Proof
Proof. What is it and why does this simple term cause such a stir among mathematics educators and mathematics students? If you were to ask a young child to prove a mathematical fact, they would be happy to show you many examples of how it works. This does not constitute a proof but it is a step in the right direction. If you were to ask a high school student or first year college student to do a proof, you will most likely be met with groans and feelings of disgust. Students at this age have probably encountered proof in a geometry class where they were expected to follow a strict format without much freedom to express proofs on their own. However, if you were to ask a mathematician about proof they would begin to tell you about how beautiful proof in mathematics can be.
Proof has always been a topic of interest for me. In high school geometry and my first year of college, I too did not understand proof. I felt like many other students, frustrated by the fact that we were asked to prove theorems that the book had already told us were true. It was as though the instructor was playing magical games on the chalkboard and all of the sudden we had a proof. However, as time progressed, I began to see the beauty of proof. Then, mathematical induction introduced me to the power of proof. In this paper I hope to address the concept of proof, how it relates to understanding and the implications for mathematics education.
BACKGROUND
In the 1950’s and 60’s proof played a significant role in mathematics education. Then in 1989, the National Council of Teachers of Mathematics (NCTM) deemphasized proof and replaced it with reasoning. Following this, mathematics educators began to see that students had difficulty with proof because they had little contact with it. In response, NCTM in the 2000 standards, elevated proof to a standard, emphasizing that it should be part of all students mathematical experiences (Knuth). Schoenfeld states proof is inseparable from mathematics. It is essential in communicating, doing, and recording mathematics (153).
Throughout most of the history of mathematics education, proof has been more of a topic of study instead of a way to understand mathematics (Knuth 73). In addition, proof has only been limited to the college bound student or the student enrolled in geometry.
Traditions are usually passed on from parent to child at an early age. In the opening lines of "The Lottery," Shirley Jackson reveals the tradition of the lottery and how all of the villagers conform to the ritual of a human sacrifice. “The people of the village began to gather in the square, between the post office and the bank, around ten o’clock…” (Jackson 373) shows how the citizens of the village are used to the tradition of always gathering for the lottery. The children in “The Lottery” were stuffing their pockets with stones before all of the parents had arrived, " Bobby Martin had already stuffed his pockets full of stones, and the other boys soon followed his example, selecting the smoothest and roundest stones" (373). This illustrates that the children were taught what to do in the event of the lottery and by being prepared it shows t...
Abstract geometry is deductive reasoning and axiomatic organization. Deductive reasoning deals with statements that have already been accepted. An example of deductive reasoning is proving the sum of the measures of the angles of a quadrilateral is 360 degrees. Another example of deductive reasoning is proving the sum of the angles of a trigon is equal to 180 degrees. From this we get, any quadrilateral can be divided into two trigons. Axioms, which are also called postulates, are statements that can be proved true by using deductive reasoning.
The second part of this memo contains a rhetorical analysis of a journal article written by Linda Darling-Hammond. Interview The following information was conducted in an interview with Diana Regalado De Santiago, who works at Montwood High School as a mathematics teacher. In the interview, Regalado De Santiago discusses how presenting material to her students in a manner where the student actually learns is a pivotal form of communication in the field (Personal Communication, September 8, 2016).
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.
Mathematical dialogue within the classroom has been argued to be effective and a ‘necessary’ tool for children’s development in terms of errors and misconceptions. It has been mentioned how dialogue can broaden the children’s perception of the topic, provides useful opportunities to develop meaningful understandings and proves a good assessment tool. The NNS (1999) states that better numeracy standards occur when children are expected to use correct mathematical vocabulary and explain mathematical ideas. In addition to this, teachers are expected
The era of his college years was also in interesting period in the realm of mathematics. Many things were and had been already changing. Mathematics was finding itself and it seemed that its rules could be fully used to find the solution to any problem. Unfortunately a known mathematician by the name of Kurt “Gödel, had proven that the axioms of mathematics never could be complete as well as consistent. This was a hard blow to many mathematicians that had been convinced that mathematics was a universal and complete system.” 2 And furthermore, there remained the question of decidability, that is, whether there was a method for deciding a mathematical statement to be provable or not.
Using literacy strategies in the mathematics classroom leads to successful students. “The National Council of Teachers of Mathematics (NCTM, 1989) define mathematical literacy as an “individual's ability to explore, to conjecture, and to reason logically, as well as to use a variety of mathematical methods effectively to solve problems." Exploring, making conjectures, and being able to reason logically, all stem from the early roots of literacy. Authors Matthews and Rainer (2001) discusses how teachers have questioned the system of incorporating literacy with mathematics in the last couple of years. It started from the need to develop a specific framework, which combines both literacy and mathematics together. Research was conducted through
...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.
Euclidean Geometry is the study of plane and solid figures based on the axioms and theorems outlined by the Greek mathematician Euclid (c. 300 B.C.E.). It is this type of geometry that is widely taught in secondary schools. For much of modern history the word geometry was in fact synonymous with Euclidean geometry, as it was not until the late 19th century when mathematicians were attracted to the idea of non-Euclidean geometries. Euclid’s geometry embodies the most typical expression of general mathematical thinking. Rather than simply memorizing basic algorithms to solve equations by rote, it demands true insight into the subject, cleaver ideas for applying theorems in special situations, an ability to generalize from known facts, and an
A somewhat underused strategy for teaching mathematics is that of guided discovery. With this strategy, the student arrives at an understanding of a new mathematical concept on his or her own. An activity is given in which "students sequentially uncover layers of mathematical information one step at a time and learn new mathematics" (Gerver & Sgroi, 2003). This way, instead of simply being told the procedure for solving a problem, the student can develop the steps mainly on his own with only a little guidance from the teacher.
Many students view mathematics as a very difficult subject since it does not only focusses on numbers but also in letters. Mathematics does not only require the students to come up with an answer but it also requires them to show the solutions on how they arrived at the answer. While in elementary, students were already taught on how to solve problems in a step-by-step procedure starting with what is asked in the problem, what are the given, make a number sentence or formulate an equation and solve the problem. These procedures are called problem-solving which cannot only apply in mathematics but also in other areas such as in Science, businesses and most
Organisational change can cause stress for employees at all levels of an organisation, one of the main causes of increased employee stress during organisational change is employees’ perception of organisational change as a threat; many employees feel that there may be a threat to their job security, their status, or their ability to achieve if the conditions of their work are altered (Dahl, 2011). Employees may face changes in their written contracts and also in their implicit psychological contracts during organisational change, the change to these unwritten contracts can result in increased stress due to feelings of anger or betrayal by employees as they feel that they no longer know what to expect from their employer (Robinson & Rousseau, 1994). In addition, if employees do not feel that the organisation acts in a fair and just way they are more likely to
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...
Throughout out this semester, I’ve had the opportunity to gain a better understanding when it comes to teaching Mathematics in the classroom. During the course of this semester, EDEL 440 has showed my classmates and myself the appropriate ways mathematics can be taught in an elementary classroom and how the students in the classroom may retrieve the information. During my years of school, mathematics has been my favorite subject. Over the years, math has challenged me on so many different levels. Having the opportunity to see the appropriate ways math should be taught in an Elementary classroom has giving me a
Euclid proved his concepts logically, using definitions, axioms, and postulates. Proclus Diadochus wrote a commentary on Euclid's Elements that kept Euclid's works in circulation. It is believed that Euclid set up a private school at the Alexandria library to teach Mathematical enthusiasts like himself. There are other theories that suggest that Euclid went on to help these students write their own theories and books later in life.