Traditional Math vs. Reform Math

The stigma about mathematics comes from traditional mathematics. Traditional mathematics refers to the math that we all learned through all our school years. Some, in fact most, happily reflect on this period of time as something in the past that will stay in the past.

Traditional mathematics is defined as studying the foundations of math through repetition and practice.


The idea behind it is logical: we must learn the basics of math thoroughly before moving on to more advanced and applied math. The repetition is what draws the most groans. But learning mathematics does not have to be this tedious. Today, many mathematicians are proposing a new way of teaching that appeals more to the creative mathematics we have been discussing.

It is known as reform mathematics, and it emphasizes the practical application of math by teaching through creativity, and therefore reveals the true value of math in a memorable and more educational way.



A creative approach to mathematics focuses on independently solving real world problems. In other words, the math should serve the problem, the problem shouldn't be formed to fit the math.

Be that as it may, there are some aspects of traditional mathematics that shouldn't be ignored. For instance, the nature of mathematics demands some mechanical learning. The systematic analysis skills we practice are excellent exercise for our ability to deal with logic puzzles.

So if we acknowledge the importance of knowing the basics of math, as well as accept the need to apply math creatively to the real world, we will have developed a much fuller understanding of mathematics that dissuades us from its stereotype!

'Nothing' On the Verge of Philosophy

Gradually the systematic and computer-like character of mathematics is being humbled to the creativity that emerges through the field. It can be astonishing just how much room mathematics has for creativity amongst its hidden ambiguities and mysteries. For example, the concept of nothing can be explored by creatively analyzing the number zero.

The capability of numbers to intersect reality and the abstract is what makes them confounding, but nonetheless, very useable. This same ability gives rise to paradoxes, such as the nature of the number zero. Of all the numbers, zero is one of the most difficult to define, and the meaning of zero can be lost as it easily escapes concrete association to any object. Be that as it may, mathematics could not exist without zero, but zero is more than just its function: it transcends mathematics.

Zero is a representation of something very simple: nothingness. Zero means a lack of value, for, as William Byers puts it, zero’s “presence implies an absence” (Byers 101). However, zero itself is not nothing. It is a concept, and therefore it is something, but by meaning, it is nothing. Despite zero’s ambiguity, it has a variety of functions. For instance, one function of zero amongst other numbers is to serve as a placeholder in the number line, the point of balance between positive and negative numbers. Another use of zero is to allow the outcome of an operation to have no value, such as two subtracted from two. Zero should be recognized for what it does and also what it means.

The nothingness about zero is fascinating, and so is what it does to challenge our vocabulary, and our conceptions about symbols, meanings, and math. It inspires one to look outside of mathematics for presences that imply absences, to look for zeroes in our world. However, zero does not seem to exist elsewhere; it exceeds reality and represents pure emptiness, which does not exist. Even clear space is occupied by air, a sort of matter. Also, zero does not demand anything to replace it, such as -1: it just is. Such a solid concept is so profound, and the fact that it is in itself a contradiction is even more astonishing.

Thus, zero demonstrates the possibility for logic and contradiction to coincide in a standard system. It also serves as an idea to allow us to contemplate nothingness in general, and contradiction in life. For instance, we have a fear of nothingness, such as in death, but when the concept of zero is applied, it is seen that nothing is nothing to be scared of, for “the content of that fear is unknown” (Byer 99). Thus, zero has the ability to serve as a vehicle for philosophical musings. We also need the concept of nothingness for our rationality and language as a foundation: things cannot be built upon unless it is understood that something did not exist beforehand. Numbers are remarkable and should be recognized and embraced for their functionality and indefiniteness, as zero has shown how it is fundamental for its use in math, but also for our comprehension of the nothing.

Works Cited

Byers, William. How Mathematicians Think. New Jersey: Princeton University Press, 2007. Print.

Gowers, Timothy. Mathematics: A Very Short Introduction. New York: Oxford University Press, 2002. Print.

Voting Theory

Mathematics shows up sometimes in the unlikeliest of places, such as voting. But how could voting for something use math? All we do is cast a vote and it is counted up? The math lies in creating different methods of voting.

Voting theory explores the different methods we can use to set up a voting system. The most popular form of voting we all know is the one mentioned before: where each voter casts one vote for their top choice, and the winner is the one with the most votes. This is known as Plurality. But there are many other ways of voting! For instance, one type of voting is called Approval Voting.

In Approval Voting, each voter votes one vote of approval for as many candidates as he or she chooses. The one with the most votes wins. This system works off the idea of having the outcome be something that all voters are willing to let happen.

