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Why Critical Thinking?

"As grads look to the future, they're all thinking one thing: Hire us...In general, students in fields that require critical thinking skills, problem-solving, and face-to-face contact will fare best in this new economy, no matter where they look for jobs", said Jim Kurre, associate professor of economics at Penn State Behrend and director of the Economic Research Institute of Erie.

Erie Times News, PA - May 20, 2008

"Employers report that such applied skills as critical thinking, teamwork, and effective communication are essential to the preparation for today’s workplace"... Tom Pauken

East Texas Review, TX - Jun 11, 2008

Studying the following articles and pages will help you build a stronger understanding of the core concepts in critical thinking

• Becoming a Critic Of Your Thinking
• Glossary of Critical Thinking Terms
• Universal Intellectual Standards
• Valuable Intellectual Traits
• Distinguishing Between Inferences and Assumptions
• Thinking With Concepts

In addition to the basic review of the definition and concept of critical thinking, the following pages and articles are recommended reading for the college, university or pre-collegiate student.

Looking To The Future With a Critical Eye: A Message for High School Graduates Developing as Rational Persons: Viewing Our Development in Stages How to Study and Learn (Part One) How to Study and Learn (Part Two) How to Study and Learn (Part Three) How to Study and Learn (Part Four) The Art of Close Reading (Part One) The Art of Close Reading (Part Two) The Art of Close Reading (Part Three) Learning the Elements and Standards online

Skill #1: Interpretation

What it Means: Having the ability to understand the information you are being presented with and being able to communicate the meaning of that information to others.

Throughout your career you will be presented with a variety of information in many different types of situations. Imagine you are looking at demographic information, hoping to target a different group of customers for a new product. Interpretation skills will enable you to better decode the information and add clarity to what you have discovered - which in turn will help you  better understand any potential new customer opportunities for your company.

Application Exercise: Write down 10 emotions you feel can be paired with facial expressions, e.g. smiling equals happiness. Then give your list to a friend and ask them to randomly recreate each facial expression as you attempt to interpret the correct emotion simply by looking at their face. You will learn how to identify thekey bits of information you need in order to connect the dots. How many can you correctly identify?

Skill #2: Analysis

What it Means: Having the ability to connect pieces of information together in order to determine what the intended meaning of the information was meant to represent.

Imagine you are reading a companywide memo that is discussing making changes in order to address recent movement within the business landscape. Having this skill will better provide you with the ability to “read between the lines” and help you understand how this will impact the overall strategy you have in your position.

Application Exercise: To practice this skill, try to determine the meaning behind this Chinese Proverb: “Be the first to the field and the last to the couch.” Are you able to identify the intended meaning behind this statement? While we all analyze information a little differently, you should have been able to determine that this proverb is referencing having a hard work ethic. Go here to practice some more in order to further develop this skill.

Skill #3: Inference

What it Means: Having the ability to understand and recognize what elements you will need in order to determine an accurate conclusion or hypothesis from the information you have at your disposal.

Picture yourself as a business manager; you are looking at the latest sales forecast and you see sales have decreased. It’s important for you to be able to understand what additional information you may need in order to determine why that happened, including identifying internal issues, external competition or even economic conditions. The ability to understand the information you already have and determine what you may still need to find the best solution is an important skill for you to have no matter what career field you are in.

Application Exercise:  Try watching an episode of a weekly crime show. Focus on absorbing the clues in an effort to determine the mystery of who completed the crime during the episode. These shows do a fantastic job of dropping clues and bits of information along the way that can help you make an educated guess prior to the end of the episode. Were you able to identify it correctly?

Skill #4: Evaluation

What it Means: Being able to evaluate the credibility of statements or descriptions of a person’s experience, judgment or opinion in order to measure the validity of the information being presented.

Imagine you are leading a focus group at work to determine how your customers view the organization’s products. You ask the group a few questions that uncover several negative opinions about certain items you sell. You will need to use this skill to evaluate those responses, so you can determine if the information you received is valid and whether or not it needs to be further looked into.

