# Research Proposal: Geometry Proof

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Geometry Proof

Geometry as a subject learned in school has a primary purpose, and that is to improve the ability of students to reason logically. Logical reasoning is one of the most vital things that a student can learn, not only for mathematics, but for many of the issues that he or she will face in life (Discovering, 2009). Without including logic, there is little that can be reasoned out in the world and/or properly addressed and decided (Discovering, 2009). While some people use only their intuition and feelings to make decisions about the course that their life will take, this is generally not a good approach overall, and it is an approach that cannot be used for many facets of a person's life as they grow older, leave school, and assume their place in life and in their career.

There are two basic kinds of reasoning that are important for all people, no matter what they are doing with or in their lives, and that holds true for both education and personal/career experience (Discovering, 2009). These two kinds of reasoning are inductive and deductive. Inductive reasoning is used to identify visual and numerical patterns so that the student can make predictions based on those patterns. Deductive reasoning explains why the patterns are true. In geometry, students learn about angle measurements in both intersecting and parallel lines and make assumptions about the measurements and their relationships to each other. Logical arguments are given so that a student may learn how to explain why these conjectures are true (Discovering, 2009).

Inductive reasoning involves cause and effect based on personal experiences of the student. For example, every time you go outside in the rain without an umbrella, you get wet. The inductive reasoning used to draw the conclusions from repeated experiences is inductive, however in mathematics the answer is not always so concrete (Discovering, 2009). For this reason, mathematicians can use inductive reasoning to find out what could be true, and because there is an element of possibility involved, inductive reasoning is not foolproof, so mathematicians do not generally use inductive reasoning to explore factual problems. In a sense, inductive reasoning is a fancy type of guesswork, leading to conjectures, which can only become mathematical facts if someone can prove that it is true in all cases (Discovering, 2009). This true proof is called deductive reasoning.

Deductive reasoning is also known in mathematics as proof. It is a form of reasoning from given facts using logical steps to arrive at a proven conclusion (Discovering, 2009). Mathematicians use proofs in many different ways, to serve different purposes. A proof can be used to prove a conjecture true or false, in all cases, not just for the examples they may have examined (Discovering, 2009). Proofs are often the answer to the most popular childhood question: Why? The use of proof in mathematics, particularly geometry, can help students to explain theories and deepen their understanding of not just math, but of the world around them (Discovering, 2009).

The most common format for explaining proofs is to have two columns drawn, one containing a set of statements and the other a column of reasons. This is called a formal proof. The reasons are supposed to justify the statements, however, most younger students are overwhelmed by this particular approach to proofs. It is recommended that proofs in column form not be studied until several other ground-rule type theories have been explored. The best way to approach the subject of proofs in young children is to simply use the informal proof, in deductive argument form as referred to above. In this style, the argument is simply written in paragraph form so that the student reads and processes the evidence given (Herbst, 2002).

When teaching proofs, particularly in geometry, students often have trouble understand the need for proof at all. Experts in the mathematical field state that it is not that students have a problem with slow cognitive development, but rather that they may not grasp the function of proof right away. The meaning and usefulness of proof in geometry does not come until later, when the student is shown the results of logical truths in geometric problems (Herbst, 2002). It can be challenging for a teacher to explain the usefulness and function of proofs in the classroom, but in a nutshell, the main purpose that students should be taught is that the function of a proof is simply to verify the correctness of mathematical problems. Teachers can apply activities that allow the student to see that proof as a concept is used mainly to remove doubt in an argument or problem.

Both students and teachers have some common conceptions of proof in mathematics, even if they do not yet have a lot of knowledge about a specific type of mathematics or know exactly where it will be leading them in life (Herbst, 2002). One of the difficulties that students have with the concept of proof is the belief that non-deductive arguments constitute a proof. Some other common student beliefs about what constitutes a mathematical proof include Ritual (it's a proof only if it is in line with specific mathematical convention); Authoritative (presented by or approved by an established authority); Inductive (if it holds for one example, or perhaps several, it is sufficient to demonstrate its validity); and Perceptual (Using a diagram, a certain property can be proven as true) (Herbst, 2002).

It is debated as to why students hold these incorrect conceptions of proof, and the most logical answer is that they are applying to mathematics what is acceptable in other subjects, as well as their own lives. For instance, drawing a general conclusion by examining many different experiences in a subject is considered entirely appropriate in the social sciences. But this is not always the case in mathematical proofs (Herbst, 2002). Proofs are very specific, and they do not allow for the 'leeway' that is often seen in other types of classes. In other words, mathematics is an exact science, not one where an individual can make educated guesses or where he or she can assume that something is correct without first determining whether it is actually correct and provable instead of only believed to be accurate.

Teachers, on the other hand, seem to believe that only proof provides mathematic certainty. This concept is foreign to students, who rely on conjectures just as strongly as proof in their beginning studies in geometry class. Teachers often find that the most difficult part of the process for writing a deductive argument is to find the very basic logic of the argument and what information to needs to be included. Students are taught reasoning strategies, or ways of thinking that help to construct a deductive argument, and most often those strategies are being taught via computer geometric software (Herbst, 2002).

As a visual system for teaching proofs, many programs include helpful aids for allowing students to work through proofs and logical reasoning on their own. For example, in one software application, a set of playing cards and a playing field electronically displays on a computer game screen. The playing field includes two boxes labeled "GIVEN" and "CONCLUSION" for entry of a statement and a reason (Herbst, 2002). By selecting from a certain menu or submenu contained in a window, a card may be reviewed.

To set up the playing field, a mathematical statement, displayed on a set-up card of a group of set-up cards containing each statement from the universe of statements known, is entered into the boxes. The statements may be custom labeled, and geometric figures associated with the theorem may also be electronically drawn on the screen by the student.

After set-up is complete, the user then chooses from a set of playing cards, each card displaying a…
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