Category: Mathematics education

Posted in Lesson Study, Mathematics education

How to facilitate a lesson study group

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The long term goal of lesson study as a professional development model is to enhance teachers’ content and pedagogical content knowledge and develop their capacities for designing and studying (researching) their own lessons. The expected output of a Lesson Study group is to develop a curriculum material in the form of a lesson plan. The process of developing the plan becomes a context for teachers to gain insights about how students think and learn the topic chosen and the discipline in general.

In my earlier  post about Lesson Study I described what Lesson Study is and the Collaborative Lesson Research and Development project of UP NISMED about Lesson Study to find out if it will work in Philippines given its school realities. In this post I will describe my experience in facilitating a lesson study group of mathematics teachers all teaching Intermediate Algebra (Year 8 math). Our CLRD project introduces the first lesson study in their school.  The discussion below shows how I “scaffolded” teachers learning of the LS process through a series of questions.

Like any project, Lesson Study starts with a goal.

1. Goal setting:

Although I wanted teachers to try the strategy Teaching through Problem Solving (TtPS), I didn’t want to impose it on them. So during the first meeting to identify the goal for our lesson study, I started with the following questions:

  1. What are some of the problems do you encounter in your mathematics class?
  2. What are some of your teaching problems in mathematics?
  3. What are some of your students learning difficulties in mathematics?
  4. What are some of the things you wish your students can do in your mathematics class?

My first question was too general.  Identified concerns were about lack of textbooks, materials, absenteeism, students’ personal problems, lack of motivation, etc. These are problems that lesson study cannot solve except perhaps the problem on motivation. The second question was equally disastrous. I received a blank look. They don’t have teaching problems. It’s the students who have problems. Hence the third question. The students’ problem is that they are not learning their mathematics. This wasn’t very helpful. It’s too general for the purpose of lesson study. So I asked the fourth question. And Voila. The teachers said they wish their students could think! This was my cue. So I said, “that’s great, let’s put our heads together and design a lesson that would engage students in thinking and reasoning”.

2. Selecting the topic:

My questions:

  1. What topic would you like to make a lesson about?
  2. What are the important ideas and skills should students learn about in this topic?
  3. What about mathematics will students learn from this lesson?
  4. Why should students learn this topic? Can we just skip this lesson?

The first question was received with excitement. Everybody was talking. It only took a couple of minutes for them to agree on one topic. However, when I asked why they like the topic they said that it’s because they already have activities for it and students find learning the topic easy. While there isn’t anything wrong with this one I encouraged them to think of a topic that the students find difficult to understand or that which teachers find difficult to teach. I explained that there will be about 5 to 7 heads that will work on the plan so they might as well take advantage of it and select a topic that they find problematic and solve it together. And they did.

Questions for selecting teaching approach/strategy

  1. What kind of mathematical task will make students think?
  2. When do you give problem solving tasks and how do you get your students to do problem solving?
  3. Would you like to try teaching the unit using a problem that you give at the end? Would you like to try to develop a lesson using TtPS?

I got what I wanted with the first question but there was a “but”.  The group said “of course, it’s problem solving but students don’t like to solve problems”. Hence I asked the second question. As I have expected, problem solving is given at the end of the unit and they admitted that most of the time they skip that part for lack of time. When they do have time, they will solve a sample problem first and then ask students to solve a similar problem to practice the method of solution. So I asked the group “Do you think the students are really thinking here?” They said “a little because they only need to follow the solution”. So when I asked if they would like to try TtPS they said “we could try”. These teachers attended an in-service training with us about TtPS but admitted that they did not use it in their teaching for reasons ranging from lack of resources, time constraint, and that it is hard to make a lesson using one. I said that with 5 to 7 heads working on a plan using TtPS, they just might be able to make one.

3. Designing and Implementing the lesson plan.

Here are the steps they we went through in developing the plan:

  1. We selected a problem found at the end of the unit.
  2. The teachers solved the problem in different ways. I asked them to try solving the problem intuitively and using students previously learned knowledge.
  3. The teachers tried the problem in the class to know students difficulties with it. Decided it needed an introductory activity to help students visualize the situation.
  4. Wrote the teaching plan. Tried it out. Discussed the result. Revised the plan. Implemented it again.

