The Role of the Proof in Math
The notion of proof has long played a key role in the study of mathematics. It is in my opinion the role of proof that separates mathematics from the sciences and other fields of study. It is the existence of proofs that give mathematicians the confidence that their work is credible and thus allows them to continue to build upon prior work without the need to second guess what has previously been accomplished.
Based upon this observation, it becomes natural to ask the questions pertaining to the use of proof in learning and understanding mathematics. If the concept of proof is so important to the field of mathematics, then is it possible that by writing proofs and studying proofs that an individual will be better equipped to understand the mathematics for which the proofs pertain? And if this is possible then when should a person be first exposed to proofs and at what level? In this paper I will give my views pertaining to these questions, as well as, a few more of my views pertaining to some other topics related to these questions.
Before discussing the virtues of proofs as a means of learning and understanding mathematics, I feel that it is first necessary to begin with a brief discussion of the functions of proof within mathematics. Following I will give a list of the functions of proof that I have comprised from three sources (Hanna [2], Knuth [3], Tucker [6]):
1.verification, the act of arguing that a statement is true
2.explanation,providing reasons for why a statement is true, which in turn
may lead to understanding
3.systematization,organizing statements and definitions into a system of
axioms, lemmas, theorems, etc.
4.discovery,creating knowledge and new results
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...ducation, V178 N1, pp. 35-45
[2]Hanna, Gila (2000), “Proof, Explanation and Exploration: An Overview,” Educational Studies in Mathematics, V44, pp. 5-23
[3]Knuth, Eric (2002), “Secondary School Mathematics Teachers Conceptions of Proof,” Journal for Research in Mathematics Education, V33, pp.379-405
[4]Lester, Frank K. (1975), “Developmental Aspects of Children’s Ability to Understand Mathematical Proof,” Journal for Research in Mathematics Education, V6 N1, pp. 14-25
[5]Selden, Annie and Selden, John (2003), “Validations of Proofs Considered as Texts: Can Undergraduates Tell Whether an Argument Proves a Theorem?,” Journal for Research in Mathematics Education, V34 N1, pp. 4-36
[6]Tucker, Thomas (1999), “On the Role of Proof in Calculus Courses,” Contemporary Issues in Mathematics Education, MSRI Publications, Cambridge University Press, Cambridge, UK
Imagine you came up with an idea, a wonderful idea of your own, but no one believes it is yours because you have lived in the shadows of your father for so long. You are unsure of what to do? Or of how others may react to it? In Proof, directed by John Madden, Catherine is the daughter suffering in the shadows of her father, now dead, when she told her “lover” and sister, that she wrote the proof, a proof that is brilliant. However, they believe her father, Robert, wrote it not her. Robert, whom at a young age had transformed the whole mathematical field and provided new research jobs in the math field. Madden’s Proof adaptation is both faithful and unfaithful to mise-en-scène in the beginning, middle, and end of the selected scene.
The number of concussions in professional and amateur football has been rising and has sparked much controversy in recent years. These concussions are most likely linked with disease and even the deaths of some pro and semi-pro football players. New research is attempting to solve the problem but the issue is still prevalent in football today.
Football is America’s favorite sport. It is a fast-paced, hard-hitting game. Every week thousands of men and boys all across the country take part in football and every week these men and boys receive violent hits during the game. Frequently, as a result of these violent hits, the player receives a concussion. However, the long-term effects of concussions on players are not fully understood. New research shows that even a slight concussion in a football game can have lasting effects on a player. As a result of this research, children under the age of fourteen should not play tackle football.
"Incidence of Concussion in High School Football Players of Ohio and Pennsylvania." Journal of Child Neurology. Web. 28 Sept. 2011. .
"Epic" heroes such as Sundiata simply cannot compete with someone like Gilgamesh. He is already a strong, powerful king when Sundiata is crawling around on all fours. The first page in Gilgamesh already shows his power. It states "a goddess made him, [Gilgamesh] strong as a savage bull." However, strength is not all he has. Gilgamesh is the king of Uruk, proven by the text, "Gilgamesh the king." This is a very good position, as Uruk is a powerful and large city, and "in Uruk he [Gilgamesh] built walls, a great rampart." This proves that he must have done many great deeds. A final clear indicator of Sundiata's early weakness is the f...
