Math in Singapore 2007

Back to Singapore in 2009

2009-02-23T13:41:00.002-05:00

Hello,

We have planned another weeklong Singapore mathematics summer program for 2009.

For details, please visit www.smathresources.com

Thanks!

Patsy

July 12, 2007: Visit to NIE

2007-08-19T12:19:00.000-04:00

In the afternoon, we visited the National Institute of Education (NIE), located on the campus of Nanyang Technological University, and responsible for pre-service teacher training throughout the country. We met with members of the Mathematics and Mathematics Education (MME) Academic Group of the NIE. Our host for the afternoon was Dr. Khoon Yoong Wong, Head of the group.

Dr. Wong described the role of the MME group. (See his PowerPoint presentation.) They provide teacher education in mathematics at several levels: primary, secondary, pre-university and graduate levels. Members of this group are involved in both research and teaching. They work closely with the Singapore Ministry of Education (MOE) and with schools, through their community service component.

There were many staff members in attendance. They included mathematics educators, mathematicians (some of whom teach), teaching fellows and support staff. We met in one of the mathematics laboratories associated with the program. This laboratory contained many sets of manipulatives, accessible to the students as part of their training. The facilities also include computer laboratories.

Professor Wong discussed the various programs offered by NIE, leading to careers in education. Pre-service programs are offered at various levels. To teach at the Primary level, one can pursue a two-year course of study leading to the Diploma in Education. Pre-requisite for this is an A-level or Polytechnic Diploma. To teach at the Primary, Secondary or Junior College (JC, equivalent to our Junior and Senior years in high school) level, one can study full time for 4 years, to obtain a Bachelor of Science in Education degree. For people who already have Bachelor’s degrees, there is a Post Graduate Degree in Education. The programs focus on Primary or Secondary studies.

NIE supports the MOE Professional Development Continuum Model by offering coursework for in-service teachers, as well, leading to advanced degrees. For example, a Primary school teacher with a two-year diploma can take coursework leading to a BA or BSc, then a Masters and finally a PhD. Similarly, graduate level programs are offered for Bachelor’s and Master’s degree holders.

There are also ad hoc courses in mathematics and pedagogy, which support MOE Professional Development plans for Singapore teachers.

{Notes on financing teacher education: Trainee teachers and graduates pursuing advanced degrees are employees of MOE. They receive a salary-like allowance (somewhat less for undergraduates or those in the diploma program). Once they begin to teach in schools, their salaries are raised. There are no tuition fees, since they are employees and MOE pays for their studies. Through MOE, there is an application process to become a teacher and to receive all this education. Some applicants are sent to schools for ½ year or more, as contract teachers, prior to entering NIE. Others enter directly, having completing their 'A' level exams or their diploma studies or their undergraduate studies.}

A number of attendees from the MME group are published leaders in the fields of Mathematics Education and Mathematics research. MME staff members founded a professional society, the Association of Math Educators, with a website which publishes a journal on teaching and learning and a newsletter, Maths Buzz. This organization also organizes an annual conference each year.

Professor Wong presented the curriculum framework pentagon, seen in the schools we visited, as the core of Mathematics Education in Singapore. This framework was developed in 1989-90. He also showed a diagram of five intertwined strands of proficiency from “Adding it Up”, one of whose authors and the committee chair was Jeremy Kilpatrick. This author visited Singapore several times. The work was published in 2001 and makes note of these strands: Adaptive Reasoning, Strategic Competence, Conceptual Understanding, Productive Disposition and Procedural Fluency.

After a Q&A period, and appropriate farewells, we left the MME group.

We headed to the SIE bookstore, where we found a number of books used in the education of teachers, most notably, the Singapore Mathematics Education Series Resource Books, written by MME lecturers. We were able to purchase the Primary School version and await the soon to be published Secondary School Mathematics, newly revised edition.

Post created with input from Perla and Patsy

Friday July 13 Visit to Ku Chuan Presbyterian School

2007-08-14T16:38:00.000-04:00

Friday, July 13, 2007
Our last business day in Singapore was spent observing at Kuo Chuan Presbyterian Secondary School (KCPSS). KCPSS is a middle tier school, consisting of 7 – 10th grade Express and Normal Academic students. Interestingly, although the school has a religious affiliation, only about 30% of the students are Christian, with the remainder being predominantly Buddhist and Taoist and some Muslims. It is interesting that the mission schools in Singapore are supported by the government and overseen by the Ministry of Education. There is obviously no separation of church and state. In fact, schools we visited had posters describing religious tolerance displayed in the classrooms.

After watching ta video of the history and mission of the school, we were led to the science special project area where a group of students had been raising fish, rabbits, hamsters and tortoises. The students had also researched the best ways to keep birds and mosquitoes from accessing their outdoor project area, and have therefore hung CDs from the building (to keep the birds away) and grown plants which smell like Citronella. The students involved in the project were eager to share their knowledge with us. The school began this project because they found that students who had trouble expressing themselves tended to open up when they cared for pets.

We were then led to a Sec 3 (9th grade) A-Maths (Advanced maths) class which was factoring quadratic equations and reviewing test-taking strategies. Students in secondary school have the choice of taking one math class per day, E-Maths (Express maths), which everybody takes, or taking both E-Maths and A-Maths. Those students who have shown proficiency in math and who choose to do so take two periods of math each day. The A-Maths class seemed to be expected to work at an incredible pace. The teacher’s emphasis on test preparation and efficient methods of quickly answering the various categorical problems students were likely to face on their Grade 10 exam (upcoming in October 2007) was noteworthy due to the length of time in the class dedicated to the demonstration (30 minutes), the teacher’s explanation for the use of the strategy (building student strategic competence?) , demonstration of a series of progressively more difficult applications of the strategy, and an immediate student practice session using worksheets printed in the school’s photocopy center. A scan of students at their desks revealed a wide variety of erasers, white-out tape dispensers (both used to “un-do” mistakes immediately), and scientific calculators – some with graphing functionality.
As we circulated through the school, we each remarked on the strong “student-centered” quality of the school, including its open multi-use spaces for both structured and unstructured student interaction, the posting of student work and academic recognitions in public spaces, and the informal interactions between teachers and students. A strong commitment to student experimentation was evident in the spacious laboratories and adjoining materials storage room supervised by support staff, who also prepare the materials and supplies for experiments and wash the used supplies.

After observing the A-Maths class we had tea in the library and spoke with several staff members. Tony Low, the principal, informed us that the biggest problem he sees in his school is what he calls “lack of motivation.” When asked to define “lack of motivation”, Mr. Low said that many of the secondary students have no goals for the future. He would like for all secondary students to have a career plan in mind in order to focus their efforts. If a student does not look to the future, the school has a career counselor work with the student to help give the student ideas of what fields are available. Mr. Low indicated that many students change their mind about their career choice as they mature, but that it is best for them to have an idea at all times so that they are motivated. When we had a chance to meet with students and carry on informal conversation, about half of them seemed to have a career path in mind. I sincerely doubt that one-half of American students in 7-10 grades have a career path formulated.
Two girls I spoke with (Sec 4 – 10th grade) are nervously preparing for their O level tests which will determine their educational path following secondary school. They hope to score well which will allow them to attend junior college. They shared a common ultimate goal: to attend medical school. Both of them receive “tuition” (private tutoring) once a week for three hours in maths, and have been receiving such tuition for several years. I asked them if they considered themselves strong math students. Both of them seemed a bit embarrassed, and laughed a little bit while shaking their heads.
Tony Low is an extremely impressive principal. He informed us that he occasionally canes students for severe behavioral infractions. These canings are generally done in the school office, although occasionally they are done in the classroom if the infraction occurred there. Every effort is made, after the caning, to be sure that the caned student feels welcome again in the school/classroom since he/she has paid the price for the wrongdoing. Mr. Low has high expectations while seeming to care about the well-being of his students. He was keenly aware of the rather recent phenomenon of computer gaming addiction among pre-teens and teenagers. He led several of us to a school computer lab containing desktop PC’s loaded with approved gaming software. Mr. Low explained that students could earn specified access to this resource based on good classroom performance. He outlined the school’s strategy of parent education, counselor availability and controlled access to approved games as a three-part strategy to address the problem in the students.
During the extended discussion with administrators and teaching staff at KCPSS, we learned that the school is moving forward with ambitious plans to replace the computer lab with a wireless environment to accommodate the large number of students who have their own computers. This was seen as a great asset to classroom instruction and student learning. At this time, the school’s computer labs are open to students each afternoon during the week. In at least one aspect of schooling, KCPSS was similar to many U.S. secondary schools- its challenge to engage faculty members as active participants in a professional learning community (PLF) , including openness to classroom visitation, colleague feedback, and instructional improvement through reflections on student work and lesson delivery. In contrast to the strong evidence of PLF in the primary schools we visited earlier in the week, KCPSS teachers and administrators admitted that they have considerable work still to do in this area of school development.
To sum up our observations of education in Singapore vs. education in the U.S., it seems fitting to say that in Singapore, education is considered an investment, while in the US, education is considered an expense. As several Singapore educators mentioned during the week, “In Singapore, our people are our only natural resource. We must develop them.”

