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This learning object introduces common privacy intrusions on the Internet and then introduces the fundamentals of network security.

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Educational Technology and Mobile Learning

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Math teachers would love what I am sharing with them today. Below is a wonderful visual featuring some of the brightest mathematicians of all time. However, the word mathematician in the title of this visual is a little bit tricky. In ancient times and before the categorization of knowledge into different sciences and branches "logic" and philosophy were the reigning fields of human knowledge under which everything else is subsumed and studied including math, astronomy, biology, physics, anatomy...etc. Therefore the figures featured in the list below were not only mathematicians in the literary sense of the word but were also great philosophical thinkers ( Renes Descartes ) and Physicists ( Archimedes ). They are very much popular for their prominent contribution to the field of mathematics.

I thought you might want to print off this visual and hang it on your classroom wall. This graphic is created by Studygeek and I learned about it from visualizingmath.

Check out the full graphic in this page.

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A Beautiful Math Poster Featuring The Greatest Mathematicians of All Time

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Math teachers would love what I am sharing with them today. Below is a wonderful visual featuring some of the brightest mathematicians of all time. However, the word mathematician in the title of this visual is a little bit tricky.

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Mrs. Milkowich's Wiki - Famous Mathematician Notebook Cover Project

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Healthcare Whiz: Promoting Healthcare Consumer Education

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In addition to fee-for-service, which continues to be the dominant reimbursement model in the United States, other methods exist and new models are emerging as the healthcare industry attempts to deal with sky-rocketing medical costs. Here we'll look at capitated reimbursement models and accountable care organizations.

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Stock Illustration - Man juggling math symbols under numbers with healthcare symbols in distance

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Philosophical thought in 19th century Europe inspired citizens of nation-states to develop a sense of nationalism. Poland existed as a nation but not as a state. As World War I came to an end and Poles fought against their Germanization and Russification, developments in mathematics helped establish Polish identity. During the interwar period of 1918-1939, Poland won its struggle for international recognition with the help of Zygmunt Janiszewski, a talented mathematician. Janiszewski's politically informed proposal, with the help of various schools of mathematics within Poland, led to the creation of a specialized and internationally recognized journal: Fundamenta Mathematicae. The journal concentrated on set theory and functional analysis. The many research journals that currently encourage the submission of specialized articles, use the powerful branches of mathematics that were developed in Poland during the interwar period.

Zygmunt Janiszewski proposed the use of mathematics as a means for Poland to gain international recognition. In his 1918 publication "O Potrzebach Matematyki w Polsce" ["On the Needs of Mathematics in Poland"] in Nauka Polska, Janiszewski states his goals to the Polish mathematical community as follows: "to distinguish [itself] and to win an independence" (1). Janiszewski's publication of Guidebooks for Self-Instruction, further supported his ideas. He produced these books in response to Russian restrictions on teaching mathematics in secondary schools (2). Janiszewski exemplified his main idea in the following statement: "it is necessary to transform strictly scholarly periodical literature into more specialized ones: for example, one journal would be devoted to number theory and algebra" (3). He foresaw Poland's political success through specialization in fields of mathematics stating: "We must seize radical means, examine the basis of failure, we must create a research center in Poland! We can attain this only by concentrating the majority of our mathematicians into the work of a single branch of mathematics" (4). Revolutionary in nature, the publication of this article caused a commotion among the community of foreign scholars, since they likened specialization to a force that could ruin the reputation of Poland (5). In rejecting these warnings, Janiszewski asserted that Poland's existence would continue through the ideas of talented Polish mathematicians.

