Posted in Curriculum Reform, Mathematics education

Curriculum change and Understanding by Design, what are they solving?

Not many teachers make an issue about curriculum framework or standards in this part of the globe. The only time I remember teachers raised an issue about it was in 1989, when the mathematics curriculum moved from compartmentalized (elem. algebra, intermediate algebra, geometry, adv. algebra & statistics) to spiral-integrated approach. The reason behind the change was the poor performance of the students. Many teachers didn’t like the change in the beginning not only because it’s the first time that the mathematics curriculum is organized that way, hence new, but also because it demands re-learning other areas of mathematics which they have not taught for years.  Also, teachers were not taught mathematics in high school nor in college that way. But the curriculum was pushed through just the same and eventually teachers complaints about it died down. Why? No one knows. They just continue teaching what they know in the way they think best.

Sometime in late 2001 or was it 2002, the then secretary of DepEd made a phone call to one of the country’s math education consultants. The country’s students seem not getting any better. Something’s got to be done about it. So one day, in 2002, the country’s basic math community woke up with a new curriculum, back to the compartmentalized system. The identified culprit according to the sponsor of the compartmentalized curriculum was that teachers are not that capable yet to implement the spiral-integrated curriculum that is why the still low students’ achievement. Clearly teachers need upgrading in their content knowledge and pedagogical knowledge and they need a lot of support resources for teaching.  The solution made to this problem? Change the curriculum. In fact not only to change it back to where it was but DepEd reduced the content further to minimum competencies consisting of learning of facts and procedures, a sprinkling of problem solving and an inch thick of content for mathematics. Did the teachers like it? Did it work? No one knows. They just continue teaching what they know in the way they think best.

It’s 2010. The minimum learning competencies lived up to its name. It provided minimum knowledge and skills. The students’ achievements did not get any better.

By June this year, the Math 1 (Year 7) teachers will be making their lesson plans based on UbD. UbD or Understanding by Design is the title of a book which proposes a new way of doing curriculum planning. In the school level, its in the way the teachers will be preparing their lesson plans. UbD is based on backward design. The main difference between backward design and the usual way of writing the lesson plan is that you spend time first formulating how you will assess the students based on your identified goals (aka enduring understanding and essential questions using UbD lingo) before thinking about the activity you will provide the class and how you will facilitate the learning.  I’ve yet to see and read a report from the proponents and users of UbD for evidence that it really works. And working in what aspect? in which subject area? and, whether it is better than the usual way teachers prepare their lesson plan?  Some schools who have tried it reported that at first, teachers had a lot of difficulty in making a UbD-based plan but they eventually got the hang of it. Are they teaching any better? Are the students doing well? Silence. I’m asking the wrong questions. For indeed, a great distance exist between way of preparing lesson plans and students’ achievement. So why are schools all over the country mandated to adopt UbD? I don’t know.

But this is what I know.  I know that teachers need support in upgrading and updating their knowledge of content and pedagogy.  I know that teachers teach what they know in the way they know.  These are things that cannot be addressed by simply changing the curriculum or changing the way of preparing the lesson plan, much more its format. The book The Teaching Gap which reports the TIMSS 1999 video study tells us what we should focus our attention and resources more on:

“Standards [curriculum] set the course, and assessments provide the benchmarks, but it is teaching that must be improved to push us along the path to success” (Stigler & Hiebert, The Teaching Gap, p.92).

I couldn’t agree more to this statement. I’m not very good at memorizing so to commit it to memory I paraphrased Stigler & Hiebert’s statement to: It’s the teaching, stupid.

Click here for my other post about UbD.

Posted in Curriculum Reform

What is mathematical literacy?

Defining mathematical literacy

The Program for International Student Assessment (PISA) of the OECD describes mathematical literacy as:

“an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgments and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen” (OECD,1999).

Mathematical literacy therefore involves more than executing mathematical procedures and possessions of basic knowledge that would allow a citizen to get by. Mathematical literacy is mathematical knowledge, methods, and processes applied in various contexts in insightful and reflective ways. According to de Lange, mathematical literacy is the overarching literacy that includes numeracy, quantitative literacy and spatial literacy. Each of these type of literacy empowers the individual in making sense of and understanding aspects of the world and his/her experiences.

De Lange’s tree structure of mathematical literacy.Spatial literacy empowers an individual to understand the three-dimensional world in which he/she lives and move. This necessitates understanding of properties of objects, the relative positions of objects and its effect on one’s visual perception, the creation of all kinds of three-dimensional paths and routes, navigational practices, etc. Numeracy is the ability to handle numbers and data in order to evaluate statements regarding problems and situations that needs mental processing and estimating real-world context. Quantitative literacy expands numeracy to include use of mathematics in dealing with change, quantitative relationships and uncertainties. Click here for deLange’s paper on this topic.

Implications to curriculum and instruction

To identify and understand the role that mathematics plays in the world is to be literate about mathematics and its applications. This means that individuals need to have an understanding of its core concepts, tools of inquiry, methods and structure.

To be able use mathematics in ways that meet the needs of one’s life as a constructive, concerned, and reflective citizen necessitates learning mathematics that is not isolated from the students’ experiences.

To be able to use mathematics to make well-founded judgment demands learning experiences that would engage students in problem solving and investigation as these would equip them to use mathematics to represent, communicate, and reason, to make decisions and to participate creatively and productively in the functioning of society.

These show that mathematical literacy requires learning mathematical concepts and principles that would be applicable to the individual and society’s life and activities; equip individuals the necessary skills in using mathematics to reason and make decisions; enable individuals to get a sense of the nature and power of the discipline in order to understand its role in the world.

To teach mathematical literacy, curriculum and instruction should therefore include these 3 R’s:

  • Relevant mathematical concepts, principles and procedures
  • Real-life context which can be investigated and modeled mathematically
  • Rich mathematical tasks that fosters conceptual understanding and development of skills and habits of mind

Check out these great books on mathematical literacy:

Posted in Lesson Study, Mathematics education

How to facilitate a lesson study group

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?

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.