Infinity

1. Identifying heroic qualities

What heroic human qualities are central to the topic? What emotional images do they evoke? What within the topic can best evoke wonder?

In order to help students connect emotionally to the material, teachers need to first identify their own emotional attachment to it. What heroic human quality or emotion––courage, compassion, tenacity, fear, hope, loathing, delight, etc.––can we identify in the topic? These human qualities help us––and our students––see the world in human terms and give human meaning to events and ideas in all disciplines. We “humanize” each topic not to falsify it confuse but to infuse the world with human meaning. Again, this first task is the most difficult part of planning the lesson or unit. We are asked to feel about the topic as well as to think about it; indeed, we are asked to “perfink” (David Kresch’s term for perceiving, feeling, and thinking together) about it.

Main heroic quality: weirdly mysterious infinity
Alternative(s): limitless imagination, interest in aliens and future, curiosity about our Universe, enjoyment in paradoxical thinking

Images that capture the heroic quality:
The image that can capture imagination of students is the weird “Hotel Infinity”. It can be drawn on a large sheet of paper, on an overhead, or on a blackboard with lines disappearing in perspective and limitless windows. We can imagine a group of aliens coming to stay in the hotel--a lot of them. They form a line at the front door, patiently waiting to be assigned a room number. The first task is easy for the hotel manager since he has infinitely many rooms; he just assigns one room to each alien. But once he is finished with the long line, a latecomer from the planet Earth arrives…

2. Organizing the topic into a narrative structure

2.1 Initial access

What aspect of the topic best embodies the heroic qualities identified as central to the topic? Does this expose some extreme of experience or limit of reality? What image can help capture this aspect?

For the first lesson of a unit or the opening part of a single lesson, teachers are asked to search their own imagination for images that catch the heroic quality that will provide the dramatic structure for the unit. Remember, it is as important to feel the heroic qualities as well as think about them.

Exotic/extreme content that best embodies the heroic quality:

There are many ways infinity can catch our imagination. Everyone has wondered if the universe is infinite, for instance. It is an easy mistake to conclude that it must be, reasoning that if it wasn't then it would have to have a boundary, and then what would be on the other side of that boundary? The answer to this is that the universe may be like a sphere. The surface of a sphere doesn't have a boundary, but it is certainly finite in area. (The universe, in other words, could be like a three-dimensional surface of a sphere that can be imagined as existing in four-dimensional space – most cosmologists think it may well be something like that.) Another common line of thought runs like this: if the universe is infinite, then it must have an infinite number of stars and planets, and therefore infinite number of possible worlds eg. a planet just like this one except that everyone has three hands (or talks backwards). The trouble here is in thinking that an infinite set must contain everything. However, a little thought shows that this needn't be true.
At one time there seemed to be a strong argument against the number of stars being infinite. A simple calculation shows that if it is, then the whole night-sky should be ablaze with light. The surfaces of the stars are, on average, as bright as the surface of the sun, and if they are infinitely numerous it can easily be shown that any line of sight will eventually terminate on a star, so that the whole sky will shine as brightly as the sun. It can be argued that the most distant stars might have their light dimmed by passing through gas or dust on its way to us, but if this were the case, the gas itself would be raised to such a temperature that it too would shine with this same brilliance.
This problem was known as "Olbers' Paradox", after Heinrich Olbers (1758-1840). But it has now been resolved; even if the universe were infinite, we know that its expansion would provide an explanation for the darkness of the night-sky. Distant stars are dimmed not because of intervening matter, but because they are moving away from us, and the wavelength of their light is increased, and its energy reduced, by this motion. So Olbers' effect does not now present an obstacle to those who believe in an infinite universe. But here again it is surprising that taking the number of stars to be infinite is an assumption that can be adopted or discarded at pleasure, without considering whether it should be ruled out on logical grounds.
Finally we can ask ourselves whether time is finite (at least looking backwards). If it is and it all started with a Big Bang from a single point then what was “before” the Big Bang? What gave an initial impetus to the whole thing and what was the source of even a single molecule that perhaps expanded?


2.2 Composing the body of the lesson or unit

How do we organize the material into a story to best illustrate the heroic qualities? Sketch the story, ensuring that the qualities will be made clear by the narrative.

The principal heroic quality should provide the drama and conflict in the story. Remember, the heroic qualities should be those that most effectively convey the content of the topic.