The publication of Fundamenta Mathematicae in 1920 sent shockwaves through the international mathematical community. In a form slightly different from Janiszewski's proposal, Stefan Mazurkiewicz and Wacław Sierpinski completed and published the first volume of Fundamenta Mathematicae with articles on set theory, written solely by Polish authors. This highly specialized journal gained immediate international acceptance. In September 1921 Raymond Clare Archibald, an editor for the American Mathematical Monthly, stated, "Of the ten mathematical periodicals started since January 1919, none are of such notable importance for mathematical research as Fundamenta Mathematicae " (6). Not only did subsequent volumes contribute to an increased number in articles from abroad, but they also included the development of two additional specialized branches of mathematics - topology and functional analysis (7). The mathematical community responded with caution to these developments in specialized mathematical concepts; nonetheless, Janiszewski's plan proved to be successful in Poland.

Mathematicians outside the circle of Janiszewski, Mazurkiewicz and Sierpinski were skeptical of the longevity of highly specialized mathematics and journals. Henri Lebesgue, a French mathematician, doubted that the journal could attract specialized material over an extended period; however, prominent foreign schools of mathematics soon recognized the importance of Fundamenta Mathematicae. (Although Lebesgue had little faith in the future of the journal, he submitted an article for publication because he believed in the development of specialized mathematics.) By the twenty-fifth volume of Fundamenta Mathematicae, The Bulletin of the American Mathematical Society stated that the journal's continuation represented "a notable event in the mathematical life of the whole world" (8).

Janiszewski proposed specialization of mathematics in "On the Needs of Mathematics in Poland" in response to the loss of national pride that Poland experienced through its century long occupation. His novel idea to push for international acclaim in mathematics as a way to gain independence required that Polish mathematicians master specialized fields in mathematics and eventually publish their ideas in an international journal. Fundamenta Mathematicae brought Polish mathematics to the international community. The works of Polish mathematicians in Fundamenta Mathematicae not only led to a phenomenon that is seen in most mathematical journals today but also led to a revival in Polish national culture.

Footnotes

1. Zygmunt Janiszewski, "O Potrzebach Matematyki w Polsce," Nauka Polska, Mianowski Foundation, 1918.

2. Roman Duda, "On the Warsaw Interactions of Logic and Mathematics in the Years 1919-1939," Volume 127, Annals of Pure and Applied Logic, 2004, p. 284.

3. Janiszewski, "O Potrzebach."

4. Sister Mary Grace Kuzawa, CSFN, Modern Mathematics, College & University Press Services, 1968.

5. Edward Marczewski, "Swieto Matematyki Polskiej," Wiadomosci Literackie, 1936.

6. Kuzawa, "Modern," p. 93.

7. Jacob Tamarkin, "Twenty-Five Volumes of Fundamenta Mathematicae," Bulletin of the American Mathematical Society, 1936.

This article is reprinted by permission from Good News magazine, a publication of the American Institute of Polish Culture.

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Mathematics and Polish National Identity

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In 1918, the noted Polish mathematician, Zygmunt Janiszewski argued that Poland's existence would continue through the ideas of talented Polish mathematicians. Joseph Pomianowski agrees, noting that Janiszewski's Fundamenta Mathematicae contributed both to mathematics and to the revival of Polish national culture.

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Mastin, L. (2010). 19th Century Mathematics. Retrieved from http://www.storyofmathematics.com/19th.html. Lewinter, M., & Widulski, W. (2002). The saga of mathematics: A brief history. Upper Saddle River, NJ: Prentice Hall. Shanholtzer, M. B. (2015). Integrated electronic health records: A worktext for Greenway Medical Technologies' PrimeSUITE® (2nd ed.). Columbus, OH: McGraw-Hill. Gartee, R. (2011). Electronic health records: Understanding and using computerized medical records (2nd ed.). Upper Saddle River, NJ: Prentice Hall. Green, M. A., & Bowie, M. J. (2011). Essentials of health information management: Principles and practices (2nd ed.). Clifton Park, NY: Delmar/Cengage Learning. NRICH Team. (2015). The Secret World of Codes and Code Breaking. Retrieved from http: //nrich.maths. org/2197. Raines, C. (2015). How to Use Math in Health Care Careers. Retrieved from http://work.chron.com/use-math-health-care-careers-26310.html. History of Math: Mathematics in the 19th Century. 2011. Retrieved from https: historyofmath. wordpress.com /2011/11/16/mathematics-in-the-late-19th-century/.

