Posted in Curriculum Reform

Enduring understanding

To know the big picture ideas, to know the enduring understanding students are supposed to learn are indeed very important in planning and teaching a lesson. However, for teachers to be able to identify and articulate the enduring understanding for a particular content topic requires knowledge of the following:

  1. knowledge of the nature of the discipline;
  2. a deep content knowledge;
  3. knowledge of the connections among the different content topics
  4. some knowledge about the connection of your discipline with other discipline or subject area;
  5. knowledge of the relevance of your discipline to real-life

All these should already be partly articulated and reflected in the standards or curriculum framework to serve as guide to teachers when they design their lesson plans. If the curriculum framework is just a list of topics or some general statements then that’s bad news.

One can argue of course that teachers are expected to already know all these (the 5 items I listed above) and hence know the enduring understanding in their discipline. But the reality in this part of the world is that majority of our teachers still need more help in these aspects. This is my reason why we have to have a curriculum framework or Standards that supports the demands of articulating the enduring understanding expected in each unit before asking teachers to plan their “ubdized” (got this term from one reader of this blog) lesson.

Textbooks, which market themselves as “UbD-based”, or “UbD-compliant” should also be required to state the big ideas for the entire course and for each chapter or unit. Statements of enduring understanding and essential questions can also precede each chapter. Teachers can just add their own or state it in their own way when they make the lesson plans. It is not spoon-feeding the teachers. We just want them to have something to start with especially if the textbooks are their only resource.

Textbooks authors are supposed to have a clear big picture idea of what they are trying to teach in the textbooks and so why not require them to put it there. They have no business writing one if they don’t know the enduring understanding that students are supposed to learn. With all these in place, teachers will have more time to plan and design the lessons targeting these big picture ideas. They will also have more time to study their students’ difficulties and misconceptions about the topic and think of ways of addressing them. Most importantly, teachers will have more time to study the topic they are going to teach and how this content topic relates with previously learned concepts and future concept so they can find the right activity/ task and use appropriate assessment process. These are what can make or unmake a lesson, not whether or not the teachers use the backward or forward design in lesson planning.

This is my fifth post about this topic. Click here to link you to my other posts on UbD and backward design.

PS1. Having identified the enduring/essential understanding does not guarantee you’re going to have a good lesson plan or a good lesson implementation.

PS2. In one of the centennial lectures, part of the activities of the University of the Philippines centennial celebration, the speaker for education-related issues said that no one in this country is paying attention to learning. Indeed. We talk about lesson planning, we talk about curriculum frameworks and syllabus, we talk about multiple intelligences, …. we talk about essential understanding … we talk about everything except how pupils learn specific content topics.

Posted in Elementary School Math, Number Sense

Who says subtracting integers is difficult?

Subtracting integers should not be difficult for most if they make sense to them. In first grade, pupils learn that 100 – 92 means take away 92 from 100. The minus sign (-) means take away or subtract.

After two or three birthdays, pupils learn that 100 – 92 means the difference between 100 and 92. The minus sign (-) means difference. The lucky ones will have a teacher that would line up numbers on a number line to show that the difference is the distance between the two numbers.

After a couple of birthdays more, pupils learn that you can actually take away a bigger number from a smaller number. The result of these is a new set of numbers called negative numbers. That is,

small numberbig number = negative number

The negative numbers are the opposites of the counting numbers they already know which turn out to have a second name, positive. The positive and the negative numbers can even be arranged neatly on a line with 0, which is neither a positive nor a negative number, between them. The farther left a negative number is from zero the smaller the number. Of course, the pupils already know that the farther right a positive number is from zero the bigger it is. It goes without saying that negative numbers are always lesser than positive numbers in value. This is easier said than understood. When I tried this out, it was not obvious for many of the learners I have to give examples of each by comparing the numbers and defining that as the number gets further to the left the lesser in value.

Now, what is 92 – 100 equal to? The difference between 92 and 100 is 8. But because we are taking away a bigger number from a smaller number, the result must be a negative number. That is 92 – 100 = -8. Notice that the meaning of the sign, -, before 8 is different from that between 92 and 100.

