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.

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.

Division of Fractions

I have just completed nine periods of mathematics lesson on this unit with a class of primary five students who struggle with mathematics. I am teaching dividing fraction a/b by a whole number c which is often done using the invert-and-multiply method. Instead of using this method which I believe contribute to these students' difficulties with mathematics, I use an alternate method which based on simple reasoning ... if 4 divided by 4 is 1 then 4 fifths divided by 4 is 1 fifth. In cases when this is not obvious such as 2/3 divided by 4 we talked about changing 2/3 into 4/6. 4 sixths divided by 6 was then "easy beezy" as one student put it. It gave the students plenty of chance to revise equivalent fraction, which is fundamental to operations involving fractions.

I particularly enjoyed the third lesson when I gave each pair of students a fraction. If they thought that their fraction has the same value as the one I put up, they put it next to mine. In other words the board was filled with equivalent fractions which the students can then use to do various division.

e-Lecture on Assessment for Secondary Mathematics

In this lecture, we cover the key ideas in test construction of a mathematics test for secondary students.

It is for pre-service teachers in Singapore.

http://videoweb.nie.edu.sg/captivate/AGs/MME/QCM521_531/test_construction/test_construction.htm

Research in Mathematics Education

It was an interesting opportunity for me to teach a graduate course at Chiangmai University in March 2009. The 26 students are all studying for a master degree in mathematics education. Some part-time students are doing coursework and need to do an independent study and the rest are full-time students who have to write a thesis. The three-day course aimed to help students with the construction of a research problem.

We spent some time identifying problems related to mathematics education. As the students are all school teachers in primary (grades one to six), middle (grades seven to nine) and high schools (grades ten to twelve), it was not surprising that most of the problems are classroom-related. We spent some time for the students to understand that the research problem is a culmination of their own personal experience and the literature. Their experience with the local situation as well as literature that describes the situation in Thailand (e.g. the TIMSS report) as well as previous studies done in Thailand and elsewhere on related topics are used to generate the content of their research problem.

For example, a student wrote that the primary-level students lack the ability to solve word problems in the topic of time. For example, third graders have difficulty in solving word problems that give the starting time of an activity, the duration of the activity and require students to find the end time of the activity.

This is a problem from the local situation. However, the research problem was still not articulated. For example, the research problem could be to investigate the types of difficulties students have when solving such word problems. A related research problem could be to use the findings from the earlier investigation to design instruction or remediation to help students develop the ability to solve word problems involving time, and to investigate features of instruction or remediation that are able to help students develop this ability.

Teaching Mathematics Education Students in Chiangmai University

Why are the students so cheerful in class? Because they are solving simultaneous equation with three unknowns - that's why? It is impossible that students were so cheerful in the mathematics class?

Each student raised a card to their forehead thus was not able to see their own card. Subsequently, each student stated the sum of numbers on their friends' foreheads. Say, A had 7, B had 9 and C had 3. A would have said 12, B 10 and C 16. If the value of the card that A had is a, B is b and C is c then b+c=12, a+c=10 and a+b=16. However, in this case A would have known the values of b and c. The only unknown is a. For secondary students this activity can be used to introduce the idea of solving equations. For primary students, it is a good exercise in addition and subtraction and, more importantly, reasoning.

"Despite resources that are unmatched anywhere in the world, we have let our grades slip, our schools crumble, our teacher quality fall short and other nations outpace us," Obama told the U.S. Hispanic Chamber of Commerce. "The future belongs to the nation that best educates its citizens, and my fellow Americans, we have everything we need to be that nation," he added. The U.S. leader painted the education drive as part of a broader push to promote economic growth in the face of a deep recession and the nation's worst financial crisis in decades.

"In a 21st century world where jobs can be shipped wherever there's an internet connection, where a child born in Dallas is competing with children in Delhi ... education is no longer just a pathway to opportunity and success, it is a prerequisite," he said.

These are words from the President of the USA. In the special lecture I had with all the undergraduate and graduate students of mathematics education in Chiangmai University, I hope the 100 of them in the room can teach 40 students well. In their many years as a teacher of mathematics, they are in the position to bring up a whole generation of citizens who can compete in this type of global economy.

We emphasized on the problem-solving approach, the focus on intellectual competence as well as the concrete/pictorial/abstract approach to teaching students. I was asked where teachers can find problems such as the ones I used in the session. I replied that the problems themselves are not as important as how the teacher uses it. I had demonstrated in the lesson how even computational procedures such as 2/3 divided by 4 can be taught to help students develop and improve their intellectual competence.

In any case there are many good problems on the internet and in moments of inspiration some teachers can come up with interesting problems. If you are one of them, please share na. (na is a particle in Thai to gently persuade someone to do something.)

Temasek Foundation Singapore & Population and Community Development Association Thailand Project

Think of two numbers e.g. 4 and 5. We will do three steps.

In a variation which I did with second graders in New York, they picked two numbers from a stack of cards. They showed the first number but not the second one. Before that they had done the three steps with the two numbers.

Use the 4 and 5 to make 45. Use the 4 and 5 to make 9 (4 + 5). Finally find the difference between 45 and 9.

After demonstrating that if they tell me the first number then I will be able to read their mind for the second and do all the three steps and giving the right answer each time, we began to record the findings.

Many of the participants saw the multiple of nine trick. As I pressed for more observations, another interesting one is how the second digit in the answer is derived by subtracting the first number from ten.

I was reminded again that asking students to explore more ways to often result interesting and outstanding responses.

I did this problem with the teachers group as well as the principals group and on both days we got equally interesting responses except I was surprised that no one in the teachers asked for the reason why the trick worked. One did in the pricipals group.