Tag: George Polya

Posted in Teaching mathematics

George Polya’s Ten Commandments for Teachers

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1. Be interested in your subject.

2. Know your subject.

3. Know about the ways of learning: The best way to learn anything is to discover it by yourself.

4. Try to read the faces of your students, try to see their expectations and difficulties, put yourself in their place.

5. Give them not only information, but “know-how”, attitudes of mind, the habit of methodical work.

teaching math

Mathematical Discovery on Understanding, Learning, and Teaching Problem Solving

6. Let them learn guessing.

7. Let them learn proving.

8. Look out for such features of the problem at hand as may be useful in solving the problems to come – try to disclose the general pattern that lies behind the present concrete situation.

9. Do not give away your whole secret at once—let the students guess before you tell it—let them find out by themselves as much as feasible.

10. Suggest it, do not force it down your throats.

I got this from the plenary talk of Bernard Hodgson titled Whither the mathematics/didactics interconnection? at ICME 12, Korea, where he highlighted the important contribution of George Polya in making stronger the interconnection between mathematics and didactics and between mathematicians and mathematics educators.

If it’s too hard to commit the 10 commandments to memory then just remember the two rules below which is also from Polya. Combine it with Four Freedoms in the Classroom and you are all set.

George Polya on teaching math

How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library)

Posted in Number Sense

Math Knowledge for Teaching Addition

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This post is the second in the series of post about the Math Knowledge for Teaching (MKT) where I present task/lesson that teachers and interested readers of this blog can discuss. The first is about Tangents to Curves, a Year 12 lesson. This second post is for young learners.

The task

How many small cubes make up this shape?

cubes

This is a pretty simple task.  Any Grade 1 pupil will have no difficulty giving the correct answer. All they need to do is to count the cubes. Yesterday, in one my workshop with teachers about lesson study, we viewed a Japanese lesson using the same task but was used in such away that children will learn not just counting.

The lesson

Before this lesson the class already learned that putting together concept and the symbol + and =.

The pupils were given small cubes to play with on their tables. After a minute, the 2x2x2 cube was shown on the TV screen and the teacher asked the class to predict how many small cubes make-up the shape. Some used their cubes to make a similar shape without the teacher encouragement to do so. The cubes were only there to help those who might have trouble imagining the bigger cube were some parts are not shown. The pupils counted the visible cubes one-by-one and then those not seen in the drawing (a drawing of the cube is posted on the board). But, the teacher was not just after the answer 8, he was after the learners’ counting strategy. So he asked: Can you use the + sign to show us your counting strategy? Some of the students answers were: 4+4 = 8, 2+2+2+2 = 8, 6+1+1=8. But, the teacher was not only after this, he wanted the class to realize that this number expressions may have come from a different way of looking at the cube. He started with those who wrote 4+4 to show the class how this counting was done. There were two different strategies: halving the cube vertically and the other horizontally which the students demonstrated using the cubes. All throughout the teacher was asking the class, “Can you follow the thinking? “Do you have a different idea?” “Who has another idea?”

After the summarizing the different ideas of the pupils in the first task, the teacher gave the second task:

What is your idea for counting the small cubes in this shape? Show your idea in numbers and symbols.

cubes

The shape was projected on the TV screen as the teacher rotated the shapes. The pupils came-up with different combinations of visible and not visible cubes like 7+3 = 10, 4+6 = 10, etc. They were invited to explain these expressions and their thinking using the drawing on the board. The teacher did not have any difficulty getting the answer he wanted from the pupils: “We already know that this shape (the big cube) is 8 so we just add 2  (8+2 = 10).

Questions for Teachers Discussion/Reflection:
  1. What about numbers will the pupils learn in the lesson?
  2. What is the role of technology and visuals in this lesson?
  3. What about mathematics is given emphasis in the lesson?
  4. What mathematics teaching and learning principles underpin the design of the lesson?

Remember this quote from George Polya: What the teacher says in the classroom is not unimportant, but what the students think is a thousand times more important.

math knowledge

For further reading:

Engaging Young Children in Mathematics: Standards for Early Childhood Mathematics Education (Studies in Mathematical Thinking and Learning Series)

Posted in Curriculum Reform, Mathematics education

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