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In the mid-19th Century, the British mathematician George Boole devised an algebra which is now called Boolean algebra or Boolean logic. Within this algebra, the only operators were AND, OR and NOT, and which could be applied to the solution of logical problems and mathematical functions. He also described a kind of binary system which used just two objects, "on" and "off" (or "true" and "false", 0 and 1, etc.), in which, famously, 1 + 1 = 1. Boolean algebra was the starting point of modern mathematical logic and ultimately led to the development of computer science.

The concept of number and algebra was further extended by the Irish mathematician William Hamilton, whose 1843 theory of quaternions (a 4-dimensional number system, where a quantity representing a 3-dimensional rotation can be described by just an angle and a vector). Quaternions, and its later generalization by Hermann Grassmann, provided the first example of a non-commutative algebra (i.e. one in which a x b does not always equal b x a), and showed that several different consistent algebras may be derived by choosing different sets of axioms (Mastin, 2010, 19th Century Mathematics).

The Englishman Arthur Cayley extended Hamilton's quaternions and developed the octonions. But Cayley was one of the most prolific mathematicians in history. He was a pioneer of modern group theory, matrix algebra, the theory of higher singularities, and higher dimensional geometry who anticipated the later ideas of Klein, as well as the theory of invariants.

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Joseph Fourier's study, at the beginning of the 19th Century, of infinite sums in which the terms are trigonometric functions were another important advance in mathematical analysis of this time period. Periodic functions that can be expressed as the sum of an infinite series of sines and cosines are known today as Fourier Series, and they are still powerful tools in pure and applied mathematics.
German prodigy Carl Friedrich Gauss, sometimes called the “Prince of Mathematics”, received his education at the prestigious University of Göttingen during this time. Some of Gauss’ ideas were a hundred years ahead of their time, and they touched on many different parts of the mathematical world, including geometry, number theory, calculus, algebra and probability. He is widely regarded as one of the three greatest mathematicians of all times, along with Archimedes and Newton. He also claimed to have investigated a kind of non-Euclidean geometry using curved space but, unwilling to court controversy, he decided not to pursue or publish any of these mathematical ideas. This left the field open for János Bolyai and Nikolai Lobachevsky who were considered respectively, Hungarian and Russian, and who both independently explored the potential of hyperbolic geometry and curved spaces.

The German Bernhard Riemann worked on a different kind of non-Euclidean geometry called elliptic geometry, as well as on a generalized theory of all the different types of geometry. However, Riemann soon took this even further by breaking away completely from all the limitations of 2 and 3 dimensional geometry, whether flat or curved, and began to think in higher dimensions. His exploration of the zeta function in multi-dimensional complex numbers revealed an unexpected link with the distribution of prime numbers, and his famous Riemann Hypothesis. After 150 years, it remains one of the world’s great unsolved mathematical mysteries and the testing ground for new generations of mathematicians.

Charles Babbage in 19th Century England designed a machine that could automatically perform computations based on a program of instructions stored on cards or tape. His large ‘difference engine’ of 1823 was able to calculate logarithms and trigonometric functions, and was the true forerunner of the modern electronic computer. Although never actually built in his lifetime, a machine was built almost 200 years later to his specifications and worked perfectly. He also designed a much more sophisticated machine he called the ‘analytic engine’ which was complete with punched cards, printer and computational abilities commensurate with modern computers.

Englishman, George Peacock, is usually credited with the invention of symbolic algebra, and the extension of the scope of algebra beyond the ordinary systems of numbers. This recognition of the possible existence of non-arithmetical algebras was an important stepping stone toward future developments in abstract algebra.

