## Saturday, June 28, 2008

### A Sample of Summer Math Problems in China for Incoming 6th Graders

At the close of my previous posting about the embarrassingly low cutoff score (30 out of 87, or 34.5% was scaled to be a 65) in the June Regents Integrated Algebra exam, I promised to post a small sample of math problems posed to graduated fifth graders in China. First, some brief background.

Primary schools in China run from grades 1-6, middle school 7-9, and high school 10-12. Each summer, elementary school students in Suzhou (a city of 1.5 million located about 50 miles west of Shanghai) receive a 60-75 page, multi-subject workbook they are expected to complete during their vacation break. The problems below are translated directly from the 2003 summer workbook for soon-to-be 6th graders. They are just a small sample from that year's assignment, but they are indicative of the level of mathematics expected from China's incoming sixth graders.

1. Two gear wheels interact with one another. One gear wheel has 72 teeth, and the other has 28 teeth. If both gears have marked starting points that are together when the wheels begin turning, how many times will each wheel rotate before the two marks meet again?

2. Three identical cubes are placed side by side to form a rectangular cube. The surface area of the rectangular cube is 400 square cm. less than the surface area of the three cubes standing separately. What is the length of one edge of the small cubes?

3. A rectangular cubic box is to be made of wood with measurements length 50 cm., width 40 cm., and height 18 cm. How many square meters of wood are needed to make the box?

4. A rectangular piece of cardboard has a length of 32 cm. Squares of 4 cm. on a side are cut out of each corner of the rectangle so that the remaining piece can be folded into an open-topped box. If the volume of the box is 768 cubic cm., what is the area of the original piece of cardboard?

5. Last year, the ages of both a mother and her son were prime numbers. This year, the product of their ages is 532. What are their ages this year?

6. Solve the following problems.

a. 6 – (9/20 + 0.15) = ________

b. 7/16 + 8.15 – 6.9 = ________

c. 7.5 + 3.7 – 3.2 – 5/6 = ________

d. 3/7 +2.16 + 4/7 + 7.84 = ________

e. 2 – (5/6 + 4/15) = ________

f. 1.956 + 4/25 – (1.176 + 3/4) = _______

7. Solve for x.

a. 1 – x = 1/4 + 2/5

b. x – 3.785 = 5 – 1/2

c. x + 8/9 = 11/12

d. 3x – 4.93 = 8.75

As a math major in college and former high school math teacher, I am particularly enamored of the mathematical aesthetics in the first question above. The rotating gear wheel question is a simple but highly elegant variation of a common problem in fraction arithmetic, as is the mother/son age problem in Question 5 (although not quite as elegant a problem formulation). A hint: unlike nearly all schools here in the U.S., Chinese primary schools emphasize the use of prime factorization for fraction arithmetic. Most primary school math teachers here would likely not even know what prime factorization is. Note also the shifting of units of measure in Question 3 and the constant mixing of fraction and decimal number representations in Questions 6 and 7b.

How might our 9th grade Integrated Algebra exam takers last week have fared with these 5th graders' questions?

Primary schools in China run from grades 1-6, middle school 7-9, and high school 10-12. Each summer, elementary school students in Suzhou (a city of 1.5 million located about 50 miles west of Shanghai) receive a 60-75 page, multi-subject workbook they are expected to complete during their vacation break. The problems below are translated directly from the 2003 summer workbook for soon-to-be 6th graders. They are just a small sample from that year's assignment, but they are indicative of the level of mathematics expected from China's incoming sixth graders.

1. Two gear wheels interact with one another. One gear wheel has 72 teeth, and the other has 28 teeth. If both gears have marked starting points that are together when the wheels begin turning, how many times will each wheel rotate before the two marks meet again?

2. Three identical cubes are placed side by side to form a rectangular cube. The surface area of the rectangular cube is 400 square cm. less than the surface area of the three cubes standing separately. What is the length of one edge of the small cubes?

3. A rectangular cubic box is to be made of wood with measurements length 50 cm., width 40 cm., and height 18 cm. How many square meters of wood are needed to make the box?

4. A rectangular piece of cardboard has a length of 32 cm. Squares of 4 cm. on a side are cut out of each corner of the rectangle so that the remaining piece can be folded into an open-topped box. If the volume of the box is 768 cubic cm., what is the area of the original piece of cardboard?

5. Last year, the ages of both a mother and her son were prime numbers. This year, the product of their ages is 532. What are their ages this year?