Many famous athletes have had life altering occurrences as a result of incurring a concussion from the sport they play. While this is mostly associated with football, there are many other sports in which concussions can and do occur. Awareness of the activities which may cause a concussion as well as ensuring that symptoms of the concussion are identified are critical to ensuring the safety of sports participants. Currently, 47% of all athletes fail to report that they are having symptoms of a concussion. In 2012, 3,800,000 concussions were reported which was double the amount reported just ten years earlier. Due to increased public awareness, this number has increased over time but is still a very low rate for such a serious, life threatening injury. Although, it is simply the nature of sports that concussions will occur, prevention is definitely something which can be enhanced through improved coaching of sports techniques and having athletes educated to the dangers associated with a concussion. Interestingly enough, not all concussions are caused at the actual competitive sporting event with one third occurring at a sports
American football, famously known to be associated with minor brain injuries, will later “Lead to long term neurological damage”, according to Sporting and Athletic Goods Manufacturing. With high school football, these kids are practically twice as possible to acquire a concussion still making it a dispute to whether continual harm results to brain disease. Disregarding a player’s apprehension, a focus that still lays a concern is principles amongst athletes rejecting to self-address on concussion. The author ...
Today, many professional, college and high school athletes throughout our country suffer from a common injury. It is an injury with serious side effects that can permanently change their lives. This injury is a concussion. Concussions are a constant threat in the game of sports. Coaches and athletes were under the assumption that a head injury that didn’t require a trip to the hospital could be ignored. We have been raised in a culture that celebrates hard knocks as a rite of passage, we don’t think twice about the bandages around our heads. (Carroll and Rosner 11). According to the Disease Control and Prevention Center, there are an estimated 1.6 to 3.8 million sports related head injuries in the United States each year. With the increasing number of concussions, the issue is relevant in today’s society. Concussions are a traumatic brain injury. In order to understand this growing issue, we need to become aware of what a concussion is, the testing and treatment that is used and the lifelong effects that concussions can cause. It is important for athletes to know this vital information about concussions in order to keep safe.
Concussions today are among the most severe injuries in sports, especially in the game of football. Concussions can almost happen at any place or at anytime just depending on the situation you are in. 5-10 % of all athletes will experience a concussion during their time of play in whatever sport they are participating in. ‘’A typical high school football player receives about 650 hits to the head per season, according to researchers at Purdue University and the University of Michigan. The impacts of those hits are the equivalent of what a seat-belted passenger experiences in car accidents ranging from 15-to-35 mph, according to University of Nebraska professor Timothy Gay, author of “The Physics of Football.”Needless to say, that is a lot of contact that most likely could leave the athlete vulnerable to brain injury’’(Baumbatch) Football is the leading sport in concussions, therefore players should be especially aware of the dangers of concussions. The re...
From long practice hours, hot summer workouts, and many Friday nights, my personal observation of this dangerous sport is exceptionally prevalent. My initial experience of the damage that football brings came my eighth grade year when I witnessed a senior football player on my team try and eat a phone on the ride home after receiving a concussion in the third quarter of the game. Which is a prime example to defend the fact that football related injuries to the head result in people not “being all there.” Not only have I seen someone try and eat a phone, but I have also witnessed head injuries resulting in my own friend randomly yelling at me after a game for no reason, and also a friend trying to jump down a full flight of stairs thinking he was starring in a movie. The fast paced, high intensity contact that comes with playing football is nothing to think flippantly of when it plays a role on brain trauma, and the results of brain trauma.
I also learned that mathematics was more than merely an intellectual activity: it was a necessary tool for getting a grip on all sorts of problems in science and engineering. Without mathematics there is no progress. However, mathematics could also show its nasty face during periods in which problems that seemed so simple at first sight refused to be solved for a long time. Every math student will recognize these periods of frustration and helplessness.
Skemp, R (2002). Mathematics in the Primary School. 2nd ed. London: Taylor and Francis .
...ett, S. (2008) . Young children’s access to powerful mathematical ideas, in English, Lyn D (ed), Handbook of international research in mathematics education, 2nd edn, New York, NY: Routledge, pp. 75-108.
Towers, J., Martin, L., & Pirie, S. (2000). Growing mathematical understanding: Layered observations. In M.L. Fernandez (Ed.), Proceedings of the Annual Meetings of North American Chapter of the International Group for the Psychology of Mathematics Education, Tucson, AZ, 225-230.
It is not only in my own writing that my awareness of math has been heightened. While reading articles for classes, on news websites, or blogs, I find myself paying more attention to the flow of the author’s argument. We’ve learned that in proof writing it is important to be clear, concise, and rigorous and the same applies to an argument within a paper. I’ve come to realize that if an author is trying to convince me of their point, then they also need to show me why their point is true or important. In this way, I’ve become more critical of an author’s argument; rather than just believing everything that they write, I more closely evaluate the progression o...