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: Basics of Singapore Math

Minutes of the Workshop transcribed. Speaker Ms. Pauline Mangulabnan

KATHA 2014: Day 1:

Basics of Singapore Math

Ms. Pauline Mangulabnan

De La Salle University Manila

July 12, 2014

Minutes of the Workshop

1. Introduction. The speaker first asked why is Math called Mathematics to the teacher-participants. It is exactly a science which provides a systematic solution for a set of problems. It is essentially part of our everyday lives. How exactly? In China, math is used for trade (calculation), in Egypt the Egyptians made the pyramids using practical Math. Ancient civilizations used math all the time, while the Greeks and Romans formalized the theoretical Mathematics. Mathematics is basically a Greek word “Mathematikos” meaning learning and mental discipline. The speaker asked the participants what is math for them. One answer was that math is one subject that pervades life at any age, in any circumstance. Thus its value goes beyond the classroom and school.

2. It’s all about thinking. Why do young Filipinos have to learn Math as Math as a school subject? Therefore it must be learned very comprehensibly and with much depth, focusing on problem solving and critical thinking. It is recognized that Math is an excellent vehicle for the development and improvement of a country. Math offers the students the opportunities for creative work, moments of enlightenment and discovery.

3. K to 12 and Math. There are two goals of K to 12 for Math: critical thinking and problem solving. Students must 1) learn necessary math knowledge 2) effectively carry out math processes 3) understand concepts and connections and 4) transfer learning through performance.

4. 2 Approaches to Chief Problem. The speaker said that the teachers must retell the problem and ask the students how they understand the problem. Problem solving sets must be localized to what the students know. Instruction and Assessment should be in the same language used. In teaching, it is important to stimulate the student’s world and give him a view of the world that lies beyond his own. To teach math is to bring out the children’s desire to learn and to think, according to Toshiyuki Makata. The speaker also said to “teach the formula but teach it last.” She also asked how do teachers develop thinking/solving not by teaching them the formula but by showing them the patterns and coming up with formulas. Teaching mathematics is broadening one’s world. The speaker finally gave an activity to the teacher-participants.

Katha 2014: Heuristic Alert: Bar Modeling

Minutes of the Workshop transcribed. Speaker Ms. Pauline Mangulabnan

KATHA: DAY 5

Heuristic Alert: Bar Modeling

Ms. Pauline Mangulabnan

De La Salle University

August 16, 2013

Minutes of the Workshop

1. Introduction. The speaker first distributed problem set materials to the teacher-participants and asked them to solve the problems using bar modeling. The speaker did not use any verbal problem algebraic expressions in the material, only bar model graphs for the teachers to answer. She also suggested the use of paper folding in the problem sets for percentages, fraction and ratio integration.

2. Understanding Algebra using Bar Modeling. This approach is applicable to both elementary and high school students where algebra can be easily understood. Big mathematical ideas are required for this approach which makes the students easily understand numbers and variables, make sense of algebraic expressions and visualization connections. The speaker noted that the teachers should not assume that the students’ knowledge is not of the same level as them.

3. Principles of Modeling Construction. The principle dated back in the 1980’s in Singapore, Japan and Korea, where students are able to understand the concepts in math and avoid memorizing formula. The students are required to draw bars to help them answer the questions and let them discover the formula for the problem. This approach may help bridge the mathematical knowledge from primary to secondary students. It helps them to understand the relationship of the given data and deduce the solutions by themselves. It also avoids problem complications.

3.1. Fractional Equations. This bar modeling may also be used in fraction-related problems which may also be integrated in ratio, comparison, multiples and percentages, all of which may be utilized together even though these are considered as difficult topics.

3.2. Problem Set. Problem sets using the bar modeling is also effective for teaching math because the students can easily read and understand the problem, make them underline the question and determine who or what is tackled in the problem. The students may draw unit bars. The speaker discussed a few problem sets on fractions and ratio using the bar modeling method.

The use of bar modeling helps the visually-stimulated students. At the same time they can proceed to more difficult questions; that is, bar modeling can hit two birds with one stone.

3.3. Practices and Conventions. Bar modeling may also be utilized in representing values by units. There are four labels in which bar modeling is materialized: 1) function label 2) quantity label 3) unit label and 4) total label. The speaker gave more problem sets to the participants and solved and discussed each one. The concept of manipulation of units is also being utilized in this method where the teacher can scaffold the questions. Other methods include the use of dashed lines, model length and classifying units by shading.

4. Types of Bar Modeling. There are different types of bar modeling. This include 1) part-whole model in topics such as derivation, representation and comparison 2) comparison model in topics such as fractions, percentages and representations 4) Stack Model in algebraic problems 5) Area Model for length 6) Remainder Subset Model 7) Container and Content Model 8) Multi-step problem in adaptive expertise of letting the students think they own Math; this is also for operations-integration in one problem and division problems. This is also used to make questions more difficult to achieve student critical thinking. 8) Comparison Model where abstract thinking is shifted to concrete ones. Performance questions are required to improve the difficulty of the question. This is also used to  analyze multiple questions. The speaker discussed more problems on this comparison. This is essential to reinterpreting abstract algebraic equations into modeling. The use of tick marks may also be helpful for the problem solving. The speaker emphasized that in high school, both concrete and abstract methods are required for them to fully comprehend the problem sets. It is always necessary to employ checking of answers and when one teacher offering another way of solving the problem. Teachers may present alternative solutions without compromising the students’ own solutions. The speaker illustrated more examples of bar modeling problem sets and emphasized the constant use of tick marks for the same value of the unit in question. This method may be applied for age problems. One teacher-participant raised a concern on the difficulty of the procedure, while others seemed to have a problem on the transition from concrete to abstract algebraic expression. The speaker concluded that it’s all in the method of the teachers using the bar modeling which is easily understood by the students. 

