Mathematics and the Court System?

In the field of mathematics and statistics many people feel that the only ethical issues that apply are those dealing with plagiarism, fudging calculations, and/or ethical issues associated with teaching mathematics in k-12 schools.  However, there are many other areas in the field where ethical issues come into play.  For example, in the field of statistics there is a huge debate that deals with who should interpret statistics associated with court cases, specifically DNA profiling.  The question is should the courts be the ones to interpret the data that the statistician has collected/formulated or should the statistician provide a final interpretation/conclusion (i.e. which person is more ethically correct to choose/pick). 

To an outsider or a person in the field of science and mathematics it seems that the answer to this question is fairly obvious; the statistician would be the most qualified person to interpret the data and offer a final interpretation/conclusion.  However, many people are skeptical of statistics and fear that the results of a particular set of data can be analyzed in such ways that, even though accurate, provide a false picture of what the data actually represents.  This can be seen in the classic saying that “Statistics are like bikinis. What they reveal is suggestive, but what they conceal is vital.” 

Personally, as a person in the field of mathematics, I feel that it would be unethical/immoral to have a court interpret the statistics because in my eyes they are not the most qualified person to do so and therefore would open the door for more mistakes and misinterpretations of the data that could severely affect the person on trial in a negative way. 

Below is an excerpt from an article in Mathematical Association of America (MAA) that deals with the state of California’s position on this issue.

“On August 16, 2006, the California Supreme Court made it official: in certain legal cases that hinge on statistical calculations, it is not the business of professional statisticians to decide how to evaluate the statistical data and to judge what method is most suited to analyze that data. From now on, in California at least, the courts will decide what statistical analysis is appropriate and what is not.”

Now that we have a general idea about the ethical debate of deciding who should analyze/interpret the data we can look into some of the statistics that go into suspect identification and conviction based on DNA profiling (the specific issue at hand).

As a standard, the FBI looks at 13 regions of the DNA strand to match completely in order to reduce the error of false identification.  The idea behind this is that since the probability that anyone would match a specific DNA strand at any one point is (1/10), that by using the product rule for multiplying probabilities, taking 13 regions would give us a chance of matching a given DNA sample at random in a population are about 1 in ten trillion (i.e. 1 / 10,000,000,000,000).  This figure is known as the random match probability (RMP). 

This number leaves little room for error when profiling DNA samples, but to attain this number a qualified professional needs to understand which formulas to apply and how to interpret the results based off of the chosen formulas.  Something I feel a court is highly unqualified to do.

Overall, it seems that the courts want to uphold the famous saying “guilty beyond a reasonable doubt”, and by trying to do this they are attempting to eliminate any possible bias that may come from the evidence/statistics they consider.   This seems reasonable, but I feel that a person needs to weigh the consequences of each action (i.e. consequences associated with letting courts select formulas and interpret statistics vs. consequences of letting a statistician do the same jobs).  I feel that the probability of wrongful conviction greatly increases when untrained professionals handle the statistics and feel that training statisticians on the ethics associated with the courts would suffice to make up for any biases the courts fear. 

If you’re interested in the California case feel free to read this awesome article at

http://www.maa.org/devlin/devlin_09_06.html

“STEM Education”

It has long been known that the U.S. education system falls behind other developed countries in the areas of math and science.  To be precise, according to the Programme for International Student Assessment comparisons from 2009, the U.S. ranks 17^th out of 34 in science literacy and 25^th out of 34 in math literacy.  

A recent movement in education has been to focus on the STEM (Science, Technology, Engineering, and Mathematics) fields.  One of the unique characteristics of this new program/project is that it not only focuses on providing more materials/experiences for students in the STEM fields, but it also encourages professionals, college students, and community members to become involved in promoting/participating in STEM fields.  In this way STEM education is looking to reform the way schools and communities think about their education system (i.e. which subjects they value) and the way they allocate funds, time, and materials to certain subjects taught in the K-12 education system.  For example, new computers or “Smartboards” would most likely be seen as more valuable to a school compared to new stove tops or microwaves for a home economics class.

As an undergraduate student in the field of Mathematics I have personally benefitted from the STEM education movement through scholarships and internships offered to me by programs that support teachers entering or currently in a STEM field.  The internships allowed me to gain hands on experiences in the fields of Agriculture and Biosystems Engineering and Math education.  Along with the internships the scholarship also provided me with supplemental educational experiences such as attending the 2011 k-12 STEM Colloquium held at the University of Minnesota (U of M).  It was here, at the Colloquium, when I realized that STEM education was not only affecting k-12 education systems, but it was also influencing colleges and the types of programs they were offering for future STEM majors.  For example, the U of M is one of the first colleges to have a declared STEM education major.

Currently Congressman Michael Honda and many others have seen the benefits of STEM education and are making movements to keep this project/program running.  Representative Honda is currently introducing the STEM Education Innovation Act of 2011 in the U.S. House of Representatives.  This bill is set out to accomplish three main things:

1.       Create an Office of STEM Education in the Department of Education headed by an assistant secretary of STEM education

2.       Support a state consortia on STEM education to shape STEM best practices and increase participation of underrepresented communities in STEM disciplines

3.       Education Innovation Project: provides grant funding to outside entities (i.e. for-profit companies, foundations, nonprofits, and institutions of higher learning)

Representative Honda is clearly not only motivated by competition with other countries in the areas of math and science, but is also aware of the economic benefits of creating a society that is competent in the STEM.   He notes that “STEM workers are among the highest-paid and fastest-growing segment in the nations” and “that by 2018, 5 percent of all jobs in America will be STEM jobs”. 

As of now STEM education programs can be seen popping up all around the U.S. and are continuously growing and molding to the advancing technological society.  Seeing how the program/initiative has already touched/impacted Fargo, ND; it will be interesting to see where it goes next!

