<![CDATA[My Site - Blog]]>Tue, 22 Dec 2015 21:06:00 -0800Weebly<![CDATA[History of Math´╗┐]]>Wed, 04 Nov 2015 16:00:16 GMThttp://khadijahshaaf.weebly.com/blog/history-of-mathWhen asked about the history of mathematics it was quite embarrassing not to have an answer and I am a math education major. Throughout my studies in mathematics I have never specifically studied the history or origins of mathematics, only content in different areas of  math. So where did it all start? What was the sequential order of mathematical findings? Who has contributed to some of the greatest findings in mathematics?
As the video above stated within the first minute, we cannot start without knowing or considering the question, "What us mathematics?" Depending on what people think the definition of mathematics is will determine where to start on our timeline. Let's start at  the very beginning of time with prehistoric mathematics and work our way to 20th century mathematics.
  • Prehistoric Mathematics: Early man kept track of regular occurrences such as the season and phases of the moon. This is just simple counting and tallying techniques.
  • Sumerian/Babylonian Mathematics: Sumerian mathematics developed as a response to bureaucratic needs. Sumerians and Babylonians were the first people to assign symbols to groups of objects in an attempt to make the description of larger numbers easier. Babylonians developed a circle character for zero.
  • Egyptian Mathematics: The Egyptians created a decimal numerical system based on our ten fingers. There was a desire for the development of a notation for fractions. There is evidence that the Egyptians knew the formula for volume of a pyramid.
  • Greek Mathematics: Most of Greek mathematics was based on geometry. Thales established Thales Theorem. Pythagoras becomes a legend and is credited with coming up with the Pythagorean Theorem. The most important contribution of the Greeks was the ideas of proof.
  • Hellenistic Mathematics: Euclid invents Euclidean geometry. Archimedes produced formulas to calculate the areas of regular shapes. Diophantus of Alexandria was the first to recognize fractions as numbers.
  • Indian Mathematics: Zero is used as a number as opposed to just a place holder. Golden Age Indian mathematicians made fundamental advances in the theory of trigonometry.
  • Islamic Mathematics: The Islamic Empire used extensive, complex geometric patterns to decorate buildings. Al-Khwarizmi introduced the fundamental algebraic methods of reduction and balancing.
  • Medieval Mathematics: Fibonacci spread the use of the Hindu-Arabic numeral system throughout Europe. Fibonacci is best known for the Fibonacci Sequence.
  • 16th Century Mathematics: The equals, multiplication, division, radical, decimal, and inequality symbols were introduced and standardized. Tartaglia demonstrated a general algebraic formula for solving cubic equations. 
  • 17th Century Mathematics: Napier invented the logarithm, which was later improved by Napier and Briggs. Descartes developed analytic geometry and Cartesian coordinates. Fermat formulated several theorems that extend our knowledge of number theory. Pascal is famous for Pascal's Triangle of binomial coefficients. Newton and Leibniz developed the idea of infinitesimal calculus. 
  • 18th Century Mathematics: Lagrange is credited with the four-square theorem and Lagrange's Theorem. Legendre made important contributions to stats, number theory, abstract algebra and mathematical analysis. Lambert was the first to introduce hyperbolic functions into trigonometry. Euler produced the Euler Identity formula and Euler's Formula.
  • 19th and 20th Century Mathematics: Was a continuous trend of furthering other mathematicians findings, increasing generalization, and increasing abstractions.

All the discoveries and findings listed above still does not cover everything that happened within these time periods, but it is a start. Mathematics has been something that has continuously progressed over the years and will continue to do so. As seen from above, mathematics was and still is something we need to operate in our everyday living. It is how mankind has made sense of the world and logical reasoning of the way life began. 
Important Mathematicians:
  • 624-546 BCE-> Thales -> Greek -> Early developments in geometry, including work on similar and right triangles
  • 570-495 BCE ->Pythagoras -> Greek -> Expansion of geometry, rigorous approach building from first principles, square and triangular numbers, Pythagoras’ theorem.
