Calculators: Good Or Bad?

Being that technology is an ever-present part of our society, it has also become part of the mathematics classroom. From calculators, basic ones (4-function) to ones that use Computer Algebra Systems (graphing calculators) to software such as Geometer's Sketchpad, the use of technology is being implemented increasingly in today's classrooms. However, there is much debate on whether the technology helps or hurts students. In the attached article, this topic of debate is discussed with research to back it up.

The article is about studies that have been conducted to show if calculators are helpful or detrimental to students. It explains that some hesitancy comes from fear, as well as ignorance of not knowing how to use them. It also gives parameters of certain studies done, as well as their results. The article does show that not only can students benefit from calculators, but they can also develop a better attitude towards mathematics. The article also mentions that the standard pencil-and-paper method can be enhanced with the use of a graphing calculator, and that understanding of concepts can deepen with a calculator. Since one of the bigger concerns in the classroom are achievement tests, this article reports that scores on achievement tests for algebra and calculus are higher with the use of a calculator.

I found this article pretty fascinating, since I wasn't introduced to a graphing calculator until I reached college. That may also be due, in part, to the fact that I graduated high school in 1996 when technology wasn't emphasized nearly as much as it is today. In my personal opinion, I do think that calculators can only help in the classroom. Not only do they help cut down on time spent doing long calculations, but actually being able to see the result helped me understand graphs better.

To view the article, click here: http://mathforum.org/technology/papers/papers/dunham.html

Have Fun With Your Students

Since the last few entries on my blog have been about my research paper, I wanted to change it up a little bit. I'm sure that not everyone wants to read about how fractals are being used in medicine, so I thought I'd take a different direction with this post. I wanted to show that teaching math doesn't always have to be boring and mundane. Teachers are always thinking of creative ways to get the students to understand the material. Not only do teachers have to be creative, but they have to make sure that they are getting the point across. As I was searching YouTube, I came across a video that showed me how creative some teachers can be, and how they are willing to go the extra mile in the classroom.

This video (https://www.youtube.com/watch?v=XKviYiZhtZY) shows a teacher explaining a problem regarding 1=2, and uses plenty of different tactics for the students to understand the problem. Granted, the steps of the teacher completing the problem have been skipped over, but the way he uses himself as an example is quite interesting. I also chose this video because it also shows that teachers can have some fun with the students, and that teachers are also willing to let loose, and go with the flow. The video was shot around Halloween, and it just shows how the teacher got into the spirit of the holiday, all while giving a lesson.

Fractals And Breast Cancer

Being that we've had a few weeks without class, it gave me time to do more research. My paper seems to becoming shorter on topics regarding fractals in medicine, but that is only because I have found so much information regarding my first two topics. I have already discussed (in previous posts) about using fractals to diagnose cervical cancer. In this week's post, the article that is attached is about how they are using fractals to detect breast cancer at an early stage. The article (link below) talks about how using a self similar fractal method to find ways and processes to diagnose breast cancer at an early stage, and even discusses how patients can be warned prior to the onset of breast cancer. The "warning" comes from the doctors reading the mammograms and using the fractal method to find specific patterns of microcalcifications and determine whether or not they may, in fact, be cancerous or pre-cancerous lesions. The article can be found at:www.ijcaonline.org/volume10/number4/pxc3871981.pdf

Fractals And Brain MRI's

So last week in class we started watching "Waiting for Superman", which was a documentary about charter schools. The movie-makers stand on it seemed to be that he is pro charter schools and the people he interviewed seemed to also think that charter schools are the end-all be-all of education.  While watching the movie, some of the claims came across as totally unbelievable, and others were shockingly true. It's sad to see statistics of our education system, yet nobody seems to care enough to try to change it in every school district.

Anyway, getting back to the purpose of the blog, I must discuss the progress of my paper. I have been conducting research for it and have come across some pretty amazing things that fractals are being used for in medicine. I have also come across things that are so involved and on such a higher level than I am, that I might as well be trying to read Japanese. Nevertheless, the information I am coming across is showing me that two separate fields (science and math) can come together and have amazing outcomes. Here is another article I came across that has to do with fractals and MRI's of the brain :
https://docs.google.com/viewer?a=v&q=cache:p-nYwvapcGEJ:citeseerx.ist.psu.edu/viewdoc/download?doi%3D10.1.1.180.899%26rep%3Drep1%26type%3Dpdf+how+do+fractals+help+identify+cancer&hl=en&gl=us&pid=bl&srcid=ADGEESiq7emV1R_SraULnMNeNdD_S6bO6cJXxejvdHzuM6GitcNDldDjAzNk6698FheVW0nilL-kiAjwG2Az-x97pbXg_QhB5ns8e0W8wePxOZtWL0oY6J_9GgL86CtYQNF-IqoPadDS&sig=AHIEtbRlQR5Ud98K8DvvntRkDwC0syX-XA

It's amazing what can be accomplished in both the worlds of science, medicine, and mathematics. I can't wait to research more and come across other interesting stuff! :-)

Paper Outline (Potential) & Fractals And Osteoporosis

In keeping with the theme of using our blog to update our status for our final paper, I have decided to include my outline for my final paper, as well as an article that I will be referencing in the paper. However, my outline is subject to change, as when I begin writing my paper, topics might be removed or added, dependent upon the amount of information I can gather regarding specific topics.  Here is the outline I have prepared for my paper:

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Topic: Fractals and medicine

A. What is a fractal?

            1. Definition of a fractal

            2. Mathematical explanation of a fractal

B. Can fractal be used in medicine?

            1. Explanation of how fractals can be related to medicine

            2. Applications of fractals in medicine

C. Fractals and cervical cancer

            1. Explanation of cervical cancer

2. Explanation of how fractals can be used to grade cervical cancer

D. Fractals and the heart

            1. How fractals are used for cardiac arrhythmias

            2. Fractals and other heart related issues

E. Fractals and mammographies

            1. Explanation of mammography

            2. How fractals are used in breast cancer detection

F. Fractals and osteoporosis

            1. Explanation of osteoporosis

            2. How fractals are used for osteoporosis

G. Will fractals in medicine be an ongoing trend?

            1. Discussion on the history and future of fractals in medicine

H. Conclusion

Also, you will notice that I have chosen to talk about how fractals are used to help in the diagnosis of osteoporosis. Again, I will be going into detail about this in my paper, but the gist of it is that they use fractals to determine if the bone is healthy, perimenopaused, or osteoporotic. If you would like to take a gander at the article, here is the link:  http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0CDQQFjAB&url=http%3A%2F%2Fwww.fmi.uni-sofia.bg%2Fcourses%2Fbiomed%2Fpattern-recognition%2Flacunarity.pdf&ei=zX98UPX8MMan0AGQv4C4DQ&usg=AFQjCNFdDkj0TcPsST9dUI0EObYXFdXkMw&sig2=Bzz1HgulR30mCTh89pIIKQ

Fractals

In continuing with the research theme for my paper, I stumbled upon this website, http://www.imaging.robarts.ca/~lgardi/html/what_are_fractals.html that explains what fractals are, where they can be found, and some links that they have to medicine. Since I am heading in the direction of how fractals are used in medicine for my paper, it is imperative that I gather the most basic information about fractals as well as the more detailed information. Not only can fractals be used to grade cancer, but there is a suggestion that the more intact your personal fractals are, the better you are mind, body, and spirit.

There are many different ways to use fractals in the medical world, so I will try to make my paper touch on some of the more important aspects of that, as well as some of the less conventional ways there are used. All the while, I will be included mathematical terms, equations, etc... to give a better understanding of how they are used.

Here is a video (simulated) that shows how the human bronchi can be called fractals: http://www.youtube.com/watch?v=jsb6iqbNC8s

Fractals And Cervical Cancer

In continuing with the theme of preparing for my final paper, I have chosen and article for my article review that uses fractals to find the grade of cervical cancer. Not only is cervical cancer the second most common cancer in women, but it is the third leading cancer related cause of death in women. So, I found this article interesting on so many levels. Not only are they using mathematics to identify the severity of cervical cancer, but the methods used and equations used don't seem that complex. If you would like to take a look at the article, as well as my 1 page review, feel free.   Journal Article 
Here is my 1 page summary of the article:

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Fractal Approach To Identify The Grade of Cervical Cancer

            The above named article was taken from the World Scientific Journal: Fractals, Vol. 19, No. 1 (2011) 125-139. This particular article discusses certain methods that involve using fractals to diagnose grades of cervical cancers. But, in order to understand what is being discussed, it is important to understand the two main topics: fractals, and cervical cancer.

            Fractal, by definition, is a geometric pattern that is repeated at ever smaller scales to produce irregular shapes and surfaces that cannot be represented by classical geometry. Cervical cancer is described as a disease caused by the abnormal growth and division of cells that make up the cervix. Since cancer arises from abnormal cell growth, there have been developments using mathematics to identify the grades. The most common method used for this is the Box Counting Method.

            The Box Counting Method is used in conjunction with MATLAB programme to find the dimensions of the cell variances for normal cells and abnormal cells.  This particular method is used because it can be applied to various sets of dimensions and patterns with or without self-similarity. The following equation is used to find the fractal dimension (D): D= , where  is the number of boxes of size S needed to cover the structure. The precancerous changes of the cells in the cervix are described by fractal analysis.

            To compute the actual fractal dimension, there are a few equations to use along with an algorithm (this will be discussed further in my actual paper). The image analysis using fractal approaches to characterize the growth of cancerous cells consists of covering an area with same-size, non-overlapping boxes. The number of boxes needed to cover the area is then plugged into D=, where D is the fractal dimension, K is a constant, and r is the size of the boxes. The relationship between r, N(r) is then used to plot points in a logarithmic scale to obtain D. This relationship helps to indicate the degree of complexity or dimensions of the fractal.

The Box Counting Method is then used with HarFA software to show the actual dimension (complexity of a fractal like structure/the cells growth) and intensity of the cell varies for normal and abnormal cells. Continuing, there is an Exponential Growth Model, Contact Model, Epidemic Model and Lacunarity (gap/size distribution of holes), that all help in determining the grade of the cancer. The Lacunarity finds the distribution and size of empty domains. For fractal dimensions, the higher the dimension, the higher the grade of cancer, and vice-versa.