Tricia and Mike

Thursday, July 12th, 2007: Visit to Geylang Methodist Secondary School

2007-08-13T02:16:00.000-04:00

Due to a scheduling change, we were able to pay a morning visit to the brand new Geylang Methodist Secondary School (GMSS). Founded in 1924, GMSS is a middle-ranked Methodist government school with 1,300 students in 35 classes distributed between express and normal tech levels. With a highest possible score of 280 points on the Primary School Leaving Exams (PSLEs – administered in math, science, English and mother tongue), the express students at GMSS scored between 189-240 points. The normal tech students scored below 111 points, with a range of 71-111 points. They have a 70% pass rate on the “O” levels for Normal Technical schools (the bottom strand); the national average for students at this level is 55%.

The current facility had been occupied for just two weeks at the time of our visit. Schools undergo minor or major renovations approximately every five to seven years. When buildings are to be replaced, as occurred with GMSS, the entire school population relocates to another facility while a new building is constructed (over a period of two years).

Introductions
Our hostess, Mdm Foo Kum Fong, whose office is at GMSS, is a master teacher of mathematics for one of the clusters. There are 160 schools serviced by 19 master teachers who travel to the different schools in their cluster. Of the two mathematics master teachers one is Mdm Fong and the other specializes in primary school.

First Observation
After short introductions in the boardroom we went to observe, for 15 minutes, a sec 1 (grade 7) mathematics Express 4 lesson (the lowest of the higher level academic track, covering the entire curriculum, 7th-10th, in four years) on ratios and improper fraction ratios. The class textbook was Discovering Mathematics by Chow Wai Keung. The students were, overall, rowdier than those we observed at the primary schools, but were still fairly well on task. The class began when the teacher asked us (the visitors) to introduce ourselves to the students. He then used the numbers of visitors, three male visitors and ten female visitors, to create a ratio: the number of male visitors to female visitors is 3:10. The teacher asked “Have you worked with ratios before?” and the students answered “yes.” Then he asked “Do you remember what it is?” and the students answered “no.” The class reviewed ratios using the worksheet below, and some students shared their answers on the board as they went through the worksheet. One procedure the teacher gave for simplifying 2/5 : ¾ is
2/5 : ¾ = 2/5 x 4/4 : ¾ x 5/5 = 8/20 : 15/20 = 8:15 since the denominators are the same. Several of us had not seen ratios with three quantities (A:B:C) and problems using them in the United States. For example, one of the problems on the worksheet was: P:Q = 6:7 and Q:R = 3:5. Find P:Q:R.

Second Observation
From the sec 1 class, a group of us moved to observe a normal tech 3 class (students that are on track to learn a technical trade) working on factoring quadratic equations. We were warned that these students would be rowdy, but we did not find them rowdier than the rest of the students in the time we were there. In this 35-minute lesson, the teacher, Mr. Joseph Lim Tuan Zheng, showed an alternative method for factoring a quadratic expression, "the cross method," using direct instruction, step by step. The teacher gave us a lesson plan detailing the knowledge of the students prior to the lesson and the lesson focus.
___________________________________________________________
(From the teacher's lesson plan)
Knowledge Prior to the Lesson:
-Students have just learnt Expansion and Factorization of Quadratic Expression by Grouping Method.
-They have learnt the 4 operations of positive and negative integers.

Lesson Focus:
1. My Target Audiences are students who are weak in Mathematics but they are willing to learn. 2. Algebra is a topic students find hard to understand. They get confused along the way.
3. I will break the Cross Method into parts so they understand how to do it.
4. The challenge is for the whole class of 42 students to follow the steps and not get lost along the way.
These Normal Technical students get restless easily and have very short attention span. Questioning their understanding of the topic after 5 minutes and asking them to do on the board will keep them on task and awake.
___________________________________________________________

The teacher taught the "cross method" using a visual model. He connected each term in the quadratic expression to its corresponding place in the model using circles, squares and triangles (see problem 1 in the worksheet below). The second example (problem 2) and the rest of the problems paralleled the first, with the exception that the shapes all became squares to be filled in. The students were to repeat the steps of the procedure by following the locations of the squares. There were numerous steps involved, and the teacher asked questions while working through the steps. For example, for problem 1, when the teacher determined that the quadratic expression could be factored into (v+1)(v+1), the teacher asked how this should be written and the students answered (v+1)^2. He also asked if the order of (a+12)(a-2) could be switched, and explained that -8-1=-9 by drawing one minus sign and eight minus signs, counting them and getting 9 minus signs (or 9-), which we "call -9."

For problem 7, the teacher, with input from the class, went through a couple of incorrect or incomplete ways of factoring 9t^2+24t+15 before arriving at the correct and complete factorization. First (9t+5)(t+3), followed by (3t+3)(3t+5). When the teacher said that the latter was incorrect, the students noticed that a 3 could be factored out, resulting in 3(t+1)(3t+5). The teacher used this opportunity to show the students that the 3 could be factored out of the original expression, "making the expression smaller and easier to factor." After they went through the process of factoring 3(3t^2+8t+5) into 3(3t+5)(t+1), the teacher asked for a vote: "Which process is easier (factoring a common term first, or using the "cross method" first and then finding and factoring common terms)?" Then he told the students to "Stick to ONE RULE: take out the common factors first."

When the quadratic equation involved a coefficient greater than one for the x^2 term (starting on problem 6), body language indicated that some students had lost their way. When the third term required more choices, body language again seemed to show student confusion. Students shared their solutions on the board and those who needed help could look at the answer. Students were asked to work in pairs on the worksheet. Some did, while some of them worked individually and others simply copied each other’s results. Many students waited for the teacher to come around and answer their questions.

Third Observation
Two people in our group observed a sec 3 express level 3 math class working on similar triangles. They reported there were four boys with behavioral issues in the classroom, which led us to wonder how discipline is handled in Singapore schools.

Music Lesson
On our way back to the boardroom we passed by a music class; each student had an everyday object that could be used for percussion (pots, pans, buckets, washboards, etc.). Seated in a circle, they took turns making a sequence of sounds for the others to imitate. This music lesson was part of the theme for the week, which was respect. In this class, respect was shown by listening and following the lead of each student.

More Information
Back in the board room we continued our conversation with Mdm Foo Kum Fong and the school principal Lim Yan Hock. We learned that sec 3 students take 10 subjects, including history (two periods/week), geography, English (six periods/week), literature, mother tongue, music, PE, art, chemistry, biology, physics, and mathematics (six periods/week). Classes begin at 7:25 am and end at 1:50pm. Secs 1 and 2 take general science, which includes chemistry, biology and physics. Sec 4 (grade 10) students encountering difficulty are required to stay after school for help from 2:30 to 4:00pm four days a week. Ninety-six percent of the students continue with school beyond sec 4. We also found out that there is a very small turnover of teachers. Last year three out of 75 teachers left.
Origami made out of paper plates by Mme. Foo.

Post created with input from Tobe, Cassie, Tricia and Patsy

Wednesday, July 11, 2007: Visit to Guangyang Primary School

2007-08-11T17:02:00.000-04:00

On Wednesday we left bright and early (6:50 am) to visit Guangyang Primary School. When we arrived, the children were sitting in the cafeteria area, reading books. Then they proceeded to stand in formation in the school yard (7:25 am), where we participated in the student-led ceremony that included raising the flag, reciting the national pledge and singing the national anthem. A group photo was taken with the school staff. (At the end of the day, each of us received an 8 ½” x 11” laminated copy of this picture.) On the day of our visit there also was a delegation from China, who had been there for several days, with the students attending classes.

