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.

http://dsc.discovery.com/news/2009/09/11/shuttle-landing.html

http://www.nasa.gov/multimedia/imagegallery/image_feature_1472.html

Positives: very visual, showed how all disciplines relate to each other, space is interesting

Negatives: too much information on the first few physics slides. Cognitive overload

Interesting: I had some good undemo ideas from testing the heartrate by sitting and standing and clenching leg muscles.

How can my class demonstrate democratic citizenship?

Simmt suggests in Citizenship Education in the Context of School Mathematics that mathematics plays an integral role in social, political and economic structures. Knowing mathematics would be a powerful skill for one to integrate into society. Simmit wants our youth to "understand and critique the formatting power of mathematics in society". She suggests instead of teaching math as fact, skills and processes, teach math as a creative process for exploration and give students responsibility for explainations.

In my classroom, depending on what type of class I will be doing, it is easier to do the latter suggestion than the first because of time constraint. For the first suggestion, Simmit suggests posing open ended problems. My fear that the students would not be comfortable enough with the mathematics yet to take on such a challenge. However, I can see myself dedicating 3 or so teaching days to open ended group problem solving to help train their minds for how mathematics can be done. I am more opened to giving my class responsibility for explainations. I may do something instrumentally and ask why I may have done that. Another way to engage the class in the dicussion of explaination is to show two ways how students do the question (in which the results may or may not differ) anad ask the class which solution is correct (or are both correct) and to show which parts are correct mathematically and which are not.

Taste of Pi

From the goodies of the math listserv:
http://www.math.sfu.ca/atasteofpi/
Taste of Pi!

Even thought "registration" is over, I e-mailed them saying I was a student teacher and would like to attend so I got in.

What-If-Not-Approach

The What If Not Approach is a powerful way of thinking about mathematics concepts. Instead of just focusing on one certain "thing" about a topic, it looks at other concepts around the topic. This is where I have a problem. I am not that creative at looking at things a different way. To me, the What If Not Approach is a question one poses to start the exploration on a topic. I am not used to asking the question, “What if not?” I find that I get stumped when I ask myself that question and draw a blank. It's like a stumbling a block. To quote Erwin, “students might not have the mathematical ability to grapple with "cycling" two negated statements and combining them into an alternate statement because it doesn't seem natural.” This was basically my problem. For example, in the geometric board, I was not able to see anything about the amount of pins there would be if the board was a different shape. It didn't even occur to me that is what would happen. I was thinking maybe the same square but the diagonal of the square would be the diameter of the circle.
In our micro-lesson, we sort of have a “what if not approach”. We are not simply, having the students graph some points and fit it to a straight line. We are exploring the possibility of what if the relation and graph were not linear. Students will either explore a y=1/x, y=x^2 or y=mx+b relationship and report back to the class their findings. That way, the students will see that graphing is a technique suitable for all type of relationships between variables.

10 Questions / Comments for the Authors

1) It's a hard exercise to come up with all those ideas after seeing one line of math. I certainly wasn't able to think up of all those questions and ideas on x^2 + y^2 = z^2

2)What are the different things students think about while playing with the geometric board?

3) How has the study / resources of this topic changed from the first edition of this book to the third?

4) How clear must the question be when we pose it?

5) Should we enlighten the students what questions to ask when we give them a question?

6) How much exploring should we allow for a student?

7) If the student has problems with the basic operations (addition/subtraction/multiplication/division), how much should the student explore on the particular math topic?

8) How long should we leave a child with a geometric board? How much exploration (guided by questions) should a student do?

9) I like how when approaching the topic one:

1. makes the observations
2. asks the questions
3. comes up with the conjecture

10) I like how the authors pointed out internal versus external exploration. Often we focus on internal exploration but don't ask the external questions that uses the skills from a certain topic. A lot of external questions were presented when exploring the Pythagorean triple.
.

Timed Writing: Into the future

After teaching 2000 students....
Maybe you are the worse teacher, the best teacher, the average teacher?

