Cheah Meng Kit KEW080006

1. What do you understand by
i. Thinking
Thinking is a way of reasoning or in other word judgment. Thinking is to have a conscious mind, to some extent of reasoning, remembering experiences and making rational decisions. Thinking is also a conscious and goal-directed mental activity.


ii. Critical Thinking
Critical thinking is the use of reason to solve a problem, make a decision or form an opinion. It is an intellectually disciplined process of applying, synthesizing and conceptualizing information, examining underlying assumptions, interpreting and evaluating evidence, imagining and exploring alternatives and developing reflective criticism to reach a conclusion. Critical thinking is also convergent thinking. It assesses the worth and validity of something existent. It involves precise, persistent and objective analysis.


iii. Creative Thinking
Creative thinking is divergent thinking. It generates something new or different. It involves having a different idea that works as well or better than previous ideas.


iv. Mathematical Thinking
Mathematical thinking is a cognitive approach to a problem that is both logical and mathematically sound. This definition allows us to approach it in a way that is conducive to solving mathematics problems while not restricting us to saying that there is only one correct solution or that we must use the shortest and quickest method possible. Mathematical thinking involves sorting objects by different traits, identifying and copying simple patterns, using words to communicate an understanding of numbers and relationships and so forth. Deductive reasoning and inductive reasoning are very important in mathematical thinking. Deductive reasoning is valid if the conclusion must be true when the premises which are the reasons given to support that conclusion are true. One classic example of deductive reasoning is that found in syllogisms like the following:

Premise 1 : All humans are mortal.
Premise 2 : Socrates is a human.
Conclusion : Socrates is mortal.

The reasoning in this argument is valid because there is no way in which the premises, 1 and 2, could be true and the conclusion, 3, be false. On the other hand, induction reasoning is a form of inference producing propositions about unobserved objects or types, either specifically or generally, based on previous observation. It is used to ascribe properties or relations to objects or types based on previous observations or experiences, or to formulate general statements or laws based on limited observations of recurring phenomenal patterns. Inductive reasoning contrasts strongly with deductive reasoning in that, even in the best, or strongest, cases of inductive reasoning, the truth of the premises does not guarantee the truth of the conclusion. Instead, the conclusion of an inductive argument follows with some degree of probability. The conclusion of an inductive argument contains more information than is already contained in the premises. A classic example of inductive reasoning comes from the empiricist David Hume:

Premise : The sun has risen in the east every morning up until now.
Conclusion: The sun will also rise in the east tomorrow.


v. Lateral Thinking
Lateral thinking is to distinguish the kind of artistic creativity from the thinking involved in creating new perceptions and new concepts.

References:
1. Dictionary.com. (2010). Thinking. Retrieved February 24, 2010, from http://dictionary.reference.com/browse/thinking

2. Wikipedia Foundation, Inc. (2010, January). Thought. Retrieved February 24, 2010, from http://en.wikipedia.org/wiki/Thought

3. Farlex, Inc. (2010). Thinking. Retrieved February 24, 2010, from http://www.thefreedictionary.com/thinking

4. Wikipedia Foundation, Inc. (2010, February). Critical thinking. Retrieved February 24, 2010, from http://en.wikipedia.org/wiki/Thought

5. Varnado Elementary. (2009). Types of thinking. Retrieved February 24, 2010, from http://www.ves.wpsb.org/focus/typesofthinking.html

6. About.com. (2010). Deductive and Inductive Arguments. Retrieved February 24, 2010, from http://atheism.about.com/od/criticalthinking/a/deductivearg.htm

7. Wikipedia Foundation, Inc. (2010, February). Reasoning. Retrieved February 24, 2010, from http://en.wikipedia.org/wiki/Reasoning#Deductive_reasoning





2. Have you ever being taught critical thinking in school or before entering the faculty of engineering?
No.





3. What do you understand by ‘analysis’?
Analysis is a detailed examination of something in order to interpret or explain it. It is also the process of separating something into its constituent elements. Analysis is often contrasted with synthesis. In other words, analysis is the examination and evaluation of the relevant information to select the best course of action from among various alternatives. In mathematics, analysis is the part of mathematics concerned with the theory of functions and the use of limits, continuity and the operations of calculus.


What do you do when you analyze
i. Argument
We must make sure the premises and conclusion of the argument. Besides, the argument is whether a deductive argument or an inductive argument. A deductive argument asserts that the truth of the conclusion is a logical consequence of the premises; an inductive argument asserts that the truth of the conclusion is supported by the premises. If the argument is a deductive argument, we need to decide whether it is valid or invalid, and sound or not sound. An argument is valid if and only if the truth of the conclusion is a logical consequence of the premises and its corresponding conditional is a necessary truth. A sound argument is a valid argument with true premises.

