## Thursday, January 29, 2009

### DCM201 MicroTeaching on Subtraction of Mixed Numbers

It is promising that I see at least one very good lesson in every microteaching session. teaching is a complex activity that defies description in words. one has to see it being played out before one's eyes. I hope the trainee teachers in my group who are observers are capable of seeing good examples of instruction. This is why I assign them the roles of 'students' and 'observers'. On the days they are the 'observers' they can divert all their attention to witness the complex activity of teaching without being distracted.

In Rendra's ;esson, we saw the use of PowerPoint to model the subtraction process. I wished he had stuck to this. Unfortunately, he moved to doing the subtraction in abstract symbols too soon. In Naz lesson, which was brilliant, I saw him being able to handle the complexities of being a teacher bringing a lesson plan to life. In particular, the pictorial-abstract linkage was well done. He used the 'models' to explain the symbolic manipulation. The subtraction process can be made obvious using suitable straegies that trainee teachers should be able figure out. In Hafeez' lesson, the idea of regrouing was not modelled by the pictorial representation. The use of a strory was good. especially if the teacher can act the situations out by making voices for Mr Gopan and whoever else in the story. Also, the pizza was impressive,

## Monday, January 26, 2009

### DCM202 Challenging Problems (1)

The problem is to find the tens digit when 2 is multiplied 2009 times. Most of the students were able to solve simpler problems to see a pattern. A strategy suggested was to try simpler cases until we get the same digit for the tens and ones digit. I was glad I gave the students time to establish the fact that the pattern will definitely repeat. That means they were expected to be able to explain the pattern. I felt that it was insufficient for students to solve the problem based on pattern observation alone. They should be able to show that the pattern is guaranteed.

## Thursday, January 22, 2009

### DCM201 Division of Small Numbers

Keng Sing's lesson using When The Door Bell Rang was enjoyable with cookies and plates for the children to act division out as sharing. It could be improved by applying the theory of variation. In all the examples the children were asked to act the situation out. In the later stages, the children could be asked to draw and use the cookies to confirm their ideas.

Rebecca's use of the cut-outs was suitable. I have suggested that the writing of the division sentence happens sooner in the lesson. And that was what Wei Wen did. In her lesson on division with remainder, she made careful links between the concrete with the pictorial and abstract represents well. One suggestion is to introduce the idea of quotient a little later.

## Tuesday, January 20, 2009

### DCM201 Multiplication of Large Numbers

Multiplication of large numbers begin in Primary 3 with multiplication of a 2 digit or 3-digit number and a 1-digit number followed by that of a 4-digit number and a 1-digit number. Later, students learn how to multiply a 2-digit number with another 2-digit number. The last two are often taught in Primary 4.

Khai taught 3D x 1D. I am pleased that he used the base ten blocks to model the process of 231 x 3 as representing 231 as 200, 30 and 1. Later he modelled the multiplication process of the hundreds, tens and ones. It is satisfying to see part of Bruner's Theory in action. One way this lesson could be improved, I told Khai, is to separates cases that requires renaming (365 x 5) from those that do not (231 x 3) and teach them in separate lessons.

In another lesson of 4D x 1D, I was excited when I saw the teacher teaching children how to use 123 x 2 to do 2123 x 2. The simplicity of it was beautiful - the new lesson simplify requires Primary 4 students to do the multiplication of the thousands (2000 x 2) and add this to 123 x 2 (which was done the year before already). It would have been great if this was done for the entire lesson. In this lesson it was done too briefly for it to have much impact on children learning. A good example of Teach Less Learn More (students learn to make connections by linking 123 x 2 to 2123 x 2).

In the third lesson, Lawrence did an excellent job applying Bruner's Theory in linking concrete, pictorial and abstract representations of 2D x 2D. You got to be there to appreciate fully what I said but breaking 34 x 42 into 4 x 42 and 30 x 42 using the area model would help children develop visualization as they learn the algorithm.

It was a good session. All the three demonstrates the basic competencies of competent teachers. They were the first three, so you can understand the nerves. It was good to see how the rest of the students who acted as students and as observers in a 'pseudo' lesson study supported their colleagues. I look forward to 27 more lessons that will unfold in the next few weeks preceeding the practicum.

