i is an imaginary number: A Taste of Pi (Part I)

Comments: I really liked this Saturday morning event so far. I observed how to teach math and observed the students reactions and interactions. There were a good number of females. I took four pages of notes. The topic of yesterday's talk were:
Speaker: Dr. Jonathan Jedwab, Department Of Mathematics
Title: "Games, counting, and curvy lines"
Speaker: Dr. Andrew Blaber, Department of Biomedical Physiology and Kinesiology
Title: "Weightlessness and fainting astronauts"
Date: October 17, 2009Time: 9:00 -12:30
Place: IRMACS Presentation Studio

I wrote my observation notes relating to teaching on the left column. These notes will be written in italics.

We started off with having a student introduce the speaker. I thought this was interesting and reminded me of the student conferences. It gives students a sense of ownership. I like how the introducers adlibed. The introduction talked about the speaker's history, what influenced them to go into this profession and their hobbies. It's a bit more than what I would hear at a professional conference but this makes the speakers more relatable and human.

Dr. Jedwab introduced the game of Set. This game had figures on cards. There were four attributes (number, colour, shading and shape). We saw the cards. The slides were very nice. He introduced the rules.

The Math of Set
Dr. Jedwab introduced the vector representation for the cards. (Each card can be represented by (i,j,k,l) where each of the letters are either 0,1 or 2.) He also introduced modulo 3 (in words... no slide) by saying it's just the remainder after dividing by 3 and giving an example. (That would be a teaching choice so the talk progresses.) At this point there was a big yawn.

3 cards form a set iff their vector representation x,y,z:
x + y + z = (0,0,0,0) (modulo 3).
If the attribute were the same, the sum would be a multiple of 3; if the attribute was all different then they would be 0,1,2 which add up to three.
He gave examples of vector representation of three cards in a set. Perhaps he should get the students to do this part.

He talked about the history of how it was created (coding information for researching epilespsy in German Shepard Dogs).

He presented the scenario: if there were only three cards left on the table and all other cards have been put into sets, what do you do? Student: You yell set. The sum of all the other cards satisfy (0,0,0,0). We need to know the sum of all 81 vectors modulo 3. I think he can have students come up to show different ways to show this sum.
Example: Partion 27 sets by fixing the last 3 digits
1 1 2 2
0 1 2 2
2 1 2 2

My answer: look at how you would generate sets orderly. Examine the x column only. You will see that you can group it in 0, 1, 2. Or 0, 1, 2, will always alternate and thus the x column will sum up to 0 modulo 3.

0 0 0 0
1 0 0 0
2 0 0 0
0 1 0 0
1 1 0 0
2 1 0 0
0 2 0 0
1 2 0 0
2 2 0 0

How many sets are there?
By example If you fix 2 cards, you uniquely determine the third one. (OK, he just gave us to that right off the bat. I would have hoped for students to come up with other answers for this.)
How many ways are there to pick 2 cards from 81?
81*80/2.
He showed the over counting 3 times.
So it's actually
81*80/(2*3). (What a nice introduction to combinatorics.)

He showed us an example of 20 cards and have us find sets in there and there turned out to be none. What about 21 cards and no sets? What's the maximum? It turns out the maximum amount of cards you can have without finding any sets is 20. ( I felt a bit of the What if not theory in this section.) 21 cards must have one set.
He showed how you must prove this fact. He used a smaller case example to show.
Constrict the cards to two attributes. There can be four cards maximum with no sets.
We must show
a) There exists four cards in which no there are no set
b) For five cards, there must be a set.

He showed four cards that didn't form a set: (I'm going to just put down their vector represetation)
(0,0) (0,1) (0,2)
(1,0) (1,1) (1,2)
(2,0) (2,1) (2,2)
(insert pic)
He used lines to represent a set. (Questions that arose was, does this affine geometry influence this game of set or does the rules influence these lines? It turns out we can just look at the geometry and its self consistency to get the same picture. You can also just have the rules define the lines. They kind of asked the chicken or the egg question here. However, it can be either or and both is just as legit.)
3 lines form a set iff they fall on a line. Very visual
Here we start off with object and we assign numbers and then we assign it to geometory.
Object -> Numbers -> Geometry Just like history mentions Dr. Jedwab.

Now we want to show any 5 points implies that they must lie on a point.

Pick a point (bottom right corner). 4 lines go through this point. One of these has 1 point. He's going too fast. (I couldn't follow so I think I'm going to ask him to type up his arguement). I think he could have placed a coin on that point.

