Over the past weeks the lessons that you gave on May 8 have been present in my mind all the time. Dr. Fung can tell you how disappointing the subsequent conference in Taipei was for me. Your four lessons were the real highlights of my journey to Asia. I am happy to give some comments.
As far as nets of a cube are concerned I have never seen such a clear derivation of all 11 nets. The lesson was a very good exercise for students both in spatial and logical thinking. The idea of classifying these nets according to the length of the longest strip of squares is very powerful. It can also be used for deriving the nets of an open cube (with five squares). This might be a good introduction to the nets of a (closed) cube (in case one has enough time). Of course one can also derive all 12 pentominoes and 30 hexominoes this way. This is very fine.
The lesson on the chords of a circle was also very interesting for me. I think it pays off to let the students measure the length of various chords and compare the results. The diameters as the biggest chords are well recognized and internalized that way. The investigation can be re-visited in various ways. For example, the triangle inequality can be employed, and in later years the length of the chords can be expressed by means of trigonometry. I wish that German teachers could have seen this lesson, as in Germany geometry is receiving less and less attention. Trigonometry has more or less disappeared from the curriculum. Also I was struck by the change of the length of a chord which is rotated around one vertex. The change is rapid when the chords are short and becomes slower and slower when the length reaches its maximum. Thank you, Ms. Yiu, for presenting such a stimulating topic.
The third lesson on measurement was interesting for me not so much for mathematical reasons as in Germany teachers would proceed roughly the same way. However, how you, Ms. Cheung, have succeeded in managing a whole class of small kids for 40 minutes. This is considered as a “mission impossible” in Germany. In our country it is the standard view that first and second graders are simply not able to listen to a teacher for more than 5 minutes. For this reason a method is widespread in this country in which the students are given a plan of tasks at the beginning of the school day and are free to work on these tasks according to their own choice. Students working on the same problem over one lesson with clear directions from the teacher as well as discussions in the whole class are very rare in German primary classrooms. Educators tend to denounce such kind of teaching as „teacher-centered“ what I think a big mistake.
Finally, fractions: Again the comparison with Germany has been striking for me. In our country fractions are considered as a difficult subject that therefore has to be postponed to grade 6. In the years before most curricula allow only for „halves“, „quarters“ and „fifths“. In our project we try to follow the advice given by Alfred N.Whitehead: Difficult subjects should be introduced as early as possible in order to give students more time to get familiar with them. However, we are offered resistance when we try this with fractions. You in Hong Kong can be happy that your curriculum is very advanced in this respect. In talking to Dr. Fung I have learned that the Chinese language has some definite advantages. We will see how we can cope with the German situation in future. In this respect the lesson I have seen has been very enlightening for me.
I would like to thank you all very much for giving me a chance to participate in your teaching. I have been greatly impressed and I cannot but congratulate you for what you are doing. The collaboration with Dr. Fung in the TFM project is obviously very successful. It is a model of how teachers as reflective practitioners and researchers should cooperate.