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: 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)).

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: Lesson Study Part 2

Minutes of the workshop transcribed. Speaker Dr. Allan Canonigo

 KATHA 2014: Day 5 

Revisiting Lesson Study

Mr. Allan Canonigo

NISMED, UP Diliman

August 9, 2014

Minutes of the Workshop

1. Review. The speaker reviewed the teacher-participants on their understanding of the Lesson Study based on the previous lectures and some raised the definition of Lesson Study as a strategy to improve research teaching, doing research on one lesson/objective, and where observers or experts would be invited for the evaluation of their respective lesson studies. In conclusion, the speaker clarified that Lesson Study is a systemic cycle towards improvement of teaching.

2. Discussion of Lesson Study Misconceptions. The speaker also clarified some points raised by the participants. 1) Lesson study is not a teaching strategy technique. He pointed out that we can never make Math easy for the students. However it becomes more difficult when teachers can’t think of ways on how to improve the student performance on math. Teachers should not say that why their students do cannot get the formula or the problem right because it would provide more stigma to the students on math. 

3. Ways on Studying Math using Own Content. The true meaning of lesson study lies in the will power of the teachers to become better one where its effect and application would be long-term. 

3.1. Research in Lesson Study. Lesson study involves studying and planning together the teachers’ respective lesson studies for a particular strategy to implement and improve on. This is to improve the student performance which should be learner-centered. Teachers should be anticipating how students are learning and not how to teach the study/topic; that is, they should improve the thinking skills of the students.

3.2. Student-thinking. Why do these certain solutions work for some of the students? This should be anticipated and appreciated and considered by the teachers; that is, they should not limit the students to one solution only.

3.3. Observer Invitation. The trick in this strategy is to observe the teacher, not dictate what he or she should do. Learning from one another should be done as equals, not as bosses or subordinates; otherwise it would not be an effective lesson study. They should work together and try to change the culture of the Filipinos where ego is implemented every time there is a teacher invitation because of unpreparedness.

3.4. Preparedness. The teachers should encourage the students to think critically. There is this teacher culture where the fear of the teacher to explain the many solutions generated by the students overpowers the catalyst for better critical thinking of the students. They should help the students out for the lesson study improvement and claim that there is always a room for improvement. After the lesson study implementation, post-learner/discussion, suggestion and recording sessions should be done.

4. Lecture Proper: Activity. The speaker divided the participants into elementary and high school group and by district. He focused on the Understanding by Design (UPD), a part of the K to 12 goals in Math where problem solving involves critical thinking in the students. The speaker related the topic in real life where reality is without any easy solutions or formula. Problems are best solved by looking for answers and struggle is part of the equation. 

4.1. Reflection. The speaker gave three questions to the participants to answer and reflect and discuss them. 1) What kind of lectures at school you want to see implemented in your own class? Provide evidence. The participants said they want the students to be critical thinkers, problem solvers, competent and independent and active learners in math. 2) What kind of lectures is implemented at school? Students are dependent, frustrated and stigmatized by math, poor study habits and poor comprehension of the problems. 3) How do you think would you be able to bridge the gap between the aspirations and the reality of this particular dilemma? Efficiency of the lesson study, developing critical questions and applying the 3Es: Explore Experiment and Experience. Some teachers raised issues on the number of students, teacher mood, insignificance of the attendance and zero critical thinking in exams. 

4.2. Teachers learn in lesson study. The teachers should focus on how students learn and observe them in the right path. They have to analyze and observe students and use this information on revising student lesson study and learning.

5. Reasons for Lesson Study. This is for teaching improvement, instructional materials improvement, professional learning community and finally scholarly inquiry due to research.

5.1. Professional Research Study Equipment. The lesson study should be devised based on the words that work best for both the teachers and the students. There should always be someone to evaluate you because it is very hard to evaluate oneself because there are not wider perspectives. The long-term goal was discussed by the speaker based on his experiences in Cabanatuan City workshop. He said that sample teaching for problem solving or content is important, where you first teach the problem, solve it in different ways and use the problem critique.

6. Professional Development. The speaker started with a chart differentiating the traditional and research based lesson studies: traditional approaches include getting the answer, expert observation, trainer-teachers and hierarchical way of teaching, while research based studies include posing the question first, engaging the participants, implementing the teacher training and a reciprocal way of teaching method. 

6.1. Differences. There are differences in lesson planning, curriculum planning, demonstration where a different fate lies: reinvention learning, learning centers, continuous improvement and usefulness loop.

6.2. Research Lesson Planning Questions. The teachers should first decide the topic, today’s lesson, the plan, the activities, and the fact on how the students would understand the topic on hand. It is important that the teachers should know the knowledge of the children beforehand they teach the lesson because there might be miscommunication of sequence of the lessons.

6.3. Problems. Problems posed by the teachers include the explain ability of the lessons, no cooperation from other teachers, sustainability, government support, fund for books and other resources, funding for the experts and incentives for the students. Most of all the problem lies in the time for the teachers to work together. Questions were also raised on the teaching competency of the teachers, the spiral loop concept where teachers build on by activating previous knowledge on prior topics in math. The topics should also be connected and finally researchers helped by teachers can be done by etic realizations and suggestions. 