From the point of view of an admiring student:
What a great day in class today. I love Math. Ms. W is right! Math is fun. I love how she shows us all these interesting ways to do and to think about math. I never understood multiplying fractions until now... She showed us a few to think about fractions today! She's always so funny but clear. Also she makes sure we understand b efore we go to the nexxt important step. I'm never lost in that class and I'm very focused. If only all the teachers have this engaging personaility.

From the point of view of a student that hates me
What is the teacher doing? Why do I have a woman teacher for math? All the other teachers in math are men. Math is in the morning and I'm too tired to do math at 8. All I ever hear is blah blah blah, I think I sleep in that class a lot. She likes to pick on students too much. Why does she ask so many questions? Doesn't she know I don't care about this subject? I'm pretty sure she does. She tries too hard. How does she have so much energy in the morning. I wish she'd stop picking on me. That's so embarasing, I don't want to look stupid in class. I don't want people to notice me. I hate math at 8 pm in the morning. Just give me my homework and I'll eventuallyy figure it out. Eventually.... Actually, I hate homework. Good thing she doesn't always check. Why did I get a C last time on my test? I did everything right. I think she just marked it wrong. I hope I can get a petition for the students to fire her... haha. Math is not fun... Math is evil!

I am most afraid of making mathematics boring and having students trying to kick me out of school.

I aspire to be someone that makes math fun.

Dave Hewitt's Video Response

Dave Hewitt has a very interesting way of teaching students. I thought in this way, the students can try to guess what is going on and then have an immediate reassurance of being correct. However, to be honest, I had no idea what he was doing at first. I thought he was showing distances at first with the counting. I also thought that maybe he was marking a spot as 4 or 5. After I saw that he was just trying to get the kids to count with him on a large number line, I understood what he was doing. Then he had the students start at different places like 700 or 1,000,0000 and have them count forwards and backwards from that spot. This was probably review for the students. The finale was when he started at 0 and then had the student go forwards and backwards from that number. I think this showed that minus 1 was just 1 less than 0.

I thought teaching grade 9 was interesting. He had "taught: them how to solve for a variable. My only problem would be what if a student answered but the question was wrong. Also not everyone was participating in the speaking portion of it. However, he had the students repeat it many times and had them pick out certain words to highlight the patterns of it. Then he had the students highlight another point. Thus he's guiding the students to speak the material he was teaching. This was very interesting. The other positive I found in this section was that he would be able to have the students speaking to learn it and reinforce the concepts. Thus, the material is not only heard, but also spoken and read.

Battleground Schools

Susan Gerofsky enlightens the reader in her article in Battleground Schools about the curriculum changes throughout the years.


Key negative ideas society holds about mathematics that affected these changes:
-is inhuman-only appropriate and necessary for small elite(flip side means it's not appropriate nor necessary for the general public)
-those who like mathematics are weird, different and do not fit into society
-it is acceptible and not shameful to be incapable of doing and understanding mathematics.

Progressive Reform (1910-1940)
Goal: Meaningful mathematics curriculum for preparation of democractic, industrial society.
Chicago Movement wanted to unify algebra and geometery, and bring pure and applied math to the curriculum. Dewey wanted students to also do and experiement mathematics. Students would be more involved and active in the classroom.


The New Math (1960)
Goal: Train mathematical students to beat the Russians.
Mathematics became more abstract. Set theory, abstract algebra, linear algebra, calculus were taught in the public stystem. However, teachers and parents were not qualified to teach this material.


NCTM Standards (1990-now)
In the 1970's and 1980's, there was much standardized testing. National Council of Teachers of Mathematics (NCTM) developed their own set of standards. The changes emphasized problem-solving, ability to represent mathematical ideas in different forms, technology such as graphing calculators, and ability to communicate mathematical concepts. They also wanted to develop relational and instrumental skills and an appreciation of the power and beauty of mathematics.