  • 428-348 BCE -> Plato -> Greek -> Platonic solids, statement of the Three Classical Problems, influential teacher and popularizer of mathematics, insistence on rigorous proof and logical methods
  • 300 BCE -> Euclid -> Greek -> Definitive statement of classical (Euclidean) geometry, use of axioms and postulates, many formulas, proofs and theorems including Euclid’s Theorem on infinitude of primes
  • 287-212 BCE -> Archimedes -> Greek -Formulas for areas of regular shapes, “method of exhaustion” for approximating areas and value of π, comparison of infinities
  • 200-284 CE -> Diophantus -> Greek -> Diophantine Analysis of complex algebraic problems, to find rational solutions to equations with several unknowns
  • 598-668 CE -> Brahmagupta -> Indian -> Basic mathematical rules for dealing with zero (+, - and x), negative numbers, negative roots of quadratic equations, solution of quadratic equations with two unknowns
  • 780-850 CE -> Muhammad Al-Khwarizmi ->Persian -> Advocacy of the Hindu numerals 1 - 9 and 0 in Islamic world, foundations of modern algebra, including algebraic methods of “reduction” and “balancing”, solution of polynomial equations up to second degree
  • 1170-1250 -> Leonardo of Pisa (Fibonacci) -> Italian -> Fibonacci Sequence of numbers, advocacy of the use of the Hindu-Arabic numeral system in Europe, Fibonacci's identity (product of two sums of two squares is itself a sum of two squares)
  • 1350-1425 -> Madhava -> Indian -> Use of infinite series of fractions to give an exact formula for π, sine formula and other trigonometric functions, important step towards development of calculus
  • 1499-1557 -> Niccolò Fontana Tartaglia ->Italian ->Formula for solving all types of cubic equations, involving first real use of complex numbers (combinations of real and imaginary numbers), Tartaglia’s Triangle (earlier version of Pascal’s Triangle)
  • 1501-1576 -> Gerolamo Cardano -> Italian -> Published solution of cubic and quartic equations (by Tartaglia and Ferrari), acknowledged existence of imaginary numbers (based on √-1)
  • 1522-1565 -> Lodovico Ferrari -> Italian -> Devised formula for solution of quartic equations
  • 1596-1650 -> René Descartes -> French -> Development of Cartesian coordinates and analytic geometry (synthesis of geometry and algebra), also credited with the first use of superscripts for powers or exponents
  • 1601-1665 -> Pierre de Fermat -> French -> Discovered many new numbers patterns and theorems (including Little Theorem, Two-Square Thereom and Last Theorem), greatly extending knowlege of number theory, also contributed to probability theory

<![CDATA[Love and Math]]>Mon, 19 Oct 2015 17:47:24 GMThttp://khadijahshaaf.weebly.com/blog/love-and-math
I finally finished reading the book, Love and Math, by Edward Frenkel who is a professor of mathematics at the University of California, Berkeley. In this novel the author addresses a host of different mathematical concepts, some of which I have heard of before and others I never knew existed. Having a background in mathematics is not necessary in order to get through the book, but it was definitely beneficial while reading the novel because Frenkel talks about a lot of concepts that you'll have to follow along with in order to understand the point he is trying to make.
Typically, I do not and would not read a book such as Love and Math.  Yes, I am a Math major with emphasis in Secondary Education, but books that are heavily loaded with mathematical concepts, proofs, and formulas is not something I like to read in my spare time. However, this novel was the difference maker for me. Although Frenkel did present several mathematical ideas, the novel was not solely about math. Frenkel did an amazing job of describing becoming a mathematician back in his day, especially saying as though he was Jewish. He also did a bunch of referencing back to previous chapters in the event that the reader forgot what the topic was that he had discussed prior to bringing it up again in other parts of the book. In addition to all of these things, Frenkel did a great job describing why he loved math and how it became apart of his life. Love and Math took me on a mathematical journey that I've never seen before and in a manner that I could actually understand. Also, Frenkel somehow made connections to our current understandings of math. This was my favorite part about the book because I could relate or point out things that I have seen or even worked on in my years studying mathematics. 
Overall, Love and Math was a good read. I would certainly recommend people to read this book. However, one suggestion I would make is that the reader have some sort of mathematical knowledge or background; otherwise, there are times in the book where it is possible one could get lost or struggle through that particular chapter. If the reader does not have any mathematical knowledge or background, I would keep the internet up while reading in the event that the reader needs to look something up in order to understand what is going on. 
<![CDATA[Nature of Mathematics]]>Sun, 11 Oct 2015 11:24:37 GMThttp://khadijahshaaf.weebly.com/blog/nature-of-mathematicsAfter reviewing several definitions of mathematics from Webster's dictionary, peers, Wikipedia, universities, college professors,  and journals, the definition has boiled down to math being an abstract science or the study of numbers, space, quantity, change, structure, and patterns. Out of all the definitions that I read, math is referred to it as being "a science..." or "an abstract science." According to Google, science is the intellectual and practical activity encompassing the systematic study of the structure and behavior of the physical and natural world through observation and experiment. With this being said, why is there such a debate or opposing opinions on whether or not math is a science?  
Having a definite answer to whether or not math is a science is quite difficult for me to decide. However, I do believe that you cannot have one without the other. Science and math plays a vital role in our daily lives. Both topics are unique in their own way, but they have given us meaning to life. In other words, without science and math, we would not be able to have logical explanations of things or understand how the world around us works. I think the two subjects work hand in hand to supply us answers and supports to theories or questions we have about our world. 