Katha 2014: Lesson Study as a professional development model

Minutes of the workshop transcribed. Speaker Sir Levi

KATHA 2014: Day 3:

LESSON STUDY as a professional development model

Mr. Levi

Minutes of the Workshop

Mathematical knowledge should be redefined and reconceptualize for the students to be able to grasp its essential components and comprehend and appreciate at the same time the field that has been long way conditioned to be stigmatized. The speaker first presented the Characteristics of Mathematical tasks in the cognitive level of a low level task, compared to a high level task.  The speaker proposed a task analysis and problem set for the participants by letting them think of high-order mathematical questions and problems for their students.

During the evaluation of the task, the teachers said that their main goal is to retain the most basic concept in the students’ mind and develop their critical thinking. One of the teachers presented a dilemma where learners tend to think that Math is already difficult as it is. The speaker presented a solution: LAS; that is, this is to imbibe the students to Love All Subjects and impose it as a school policy. Another teacher explained that there is a difficulty in the basic operations in the students. Another raised the issue of planning the lesson study for the day, if the topic should be one by one or by integration. The speaker explained that it would be better if the teachers first give the problem set to the students to reflect on before presenting the lesson for the day. One teacher said that students also tend to forget the lesson of the previous session, and the speaker explained that this is because the students are honed to memorize the topic instead of understand its basic concept. It would be better, furthermore that the teachers let the students discover the solutions and conclusions to the previous problem sets so that they will be able to remember the lesson better.

More solutions to the problems presented by the teachers during the evaluation include letting students discover multiple solutions to the problem set, make math learning fun for them and give them small incentives once they improve on their skills.

The speaker also presented general truths regarding Math teaching and lesson study: 1) Heterogeneous classes are very hard to handle. This is to mean that not all students will be able to grasp the basic concept all at once. 2) Teachers are forced to fit to the basic and traditional curriculum standards, which will be very hard to penetrate especially because improving lesson study is exactly the opposite method of the traditional lesson study planning. 3) Student attitude is typically reluctant with low confidence over a problem set because of the conditioned stigma and the deemed difficult problem sets. This is one problem that should be specifically addressed. 4) Teachers have no guts or the opportunity or the time to do proper mathematical teaching.

The speaker proceeded to discuss the Problem Set Analysis, which would help scaffold the lessons for the slow learners and be expanded for the advanced students. He proposed that there should be collaborative groups for the students, outline expectations and identify one lesson topic for each group, identify exercises with potentials, access the difficulty of the questions and illustrate samples of deep and challenging problem sets to the students. Given the traditional books in school, the teachers may revise the textbook questions to problem task analyses. The may also discuss concepts of different math topics without really using jargons and technical terms. Finally, it would be effective for the teachers to let the students arrive at the conclusions of patterns and discovery of solutions. That is, they have to let the students make sense of the problem, reason abstractness, and construct their own concepts.

The speaker followed the discussion with Performance Scores. This topic includes the rubrics, descriptions of benchmarks and evaluation of the level of proficiency of the student in their mathematical skills. The speaker presented this lesson study by providing problem sets to the teachers in five ways:

1) Comparing Numbers by 100 000. This activity may be discussed using number cards and use elements of surprise as well. This can be applied to Grade 2 and above. This activity identifies and enhances the learning capacity of the students and at the same time scaffolding the lesson plan to avoid curriculum congestion, which is a number one problem in the traditional curriculum. It would also be better if the teacher uses the lesson plan as part of the puzzle for the lessons of the week; that is for integrated learning.

2) Exploring Numbers with 10 000. Teachers may use guide questions to scaffold and extract patterns and similarities from students based on their own concepts. The teachers should also let the students generalize and find their own solutions and patterns. This may be applied to different topics in subtraction, place value and addition as well, which can all be integrated into one lesson study for the week.

However one teacher raised the issue of the complexity of the integrated lesson study which he thought would confuse the students just as much as the teacher would be confused especially on his lesson study for the week. But then the speaker pointed out that was exactly what the old curriculum had been inhibiting the teachers and students to think in high order critical thinking,no integration and connections on the topics in mathematics.

3) Looking at Multiplication Errors. Teachers may give out questions for the students and applying the technique of Cuisinaire Rods; that is, using fractions, grouping the students by problem sets and letting them come up with their own generalizations.

4) Fractions as Percentages. Teachers may apply paired grouping in this lesson study and draw models to illustrate fractions to integrate with percentages (using proper and improper fractions). This may also be the time to scaffold terms used previously.

One teacher raised a question on the structure of the lesson plan and how to apply the lesson proper. The speaker said that the ultimate end product of lesson study is assessment and evaluation of the student’s knowledge in the different math problem sets presented for the day. This he said is termed as the problem-structured approach.

5) Average and Making Connection. Teachers may illustrate integration of the topics in math to both cater slow learners and advanced students; also by using extension questions. Teachers may also use pattern mobiles by using cut-out shapes of the different topics in math for the week.

Finally the speaker left the participants with an activity on formulating high level tasks for their classes for one week, by using the lesson study problemstructured approach, guide questions, multiple choice questions and report on its implementation.

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.