6. Solve the following problems.

a. 6 – (9/20 + 0.15) = ________

b. 7/16 + 8.15 – 6.9 = ________

c. 7.5 + 3.7 – 3.2 – 5/6 = ________

d. 3/7 +2.16 + 4/7 + 7.84 = ________

e. 2 – (5/6 + 4/15) = ________

f. 1.956 + 4/25 – (1.176 + 3/4) = _______

7. Solve for x.

a. 1 – x = 1/4 + 2/5

b. x – 3.785 = 5 – 1/2

c. x + 8/9 = 11/12

d. 3x – 4.93 = 8.75

As a math major in college and former high school math teacher, I am particularly enamored of the mathematical aesthetics in the first question above. The rotating gear wheel question is a simple but highly elegant variation of a common problem in fraction arithmetic, as is the mother/son age problem in Question 5 (although not quite as elegant a problem formulation). A hint: unlike nearly all schools here in the U.S., Chinese primary schools emphasize the use of prime factorization for fraction arithmetic. Most primary school math teachers here would likely not even know what prime factorization is. Note also the shifting of units of measure in Question 3 and the constant mixing of fraction and decimal number representations in Questions 6 and 7b.

How might our 9th grade Integrated Algebra exam takers last week have fared with these 5th graders' questions?

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## 12 comments:

They would not have been able to answer most of them.

We are not teaching mathematics anymore. Even the top kids are not learning what they should because the mathematics is watered down to reach everyone.

On the other hand, not everyone in China goes on past elementary school so more advanced topics can be taught earlier on. I'm willing to bet plenty in China cannot pass that test. My school has a bilingual Chinese class for integrated algebra full of kids who could not even score 20 points on that regents.

Actually, education through middle school (9th grade) is mandatory in China. The problems arise in the sizable disparity between educational opportunities in the large cities and more developed areas compared to the smaller villages and poorer provinces. This gap between rich and poor, urban and rural, is a widely acknowledged (if not always openly admitted) problem in China.

My guess would be that many of your students, as is frequently the case among Chinese immigrants coming into NYC, come from small towns and villages in Guangdong and Fujian Provinces where the educational opportunities are fewer. In Suzhou, as in Shanghai, the standards and expectations from students are very high and the facilities range from adequate to outstanding.

Don't get me wrong, most of our Asian students are very good. I just wanted to point out that Chinese education does not reach everyone. One of my students emigrated to the US because he could not get an education in China.

Actually many students are in that boat, and not just from China either. We still have the rep of being a land of opportunity, and with the plummeting dollar, there are plenty. Not as many for those of us who actually live here, though.

Hopefully the next prez will address that. Otherwise, I think cynical and superficial education "reforms" will be among the least of our problems.

I find these problems too difficult and complicated for 6th Graders. I'm not sure that I would manage to solve them although I'm good at math:)

If we trained and expected our students to answer these questions, many could. Everyone is not the same! Our brightest students are the ones being hurt by content and scoring designed to narrow gaps and prove everyone is the same. Many of our students can handle the questions presented if we trained them correctly. This would eventually lead to a stronger teaching staff. I am a high school math teacher and I can easily answer all the questions posted. However, my teacher friends who are originally from China tell me that after teaching in New York, we would all need to brush up for a high school teaching position in Taiwan.

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Not bragging, I would feel these question really easy for chinese 5th grader. I went to China's school for all my junior high and elementary and partially in senior high, and then I came to the US for my senior and then college study. I felt that the math in China goes way difficult than it should be, but in the state, it goes the other direction, too easy. Here, i'm talking about the comparison of difficulty of the problem in China and US, not the topics taught in these two countries. The topic are, for most part, similar, from basic arithmetics to calculus. Math education in China just goes in much deeper. And another factor is chinese student from 1th grade to 12th grader has to take math course every semester, it's required.

Looking at these questions as a junior high school student, I find that they seem more like brain teasers you would find in a puzzle book, the numbers seem a little awkward to work with even for me. However, I find that by looking at the questions a second time they require a certain amount of critical thinking and problem solving that is lacking in education today. I feel that is a reform that is much needed, if I was a sixth grader I would certainly struggle with these questions. But I don't think that we should be assuming that these children are not trying hard to solve these problems. I. Think that they must be pushing themselves in a very respectable manner and I think that students should be encouraged to learn math beyond the classroom. I love problem solving but math gets boring if it's comes in the same format. I like variety and critical thinking and that is what these questions seem to be pushing at. Math should not be taught just so children can tick a box and be done with a skill, but so they can learn to break down and analyze a problem with mathematical thinking.

And please pardon the grammatical errors I didn't prof read my comment

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