International Open Bank of Mathematical Problems and Indonesian Perspective Fadjar Shadiq FJR: 2015 Russia

PowerPoint Presented on Open Environment for Worldwide Mathematical Education Moscow, Russia 8 to 11 of April 2015 FJR: 2015 Russia

Personal Identity Place and Date of Birth: Sumenep, Education: Unesa (Surabaya Teachers Colleage) and Curtin University of Technology, Perth, WA Teaching Experience: SHS Mathematics Teacher. Instructor and Teacher Trainer Name: Fadjar Shadiq, M.App.Sc (0274)880762; & FJR: 2015 Russia

Gagne A.Facts B. Concepts C.Principles D. Skills FJR: 2015 Russia

Gagne, Facts FJR: 2015 Russia The symbol ‘  ’ is an abstraction for …. A.‘is an element of’ B.‘is not an element of’ C.‘is a subset of’ D.‘the empty set’

Gagne, Concepts FJR: 2015 Russia Which of the following mappings represent a function? A.m only B.f, j and m only C.m and h only D.m and j only f g h j k m

Gagne,Principles FJR: 2015 Russia Which of the following triangles has a hypotenuse of length  5 cm? 3 cm 2 cm 1 cm 2 cm 1 cm 4 cm A B CD 3 cm

Gagne, Skills FJR: 2015 Russia ½ + 1/3 = …. A.1/5 B.2/6 C.2/5 D.5/6

FJR: 2015 Russia Teaching and Learning of Mathematics Purpose/ Objective Assessment

The relationship of assessment and teaching process. The area of the most left hand side figure is 1 unit area. Among the four figures P, Q, R and S, which figures have 10 unit areas? FJR: 2015 Russia

The relationship of assessment and teaching process. The perimeter of the following figure is…. A. 19 cm B. 28 cm C. 29 cm D. 38 cm FJR: 2015 Russia

Even dan Ball (2009:1): “... teachers are key to students’ opportunities to learn mathematics.” Students should be the focus. Therefore we need teacher whose actions promoting student learning. FJR: 2015 Russia

In the past Mathematics is known as deductive- axiomatic subject. Children only as follower. FJR: 2015 Russia

Postulates/Axioms in Algebra Vance (19..) : Closure: a + b  R and a.b  R. Associative :a + (b + c) = (a + b) + c a.(b. c) = (a. b). c Commutative: a + b = b + a, a.b = b.a Distributive: a.(b + c) = a.b + a.c (b + c).a = b.a + c.a Identity: a + 0 = 0 + a = a, a.1 = 1.a = a Inverse: a + (  a) = (  a) + a = 0 and a.1 = 1.a = a FJR: 2015 Russia

PROVING Prove:  b + (a + b) = a Proof:  b + (a + b) =  b + (b + a)Commutative = (  b + b) + aAssociative = 0 + aInverse = aIdentity  =  5 + (3 + 5) = 3 FJR: 2015 Russia

Lakatos was quoted by Burton (1992:2) states: “Deductivist style hides the struggle, hides the adventure. The whole story vanishes; the successive tentative formulations of the theorem in the course of the proof- procedure are doomed to oblivion while the end result is exalted into sacred infallibility.” FJR: 2015 Russia

Children only as follower. How to help them to: Be Innovative Be Creative FJR: 2015 Russia

George Polya (1973: VII): “Yes, mathematics has two faces; …. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science.” FJR: 2015 Russia

Pythagoras What can you say regarding this picture Source: NCTM FJR: 2015 Russia Bruner: Discovery Learning is Learning to Discover

De Lange (2005) stated: “Mathematics could be seen as the language that describes patterns – both patterns in nature and patterns invented by the human mind.” What is Mathematics? FJR: 2015 Russia

The Four Important Questions: “How to Help Our Students to Learn Mathematics: 1.Meaningfully  easily? 2.joyfully? 3.to use their heads (think)? 4.to be an independent learner?” FJR: 2015 Russia

1.Observing 2.Questioning 3.Experimenting 4.Reasoning 5.Communicating Scientific Approach in Indonesia FJR: 2015 Russia

The PSA (Japan) and the Scientific Approach (Indonesian) 1. Problem Posing 2. Independent Solving (The first 4 steps on SA: FS) 3. Comparison and Discussion (the last step on SA: FS) 4. Summary and Integration. Source for PSA: Masami Isoda FJR: 2015 Russia

 Start with Task/Activity  Open Ended  L Let Students to Explore  see Math Attitudes (Mindset)  I Inductive, Analogy, Deductive, and others  see Math Methods in General (Source: Isoda & Katagiri, 2012:50-52) How do We Help Our Students to Think? FJR: 2015 Russia

 How to ensure that the teaching and learning of mathematics will be focused on student centre approach  to help our students to be independent learners?  Can the assessment will reinforce the teaching and learning of mathematics will be focused on student centre approach  to help our students to be independent learners? The Questions FJR: 2015 Russia