Sketch the overall structure of the lesson/unit:
With the picture of the Hotel Infinity at the background, the teacher may tell the following story. Hotel Infinity is an enormous hotel at the center of our Galaxy. Most of its rooms extend through a black hole into a higher dimension. The room numbers start at 1 and go forever. There is a Room 2, a Room 1,001, a Room 1,000,000 and even googol(1 followed by 100 zeros).
One day, when every room was occupied, a pilot, on his way to another galaxy, arrived and asked for a room. He looked a bit anxious as he could see that so many of the rooms seemed already full. The manager of the hotel said: “Don’t worry. I can always find a room for you.”
The Hotel Infinity was a bit odd in a number of ways, as you might expect. For one thing, every room had a loudspeaker in it, and everybody who checked into the hotel agreed to obey whatever commands were given by the management.
So the manager simply asked the occupants to move to the room with a number one higher than the one they were currently in. So the girl from Room 1 moved to Room 2, the couple from Room 2 moved to room 3 etc. It was a mad flurry of rushing people, changing rooms in the infinitely long hallway of the Hotel Infinity: 1‡2, 2‡3, 3‡4, …, n‡n+1,…
That left Room 1 vacant for the pilot.
The next day, five couples on their honeymoon showed up and they asked for five separate rooms. Can Hotel Infinity take care of them?
Students should not have any difficulty providing the manager’s solution. Ask the class how he would have managed it. Yes, easy--the manager can move everybody to a room that is five higher: 1‡6, 2‡7, 3‡8, …, n‡n+5,…
This would leave rooms 1 through 5 vacant for the five couples.
On the weekend, before anyone had yet moved out of the hotel, an infinite number of bubble gum salespeople came for a convention. Can we find a room for all of them? Ask the students how they would manage that.
The manager found a solution. Using the intercom he announced, “ Sorry to interrupt you again, but if you are in room n would you please move to room 2*n.” Since everybody at the hotel had at least two fingers and multiplying by 2 wasn’t much of a problem, the task was done swiftly: 1‡2, 2‡4, 3‡6,…, n‡2*n
That left all odd numbered rooms empty, and the sales people moved to rooms 1, 3, 5, …
But the manager’s troubles didn’t end there! The next day
there was a big religious convention and each of infinitely many civilizations sent a starship full of the faithful . Actually there were infinitely many creatures on infinitely many starships and they all wanted a separate room.
No problem said the manager. Just give me a minute to figure it out. Can the students work out his solution, or another one?
First, the manager asked all hotel guests to move to room with a number 2n so a person Room 1 moved to 2 and so on: 1‡2, 2‡4, 3‡8, 4‡16,…n‡2n
That left all odd numbers free and many others as well.
Fortunately he remembered that a Greek mathematician from the planet Earth, Euclid, proved that there are infinitely many prime numbers.
(an elegant proof by contradiction could be included here for more advanced students). So the manager announced to the pilot of the first starship, “Your folks will use prime number 3 and its powers only. You will get therefore Rooms 3, 9, 27, 81, etc. There are infinitely many of them, so there will be plenty for your people. To the pilot of the second starship the manager said, “Your folks will use prime number 5 and its powers only. You will therefore get Rooms 5, 25, 125, 625 etc. There are infinitely many of them too, so there will be enough for your people.”
For every starship there is a unique prime number, and since all powers of prime numbers are different everybody gets his, her, or its own room. Moreover, after that operation, an infinite number of rooms are still unoccupied. For example Rooms 6, 10, 12, 15 are empty. In fact, every room that can be written as a product of two different prime numbers is left empty because it is not a power of 2, or a power of a prime number.
6=2*3, 10=2*5,…, 1000=2*2*2*5*5*5.


2.3 Humanizing the content

What aspects of the story best illustrate the human emotions in it and evoke a sense of wonder? What ideals and/or challenges to tradition or convention are evident in the content?

Think of how a good movie or novel makes aspects of the world engaging. Obstacles to the hero are humanized in one form or another, almost given motives; they are seen in human terms. To do this, we don’t need to falsify anything, but rather we highlight a particular way of seeing it––because this is precisely the way students’ imaginations are engaged by knowledge.


What content can be best shown in terms of hopes, fears, intentions or other emotions?

One of the things that may capture student’s imagination is paradoxical thinking. What is initially contradictory to our intuition, suddenly after closer examination (and with a help of our logical and unflappable hotel manager, starts to make sense. Students should be amazed that our common arithmetical practices founder for sets that have infinitely many elements, and that infinity plus infinity is still infinity.

2.4. Pursuing details

What parts of the topic can students best explore in exhaustive detail?

