Ateneo de Manila Summer Institute

This is the third year I have been here to teach the summer institute. This year I am teaching the middle school program where the teachers are teaching Grades 4 to 7. We began after a brief opening ceremony.

In the opening session, I tried to discuss Bruner's enactive, iconic and symbolic representations using division of whole numbers by fractions and long division algorithms. The teachers asked questions related to teaching of division of fractions by fractions, multiplication of decimals and operations involving negative integers. We ended the day with a game of 'either one or three'- in a game where two players take turn to remove exactly one or three paper clips from apile of clips and the winner is the person who remove the last clip(s) - what is the winning strategy?

What if the rule is changes to wither 'one or two'?

We begin Day Two with Dienes' Theory of Variation ... but not before I pose the class the Fido Problem. Tell you more later.

Problem-Based Approach

BBS April 2010 Singapore Math in Indonesia by BBS Maths Consultant Dr Yeap Ban Har

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The open-ended approach common in the Japanese mathematics classroom uses a single problem in a mathematics lesson. In training teachers at Bina Bangsa School - a family of schools in Indonesia that uses the Singapore curriculum - I introduced the idea of teaching new concepts, doing drill-and-practice as well as getting students to apply what they learn using a well-selected or well-crafted problem. Other than workshops and lectures, teachers also get to see such problem-based lessons - there was one on learning area (primary three) and one applying Pythagoras theorem to find distance bewteen points (secondary two). Teachers saw examples, they worked through afew such problems, I modelled a few of such lessons with them and one with students,they worked with other colleagues to design one such lesson which hopefully they will try out and see how studenys respond to the lesson).

In designing such lesson, teachers select a problem and solve the problem themselves to understand the processes and challenges involved, as well as to see the mathematics in the problem. They decide if the problem can be used to introduce a topic or to provide drills-and-practice or to provide opportunities for students to apply what they have learnt.

They also considered how students will solve the problem and how to use the solutions to help students construct knowledge.

Fraction Addition: Dienes' Theory of Mathematical Variability

In the final lesson with my first year diploma students (they will go on to read at least two more mathematics education courses as part of their pre-service training), we had a chance to review the test they took and key learning theories.

I used examples from fraction additions to discuss theory of variability (Zoltan Dienes). How is 2/5 + 2/5 different from 2/3 + 2/3? Both are symbolic representation of fraction additions. But mathematically they are not the same. Dienes refer to this as mathematical variability. In 2/3 + 2/3, students need to have the knowledge of mixed numbers and improper fractions. Such knowledge is not needed in doing 2/5 + 2/5. These two tasks possess mathematical variability.

In Singapore, these are taught in different grade levels. 2/5 + 2/5 is taught in Primary 2 while 2/3 + 2/3 are taught in Primary 4.

How about 1/8 + 1/8? This is taught in Primary 3. Why is addition of like fraction (2/5 + 2/5, 1/8 + 1/8 and 2/3 + 2/3) are considered not the same and are taught in different grade levels? According to Dienes, there is mathematical variability among the tasks. The Singapore approach seems to suggest the need for students to be systematically introduced to each variation.

In 1/8 + 1/8, students need to know equivalent fractions. Mathematically the three tasks are not the same and this is mathematical variability that Dienes wrote about.

Is 5/8 + 5/8 a variation of the three tasks given above?

Fractions: Equal Parts

The mathematics education course focuses on curriculum structure, teaching strategies including ICT and students' responses (errors). For example in the Diploma programme, the first course (of three or four) includes Teaching of Whole Numbers, Teaching of Fractions, Decimals and Percent, Teaching of Ratio and Teaching of Rate and Speed.

Student teachers also learn the Singapore curriculum framework, learning theories, scheme of work and lesson planning, Ministry of Education initiativces including the role of calculators and ICT.

In a lesson on Teaching of Fractions, teachers were presented with this problem - is a rectangle cut into four equal parts by its two diagonals?

There were student teachers who said that the parts are not all equal because the triangles are not 'the same' meaning not congruent. Subsequent whole-group discussion with paper rectangles and a pair of scissors led to four responses that are shown at the top of this page.

There was a response that suggested cutting each the the four parts into two triangles which are equal parts as they are congruent triangles. Each of the four parts is then clearly 2 eighths. See Photograph 1.

A second response is based on using the formula to calculate the area of triangle. On Photograph 2, it can be seen that B = 2h and b = 2H. The area of the two non-conguent triangles can be shown to be the same.

Another response includes cutting each of the four parts into two congruent triangles and rearranging the pieces to form congruent triangles. Thus, the triangle shaded black and the one shaded red in Photograph 3 can be rearranged to form congruent triangles.

Finally there was another response that is based on using two congruent triangles to form a rhombus. This resulted in two congruent rhombii. If the two rhombii are equal, it follows that half of one rhombus (one of the triangles) is equal to half of the other (the other triangle).

