Mathematics

= = =**__Mathematics__**=

__Day 1–__ Math in the school context: mathematical claims, Buchberger's model for mathematical discovery
- Math is something we as students and anyone who went to school has been exposed to for at least a good 9-10 years at least. - Teachers, even in lower grades, ask students to prove that things in math are true. For example, in lower school we were asked to prove that 1+1=2 by taking an item and placing it next to another and seeing that there are, indeed, two of them. - In later school years we we/are asked to prove something in math using a mathematical proof. An example of this is proving that all angles of a triangle are 180*. - Buchberger's model for mathematical discovery contends that mathematical discovery takes place in 3 stages. Stage 1 is the "phase of experimentation," stage 2 is the "phase of exactification," and stage 3 is the "phase of application."

__Day 2–Math:__ Conjecture and proof
- A conjecture is a conclusion made from a reasonable number of cases, but it is not 100% certain that it is true. - A proof is a collection of logically valid steps which support a claim and serves as justifying that the claim is true. - In order for a conjecture to be made, it is necessary to have a strong number of results. - Proofs are what make math so unique since in most other subjects, things cannot actually be proved. __Day 3–Math: Discovered or invented? Math and reality.__ - Math is something that is discovered. Manipulation of objects is something that supports this claim. - Math is also discovered since it is something that exists in the natural world, i.e. Fibonacci Sequence. - Math is also found in other aspects of nature such as leaves and shells. - Math is pretty much how it seems. If two lines do not appear to intersect, they don't, and that is the conclusion that we have come to.

__Day 4–The nature of math, math as a creative art__ - Many artists take advantage of math to improve or make their artwork unique. - Many mathematicians view math as an art form. - Many would consider math to be a very creative thing since it takes much thought and creativity to come up with solutions in math often times.

__Day 5–More about axioms, math as a human endeavor__ - It is not correct to say that one can make up a set of axioms and from that come up with any claims they wish, it is more complex than that. - Math is viewed by many as a language since it has signs and symbols that can represent very abstract ideas that are not easy to convey using words. - Math is also used in everyday life in almost every career; finance, business, medicine, law, music, art etc.