Teaching mathematics is not simple. It is needed
some methods, creativity, and knowledge. It is also needed some experience from
applying mathematics. To understand about it, then we should study about
mathematical thinking. Mathematical thinking is said by Shigeo Katagiri in
2004. It include 3 matters. First, Mathematical
attitudes. Second, mathematical thinking related to mathematical methods.
Third, mathematical thinking related to mathematical contents.
Let see
the explanation and the example about the 3 matters of mathematical thinking in
Marsigit’s book, that is in Mathematics 3
for Junior High School Year IX.
1. Mathematical Attitudes
Mathematical attitude is different
with attitude toward mathematics. Mathematical attitude, they are more
specific. Attitude toward mathematic, they are more general.
Attempting
to Grasp One’s Own Problems or Objectives or Substance Clearly, by
Oneself
- Attempting to have questions
-
Attempting to
Maintain a Problem Consciousness
-
Attempting to
Discover Mathematical Problems in Phenomena
*Example:
page 35 number 1
A frame of picture is in the form of rectangle by
the size of outside edge 30cm x 20cm. if the edge of the picture frame is
attached by additional frame of 5 cm wide, decide whether the outside edge of
rectangle of the picture is similar with the inside edge of rectangle. Explain
your answer.
Attempting
to Take Logical Actions
-
Attempting to Take Actions that Match the Objectives
-
Attempting to Establish a Perspective
-
Attempting to Think Based on the Data that Can Be Used, Previously Learned
Items, and
Assumptions
*Example:
page131 number 1.a and 1.b
A
six-sided dice was rolled once. Find (a) the probability of obtaining the
even-spot side or the odd-spot side. (b) the probability of obtaining the
even-spot side or the side with spot greater than 3.
Attempting
to Express Matters Clearly and Succinctly
-
Attempting to
Record and Communicate Problems and Results Clearly and Succinctly
- Attempting to Sort and Organize
Objects When Expressing Them
*Example:
page 141 number 16
A
cone has a slant height of 7cm and radius 3,5 cm. The surface area of the cone
is …
Attempting
to Seek Better Things
-
Attempting to
Raise Thinking from the Concrete Level to the Abstract Level
- Attempting to Evaluate Thinking
Both Objectively and Subjectively, and to Refine
Thinking
-
Attempting to Economize Thought and Effort
*Example:
page 213 number 20
The
sum of 8 – 4 + 2 - … - ¼ is …
2. Mathematical Thinking Related to
Mathematical Methods
It cover all of your past and
present experiences in doing mathematics plus its theoretical review.
Inductive Thinking
Page 195 number 1.a:
Find the 21st term of the following
arithmetic sequences. a) 3, 7,11,15, …
Analogical
Thinking
Bab 1 page 49
number 11:
The proportion between the length of two similar
sides is 2:3.If the diagonal length of the small rectangle is 30 cm, so the
length of the diagonal of the bigger rectangle is …
Deductive
Thinking
Page
231 number 11:
In
a class, 25 people join in basketball training, 35 people join in table tennis,
and 15 people join both of them. If 3 people in the class do not join in any
activities, the amount of students in the class is …
Integrative
Thinking
Page 133number 5:
The mean of the art test score from a certain group
of students is 75. After a new member join the group, the average turns into
73. The art test score of the new member is …
Developmental
Thinking
Page
45 number 3:
Two
flag poles have their length of shadow respectively x m and (x+12) m. If the
length of the shorter pillar is 1/3 the length of the higher pillar, calculate
x.
Abstract
Thinking
Page 17 Example:
Show that ΔPQY congruent ΔRQY.
Thinking
that Simplifies
Page
170 number 6:
(-a^-3)^3
(-a^5b^5)^2
Thinking
that Generalizes
Page
179 number 1:
Find
the sum of 2+4+6+8+ …
Thinking
that Specializes
Page 47 number 5:
ΔDEF and XYZ are congruent. The length of YZ is ...
Thinking
that Symbolize
Page
225 number 25:
The
values of a and b of the sequence 41, a, 55, b, …
Thinking
that Express with Numbers, Quantifies and Figures
Page
213 number 3:
In
5 weeks, Budi has been training to face the marathon competition. Each week he
has to run twice time further than the week before. In the 3rd week
he runs 3km. Determine the total distance for Budi in five weeks training.
3. Mathematical Thinking Related to Mathematical
Contents
Clarifying Sets of Objects for Consideration and
Objects Excluded from Sets, and Clarifying Conditions for Inclusion (Idea of
Sets)
Page 15 Example:
Let ΔAB Congruent
ΔDEC as shown in the figure. Determine the congruent sides and angles of
both triangles.
Focusing on Constituent Elements (Units) and Their
Sizes and Relationships (Idea of Units)
Page 77 number 5:
An orange is sliced athwartly in equal size. It is
found that the orange diameter is 7 cm (the orange is assumed as spherical
shape). Determine the volume of the half of orange.
Attempting to Think Based on the Fundamental
Principles of Expressions (Idea of Expression)
Page 169 number 1:
Rationalize the following roots. 1/2√5 - √3
Clarifying and Extending the Meaning of Things and
Operations, and Attempting to Think Based on This (Idea of Operation)
Page 237 number 39:
A ladder has steps height from the ground of 15 cm,
25 cm, 35 cm. … If the ladder has 25 steps, the height of the last step from
the ground is …
Attempting to Formalize Operation Methods (Idea of
Algorithm)
Page 235 number 29:
The solution of 3x + 2y = -5 and 4x – y = 19 is p
and q. The value of p + q is …
Attempting to Grasp the Big Picture of Objects and
Operations, and Using the Result of this Understanding (Idea of Approximation)
Page 141 number 14:
A biscuit can has diameter 20 cm and height 10 cm.
The surface area of the biscuit can is …
Focusing on Basic Rules and Properties (Idea of
Fundamental Properties)
Page 155 Example:
Covert the following negative exponents into
positive exponents. 1) 2^-5 answer is ½^5
Attempting to Focus on What is Determined by One’s
Decisions, Finding Rules of Relationships between Variables, and to Use the
Same (Functional Thinking)
Page 35 number 5:
At the TV screen, it was seen a monument of 10 cm
high and 4 cm wide. If the real width of the monument is 10 m, what is the real
height of the monument?
Attempting to Express Propositions and Relationships
as Formulas, and to Read their Meaning (Idea of Formulas)
Page 185 Example number 2:
The 25th term of the square numbers is
n^2 = 25^2 = 625
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