You can tell by the process we went through that lesson study is highly rooted in the principle of social constructivism.

I recommend this book by Catherine Lewis. It’s a valuable resource for conducting your own Lesson Study. I met the author in two separate Lesson Study conferences. She was keynote speaker in 2010 World Association of Lesson/Learning Study and she was also speaker in the APEC Tsukuba Conference V in Japan. She is actively promoting LS in the US.

Posted in Curriculum Reform, Mathematics education

What is mathematical investigation?

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Mathematical investigation refers to the sustained exploration of a mathematical situation. It distinguishes itself from problem solving because it is open-ended.

I first heard about math investigations in 1990 when I attended a postgraduate course in Australia.  I love it right away and it has since become one of my favorite mathematical activity for my students who were so proud of themselves when they finished their first investigation.

Problem solving is a convergent activity. It has definite goal – the solution of the problem. Mathematical investigation on the other hand is more of a divergent activity. In mathematical investigations, students are expected to pose their own problems after initial exploration of the mathematical situation. The exploration of the situation, the formulation of problems and its solution give opportunity for the development of independent mathematical thinking and in engaging in mathematical processes such as organizing and recording data, pattern searching, conjecturing, inferring, justifying and explaining conjectures and generalizations. It is these thinking processes which enable an individual to learn more mathematics, apply mathematics in other discipline and in everyday situation and to solve mathematical (and non-mathematical) problems.

Teaching through mathematical investigation allows  for students to learn about mathematics, especially the nature of mathematical activity and thinking. It also make them realize that learning mathematics involves intuition, systematic exploration, conjecturing and reasoning, etc and not about memorizing and following existing procedures. The ultimate aim of mathematical investigation is develop students’ mathematical habits of mind.

Although  students may do the same mathematical investigation, it is not expected that all of them will consider the same problem from a particular starting point.  The “open-endedness” of many investigation also means that students may not completely cover the entire situation. However, at least for a student’s own satisfaction, the achievement of some specific results for an investigation is desirable. What is essential is that the students will experience the following mathematical processes which are the emphasis of mathematical investigation:

  • systematic exploration of the given situation
  • formulating problems and conjectures
  • attempting to provide mathematical justifications for the conjectures.

In this kind of activity and teaching, students are given more opportunity to direct their own learning experiences. Note that a problem solving task can be turned into an investigation task by extending the problem by varying for example one of the conditions. To know more about problem solving and how they differ with math investigation read my post on Exercises, Problem Solving and Math Investigation.

Some parents and even teachers complain that students are not learning mathematics in this kind of activity. Indeed they won’t if the teacher will not discuss the results of the investigation, highlight and correct the misconceptions, synthesize students’ findings and help students make connection among the math concepts covered in the investigation. This goes without saying that teachers should try the investigation first before giving it to the students.

I think mathematical investigation is constructivist teaching at its finest. For a sample lesson, read Polygons and algebraic expressions.

The book below offers investigation “start-up” for college students.

Posted in Curriculum Reform, Mathematics education

Understanding by Design from WikiPilipinas

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I think the following entry from WikiPilipinas needs revising. “Learning of facts”? Check also the last statement.

“Teaching for understanding” is the main tenet of UbD. In this framework, course design, teacher and student attitudes, and the classroom learning environment are factors not just in the learning of facts but also in the attainment of an “understanding” of those facts, such as the application of these facts in the context of the real world or the development of an individual’s insight regarding these facts. This understanding is reached through the formulation of a “big idea”– a central idea that holds all the facts together and makes these connected facts worth knowing. After getting to the “big idea,” students can proceed to an “understanding” or to answer an “essential question” beyond the lessons taught.

