Mathematical Thinking at Monday Morning

Name     : Ika Dewi Fitria Maharani      
Mathematics Education 2011
NIM     : 11301241009

This Monday morning (March 12th, 2012 : 09.00 am) my class, Mathematics education 2011 got the lesson from My English lecture, Mr. Marsigit. He is a great and funny lecture. He always have various way to teach english, and the way of teach make english more interesting I think. The day he teach about Mathematical Thinking (Shigeo Katagiri, 2004).  Then he wrote in white board. This is my note from him ^_^
Mathematical thinking are :
1.Mathematical Attitude
2.Mathematical Method
3.Mathematical Content
Then he wrote in white board. This is my note from him ^_^
1.    Mathematical Attitude
It is different with attitude toward mathematics
Mathematics attitude  They are more specific
Attitude toward Mathematics They are more general (every people had been heard of about mathematics)
Example :
a.    Curiousity  of math problems
b.    Willingness to ask about mathematics problems



2.    Mathematical method
It covers all of you past and present experience in doing mathematics plus its theoritical  review .
Example :
a.    Basic Logic
-to compare
-to order
-to relate
-to calculate
b. Arithmatics operation
-to add
-to subtract
-to multiplied
-to divide
-to make operation
-to make higher level of arithmatics
c. Mathematics Approach
-Inductive (SpecificGeneral)
-Deductive (Generalspecific)
-Constructing
-Conjuntion
-Proving
-Looking for
-Looking for the pattern
-Problem Solving
-Contectual
-Realistic Mathematics (PMRI) from Belanda

3.    Mathematical Content
a.    4=4 identity, equivalent
Four is equal to four
Four equals four
b.    4>3 un-identity, un-equivalent
Four is greater than three
Four is more than three
c.    4≥4
Four is equal to four or Four is greater than four
d.    a˂b
a is less than b
a is smaller than b
e.    a≤b
a is smaller than b or a is equal to b
f.    4x+3 algebraic term
g.    4x+3=0 algebraic equation
4 = content/coeficient 
X = variable
What is the main problem of everything related to algebraic equation ?
The main problem is to find the solutions

ax+b linier equation –x
ax^2+bx+c quadratic equation –x
example : find the solution of the equation 4x+3=0
Solution :
We have the equation 4x+3=0
Substract both sides with -3, then we get :
4x+3-3=0-3
4x=-3
Divide both side by 4, then we get :
4x=-3
_____ : 4
X = -3/4
Then we get the solution as x equals -3/4
X=-3/4

           Then he asked us to search Mathematical Thinking from Google. So, This is my result of  Mathematical Thinking by Sigeo Katagiri. There are types of Mathematical Thinking. The types are :

1.] Mathematical Attitude
1.)  Attempting to Grasp One’s Own Problems or Objectives or Substance Clearly, by
Oneself
1. Attempting to have questions
Example : what
What is the real height of the moment? (at page 35)
2. Attempting to Maintain a Problem Consciousness
Example : Exercise
Do the following exercise correctly. (at page 137)
3. Attempting to Discover Mathematical Problems in Phenomena
Example : Mention
Mention the couples of propotional side. (at page 53)

2.) Attempting to Take Logical Actions
1. Attempting to Take Actions that Match the Objectives
Example: should be
Then the sides of the buildings should be in form of curve. (at page 55)
2. Attempting to Establish a Perspective
Example: Considering
By considering the requirements of the similarity of two planes figure. (at page 31)
3. Attempting to Think Based on the Data that Can Be Used, Previously Learned Items, and
Assumptions
Example: Assume
The perimeter of the surrounding watermelon is 62,8 cm. (The watermelon is assumed as a sphere) (at page 81)

3.) Attempting to Express Matters Clearly and Succinctly
1. Attempting to Record and Communicate Problems and Results Clearly and Succinctly
Example: Symbols
The slant height (s) of the cone can be determined as follows : (at page 68)
2. Attempting to Sort and Organize Objects When Expressing Them
Example: Table
Classified data frequency distribution table of the data above is as follow : (at page 101)