The student teachers were told that in mathematics classrooms, teachers use different models to show fractions. In the example, we use the area model - where we use area to represent fractions. There is also the length model (e.g. the bar model or line model) where length is used to represent fractions. Other models include the volume model (e.g. using a cylinder of water to show fractions).

In terms of curriculum structure, we learn that equal parts is an important concept in understanding of the fraction notation. In terms of common misconceptions, we see that some students may have the wrong idea that equal parts refer to congruent parts rather than parts with equal area.

In Singapore curriculum, fraction is introduced formally in Primary 2.

Becoming a Teacher in Singapore

There are plenty of opportunities for professional development for teachers in Singapore. These four teachers from four different priamry schools were on teacher work attachment (TWA) scheme with Teachers Network and went to Cambodia in December 2009 to share with Cambodian teachers and educators interesting ways to teach primary mathematics. The second batch of teachers that went on TWA to Cambodia in June 2010 captured their experience at http://twacambodia2010.blogspot.com/

How does one become a teacher in Singapore? You can finish the A Levels (Grade 12) or graduate with a polytechnic diploma and join a two-year course at the National Institute of Education (NIE) to be certified to become a teacher. You graduate with a Diploma in Education. There are opportunities to obtain a university degree if you do well at the end of the two years (that means you study for another two years for the bachelors degree) or return to school after teaching for a couple of years.

From 2012, Ministry of Education (MOE) will stop taking in non-graduate teachers so this programme will presumably cease to exist. As of 2009 about 50% of teachers in primary schools are not university graduates. In secondary schools, most teachers are university graduates.

If your grade twelve national examination (the A Levels) result is good or you have done extremely well in your polytechnic course, you gain admission into a four-year degree programme where you graduate with a bachelors degree in science or arts with education.

You can also come in for a one-year course (PGDE) if you already have a university degree.

As far as mathematics education courses are concerned, they are the same in all three programmes. It makes sense because everyone, whether you have a university degree or not, will be doing the same job - teach children mathematics.

For secondary school teachers, the diploma option is not available as MOE takes in only university graduates as secondary school teachers. Secondary teachers either read the one-year PGDE programme or the four-year degree programme.

Secondary teachers are trained to teach two subjects, unless you are teaching in junior college (grades eleven and twelve). In that case, you do Teaching of Mathematics (Secondary) and Teaching of Mathematics (JC). There is also Teaching of Lower Secondary Mathematics for those do not not have the prerequisite to teach upper secondary mathematics.

Primary teachers are trained to teach mathematics with one (less common for now)or two (presently more common) other subjects.

Teacher Education in Singapore

The new focus of this blog is to share what we do in teacher training in Singapore. I will focus on the teacher education for primary school (elementary school) teacher who teach in grades one to six upon graduation.

The examples are limited to the courses I teach.

In this introductory entry, I write about some background information that is useful in understanding teacher education in Singapore.

All the 150+ primary schools in Singapore are all public schools. We do not have a private education sector. The variations that are now available in the secondary and junior college levels (grades seven to twelve) are not available in the primary levels.

Formal schooling starts in grade one. Students turn seven sometime during grade one. Pre-school education is varied and a small number of children do not attend kindergartens. Compulsory education is only for grades one to six. However, it is extremely rare for teenagers not to attend school although it is not compulsory to do so.

All our primary school teachers are public school teachers employed by the Ministry of Education before they study to become teachers. This is not common. In most places people study in faculties of educations in various universities in the country and they apply for a teaching job after they graduate with a teaching degree (e.g. bachelor of education). In Singapore, people apply for a teaching job before they study to become a teacher. Once they get the job, they will all study, at present, at one place - National Institute of Education (NIE) which is housed in one of the four universities in the country.

The photo shows one of my classes conducting a lesson for their colleagues. They have designed an activity-based lesson to teach pie charts. I was pleased they attempted to put into practice what they learn in the course on engaging pupils with meaningful activities.

AKM101 Lesson 4

We had four lessons and for the next few weeks we will be having online lessons. Today, we re-examine the issues of teaching mathemeatics versus teaching children. I am glad to hear many views including if the national examination assess good habits of mind and spirit of innovation and enterprise. It is popular rhetoric that all the things outlines in various initiatives are fine and good but the examinations do not value them and hence teachers do not place emphasis on them. That is what this course is about - for the participants to re-examine this beyond the popular rhetoric, to reconsider one's present point of view and points of views of the other participants and to examine the issues more deeply. Thus two questions - one, how to help students reach a certain level of achievement that will allow them to progress towards becoming a productive citizen of the country and the world and, two, as teachers, should we go all out to ensure highest grade possible, even at the expense of teaching the child values and developing his or her confidence to face the world or moulding their personality as a well-adjusted youngsters?