HUE City, Vietnam Hue University’s College of Education32 Le Loi St.

APEC-Khon Kaen International Symposium August 2007

A LESSON THAT MAY DEVELOP MATHEMATICAL THINKING

OF PRIMARY STUDENTS IN VIETNAM

Dr. Tran VuiHue University, Vietnam

APEC-Khon Kaen International Symposium August 2007

FIND TWO NUMBERS THAT THEIR SUM AND

A RESTRICTED CONDITION ARE KNOWN

APEC-Khon Kaen International Symposium August 2007

12 bottles containing 33 liters

Find and such that:

+ = 12.

and

2 + 5 = 33 (Condition)

x y

x y

Teachers’ Mathematics Background

Teachers’ Mathematics Background in Solving System of Equations

x + y = 12 (1)

2x + 5y = 33

2x + 2y = 24

2x + 5y = 33

3y = 9

Time (1) by 2 means that:

If all ... are ... , ... then

SOCIAL-CULTURAL CONTEXT

MATHEMATICAL THINKING

(UNIVERSAL)

TMTSMT

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

2006

2001

2000

1995

CHANGING CURRICULUM & TEXTBOOKS

1985

Current Cur. Less Academic, Skills, Techniques

Old Cur.: Academic, Logic, Proof, Algorithms

Reform Cur. Problematic Situation, Mathematical Thinking through PS.

5-year Pilot Study

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.

WE BELIEVE FOLOWING PRINCIPLES

Challenging between Teachers and

Students

Teaching algorithms, procedures, techniques, rules to solve difficult problems (Practicing)

Finding answers for structured problematic situations (Solving)

Mathematical Thinking Process

Con

ten

t K

now

led

ge

Refor

m C

urricu

lum

VIETNAMESE REFORM CURRICULUM

15%-25%

Step by Step REFORM

COMM

UNICATING

INVESTIGATING

P.SO

LVIN

Gexercising

exploring

LOGIC

AL

REASO

NIN

G

MATHEMATICAL thinking

St.

CREATIVE

critical

basic

recall Passive

Active

Student

HIERARCHY OF MATHEMATICAL THINKING

Problem Solving

Practicing Skills

&

Algorithm

Stephen Krulik, 1993

OUTPUTinput processing

MATHEMATICAL THINKING

Observing

Inquiring

Recalling

Summarizing

Symbolizing

Exploring

Analyzing Applying

Logic ReasoningInducingDeducing

Problem SolvingInvestigatingGeneralizing

Reflecting

Evaluating

Questioning

Synthesizing

understanding manipulating generating

BACK TO THE RESEARCH LESSON

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).

PRACTICING

SETTING THE PROBLEM IN A REVERSE WAY

A toy train has two types of wagon: 2 cm-wagons 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 1Open-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 2Inducing

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 3Generalization

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).

OBSERVATIONS

GENERALIZATION

INDUCTION

INDUCTION

Creative thinkingInductive, generalizing, conjecturing...

GOAL 1

GOAL 2

STARTDivergent

ANALYSIS OF QUIZApplication

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 QUIZApplication

2

5

3 : Difference

The number of 5-liter bottles: (33 - 122) ÷ 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.

INVESTIGATING TASKS

SOLVING “PROBLEMS”

PRACTICING EXERCISES, SKILLS

EXPLORING & RECALLING FACTS, PRINCIPLES, PROCEDURES

MATHEMATICAL THINKING DEFINED IN VN CURRICULUM

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