Katha 2014: Singapore Math and PSA

Minutes of the workshop transcribed. Speaker Ms. Pauline Mangulabnan.

KATHA 2014: Day 2: 

PSA Japan

Ms. Pauline Mangulabnan

De La Salle University Manila

July 26, 2014

Minutes of the Workshop

1. Introduction. The speaker first presented a photo of a classroom with students for the teacher-participants to interpret it in different ways. She explained that teaching plans should not produce monotonous teaching, but should be a one-to-one teaching with one-to-one assessment per student. This should end with a post-lesson discussion in class.

2. Lesson Study. Lesson Study is a professional development model where the teacher practice their research in Japan (called jugyou kenkyuu). This is a collaborative effort where research is put into practice and the lesson study is a partnership of theory and practice. It is also a collaborative effort and a systemized cycle in teaching.

2.1. Lesson Study Cycle. The lesson study cycle should start with a identifying a goal or a problem. This is a Japanese way of teaching students to achieve a motivation for each student for advancement in their learning. This is also a transition for primary to secondary learning which is a number one dilemma according to one of the teachers who raised the issue.

For Mathematics, the teachers should let the students enjoy the fun of discovering and achieving knowledge through their own unique methods. The speaker said it should be remembered that there are 6 years of elementary schooling and another 3 years of transition middle schooling and this should be addressed properly and in the most efficient way possible. The grade levels in Grade 1 for example, should be able to master addition, by Grade 4 arithmetic equations, by Grade 6 Division and Multiplication.

There should also be a collaboratively developing lesson time and teacher meeting for a prelesson preparation. That is, there should be a dryrun of the lesson study plan for the other teachers to be able to evaluate each others’ methods and further improve their lesson study. This also follows that the implementation of the lesson plan may include inviting other teachers for observation and assessment, which all results to collaborative effort.

2.2. Research Lesson. Research lesson involves a very specific and detailed lesson study where students must be very engaged in the lesson of the day. Students must also be assessed and there is an importance of teacher intervention. Teachers must devise activities and lesson for the students and let them discover the solution. 

3. Implementation. Implementation of the PSA and Lesson study plan include post-lesson plans and collaborative efforts on school-based institutions.

3.1.Postlesson Plan. There should be a debriefing of the lesson discussion and revising the traditional cycle. There should also be future directions and research trajectories. There must also be detailed lesson plan with collaborative efforts, where teachers also become researchers to be able to create a formative assessment for the students.

3.2. Process of Lesson Study. Training teachers is a number one must in the process of the new lesson study. Research should also be developed in educational institutions and demonstration of extra-curricular studies for the students. Steps on the lesson study include analogies on experiment, practice, observation, seminars and research. The teachers should also look into the context of classroom and society development. The hierarchy includes the lesson study which is deeply ingrained in classroom performance. Teachers should be in full power on knowledge (bottom to top model) and initiatives must come from the teachers themselves. 

Research also informs lesson studies, where the lesson study plan for the day should also be considered as part of the teachers’ research. Questions were raised on the collaborative efforts and multiple or integrated topics and how to write the study plan. The speaker explained that this should be the way to go because the monotonously traditional lesson study that has been imbibed in the cultural education in the Philippines for many years has been the core problem in the first place.

3.3. Replicability of LS in the Philippines. A video on Japanese teaching was shown to the teachers where the there is an induction teaching for the teachers. In Japan, there is teacher collaboration for 10 years. There is mandatory training and voluntary leadership trainings for lesson studies in Japan to achieve the advocacy of practical learning in the country. Constructive criticisms are required for teacher observation for improvement of the lesson study.

3.4. Lesson Study Steps. First is the lesson study planning, that is, how the lesson is presented in class and how students can learn in par with the objectives of the teacher for the day. There should be a set selection of subjects and appropriate test materials which may be unique. There should also be specific steps for lesson plans and questions should be raised for the students to answer. Teachers may seek advice from their principals and other teachers.

Lesson study is also observed by other teachers and students themselves. It is a fact that the student’s mind is unpredictable and reveals that collaborative interaction between the teacher and the student is a must.

Reflections of the lesson study include overview of the lessons, constructive criticisms from other teachers and suggestions of concrete opinions for lesson improvement. This develops a keen eye on the part of the teacher to improve his lesson study in classrooms.

4. Implementation Structure. The speaker presented the participants with a math activity on filling in the blanks on a problem set by pairs. This is to make math more enjoyable both for them and for the students themselves. The teachers should also be more open to others’ opinions and suggestions. They should also explore new methods on how to approach the lesson to engage as many students as possible. This should result to the enjoyment of both the students and the teachers. Again, lesson study is equivalent to research study for the teachers.

5. Problem Solving in Mathematics. Assessment practices should be implemented to develop a conceptual understanding of the topic in math. There are stages of the Structured Problem Solving: 1) posing the problem; 2) autonomous problem solving 3) whole class participation 4) summing up 5) exercises and finally 6) evaluation that there are no perfect solutions.

6. Final Activity. The speaker gave a final activity for the participants by pair on how to assess a problem set. There are theories on learning: 1) behaviorism, or assessing learning via change of behavior, that is, the knowledge is incremented; 2) cognitivism or assessing the order of thinking and takes time to assess the knowledge of the student and 3) social, or the change in participation. The speaker also gave the participants homework on dividing fractions and how to come up with different explanations and solutions.