 The practice of examination will impact on practice of the teaching and learning of mathematics in the classroom.  We have to change the practice of examination in such a way to ensure that the practice of teaching and learning of mathematics will help our students to be independent learner and creative and innovative citizens.  SEAMEO QITEP in Mathematics support the effort from Russia concerning International Open Bank of Mathematical Problems. The Alternative Answer FJR: 2015 Russia

On the 1 st of January 2015, Anto save his money Rp ,00. Then on the 1 st of every next month he save Rp ,00, without interest. 1Calculate the amount of his money on the 2 nd of every next month. 2Calculate the amount of his money on the 2 nd of the next month after he save his money 21 times. 3Calculate the amount of his money on the 2 nd of the next month after he save his money 101 times. 3Calculate the amount of his money on the 2 nd of the next month after he save his money n times. On the 1 st of January 2015, Anto save his money Rp ,00. Then on the 1 st of every next month he save Rp ,00, without interest. 1Calculate the amount of his money on the 2 nd of every next month. 2Calculate the amount of his money on the 2 nd of the next month after he save his money 21 times. 3Calculate the amount of his money on the 2 nd of the next month after he save his money 101 times. 3Calculate the amount of his money on the 2 nd of the next month after he save his money n times.

FJR: 2015 Russia  A B C Find out the area of these triangle.

FJR: 2015 Russia Apply the quadratic equation formulae x 1,2 = In finding out the root of these quadratic equation a. x 2 – 2x + 1 = 0 b. x 2 – 2x – 3 = 0 c. x 2 – 2x + 3 = 0 Investigate or explore your results. Apply the quadratic equation formulae x 1,2 = In finding out the root of these quadratic equation a. x 2 – 2x + 1 = 0 b. x 2 – 2x – 3 = 0 c. x 2 – 2x + 3 = 0 Investigate or explore your results. This problem can be used before students learn ‘discriminant.’

Find the Shaded Area  Closed Open Ended Question  The Importance of Creativity and Innovation. 20cm 40cm A B C D E F H K L FJR: 2015 Russia How to Find?  Open.

(B) How to Teach Median (N = 22)? Source: Shadiq (2011) Find a vertical line to divide the number of the data into two equal parts ,59,514,5 19,524,5 29, = =8 Need 3 more data to reach 11 or 1/2 n FJR: 2015 Russia

How many squares are there in this diagram? (Isoda & Katagiri, 2012:31) How do you teach your students? What are the advantages? Disadvantages? How to improve the method? FJR: 2015 Russia Investigation

How many squares are there in this diagram? The Preferred Method (Isoda & Katagiri, 2012:31) 1.Clarification of the task #1  All of the squares 2.Clarification of the task #2  Let them to think the best way of counting (better and easier) 3.Realizing the benefit of sorting 4.Knowing the benefit of encoding (naming) 5.Validating the correctness of result 6.Coming up with a more accurate and convenient counting method FJR: 2015 Russia

The first pattern consist of three matches. How many matches are there in the tenth and hundredth pattern? FJR: 2015 Russia Investigation

How many cubes are needed in building number 4, 10, and 100? FJR: 2015 Russia Investigation

F is a midpoint of BC. ABCD is a square. If the area of quadrilateral CDEF is 45, then the area of triangle BEF is.... a. 7,5 b. 9 c. 10,5 d. 12 e. 13,5 A B C D F E G H Geometry Problem FJR: 2015 Russia This problem can be used after students learn similarity

Algebra Problem Find all the sets of consecutive natural numbers which the sum is This problem can be used after students achieved the formula of the sum of n term of Arith Series. FJR: 2015 Russia

 The practice of examination and assessment can be used to ensure that the practice of teaching and learning of mathematics will help our students to be independent learner and creative citizens.  The various kinds of assessment can be used: (1) to assess facts, concepts, principles, or skills achieved by our students; (2) problem (before or after students learn the knowledge, closed or open) and (3) exploration/investigation. Conclusion FJR: 2015 Russia

Even R.; Ball, D.L. (2009). Setting the stage for the ICMI study on the professional education and development of teachers of mathematics. In Even R.; Ball, D.L. (Eds). The Professional Education and Development of Teachers of Mathematics. New York: Springer Burton, L. (1992). Implications of constructivism for achievement in mathematics. In 7th International Congress on Mathematical Education (ICME-7). Topic Group 10; Constructivist Interpretations of Teaching and Learning Mathematics. Perth: Curtin University of Technology. Isoda, M. & Katagiri, S. (2012). Mathematical Thinking. Singapore: World Scientific. Polya, G. (1973). How To Solve It (2nd Ed). Princeton: Princeton University Press. Reference FJR: 2015 Russia