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: , 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 strategy 2. To allow different ways of solving a problem 3. To see what could we do to make hypothesis 4. To see what could we do to make a conjecture 5. To observe the beauty of mathematics 6. 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 people Let 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 B C D E 2 1 D E C D E 1 D E

Data (choose from) …(sequence) 1x1 1x2 1x3 1x4 1x5 2x1 2x2 2x3 2x4 3x1 3x2 3x3 4x1 4x2 5x1

Presenting data The number of delegations that could be formed appear in the following 1 st sequence ? ? ? … ?

If from this 1 st sequence the n th term is Un then we can find a formula for Un, where Un = n (n+1)(n+2)/6

order number chose 3 from … … … 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 n + 2 = m n + 1 = m – 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