Above is a video more so for humor as opposed to truth, but this is the battle I have with figuring out is math a science or is science a math. Understanding Science is website that was developed by the University of  California Museum of Paleontology to help with the understanding of science and how it works. According to Understanding Science, there is a science checklist of all the things science aims to do or accomplish and this list is compared to mathematics. The list consists of science focusing on the natural world, science aims to explain the natural world,  it uses testable ideas, relies on evidence, leads to ongoing research, and researcher behave scientifically. In the article, it is stated that mathematics does not necessarily focus on the natural world, but "mathematical abstractions arise directly from the natural world." In addition to the previous statement, mathematics helps us understand and explain the natural world. The example given is the fact that Isaac Newton''s laws of motion was made possible by the advances he made in calculus. Futhermore, science uses testable ideas; whereas math consists of testable ideas just not against evidence in the natural world, as in biology, chemistry, and physic. In math we can easily test a conjecture of whether or not an odd number plus an odd number equals an even number by computation. If we find one example where this is not the case, the conjecture would be invalid because we have that strong opposing evidence. In science, there are many factors to consider when posed with a new hypothesis. One counterexample is not enough to say a hypothesis is true because scientific ideas can never be absolutely proven. Moving forward, researchers in math and science behave scientifically. To put more simply, scientist and mathematician build upon each others work, share ideas/results, respond and act respectively on criticism, and are honest in their work (they give credit where  credit is due).
Above is a video that asks a panel of individuals, is math invented or discovered? Before actually watching the video, I automatically assumed that math was invented. I then asked myself, "is science invented or discovered?" Based off of my knowledge of science being the natural world around us, I assumed it was discovered. Yes, math is also a make up of the world around us, but it typically deals with some sort of computation or abstract idea. This new question gave me a different perspective when looking at the original question, is math a science? I still have not made an finalized decision, but I definitely have considered the question in a different point of view. Check out the video! Do you think math is discovered or invented? Do you think science is discovered or invented? Does this change your opinion at all about math being a science or vise versa?
Noted 12/12/15 -After several weeks of discussing is math a science, I've concluded that math is not a science. Math and science have distinct components, which separates the two. However, I still believe math  plays a role in science and there would be no way to complete science without math (Example: physics and chemistry).
<![CDATA[Doing Mathematics]]>Sun, 27 Sep 2015 10:55:56 GMThttp://khadijahshaaf.weebly.com/blog/doing-mathematicsThe biggest questions that lie within doing mathematics is "How do you do math?" and "What is considered doing math?" Being a future educator in the field, one would think I had an immediate answer; however, this is not the case. I actually struggle with answering such a broad questions because of the variety of responses that could be accepted as an answer to these questions depending on the circumstances. For instance, how do you do math could involve doing math research, doing math functions, doing math fractions, doing math expressions, doing math percentages, and the list could continue. Additionally, trying to predict what is considered doing math is circumstantial because someone could say they did math today just because they drew a triangle, went to the mall for bargain shopping, or created a proof for the number zero existence. 
Above is a picture of how some individuals "do" math. Although I found this to be humorous because it explains me working through a challenging problem, I think it is a true statement. No, I don't consider reading a problem and crying doing math, but this is as far as some people get when attempting to solve a math problem. 

On the other hand, the picture above shows that some mathematicians do math by solving various calculations. These can be complex equations, simple equations, geometry problems, proofs, and apparently how to efficiently catch a mouse. All in all, the answer to the big questions depends on the math related topic. 
So, how do I do math? And what is considered doing math to me? To answer the preceding questions, I took a day to evaluate my daily experiences of doing math and at the end of the day I determined whether those things would be considered doing math. Until September 24, 2015, I had never paid attention to how much math I did outside of my math courses. On Thursday I  just had a Climate class and work at Dick's Sporting Goods and my day started off with me doing math. After waking up for class that morning, I predicted the time it would take me to shower, brush my teeth, get dressed, do my hair, drive to campus, find parking, and make it to class on time. Every morning, I am positive most people calculate a similar process to make it to where ever they are headed. This is considered doing math to me because you have to add up all the time it takes to do a normal morning routine and traffic to make it to your destination on time.  
