In standard study environments, there are the teachers, who know a certain piece of knowledge, and the students, who don't know it. The role of the teacher is to transfer his knowledge to the student. The teacher had learned this from his teacher, and so forth. But the first person discovering something in mathematics did not learn this from a teacher. How did he discover this? Nowadays, research mathematicians work to discover new knowledge in mathematics that isn't known by anyone else. So it's not true that one needs a teacher to learn. But can we expect just anyone to discover something on his own with no help? Perhaps not, but as long as we teach by transferring knowledge, we preserve the feeling students have that their role is passive and they are not capable of discovering on their own. But if instead, the teacher feels that his role is only to ask questions, and choose them wisely so that they fit the student's way of thought, and do not hint at a solution at all, the student will be able to discover whatever needs to be discovered. And in this setting, there isn't really a teacher and a student, but rather two people studying together. One is studying how to discover a certain piece of knowledge, and the other is studying how different people comprehend this same piece of knowledge -- and by allowing the supposed student to attempt his own ways instead of the supposed teacher imposing his way, new ways to explain the same knowledge will be discovered.
This is the first of a series of discussion-formed introductions to mathematics based on this idea. These all grew out of discussions we had and are having at the School of Mathematics.
Participants: Avital, Gregory, Harlan, Samuel
Avital: Hallo.
Samuel: Hello.
Gregory: Hey good morning!
Avital: So say you have a fruit bowl and five pieces of fruit. Apple, orange, pear, guava, and kiwi. Say. Or maybe some other fruits. I don't know.
(APPLE) (ORANGE) (PEAR) (GUAVA) (KIWI)
Gregory: Yeah...
Avital: And you take two of them, any two, and put them in a lunchbox. How many different lunches could you eat in this way?
Samuel: I don't know about you guys, but this isn't how I make my lunch every morning....
Avital: Fruit here is just a demonstrative example. This same discussion would arise in many different scenarios. For example a chemist might have 5 compounds and want to discover which pairs react in interesting ways. In this case, he needs to determine how many tests he will need to run. Another example would be a "round robin" between 5 basketball teams, where each team must play against each other team exactly once. Here, one would to know how many games will be played in total.
Gregory: Sweet, but getting back to fruit. Let's write out all of the pairs.
Samuel: Let's designate them by their initial:
(A) (O) (P) (G) (K)
Harlan: Then we'll have A&O, A&P, A&G, A&K, O&A—
Gregory: Wait, you already ate that one.
Harlan: Oh yeah. A&O, A&P, A&G, A&K, O&A, O&P, O&G, O&K, P&A, P&O, P&G, P&K—
Avital: What method are you using to list these pairs?
Harlan: I started with all the A&something, then went to all the O&something, etc.
Samuel: So let's write the whole list again.
Harlan: Ok, let me write it more clearly.
Avital: So how many are there?
Harlan: 4 + 3 + 2 + 1 makes 10.
Gregory: This table has a lot of structure... the pairs we've crossed out seem to form a triangular pattern.
Harlan: I notice something else: We crossed out one pair on the second row, two pairs on the third row, three pairs on the fourth row and four pairs on the fifth row.
Gregory: You're right. But you can also look at the pairs we included instead of the pairs we crossed out. We have four pairs on the first row, three pairs on the second row, two pairs on the third row, one pair on the fourth row and no pairs on the fifth row. And these also form a triangle!
Samuel: Let's mark-up our table accordingly.
Harlan: These are interesting observations... what's next?
Samuel: Now, what if we had a larger basket of fruit, say 6 or 7 or 17 fruits. How would we count the number of lunches?
Gregory: Well, maybe before we go and do all that work we should look at the case of 3 or 4 fruits.