In my last post, I started talking about my visit to RDFZ, China’s top high school. In this post, I’ll talk about some of the classroom observations I did while I was there.

My second morning at RDFZ, I was invited to observe Mr. Zhang’s Grade 11 class at 8am followed by Mr. Wu’s Grade 10 class at 8:50am. It appears that 40 minutes is the standard class length.

Mr. ZHANG (48 students)

I was not told what the material for this lesson was ahead of time. Actually, I may have been told, but in Chinese. Either way, I didn’t know what I was in for.

The lesson started rather slowly. (He was using the second chalkboard from the left in the photo above.) He started with a square, and then a rectangle, and then a parallelogram, and then a trapezoid. It hardly felt like 11th grade material. I figured out based on the dissections he was drawing that they were talking about area. He then drew a circle, sliced it up, and rearranged the slices into a pseudorectangle to derive the familiar $A=\pi r^2$ area formula. That is good mathematics, but I wasn’t excited yet. Until the next bit: he took that same sliced up circle, approximated each with an isosceles triangle, and found the approximation $$A\approx \dfrac{1}{2}\sin\left(\dfrac{2\pi}{n}\right)\cdot r^2\cdot n.$$ Noting that the triangles approximated the slices better as $n\to\infty$, he reasoned that $$A=\lim_{n\to\infty} \dfrac{1}{2}\sin\left(\dfrac{2\pi}{n}\right)\cdot r^2\cdot n.$$ He gave them some time to work out the limit on their own, and sure enough, they used the fact that $\lim_{x\to 0} \dfrac{\sin(x)}{x}=1$ to find that, once again, the area of a circle is $\pi r^2$.

That last bit impressed me for a number of reasons. First, I had never seen that derivation before. I’m sure that the reason is because by the time I learned how to compute the limit of $\sin(x)/x$, the area formula for a circle was child’s play. (Not that it wasn’t for the RDFZ students, though.) But it did give me a new idea how to approach that ad-hoc limit with my calculus students. Second, when he asked his students to work the formula out, they just…did it. There was no questioning or whining or even pausing. They just got right to work and did it. How did he manage such wizardry?

Mr. Zhang then proceeded to the main idea of the lesson, computing integrals as limits of Riemann sums. He worked an example of computing the area of the triangle with vertices $(0,0)$, $(1,0)$, and $(1,1)$ (with a right-hand equipartition, in calculus parlance), getting the expected value of $\dfrac{1}{2}$. After defining $\int_a^b f(x)\,dx$, he stated that this integral was a limit of Riemann sums, and then used that idea to compute $\int_0^1 x^2\,dx=\dfrac{1}{3}$. *His students applauded!* With the few minutes he had before class ended, he mentioned left-hand sums and how they could be used to “squeeze” out the value of the limit, and then said something quickly about trapezoidal sums before the bell rang.

Mr. WU (40 students)

I was shepherded into Mr. Wu’s classroom, arriving before the teacher himself. His students were washing the board for him. When class started, the students stood and bowed to the teacher.

I was lucky that Mr. Zhang’s class had used so many familiar pictures so that I could follow along. My luck ran out in Mr. Wu’s class. I had a hard time making out much of anything. It was clear that the subject was sequences and series. From what I could tell, he gave the students various explicit and recursive formulas and asked them qualitative questions about the sequences they described. There was very little calculation happening, and very few symbols for me to use for context. Mr. Wu later told me that his students were at the end of a unit and they were reviewing for an exam, which is the main reason for the heavy emphasis on concepts as opposed to computation.

I was proud of myself, though: I managed to figure out that “$n$ ja yi” meant “$n+1$”, which increased my Chinese vocabulary at the time by about $40\%$.

The language gap was too big a barrier for me to understand most of the pedagogical decisions, but I did find the questioning techniques intriguing. In China, being called on in class is a process. The question is posed, the student is selected (cold, no hands raised), and then he stands and answers without delay. Before the student can return to his seat, the teacher will ask him 2 or 3 more follow-up questions. During a 40 minute class, nearly 10 students were called into a mini-conference with the teacher.

In my classes, I make sure to call on every student every session, but it is rare for me to make a conversation out of it, especially in front of the rest of the class. The Chinese teachers sacrifice having every student getting involved each lesson in order to leave time for more meaningful discussions. I am told that a considerable amount of educational research in China goes into the use of these dialogues for feedback. Once again, I am left wondering whether this is another calculated tradeoff, like the class size vs. teacher time I mentioned in my last post, that is worth thinking more about.