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Post  LEE JIAN HSIEN KEW080012 on Tue 02 Mar 2010, 6:45 am

What do you understand by
(i) Thinking
(ii) Critical thinking
(iii) Creative thinking
(iv) Mathematical thinking
(v) Lateral thinking

(i) Thinking is processes, judgement, reflection that allows beings to model the world and to represent it according to their objectives, plans, ends and desires. Thinking is the act or practice of one that thinks; thought. It is also a way of reasoning or judgement. Accoding to Collins English Dictionary (6th ed., 2003) thinking is defined as opinion or judgement; the process of thought.

The word thinking is often used loosely to refer to almost anything that goes on in our heads, it can be day-dreaming, imagination, guessing, remembering or understanding. In psychological theory and research, however, the term has acquired a more restricted meaning and has become identified with problem-solving.

(ii) Critical thinking is a set of conceptual tools with associated intellectual skills and strategies useful for making reasonable decisions about what to do or believe. This is convergent thinking. It assesses the worth and validity of something existent. It involves precise, persistent, objective analysis. When teachers try to get several learners to think convergently, they try to help them develop common understanding.

(iii) Creative thinking is the process which we use when we come up with a new idea. It is the merging of ideas. This is divergent thinking. It generates something new or different. It involves having a different idea that works as well or better than previous ideas. Brainstorming is one form of creative thinking: it works by merging someone else’s ideas with our own to create a new one. Creative thinking process can be accidental or deliberate.

(iv) Mathematical thinking is a cognitive approach to a problem that is both logical and mathematically sound. This is the an important thinking for solving mathematics problem, of course.

(v) Lateral thinking is a way of thinking “around” a problem. Lateral thinking generates the ideas and vertical thinking develops them. Lateral thinking is then sometimes used as a synonym for creative thinking or idea discovery.

Edward de Bono coined the term ‘lateral thinking’ to describe the shift needed to take you to a different, more creative mode of thinking. In terms of digging a hole, ‘vertical thinking’ would represent digging more deeply in the same place and staying within the same perceptual framework. By contrast, lateral thinking represents the willingness to move to a different position and a different perceptual framework and start digging a fresh hole.

Have you ever been taught critical thinking in school or before entering the faculty of engineering?

No. Critical thinking is not in the syllabus of primary and secondary school of any subject. Perhaps teachers were teaching critical thinking indirectly, or else the decisions that I have made so far were not based on thinking critically.

Question 3
What do you understand by ‘analysis’?
What do you do when you analyze
(i) Argument
(ii) Reasoning
(iii) Claims
Could you analyze this problem?
Solve the equation 4x^3-x^4=30. If there is no solution give reason(s).

Analysis is the (1) the separating of any material or abstract entity into its constituent elements, (2) a process as a method of studying the nature of something or of determining its essential features and their relations, (3) a philosophical method of exhibiting complex concepts or propositions as compounds or functions of more basic ones, etc.

(i)Analyze based on the truths and myths on what comes up from our mind. Analyzing arguments based on fact-check process if there is any statistics.
(ii) Reasoning is the process of forming conclusions, judgements, or inferences from facts or premises. Analyzing reasoning should be again based on facts, statistics, etc that could support the conclusion/judgement/inference.
(iii) Claim is an assertion of a right or an alleged right, or an assertion of something as a fact. We should analyze claims based on source of the claims, whether it is reliable source. Also, analyze based on facts, information, experiences that have been gathered. Observation is another important aspect.

First of all, we see that the equation 4x^3-x^4=30 can be written in one of the many forms:-

From the first form, we can further factorize to
x^3 (4-x)=30
For x=0 or x=4, the equation will yield 0=30 which is a contradiction. Thus, x=0 and x=4 is not a solution.
For x<0, the term x^3 will yield negative value on the LHS. Thus, The solution is not lying on x<0.
For x>4, the term (4-x) will yield negative value on the LHS. Thus, the solution is not lying on x>4.
Thus, we can guess that if there is a solution for this equation, then the solution should be in the range of
Now, let f(x)=x^4-4x^3+30
From the mathematical theorem, we know that if the solution of f is in between x_1 and x_2 then f(x_1 )f(x_2 )<0.
x f(x)
1 27
2 14
2.5 6.5625
2.99 3.00179201
3 3
3.01 3.00180801
3.5 8.5625

Notice that f(x)>0,∀(0<x<4). We thus conclude that there is no solution for 0<x<4.
Summarizing, we found that the solution is not lying on
x=0 and x=4
We can now safely conclude that there is no solution for the equation 4x^3-x^4=30.

