Transcript Document

APEC-Khon Kaen International Symposium August 2007
Hue University’s
College of Education
32 Le Loi St.
HUE City, Vietnam
APEC-Khon Kaen International Symposium August 2007
A LESSON THAT MAY DEVELOP
MATHEMATICAL THINKING
OF PRIMARY STUDENTS IN VIETNAM
Dr. Tran Vui
Hue University, Vietnam
APEC-Khon Kaen International Symposium August 2007
FIND TWO NUMBERS
THAT THEIR SUM AND
A RESTRICTED CONDITION ARE KNOWN
12 bottles containing 33 liters
Teachers’ Mathematics Background
x
Find
y
and
such that:
+
= 12.
and
2
x
 5 = 33 (Condition)
+
y
Teachers’ Mathematics Background in
Solving System of Equations
Time (1) by 2 means that:
If all ... are ... , ... then
x + y = 12
2x + 5y = 33
(1)
2x + 2y = 24
2x + 5y = 33
3y = 9
MATHEMATICAL
THINKING
(UNIVERSAL)
SMT
TMT
SOCIAL-CULTURAL CONTEXT
How Do TEACHER
DEVELOP
STUDENTS’ Mathematical Thinking
In a CLASSROOM Setting?
(with Teacher’s Background in Solving System
of Equations and Beliefs)
SOME BACKGROUNDS ON
THE REFORM OF
VIETNAMESE EDUCATION
BEFORE DISCUSSING THE RESEARCH LESSON
CHANGING CURRICULUM & TEXTBOOKS
1985
2000
Old Cur.: Academic,
Logic, Proof,
2006
Algorithms
1995
Current Cur. Less
Academic, Skills,
Techniques
2001
5-year Pilot
Study
Reform Cur.
Problematic Situation,
Mathematical Thinking
through PS.
WE BELIEVE FOLOWING PRINCIPLES
Students’ mathematical thinking can
be:
- defined,
- taught,
- observed,
- tested,
- evaluated, and
- reported
THROUGH its products: the students’
works, talks and representations
when they solve mathematical
problems.
Challenging between
Teachers and Students
Teaching algorithms, procedures,
techniques, rules to solve difficult
problems (Practicing)
Finding answers for
structured problematic
situations (Solving)
VIETNAMESE REFORM CURRICULUM
Content Knowledge
15%-25%
Step by Step
REFORM
Mathematical Thinking Process
St.
MATHEMATICAL
thinking
HIERARCHY OF MATHEMATICAL THINKING
Active
Problem
Solving
CREATIVE
critical
Practicing
Skills &
Algorithm
Stephen Krulik, 1993
Student
basic
recall
Passive
MATHEMATICAL THINKING
understanding
input
Observing
Inquiring
Recalling
Summarizing
Symbolizing
Exploring
manipulating
processing
Analyzing
Applying
Logic Reasoning
Inducing
Deducing
Problem Solving
Investigating
Generalizing
generating
OUTPUT
Reflecting
Evaluating
Questioning
Synthesizing
BACK TO THE RESEARCH LESSON
PRACTICING
At the end of Grade 4, students know how to solve and
express solutions of problems having three operations
of natural numbers.
Example. A toy train has 3 wagons with the length of 2
cm, and 2 wagons with the length of 4 cm. Find the
length of the train?
Answer.
3 2+
2  4 = 14 (cm).
SETTING THE PROBLEM IN A REVERSE WAY
A toy train has two types of wagon: 2 cmwagons and 4 cm-wagons. This train has the
length of 14 cm including 5 wagons. Find the
numbers of 2 cm-wagons and 4 cm-wagons of
the train.
Find
and
such that:
+
= 5.
and
2
+
 4 = 14 (Condition)
ANALYSIS OF INTRODUCTORY TASK
Open-ended Task
Use 2 cm-cards and 4 cm-cards to make a toy train of 5
wagons?
PLAYING AROUND AND OBSERVING
Pupils can arrange the cards to make a train, use the
strategy "guess and check" to get many answers
MAKE A SYSTEMATIC LIST
N. of reds
0
1
2
3
4
5
N. of blues
5
4
3
2
1
0
The length
20
18
16
14
12
10
THE RELATIONSHIP BETWEEN THE LENGTH
AND THE NUMBERS OF REDS AND BLUES
If the number of red wagons increases one,
then the length of the train decreases 2 cm.
If the length of the train is given then we
can find exactly the N. of reds wagons and
N. of blues.
The length of the train is understood as a
restricted condition
T: How many red wagons and blue
wagons in your train?
S: 3 and 2. We have 3  2 + 2  4 = 14 cm.
ANALYSIS OF TASK 1
Open-ended Task
Make a train with the length of 16 cm.
