I taught this lesson in Manila to a group of high school teachers using the New Syllabus Mathematics. The problem posed is to find the probability of drawing two cards with the same number, one from a stack of cards labelled 1 to 4 and another from another stack of cards, also labelled 1 to 4. Using the responses from eight groups, we were able to solve this problem that involved combined events using the knowledge of basic probability of single event. The responses included the use of listing, tree diagrams and table. I was quite satisfied we managed to use the responses to explain the multiplication rule.

## Saturday, May 23, 2009

## Saturday, May 16, 2009

### Education in the US

Secretary of Education Arne Duncan

Last week I went to Berkeley County, West Virginia, to begin an open, honest conversation about education reform.I wanted to hear ideas about how we can accomplish President Obama's goal of providing every child in America a complete and competitive education, from cradle through career.As we prepare for the reauthorization of No Child Left Behind, I want to hear from classroom teachers and other educators, parents and students, business people and citizens. What's working, and what's not? What do we need to do that we're not doing, and what do we need to stop doing - or do differently?

In the coming weeks, I will ask questions here. Topics will include raising standards, strengthening teacher quality, using data to improve learning, and turning around low-performing schools. I will be reading what you say. So will others here at the U.S. Department of Education.Today, I want to start with a simple set of questions: Many states in America are independently considering adopting internationally-benchmarked, college and career-ready standards. Is raising standards a good idea? How should we go about it?

Let the conversation begin!

Arne Duncan

## Friday, May 15, 2009

### Art & Mathematics

Another interesting problems we solve at the Ateneo Summer Institute is the one in which the teachers created interesting pictures like rat, cat and flowers on geoboard. Later, I asked the teachers to predict which of the picture occupies the largest area. Interestingly, in both groups, teachers suggested counting 'dots left over' as an indicator of area. We later tested this theory by finding the area by counting squares.

Some of the pictures were easily dealt with because they were made up of squares and half-squares. Others were more challenging. I hope the teachers got the idea of how to sequence discussion in students' creation in a way they are kept challenged all the time. We moved from cases which were easy to find the area to cases which demanded more visualization.

## Monday, May 11, 2009

### Ateneo Summer Institute

This is the second year I am teaching this course for early grades teachers (kindergarten to third grade). In today's session, we explore why we teach mathematics. We did a problem that also includes drills-and-practice based on subtraction. Typically subtraction algorithm is done in grades one to three all over the world. I hope the teachers were able to understand that if all the child is able to do is to do subtraction then we have failed. The teachers told me a a cheap calculator cost about 50 pesos retail, that's a little more than USD1. I am sure the wholesale price is even lower. The teachers agreed that a 50-peso calculator can do subtraction rather well. So, if a drill-and-practice exercise enable childrent o do subtraction perfectly then they are just as valuable as a 50-peso calculator. i tried to convey the message that helping children develop thinking and other human competencies are equally important, if not more improtant than developing the ability to perform the subtraction algorithm. A teacher said the activity took more time. I hope we will see that the time invested (50 minutes) is well-spent. In fact, I argue that this is necessary. Unless a task is an extended one, the thinking involved is not very deep and, hence, not significant.

In case you are not in the class today, the problem was to use one set of digit tiles (0 to 9) to make two 4-digit numbers such that the difference is as small as possible. In both classes, we had 25 as the smallest possible difference. The teachers were impressive and were able to make some connections - such as the thousands digits are consecutive. In the end, they found three possible solutions 4xxx - 3xxx = 25, 5xxx-4xxx = 25 and 6xxx - 5xxx = 25. Try it and let me know what other insights you manage to have.

A teacher asked how teachers can formulate such problems. A short answer is to attend courses like these as well as to lookt hem up in books and on the internet. A long answer is to change our beliefs what mathematics teaching is all about. Once that happens, all the tasks we create or choose will have features that allows students to develop good thinking abilities and habits.

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