Posted in Curriculum Reform, Mathematics education

What is Lesson Study?

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.

Posted in Assessment, Curriculum Reform, Elementary School Math, High school mathematics, Number Sense

Assessing conceptual understanding of integers

Assessing students’ understanding of operations involving integers should not just include assessing their skill in adding, subtracting, multiplying and dividing integers. Equally important is their conceptual understanding of the process itself and thus need assessing as well. Even more important is to make the assessment process  a context where students are given opportunity to connect previously learned concepts (this is the essence of assessment for learning). Because the study of integers is a pre-algebra topic, the tasks should also give opportunity to engage students in reasoning, number sense  and algebraic thinking. The tasks below meet these criteria. These tasks can also be used to teach mathematics through problem solving.

The purpose of Task 1 is to encourage students to reason in more general way. That is why the cells are not visible. Of course students can solve this problem by making a table first but that is not the most ideal solution.

adding integers
Task 1 – gridless addition table of integers

A standard way of assessing operations involving integers is to ask the students to perform the operation. Task 2 is different. it is more interested in engaging students in reasoning and in developing their number and operation sense.

subtracting integers
Task 2 – algebraic thinking and reasoning in numbers

Task 3 is an example of a task with many possible solutions.  Asking students to find a relation between the values in Box A and Box B links operations with integers to the study of varying quantities or quantitative relationship which are fundamental concepts in algebra.

Task 3 – Integers and Variables

More readings about algebraic thinking:

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Posted in Curriculum Reform

Mathematical habits of mind

Learning mathematics is not just about knowing, understanding, and applying its concepts, principles and all the associated mathematical procedures and algorithms. It’s not just even about  acquiring the capacity to solve problem,  to reason, and to communicate. It is about making these capacities part of students’ thinking habits. It is only then that one can be said to be mathematically literate.

The test for example that solving problem is no longer just a skill but has become part of students thinking habit is when students are doing it without the teachers still having to ask “Can you explain why you solve it that way?” or “Can you do it another way?” Those should be automatic to students.

“A habit is any activity that is so well established that it occurs without thought on the part of the individual.”

Here’s is a list of important mathematical habits of mind that I believe every teacher should aim for in any mathematics lesson.

Habit #1: Searching for Patterns

Students should develop the habit of

  • generating cases and generalizing patterns
  • looking-out for short-cuts that arise from patterns in calculations
  • investigating special cases, extreme cases from patterns observed

Habit #2: Reasoning

Students should develop the habit of

  • explaining the positions they take
  • providing mathematical evidence/justification for the conjectures or generalizations they make
  • testing conjectures by generating cases both special and extreme
  • justifying why a generalization will work for all cases or for some cases only

Habit #3: Solving and posing problems

Students should develop the habit of

  • always looking for alternative solutions to problems
  • extending problems and solutions to more general case
  • solving problems algebraically, geometrically, numerically
  • asking clarifying and extending questions

Habit # 4: Making connections

Students should develop the habit of

  • Linking algebra, number, geometry, statistics and probability
  • Finding/devising equivalent representations of the same concept
  • Linking math concepts to real-world situation

Habit #5: Communicating mathematically

Students should develop the habit of

  • using appropriate notation and representation
  • noticing faulty, incomplete or misleading use of numbers

Habit #6: Reflecting and self-directing learning

Habit is a cable

All these are only possible  in an environment where students are engage in problem solving and mathematical investigation tasks.

If you want to know more about mathematical thinking, the books below are great read.

Posted in Assessment, Curriculum Reform, Mathematics education

Assessment for learning – its genealogy

IN the beginning there was only diagnostic and summative assessment. Diagnostic assessment was supposed to share power with summative assessment in the classroom but never really attained equality with it not because teachers did not want to give diagnostic assessment but because stakeholders (parents and state) are more interested with statistics and well-defined label of students’ level of learning as measures of return of investments.

One glorious day, the education community had a dream. It dreamt that in the teaching and learning process, the students have much to contribute especially in the what and in the how they will learn! Thus formative assessment was conceived and born. Formative assessment shifted the focus of assessment from simply a process for collecting information about the learner to it being an integral part of the teaching- learning process. But the education community discovered a bug in the formative assessment.  At the end of the day, it couldn’t tell whether there was really learning that occurred or not because the teacher did not have the data of students’ initial understanding.  Actually they do only that the collection of the data is not “scientific” enough for educators. Thus, diagnostic assessment was resurrected and assessment for learning, was born.