Jozua Sabandar

Widyatama University, Bandung - Indonesia

I don’t know what may appear to the world but to myself I seem to have been only like a boy playing on the sea shore and diverting myself in now and then finding a smoother

peble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me ( Newton)

Do one at a time

What we need to do with the teachers and the learners to become long life learners??

To model to them how to appreciate mathematics , emphasize on characters , sense making of math and how useful and beautiful math is

Teaching and Learning Mathematics related

to Long Life Learning should allow, among others, the math teachers work on math problems in such a way that the learners and the teachers as well could sense the presence of mathematics characteristics and its impacts on their characters and see how beautiful, useful, meaningful mathematics is, that they want to teach and learn math more in the future

What we need to do now with our teachers is, among others, to model to them how to appreciate mathematics and its effects on characters and how to become long life learners??

Very often mathematics teachers will stop the learning process once their students already solve problems , and everybody is happy.

It is good but certainly it is not good

enough… which means that they should do better than that.

They should refer to one of the

characteristics of mathematics that mathematical problems could be solved in more than one way,,… they should be flexible in dealing with mathematics problems

Observe the following situation and its problem

A delegation consists of 3 individuals will be selected from a group of people

How many delegations can be formed, if the delegation is selected from a group of:

1. 3 2. 43. 5 4. 6 5. 7, 8, 9 … or n people.

These consecutive questions (tasks ) are intended for certain purposes. These questions are deliberately posed, among others are:

1. to allow informal strategy2. To allow different ways of solving a problem3. To see what could we do to make

hypothesis4. To see what could we do to make a

conjecture5. To observe the beauty of mathematics6. To see connections among mathematical

concepts

-Applicability and effectiveness,- abstraction and generality,

-simplicity, -Logical derivation,

-Axiomatic arrangement,- precision,

-correctness. (Ebrahim, 2010).

1.To select from 3 people Let the group has A, B, C , then there will be only one possible choice….. namely

. A B C

B. To select from 4 people

Let the group has A, B, C, D …then there will be 4 possible choices….. namely

A B C A C D ABC, ABD, ACD, BCD

A B D B C D

4 choices

3.To select from 5 peopleLet the group has A, B, C, D , E then there will be 10

possible choices….. Namely

A B C A C D A D E ABC A B D A C E ABD A B E ABE

ACD B C D B D E ACE B C E ADE

BCD C D E BCE

BDE

CDE

10 choices

4. To select from 6 people A B C A C D A D E A E F

A B D A C E A D F A B E A C F A B F

B C D B D E B E F B C E B D F B C F

C D E C E F C D F

20 choices D E F

Tree diagrams A B C 1 B C

A B C D C D 2 1 B C D 1 C D

A B C D E C D E D E 3 2 1 B C D E 2

1 D E C D E 1 D E

Data3 4 5 6 7

(choose from) 1 2 1 3 2 1 4 3 2 1 5 4 3

2 1 1 2 1 3 2 1 4 3

2 1 1 2 1 3

2 1 1

2 1

1 1 4 10 20 35 …

(sequence)

1x1 1x2 1x3 1x4 1x5 2x1 2x2 2x3 2x4 3x1 3x2 3x3 4x1 4x2 5x1

Presenting dataThe number of delegations that could be formed

appear in the following 1st sequence

1 4 10 20 35 56 84 ..?

- - - - - - -

3 6 10 15 21 28 36 ...?

- - - - - - 3 4 5 6 7 8

..? 1 1 1 1 1

… ?

If from this 1st sequence the nth term is Un then we can find a formula for Un,

where Un = n (n+1)(n+2)/6

order number chose 3 from 1 1 3 2 4 4 3 10 5 4 20 6 5 35 7 6 56 8 7 84 9 … … …

n n(n+1)(n+2)/6 m(m-1)(m-2)/6 m! n = 1, 2, 3,… m = 3 , 4, 5, . . . (m-3)!.3!

Look at the following table

Observe the following correspondence

1 ------- 3 2 ------- 4 3 ------- 5 4 ------- 6 n

+ 2 = m 5 -------- 7 n

+ 1 = m – 1 . . . . . . . . . . . . . . . n

= m – 2 n -------- m

(= n+2 )

By doing this, the teachers and the

students already articulated the Polya’s heuristics principle as well as Newton’s view about science

This attitude (flexibility) certainly will support the learners ( teachers and students) to open another opportunity to accommodate their new learning/knowledge into their cognitive structure or to retain the new knowledge , or even to strengthen their existing knowledge

Otherwise they might lost such a good momentum

Participants Impressions In a number of presentations, the audiences’

( teachers, student teachers, teacher trainers) responds are typically the same.

I will try to do this with my students and I think my student will like it. Enjoy the beauty of math

I wished I experienced this stuff when I was a student

Thanks God, that I could see how wonderful mathematics is.. I have never seen this before and I hope my lecturers should know how to do

it

  Very good , accurate, systematic and

impressive I really need how to solve something

in different way Alhamdulilah, in this session we really

got a relatively new style in presenting a topic

The teaching material is very useful

and meaningful for the participants

  

Show us a meaningful approach how a math teacher always use different strategies in solving math problem.

We are very happy and it is very pleasant

to have another presentation from you next time.

We got something new through this

training This lesson is very impressive and we

learned new things Very happy to have you here with us and

got something new that we could refer to in designing mathematical instruction. Thank you so much

A Simple Design

Teachers’ role and Learners’ engagement

Beauty, Wonder of math Math, good character

Further learning

1.Math Topics/ Problems

2.Characteristics of Math

3.Learning Objective

Tips for math teachers

Be a good teacher: good teaching facilitator Motivator, helpful, Present the beauty of math, through doing

math Be a good designer of teaching material Have skill in using technology Be aware of moments that give teacher

opportunity to respond regarding characters to learn something beyond what has been learned