Alternatively, by using differentiation method,
Let f(x)=x^4-4x^3+30
f^' (x)=〖4x〗^3-12x^2=0
〖4x〗^2 (x-3)=0
When x=0,f(0)=30
When x=3,f(3)=3

f^'' (x)=〖12x〗^2-24x
f^'' (0)=0 and f^'' (3)=36>0
∴(3,3) is a minimum point. But for x=0, we must investigate the sign of f^' (x) on either side of the point in order to know whether it is a maximum or minimum point or point of inflexion.
Values of x –1 0 +1
Sign of f^' (x) – 0 –
Change of slope of tangent \ – \

From the table above, we conclude that (0,30) is a point of inflexion.
Now, we realize that the graph is decreasing from y=+∞ at x=-∞ to y=30 at x=0. Then the graph continues to decrease to the point (3,3) and thereafter increasing towards +∞. Notice that the graph does not cross at x-axis for the range of (-∞,+∞).

Thus, we conclude that there is NO solution for the equation 4x^3-x^4=30.

Question 4
What do you understand by
(i) Inductive reasoning in mathematics. Why it is useful to reason inductively in mathematics?
(ii) Inductive argument. Give example.
(iii) Deductive argument. Give example.

(i) Inductive reasoning is a type of reasoning that involves moving from a set of specific facts to a general conclusion. In mathematics branch, there is a tool of proof called mathematical induction which is used to prove the validity of a more general conclusion giving a specific case. Thus, induction is important in mathematics in terms of proofing.

(ii) Arguments can be separated into two categories: deductive and inductive. 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:

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) 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 a classic 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.

Taken from Deductive and Inductive Arguments: What’s the Difference? at http://atheism.about.com/od/criticalthinking/a/deductivearg.htm

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

Both analyze from a more specific case to a more general conclusion.

Question 6
Where do you find similarities in application of mathematical thinking in mathematics and in real life situation?

We encounter mathematics in our life every day, whether intentionally or not. Both in mathematics and real life, we are using mathematical thinking to solve problems. There might have more solutions for each problem.

Question 7
How you ever being taught how to think mathematically in school or in your engineering mathematical course?

There is no one subject specifically taught me on how to think mathematically. However, I think my teacher has taught me implicitly, so that I can pass my mathematics subject. In varsity level, our mathematics lecturer assumes the students to have mathematical thinking itself in their mind, thus, no emphasis is given in teaching this type of thinking.

Question 8
Do you think a more depth knowledge of mathematical thinking will help you in your engineering mathematics courses? In what way do you think?

Of course it will. Mathematical thinking is essential in solving mathematical problems. Without mathematical thinking, I am sure I could not solve mathematical problems in smooth.

Question 9
What do you understand by problem solving in mathematics?

Explicitly, solve a mathematical problem. Solve problems by variety of theorems, definitions, rules, etc that are available, coupled with mathematical thinking, to find a solution for a mathematics problem.

Question 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?

Both critical and mathematical thinking are important not only for engineering students but also for other courses of students. In fact, any good types of thinking should be in the students’ mind. Critical thinking is important when comes to reasoning and decision-making. Mathematical thinking is paramount importance in solving advanced mathematics problems.

Question 11
Is there any tangent parallel to the x-axis for the curve y = ln x?

After differenting y, we get
dy/dx = 1/x
Equating dy/dx to zero,
1/x = 0
1 = 0
which is a contradiction. Thus, there will be no tangent for the curve to be parallel to the x-axis.


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