Guangyang Primary School was originally established as a Chinese School (Kiong Yong High School) in 1918. The primary school branch was converted into a government school in 1986 and it moved to its current location in 1993. As with the other schools in Singapore, there is obvious support (from the Ministry of Education) for both the physical and educational well-being of the school. In Singapore, the ministry “puts it money where its mouth is.” In other words, they say that they are concerned with education and then put in the necessary financial structures to support this concern. The necessary structures are in place to provide a learning and teaching environment that is motivating, responsive and creative. Guangyang is truly beautiful (both inside and out). It has amazing facilities. There are games painted all over the play area and many kiosks with computers throughout the campus.
There are also positive messages and lessons everywhere around campus. For example, the school’s habits of mind-persisting, questioning and posing problems; creating, imagining and innovating; and taking responsible risks-are written on the building. There are signs about correct English usage and about healthy eating. Even the children’s clothes carry messages: the school’s core values-courage, diligence, honesty, and loyalty are written on the younger children’s sleeves.

Special Rooms

There are a number of beautiful rooms dedicated to different student programs. The first is a hands-on math/science remediation center. This center was designed for students who struggle in mathematics. The room is full of fun math games, (including an enormous abacus) and manipulatives, and struggling students are pulled out of P1 (Primary 1, equivalent to grade 1) to engage in mathematics by playing games. It is refreshing to see struggling students motivated by these activities as compared with many of the “drill and kill” methods that are often used in U.S. classrooms.

A second room is dedicated to the media arts and music. This Prodikey room contains a beautiful wooden floor that acts as a staging area for video productions. The room itself is painted in vibrant colors. On one side there are state of the art computers and cameras – all with the purpose of engaging students in digital/video arts. On the other side there are keyboards attached to computers for learning and composing music.
A third room serves as a space for students to develop their entrepreneurial skills. Again, it is beautifully decorated. It contains a real-life cash register for keeping track of money. Different student groups can rent out the room and use it to sell things to raise funds. From recycling to fundraising, it is clear that students can learn a great deal about the world around them by interacting in this room.

“Village of Knowledge” is the name of the library at Guangyang Primary school. As you enter the room, you feel that you are crossing into a beautifully trellised garden. How beautiful to see so much effort and intention placed into the center of knowledge—the library. Many, many books are available to students—both in English and in their mother tongues. A daily puppet show is held in an enchanting “stage” near the back of the library—P4 and P5 students perform for P1 students during recess. The daily entertainment is probably designed to help get students more interested in reading.

The school also offers students opportunities for performing in theater, music, and exhibiting their art.

Touring the school, one can see that teachers feel that they are a part of the vision and the success of the school. Teachers are encouraged to support the student special projects.

Third Grade (P3) Research Lesson
The use of lesson study for teacher professional learning is relatively new in Singapore (pilot efforts began in 2004). Guangyang Primary School was introduced to lesson study in 2006. We observed a P3B (primary 3 (grade 3) medium ability) lesson on equivalent fractions developed by Mr. Andrew Leung, Ms. Mubina Faizie and Mrs. Vernice Kong and taught by Mr. Leung. The students were in a different setting (to accommodate the number of observers), were not used to having visitors, and were intimidated by our presence. According to Mr. Leung, they were unusually quiet and not very interactive. (We later discovered that Mr. Leung is also Dean of Discipline.)

Each of the observers received a copy of the lesson plan, an observer’s template with specific points to notice and evaluate including space to record observations, and a seating chart. The unit objectives on the lesson plan were: 1) recognize and name equivalent fractions; 2) list the first 8 equivalent fractions of a given fraction with denominator not greater than 12; 3) write the equivalent fraction of a fraction given the denominator or numerator; and 4) feel motivated to study the next lesson. The lesson plan also included current characteristics of the students, learning plan for the unit, students’ prior knowledge, a very detailed script, anticipated student thinking and activities, points to notice and evaluate, and a list of materials, strategies and purpose. Each group of three visitors was assigned a table of two students to observe. We sat in chairs around the classroom and moved closer to the tables when the students worked in groups.

Mr. Leung began with a review of the previous day’s lesson using circle magnets (cut into slices that could be separated) on the board. He reminded the students how they had thought about sharing cake equally with a friend. He told them that the ½ cake slices were too big and asked them what they could do. The student responses led to cutting the cake into different numbers of equally-sized slices and the realization that no matter how many pieces the cake was cut into, if each of the two people received an equal number of slices, each would still get half of the cake. For each cut-up disc, he wrote the fraction associated with one person’s share of the cake (1/2, 2/4, 3/6, 4/8, 5/10, 6/12). During this discussion, Mr. Leung drew a circle, cut it into two unequal slices and questioned the children in order to establish the rule: “Every piece must be of equal size and each person must have the same number of slices.” The children answered the teacher’s questions throughout this session (“Are they all equal pieces?”; “Are you getting more cake or the same cake?”; “Now the slices are too big, what would you do? How would you do it?”).

After reviewing the cake lesson, Mr. Leung asked the students “Does anyone like chocolate?” and told the children that now they were going to share bars of chocolate. He distributed six strips of paper to each pair of students along with a worksheet. As he demonstrated he asked them to fold one of the strips of paper into two halves, draw a line along the crease, shade the part that one person would get (“shade like me or creatively”), and attach the strip to their worksheet with sticky putty. Mr. Leung then asked “What do I want you to do next?” and answered “break the chocolate into four equal pieces.” They folded a strip of paper into four equal pieces, traced along the crease, and shaded two fourths as the teacher told them that “the shaded parts must be on the same side.” Mr. Leung showed one of the pair’s resulting strip, commented that “they are fair to each other because they each get the same amount,” and asked if anyone’s work looked different. Some students corrected their work (one of the pairs had only shaded one fourth, for example). The teacher then led the students to repeat the process for folding into and sharing six pieces, eight pieces and ten pieces. At certain points he justified the process by pointing out to the students that, “just like the cakes,” the pieces were still too big and “you can’t put the whole thing into your mouth.” Some students filled in the “number of shaded parts,” “total number of parts” and “fraction of shaded parts” in the worksheet as they worked, while others left them blank.

A large part of the lesson was spent on paper folding strategies (an unanticipated aspect of the lesson not discussed in the very detailed lesson plan). Mr. Leung showed the students how to fold the strip into six equal parts. First he folded it into three parts: Without creasing, he folded one end of the strip approximately two thirds of the way on one side of the strip, folded the other edge onto the other side of the strip (in an accordion-like manner), adjusted the three resulting parts so that they would be equal, and creased them. He then asked how to get six out of three parts. Some students folded each third separately into two parts, while others folded the overlapping thirds in half. One group folded into three pieces, shaded one third, shaded another sixth, by stopping at the halfway mark, and then folded each third into two parts.

Mr. Leung commented that folding into ten parts “is the trickiest” and asked the children if anyone would be able to “break ten pieces from one bar” (approximately, since “in real life chocolate the lines don’t break exactly”). One of the girls immediately took her ruler out and started to measure. Other children started using the “accordion” strategy they used for folding into thirds. Many students waited. The teacher proceeded to show them a clever trick for folding the strip into five equal parts. Before creasing, he folded one side of the strip over until the remaining piece appeared to be approximately one half the length of the visible part of the folded over piece. He doubled the remaining piece back over the folded piece to show this. He then permanently creased the longer piece. and then folded it in half again. When he opened the strip it had five equal parts. He accordion-pleated the five parts and then folded the whole strip in half. When he opened the strip, there were now ten equal parts.

Mr. Leung noticed that class was almost over so he had the students vote to decide if they should fold any more strips (the lesson plan had them dividing the paper into twelve parts, but they voted that they wanted to stop folding). He shared one pair’s work and led a discussion about the shaded parts. The class pointed out that the shaded parts are equal and that each person is “getting the same chocolate” in each of the situations. Mr. Leung used this discussion to introduce the term “equivalent fractions” and wrote: 1/2 = 2/4 = 3/6 = 4/8 = 5/10. Since the class time had ended, he gave them a worksheet to complete at home (some of the students completed the worksheet right away). The part of the lesson plan that was not completed had the students working with fraction tiles to complete the worksheet and writing a journal entry on what they learned.

