To elaborate on this more, we need to define the terminology first. Discovered is the thing always exist. Someone found it. Invented is the thing did not previously exist. Someone created it. What about mathematics? Mathematics is a tool, a model. It is something that we can use to describe or model parts of reality, or any other system based on quantifiable things. So, is mathematic being created or discovered?

People have debated about this over the years and even now. Some people say that mathematic is being created and some say that people discovered math. As math is a model to describe things, and a model is created, so is math. Mathematics itself did not exist before the first mathematician. But, from this can we concluded that math is ultimately being created? I don’t think so. I do think mathematics is also discovered.

Looking at this example might trigger our mind to think more about it. Eclipse is the example. Have you ever thought how human being can determine when eclipse is about to happen very precisely? By using a set of statistic and also calculation, mathematicians are able to predict when the next eclipse would occur. The idea to calculate the eclipse comes from the existing phenomenon. My point is, mathematicians are able to invent a model to describe the reality by observing the existing situation, which is being discovered. So, I believe mathematics is invented but the discovery of new things help it to be invented. Mathematics is both invented and discovered.

...“Is mathematicsdiscovered or invented?” To commence with this essay, we must first understand a few key words used in this statement and question. ‘Mathematics’ is generally believed to be the body of knowledge centered on concepts such as quantity, structure, space, and change, and also the academic discipline that studies them. Whereas ‘discover’ and ‘invent’ means to find information, a place or an object, especially for the first time and to design or create something which has never been made before respectively. Therefore this question can be rephrased. The body of knowledge centered on concepts such as quantity, structure, space, and change is being found or created?
Actually, this is a great philosophical question. Mathematics is like a religion. Though it appears to be a very precise and concrete subject with many proofs to support with, for example like the Pythagoras’s theorem to support the a relation in Euclidean geometry among the three sides of a right triangle, it is not always how it seems. We never know whether the so called proof is correct. What if one day somebody simply proved this theory wrong? There could be many fallacies behind.
Generally, there are two streams of people who believe in different sides of mathematics. One believes that there are universal principles that just "are" and what we as humans are doing are to discover them one by one. And obviously...

...Areas of Knowledge: Invented, Discovered, or Both?
One cannot say with certainty that specific areas of knowledge are invented nor can one say with certainty that areas of knowledge are discovered. Two areas of knowledge that deal with this topic are mathematics and natural science. In my opinion, I believe that mathematics is an invented area of knowledge with some discovery whereas science is a discovered area of knowledge with some invention.
Certain topics and categories of mathematics might be created based on the need during the time. For example, accountancy was invented due to the fact that people needed a way to keep track of profits (Friedlob and Plewa, 1). Accounting is a form of math used to record data regarding business-like matters. The earliest form of accountancy was in Mesopotamia. It arose more than 7,000 years ago (Friedlob and Plewa, 1). This form of math was used among people in order to record the growths of their crops and also keep track of their herds of animals. The people during the time used clay coins, or tokens, in order to keep track of the information (Friedlob and Plewa, 1). This form of math is an example of how math is invented. Accountancy was invented based on a need of a process or way to keep track of certain information.
Not only are subtopics in...

...History of mathematics
A proof from Euclid's Elements, widely considered the most influential textbook of all time.[1]
The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past.
Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available arePlimpton 322 (Babylonian mathematics c. 1900 BC),[2] the Rhind Mathematical Papyrus (Egyptian mathematics c. 2000-1800 BC)[3] and the Moscow Mathematical Papyrus (Egyptian mathematics c. 1890 BC). All of these texts concern the so-calledPythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.
The study of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greekμάθημα (mathema), meaning "subject of instruction".[4]Greek mathematics greatly refined the methods (especially through the introduction of deductive reasoning andmathematical rigor in proofs) and expanded the subject matter of mathematics.[5] Chinese...