4.) Attempting to Seek Better Things
1. Attempting to Raise Thinking from the Concrete Level to the Abstract Level
Example: Sketch
Two flag poles have their langth of shadow respectively x m and (x+12) m. (at page 45)
2. Attempting to Evaluate Thinking Both Objectively and Subjectively, and to Refine
Thinking
Example: Method
First method:
23 x 25 = 2x2x2 x 2x2x2x2x2 = 2x2x2 x 2x2x2x2x2= 28
Second method:
23 x 25= 23+5=28 (page 151)
Seek what is better, first method or second method?
3. Attempting to Economize Thought and Effort
Example : Pattern
The pattern of  square numbers is 1,4,9,16,…
The figure of the pattern of  square numbers are given as follows. (at page 182)

2.] Mathematical Method
1. Inductive Thinking
Example : Induction
Mrs. Tuti tastes 1 spoon of soup from a bowl of soup. (at page 133)
2. Analogical Thinking
Example: correspond
Due to the three corresponding angles on triangle PQR and triangle XYZ are in the same size so triangle PQR and triangle XYS are similar (at page 39)
3. Deductive Thinking
Example: Probability
If from 1 pack of cards is taken 1 card randomly (at page 135)
4. Integrative Thinking
Example : Sum
The sum of the beads that will be received by Desta in March 16st (at page 205)
5. Developmental Thinking
Example : Probability
The probability of mutually exclusive events is as follow : (at page 127)
6. Abstract Thinking
Example : Exponent
Exponent zero, a exponent zero equals 1(page 153, where a not equals zero.
7. Thinking that Simplifies
Example : Data in table
Find the median of the single data frequency distribution table below. (at page 111)
8. Thinking that Generalizes
Example : Pattern
n=1, the 1st order=2(1) - 1=1
n=2, the 2nd order= 2(2) - 1=3
n=3, the 3rd order= 2(3) - 1=5. (at page 177)
9. Thinking that Specializes
Example : Side and angle
Indicating that two triangles are congruent in the case of (side, angle, side) (at page 19)
10. Thinking that Symbolize
Example: Example
if x^2 = a (at page 157)
11. Thinking that Express with Numbers, Quantifies and Figures
Example: Quantifies
JK = MN = 3(at page 23)

3.] Mathematical Content
1. Clarifying Sets of Objects for Consideration and Objects Excluded from Sets, and
Clarifying Conditions for Inclusion (Idea of Sets)
Example : Consider
Consider triangle JKL and triangle MNO (at page 23)
2. Focusing on Constituent Elements (Units) and Their Sizes and Relationships (Idea of
Units)
Example: Unit
 A box constain 4 red marble, 5 white , and 6 green ones. (at page 135)
3. Attempting to Think Based on the Fundamental Principles of Expressions (Idea of
Expression)
Example: expression
The expression of a^n can be written as follow  (at page 151)
4. Clarifying and Extending the Meaning of Things and Operations, and Attempting to
Think Based on This (Idea of Operation)
Example : Properties
The properties of square root (second root) is given as follow (at page 157)
5. Attempting to Formalize Operation Methods (Idea of Algorithm)
Example : Arranged
Arrange the classes from the lowest to the highest ; 79,49,48,...,88 (at page 91)
to arrange the data we must consider that 49 is higher than 48, 88 is higher than 79, etc . We can call it,the idea of Algoritm.
6. Attempting to Grasp the Big Picture of Objects and Operations, and Using the Result
of this Understanding (Idea of Approximation)
Example : Chart
Consider of circle chart explaining favorite sport from 1.000 people below (at page 99)
7. Focusing on Basic Rules and Properties (Idea of Fundamental Properties)
Example: Rationalize
Rationalizing the roots form. (at page 163)
8. Attempting to Focus on What is Determined by One’s Decisions, Finding Rules of
Relationships between Variables, and to Use the Same (Functional Thinking)
Example: Relation
The relation between the velocity and height of the man from the ground is formulated by v equals root of 2 multiplied by g and h... (at page 149)
9. Attempting to Express Propositions and Relationships as Formulas, and to Read
their Meaning (Idea of Formulas)
Example: Formula
The relation between the velocity and height of the man from the ground is formulated by v equals root of 2 multiplied by g and h... (at page 149)