Tag: teaching through problem solving

Posted in Assessment

Teach and assess for conceptual understanding

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To teach for conceptual understanding requires assessing for conceptual understanding. The principles that guide teaching for understanding must be the same principles that should guide assessment. What are some of these?

  • Open-ended, problem solving tasks
    To teach for conceptual understanding, it is not enough that students engage in problem solving task. The tasks should be (1) open-ended which means that it can be solved in many ways using a range of concepts; (2) accessible, that is , not too easy or too difficult but just beyond the students ability; (3) can be extended by changing conditions in the problem so that it can be used for building concepts and for making synthesis and generalization; and, (4) the task should encourage creativity in the problem solver. These, together with right amount of scaffolding from the teacher and assessment tasks possessing the same characteristics is a perfect recipe for understanding mathematics conceptually.
  • Activities that promote mathematical communication
    Mathematics is a language that enables us to communicate ideas with conciseness, clarity and precision both in oral or written form. Students learning experiences should always aim at developing this capacity. They should be given opportunities to talk about mathematics, to speak mathematics, and communicate mathematically through its written symbols.  These are possible with the right mix of collaborative and individual work. Click this link for sample. This also implies that assessment should focus not only on the knowledge the students are acquiring but also on their skill on communicating this knowledge.
  • Tasks that build on students’ previous knowledge
    Teaching should build on the knowledge that students already have. This does not mean simply putting something on top of what they know. Knowledge has to be connected with other knowledge from within and from without. The more connections there are, the more robust is the understanding. Conducting formative assessment can provide teacher with information on how to structure the lesson to help students make connections. Another strategy which I highly recommend is to teach via problem solving. Click here for sample lesson.
  • Discussions that respects reason
    Mathematics is a way of thinking logically and methodically. As such, classroom culture that respects reason must be created both in the teaching and in assessing. Group or whole class discussion and assessment rubrics should give appropriate feedback to the students as to the way they reason and build on each others reasoning or on each others opinion.
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 Algebra, High school mathematics, Math investigations

Teaching combining algebraic expressions with conceptual understanding

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In Math investigation about polygons and algebraic expressions I presented possible problems that students can explore. In this post, I will share some ideas on how the simple investigation of drawing polygons with the same area can be used as an introductory lesson to teach operations with algebraic expressions with meaning and understanding. Like the rest of the lessons in this blog, this lesson is not so just about learning the math but also making sense of the math and engaging students in problem solving.

The lesson consists of four problem solving tasks to scaffold  learning of adding, subtracting, multiplying and dividing algebraic expression with conceptual understanding.

Problem 1 – What are the different ways can you find the area of each polygons? Write an algebraic expression that would represent each of your method.

The diagram below are just some of the ways students can find the area of the polygons.

1. by counting each square
2. by dissecting the polygons into parts of a rectangle
3. by completing the polygon into a square or rectangle and take away parts included in the counting
4. by use of formula

The solutions can be represented by the algebraic expressions written below each polygon. Draw the students’ attention to the fact that each of these polygons have the same area of 5x^2 and that all the seven expressions are equal to5x^2 also.

Multiple representations of the same algebraic expressions

Problem 2 – (Ask students to draw polygons with a given area using algebraic expressions with two terms like in the above figure. For example a polygon with area 6x^2-x^2.

Problem 3 – (Ask students to do operations. For example 4.5x^2-x^2.)

Note: Whatever happens, do not give the rule.

Problem 4 – Extension: Draw polygons with area 6xy on an x by y unit grid.

These problem solving tasks not only links geometry and algebra but also concepts and procedures. The lesson also engages students in problem solving and in visualizing solutions and shapes. Visualization is basic to abstraction.

There’s nothing that should prevent you from extending the problem to 3-D. You may want to ask students to show the algebraic expression for calculating the surface area of  solids made of five cubes each with volume x^3. I used Google SketchUp to draw the 3-D models.

some possible shapes made of 5 cubes

Point for reflection

In what way does the lesson show that mathematics is a language?

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.