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.
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.
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”.
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.
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