It is very important to identify the argument correctly. Each premise and the conclusion are only either true or false. The sentences composing an argument are referred to as being either true or false, not as being valid or invalid. However, deductive arguments are referred to as being valid or invalid, not as being true or false. Some authors refer to the premises and conclusion using the terms declarative sentence, statement, proposition, sentence, or even indicative utterance. The reason for the variety is concerned about the ontological significance of the terms, proposition in particular. Whichever term is used, each premise and the conclusion must be capable of being true or false and nothing else.


ii. Reasoning
One approach to the analysis of reasoning is to identify various forms of reasoning that may be used to support or justify conclusions. The main division between forms of reasoning that is made in philosophy is between deductive reasoning and inductive reasoning. Formal logic has been described as 'the science of deduction'. The study of inductive reasoning is generally carried out within the field known as informal logic or critical thinking. For deductive reasoning, reasoning in an argument is valid if the argument's conclusion must be true when the premises which are the reasons given to support that conclusion are true. However, induction is a form of inference producing propositions based on previous observation. Therefore, for inductive reasoning, the truth of the premises does not guarantee the truth of the conclusion. Instead, the conclusion of an inductive argument follows with some degree of probability.


iii. Claims
First, we must try to summarize the claims. The person may or may not state this directly, but we should always state our idea of his or her main claim in our analysis as a complete sentence. Besides, we must identify and evaluate the reasons the person give for making the claims. Are they really good reasons? Are they relevant to the claims? Sometimes the person presents only one or two reasons, often spending much time developing and supporting just one reason. After that, we need to identify, analyze, and evaluate the evidence given in support of the reasons. What kinds of evidence are given whether they are data, anecdotes, citations from authorities, case or research studies? Is the evidence good which means sufficient, accurate, relevant or credible? We should question evidence in terms of both quality and quantity.


Could you analyze this problem? Solve the equation 4x³ - x⁴ = 30. If there is no solution, give reason/s.
Given 4x³ - x⁴ = 30
In order to solve this problem, a graph can be used.

Let f(x) = 4x³ - x⁴ and g(x) = 30

For f(x) = 4x³ - x⁴,
x-intersection point
f(x) = 0
So, x³(4 - x) = 0
x = 0 or x = 4

y-intersection point
x = 0
So, f(x) = 4(0) – (0)
= 0

Identify max, min or inflexion point
f(x) = 4x³ - x⁴
f’(x) = 12x² – 4x³
f’’(x) = 24x – 12x²

Let f’(x) = 0
12x² – 4x³ = 0
4x²(3 - x) = 0
x = 0 or x = 3

For x = 0,
f’’(x) = 24(0) – 12(0)
= 0
Therefore, x = 0 is an inflexion point.

For x = 3,
f’’(x) = 24(3) – 12(3)²
= -36
Therefore, x = 3 is a maximum point.






In conclusion, there is no intersection between the f(x) = 4x³ - x⁴ and g(x) = 30. Therefore, it is impossible to have a solution for this equation 4x³ - x⁴ = 30.

References:
1. InvestorWords.com. (2010). Analysis. Retrieved February 24, 2010, from http://www.investorwords.com/208/analysis.html

2. Wikipedia Foundation, Inc. (2010, February). Argument. Retrieved February 24, 2010, from http://en.wikipedia.org/wiki/Argument

3. Wikipedia Foundation, Inc. (2010, February). Reasoning. Retrieved February 24, 2010, from http://en.wikipedia.org/wiki/Reasoning





4. What do you understand by
i. Inductive reasoning in mathematics. Why is it useful to reason inductively in mathematics?
Inductive reasoning is used when you need to draw a general conclusion from specific instances. For example, a detective puts together specific clues to solve a mystery. Inductive reasoning is when you move from a set of examples to a theory that you think explains all the examples, as well as examples that will appear in the future. The simplest kind of induction looks like this:

The sun came up this morning.
The sun came up the day before that.
The sun came up the day before that.
.
.
Therefore, the sun comes up every day, and will come up tomorrow too.

