced to mathematics, each problem had to be approached as a new, unique situation. Problems like the length of rope needed to create a 4-sided fence or the angle to which the peak of a roof had to be constructed were seen as unique problems that had to be solved in context. However, with the development of abstraction, these every day problems could be taken out of their real world situation and be solved using visual and mathematical representation.

Another technique that was contributed was generalization. The concept of generalization allowed mathematicians to make general claims about situations that were true in all cases. An example given by Violatti was the Pythagorean Theorem. For example, if an applicable real-world situation could be abstracted as a 3:4:5 right triangle, the theorem would apply to this abstraction. However, with the use of generalization, mathematicians can now claim and prove that theorems, such as the Pythagorean Theorem,  can be applied to all situations. As math students, we see the use of generalizations every day in our studies. We have the tools to prove statements with universal quantifiers and for multiple cases thanks to the progress that the Greeks made towards modern proof technique.

The last technique that was mentioned by Violatti was deductive reasoning. Since the technique of generalization allows us to prove claims for general cases of mathematical situations, we are then able to use those proven claims to produce desired conclusions. This method of deductive reasoning that was established by the Greeks has been used to perpetuate every field of mathematics since it allows us to use what we already know in order to create new conclusions and mathematical relationships.

Without the contribution of new proof techniques by the Greeks, modern proof techniques would not have been established and the field of mathematics could not have progressed as far as it has.
Cristian Violatti. “Greek Mathematics,” Ancient History Encyclopedia. Last modified September 24, 2013. /article/606/.

Thursday, September 3, 2015

What is Math?

As an aspiring mathematician, I often get asked, "Lindsay, why do you like math? Isn't math just equations and numbers that you have to memorize to pass an exam?" I am then forced into defense mode in which I must defend the subject that is so near and dear to my heart. However, this is such a difficult thing to do. Sure, the quadratic equation and triangle congruence rules that we had to memorize in high school fit under the umbrella of "Mathematics," but that's not all that it includes.

It is my understanding that mathematics is a collection of tools that we use to quantify and describe the world around us. We use mathematics very similarly to how we use language. Using language, we can identify objects, convey ideas, and argue. Math can be used in the exact same way when communicating scientific ideas, defining mathematical objects, and proving theorems. The most interesting relationship between language and mathematics is that both can be utilized to describe events and objects that do not exist in the physical universe. Language can be used to create poetry which describes abstract emotions or fictional events. Similarly, mathematics can be used to describe things like how a 2-dimensional circle would look if it were projected on a 1-sided Klein bottle (like what Dr. William Dickinson at GVSU researches.) I think that, at its very root, mathematics is a form of communication.

A very large turning point for math was when mathematical concepts could be argued and verified through what we all now recognize as a proof. This began in the field of geometry and this method of logical argument showed that math concepts were concrete and could be verified. The conceptual topics that were being proved early in mathematical history were rooted in applications. Another key turning point for math was when observable events and objects were abstracted into general cases. This pushed the field of mathematics from simply being computation and quantification into a field that could describe both concrete and abstract situations.