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Hermann Minkowski was a great friend of David Hilbert and teacher of the young Albert Einstein who developed a branch of number theory called the ‘geometry of numbers’ in the late 19th Century as a geometrical method in multi-dimensional space for solving number theory problems. It involved complex concepts such as convex sets, lattice points, and vector space. Later in the year 1907, it was Minkowski who realized that the Einstein’s 1905 special theory of relativity could be best understood in a four-dimensional space, often referred to as Minkowski space-time.
Gottlob Frege was the first to explicitly introduce the notion of variables in logical statements, as well as the notions of quantifiers, universals and existentials. He extended Boole's ‘propositional logic’ into a new ‘predicate logic’ and in doing so, he set the stage for the radical advances of Giuseppe Peano, Bertrand Russell and David Hilbert in the early 20th Century.
Henri Poincare was the first to explicitly introduce the notion of variables in logical statements, as well as the notions of quantifiers, universals and existentials. He extended Boole's ‘propositional logic’ into a new ‘predicate logic and, in so doing, set the stage for the radical advances of Giuseppe Peano, Bertrand Russell and David Hilbert in the early 20th Century. He was also an engineer and a polymath, and perhaps also the last of the great mathematicians to observe an older conception of mathematics, which championed a faith in human intuition and formalism. He is sometimes referred to as the ‘Last Univeralist’ as he was perhaps the last mathematician able to shine in almost all of the various aspects of a huge, encyclopedic and incredibly complex subject.

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The wide variety of health care professionals that form part of the health care industry all carry out different tasks and functions. For example, accountants calculate the profits and losses of health care facilities, while administrators take care of the finances. Traditionally, medical centers count revenue when the patient is admitted or receives outpatient services. When accountants report revenues based on cash, they must tally insurance reimbursements, out-of-pocket payments, and government assistance. In addition, donations add to revenue, and administrators use mathematical skills to prepare budgets and reports for owners and investors (Raines, 2015, How to Use Math in Health Care Careers). As a result, they must also be familiar with penalties; civil monetary and federal criminal penalties within the health care industry. For example, health care professionals must make accurate calculations with the information already established: civil monetary penalties of $100 per violation, may go up to$25,000 per person, per year for each requirement or prohibition violated.

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Furthermore, the billing claim forms coding system is also an important process within the health care industry. The International Classification of Diseases-10th revision, Clinical Modification/Procedure Coding System (ICD-10-CM/ PCS) is the classification system used to convert narrative diagnoses and procedures into numeric codes. It became effective on October 1st of the year 2014, and it replaces ICD-9-CM. In addition and similarly to the ICD-10CM/PCS, the Current Procedural Terminology (CPT) is a coding system used to convert narrative procedures and services into numeric form. CPT is used to code procedures and services in a physician’s office. In a hospital setting, it is used for outpatient coding in locations such as the emergency room, outpatient diagnostic testing, or ambulatory surgery. These steps must be taken by the health care professionals in order to submit bills to health insurance carriers in which case a claim form must be generated for each visit. This is done by compiling the patient’s identifying information, insurance information, and the ICD-10-CM/ PCS and CPT codes into a form called the CMS-1500, which is used by physicians’ offices, or the UB-04, which is used to bill hospital claims (Shanholtzer, 2015, p. 3).

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Each PKI (arrangement that binds public keys with respective user identities by means of a certificate authority CA) user has two keys, a private key for signing documents and a public key for verifying his or her signature. A computer software program performs a mathematical calculation on the entire contents of the electronic document to be signed. The result is a unique ‘message digest’ which is then encrypted using the ‘private’ key. The digital signature is then attached to or sent with the document. When the recipient wishes to validate the signature, a similar computer program regenerates the ‘message digest’ and decodes the digital signature with the public key. Comparing the two, the program determines if the message digest isidentical to that which was originally sent. In this way digital signatures not only confirm that you are the signer but also that the document has not been altered since it was signed (Green & Bowie, 2011, p. 410).