Application Exercise: With the invention of the Internet this is a skill you can hone anytime you begin a search online. For practice, run a search on how to be a leader at work to test your evaluation skills. Are you able to identify whether the information you find is reliable and valid? Being aware of additional information you may need to make this assessment is essential. (Hint: In this example, you may need to evaluate the website’s URL, the author’s bio and potentially many other pieces of information.)

Skill #5: Explanation

What it Means: Having the ability to not only restate information, but add clarity and perspective to the information, so it can be fully understood by anyone you are sharing it with.

Imagine you are giving two presentations for new product ideas; one to the CEO of the company, and the other to product engineers. You know both groups are extremely interested in hearing what you have to say, but you will need to be able to explain these ideas in two very different ways. The CEO may only need to hear high level ideas about the products while the engineers will need more specific product details. Your ability to clearly explain your ideas while keeping in mind who you are presenting to is important for making sure the information is understood and well received.

Application Exercise: It’s important to remember not everyone has or needs the knowledge or information  you have. Practice this skill by explaining something complex that you have knowledge about to two different types of people. Use your kids and your spouse since they typically will not have the same level of knowledge of certain subjects. For example, explain to your kids and your spouse separately how a cell phone works. The explanation should be different since they have two different levels of knowledge yet both of them should receive a similar understanding of the subject.

Skill #6: Self-Regulation

What it Means: Having the awareness of your own thinking abilities and the elements that you are using to find results.

Imagine you are on the phone with a customer attempting to work through a problem they are having with your company’s software, and it’s your first week on the job. The problem they are having is complex, and yet, you still want to assist them in order to make a good impression at work.  Your ability to understand that perhaps transferring them to a coworker with more knowledge on the subject is the best way to provide the customer with a positive result.

Application Exercise: It’s important to be able to separate your own personal biases or self-interests when making decisions at work. Let’s imagine your manager asks you to write down 10 reasons why you deserve a raise prior to an annual review. After you complete your reasons, read through each and focus on identifying your own potential biases that may impact your argument. Awareness of these biases will help you generate a much stronger case for getting a raise.

So the next time someone starts talking to you about developing critical thinking skills for your career, hopefully now you will be able to tell them you know what they are, how they are used in the workplace and how to go about continually developing them for the future.

Notes : CONIC SECTIONS

CONIC SECTIONS

A conic is the intersection of a plane and a right circular cone. The four basic types of conics are parabolas, ellipses, circles, and hyperbolas. We've already discussed parabolas and circles in previous sections, but here we'll define them a new way. Study the figures below to see how a conic is geometrically defined.

Figure 1 : The four basic types of conics

In the conics above, the plane does not pass through the vertex of the cone. When the plane does intersect the vertex of the cone, the resulting conic is called a degenerate conic. Degenerate conics include a point, a line, and two intersecting lines.

The equation of every conic can be written in the following form: Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 . This is the algebraic definition of a conic. Conics can be classified according to the coefficients of this equation.

The determinant of the equation is B 2 - 4AC . Assuming a conic is not degenerate, the following conditions hold true: If B 2 -4AC > 0 , the conic is a hyperbola. If B 2 -4AC < 0 , the conic is a circle, or an ellipse. If B 2 - 4AC = 0 , the conic is a parabola.

Another way to classify conics has to do with the product of A and C . Assuming a conic is not degenerate, the following conditions hold true: if AC> 0 , the conic is an ellipse or a circle. If AC < 0 , the conic is a hyperbola. If AC = 0 , and A and C are not both zero, the conic is a parabola. Finally, if A = C , the conic is a circle.

In the following sections we'll study the other forms in which the equations for certain conics can be written, and what each part of the equation means graphically.

Problem : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: -3x 2 + y + 2 = 0 ?

It is a parabola.