One of my initial concerns about UbD in my previous post is about not checking first if the bandwagon we jumped in to will run in our roads although  I received a comment that said the DepEd did pilot it and are confident that it can. The results of the pilot I believe are not for public consumption. We just have to believe their word for it. But with this post at WikiPilipinas, I don’t know if it is clear to us what the wagon is.  Here’s the next paragraph:

Through a coherent curriculum design and distinctions between “big ideas” and “essential questions,” the students should be able to describe the goals and performance requirements of the class. To facilitate student understanding, teachers must explain the “big ideas” and “essential questions” as well as the requirements and evaluative criteria at the start of the class. The classroom environment should also encourage students to work hard to understand the “big ideas” by having an atmosphere of respect for every student idea, including concrete manifestations such as displaying excellent examples of student work.

But I love the description of traditional method of constructing the curricula in the following paragraph. Very honest. But I can’t agree about the analogy with Polya’s.

The UbD concept of “teaching for understanding” is best exemplified by the concept of backward design, wherein curricula are based on a desired result–an “understanding” or a “big idea”–rather than the traditional method of constructing the curricula, focusing on the “facts” and hoping that an “understanding” will follow. Backward design as a problem-solving strategy can even be traced back to the ancient Greeks. In his book “How to Solve It” (1945), the Hungarian mathematician George Polya noted that the Greeks used the strategy of “thinking backward” by knowing what you want as a solution in order to solve a problem.

If I remember right, G. Polya wrote “look back” as the last step for solving a problem. It means you reflect on your solution and answer in relation to the problem. But wait, there is a problem solving strategy called “working backwards” which is probably what is meant here but as an analogy to backward design? Uhmmm …

Oh, by the way, “backward design” is a problem solving strategy?

Not that I’m happy we’re adapting Understanding by Design but who cares if I’m happy with it or not. There isn’t anything I can do in that department but just to help now to make sure we make the most of it. It is is a multimillion peso project. That’s our taxes. The one in WikiPilipinas is by far the only resource in the net for UbD Philippines. If you happen to know other related sites, please share.

Here’s one research about UbD in Singapore. Here’s my other UbD related post

Posted in Curriculum Reform, Mathematics education

What is Lesson Study?

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Studies show that the way to go to implementing effective and sustainable educational reform will be through an inquiry-type professional development program and while the teachers are in action. One of these professional development models that has proven effective in Japan and is now being implemented and widely used in many countries is Lesson Study. It is also one of the identified factors for Japan’s high achievement in TIMSS.

Lesson Study engages teachers in creative and collaborative work in developing and researching a lesson through a “design-tryout-reflect-revise” cycle until it reaches a form to which they believe would be exemplary to them and to other teachers. It assumes that by investigating the teaching and learning process in the context of designing and implementing a lesson, it could provide teachers with experiences that has the potential to deepen their content knowledge and their knowledge about how students learn concepts and skills, develop teachers skills in designing and facilitating lessons, and most importantly develop the skills, habit and confidence in investigating their own lessons.

In the Philippines, the first Lesson Study project was implemented in 2006 by UP NISMED with selected mathematics classes. The project was called Collaborative Lesson Research and Development (CLRD) to give emphasis on the collaborative nature of designing and researching the lesson, something that is not yet a popular practice among teachers in our country. The objectives of the project were 1) to equip teachers with skills in designing mathematics lessons that engage student in mathematical thinking processes; 2) to enhance teachers’ knowledge of content and pedagogy as they study how their students think, learn, and reason; 3) to develop a lesson study model that is adaptable to Philippines classroom realities directly affecting teaching and learning of mathematics which include among others large class sizes, inadequate content and pedagogical content knowledge of teachers and insufficient materials and resources; and, 4) to gain insights about how teachers implement reform-based teaching strategies in their classes. The unique feature of this lesson study project in the Philippines is the focus on developing teachers capacity in designing lesson and teaching mathematics through problem solving, something that is also not yet a common practice of teaching mathematics in our classes.

The first step in doing lesson study is to articulate the goals for doing the lesson study. Click the link to read how I facilitated a group of teachers to identify their goals. It was their first time to do a lesson study. I reported the result of this study in the post Lesson Study for Teaching through problem solving.