Welcome Address and Presentation
After the research lesson, Principal Mdm Kit Gek Wah, the vice principal Hanafi Asmore, and the Head of the Math Department Mdm Lim Siew Hua, gave a presentation that included an overview of the education system in Singapore (mission, vision, philosophy, special features, and a schooling flowchart); specific information about Guangyang Primary School (mission, vision, core values, habits of mind and strategic thrusts); an overview of primary mathematics education in Singapore (aims, maths curriculum framework (the pentagon we had already seen a couple of times), schedules, curriculum and its spiral approach, and the teaching approach from concrete to pictorial to abstract); and information about the specific school programs and support systems. They also talked about assessment, professional development for teachers, workshops for parents, and pedagogy. Edmund asked for a copy of the powerpoint presentation and instead, they burned us each a CD.

First Grade Observations
We were divided into two smaller groups, and each group observed the same first grade lesson on picture graphs but crafted differently by each teacher. The students in both classes were having fun and were excited to participate, and we could feel the joy!

In one of the first grade classrooms, the lesson began with the class singing a song about fruit salad. The students then were asked to vote on their favorite fruit. This voting turned into a well designed lesson on graphing. As students voted for a given fruit (one of 4 options—durian, mangosteen, apple or orange), they were handed a small printed replica of that fruit. After all votes were in, the teacher asked students how the fruits might be organized in a graph. Students offered their input, and a beautiful graph was constructed, labeled and organized on the board (see Figure 1). Students placed their fruits on the graph in the corresponding areas. Next, students were asked to make statements about the graph such as, “the most students like apple.”

In preparation for small group work, the teacher discussed explicitly with the students the norms of behavior. Students moved into groups of four, and a number assigned to each student (attached to the front of each shirt with a clothes pin) was correlated with a specific task. Reading the detailed instructions together, they constructed graphs similar to the “class graph." The materials provided by the teacher were very supportive of the students understanding of the idea of graphing. Most impressive were the generalized statements that these young students were able to make about their graphs. The students’ spelling was quite good (those who couldn’t write well were helped by their partners).

In the adjacent first grade classroom the lesson began after the children bowed and recited “welcome sirs and madams." The teacher asked the students to consider the recycled items she had collected in a basket to answer the question “Which items that I collected are more?” The class decided that they should sort the items into groups in order to compare them. The children took turns selecting the items of each type (paper, glass, metal plastic) and the teacher placed them in separate sections within a paper organizer that she had attached to the board using magnetic strips. The teacher mentioned the need to label the graph and asked the students “what does it mean to label?” She then labeled the graph as the children spelled the words and wrote “This graph shows the recyclable things in Mdm Low’s basket” above the graph. Once the graph was done, the teacher asked the students to compare the number of items in the different categories by asking different questions, including “How many more metal things do we have than plastic?” The children clapped with excitement as they discovered the different relationships quickly by just looking at the chart.

For a second example, as with the other first grade class, the children were asked to select their favorite fruit from the available choices (bananas, watermelon, pineapple and durian). The teacher moved the paper organizer from the recyclables graph to the right side of the board (recycled) and used pictures of the fruits as labels for each section. Then the children took turns placing a blue sticky note in the section corresponding to their favorite fruit. After the teacher organized the sticky notes within each section, the class counted together and determined “which fruit is the most popular?” “Least popular?” “How many more like watermelon than bananas?”

Then it was time for work in pairs. The children went to their tables, where a box of materials awaited. Before the children were allowed to touch the materials, the teacher used a document camera to discuss what they would be doing (see Fig. 6). Each box of materials included a container with objects of different shapes (circles, triangles, stars and squares) and colors (yellow, pink, blue and green), a felt piece to keep the plastic objects together on the table, a felt tip pen, an organizing chart with space for labels, and a worksheet. The children discussed the different ways that the objects could be sorted (by color or by shape) and proceeded to organize them by shape (the teacher said that next time she would let them do the graph on colors). Then they labeled their graphs, answered the comparison questions on the worksheet, and discussed them as a class using a worksheet on the overhead.

Observing another lesson
At 11:00 am, we were invited to observe another regular class of either a P3 or P4 lesson. The P3 lesson was on mass and estimation. The P4 lesson was on adding arithmetic sequences of numbers using Gauss’s method, which the teacher referred to as an extension activity.

Senior teacher Mrs. Ai-Choo Han taught the P4 lesson. There were several older students who had recently arrived from China in this class. The lesson began with the task of finding 1+2+3+…+98+99+100. The teacher said that she was “not interested in the answer,” but in “how you do it.” Many of the children began to work right away, while others wrote nothing or simply made a guess (one child wrote 500 down as the answer). Some of the children immediately paired numbers (100+1=101, 99+2=101), leading them to 101 times 50 = 5050 (some said that they had been taught the strategy at home or in extra tutoring classes). Other strategies included:
1) writing down and adding all the numbers, ten at a time, and then adding the sum from each group.
1+2+3+4+5+6+7+8+9+10=55
11+12+13+14+15+16+17+18+19+20=155
21+22+23+24+25+26+27+28+29+30=255
31+32+33+34+35+36+37+38+39+40=355
41+42+43+44+45+46+47+48+49+50=455
51+52+53+54+55+56+57+58+59+60=555
61+62+63+64+65+66+67+68+69+70=655
71+72+73+74+75+76+77+78+79+80=755
81+82+83+84+85+86+87+88+89+90=855
91+92+93+94+95+96+97+98+99+100=955

55+155+255+355+455+555+655+755+855+955=5050

2) adding the numbers (using the usual U.S. addition algorithm format) one at a time:
100+99=199
199+1=200
200+98=298
298+2=300
300+97=397
397+3=400
Etc.

The teacher made quick comments about some of the student solutions and taught the students “the pairing-off strategy” using a powerpoint presentation, “with the help of two friends from overseas—Mulan and General Shang.” Then the students worked on problems presented on the powerpoint slides, interspersed with similar problems on the worksheet. For example, the class worked together on the powerpoint examples 1+2+…+19+20 and 1+2+…+49+50, then students worked individually and then together on 1+2+3+…+38+39+40. The next powerpoint example and worksheet problem were to add consecutive numbers that did not begin with 1; then a set of even numbers beginning with 2; then a set of even numbers not beginning with 2; and finally, an arithmetic series of numbers with a common difference of 3, beginning with an integer larger than 3.

After the children had an opportunity to complete the explanation section of the worksheet, the teacher went back to 1+2+3+…+98+99+100 and introduced the German mathematician Gauss. She explained that this problem was given to Gauss by his elementary school teacher in hopes that it would keep the class occupied for a while; Gauss solved the problem in minutes by adding each of the terms in 1+2+3+…+98+99+100 to their corresponding term in 100+99+98+…+3+2+1 resulting in 101+101+101+…+101+101+101, which led him easily to the answer (one version of the story). Mdm. Tan then showed the first four triangular numbers and asked the students to predict the number of dots in the 10th term. (One of the girls in the back proceeded to add 1+2+3+4+5+6+7+8+9+10=55.)

The lesson concluded with a general lecture on patterns and their importance, with examples provided by different sequences on the powerpoint (1,4,16,25,36,49,64,…; 2,4,8,16,32,64,128,256,…; 0,1,1,2,3,5,8,13,…;1,8,27,64,125,216,343,512,729,…) The teacher said that “mathematicians are like detectives” and look for clues to make meaning of patterns and challenged the children to “open your eyes and look for patterns.”

The P3 lesson was an experiment conducted by the students. Direct instruction was limited to a brief discussion of the concept of mass. The students performed various experiments in measuring common items, using the scales provided by the teacher. They recorded the measurements, working in groups of ?. Books, pens, and other classroom items all came into the scale, as they tried to estimate weights (no distinction was made between mass and weight). They were all actively engaged in the lesson and very enthusiastic about their task.

The students were greatly surprised at how far off some of their estimates were. Some students came very close in their estimates and were also surprised. Probably, the most interesting thing about this lesson was that the children were very willing to take chances and to learn from them. Some students began experimenting with multiple items on the weight scales. It was fun for them and a great learning experience. So the long term goal of making the study of Mathematics enjoyable was certainly supported in this lesson.