Problem : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: 2x 2 +3xy - 4y 2 + 2x - 3y + 1 = 0 ?

It is a hyperbola.

Problem : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: 2x 2 -3y 2 = 0 ?

It is a hyperbola.

Problem : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: -3x 2 + xy - 2y 2 + 4 = 0?

It is an ellipse or a circle.

Problem : Is the following conic a parabola, an ellipse, a circle, or a hyperbola: x = 0 ?

It is a degenerate conic. x = 0 is a line.

PARABOLA

As we saw in Quadratic Functions , a parabola is the graph of a quadratic function. As part of our study of conics, we'll give it a new definition. A parabola is the set of all points equidistant from a line and a fixed point not on the line. The line is called the directrix, and the point is called the focus. The point on the parabola halfway between the focus and the directrix is the vertex. The line containing the focus and the vertex is the axis. A parabola is symmetric with respect to its axis. Below is a drawing of a parabola.

Figure %: In the parabola above, the distance d from the focus to a point on the parabola is the same as the distanced from that point to the directrix.

If a parabola has a vertical axis, the standard form of the equation of the parabola is this: (x - h)2 = 4p(y - k), where p≠ 0 . The vertex of this parabola is at (h, k) . The focus is at (h, k + p) . The directrix is the line y = k- p . The axis is the line x = h . If p > 0 , the parabola opens upward, and if p < 0 , the parabola opens downward.

If a parabola has a horizontal axis, the standard form of the equation of the parabola is this: (y - k)2 = 4p(x -h) , where p≠ 0 . The vertex of this parabola is at (h, k). The focus is at (h + p, k) . The directrix is the line x =h - p . The axis is the line y = k . If p > 0 , the parabola opens to the right, and if p < 0 , the parabola opens to the left. Note that this graph is not a function.

Let P = (x, y) be a point on a parabola. Let l be the tangent line to the parabola at the point P . Let  be a line segment whose endpoints are the focus of the parabola and P . Every parabola has the following property: the angle θ between the tangent line l and the segment  equal to the angle μ between the tangent line and the axis of the parabola. This means (in a physical interpretation) that a beam sent from the focus to any point on the parabola is reflected in a line parallel to the axis. Furthermore, if a beam traveling in a line parallel to the axis contacts the parabola, it will reflect to the focus. This is the principle on which satellite dishes are built.

Figure 2 : The reflective property of a parabola: θ = μ

ELLIPSE AND CIRCLE

An ellipse is the set of points such that the sum of the distances from any point on the ellipse to two other fixed points is constant. The two fixed points are called the foci (plural of focus) of the ellipse.

Figure 3 : The sum of the distances d 1 + d 2 is the same for any point on the ellipse.

The line segment containing the foci of an ellipse with both endpoints on the ellipse is called the major axis. The endpoints of the major axis are called the vertices. The point halfway between the foci is the center of the ellipse. The line segment perpendicular to the major axis and passing through the center, with both endpoints on the ellipse, is the minor axis.

The standard equation of an ellipse with a horizontal major axis is the following:  +  = 1 . The center is at (h, k) . The length of the major axis is 2a , and the length of the minor axis is 2b . The distance between the center and either focus is c , where c 2 =a 2 - b 2 . Here a > b > 0 .

The standard equation of an ellipse with a vertical major axis is the following:  +  = 1 . The center is at (h, k) . The length of the major axis is 2a , and the length of the minor axis is 2b . The distance between the center and either focus is c , where c 2 =a 2 - b 2 . Here a > b > 0 .

The eccentricity of an ellipse is e =  . For any ellipse,0 < e < 1 . The eccentricity of an ellipse is basically a measure of the "ovalness" of an ellipse. It is the ratio of the distance between the foci and the length of the major axis. If the foci are very near the center of an ellipse, the ellipse is nearly circular, and e is close to zero. If the foci are relatively far away from the center, the ellipse is shaped more like an oval, and e is closer to one.

