Katha 2014: Understanding Math: Elementary CPA

KATHA 2014: Day 4:

Understanding Math: Elementary CPA

Ms. Pauline Mangulabnan

De La Salle University Manila

August 2, 2014

Minutes of the Workshop

1. Introduction. The speaker started the discussion by letting the teacher-participants construct tests, especially problem solving sets, because such tests are considered as one of the weaknesses of both the teachers and the students. She also conducted a seatwork on the examination of how Japan writes problems in Math; that is, by rewriting the problems into multiple choice word tests. She then conducted an activity of grouping the teachers by year level and discuss three questions on process and understanding.

2. Problem Solving Heuristics. It is easy to create poor processes in Math but very difficult for understanding. There should be a twin focus to Math: skills-content and attitude. Values should be developed hand-in-hand with problem solving strategies. The main goal is to understand that Math goes beyond formula and procedures, and the Philippines has a culture of intellectualism. According to Paul Zeits, there are two approaches to Math teaching: exercises and problem. In Exercise, it tests the students’ mastery of a narrowly focused technique. Problems can not be answered immediately. It demands much thought and resourcefulness is a must before proceeding to the right approach. This is the most common approach in the Philippines. The speaker said that “To solve a problem is to find a way where no way is known,” according to George Polya.

3. Through PS. This acquires ways of thinking and habits of persistence and curiosity, and finally confidence in unfamiliar situations. Polya’s Four Step Approach include 1) Understand 2) Devise a plan to solve the problem 3) Carry out the plan and 4) Reflect. However there are common difficulties which include the inability to read and comprehend the problem, misinterpretation of the conditions of the problem, lack of strategy knowledge, inability to translate or formulate the math form of the problem and computational errors, carelessness and imperfect mathematical knowledge. There are three concepts of PS lessons: teaching through, about and for PS.

4. Activity. The speaker conducted a task for the participants and handed them a worksheet to be accomplished individually which would serve as their output.

Katha 2014: Understanding Math: CPA and Mental Computations of SG

Minutes of the Workshop transcribed. Speaker Ms. Pauline Mangulabnan

KATHA 2014: Day 3:

Understanding Math: CPA and Mental Computations of SG

Ms. Pauline Mangulabnan

De La Salle University Manila

July 26, 2014

Minutes of the Workshop

1. Introduction. The speaker started with the review of the past lectures. The aim of the lecture was to identify CPA in Math and what are the principles of Singaporean Math. One is to boost the self-esteem of the students in Math and be able to hone their mathematical skills. In the Philippines, math skills do not reflect high self-esteem. In Singapore on the other hand and other East Asian countries, math skills are important.

2. Singapore Math. According to Dinyal, the success of the SG curriculum is on the following: 1) intended curriculum, where the SG Ministry of Education studied the curriculum thoroughly. This paves way to the sayings such as “It’s okay for them to count with their hands” and “Self-esteem is a double-edged sword.” The intended curriculum has a differentiated approach but not different in content. It is carefully sequenced in terms of the range of topics in a spiral curriculum. Textbooks reflect multi-step problem sets using the CPA approach. 2) Implemented Curriculum. This is the centralized system in grades 1 to 4, where the students should be exposed to such system for at most five years. There is also the worksheet culture where teachers reinvent the curriculum. 3) Attained Curriculum. This is where the benchmarking of grades is included: In grade 4 for example, in Singapore it should be 71%, Hong Kong 67%, Japan 62% and China 50%. There is a large amount of time to be devoted to doing Math, at least 20% of the curriculum. Philippines has the highest time allotment for Math, but then it is not in the length of time but how we make use of the time given.

3. SG Math. There should be a right attitude for Math and an emphasis on visual thinking. SG problems can be simple and non-routine, where real world problems that are not well-defined must be included, open-ended and complex in nature. The focus on critical thinking is essential because it emphasizes mental computations, ensures conceptual understanding and de-emphasizes procedural memorization. CPA basically means Concrete-Prictorial-Abstract according to Jerome Brune’s Theory of Representation. The speaker added that “Teaching for learning is not a waste of time.”

4. How does Math start. Math starts in creation, communication, and use of intuition. In early kindergarten, introduction to word problems using nursery rhymes, games and fairy tales may be used. Concepts of absence and presence, syllables in lines, drawing scenarios with money in quantity and valuing what you have in life skills may also be used in this approach. The speaker also said “Don’t go straight to ABC’s, to XYZ; make use of pictures first especially in algebra.” Games like Piko may be utilized to create critical thinking problems. The speaker added that “Mathematics is a subject not to be memorized but to be understood.”

5. Number Bonds. Number bonds are one concept applied in Singapore Math also called as the Math family. Mental computation without memorization relies on concepts rather than in formula. Issues were raised that this kind of approach is not found in textbooks. The speaker clarified that they still have the same competencies but different approaches. The teachers have to identify which part of the curriculum can be applied. Reactions from the participants also included the “discovery approach”, but there may be time constraints with the number of competencies used; therefore it should be arranged in such a way that it can be accommodated. Another question was raised on if the children are then thinking critically, what would be its consequence. The consequence would be way beyond the teachers’ expectation, the speaker said. Concerns were also raised and included that elementary teachers are generalists. They could be assigned to different subjects almost every year, which may mean that there would be a loss of expertise, but the speaker perceived it as a new learning for them to apply this kind of approach.

Katha 2014: Teaching Geometry

Minutes of the Workshop transcribed. Speaker Mr. Robert Contreras

KATHA: DAY 5

Teaching Geometry

Mr. Robert Contreras

University of the Philippines Diliman

August 16, 2013

Minutes of the Workshop

 1. Introduction. Conceptual understanding in Geometry is essential for the students to understand and comprehend the subject better. There are necessary conditions on the different jargons and terms to be used in Geometry.

2. Geometry. The speaker said that Geometry is a tool for understanding and interacting with space. It also describes the world we live in and are axiomatic; that is it is rule-based.

2.1. Deductive System. This may be an analogy for a building where postulates are the base of the foundation of geometry.

2.2. Van Hiele Model. This model is learner-centered and is proposed in high school geometry in 1957 by teachers in Netherlands. It is a learning model with a holistic perception and develops a geometric thinking. It also refines the understanding of geometrical proofs.

There are three levels of the Van Hiele Model which are a validation of the CPA model: 1) Level 0 or Visualization. This is where the students perceive geometric shapes as total entities without attributes. This means that they are only aware of space as it is. 2) Level 1 or Analysis Model. This is where parts of the shape are already recognized by the students and they can discern characteristics of the shapes.  3) Level 2 or Abstraction. This is where figures are identified as having parts and are recognized by parts. Students can already discern the properties of the shapes and establish and analyze figures. Finally they can also subset classifications. 4) Level 3 or Deduction. This is the level where there is already the significance of deducing the figures to prove something and construct proofs based on the figures.  5) Level 4 or Rigor. This is where geometry is seen as an abstract form, i.e. in spherical geometry where parallel lines do not meet. Postulates are also recognized as usual functions of the body. These are the major characteristics of the Van Hiele Model, where it is sequential and in a reduction of level.

3. Implications to Teaching. This model entails mastery of polygons, grasping shapes as in CPA approach and the basis for each one. This can also be improved into shape teaching where teachers can create robots, counting how many are used, hopscotch, sorting cut out shapes and letting the students list the properties. The speaker illustrated different examples on how teachers can improve teaching Geometry through discovery games. He also provided samples on properties of shapes of everyday objects and property cards (What am I (shape)).