I think let's look at the bottom right corner. Pick that point. Now choose a point on each of the lines going through it making sure you don't make a line with those other points. Do this for 3 other points. Now you notice if you place one more, three of them must form a line.

This is the Pidgeon Hole Principle. Real life! n holes, n+1 objects=> One hole must have 2 objects. (Anecdote about having a name to this principle. However, this is the one of the few things we really know and not just made up.) Very powerful.

Mathematicians solve 1 problem, wrap it up and then try to generalize.

3D version (3 attributes).
a) 9 cards
b) any 10 cards must contain a set
(I'm going to just transcribe my notes...)
Lines and planes. Every plane that goes through a point looks like lines on the 2D version. Every plane has 2 points (?) at least fix two points... all planes in 2 points => 2 planes. 7 more points not on this plane / line => one of other plane. 2 more points = > 5 points on one plane => 1!
Anyways I bought his arguement. It's kinda about placement and sort of follows the 2 variable case. He went a lot faster with this one. He said he would be fishy about this one.
1h has gone by.
Affine gemeotry

4, 10, 20 are the maximal caps

5 variable (2002) - 45 is max. It's an obvious question to ask. They used Fourier Transform.
6 variable (2008) - 112 is max. Talked about how research is new and how one goes into a hole for 3 years and come out with an answer. Talked about pure math (like number theory) where it wasn't tainted but just research for the beauty but now it's the backbone for e-commerce.

Good Question asked by audience. If motivation is Set then it defines the line. If motivation is geometry then the line is defined by the thing that makes sense.

He's being fishy: What we know has run out. we must create new math! => Needs to make sense. This means that it has to be internally consistent.

Postive:

talked about history of math

audience participation

jokes

dancing

principles of how to show max (noted mistakes grad students make in not showing the two)

added a lot of relevant real life annecdots

made math fun

showed what math research does / how math is developed in the past and currently

very visual and colourful

Minus:

some parts could be left for students to figure out instead of giving out the answer (of course he had to do this because of time constraints

the punchline (of the 2 variable case) was given too fast and there were too many parts. I think it would have been better if this was written up and posted on a slide

Interesting:

I would like to introduce this game and use it to present the variety of concepts that were presented. I think I would give the students time to go home and think about the problems and the different ways of solving it.

So that was my three pages of notes.

Astronauts and Weightlessness and Fainting

Basically this was a talk about physics, "biology" and space using math as a tool. Dr. Blaber's research was testing astronauts in weighlessness and when they got back to earth on blood pressure, heart rate and blood flow in the brain. He showed us the tools and the graphs being generated by measuring his graduate student. (Go guinea pigs.)

He introduced weightlessness with a lot of physics. I thought there was too much text on the slide. He talked about hairs in ear sensing where gravity is. I thought htis was interesting.

Then he talked about pressure in the capilaries. I liked his diagram with the backup flow. He related Blood Pressure (P), flow (Q), and resistance (R) as

delta P = Q R

which is like

V = IR

At this point the chairs was so comfortable I noticed a few students falling asleep.

This system depends on heart rate, stroke volume, and resistance (how big the arteries are... there's things that control the size of them).

The blood flow's velocity doesn't go to 0 in the brain. If it does then you faint or have a stroke.

(Neat) Resonance: Can get standing rate of with 6 breaths per minute. This is for a think blood flow int the brain. This is why people have breathing exercises.

There was talk about hyperventilation. It stops the blood flow or slows it down?

If CO2 goes up, the capilaries dialte and eliminates Co2 faster than it is being produced. (Breath out faster)

Pressure at tow is = 2 * Pressure at heart.

Then he showed how the blood pressure at his grad student's finger changes as he changes the position of his hand (above head, at heart level and as low as he can sitting). This was very neat.

We saw how the blood flow changes after his student stood up from sitting in the chair for over half an hour. The blood flow dropped dramatically and the heart rate went up (fastest response). The heart beats was broadcasted in number and audio form.

Good Undemo Idea: He had his student clench his leg muscles. This is what they tell pilots to do when they reach high g. (This relates to blood flow and constricting it in the legs so blood doesn't rush to the legs I think.)

Dr. Blaber talked about the astronauts he worked with and showed pictures. Students were very interested in this. He also then talked about witnessing the shuttle landing and how the shuttle slows down. There was also talk of the Colbert treadmill on the shuttle and Buzz Lightyear returning after being in space for 15 months. Everyone was attentive as soon as he started talking about space.