Another form of voting is called "Vote for n," a modified version of Approval Voting. In this system, each voter is only allowed a certain amount of votes to allocate to candidates, and the most votes win. This forces the voter to put more thought into which candidates they especially support.

The last voting system we will address is called Borda Count. Borda Count is the most complex of these voting methods. Here, the voter assigns 1 point for their first choice, 2 points for their second choice, and 3 points for their third choice. So contrary to the other systems, this winner wins by the least amount of votes. This system most accurately reflects the varying preferences and rankings of each voter.

Each of these voting systems are capable of producing a different winner, even if the voters and what is being voting on stays constant. The different voting systems show how creativity can be used to skew the results and manipulate the numbers of voting to achieve a certain end. Thus, voting theory serves as an excellent example of how mathematics can be creative!

Careers in Math

According to whenwilliusemath.com, a quantitative financial market analyst usually comes from mathematics or physics backgrounds rather than finance related fields. In another category of the business/finance field, stockbrokers use algebra and geometry to calculate how many shares of a stock a client can buy. Besides using math to do basic calculations, stockbrokers also use more complex concepts like PE Ratio, Alpha, and Beta to find out if a stock is overpriced or to find out what the risk level is if investing in certain stocks. (I had to look this up, but a PE Ratio is the price/earnings ratio that determines how expensive a stock is.) Apparently, game theory is also a useful math concept to know if you are a stockbroker most likely because it helps you analyze economic situations. People in the computer science field use graph theory and combinatorics, which are both principles that we learned about in Motion and Mechanics class.

Under the section Math in Real Life, I chose to read the category about Business Schools and mathematics. According to whenwilliusemath.com, mathematics majors frequently outscore other majors in the graduate business school entrance exam, and being a math major is actually a really good preparation for getting your MBA. According to Dmitri Kuksov, a professor at the Olin School of Business, "math... is possibly the best undergraduate background to get into business schools" (whenwilliusemath.com). Steven Wheelwright, a former professor at the Harvard Business School, stated that math teaches the problem solving and analysis skills needed in order to succeed in the business world, especially in areas that involve data analysis and statistics (whenwilliusemath.com). Reading this article makes me, as a business major, wonder if I should switch to a math undergrad major before applying to a graduate business school.

Millenium Prize Problems

The Millennium Prize Problems are seven unsolved mathematics problems that The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) chose in celebration of the transition to the new millennium. In the year 2000, CMI chose these seven problems and decided to offer a prize of $1 million per problem to the person who could find a solution. Obviously, these mathematical problems are extremely difficult: only one award has been given, and it took ten years for that to happen! The first – and only, so far – person to solve a millennium problem is Dr. Grigoriy Perelman, who resolved the Poincaré conjecture. Apparently, Dr. Perelman was awarded and declined both the Millennium Prize and the Fields Medal for his proof. While Dr. Perelman managed to resolve the first of the millennium problems, other mathematicians were not so successful. In August 2010, HP Lab researcher Vinay Deolalikar circulated a claimed proof of P ≠ NP in resolution to the P versus NP millennium problem. Even though the proof received much attention from the press and on the internet, and even though the proof was well-written in a historical context, academics spurned this proof because its was seriously lacking in some of its conceptual arguments. According to Finite Model Theory expert Neil Immerman, the proof’s “domain… does not express all of P” (Lipton).

      The millennium problem I found most interesting was the P versus NP problem because it’s an important problem in computer science (and I like anything that has to do with computer science). According to the Clay Mathematics Institute, a P versus NP problem essentially asks if any mathematical problem exists such that its answer can be easily checked with the help of a supercomputer, but is impossible to solve by any direct process. 

The “P” signifies that a problem is easy to solve while the “NP” signifies that a problem is easy to check. The CMI website gives an example of an NP problem in which you have to create a list of dorm-mate pairs from a pool of 400 students, but you can only choose 100 students and there are certain restrictions as to who can room with whom. This problem is NP, or rather “easy to check”, because if I made a list of 100 students, my friend could easily check if the list fulfilled all the requirements of a solution to this problem. Apparently there are more solutions to this problem than there are atoms in the universe, which makes it seem impossible that we could ever build a supercomputer smart enough to go through and write out every list possible.

Works Cited

Lipton, Dick. "Fatal Flaws in Deolalikar’s Proof?" Gödel's Lost Letter and P=NP. 12 Aug. 2010. Web. 1 Nov. 2010. <http://rjlipton.wordpress.com/2010/08/12/fatal-flaws-in-deolalikars-proof/>.