While it is easy to give students a project to carry out, it is a little harder to think about what aspect of the topic they might be able to exhaust, i.e. be able to find out nearly everything that is known about it. But there are such parts in every topic, and the security and sense of mastery that comes from knowing nearly as much as anyone about something is a great stimulus to inquiry. Think of something that is intriguing, that can be seen from a variety of different perspectives, or that is alluded to but not examined in detail in the content or in your teaching of it (referring to your notes from 2.2 and 2.3 above should help!).

List those aspects of the topic that students can explore exhaustively:

There are a lot of topics, almost infinitely many!. that students may explore regarding infinity. Here are just a few:
* The arithmetic of very big numbers like googol. This arithmetic is somewhat analogical to infinity but there are important differences, eg. googol+1=googol, similarly to infinity, but googol+googol =2googols, which is different from infinity

* How about dividing sets instead of adding them? A wonderful Zeno’s paradoxes could be explored here. For example, the paradox of Achilles chasing the tortoise is an excellent introduction to sequences and another example of narrative story that can be used in high school mathematics

* For more advanced students exploration of aleph-zero and continuum with Cantor‘s amazing proofs could be considered.

* Comparing Leibnitz theory of the infinitesimally small and monads that have major philosophical connotations with Newton’s notion of fluxions. Both theories gave birth to modern calculus.

3. Conclusion

How can one best bring the topic to satisfactory closure? How can the student feel this satisfaction? How can we evoke a sense of wonder about the topic?

One wants to end a topic in an “heroic” way, which can have two forms. The first form is to re-examine the images we started from and review the content through the lenses of other heroic qualities, including some that might give an opposite or conflicting image to that of our earlier choice. The second form is to show how the romantic association the student has formed can help them understand other topics in a new way. Or one can use both, of course. In concluding we will also want to reflect back on the topic bringing out why we should feel wonder or awe about it.



Concluding activities:
We may ask ourselves if our Universe is finite or infinite and in what sense. Perhaps have a Socratic type of dialog with two sides taking opposite points of view and providing some supportive arguments based on our extended understanding of infinity.
Perhaps the following song can be used to review the main ideas

"Hotel Infinity"

lyric © 2000 Lawrence Mark Lesser
May be sung to the tune of
the Eagles’ “Hotel California” (by Don Felder, Don Henley, Glenn Frey),

On a dark desert highway -- not much scenery
Except this long hotel stretchin’ far as I could see.
Neon sign in front read “No Vacancy,”
But it was late and I was tired, so I went inside to plea.

The clerk said, “No problem. Here’s what can be done--
We’ll move those in a room to the next higher one.
That will free up the first room and that’s where you can stay.”
I tried understanding this as I heard him say:

CHORUS: “Welcome to the HOTEL INFINITY --
Where every room is full (every room is full)
Yet there’s room for more.
Yeah, plenty of room at the HOTEL INFINITY --
Move ‘em down the floor (move em’ down the floor)
To make room for more.”

I’d just gotten settled, I’d finally unpacked
When I saw 8 more cars pull into the back.
I had to move to room 9; others moved up 8 rooms as well.
Never more will I confuse a Hilton with a Hilbert Hotel!

My mind got more twisted when I saw a bus without end
With an infinite number of riders coming up to check in.
“Relax,” said the nightman. “Here’s what we’ll do:
Move to the double of your room number:
that frees the odd-numbered rooms.” (Repeat Chorus)

Last thing I remember at the end of my stay--
It was time to pay the bill but I had no means to pay.
The man in 19 smiled, “Your bill is on me.
20 pays mine, and so on, so you get yours for free!”

4. Evaluating

How can one know that the content has been learned and understood and has engaged and stimulated students' imaginations?

Any of the traditional forms of evaluation can be used, but in addition, teachers might want to get some measure of how far students’ imaginations have been engaged by the topic, how far they have successfully made an imaginative engagement with the material. In addition, the concluding exercises (above) are also evaluative in nature. Students could be asked to identify heroic qualities in stories in other disciplines to examine both their imaginative use of narrative and their understanding of the content. Heroic qualities can also be examined on moral/ethical terms.

Forms of evaluation to be used:  

* Some traditional methods of evaluation can be used initially. For example we may ask students to prove that certain infinite sets of numbers have the same number of elements by the method of 1-1 correspondence.

* As an extension we may ask the students to prove that the set of all fractions have the same number of elements as the set of integers, but less than the set of all real numbers

* Participation in a Socratic dialog and some essay type of writing where students will need to support “big “ ideas regarding infinity could be taken into consideration.

* Students could be asked to express, through some written or pictorial or movie form a conception of infinity. These could be assessed for combining imaginativeness and accuracy in capturing some notion of infinity not touched on in the unit.