Following the class visitations, we informally discussed what we had observed in the five classrooms with others in our group, and with many of the teachers, staff and student interns. Then we all had a lunch of Chinese food and continued our conversations.

Research Lesson Discussion
After lunch we began the research lesson discussion with the three teachers who created the lesson and the observers. For some of us, this was a first hands-on experience with lesson study. We received guidelines for discussing the lesson, including making clear the distinction between observation of teachers by administrators versus lesson study observation by peers. In traditional observations by supervisors, the focus is on the teacher, who is evaluated during the observation. In lesson study, the purpose of observing lessons is to focus on students and not the teacher.

Mr. Leung discussed the goals of the lesson, the parts of the goals he felt he did not reach and things he would have done differently. Then we each mentioned one area of the lesson we would like to discuss, and each observer gave his/her observations of individual students and pairs. It was noted, for example, that the work was equally shared in some pairs, but dominated by one student in other pairs. The teacher was very aware of the dynamics in each pair. Participants were advised to share their observations, the data they collected, and not their opinions. Patsy shared a quote, “Without data, chatta don’t matta.” Patsy Wang-Iverson was the “final” commentator. She began by quoting a Japanese master teacher, who recommended that “one should praise nine and critique one.” Such an approach helps focus the teacher’s attention on one improvement to make. Each participant was asked to offer thoughts for improvement/strengthening of the lesson to increase student learning (although not part of the traditional protocol).

At the parting ceremony, the principal presented each of us with the powerpoint CD and the picture of our group and their staff in front of the school. So awesome!

We dropped by the Y for a quick change of shoes and to pick up spouses, and were whisked off to Little India for some shopping and dinner. Our guide took us to Sri Veeramakaliamman Temple, Little India's busiest and oldest temple, dating back to 1881 (the present structure was completed in 1986 and is managed by an American). Our last stop was at Muthu’s Curry, one of the best restaurants in town. Nothing was too spicy; the curry was smooth and delicious. Best of all, Tricia took one for the team and ate the fish eyeball. What a gal! They are high in protein. As Valpreet (Berinder’s 18 yr old daughter) said, “Yum, Aqueous humor”.

Post created with input from Tricia, Cassie, Joi, Tobe and Patsy.

some thoughts

2007-08-10T09:57:00.000-04:00

A month has passed since our visit to Singapore, so there has been time to reflect and see what stayed with me. This has been made easier, as people ask me what I learned and what Singapore was like. I would appreciate any contributions you might make, reflecting on my ideas.

So here are a few thoughts:

The Mathematics education plan from the Ministry of Education is quite comprehensive and certainly has been communicated down to the school level. The question of implementation in classrooms varies from class to class and school to school. So what is different?

There is the clear framework of three modes of learning: Concrete, Pictorial, Abstract. This is enunciated and embraced from the top down to the classroom. Teachers understand it and use it in classroom practice.

Textbooks are used to assign work, after direct instruction and small group work have taken place. The teacher is not supposed to teach from the textbook. Lessons can extend over several days, building from day to day. Nightly homework is not a hard and fast rule across all classrooms. It is given thoughtfully, to supplement material already learned. The curriculum spirals from year to year. Inspection of the list of topics in the MOE curriculum list yields a short list of topics to be covered in a year. Problem solving is the stated overarching long term goal, developing creative, thinking and thoughtful students.

There is tracking. This occurs as a result of teacher evaluation and testing. Tracking goes in two directions. First, in order to complete the topics in the syllabus, within a given length of time, some topics are simply eliminated. Second, children advance from year to year and are given support outside the regular classroom, in order to stay on track - in whatever track they are placed. Students can jump to a more advanced track, if they show they can handle the work, so the tracking is not set in stone.

Special Needs children are placed in special school settings. The gifted children (top 1% identified by testing at the end of P3) are invited to receive a 'gifted' education in one of nine special schools. The children deemed needing Special Services are placed in settings where these can be provided. (There are many ADD and ADHD students in mainstream classes throughout the school system.)

Class size is very large, averaging around 40 students, compared to classes in the public schools in New York and perhaps in other states as well. There is a stated obligation on the part of the teachers to know each of their students and to maximize their learning, no matter what it takes, including extended day sessions.

The synthesis of cultures occurs in the educational setting, where English is the official language of the education system. Mathematics education is treated equally with English Language Arts. The third part of the required education, and also of equal importance, is the student's mother tongue. If a student's mother tongue is not Mandarin, Malay, or Tamil, s/he may select to take one of the three or may be exempt.

Prior to entering a teacher education program, the potential teacher spends six months in a school, with some limited remuneration, to make sure the teacher is really exposed to the job and therefore able to make an appropriate career decision, prior to having Singapore invest in the teacher's training.

The teacher education system requires teachers to participate in Professional Development experiences throughout their careers, to the tune of 100 hours per year. This is part of the contract and the culture of being a teacher in Singapore. The teacher is a career-long learner.

Tuesday, July 10, 2007: Solving (?) math problems with Celine Koh

2007-07-31T11:57:00.000-04:00

After Berinderjeet Kaur's and Marshall Cavendish's presentations, the day still was not over. Most of us remained for a problem-solving session (lasting until 18:30) with Celine Koh, organizer of this week-long program and a former teacher of gifted students and current mathematics tutor. It was this session that introduced concretely to those previously not familiar with Primary Mathematics, what 'Singapore Math' is all about.

Following are the problems with which we, with our fried brains, struggled, and then Celine made it seem all so simple. Answers are not provided for the benefit of those who relish a challenge.

Grade 3

A mechanic fixed 12 wheels to 5 bicycles and tricycles. How many of them were bicycles and how many were tricycles?

Adapted from school exam paper

Grade 3/4

Mrs Tan is 31 years old and her daughter is 13 years old. How many years ago was Mrs Tan 3 times as old as her daughter?

Source: Challenge Problems Set B Question no. B52

Grade 4/5

Jane used 880g of a packet of sugar to bake a cake and 1/10 of the remaining sugar to make jelly.
She then had 3/7 of the packet of sugar left.
How much sugar was in the packet at first?

Adapted from school exam paper

Grade 5/6

Tim and Sally each have some money.
If Tim spends $80 per day and Sally spends $40 per day, Tim will have $500 left when Sally has spent all her money. If Tim spends $40 per day and Sally spends $80 per day, Tim will have $1100 left when Sally has spent all her money. Find the amount of money Sally has.

Adapted from school exam paper

Grade 5/6

Four toy cars cost as much as 3 dolls. Five toy cars cost $3.50 more than 2 dolls. Clare spent $14 on equal number of toy cars and dolls. How many toy cars did she buy?

Adapted from A* Maths Problems Whole Numbers and Decimals page 11 Question 5

Grade 5/6

John has a tank of fish.
The number of guppies is 25% of the total number of fish in the tank.
He buys as many guppies as he had.
Find the percentage of the angel fish now in the tank.

Adapted from PSLE Past year Questions

Grade 5/6

A jar contained some chocolates and sweets.
At first, the number of chocolates was 60% of the sweets.
After adding in another 10 chocolates and 10 sweets, the number of chocolates becomes 80% of the number of sweets.
How many sweets were there at first?

Adapted from school exam paper

Grade 5/6

Class A and Class B have the same number of pupils. The ratio of the number of boys in Class A to the number of boys in Class B is 3:2. The ratio of the number of girls in Class A to the number of girls in Class B is 3:5. Find the ratio of the number of boys to the number of girls in Class A.

Source: Challenge Problems Set C Question no. C66

Grade 6

The number of 20-cent coins to the number of 50-cent coins in a box was 3:2 Lyn took out four 50-cent coins and replaced them with 20-cent coins of the same value. After that the ratio of the number of 20-cent coins to the number of 50-cent coins became 7:2. How much money was there in the box?

Adapted from Challenge Problems Set C Question no. C67

Grade 6

There are two bags of stones labelled A and B. In Bag A, there are 350 black stones and 500 white stones. In Bag B, there are 400 black stones and 100 white stones. How many black and how many white stones should be transferred from Bag B to bag A so that 50% of the stones in Bag A and 75% of those in Bag B are black?

Source: A* Maths Problems Percentage page 9 Question 9

Many of the above resources can be purchased from a bookstore called Popular, introduced to us by our guide. Several members of the group, led by Cassy Turner and Tricia Salerno, made trips to Popular to buy more books, resulting in the need to purchase additional suitcases to transport them home.

