Report of Lecture 6
Presenters: Lau Ting Sum Samson, Suen Wai
Date: 24th Feb, 2011 (Thursday)
Topic: How to cut a piece of cake?
1. Introduction
The presentation was about fair division of a rectangular cake. It consisted of three parts. First, the definitions of proportional and envy-free were introduced. Next, three division methods, including Divide-and-Choose, Last Diminisher and Selfridge-Conway Envy-free Procedure were presented. Finally, further extensions of the three methods, including getting a smallest piece of cake and dividing other things, were discussed.
2. Comments
*** Can consider different cases in the algorithm (e.g., if no trimming then done).
> Good suggestion. You may speak in class if you have any constructive suggestion next time. We believe that it will help other students learn.
*** Can explain proportional and envy free with specific numbers.
> Please ask us to clarify in class if you have any question. We really want everyone to understand well.
*** The atmosphere is quite good and almost half of the class can play the game.
> We believe that interaction helps you to concentrate and understand well.
*** The interaction between classmates is enough.
> We insisted to have tons of interaction. Glad that you like it.
*** If more players in the example are considered, it would be better, especially, the envy-free one.
> Well, envy-free procedures for four or more players are much complex than the procedures we introduced in class. It seems that the material we presented was a bit too difficult for some of our classmates already. We did not want to confuse them too much. If you want to know more, you may borrow some books from the library. Here is a paper about an envy-free chore division algorithm among four players:
http://www.math.hmc.edu/~su/papers.dir/chore4.pdf
*** Easy to follow.
> Thank you.
*** Lot of interaction make the atmosphere good.
> Thank you. We believe that is good for learning.
*** Daily example is common and useful.
> Thank you. We will continue to give daily examples next time.
*** Maybe, more discussion about this method to other daily life problems would be good.
> Some students suggested some useful applications in the online discussion forum. Please have a look.
*** More pictures would be more interesting.
> Good suggestion. We will include more pictures in the PowerPoint next time.
*** More explanation on proportion and envy free would be good.
> We tried very hard and used a lot of time to explain these two concepts. To understand these two concepts, students may think more about their definitions.
*** Shorter sentence / point points would be better.
> Thank you for your suggestion. We will try every possible ways to improve our presentation next time.
*** The host can lead/guide the presentation better.
> Please give specific suggestions so that we know how to improve.
*** Great presentation closely related to daily life.
> Thank you very much.
*** How to play the game is clear. Better discuss the strategy.
> Good idea. However, that is not our main focus. Hope you really understand proportional and envy-free well.
*** Procedures are too theoretical, unreadable. The steps are logical and the words in slides could be less complicated.
> Sometimes, words have to be “theoretical” if we want them to be precise. However, we will try to strive for a better balance next time.
*** Samson's explanation is clear; Suen Wai could be more engaging.
> Suen Wai actually did a lot of technical work which includes explaining the procedures and handling the PowerPoint. Suen Wai also contributed a lot in producing the PowerPoint. We tried to divide the work according to our strength. However, we have to admit that Suen Wai was quite nervous since it was her first time to present Mathematics in front of a class. Hopefully, Suen Wai will present better in the next presentation.
*** Any other example besides cake division?
> We suggested some examples under “further extensions”. Some students suggested other interesting examples in the online discussion forum.
*** More pictures are better.
> Good suggestion. We will do it next time.
*** Further extensions are interesting.
> Glad that you like them.
*** Proportion and envy free are abstract concepts, better show examples right after the definitions to illustrate the concepts.
> We actually did. Please stop us and ask us to clarify if you have any questions. Other students might have the same question, too.
*** Better use color pens to do the illustrations on the white board.
> Good idea. We will try to have more varieties in our presentation next time.
*** Interesting discussion/topic. More details in the two definitions.
> We also like the topic very much. We tried to provide details. However, we will try even harder next time.
*** The two concepts could be further clarified. Not easy to learn a lot of concepts in a short time.
> We spent a lot of time on them and invited you to ask questions about them. However, may be they are too abstract for some of you to understand. May be we will try something more concrete next time.
*** Could mention clearly the underlying assumptions, say, everyone is honest.
> Well, in fact that is not necessary. The essence of proportional is that even if all other players cooperate and play against you, you are still guaranteed to get a piece of size that you deserve (1/n of the whole cake if there are n players dividing the cake).
*** The last diminisher is great; the envy-free allocation is not so interesting.
> May be the envy-free allocation is a bit too complex. When it comes to 4 or more players, the procedure will become even more complex.
*** Overall presentation is good. Try to avoid moving around too much.
> Thank you. We will try to avoid it next time.
*** Louder would be better.
> We did not notice this problem. We will try to speak louder next time.
*** Take turn to present would be better.
> In fact, Suen Wai presented the procedures and Samson tried to discuss the properties of the games afterwards. However, the discussion of the games was much longer than expected. This may lead to an illusion that Samson was talking all the time.
*** Overall, good presentation with patient clarification.
> Thank you. We hope that you enjoyed our presentation and learned something out of it.
3. Suen Wai’s Reflection
In this presentation, I was responsible for talking about the objectives and explaining the procedures of the three division algorithms. However, I was very nervous because it was actually my first time to explain Mathematics in front of a class of students. I felt especially nervous when students asked questions. Fortunately, Samson handled the questions very well.
Despite I did not do very well in the presentation, I learned a lot from it. Before preparing for this presentation, I had no idea about fair division. At that time, I thought fair division was about everyone getting an equal share of something. However, I did not notice that it is quite difficult to achieve in reality. One example is dividing a round cake to an odd number of people. The “proportional” and “envy-free” concepts have really enlightened me. I never imagined that things can be thought this way. Also, the division algorithms are amazing. People who thought of the algorithms are really creative. The algorithms satisfy one (or even both) properties of fairness and are quite practical.
Cutting a cake may not sound very important. However, many important inventions are actually base on some very simple theorems. I hope to learn more and more interesting topics of Mathematics.
Last but not least, I really want to thank my partner Samson. He is really kind and helpful.
4. Samson’s Reflection
With my experience of teaching mathematics summer courses in two different secondary schools and receiving a lot of positive feedbacks, I thought I could explain mathematics quite well. However, many students could not understand the definitions of “proportional” and “envy-free” well in this presentation, as shown in the online discussion forum. Here are some examples:
“If a procedure is not envy-free, then I may take the whole piece of cake.”
(It will never happen if the procedure is proportional.)
“Envy-free ensures that every gets happy though out the division, hence everyone will get a piece of equal size, and thus this is a necessary property.” (Envy-free does not imply that everyone will get a piece of equal size.)
“Envy-free is very subjective” (No, it is not subjective. It is completely objective.)
I felt frustrated to read the above comments. Contrary to my past successful experience in teaching, I received a lot of negative feedbacks this time. However, I realized a main reason behind the failure after some thoughts: I did not think deeply about the needs of the audience. In fact, not many of the students are Math-major. They are not used to digest abstract concepts quickly. I should select some concepts that are more concrete and easy-to-understand to teach next time.
Next year, I am going to study Postgraduate Diploma in Education at The Chinese University of Hong Kong. I plan to become a mathematics teacher afterwards. This presentation reminds me that I should select material to teach carefully and teach at an appropriate pace according to the needs of the students if I want to become a good teacher. After all, I believe that what I have learned in this presentation is far more than what the audience has learned from me. I look forward to our next presentation.