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 Curriculum Reform, Mathematics education

My issues with Understanding by Design (UbD)

Everybody is jumping into this new education bandwagon like it is something that is new indeed. Here are some issues I want to raise about UbD.  I am quoting Wikipedia in this post but this is also how UbD is explained  in other sites.

Understanding by Design, or UbD, is an increasingly popular tool for educational planning focused on “teaching for understanding”.

Is not teaching for understanding been the focus of all curricular reforms, then and now? No curriculum reformer wants to be caught in the company of rote learning, never mind that it’s how curricula are implemented, regardless of its form, kind and  substance in many classes. Teaching for understanding is not something new.

UbD expands on “six facets of understanding”, which include students being able to explain, interpret, apply, have perspective, empathize, and have self-knowledge.

I wonder which of these facets has not been a part of what it means to understand then. I’m not sure in other subject areas but these facets of understanding such as explain, interpret, and apply does not capture what it means to understand mathematics.

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.

Back in college we attribute it to Ausubel who promoted the idea of using advance organizers.  Of course, you don’t tell your students right away how they will be assessed. They don’t have those rights, then. Also, this method only works for some topics. In mathematics if the approach is Teaching through Problem Solving or Discovery method, this is a no-no as it might limit the students thinking in exploring their own ways of working with the task at hand.

The emphasis of UbD on “big ideas” is welcome development but shouldn’t this be contained in the curriculum framework? The “essential questions”, those elusive questions that teachers have difficulty formulating since probably the time the  education community was talking about “art of questioning” are also good reminders to all of us that ‘hello, processing questions before or after any activity are what make and unmake a lesson’. But isn’t it that one can only identify the enduring understanding required and formulate good questions if he/she has a very good content knowledge (CK) and pedagogical content knowledge (PCK)?. Shouldn’t the money and time for training teachers how to design a lesson using UbD be spent instead on deepening their understanding of CK and PCK? Shouldn’t we make sure first that we have a good curriculum framework that articulates what are important for students to know and understand in each subject area and in each content topic?

The emphasis of UbD is on “backward design”, the practice of looking at the outcomes in order to design curriculum units, performance assessments, and classroom instruction.

In my part of the globe, there is a national curriculum which is a collection of SMART objectives. These learning objectives have always been stated in terms of outcomes. Weren’t they called competencies? Aren’t these competencies tell what to assess? The trouble is, our list of competencies consist of factual and procedural knowledge and very little on problem solving and reasoning which never really get taught because they are all found at the end of each chapter!

According to Wiggins, “The potential of UbD for curricular improvement has struck a chord in American education. Over 250,000 educators own the book. Over 30,000 Handbooks are in use. More than 150 University education classes use the book as a text.”

That explains everything. Everybody is hooked on the book that no one found time to do research if it works or not. Of course, on this part of the world where I come from I could not possibly have full access to current studies in educational planning and curriculum conducted elsewhere. I’m pretty sure though that we don’t have a study here yet. This is actually my issue. We’re jumping on a bandwagon created elsewhere without checking first if it will run on our roads.

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Understanding by Design recommends a structure for curriculum planning, for designing instruction. It is not surprising that this is a welcome development because of lack of the same when it comes to this area. College education and in-service programs have failed to equip teachers the knowledge and skills to identify the important ideas in their major field of study.

Click here for the proposed stages of lesson development by UbD (thanks Jimmy Wysocki). Imagine it in the hands of our classroom teachers. Imagine how their faces will look like if you tell them “these elements should be in your written lesson plans”! And when they look for resources, all they have is an anemic curriculum framework and textbooks teaching facts that can be Googled. They will follow the directives, of course, as they have always done in the past in this part of the globe. They won’t just have time anymore to study and prepare  for the actual teaching of the lesson, especially in examining how their students learn specific topic. Surely, they will have a very neat plans complete with the elements. But lest we forget, learning is still more a function of the experiences students engages in, that is the lesson, and not in the lesson plans format.

Lastly, UbD is a one size fits all for all subject areas. That’s what make it highly suspect. Click here and here for sequels of this post.