Tuesday, July 10, 2007: Presentation by Marshall Cavendish

2007-07-31T11:21:00.000-04:00

Marshall Cavendish publishes not only Primary Mathematics, U.S. edition, used by homeschoolers and some schools across the United States, but also My Pals are Here (MPH) and Shaping Maths, the two most commonly used primary textbook series in Singapore.

Thanks in large part to Madge Goldman, president of the Gabriella and Paul Rosenbaum Foundation, who has had a long-standing relationship with Marshall Cavendish, the publisher Duriya Aziz and Daio' Ng Jui Sia, CEO of the parent company TImes Publishing Limited, hosted a marvelous dinner on Saturday, July 9, at Straits Kitchen, which provided an experience that will be long remembered.

In the afternoon on Tuesday, July 10, 2007, eight Marshall Cavendish staff members, led by Khairi Abdullah (khairi@sg.marshallcavendish.com), introduced us to MPH and Shaping Maths and their pre-kindergarten and kindergarten series. They also showed us the Primary Mathematics, standards edition, a series developed for submission for California adoption. Their PowerPoint presentation is available here.

Twenty to 22 countries currently are using the Singapore edition of either MPH or Shaping Maths.

The decision to move away from Primary Mathematics in Singapore was attributed to a desire to show students more than one way (bar model in PM) to solve problems. MPH takes the middle ability approach, while Shaping Maths targets the 20% high ability students and provides fewer and less explicit steps.

Supplementary resources published by Marshall Cavendish include Handbook for Teachers (one each for primary and secondary) and The Essential Parents’ Guide for Mathematics*.

*Another parent book that may be of interest to readers of this blog is Ron Aharoni's Arithmetic for Parents.

As part of the program registration fee, participants were invited to select books from two grades. Many of us, unable to resist the full display of books, purchased many additional books, requiring us to buy a suitcase to carry all our acquisitions.

Tuesday, July 10, 2007: Teaching of Mathematics in Singapore Schools by Berinderjeet Kaur

2007-07-31T09:03:00.000-04:00

Berinderjeet Kaur, Associate Professor in the Mathematics and Mathematics Education Group at NIE, spent the better part of a day with us to answer our many questions about mathematics education in Singapore. In lieu of a scripted presentation, she invited us to pose questions, which she then proceeded to answer in a relaxed and open manner.

In addition to our conversation, answers to our many questions can be found in the papers and brochures she distributed:

1. Kaur, B., Lee, K.P., Fwe, Y.S. Initial mathematics teacher training in Singapore.

2. Kaur, B. What is the method of models?

3. Mathematics Framework & Pedagogy [Source: A Guide to Teaching & Learning of 'O' Level Mathematics 2007 (Curriculum Planning and Development Division, Ministry of Education (Singapore), 2006]

4. Kaur, B. Teaching of mathematics in Singapore schools [paper presented at ICME-10, Copenhagen, Denmark].

5. Professional development for primary mathematics teachers (brochure)

6. Professional development for secondary and JC mathematics teachers (brochure)

Berinder also showed us The Teacher's Pledge, which was presented in a calendar format.

Monday, July 9th, 2007: Visit to Cedar Primary School

2007-07-24T14:36:00.000-04:00

On Monday we met at 8:30 am to go to a school called Cedar Primary School, a progressive school with 920 students. Cedar is involved in an education partnership with a school in China, and is in the beginning stages of using Lesson Study as part of the professional development of teachers. The principal (since 2004),Shirley Ho-Woo. is a former superintendent who is a strong believer in "human dynamics.” She will be retiring at the end of the school year (November, 2007) and would like to leave a permanent legacy.

When we got off the bus we were greeted by Shirley and Elaine, the P5 HA (Primary 5 high ability track math teacher) and mathematics teacher expert in the school. We met in a well air-conditioned conference room, decorated with student artwork, and covered with awards. The Singapore Ministry of Education (MOE) gives yearly awards to schools to recognize academic excellence. The school also has its own internal awards.

The school has been in existence since 1957, but 2 years ago it moved into a brand new building which has the best IT program in the country. In fact, next year they will be opening 6 distance-learning classrooms and will be implementing several types of techno-education. Cedar is also the leading school in Singapore for Lego and Robotics and is implementing a “school within a school” idea where different-aged children come together to learn. We also learned about the education system in Singapore: one of the new initiatives of the MOE is to “Teach Less, Learn More,” to lead children to discover the objectives of a lesson through exploration and investigation. The Mathematics Curriculum Framework consists of several categories with problem solving at the center. The principal also shared the importance of mentoring teachers. New teachers have 20% of their time free to use to learn and improve their teaching—they design their own learning. All the teachers also have time each week to get together with teachers at the same grade level. Teachers attend local and overseas conferences supported by the MOE.

The school year starts in January. There are four 10-week quarters, and they get the months of December and June off. The children (starting in 3rd grade) are separated into different tracks—what they call HAMALA—higher ability (42 children per class), middle ability (40 children per class), and lower ability (25 children per class). The children in lower ability classrooms are supposed to have the most capable teachers and more instruction time. Mathematics class-time per week consists of 12 periods, each lasting half hour. The tracks for primary school are determined by student achievement on exams and teacher observation, and they currently keep children of each track together for all their classes, except for “mother-tongue.” Each child goes to language instruction according to his/her home-spoken language (Chinese, Tamil or Malay). Children with a mother-tongue that is not one of the languages taught at school are allowed to select the language of their choice. If, after a good try, they are not able to keep up with the language, they are exempted from the “mother-tongue” requirement. Next year they will establish a more elaborate “banding” scheme—children will be tracked for each subject according to their achievement in that subject (subject-banding).

The first class we observed was a high ability (achievement seems a better word) Primary 4 class doing a lesson on adding and subtracting decimals taught by Vivian Chiau. We received a lesson plan and sat in the back of the classroom. The teacher first used an interactive powerpoint presentation and projector to review addition and subtraction of decimals, lining up the decimal points, and adding and subtracting normally. She was very precise in the language she used. For example, she talked about subtracting hundredths from hundredths, tenths from tenths, etc. She reminded the children that only decimals of the same value can be added, and that, while using the addition/subtraction algorithms, we add/subtract from right to left. Many of the children were quick at the computations; there were a couple of children who were doing the computations mentally. Many hands went up within seconds after the teacher projected the numbers to be added or subtracted (they were asked to raise their hands when done). There was some use of the words “minusing” and “plusing” as verbs during the lesson.

The teacher asked questions throughout the review. For example, for the exercise “3.8 minus 1.26,” she asked “Why do we need a zero after 8?” Other questions included “Is one hundredth enough to subtract 3 hundredths (then she said that “we need to borrow from the 9”)?” and “Do you understand?” When a student gave the wrong answer for the exercise “subtract 2.03 from 10.91” (8.87) the teacher asked the class “Did you get 8.87?” and many of them responded “no” and gave the correct answer.

After a few addition and subtraction exercises (in order of increasing sophistication) the teacher gave the students very specific instructions for an application activity. Each small group was tasked with determining which of two grocery stores would give the “best deal” for a list of items needed for a party. The maximum budget was $200, but they were asked to find the best deal they could. One representative from each group very quickly and efficiently went up to receive the materials. The students were enthralled with the activity and started finding prices on the two store circulars right away. They distributed the work among the people in the groups and collaborated well. They spent approximately 40 minutes working on this task.

There were no calculators. The students made long lists of prices and added them using the (common U.S.) addition algorithm. More than one child in some groups performed the addition in order to compare with the others in the group.

Many of the students were multiplying decimals, although the lesson was supposed to be reviewing addition and subtraction. When asked how they knew how to multiply decimals, they answered that they had learned at home. Students in Singapore may receive additional lessons after school (tuition)—either through paid tutors or parental help. Practice math (and other subjects) books are readily available for parents to purchase. One of the stores that sell these materials is called “Popular” (http://www.popular.com.sg/jsp/index.jsp).

Time ran out so we did not have the opportunity to see much of the discussion of different solutions or conclusions/wrap-up (the lesson plan only said “Students present their proposal to the class”). Before the lesson ended, some of the children were beginning to compare the lists of items and prices from the different stores. One of the children shared his group’s solution—the list of items would cost in the low 100’s for each of the stores. The teacher asked “why didn’t you make use of the $200?” The child answered that $200 is too much money to spend on a party. One of the adults in the classroom (head of technology) pointed out that a package of rice contained 5kg in one of the stores and 10kg in the other. He asked the students if they had considered the different amounts when making their list.