...concrete model.
Looking on the locality of the paper, I highly acknowledge the fact that the researchers described the current state of math education in the Philippines. They emphasized the fact that we are more focused on procedural knowledge rather than the more desired conceptual knowledge. That is our disadvantage because we usually train students to perform math without understanding or making connections on what they are doing. By mentioning this, the readers would really have an idea that the paper itself could be a solution to the problem mentioned. Moreover, it makes the thesis more realistic.
To sum up everything that was tackled, I could say that the thesis served to have an important purpose in the current state of Mathematics Education in the Philippines. It is very informative and feasible. Since it is a small study because it only involved 6 average students, we could propose more studies rooting from this which would have a bigger scope such as implementing the same study but now comparing it to the results gathered from high and low performing students....

...HISTORY OF MATHEMATICS
The history of mathematics is nearly as old as humanity itself. Since antiquity, mathematics has been fundamental to advances in science, engineering, and philosophy. It has evolved from simple counting, measurement and calculation, and the systematic study of the shapes and motions of physical objects, through the application of abstraction, imagination and logic, to the broad, complex and often abstract discipline we know today.
From the notched bones of early man to the mathematical advances brought about by settled agriculture in Mesopotamia and Egypt and the revolutionary developments of ancient Greece and its Hellenistic empire, the story of mathematics is a long and impressive one.
Prehistoric Mathematics
The oldest known possibly mathematical object is the Lebombo bone, discovered in the Lebombo mountains of Swaziland and dated to approximately 35,000 BC. It consists of 29 distinct notches cut into a baboon's fibula. Also prehistoric artifacts discovered in Africa and France, dated between 35,000 and 20,000 years old, suggest early attempts to quantify time.
The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be as much as 20,000 years old and consists of a series of tally marks carved in three columns running the length of the bone. Common interpretations are that the Ishango bone shows either...

...Discuss the claim that some areas of knowledge are discovered and others are invented
By
Olivia Emans
In my opinion an invention may be defined as a new device, method, or process developed from study and experimentation whereas a discovery is an act or process of finding something or somebody, or learning about something that was not known about before.
Another way of differentiating discovery and invention is in the following: A discovery is something someone else can test for himself and find out if it is true or not. An invention is something that is true (and exists) only because of the inventor. Some people would say: Invention is of our making while discovery is what we can learn from God’s making or from the nature.
There are many different areas of knowledge, from natural sciences (most notably physics, chemistry and biology), mathematics up to areas like human sciences, history, religion and arts. As a generalisation it can be said that discoveries play a bigger role in natural sciences and even in mathematics than in areas like religion or arts. Art, be it a painting or a piece of music or certain literature, is generally invented by individuals. If a piece of art would not have been invented it simply would not exist. Laws of nature like the gravitational forces are not invented. They exist on their own and can only be discovered. Such laws...

...Mini Lab
Refraction of light
By: Brandon Simmons
SRU7, 3/21/13
Mini Lab
Refraction of light
By: Brandon Simmons
SRU7, 3/21/13
Procedure
1. Fill a drinking glass about half with water.
2. Place a pencil in the glass. Describe the appearance of the pencil.
3. Slowly add water to the glass. Describe how the appearance of the pencil changes.
Analysis
Analysis
1. How does the appearance of the pencil depend on the level of water in the glass?
Answer= The water in this picture is actually not special at all: it is ordinary tap water. This pencil is still intact, but it appears distorted because of a property of light called refraction. Refraction causes light to bend when it passes from one substance into another, in this case from air to water. You can test this property with any object, not just a pencil. You may observe refraction affecting your own body when you are at the swimming pool. The parts of your body out of the water may not line up with the parts in the water. This distortion is all due to the bending of light.
2. Where do the light waves...

...The evolution of mathematics might be seen as an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction, which is shared by many animals,[19] was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely quantity of their members.Evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also recognized how to count abstract quantities, like time – days, seasons, years.[20]
More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy.[21] The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time.
In Babylonian mathematics elementary arithmetic (addition, subtraction, multiplication and division) first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have been many and diverse, with the first known written numerals created by Egyptians in Middle Kingdom texts such as the Rhind Mathematical Papyrus.[citation needed]
Between 600 and 300 BC the Ancient Greeks began a systematic study of mathematics in its own right...