Graph Theory

"Please graph the following..." and groans already echo around the classroom. Graphing data is one of the most widely used areas of mathematics in the real world. It is unfortunate that graphs are often met with frustrated grumbling and then quickly done without truly understanding the fantastic capabilities of graphs and just how useful they can be. Reconsider, however, your definition of graphs. When you think of graphs, you imagine bar graphs and line graphs, which are used as models of relationships between things. Graph theory is the not the study of graphs, in the traditional sense. True, graph theory is a study of relationships between things, but not in the sense we are used to. The graphs that we will be discussing show an important aspect of math that mathematician Keith Devlin puts forth: "Math is about patterns. And patterns are what life is all about" (30). The identification of patterns or lack of patterns amongst the relationship between objects is key in graph theory.
Graph Theory


A graph is a collection of points called nodes that are connected by lines called edges. Here is an example of two graphs that are exactly the same, even though they look different.

Both graphs 1 and 2 have 4 nodes and 6 edges. Points A, B, C, and D are what we would call nodes and they each have a qualitative value based on how many other nodes it is connected to. This is described in terms of degrees. For example, the node A has a degree of 3 because it is connected to three other nodes. You can also have directed graphs where there is an arrow on each edge in order to represent different types of relationships. You can also have weighted graphs where each edge has a qualitative value, and this will represent different types of relationships as well.

Graphs are used effectively in many different applications.

• Computer Networking: Graphs can keep track of which computers are networked together, which were the servers, and weights and arrows could show what kind and to what a computer has access to.
• Family Tree: Graphs can organize who is related to whom in the family. Arrows can be used to show who gave birth to whom, and weights could show many things, such as the number of years in age difference between two members, or how positive the relationship is between two members on a scale of 1 to 100.
• Product Distribution: Graphs can use nodes that represent either factories or stores. Arrows flow from factories to stores, and weights show how many products are shipped from the factory to the store.

For more information on graph theory, go to http://www.cs.elte.hu/~hubenko/graph_theory.html 

For a cool program that helps you do graph theory, go to http://www.graph-magics.com/  (be aware that this program only works on PCs.)

Number Theory

"Mathematics is the study of numbers!"

A very prompt, but slightly misled response.

Numbers are a large part of the field of mathematics, but they most definitely do not define the entire field.

"Ok, well I totally understand the number part, all that adding and subtracting stuff."

But let us consider numbers and mathematics. If anyone is to understand mathematics, most would assume they understand its relationship to numbers. But numbers may surprise you.

Number Theory

Number theory is the branch of math that studies numbers. Questions that number theory raises include:

• Where do numbers come from?
• What is a number?
• How are numbers used?

The origin of numbers can be answered by considering numbers as discovered or created, or a combination of both. Certainly amounts of things have always existed, but the tools to understand them, numbers, were created by man. Numbers by themselves are very abstract. We can say, "5." But what does 5 mean? 5 of what? Immediately we try to think of 5 objects, and it becomes apparent we are impatient and cannot sit and ponder on 5 itself for very long. But let's think about it.

What is a number? Numbers by themselves are quite abstract, as they refer to nothing but a quantity. It is odd to try to think only about the quantity and not the object, even when the objects can be abstract as well (ie. colors). Our straightforward, mathematical numbers have become quite mysterious! What in fact are we using these abstract characters for?

Humans grasp numbers at an early age through learning how to count. As Keith Devlin makes the distinction in his book The Math Instinct: Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs), counting and knowing how many are two different things. The latter develops as we learn the language of operation mathematics, things such as adding, subtracting, multiplying, and dividing. Here we learn of the symbols.

Before, when we were trying to only think of what five is on its own, the symbol of the number, 5, most likely popped into our heads to fill the lack of objects. However, there is a distinction between the number and the numeral. These numerals, or symbols, developed over thousands of years, from clay tokens to the Arabic numerals we have today. The symbols are arbitrary, but they allow us to manipulate numbers and thus open up the world of math. The symbols are what give us the capabilities to do perform operations of math, without them, it would be very difficult to see relationships between numbers!

To know that one of the major components of mathematics has such ambiguous qualities should be alarming to hear, because doesn't such a systematic and direct field need stronger foundation? It turns out mathematics relies on such uncertainty. It just goes to show that math is not all straightforward and calculating, but a mystery that develops and grows with creativity!

References
Devlin, Keith. The Math Instinct: Why You're a Mathematical Genius (Along with Lobsters, Birds, Cats, and Dogs). New York: Avalon Publishing Group, Inc., 2005. Print.