February 2010 - Mathematics for Teaching

Understanding by Design from WikiPilipinas

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I think the following entry from WikiPilipinas needs revising. “Learning of facts”? Check also the last statement.

“Teaching for understanding” is the main tenet of UbD. In this framework, course design, teacher and student attitudes, and the classroom learning environment are factors not just in the learning of facts but also in the attainment of an “understanding” of those facts, such as the application of these facts in the context of the real world or the development of an individual’s insight regarding these facts. This understanding is reached through the formulation of a “big idea”– a central idea that holds all the facts together and makes these connected facts worth knowing. After getting to the “big idea,” students can proceed to an “understanding” or to answer an “essential question” beyond the lessons taught.

One of my initial concerns about UbD in my previous post is about not checking first if the bandwagon we jumped in to will run in our roads although I received a comment that said the DepEd did pilot it and are confident that it can. The results of the pilot I believe are not for public consumption. We just have to believe their word for it. But with this post at WikiPilipinas, I don’t know if it is clear to us what the wagon is. Here’s the next paragraph:

Through a coherent curriculum design and distinctions between “big ideas” and “essential questions,” the students should be able to describe the goals and performance requirements of the class. 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. The classroom environment should also encourage students to work hard to understand the “big ideas” by having an atmosphere of respect for every student idea, including concrete manifestations such as displaying excellent examples of student work.

But I love the description of traditional method of constructing the curricula in the following paragraph. Very honest. But I can’t agree about the analogy with Polya’s.

The UbD concept of “teaching for understanding” is best exemplified by the concept of backward design, wherein curricula are based on a desired result–an “understanding” or a “big idea”–rather than the traditional method of constructing the curricula, focusing on the “facts” and hoping that an “understanding” will follow. Backward design as a problem-solving strategy can even be traced back to the ancient Greeks. In his book “How to Solve It” (1945), the Hungarian mathematician George Polya noted that the Greeks used the strategy of “thinking backward” by knowing what you want as a solution in order to solve a problem.

If I remember right, G. Polya wrote “look back” as the last step for solving a problem. It means you reflect on your solution and answer in relation to the problem. But wait, there is a problem solving strategy called “working backwards” which is probably what is meant here but as an analogy to backward design? Uhmmm …

Oh, by the way, “backward design” is a problem solving strategy?

Not that I’m happy we’re adapting Understanding by Design but who cares if I’m happy with it or not. There isn’t anything I can do in that department but just to help now to make sure we make the most of it. It is is a multimillion peso project. That’s our taxes. The one in WikiPilipinas is by far the only resource in the net for UbD Philippines. If you happen to know other related sites, please share.

Here’s one research about UbD in Singapore. Here’s my other UbD related post

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 to $5x^2$ also.

polygons and algebraic expressions — 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.

3-D models using Google SketchUP — some possible shapes made of 5 cubes

Point for reflection

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

Elementary School Math, Geometry, Math investigations

Math investigation lesson on polygons and algebraic expressions

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Understanding is about making connection. The extent to which a concept is understood is a function of the strength of its connection with other concepts. An isolated piece of knowledge is not powerful.

To understand mathematics is to make connections among concepts, procedures, contexts. A lesson that has a very good potential for learning a well-connected mathematical knowledge is one which is organized around a mathematical investigation. This is because of the divergence nature of this task which revolves around a single tool or context.

Here is a simple investigation activity about polygons. Change the x by x unit to 1 by 1 unit if you will give this to Grade 5-6 students.

Investigate polygons with area 5x^2 units on an x by x unit grid.

Some initial shapes students could come up with may look like the following:

different shapes, the same area — Figure 1. Polygons with the same area

Note: This is a mathematical investigation so the students are expected to pose the problems they want to pursue and on how they will solve it. It will cease to be a math investigation if the teachers will be the one to pose the problems for them. The following are sample problems that students can pose for themselves.

What is the same and what is different among these polygons? How can I classify these polygons?

Possible classifications would be

a. convex vs non-convex polygons

b. according to the number of sides

What shapes and how many are there if I only consider polygons made up of squares?

Students will discover that while they can have as many polygons with an area of 5, there are only 12 polygons made of squares. This is shown in Figure 2. These shapes are called pentominoes because it is made up of 5 squares. I have arranged it here for easy recall of shapes. It contains the last seven letters of the english alphabet (TUVWKXZ) and the word FILIPINO without the last 2 I’s and O in the spelling.