## Friday, January 9, 2009

### Stamford Primary School

This was a workshop on mathematical problem solving. We spent most of the time on undertanding the importance of number sense, looking for patterns, articulation of thought and visualization. We also explores two ways to improve understanding of word problems - metacognitive reading and problem posing. Finally, we explore the what-if strategy as a post-solving strategy. In one of the problems, teachers are asked to look for relationship between and among a set of numbers generated in a mind-reading game. The idea is to tell kids that if they think of two digits (say, 5 and 2) and tell you only the first one, then you can rread their mind to get the secon digit and even able to use the digits to (a) form a number 52, (b) form another number 7, from 5 + 2 and finally (c) find the difference between 52 and 7 and be able to write that down on a piece of paper. All these without them having to tell you the second digit. So when the kid tells you 5, you can write 45 down. When the kid tells you 8, you write 72 and when the kid says 9, you write 81. What do you write on the piece of paper if the kid says 3. 27 is correct. How does it work. That was the problem the teachers solved. They came up with two strategies (1) multiply the number the kid says by 9 and (2) subtract 1 from the number the kid says and then making sure the sum of the digits in the answer is 9. Do you know why the trick works? Using this problem, teachers can see that problem solving involves the ability to see a pattern and generalize the pattern. Also, it involves putting our ideas into words as we try to describe our observations of how the three numbers are related.
This workshop was attended by n teachers.

## Thursday, January 8, 2009

### DCM202 Mathematics Games

When your friend says Salute, each player draws a card and outs it on his or her forehead. That means you can see everyone's card but not your own. Each player tells the sum of all the numbers he or she can see. The object of the game is to tell what number is on the card on your own forehead! After playing the game, did you realise what this game can do to help your sudents learn? Did you realise the mathematics that students can learn while playing this game? It was good to see the smiles and hear the laughter during this and other games that you played.

Throuhout the week, the students played several other games. I was hoping that they can abstract for themselves what roles games can play in the learning process. There was this game where they had to piece together 9 squares to form a larger square in such a way that the adjacent numbers are equal.

There was also this game of I Have... Who Has? where players have cards that read I Have 40. Who has 10% of 280? and another player who has it will continue I Have 28? Who has 5% of 280?. Once you have read from the card, you can get rid of it. The aim of the game is to lose all your cards.

The approach I adopted was to let the student teachers play the game so that they can abstract for themselves the importance of games in the learning process. Is it necessary for me to put in some structures so that this abstraction has a higher chance of happening. I decided on this approach as opposed to giving them a lecture to describe the different ways to use games - for drills, for developing concepts, for problem solving, for motivation and so on - and illustrated by specific examples because I want them to engage in the games so that they will remember the games and think of using them when they are in schools.

### DCM202 The Role of Calculators in Primary Mathematics

This is the first lesson with the two classes who offer this course. I have planned to do some calculator activities with the teachers and get them to reflect on the roles of the calculator in the primary mathematics classroom. The students were asked to find a pair of two-digit numbers such that AB x CD = BA x DC. As expected, solutions such as 11 x 33 and 12 x 21 came quickly. Later, solutions using three or four distict digits such as 24 x 84 and 63 x 12 were given. One class gave nine different solutions while the other gave eight. As usual the students observed relationship between the product of the tens digits and the product of the ones digit (such as 2 x 8 and 4 x 4 in 24 x 84 and 6 x 1 and 3 x 2 in 63 x 12). In 93 x 26, students observed the relationship between the ratio 9/6 and 3/2. It is good that some students could say that these two are essentially the same. It shows that they were making linkages. It was particularly satisfying when one student gave an incorrect answer 41 x 22 (which I wrote down without judgement) and another group of students using that as a basis to obtain a correct answer 42 x 12. I was thrilled when another student came up with a novel observation that in 24 x 84, doubling the 24 gives 48 and reversing the 48 gives 84 - which is the larger number. Similarly with 42 x 12, doubling 12 gives 24, reversing the digits in 24 gives 48 - and, hey, that's the larger number. I was so excited by this that I forgot to tell the class that this is a rare observation. It could well be the first time I hear it although I have done this lesson countless number of time.

At the end of the lesson, I was wondering if my students could see the role of the calculator in this activity. And how the role of the calculator in this activity is different from its role in the other activities that we did that day and in subsequent lessons.