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Encryption and Decryption Encryption is a method of converting regular text into code. The original message is encrypted by means of a mathematical formula called an algorithm. The receiving party uses a key to convert (decrypt) the coded message back into plain text. Encryption is part of access control because it prevents someone without the key from viewing or using the information. Encryption uses a mathematical algorithm to convert readable data into encoded or scrambled data. The authorized recipient decrypts the message back to its original form using a mathematical key.

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The performance, accuracy, and efficiency of a health data analysts is essential within the health care industry. Their profession/job consist of important tasks such as collecting and crunching numbers to help medical facilities reach their ultimate goal; providing each and every patient with the best high quality health care service. The data may include wait, and service or recovery times and instances of multiple appointments and visits for the same or similar conditions. Based on the figures and mathematical models, health care professional such as analyst, administrators, and physicians may implement time-saving and more efficient techniques and scheduling in order to experience improvements for both, patients and health care professionals. In addition, statistical models also help analysts identify the factors that determine the length ofrecovery time and success of recovery periods from various diseases and conditions.

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Mathematics is also critical to helping anesthesiologists establish and maintain safe solutions, and the right levels of oxygen for surgical their patients. Anesthesiologists consider variables such as patients’ weight, the desired drug or solution dosages and the amount of dilution needed so that the active chemical is not too strong. A successful anesthesia practice requires command of converting English to metric units, such as pounds to kilograms. In addition, facial and plastic surgery practices, physicians require further assistance to make sure they get the job done accurately in the OR. For this specific reason, health care professionals run efficient math-generated computer models to project the results of procedures. The database system makesall of the accurate calculations for them and as a result, they don’t take any chances or risks that will put the patient in danger.

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Within the health care environment, nurses must translate medication orders into the right doses and number of pills to administer them to the patients with accuracy. Essentially, the number of pills needed equals the dosage desired divided by the dosage the hospital has on hand. For example, if a doctor orders half a gram of a drug every 12 hours and the hospital has 250 milligram pills, the nurse must divide 500 milligrams (0.5 grams multiplied by 1,000) by 250 for two pills every 12 hours. Using these formulas often requires the nurse or pharmacy technician to convert units of measurement, such as grams to milligrams and therefore, he or she (health care professional) must know, understand, and be familiarized in order to utilize, basic mathematical principles such as addition, subtraction, multiplication, division, and conversion of measurements/quantities; the lives of his/her patients depend on it.

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Vital signs consist of an individual’s temperature, pulse rate, breathing rate and blood pressure; these serve as our bodies’ dashboard indicators. Medical assistants and nurses must know the units of measurement for these signs and how to read the instruments that measure them. For example, non-digital thermometers use a long line to represent one degree Fahrenheit and small notches for 0.2 degrees. For blood pressure gauges, each large line equals 10 mm Hg (millimeters of mercury) and each small line stands for 2 mm Hg. A health care professional must have proper knowledge of mathematical measurements and terminology in order to prevent/avoid errors that may be detrimental to the patient’s health. In the same way, BMI stands for Body Mass Index and it is a number that shows body weight adjusted for height. BMI can be calculated with a simple math formula wt/ht2 that may be equally applied to either English or metric measurements (weight/wt pounds and height/ht inches) or (weight/wt kilograms and height/ht meters) (Gartee, 2011, Chapter 5).

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Today, the combination of rapid technology, encryption, and codes, clever algorithms are essential for quality performance and functions in many health care environments. Mathematics plays a vital role in the health care industry for all health care professionals within the working environment. Medical providers must obtain reliable data and calculations to prevent, diagnose and treat medical problems. Therefore, mastery of the tools of health care, units of measurements and formulas whether they are scientific and/or financial,they promote the efficient and profitable delivery of services, and decrease the risk of medical mistakes that can turn into potential tragedies and malpractice lawsuits.