Texmaker vs. LaTex ?

In the field of mathematics most documents are typed up using some sort of LaTex system.  The advantages of this system versus using  Microsoft Word is that there are many more math symbols, it displays equations/formulas in a more appealing way, and overall makes it easier to follow the conventions of writing theorems, proofs, and lemmas. 

The version of LaTex that I’m reviewing is called Texmaker. 

Process of downloading software

Texmaker is a cross-platform LaTex editor for Linux, Macosx and Windows systems that consists of many of the same tools that are needed to form a document with LaTex.  One of the most convenient things about Texmaker is that it is easy to access/download.    A person simply has to search Texmaker in the Google toolbar and click on the first hit or a person may access the site directly at

http://www.xm1math.net/texmaker/

This site not only offers free downloadable versions of Texmaker, but also keeps their programs up to date!  Thus, all a person needs to do is check the same site/location for an updated version of Texmaker if s/he feels theirs is out of date.  As of now the latest version is version 3.2.2, which was updated January 12, 2012.

Visuals

Besides being free and easy to download Texmaker also makes writing /editing code easy.  It does this by ensuring that the window the code is displayed on or typed into is large enough to see most of the page, but still small enough so that it doesn’t take up the entire computer screen (i.e. much like a Word document).  Also any errors with the code are highlighted in about a 3-4’’ margin at the bottom of the page.  It is here that the error message displays the type of error and the location of the error within the code.  Which leads to another nice property of Texmaker, lines of code are numbered making it easier to locate and edit your document. 

Texmaker also includes Unicode support, spell checking, auto-completion, code folding and a built-in pdf viewer with synctex support and continuous view mode.

Lastly, I would like to go over what I consider to be the biggest “plus” associated with Texmaker.  This would be the tool bar located on the left side of the document.  This toolbar runs vertically down the page and displays a wide array of various math symbols (i.e. infinity, summation signs, various arrows, Greek symbols, etc.).  Thus, instead of a person having to memorize or constantly look up the code for these various math symbols, all they need to do is go over to the tool bar and search for the image of the symbol s/he wants.  These symbols are organized by different types of arrows, proving symbols, Greek letters, and an assortment of algebraic symbols such as the number π = 3.14.  Then a simple click of the mouse allows the program to automatically insert the code for the user.  Pretty sweet I would say!

Compare/Contrast usage

Essentially Texmaker and any LaTex system use the same code and conventions for setting up the code, only Texmaker makes it easier for the beginner coder by providing a  “cheat sheet” tool bar, as mentioned above, that allows a beginner to spend less time searching the internet for code and more time setting up the document.  Thus, Texmaker is a great way to go for beginning LaTexers!

Conclusions

Overall, I feel that Texmaker is very easy to use and is compatible with many different LaTex programs and computer systems.  It is very useful for beginners, especially those who need to learn the program fast.  For the experienced LaTexer, the extra tool bar on the side my seem unnecessary, but a person can change the viewing style so that the tool bar is hidden.  Hence, making the program appealing to both experienced and inexperienced LaTexers.

Mathematics… (Say What?)

Initially, when people think about the field of mathematics many think about number crunching and intense/confusing calculations involving the use of various algorithms.  However, true this may be, mathematics is much more than this and a large portion of mathematics involves little to no number crunching/calculations!

The core values in the field of mathematics are LOGIC and TRUTH. 

The field of Mathematics is based off of a set of axioms (i.e. statements) from which theorems, postulates, and lemmas can be deduced (i.e. true statements).  For example, a famous axiom set proposed by Hilbert to help develop the field of geometry looks like such:

Undefined Terms: point, line, incidence, betweenness, and congruence.

Incidence Geometry

AXIOM I-1: For every point P and for every point Q not equal to P there exists a unique line l that passes through P and Q.

AXIOM I-2: For every line, l, there exist at least two distinct points incident with l .

AXIOM I-3: There exist three distinct points with the property that no line is incident with all three of them.

One of the theorems deduced from this axiom set is known as the Crossbar Theorem which states:

If ray AD is between rays AC and AB, then ray AD intersects line segment BC.

In simpler words this theorem basically states that if we have a triangle ABC with a line that passes through the one of the vertexes such that the line passes through the interior of the triangle, then the line must intersect another side of the triangle.

It may seem redundant to actually prove this, but a large part of mathematics is proving what is seemingly obvious.  For example in an introductory proving course a person may actually prove 1 + 1 = 2 !

Overall, mathematics is about pattern and structure.  Mathematicians use logical analysis, deduction, and calculations within these patterns and structures.  Which, to state again, mathematics main emphasis is on LOGIC and deriving statements based off statements known to be TRUE (i.e. theorems, propositions, etc).  Mathematics main purpose is to extend on these theorems and propositions to find new patterns/structures and then see if these new findings relate to other fields (i.e. computer programing, biology, physics, chemistry, ecology, oceanography, etc.).  In a way, mathematicians prove true statements/ find patterns before we even know/understand how they relate to the real world or other fields. 

However, once a connection is made between the math and a practical application the ramifications are endless.  This can be seen in many fields such as:

·        Physics (quantum theory)

·        Biology (relations to the way DNA unknots itself before dividing)  

·        Computer science (algorithms behind programming languages and the use of fractals as a practical tool for compressing data on computer disks). 

There are also everyday fun practical applications that abstract mathematics gives us as well!  For example, the mathematics which go behind solving a Rubik’s cube.  Here a very complex series of steps can be reduced to relatively few simple algorithms. 

The link below goes over the math and shows the “everyday person” how to solve any 3 X 3 Rubik’s cube.

http://web.mit.edu/sp.268/www/rubik.pdf

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