When we left the classroom, the students stood and bowed to us and said “Thank you, sirs and madams.” We then went to the conference room to discuss the class (over cake and tea) and the teacher and principal joined us.

After the meeting we went to watch Elaine, a teacher with 11 years experience, teach a volume lesson to her 5th grade high ability class. She gave a very quick introduction using a powerpoint presentation and then gave the students connecting plastic blocks and three worksheets, one at a time. The students constructed cuboids shown on the sheet and answered questions about their structures’ volume (in terms of cubes) while the teacher walked around and helped individual students. Most of the students worked individually. Many of them know the formula for the volume of a cuboid: length times breath times height, so they used the formula to find the volume and then filled in the dimensions. A child in the back row filled in all the answers within a couple of minutes and then started building the structures.

There were many other student strategies. For example, one student counted the number of cubes on one face of the cuboid, one at a time, and then multiplied that number by the number of layers. Another student drew a rectangular array representing the cubes on all the layers, placed side-by-side on one level and found the number of squares in the grid by multiplying the number of rows by the number of columns.

The main goal of the lesson was for the children to see the layers of cubes creating the cuboid and how that leads to the volume of a cuboid. The third worksheet asked the children to separate the cuboid into certain numbers of groups that corresponded with the dimensions of the cuboid. Elaine used this worksheet to assess whether the children saw the layers that make up the cuboid. Their level of understanding might have an impact on their choice of cubes. Many children did not use the layers when drawing.

The class came together at the end of the lesson to talk about the answers. The teacher asked various questions: “Did everyone see that there are layers or do you just see a bunch of cubes?” “When you see a cuboid, what do you see?” “What’s the arrangement?” “How many layers can you see?” “Can you decompose?”

We had a catered Chinese lunch with many of the teachers in the school, and we found out much from them. For example, we found out that teachers in Singapore make about $24,000/year (take-home, after mandatory savings). In some schools, teachers get home four days a week by 2:00 as their school does not require teachers to be involved in so many initiatives. At Cedar, however, the teachers have a lot of additional responsibilities, and often don't leave until after 5:00. As part of their responsibility, teachers tutor students twice a week after school.
We then met with the principal, Elaine, and two men from Cedar for over 2 hours to discuss math instruction. Elaine believes that one of the reasons students have performed so well on TIMSS is that they are used to high stakes testing. She shared the desire to encourage and teach creativity to the children and pointed out that there are no Nobel prize winners in Singapore. The principal said that she is very interested in creating a relationship with U.S. teachers.

Tea and snacks were provided during this period. One of the teachers said that Singaporeans like to feed their guests. After eating AGAIN, we took our leave. After a full day, we headed for a tour of Kampong Glam, the Malay area of town. We arrived at the Malay Heritage Center 15 minutes before closing. We persuaded the museum attendant to allow us to stay until 6:15 for a quick visit. The number one pastime in Singapore is shopping, which we immediately embraced as we walked through Kampong Glam, the Malay heritage area. We passed a hookah café and entered Restoran Tepak Sireh. Tepak Sireh is housed in the building that was once the home of the Bendahara (Prime Minister to the Malay ruler). Dinner was a Malay buffet, including laksa, mee rebus (Malay spaghetti), taubu, fermented bean curd. etc. After dinner we boarded the bus and headed back to the Y to rest and prepare for the next full day of activities.

This post created with input from Cassie, Tricia, Celine and Patsy.

Special Education Overview

2007-07-15T23:25:00.002-04:00

This chart comes from Singapore's Ministry of Education and describes the educational paths available to those with special needs in the public school system. (Click on the chart for a better view!)

For those still struggling with sorting out the tracking system and regular education paths in Singapore, you can find a concise chart also at the MOE site: Education Overview

A few photos from Singapore

2007-07-07T21:23:00.000-04:00

For those familiar with Primary Mathematics, Third Edition, below are photos of the Esplanade Theatres in Singapore, architectural representations of the durian fruit. The durian is the "King of Fruit" in southeast Asia and is particularly noted for its distinctive smell.

The Great Durian

Our hotel has rules against the eating of durian in the room!

Final program for July 7-14, 2007

2007-07-06T13:57:00.000-04:00

Participants for the July 7-14, 2007 mathematics program have been arriving since Wednesday, July 4.

Following is the final program, which now includes visits to four schools and observations of six classes: three regular elementary, two secondary classes and a grade 3 research lesson.

Singapore Mathematics Summer Program 2007
Organized by SMath Resources

July 7 (Saturday) to July 14 (Saturday), 2007
Schedule of the Singapore Summer Math Program

Day 1: July 7, Sat
Welcome to Singapore
18:00 Brief meeting regarding the Singapore program
18:45 Leave for Dinner
19:00 Dinner at Straits Kitchen with Marshall Cavendish Publishers
20:15 Free & Easy

Day 2: July 8, Sun
Free & Easy
Shoppers:
Orchard Road Malls
Raffles City & Marina Malls
Vivo City Mall

Nature-lovers:
Singapore Botanic Gardens
Singapore Zoo / Night Safari
Bukit Timah Nature Reserve
Pulau Ubin

Arts & Culture-lovers:
Singapore Art Museum
Asian Civilization Museum
National Museum of Singapore

Day 3: July 9, Mon
Visit to Cedar Primary School
08:30 Meet at the lobby to leave for Cedar Primary School
08:50 Welcome to Cedar
09:00 School Tour & Introduction
10:00 Observation of Math Lesson 1
11:00 Post-Lesson Discussion of Math Lesson 1
12:00 Observation of Math Lesson 2
13:00 Lunch
14:00 Math Lesson Discussion
16:00 Reflection & Reception
16:45 Leave Cedar for Kampong Glam
17:15 Tour of Malay Heritage Centre
18:00 Tour of Kampong Glam
18:30 Dinner at Tepak Sireh
20:00 Leave for Metropolitan Y

Day 4: July 10, Tue
Math Workshop
09:30 Preview of Day 5 Program
10:00 Session 1-What do we want to learn about Singapore Mathematics?
10:30 Session 2- Learning about Singapore Mathematics
12:00 Lunch in the Café
13:00 Session 3- Process & Practice of Mathematics in Singapore
14:30 Session 4- Meeting with Singapore Mathematics Publisher, Marshall Cavendish
15:30 Tea & Selection of resources
16:00 Session 5- Discussion & Reflection
16:30 Free & Easy

Day 5: July 11, Wed
Visit to Guangyang Primary School
06:55 Meet at the lobby
07:25 Flag Raising Ceremony
07:40 School Tour
08:00 Math Research Lesson
09:00 Breakfast & Buzz Time
09:15 Welcome Address, Presentation
09:35 Introduction to the Math Program in Guangyang
10:00 Math Lesson Observation 1
11:00 Math Lesson Observation 2
12:00 Buzz Time
12:30 Lunch & Post-Lesson Discussion
14:00 Post- Research Lesson Discussion
15:30 Tea
16:00 Reflection
16:45 Souvenir Presentation & Farewell
17:00 Leave for Little India
17:30 Tour of Little India
18:30 Dinner at Muthu’s Curry
20:00 Leave for Metropolitan Y

Day 6: July 12, Thurs
Visit to Geylang Methodist Secondary School & NIE
08:30 Meet at the lobby
09:00 Geylang Methodist Secondary School Visit
& Talk with Senior Math Educator
13:00 Lunch
13:45 Leave for the National Institute of Education
14:30 Meeting with the Head of Math & Math Education Academic Group
& Dean (Academic)
15:15 Tour of MME Facilities
15:30 Tea
16:00 Talk on Teaching Math in Singapore
17:00 Leave NIE
18:00 Suntec City & Duck Tour cum Cruise
19:00 Dinner
21:00 Leave for Metropolitan Y

Day 7: July 13, Fri
Visit to Kuo Chuan Presbyterian Secondary School
08:30 Meet at the lobby
08:50 School Tour
09:00 Introduction
09:45 Math Lesson Observation
10:30 Professional Exchange & Sharing
12:00 Lunch
12:45 Souvenir Presentation & Farewell
13:00 Leave for Chinatown
13:45 Tour of Chinatown Heritage Centre
14:30 Tour of Chinatown
15:00 Tea
15:30 Leave for Vivio City / Sentosa / Hotel

Day 8: July 14, Sat
08:30 Leave for Bird Park / Singapore Discovery Centre (wet weather)
11:15 Leave for Metropolitan Y
12:00 Thank you & Farewell

Differences between U.S. state mathematics standards and the Singapore curriculum

2007-07-03T11:25:00.000-04:00

At the First International Conference* on Mathematics Curriculum (November 11-13, 2005) sponsored by the Center for the Study of Mathematics Curriculum (CSMC), Alan Ginsburg (USED) presented a paper entitled Singapore Math: Can it help close the U.S. mathematics learning gap?