Circles

A circle is the collection of points equidistant from a fixed point. The fixed point is called the center. The distance from the center to any point on the circle is the radius of the circle, and a segment containing the center whose endpoints are both on the circle is a diameter of the circle. The radius, r , equals one-half the diameter, d .

HYPERBOLA

A hyperbola is the set of all points such that the difference of the distances between any point on the hyperbola and two fixed points is constant. The two fixed points are called the foci of the hyperbola.

Figure 4 : The difference of the distances d 1 - d 2 is the same for any point on the hyperbola.

The graph of a hyperbola is not continuous--every hyperbola has two distinct branches. The line segment containing both foci of a hyperbola whose endpoints are both on the hyperbola is called the transverse axis. The endpoints of the transverse axis are called the vertices of the hyperbola. The point halfway between the foci (the midpoint of the transverse axis) is the center.

The standard equation for a hyperbola with a horizontal transverse axis is  -  = 1 . The center is at (h, k) . The distance between the vertices is 2a . The distance between the foci is 2c . c 2 = a 2 +b 2 . The line segment of length 2b perpendicular to the transverse axis whose midpoint is the center is the conjugate axis of the hyperbola.

The standard equation for a hyperbola with a vertical transverse axis is  -  = 1 . The center is at (h, k) . The distance between the vertices is 2a . The distance between the foci is 2c . c 2 = a 2 + b 2 .

Every hyperbola has two asymptotes. A hyperbola with a horizontal transverse axis and center at (h, k)has one asymptote with equation y = k + (x - h) and the other with equation y = k - (x - h) . A hyperbola with a vertical transverse axis and center at (h, k) has one asymptote with equation y = k + (x - h) and the other with equation y = k - (x - h) .

The eccentricity of a hyperbola, like an ellipse, is e =  . For all hyperbolas, though, c > a , so e > 1 . If e is close to one, the branches of the hyperbola are very narrow, but if e is much greater than one, then the branches of the hyperbola are very flat.

~ Destress Corner ~

Notes : DATA DESCRIPTIVE

DATA DESCRIPTIVE

 UNGROUPED DATA  

 MEAN  : Sum of all data / Number of data

 MEDIAN  :
1. Arrange data in order 
2. Choose middle point
3. Use the formula :

   



N = number of data

Example : 

Find the median of the set of data given : 

2.71, 3.56, 4.35, 5.48, 6.22, 8.61

Median = ( (6+1) / 2)th  
             = 3.5th observation
             
2.71, 3.56, 4.35,* 5.48, 6.22, 8.61

             = 1/2 ( 3th + 4th ) 
            = 1/2 ( 4.35 + 5.48 )
            = 4.915

RANGE : Highest data - Lowest data

QUARTILE : 



INTERQUARTILE RANGE  : Q3 - Q1   

Example : 

What is the interquartile range for the set of data?

20 , 30 , 40 *, 50 , ** 60 , 70 ,*** 80 , 90 

* Q1
** Q2
*** Q3

Therefore, interquartile range = ( 70 + 80 ) - ( 40 + 50 )
                                                = 150 - 90
                                                = 60

 GROUPED DATA 

 CLASSES OF DATA  : 

- 0, 10, 20, 30, 40, 50 are LOWER CLASS 

- 9, 19, 29, 39, 49, 59 are UPPER CLASS 

FREQUENCY DENSITY : Frequency / Class Width 

CLASS MARK : Midpoint ( ( Upper class- Lower class ) / 2 )

MODE :

MEDIAN : 

m = median
L = lower class of median
N = number of data
F = frequency before class median
fm = frequency of class median
c = breadth of class


QUARTILE :

 N = number of data
Q1 = lower/first quartile
Q3 = upper/third quartile

F = frequency before class median
fQ = frequency of class median
C = breadth of class

VARIENCE :

STANDARD DEVIATION : 

SUMMARY :