Category: Lesson Study

Posted in Lesson Study, Number Sense

Patterns in the tables of integers

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Mathematics is said to be the science of patterns. Activities that involve pattern searching is a great way to engage students in mathematical thinking. Here are some of my favorites for teaching positive and negative integers. If you are wondering why most of my posts are about integers it’s because I’m doing a Lesson Study with a group of Mathematics I (Year 7) teachers about this topic. Last week we concluded the first cycle of our research lessons on teaching subtraction of integers.

The first task students need to do with the tables is to list 3-5 observations. From there you can start asking the ‘whys’ for each observation. Sample questions are provided for each table below:

1. Adding integers

Sample questions for discussion:

a) Under what conditions will the sum be positive? negative? zero?

b) Why are there the same numbers in a diagonal?

c) How come that the sum is increasing from left to right, from bottom to top?

2. Taking away integers

Sample questions for discussion:

a) Under what conditions will the difference be positive? negative? zero?

b) Why are there the same number in a diagonal?

c) How come that for each row/column, the difference is decreasing?

3. Multiplying integers

Sample question for discussion:

a) Under what conditions will the sum be positive? negative? zero?

4. Dividing Integers

Sample question for discussion:

a) Under what conditions will the difference be positive? negative? zero?

       b) Does dividing integers still results to an integer? What do we call these new numbers?

Feel free to share your ideas/questions for discussion.

You may also want to share other  math concepts that students can learn with these tables.

Posted in High school mathematics, Lesson Study

Pedagogical Content Knowledge Map for Integers

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I’m working with a group of Year 7 mathematics teachers doing Lesson Study for the first time. The teachers chose to do a lesson study for what they believe to be the most difficult topic in this year level – integers. Part of my preparation as facilitator is to draw a map of what I know about teaching the topic. The map is more than a concept map because it includes not just related big ideas or concepts but also how  these are taught and learned. Hence, I call this pedagogical content knowledge map (PCK map).

The PCK map I present here is a product of my own readings and my own experiences of teaching the topic. This means that it may not be the same as other teachers especially the ‘teaching part’ of the map, the ones in orange colors. For example, experience and research results back my claim that the number line is a very good way of representing the set of integers but not in teaching operations. Click here for my post about this. Notice that I gave emphasis on students knowing when a negative, a positive or a zero result rather than the rules for operation. I believe that without this, a conceptual understanding of the operation involving integers will be weak. Also, experience has taught me that although integers are numbers, the teaching of it must be algebraic. The instructions should be so designed so that students are learning algebraic thinking as well. I have noted this in the PCK map.

The map is not yet complete. I intend to include descriptions of effective activities and students’ learning trajectory of the concept after my research with the teachers. Please feel free to give your comments and share experiences for teaching integers that I could look into in my study.

pedagogical content knowledge
PCK Map for Integers

Please click the link to see my PCK map for Algebraic Expressions.

Posted in Lesson Study, Mathematics education

Lesson Study in Mathematics

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From 2006 – 2009, the University of the Philippines National Institute for Science and Mathematics Education Development (UPNISMED) piloted a school-based professional development  program for teachers called Lesson  Study to two high schools and two elementary schools in Metro Manila. The project involved all the mathematics teachers of the schools. Lesson study engages teachers in creative and collaborative problem solving activity in designing a lesson that teaches mathematics through problem solving. The project is based on the following principles: (1) learners construct their own knowledge whether that learner is a teacher or a student; (2) learners learn most when they are engage in tasks that they view as significant to them and that presents a real problem for them; and (3) learning is a social activity whatever the object of the learning is.

Lesson Study in Philippines

The results of the lesson study project has been very encouraging. In terms of outputs, video lessons and lesson plans have been produced showcasing teaching mathematics via problem solving. These lessons were developed and implemented by the teachers collaboratively together with one UP NISMED mathematics education specialist per year level. The lessons produced show:

  • how to facilitate a problem solving lesson where students solve problems without being shown a solution first (the essence of the problem solving activity is preserved);
  • that a problem, traditionally given at the end of the chapter can be given at the start of the chapter;
  • that review of concepts,  traditionally a separate part of the lesson and in drill type, can be integrated to the main lesson itself;
  • that lesson can be structured that would engage students to represent ideas mathematically, solve problems in different ways, and reason out;
  • that a problem solving task can be a rich context for learning new mathematical concepts and link these with previously learned concepts.

As a result of these, and this is perhaps the most important achievement of the project, is the change in the teachers’ perception about the role of problem solving in mathematics.  During the planning meetings, the mathematics teachers I was working with expressed apprehension about the problem solving lesson they were developing. They said that “Work” problems are one of the most difficult types of algebra problems so they thought there is no way students can solve it by themselves without being shown sample problem and solution first and the even if these are shown, students still need to know how to solve rational equations. This is the reason why the problem is found at the end of the chapter! These were their impressions until they produced and implemented a lesson that challenged their own assumptions. They realized that problem solving can also be a means for learning mathematics rather than simply a reason for learning it; and, that students are more capable in solving problem on their own than they previously thought.

The teachers admitted that initially, they saw lesson study as another “burden” to them but as the project progresses they eventually appreciated it. They said that they learned a lot from each other and the post conference and discussion part became a venue for them to deepen their understanding of mathematics and how students understand mathematics. We also documented changes in the quality of teachers discussion during the post conference. Initially they were focusing on general pedagogy but towards the third cycle of the lesson implementation they were now more focused on the content and how their questions for discussion is affecting the quality of the students’ thinking.

This year we are working with another school with an improved design of the project. We just finished a three-day orientation seminar about lesson study and teaching mathematics via problem solving for the mathematics teachers of the said school. Goal-setting, the first step in the lesson study process was done during the seminar. The teachers agreed that their goal is to make students value mathematics by developing their thinking skills. Their sub goals for this year is to develop lessons that engage students in mathematical representations and solving problems in different ways. I will talk more about these in my next post.

Download full paper: Scaffolding Teacher Learning through Lesson Study.

Email me if you are interested to do lesson study in your schools (schools in Philippines only.) To give you an idea how lessons are planned and analyzed in a lesson study context view this presentation:  Planning and Analyzing Mathematics Lessons in Lesson Study

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.