"This paper compares the Singapore and U.S. primary mathematics systems to explore what the United States can learn from the Singapore system of elementary mathematics education that may help improve the mathematics performance of U.S. students. It also identifies a few areas where the U.S. mathematics system may be preferred to Singapore’s system." Comparisons are made in the areas of frameworks, textbooks, assessments, and teachers.

Proceedings from the 2005 conference will be available for sale in the near future from Information Age Publishing.

*A Second International Conference on Mathematics Curriculum: Future Curriculum Trends in Algebra and Geometry is scheduled for May 4-8, 2008 at the University of Chicago. Check the CSMC website for details that are forthcoming.

Lesson Observation Culture and Norms

2007-07-02T20:39:00.000-04:00

Singapore teachers are generally not quite used to having many visitors in the classrooms, unlike in Japan. However, it is slowly changing in certain classrooms where the teachers are involved in lesson study. In some schools, the culture of peer lesson observation has started.

Observations in Singapore classrooms are normally conducted by the reporting officers (RO) such as a department head or principal, who will do the annual or semesterly observation of a classroom teacher teaching. It has been rather evaluative in nature, though sometimes, it is formative. The lessons of trainee or beginning teachers are also observed by their mentors and supervisors. There is some anxiety associated with lesson observations.

It may be a little unsettling for some teachers and pupils in Singapore if there is a considerable amount of movement by visitors in the class. The observers in the classroom tend to sit quietly to observe.

The schools in Singapore are excited about your visit. For most of the students in Singapore, it may be the first time they had so many overseas visitors in a classroom. Having Caucasian visitors may intrigue the younger students even more.

For some of the lesson observations, the observers will be divided into 2 groups so that it will be less cramped in the classroom. Each participant will observe about 2 lessons per school.

Regarding the use of digital camera, we can ask for permission to take photos of the lesson. Please avoid using flash when taking photos during the lesson time. Please avoid taking close-up photos of specific students, especially if the photos will be viewed by others outside the progam or the general public. Thank you for your understanding.

The teachers are looking forward to the lesson observations and discussions too.

Singapore Mathematics: From the past to the future

2007-06-30T21:24:00.000-04:00

In three publications, Lee Peng Yee, Associate Professor in the Mathematics and Mathematics Education (MME) Academic Group of the National Institute of Education (sole teacher training institute in Singapore),

1. offers a 60-year (1945 - 2005) history of mathematics syllabi and textbooks in Singapore (Proceedings of The First International Mathematics Curriculum Conference, 11-13 November 2005, Chicago).

2. discusses mathematics for teaching vs. mathematics for teachers in an editorial published in The Mathematics Educator, vol. 16#2, Fall 2006.

3. explains what is new in the new O level mathematics syllabus.

Content vs. Culture

2007-06-23T15:39:00.000-04:00

Some individuals in the U.S. may be unwilling to consider the Singapore Primary Mathematics curriculum due to their perception of 'cultural' differences. However, the mathematics scope and sequence (what Scott Baldridge calls its storyline) has no cultural boundaries. This statement is supported by the common mathematics scope and sequence in grades 1-8 shared by Belgium, Hong Kong, Japan, Korea, Netherlands, and Singapore, which is summarized in a table originally developed by Schmidt et al. This particular table includes the scope and sequence of NCTM's Focal Points (FP), released in September, 2006 [Table key: blue = topics shared by A+ countries and FP; magenta = A+ countries' topics; open ovals = FP topics.].

Please note in the table at what grade topics are introduced and their duration in the curriculum. How does your curriculum compare? How familiar are teachers with the district's mathematics scope and sequence? Do they know what (and how) students are taught in earlier grades? Do they know for what future mathematics learning they are preparing the students in their class(es)?

Is there vertical articulation (communication among teachers across grades, and do teachers share a common mathematical language as well as expectations for student work?

Do answers to the above questions reflect differences in culture, or should they embody effective mathematics knowledge and instruction?

Cultural differences, however, can be found in:

1. pedagogy
2. teachers' mathematics background and knowledge
3. professional development support for teachers (its nature and duration)
4. opportunities for teachers to collaborate and observe each others' classes
5. beliefs about students' ability to learn mathematics
6. access to quality learning environments
7. parental roles in children's education

Others can add to the list by responding to this post.

This one week summer mathematics program was stimulated in part by our desire to see for ourselves what actually occurs in Singapore mathematics classrooms.

Readers of this blog are invited to submit questions you would like answered during our week in Singapore.

Glimpses into mathematics classrooms in other countries

The 28 public release lessons from the TIMSS 1999 Mathematics Video Study allow a glimpse into eighth grade mathematics classrooms representative of each country (Australia, Czech Republic, Hong Kong, Japan, Netherlands, Switzerland, U.S.). These lessons provide insight into the level of mathematics eighth graders in these countries are expected to learn as well as cultural differences, which hopefully can help us in the U.S. re-examine our own practices through an international lens.

The TIMSS four-CD set for mathematics (and five-CD set for science) are available from LessonLab. A TIMSS Resource Guide for the public release mathematics lessons is available online.

Why examine mathematics in Singapore?

2007-06-19T16:42:00.000-04:00

For some of us, interest in Singapore's mathematics curriculum can be traced to its students' consistently high performance in the Trends in International Mathematics and Science Studies (TIMSS: 1995, 1999, 2003: www.timss.org). Such performance is all the more remarkable given its status as a young nation which became independent on 9 August 1965 (http://www.sg/explore/history_independence.htm) and its past history (or lack thereof) in international assessments.

1995 was the first year Singapore participated in an international mathematics assessment. It did not participate in either the First or the Second International Mathematics Study (nces.ed.gov/pubsearch/pubsinfo.asp?pubid=92011). It did participate in the Second International Science Study (1983-86), in which its 10-year olds tied for last place with Poland, England, Hong Kong, and the Philippines (p. 95). Its 14-year olds fared a little better, managing to outperform the Philippines (p. 96). However, as demonstration of what was to come, its students in their last year of secondary school scored at the top in science (p. 97).

As part of the TIMSS 1995 study, an in-depth curriculum analysis was funded by NSF and conducted by Bill Schmidt et al. at MSU (click on overheads and datafiles; miscellaneous overheads). The scope and sequence of the gr. 1-8 mathematics curricula of the six top-performing countries were summarized in one table, revealing a common, coherent curriculum sequence.

Of these six countries (Belgium, Hong Kong, Japan, Korea, Netherlands, Singapore), only one curriculum, that from Singapore, was available in English for grades 1-8. Although 77% of Singapore students do not speak English at home (http://www.singstat.gov.sg /papers/c2000/adr-progress.pdf), classroom instruction is conducted in English. The Primary Mathematics textbooks are written in simple English, thus making them well suited also for English-language learners.

The Primary Mathematics (PM) textbooks are integrated into a textbook for preservice teachers, Elementary Mathematics for Teachers, co-authored by two mathematicians, Tom Parker and Scott Baldridge, who use the book and PM texts in their college courses at Michigan State University and Louisiana State University, respectively

Note:

1. In Singapore, 'primary mathematics' includes grades 1-6.

2. Although Primary Mathematics (U.S. edition) is the series most frequenty used in the United States, elementary schools in Singapore currently are using either "My Pals Are Here" http://www.marshallcavendish.com/marshallcavendish/education/international/catalogue/primary/maths/my_pals_are_here/index.xml or "Shaping Math"

http://www.marshallcavendish.com/marshallcavendish/education/international/catalogue/primary/maths/shaping_maths/index.xml.