NG CHEE LING KEW080015

1) What do you understand by
i) Thinking
Thinking is mental forms and processes. It allows beings to model the world and to represent it according to their objectives, plans, ends and desires.
ii) Critical Thinking
Critical thinking involves determining the meaning and significance of what is observed or expressed, or, concerning a given inference or argument, determining whether there is adequate justification to accept the conclusion as true. Critical thinking employs not only logic but broad intellectual criteria such as clarity, credibility, accuracy, precision, relevance, depth, breadth, significance and fairness.
iii) Creative Thinking
Creative thinking is the process which we use when we come up with a new idea. It is the merging of ideas which have not been merged before. Brainstorming is one form of creative thinking: it works by merging someone else's ideas with your own to create a new one. You are using the ideas of others as a stimulus for your own. This creative thinking process can be accidental (without using special techniques - pure use of intelligence and logical progression) or deliberate (using special techniques – brainstorming).
iv) Mathematical Thinking
Mathematical thinking is that when we doing a mathematical task and while we are doing it we pay attention to any thinking that we are doing.
v) Lateral Thinking
Lateral thinking is about reasoning that is not immediately obvious and about ideas that may not be obtainable by using only traditional step-by-step logic. It is more concerned with the movement value of statements and ideas. A person would use lateral thinking when they want to move from one known idea to creating new ideas.

2) Have u ever being thought critical thinking in school or before entering the faculty of engineering?
No.

3) What do you understand by ‘analysis’?
Analysis is the process of breaking a complex topic or substance into smaller parts to gain a better understanding of it. The technique has been applied in the study of mathematics and logic.

What do you do when you analyze
i) Argument
Argument is how we put together our own claims and how we link ideas. Critical analysis is about how we evaluate and understand other people's work, that is, their argument. We must analyze an argument based on factual information and previous theoretical claims (even if these are to be rejected) not on personal experience (which is too particular and not generalisable) or beliefs which are not grounded in fact and are therefore not generalisable.
ii) Reasoning
We must compare the important facts in the opinion with the new set of facts in the present controversy to identify similarities and differences. If there are similarities between the facts and the rule of law in the opinion and the present controversy, you can make a prediction regarding the outcome of the controversy.
iii) Claims
A claim is a conclusion which one can reach after analysis of resource material. It is more than a statement of a topic. When made a claim, we should try to support with by analyzing the evidence. We must be able to analyze information, synthesize it, and then offer a claim which we then support with references to the information.

Solve the equation 4x³ - x⁴ = 30. If there is no solution give reason/s.
let y = 4x³– x⁴– 30
y’= 12x²– 4x³
When y’ = 0, x = 3.
When x = 3, y = -3
As x approaches infinity, y approaches negative infinity. As x approaches negative infinity, y approaches negative infinity. It means that y does not intersect with the x-axis.
Therefore, the equation 4x³ - x⁴ = 30 has no solution.

4) What do you understand by
i) Inductive reasoning in mathematics. Why is it useful to reason inductively in mathematics?
Inductive reasoning progresses from observations of individual cases to the development of a generality. If a child puts his or her hand into a bag of candy and withdraws three pieces, all of which are red, he or she may conclude that all the candy is red. Inductive reasoning, or induction, is the process by which a general conclusion is reached from evaluating specific observations or situations.
Inductive reasoning is useful to reason inductively in mathematics as it facilitates problem solving, learning, and the development of expertise. It is fundamental to the learning and performance of mathematics, and is, therefore, an important process to investigate to gain a deeper understanding of mathematical cognition.

ii) Inductive argument. Give example.
Inductive argument is usually described as moving from the specific to the general. It is based on experience or observation is best expressed inductively. It 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:
1. Socrates was Greek. (premise)
2. Most Greeks eat fish. (premise)
3. Socrates ate fish. (conclusion)

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.
Deductive argument begins with the general and ends with the specific. It is based on laws, rules, or other widely accepted principles are best expressed deductively. 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 (conclusion). Here is an example:

1. All men are mortal. (premise)
2. Socrates was a man. (premise)
3. Socrates was mortal. (conclusion)

As you can see, if the premises are true (and they are), then it simply isn't 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.


5) What do you notice in common about inductive reasoning in mathematics and inductive argument in real life situation?

Inductive reasoning is making conclusions based on patterns observed. The conclusion reach is called a conjecture. Conjecture is an unproven statement that is based on observations. In mathematics, we are able to find and describe visual and numerical patterns by using inductive reasoning. Real-life arguments are often inductive; in real life situation inductive argument examines our ability to reach general conclusions based on perceived patterns observed in specific events.

6) Where do you find similarities in application of mathematical thinking in mathematics and in real life situation?
In application of mathematical thinking in mathematics, student are learn to gather, graph, and analyze data, make conjectures, find alternative methods, and discuss connections. In real life situation, we verify conclusions based on logical reasoning that it is true in all cases and solve real life problems using a variety of strategies in application of mathematical thinking.

7) Have you been taught how to think mathematically in school or in your engineering mathematic courses?
No.

8 ) Do you think a more in depth knowledge of mathematical thinking will help you in your engineering mathematic courses? In what way do you think?
Yes. It improves my willingness to try problems and improve my perseverance when solving mathematical problems. Besides, it makes me aware that many problems can be solved in more than one way and improves my ability to get more correct answers to problem.

9) What do you understand by problem solving in mathematics?
In solving any mathematical problems, it helps to have a working procedure. Before we can solve a problem we must first understand it. Read and re-read the problem carefully to find all the clues and determine what the question is asking you to find. What is the unknown? What are the data? What is the condition? Once we understand the question and the clues, it's time to use our previous experience with similar problems to look for strategies and tools to answer the question. After deciding on a plan, we should try it and see what answer we come up with. See clearly that whether the step is correct
and try to prove that the step is correct. Once we've tried it and found an answer, go back to the problem and see if we've really answered the question. Sometimes it's easy to overlook something. If we missed something check the plan and try the problem again.

10) Do you think critical thinking skill should be learned by all engineering students or a combination of critical thinking and mathematical thinking and why?
A combination of critical thinking and mathematical thinking should be taught as it improve engineering students' self-concepts with respect to the abilities to solve problems and make them aware of the problem-solving strategies and the value of approaching problems in a systematic manner. Moreover, it might improve students' abilities to select appropriate solution strategies and also to implement solution strategies accurately.