After class, I went to work where I am currently a cashier. Technically, I do not do much math throughout my shifts because the computer does all the calculations for me. However, on this day I noticed how many random math related questions I was asked by customers and coworkers.  For example, a customer noticed we had a 25% off sale on select Nike apparel and she asked me what would her total be if she purchased a $55 shirt and $40 pants. As always, I get excited to help people with anything math related and I have her pull out her calculator. We entered (55+40)(.75) to calculate the sale price before taxes which was $71.25 and then multiplied that number by 1.06 and got $75.53 as her total. To her surprise, method was much shorter than what she was going to do and she was highly satisfied that the price I gave her matched what the computer said when she finally checked out. Another incident occurred during this shift when a coworker of mine asked me, " hey Khadijah, aren't you a math major?" I responded yes and he said, "Cool, I have a riddle for you to solve. There are 30 cows in a field, 28 chickens, how many didn't?" I was beyond confused! How could I answer such a question? I was convinced there was missing information. Being a math major, I started brainstorming and trying to think of the most logical answer. After about 20 minutes of coming up with nothing and throwing out random answers, I finally gave up. Then my coworker says to me, "people who love math always get stuck on this question when I ask." I was thinking to myself, well yeah because there is missing information. Finally, he writes down the question instead of asking and it read, "There are 30 cows in a field, 20 ate chickens, how many didn't?" At this point I knew the answer and was so amazed when I discovered the mistake most people make when they are asked the question. As the day went on, I continued to notice the math I unconsciously do on a daily. 
All in all, my lesson learned from this is that math can be done in multiple ways and a lot of math can be done without thought. This lesson also put into perspective for me why teachers always tell students they need math and students do not agree. Well, it is indeed necessary to possibly help a customer figure out sales prices, solve a riddle,  time daily routines, cry, calculated catching a mouse, and the list goes on. In general, math goes beyond what I described above. This was just my reaction to many peers and students I have encountered that hate math, think we don't use it, or don't need it. My goal was to show how I use it just in my daily living, whether or not I'm in a math class. 
<![CDATA[Communicating Math´╗┐ematics]]>Wed, 16 Sep 2015 16:16:12 GMThttp://khadijahshaaf.weebly.com/blog/communicating-mathematicsIn my experiences as a pre-service teacher, I have found that communicating math is a unique experience because every learner brings different things to the table (i.e. prior and background knowledge). Thus, making it sometimes easier and sometimes harder to explain mathematical concepts. Below is a math activity, Count the Squares, that I used to see how it would be communicating mathematics to two friends that are around the same age as myself and have not had a math course since freshman year of college. The handout that I gave my friends had three questions. The first question asked, "What is a square?" The second question asked them to count the squares in the figure below. Then there was a break where I, the facilitator, gave each friend the definition of a square. Finally, the third question asked for a final answer on how many squares are in the figure. 

Before giving my friends the handout I provided them with pens, pencils, and markers to use to assist them in counting the squares. Both neglected to use the materials provided. Friend number one defined a square to be " four lines connecting that make a box" and got 31 squares total as her initial answer. Friend number two defined a square to be "a polygon with all right angles and equal sides" and got 43 squares total as her initial answer. Based on the definitions given, I anticipated that friend number two would have been closest to the actual answer because she had a clearer understanding of the definition of a square. 
In between question two and three, I paused my friends to give them my definition of a square which is "a parallelogram with four equal, straight sides and four interior right angles." After the definition was given I gave each friend the opportunity to recount and provide a final answer. Friend number one counted 36 squares and friend number two kept her answer at 43 squares. Over the duration of the task, neither friend felt like this was a trick question; instead, they were intrigued by it and was anxious to know how many squares were in the picture. At the conclusion of the activity, I revealed that I counted 39 squares and we watched the video below to find out the true final answer. 

Unfortunately, we were apart of the 92% of people who fail this simple test. However, we all learned something from the problem. Friend number one expressed to me that knowing my definition of a square helped her to see that there were more squares than she originally thought, but she neglected the 3-by-3 squares because they looked like rectangles. At this point, we went over the definition of a rectangle in comparison to a square and she understood her mistake. Friend number two expressed to me that she should have used the different materials I proved because she counted some squares twice and the different colors would have minimized double counting. As for myself, I counted three 3-by-3 squares and simply slowing down would have fixed my counting error. The two squares in the middle is what makes the problem most interesting to me. It adds on 10 squares and if you are not carefully counting they can be missed.
All in all, I learned that communicating math is going to be difficult. Nevertheless, teachers should try to present materials in various ways and make sure students have a clear understand of all the parts of an assignment.
Above is a picture of friend one work on the right and friend two work on the left.
<![CDATA[What Is Math?]]>Wed, 09 Sep 2015 03:32:04 GMThttp://khadijahshaaf.weebly.com/blog/what-is-mathMathematics is the study of, shapes, quantity, space, change, and logic. It is also a combination of patterns that helps us understand and function in the world we live in today. As a future educator in mathematics, it is important for me to stress to my students that math is much more than addition, multiplication, subtraction, and division.  Since the beginning of time, mathematics has been something that impacts our daily living and something we cannot live without. It is my goal to have students leave my classroom knowing how math is used in their daily lives. Teaching math is very difficult because most students I have encountered see no use in the topic and dislike it. However, I want to turn the tables and make math relatable to my students by application. ]]>