Book: The Teaching Gap

A friend and colleague lent me a book called The Teaching Gap. It looks at maths teaching in the US and compares it with maths teaching in Japan and Germany. The authors looked at heaps of video footage of maths lessons in all three countries to draw their conclusions.

It's very interesting to read. I'll just jot two points that have come up so far:

• The authors firmly believe that the difference between cultures is much greater than the difference within them. Of course, maths teachers within one country vary greatly in skill and style, but they all look fairly similar when compared with maths teachers from another country.
• An unnamed education expert, when asked to summarise the difference in style between the three countries, said: "In Japanese lessons, there is the mathematics on the one hand, and the students on the other. The students engage with the mathematics, and the teacher mediates the relationship between the two. In Germany, there is mathematics as well, but the teacher owns the mathematics and parcels it out to students as he sees fit, giving facts and explanations at just the right time. In US lessons, there are the students and there is the teacher. I have trouble seeing the mathematics; I just see interactions between students and teachers."

The US authors are critical of maths teaching in the US, and see improvement in teaching style as necessary, rather than continual focus on class sizes and equipment. Their basis for criticism is the poor results US students achieve in the worldwide (41 countries) TIMMS study. After all, Japan always scores near the top, and their typical class size is 37!

I'd love to observe some maths lessons in Hong Kong, Japan, China, Singapore, etc. Judging from the Asian students I teach, they cover advanced material significantly earlier than we do in Australia.

One more thing about gravity

Everyone knows that maths is important for science, but it's not always easy to present demonstrative examples of this to students. If a high school student studies physics or chemistry, the most advances mathematics they will need is probably solving or rearranging an equation, which is a Year 8 or 9 skill.

One of my favourite examples to talk about with students is calculus. It could be considered the crown jewel in the mathematical canon, but the fact is it was developed by Newton with an eye to his studies of planetary motion. It's a great example of a mathematical development being motivated by science. OK, that much I knew, but what about some details.

In the Feynmann book I referred to yesterday, it digs a little deeper. Kepler discovered three laws about planetary motion by observation. He looked at heaps of data that was collected on the location of planets at different times and noticed that:

• planets orbit the sun in an ellipse, not a circle
• planets sweep out equal areas in equal times, rather than equal distances

(That's the first two laws; the third I don't understand at this point.)

Newton decided that there was this thing called gravity, whose effect on two objects was directly proportional to their mass and inversely proportional to the square of the distance between them.

(Image taken from Wikipedia.)

The beautiful thing is that from this one equation, Newton could mathematically prove that orbits are elliptical and that planets trace out equal areas in equal times. This is significant because no observation or experiment is involved (except in forming the original equation, which was a hypothesis). It was through mathematics, not science, that he could show that the known facts about planetary motion were a consequence of gravity.

Newton's work on planetary motion was great science and great mathematics. As a maths teacher, I want students to appreciate the importance of mathematics, even if they're not going to make a lot of use of it in their lives. Calculus is one of mankind's greatest intellectual achievements, and many high school and university students study in in detail. It's good for them to have some idea of how it has contributed to our understanding of the universe.