Communication in Mathematics

Communication is an essential part of Mathematics. Mathematicians not only need to know how to add and subtract numbers, but also how to communicate to the world the new discoveries that are found each day. This communication starts at an early age with students in a classroom explaining the way the student discover the answer to a problem and learning the unique language of mathematics. Communication will be used in a variety of mode and settings. Students will also know how to communicate effectively using mathematical language and symbols because the ideas will be generated and shared.

Communication involves a variety of modes: pictures, written symbols, spoken language, relevant situations, and manipulative. Each way the students has to find the link between each one and how the variety of modes represent the same problem, but in multiple ways. Students use pictures, whether told to draw a picture by the teacher or the student creates a picture on their own, to create a better understanding of the problem and use this picture to help think through the process needed to take in order to find the solution. Manipulatives are objects that can be touched, moved, and stacked. This allows the students to physically see the problem being manipulated and how a small piece is related to the whole. Students are able to communicate in number and symbols, but they are also able to communicate in the spoken language of mathematics. The students can explain their reasoning and process of the result they came up with to help better understand problems related to the one the student solved. A relevant situation can be any context that involves appropriate mathematical ideas and holds interest for children, and connected to a real-life situation. This simple act will create interest in the problem for the students and allow them to invest time into the problem to come up with the solution. Using relevant situation can also show the student how the material can be used in everyday life.

Students are able to use these variety of modes in a variety of setting: small group work, whole class work, partner work, and individual work. This will allow the students to think together and discover the solution to a problem that they can all understand using the different modes of communication. With the discovery  of the solution, the students will be able to write their process of thinking using the language of mathematics and symbols. This written report or oral report of their process can help better understand the problem as a whole, and the student are then able to help others in sharing their ideas for the problem.

When students understand the problem and how they discovered the solution, this will create confidence in mathematics. When they gain the confidence, the students will use communication to share their ideas and understanding of many real world problems.  

References

Clement, Lisa. Pictures Written Symbols Manipulatives Relevant Situations Spoken Language (n.d.): n. pag. Web.

"THE FIRST FOUR STANDARDS STANDARD 2 - COMMUNICATION." New Jersey Mathematics Curriculum Framework. New Jersey Mathematics Coalition, n.d. Web. 26 Nov. 2014.

Trigonometric Identities

As math students and future teachers, we all have to memorize all the different Trigonometric Identities. With this simple diagram you are able to get 21 different Trigonometric Identities with four different tricks. If I was presented this diagram in High School (this was presented to me in my college education class), I would have been able to memorize some of the Trigonometric Identities a lot faster. Below is the Trig Identity Trick:

Now I will break down each one of the four tricks in this one diagram. The first trick is finding the Reciprocal Identities. 

We will follow the red lines in the diagram. From the start of one identity on the line equals 1 divided by the end identity of line. 

 The second trick is finding the Pythagorean Identities. 

We will now look at the triangles. We will look at the left corner of the triangle and travel around the triangle clockwise. The point of the triangle is what the Pythagorean Identity is equal to and the top two points are added together. Remember that all the identities are squared!

The last two tricks go hand in hand. The third and fourth trick is finding the Quotient Identities and relations of the other identities.

Quotient Identities

Other Relations 

going around clockwise

going around counter clockwise

This is a great tool in which students and teachers can use all through out Trigonometry. This will relieve stress in memorizing all the trigonometric identities and focus more on applying the identity to problems. This graph is a great tool to use int he classroom because it uses only the six basic Trig Identities and creates many different formulas that they students will use multiple times through out the life time of math. Instead of the students memorizing all the different formulas they just have to remember how to construct this graph and how to use it. All of the formals are within this graph!