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.