Note while the conclusions induced by induction, they may be true or they may not. For example, people who visit Seattle for short periods often find that it rains every day of the visit. They could induce (or infer, or draw the conclusion) that it rains every day in Seattle. But, this would not be true. In mathematics, inductive reasoning is often used to make a guess at a property, and then deductive reasoning is used to prove that the property must hold for all cases, or for some delimited set of cases. In other words, inductive reasoning is needed when you are given a pattern and you need to come up with the rule for the pattern. For example, you might notice that every time you inscribe a triangle in a circle so that one leg lies along a diameter of the circle, it seems to be a right triangle. You might then guess that this is always the case. That is induction. Therefore, it is useful to reason inductively in mathematics.


ii. Inductive argument. Give example.
Inductive argument is a non-deductive logic. It is reasoning using arguments in which the premises support the conclusion but do not entail it. Therefore, an inductive argument is one in which the premises are supposed to support the conclusion in such a way that if the premises are true, it is improbable that the conclusion would be false. Thus, the conclusion follows probably from the premises and inferences. Here is an example:

Premise 1 : Socrates is Greek.
Premise 2 : Most Greeks eat fish.
Conclusion : Socrates eats fish.

In this example, even if both premises are true, it is still possible for the conclusion to be false (maybe Socrates was allergic to fish, for example). Words which tend to mark an argument as inductive and hence probabilistic rather than necessary include probably, likely, possibly and reasonably.


iii. Deductive argument. Give example.
A deductive argument is one which, if valid, has a conclusion that is entailed by its premises. In other words, the truth of the conclusion is a logical consequence of the premises which means if the premises are true, then the conclusion must be true. It would be self-contradictory to assert the premises and deny the conclusion, because the negation of the conclusion is contradictory to the truth of the premises. Therefore, a deductive argument is one in which it is impossible for the premises to be true but the conclusion false. Thus, the conclusion follows necessarily from the premises and inferences. In this way, it is supposed to be a definitive proof of the truth of the claim or conclusion. Here is an example:

Premise 1 : If pornography has a detrimental effect on one’s character, it would be best to avoid it.
Premise 2 : Pornography has a detrimental effect on one’s character.
Conclusion : It would be best to avoid pornography.

As you can see, if the premises are true, then it simply is not possible for the conclusion to be false. If you have a deductive argument and you accept the truth of the premises, then you must also accept the truth of the conclusion; if you reject it, then you are rejecting logic itself.

References:
1. The Math Forum. (2010). Logic: Definitions. Retrieved February 24, 2010, from http://mathforum.org/library/drmath/view/55620.html

2. Seward, K. (2002). Beginning Algebra Tutorial 35: Reasoning Skills. Retrieved February 24, 2010, from http://www.wtamu.edu/academic/anns/mps/math/mathlab/beg_algebra/beg_alg_tut35_reason.htm

3. Wikipedia Foundation, Inc. (2010, February). Argument. Retrieved February 24, 2010, from http://en.wikipedia.org/wiki/Argument

4. About.com. (2010). Deductive and Inductive Arguments. Retrieved February 24, 2010, from http://atheism.about.com/od/criticalthinking/a/deductivearg.htm





5. What do you notice in common about inductive reasoning in mathematics and inductive argument in real life situation?
Inductive reasoning is the process of reasoning that a general principle is true because the special cases you've seen are true. For example, if all the people you've ever met from a particular town have been very strange, you might then say "all the residents of this town are strange". That is inductive reasoning: constructing a general principle from special cases. Inductive reasoning is not logically valid. Just because all the people you happen to have met from a town were strange is no guarantee that all the people there are strange. Therefore, this form of reasoning has no part in a mathematical proof. However, inductive reasoning does play a part in the discovery of mathematical truths. For example, the ancient geometers looked at triangles and noticed that their angle sums were all 180 degrees. After seeing that every triangle they tried to build, no matter what the shape, had an angle sum of 180 degrees, they would have come to the conclusion that this is something that is true of every triangle. In summary, inductive reasoning is part of the discovery process whereby the observation of special cases leads one to suspect very strongly that some general principle is true.

On the other hand, inductive arguments in real life situation are usually based on experience or observation. In effect, then, inductive arguments are all comparisons between two sets of events, ideas, or things; as a result, inductive arguments are sometimes called analogical arguments. The point of those comparisons, or analogies, is to establish whether the two sets under consideration, similar in a number of other ways, are also similar in the way of interest to the argument. Consider this example:

Mariko says, "Every time I've seen a red-tinted sunset, the next day's weather has been beautiful. Today had a red-tinted sunset, so tomorrow will be beautiful."

In conclusion, inductive reasoning in mathematics and inductive argument in real life situation are both based on experience or observation but they are not based on logical certainty.