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A few of the first people who used cryptography were Julius Cesar and 1600 years later, Queen Elizabeth to transmit secrets and important information. Nearly 2000 years ago, Julius Caesar was busy taking over the world, invading countries to increase the size of the Roman Empire. He needed a way of communicating his battle plans and tactics to everyone on his side without the enemy finding out. So Caesar would write messages to his generals in code. Instead of writing the letter 'A', he would write the letter that comes three places further on in the alphabet, the letter 'D'. Instead of a 'B', he would write an 'E', instead of a 'C', he would write an 'F' and so on. When he got to the end of the alphabet, however, he would have to go right back to the beginning, so instead of an 'X', he would write an 'A', instead of a 'Y', he'd write a 'B' and instead of 'Z', he'd write a 'C'. When Caesar's generals came to decipher the messages, they knew that all they had to do wasgo back three places in the alphabet (NRICH Team, 2015, The Secret World of Codes and Code Breaking).

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Codes and encryptions are also used today within our diversified society to perform daily routine activities that require safe and secure transactions. For example, they are useful when making online credit card payments and ATM withdrawals. Furthermore, within the twentieth century there were also experiences, operations, and research which drew on many mathematical traditions offering solutions to problems that seemed impossible. Among the many benefits, codes and encryptions provide the ability to perform millions of calculations in a matter of seconds; an advantage we constantly use and rely upon today for innumerable tasks, functions, and activities. This is a great accomplishment because the truth is that mathematics has been present in every event of human history in one way or another. It was a necessity and requirement for many individuals’ goals to complete the simplest tasks and the most complex endeavors throughout history; truly indispensable to and for all.

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In the nineteenth century, mathematics continue to advance to more complex systems and methods due to the rapid growth and incorporation of technology. Binary arithmetic for example, is essentially important because cables carry binary signals around the world representing pictures, music, texts, and all other types of media that we rely upon in our day and age (Lewinter, & Widulski, 2002, p. 237). In the same way, the use of number theory in cryptography consists of the science of coding and decoding information. This particular mathematical system/method is used by very prestigious organizations and companies to send and receive important informative details regarding a wide variety of confidential and non-confidential information. Although this mathematical system is as ancient as Rome, Julius Cesar was able to use it efficiently; the United States hasdone the same. During very important times and events such as WWII in which confidential information and codes were transmitted.

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“The High Middle Ages was a time of growth and awakening” (Lewinter & Widulski, 2002, p. 110). During this time period in Europe, many new ideas emerged which required the development products such as gun powder, mechanical clocks, and spectacles. As a result, the demand for commercial arithmetic mathematics increased, especially once the European got the news of new mathematical ideas such as Hindu-Arabic numeral system. In addition, merchants in Italy began to see how and why mathematics was now a necessity rather than an option for creative entertainment. Within these time periods in Europe, people began to realize the significant meaning and value of mathematics and education; they were essential to the development of mathematics of that particular region as they began to see a connection between mathematical knowledge and economical, academic, and social progress, success, and superiority.

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Although ancient mathematical numbering systems such as hieroglyphics may not be as convenient for us in our modern day and age, it was pretty useful and convenient for the individuals in that time period. According to Lewinter & Widulski (2002) this numbering system “enabled the Egyptians to write very large numbers describing vast quantities of food, soldiers, slaves, or livestock” (p. 4). Hieroglyphics and the dyadic Egyptian multiplication method were just a couple of mathematical systems from which basic math principles such as doubling, multiplication, and the distributive property came from, methods we continue to use and rely upon to this day because these mathematical concepts go a long way. They were the beginning of an extraordinary process through which technology would be secured and expanded. The Egyptians much like today, utilized mathematics for more than just one thing, they used it for labor, agriculture, architecture, personal, and military purposes; it continually developed in different locations and though various individuals all over the world.