What about -100 – 92? Because -100 is 100 units away from the left of 0 and 92 is 92 units away from the right of 0, the total distance or difference between them is 192. But because we are taking away a bigger number, 92, from a smaller number, -100, the answer must be negative (-). That is, -100 – 92 = -192.

And -100 – -92? Easy. Both are on the left of 0. The difference or distance between them is 2 but because -92 is bigger than -100, the answer should be a negative number. That is, -100 – -92 = -8.

We  shouldn’t have a problem with 100 – -92. These numbers are 192 units apart and because we are taking away a small number from a bigger number, the answer must be positive. That had always been the case since first grade.

Who says we need rules for subtracting integers?

Click the links for other ideas for teaching integers with conceptual understanding

Posted in What is mathematics

Mathematics is an art

Whether we are conscious of it or not, the way we teach mathematics is very much influenced by what we conceive mathematics is and what is important knowing about it. As part of our Lesson Study project with a group mathematics teachers, I was tasked to share my thoughts about the nature of mathematics and its implications to its teaching and assessment.

What is mathematics?

I have always believed that mathematics should be experienced by the k-12 learners as both practical and theoretical, as a language, as a process of thinking and, as an art.  Of these five, I have always felt the least confidence in speaking of mathematics as an art. Most of the times, my “mathematics is an art” becomes “there is art in mathematics”. The latter is much easier to discuss because teachers know this so there’s not much need for me to explain. What I do and I don’t know if I get away with it, is give beautiful examples. Here are two of them. Click image to get to the source.

http://math-art.net/2008/12/07/love-and-tensor-algebra/

Math Art | Love and Tensor Algebra via kwout

Where there is art, there is beauty. And what is the beauty of mathematics? In most cases, it’s in patterns. I would regale teachers with patterns in nature that mathematics could perfectly represent and capped my lecture with Galileo’s pronouncement that Mathematics is the language used by God to write the universe. With this, I could get away from mathematics as art to mathematics as language.

Then I came across this post titled What is an art? which defined art as a habit of thinking, doing, or making that demonstrate systematic discipline based on principles. The post described arts as about connections and that understanding the connections between things allows designers to accomplish their goals. It described  art as based on principles and not just a series or procedures or methods; that there can be many methods inside an art …  Finally, and I love this part, it said that art must be acquired as a habit, so that its practitioners become “unconsciously competent.”  I thought the post could very well be speaking of teaching and learning of mathematics especially in its giving  importance to making connection, open-ended problem solving, and the acquisition habits of mind which are favorite topics of mine when I’m invited to share my thoughts about mathematics teaching.

You may want to view this beautifully crafted video about mathematic in nature. Click link.

Posted in Algebra, High school mathematics

PCK Map for Algebraic Expressions

When I design instruction or plan a lesson I always start with making a map of everything I know about the subject. The map below is an example of a map I made for algebraic expressions. I won’t call it a conceptual map because it’s only the left part of it (the ones in black text) which deals with the concept of algebraic expressions. Those at the right (in red texts) describe what I know about the requisites of good teaching of algebraic expressions including my knowledge about students’ misconceptions and difficulties in this topic. Maybe, I should just call this kind of map, PCK Map, for pedagogical content knowledge map.

Pedagogical Content Knowledge (PCK) Map for Algebraic Expressions

I find doing the PCK Map a useful exercise because it helps me link concepts, synthesize my teaching knowledge about the topic, not leave out important ideas in the course of the teaching and of course in planning the details of the lesson especially in the selection of activity/tasks and in framing questions for discussions.  I also find it useful in evaluating my teaching of the unit.

There are two ways a PCK Map can be enriched: (1) use Google (alright, go to the library and see what experts think are important to cover in the topic, they’re also outlined in the Standard) and (2), after each lesson or at the end of the unit, write your new knowledge about the topic especially students misconceptions and difficulties and how it can be addressed next time.

Click this link to see a the lesson plan I made based on the PCK Map. The lesson is about teaching combining algebraic expressions via a mathematical investigation activity.