Reference:
1. Spencer, P. (1999). Deductive and Inductive Reasoning. Retrieved February 24, 2010, from http://www.math.toronto.edu/mathnet/questionCorner/deductive.html

2. Masher, D. (1999). Introduction to Induction. Retrieved February 24, 2010, from http://www.sjsu.edu/depts/itl/graphics/induc/induc.html





6. Where do you find similarities in application of mathematical thinking in mathematics and in real life situation?
Mathematical thinking does not mean only knowing everything related to mathematics. It is to think mathematically, to think within mathematical laws, and to be aware that it permeates everything from man’s thoughts to the depths of existence, from physics to metaphysics, from matter to energy; from body to soul, from law to Sufism. Besides, deductive and inductive reasoning of mathematical thinking is similar to the deductive and inductive argument in real life situation. Therefore, there are similarities in application of mathematical thinking in mathematics and in real life situation.

Reference:
1. Fountainmagazine.com. (2008). Mathematical Thinking. Retrieved February 24, 2010, from http://www.fountainmagazine.com/article.php?ARTICLEID=906





7. Have you being taught how to think mathematically in school or in your engineering mathematics courses?
No.





8. Do you think a more in depth knowledge of mathematical thinking will help you in your engineering mathematics courses? In what way do you think?
I think a more in depth knowledge of mathematical thinking will help us in our engineering mathematics courses. By having depth knowledge of mathematical thinking, we will be able to make deductive and inductive reasoning more easily. Both reasoning are necessary parts of mathematical thinking. Inductive reasoning is part of the discovery process whereby the observation of special cases leads one to suspect very strongly that some general principle is true. Deductive reasoning, on the other hand, is the method you would use to demonstrate with logical certainty that the principle is true. Therefore, we will be able to get the patterns, theorems or formulas to solve the mathematical problems easily by having depth knowledge of mathematical thinking.

Reference:
1. Spencer, P. (1999). Deductive and Inductive Reasoning. Retrieved February 24, 2010, from http://www.math.toronto.edu/mathnet/questionCorner/deductive.html





9. What do you understand by problem solving in mathematics?
Problem solving is an important component of mathematics because it is the single vehicle which seems to be able to achieve at school level all three of the values of mathematics which are functional, logical and aesthetic. Problem solving approach can contribute significantly to the outcomes of a mathematics education. Not only is it a vehicle for developing logical thinking, it can provide students with a context for learning mathematical knowledge, it can enhance transfer of skills to unfamiliar situations and it is an aesthetic form in itself. A problem solving approach can provide a vehicle for students to construct their own ideas about mathematics and to take responsibility for their own learning.

Reference:
1. Taplin, M. (2010). Mathematics Through Problem Solving. Retrieved February 24, 2010, from http://www.mathgoodies.com/articles/problem_solving.html





10. (a) Do you think critical thinking skill should be learned by all engineering students or a combination of critical thinking and mathematical thinking and why?
I think critical thinking skill should be learned by all engineering students or a combination of critical thinking and mathematical thinking. Critical thinking is very important in problem solving or making decision. It is an intellectually disciplined process of applying, synthesizing and conceptualizing information, examining underlying assumptions, interpreting and evaluating evidence, imagining and exploring alternatives and developing reflective criticism to reach a conclusion. Therefore, it is very important for all engineering students. On the other hand, mathematics is the most objective tool which can be used to draw a general conclusion from outcomes obtained. Mathematics is considered to be an expression of the knowledge of the All-Knowing. In the history of science, the structure and the mechanism of the universe have been explained to some extent using mathematics. Physicists have developed equations to demonstrate the structure of matter and forces in nature. An engineer who designs an artificial heart considers the equation that governs the bloodstream in a vein. An astronaut at NASA utilizes equations that describe the motion of satellites or the orbit of a spacecraft. In our contemporary world, the crucial role of mathematics is the main reason why Landon Clay, a millionaire philanthropist. In conclusion, all engineering students should be taught or guided to think critically as well as mathematically.

Reference:
1. Fountainmagazine.com. (2008). Mathematical Thinking. Retrieved February 24, 2010, from http://www.fountainmagazine.com/article.php?ARTICLEID=906


(b) Could you find a tangent parallel to x-axis on the curve y = In x? Explain your solution.





y = In x
dy/dx= 1/x

To find the tangent on the curve, let dy/dx=0.

Therefore, 1/x=0
x = ∞ or x is not defined

In conclusion, there is impossible to have a tangent parallel to x-axis on the curve y = In x.