MANIPULATING AND OBSERVING
Pupils can arrange the cards to make a train, use the
strategy "guess and check" to get many answers
MAKE A SYSTEMATIC LIST
N. OF REDS
N. OF BLUES
TOTAL
Students analysed number 16 as follows:
16 = 8  2 + 0  4
16 = 2  2 + 3  4
16 = 6  2 + 1  4
16 = 0  2 + 4  4
16 = 4  2 + 2  4
THE RELATIONSHIP BETWEEN THE LENGTH
AND THE NUMBERS OF REDS AND BLUES
• The number of red wagons is always even.
?
• If the number of wagons of the train is given
then we can find exactly the N. of reds
wagons and N. of blues.
• The number of wagons of the train is
understood as “a restricted condition”.
LOGICAL REASONING
T: If the train has 6 wagons, how many red wagons
and blue wagons in this train?
S: From the table I saw that this train has 4 red
wagons and 2 blue wagons.
T: If we do not make the table, can you explain your
solution?
S: If all 6 wagons are red, the train's length
decreases 4 cm. So I got 2 blue wagons.
T: Who can express the answer by using
mathematical operations?
S: (16 - 6  2) ÷ 2 = 4 ÷ 2 = 2 (blue wagons).
ANALYSIS OF TASK 2
Inducing
A train with the length of 50 cm including 20 wagons,
how many red wagons and blue wagons are there?
The teacher guided students to induce a procedure by
using the temporary assumption to solve the problem.
T: If 20 wagons are red, what is the length of the train?
S: 40 cm.
T: Why does the length decrease?
S: Because we replaced blue wagons by red wagons?
T: How many blue wagons did we replace?
S: 5 blue wagons.
T: How did you get 5?
S: (50 - 40) ÷ 2 = 5.
ANALYSIS OF TASK 3
Generalization
A train with the length of 100 cm including 36 wagons,
how many red wagons and blue wagons are there?
The teacher guided students to generalize the
procedure by using the temporary assumption to solve
the problem.
Students applied the procedure to solve Task 3.
N. of blue wagons: (100 - 36  2) ÷ 2 = 14 (wagons).
The number of red wagons: 36 - 14 = 22 (wagons).
INDUCTION
OBSERVATIONS
INDUCTION
GENERALIZATION
Creative thinking
Inductive, generalizing,
conjecturing...
GOAL 1
START
Divergent
GOAL 2
ANALYSIS OF QUIZ
Application
There are 33 liters of fish sauce contained in 2-liter
bottles and 5-liter bottles. The number of bottles used
is 12. Find the number of 2-liter bottles and 5-liter
bottles used. Known that all bottles are full of fish
sauce.
ANALYSIS OF QUIZ
Application
2
3 : Difference
5
The number of 5-liter bottles: (33 - 122) ÷ 3 = 3.
With this kind of teaching, teacher
helps students dig deeply into a
textbook problem and build up a habit
of unsatisfying with achieved results;
Encourage students to be interested in
investigating and seeking for another
solutions, and creative in learning
mathematics.
Teacher helps students develop their
mathematical thinking.
MATHEMATICAL THINKING DEFINED
IN VN CURRICULUM
INVESTIGATING TASKS
SOLVING “PROBLEMS”
PRACTICING EXERCISES, SKILLS
EXPLORING & RECALLING FACTS,
PRINCIPLES, PROCEDURES
FOUR MAIN ACTIVITIES IN A LESSON THAT
TEACHERS SHOULD FOLLOW TO DEVELOP
STUDENTS’ MATHEMATICAL THINKING (MOET
2006):
Activity 1. Examine and Consolidate the previous knowledge
involved with new lesson;
Activity 2. Teacher facilitates students explore mathematical
knowledge and construct new knowledge by themselves.
Activity 3. Students practice the new knowledge by solving
exercises and problems in the textbook and exercise book.
Activity 4. Teacher concludes what students have learnt
from new lesson and assigns the homework.
ENGAGING TO THE LESSON, THE PUPILS WILL HAVE
OPPORTUNITIES TO SHOW THEIR MATHEMATICAL
THINKING THROUGH:
• The ability of observing, predicting, rational reasoning
and logical reasoning;
• Knowing how to express procedures, properties by
language at specific levels of generalization (by words,
word formulas);
• Knowing how to investigate facts, situations,
relationships in the process of learning and practicing
mathematics;
• Developing
ability
on
analyzing,
synthesis,
generalization, specifying; and starting to think critically
and creatively.
The level of difficulty and complexity of a
problem is defined by the achievement
objectives in the standard curriculum for
each strand of mathematics.
The exercises in the practice lessons are
ranked:
- From easy to difficult,
- From simple to complicated,
- From direct practice to flexible and
combined applications.
It is possible and desirable
to call upon pupils’
Mathematical thinking
powers by offering
challenging and meaningful
Questions
Exercises
Problems
to work on