I was given an itinerary when I arrived in Beijing. For Friday morning, it said “Lessons on the same topic”. My assignment: “Mean value inequation”.

What those terse instructions meant was that I was going to teach a class to the RDFZ students and watch a Chinese teacher deliver the same lesson back-to-back. They handed me this book:

They told me to teach this section.

And that was it. The rest was on me. They did not tell me anything about the students’ background or any goals I was supposed to hit. They clearly wanted to see how I would approach the material with completely fresh eyes.

But this task went beyond just comparative lesson planning. I actually had to deliver the lesson, too, on a double-bill against a veteran with a considerable home court advantage. This was nothing short of an international teaching throwdown.

Challenge accepted.

**My Plan**

If you’ve read my earlier two posts on my RDFZ trip, you’ll know that I am less-than-fluent in Chinese. My first task was to figure out what the “Mean value inequation” was. I made enough of the text out to deduce that it was what we would call the AM-GM Inequality.

The AM-GM Inequality is not exactly standard high school fare in American curriculums. Competitors in math contests should know it, but the general population can experience successful secondary math educations without having heard of it. I had taught it at IMSA, but only as an extra topic and not as core content. I did not have a lesson’s worth of problems in my head to cobble into a lesson.

My only access to the internet was the desktop computer in my dorm room, which would have been considerably more useful if I could have figured out how to change the language setting from Chinese to English. Even then, I couldn’t figure out how to get it to stop auto-completing everything I typed into Chinese characters. My typical response in this situation is to Google for instructions, but what do you do if you can’t even type “google.com”?

With nowhere else to turn, I cracked the textbook. (How much does that sentence sound like a student?) I was actually delighted by two things:

- There were a lot of problems in there with fresh and interesting applications of the AM-GM Inequality.
- I was able to read them. (Or at least enough of them to get the gist.)

My lesson plan was:

- Introduce myself. Warn the class that I’d be teaching in English. Hope the students don’t freak out.
- Define arithmetic means and geometric means, compute them for several pairs of numbers. Try to infer a pattern.
- Prove the Inequality by translating into symbols as $\dfrac{a+b}{2}\geq\sqrt{ab}$ and showing it is equivalent to $(a-b)^2\geq 0$. (Don’t gloss over the fact that $a$ and $b$ are positive here!) Then, give a geometric interpretation of the inequality in which the two means represent the altitude and a median in a right triangle (with hypotenuse $a+b$).
- Question: What is the minimum sum of a positive number and its reciprocal?
- Question: What is the minimum perimeter of a rectangle with area $30$?
- Follow-up Question: What is the maximum such perimeter?

- Sketch the graph of $y=\dfrac{x(x-2)}{x+1}$. Along the way, use the AM-GM Inequality to find the local minimum.

**How It Went**

As if I weren’t nervous enough to teach in a foreign country through a language barrier, I had an unusually large audience.

At the front of the class, there is a half-foot-tall “stage” that runs the length of the chalkboard. I felt silly standing up there (I was not in need of an extra six inches of height) and was positive I was going to tumble off of it during some climactic build-up. While I was able to avoid that particular embarrassment, I did draw some chuckles by trying to clear the chalkboard with something that only looked like an eraser. Still, the students were a cooperative bunch and class was going well.

About two-thirds of the way through, I broke the fourth wall and circulated while the class worked on the perimeter problem. The aisles in Chinese classrooms are not designed for easy passage. I did manage to lumber through and even found a few students to help along the way, but I did not see any other teachers doing the same in any of my observations.

The most excitement from the students came when we were graphing $y=\dfrac{x(x-2)}{x+1}$. We first identified the easiest features of the graph, the $y$-intercept and the vertical asymptote. Then I suggested *long division* (the words brought out a chorus of buzzes) to identify the oblique asymptote and to set up the clever application of the AM-GM inequality that finds the local minimum (see the third panel below).

We finished just as the bell sounded. The students applauded (nothing special, just the typical Chinese student applause) and I breathed in relief. Nailed it!

My counterpart in this teach-off was Ms. Zhuang.

Her lesson ran essentially like this:

- She asked students to identify the local minimum of the graph of $y=x+\dfrac{1}{x}$. Then she had them guess how the pattern generalized to $y=x+\dfrac{k}{x}$.
- She tied the question into one about arithmetic and geometric means. She asked the students how $\dfrac{a+b}{2}$ and $\sqrt{ab}$ compare, being careful about the cases in which $a$ or $b$ is not positive.
- She had a student present an algebraic proof of the AM-GM Inequality, and then showed the same picture I did with right triangles to give the geometric viewpoint.
- She asked students to apply the inequality to other functions. For example, what is the minimum value of $\sin(x)+\dfrac{4}{\sin(x)}$ for $x\in (0,\pi)$? (This question is not as straightforward as it looks.)
- Finally, she passed out cards to the students with numbers on them like $a$, $b$, $\dfrac{1}{a}$, $\dfrac{1}{b}$, $\sqrt{a}$, $\sqrt{b}$, $a^2$, and $b^2$ and told the students to build and prove an inequality using those numbers, presumably by applying the AM-GM Inequality. The students then wrote their inequalities on the board. (This activity took up most of the second half of the session.)

I loved my lesson plan and was confident with my delivery, but I left Ms. Zhuang’s class feeling outdone. Her question about $\sin(x)+\dfrac{4}{\sin(x)}$ was beautiful. Why didn’t I have any beautiful questions? And how was it that the Chinese representative, whose education system I had been told unapologetically endorsed didacticism, spent half her class on exploration where I had spent none? Sorry, America, but I lost this round.

After the two classes, I met with RDFZ’s math department to debrief. I didn’t know how to take it when they told me that my “teaching style follows traditional Chinese pedagogy”, which they saw as a kind of transition from examples to generalizations to hypotheses to harder examples. Their description did not strike me as particularly Chinese, but something was almost certainly lost in translation.

I was awfully curious why my students had gotten excited at the mention of long division. As it turns out, long division is not part of the Chinese curriculum, so students will not see it until college (unless they have some kind of rogue foreign substitute teacher). The teachers turned around and asked me when we taught it at IMSA, and I had to respond that we didn’t – our students generally *came in* knowing it as sophomores, so we did little more than refresh their memories. In my week at RDFZ, I had seen students learning calculus and creatively applying fringe topics, so I could not fathom why something as simple and useful as polynomial long division was deemed too advanced. Ultimately, regardless of any arguments for its inclusion, long division was not part of the Chinese Compulsory Curriculum (yes, that is its actual name) for high school and so it was not going to be covered.

Simplistic comparisons of the American and Chinese education systems will try to ask whose students are more ready for the next level. Whose students learn more? Whose are ahead? What seems most likely to me is that the Chinese students would fail American exams for not knowing such foundational topics as long division and the American students would fail Chinese exams for their ineptitude with things like AM-GM. Questions about which is “better” involve inherent value statements, and the answers speak more about those values than educational quality. I think it is more fruitful to use the commonalities between the systems to help each side better achieve its own goals. My trip to RDFZ convinced me that there is a lot more room for that to happen.